Between order and disorder
Ultracold atoms in a quasicrystalline optical lattice
Matteo Stefano Sbroscia
This dissertation is submitted for
the degree of Doctor of Philosophy
University of Cambridge
Churchill College
August 2019
“Twixt order and disorder lies but a silver mean” was the originally intended title. But I
did not want it to appear on my CV for the rest of my life.
To my grandparents Guido Paletti and Ursula Seweron,
and to my great-aunt Rita Paletti.
Declaration
This dissertation is the result of my own work and includes nothing which is the outcome
of work done in collaboration except as specified in the text.
It is not substantially the same as any work that I have submitted, or, is being concurrently
submitted for a degree or diploma or other qualification at the University of Cambridge
or any other University.
I further state that no substantial part of my dissertation has already been submitted,
or, is being concurrently submitted for any such degree, diploma or other qualification at
the University of Cambridge or any other University.
This thesis does not exceed the word limit of 60,000 words, including abstract, tables,
footnotes and appendices, imposed by the Faculty of Physics and Chemistry.
v
Abstract
Between order and disorder:
ultracold atoms in a quasicrystalline optical lattice
by Matteo Stefano Sbroscia
Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a
wealth of intriguing physical phenomena, such as self-similarity, the inheritance of topo-
logical properties from higher dimensions, and the presence of non-trivial structure on all
lengthscales.
However, quasicrystalline materials are notoriously hard to synthesise, as defects and
impurities may greatly alter their final microscopic composition. The field of quantum
simulation with ultracold atoms offers a solution to this problem, namely the use of optical
lattices – standing waves of light. Optical lattices are free of impurities and therefore
ideally suited to study quasicrystals, enabling unprecedented access to observables that
are unattainable in condensed matter systems.
This study presents the first experimental realisation of a two-dimensional quasicrystalline
potential for ultracold atoms, based on an eightfold symmetric optical lattice. Features
pertaining to both ordered and disordered phases are observed, from sharp diffraction
peaks in the matter-wave interference pattern, to a disorder-induced localised phase emer-
ging at a critical lattice depth Vloc ≈ 1.78Erec. The localised phase seems to be resilient
against moderate interactions, which would make this the first experimental realisation
of a 2D Bose glass.
vii
Contents
Introduction 1
I Theoretical foundations 5
1 Quantum, interacting gases 7
1.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Classical, ideal gases . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.3 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1 A distinct phase of matter . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2 Useful formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Collisions and Feshbach resonances . . . . . . . . . . . . . . . . . . 19
1.3.2 Two- vs three-body losses . . . . . . . . . . . . . . . . . . . . . . . 22
2 Cold atoms 25
2.1 Atom-light interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Semi-classical treatment . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.3 Steady state forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Phase space density increase . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Quantum simulators . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.2 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Quasicrystals 49
3.1 From periodic to quasiperiodic . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Regular crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Aperiodic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Physical properties and current research . . . . . . . . . . . . . . . 56
3.2 Mathematical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 1D: sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
ix
x Contents
3.2.2 2D: tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.3 Quasicrystals as the path to complexity . . . . . . . . . . . . . . . . 62
3.2.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
II Experimental details 73
4 Design, assembly and control 75
4.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.1 Setup overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.2 Starting a lab from an empty room . . . . . . . . . . . . . . . . . . 75
4.2 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Lasers and magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Frequency Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Reaching quantum degeneracy 99
5.1 MOT chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.1 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.2 3D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.3 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Magnetic transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.1 Science cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.2 Evaporation in a magnetic trap . . . . . . . . . . . . . . . . . . . . 110
5.3.3 Evaporation in an optical trap . . . . . . . . . . . . . . . . . . . . . 113
5.3.4 Quantum degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Probing and characterisation 121
6.1 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.1 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 Fit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2.1 Atom numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
III Results and findings 131
7 Order: sharp diffraction peaks 133
7.1 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.2 Real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.3 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 Diffraction dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2.1 Multiple dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.2 Quantum walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Diffraction spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Contents xi
7.3.1 Singularly continuous component . . . . . . . . . . . . . . . . . . . 155
7.3.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8 Disorder: localisation transition 161
8.1 Theoretical aspects of localisation . . . . . . . . . . . . . . . . . . . . . . . 161
8.1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1.2 Probing localisation in real and momentum space . . . . . . . . . . 166
8.1.3 Numerical evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.2 Experimental investigations of localisation . . . . . . . . . . . . . . . . . . 175
8.2.1 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2.2 2D periodic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2.3 2D quasiperiodic lattice . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2.5 Limitations of simulation . . . . . . . . . . . . . . . . . . . . . . . . 192
9 Plans for Z experiments 195
9.1 Z setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.2 Anticonfining potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.2.1 On-site trapping frequencies . . . . . . . . . . . . . . . . . . . . . . 196
9.2.2 1D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.2.3 2D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9.3 Axicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.3.1 Mathematical derivations . . . . . . . . . . . . . . . . . . . . . . . . 206
9.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Conclusions and Outlook 213
Acknowledgments 215
Bibliography 221
Appendices 237
A Hyperfine structure 239
B More Experimental details 247
C More localisation transitions 259
D Miscellaneous 263
The noblest pleasure is the joy of understanding.
Leonardo da Vinci
Introduction
Past, present and future research in the field
The work presented in this thesis belongs to the field of quantum simulation with ultracold
atoms in optical lattices. This is contained in the macrocanonical ensemble that spans
(ultra)cold atoms, quantum gases and optical lattices, and has seen both experimental
and theoretical breakthroughs with the former leading the latter on several fronts.
The experiments described in this thesis employ Rb and K, two of the most widely used
species in cold atoms. Rb and Na were the first to be Bose-condensed in 1995 [1, 2] which
led to the 2001 Physics Nobel prize. Potassium, on the other hand, exhibits experimentally
accessible Feshbach resonances which allow investigation of strongly interacting physics,
though still contact and short-range. Other established atomic species to date are H,
He, Cs, Li, Na, Yb, Sr, Er, Dy, Ca and Cr, which have all been Bose-condensed (in
some cases multiple isotopes) along with more elements soon to join the list, such as
Fe and Ti [3]. Novel, different and alternative cooling and trapping schemes are also
being investigated, and are culminating in technical milestones such as an all-laser-cooled
Bose-Einstein condensate of Rb [4].
A field of great interest is strongly interacting gases, as interactions and correlations lead
to the emergence of new and distinct many-body phenomena (“more is different”[5]).
These range from the creation of quantum droplets [6, 7] to the unitarity and universality
regimes in Fermi gases [8] and (recently emerging) Bose gases [9, 10].
After the generation of the first degenerate Fermi gas [11], tuneable interactions enabled
observation of the BCS-BEC crossover [12] and creation and subsequent condensation of
molecules [13, 14]. These not only grant access to the field of ultracold quantum chemistry,
but also to long-range interactions, where they join the ranks of Rydberg and two-electron
atoms.
The controllable and tuneable environment provided by cold atoms enabled the experi-
mental investigation of the BKT phase [15] and theoretical proposals to realise the FFLO
phase [16], as well as empirical studies of out-of-equiliubrium phenomena from turbulence
[17] to the Kibble-Zurek mechanism [18]. Cold atoms are also being used to reproduce
effects pertaining to high-energy physics and cosmology, such as Sakharov waves [19], the
dynamical Casimir effect [20], Hawking radiation [21], and the Unruh effect [22].
These last applications fall under the umbrella of quantum simulation, Feynman’s original
idea [23, 24] to circumvent the numerical intractability of large Hilbert spaces by perform-
ing the investigation directly on a quantum platform. A system of 300 spins would require
10300 bits of storage on a classical transistor-based computer, more than the number of
protons in the Universe; simulating a macroscopic quantum device with an Avogadro’s
number of particles would therefore be impossible.
1
2 Introduction
Key protagonists of quantum simulation are optical lattices, generated from interfering a
variable number of laser beams, enhancing interactions and providing a flexible platform to
study cold atoms in systems exhibiting different symmetries and dimensionalities. These
have allowed direct observation of quantum phase transitions [25] and generation of novel
phases of matter such as supersolids [26, 27], Bose glasses [28, 29], Anderson localised
[30] and many-body localised [31, 32] states. They have also been combined with the
single-site resolution enabled by quantum gas microscopes [33, 34].
Optical lattices also offer insight into topological phases of matter, where many-body
effects may emerge even in the absence of interactions. These range from the experimental
implementations of the Haldane [35] and Harper [36] models, to artifical gauge fields and
synthetic dimensions [37, 38], also encompassing Floquet and Hamiltonian engineering in
the time domain, leading to new dynamics [39] and phase transitions [40]. Finally, optical
lattices can be used as clocks [41], where they join the metrology and ultra-high precision
measurements brigade along with long-lived Rydberg and two-electron atoms.
Quantum simulation is also perfomed on other platforms, from trapped ions (though hard
to produce in large numbers, the current record is 200 [42]) and Bose-condensed photons
[43], where thermo-optic phase imprinting [44] provides an analogue to optical lattices,
to photonic crystals, waveguides and solid-state systems probed by scanning electron
microscopy.
From talks and grant proposals, the goal of cold atoms research seems to be providing
a platform for quantum computation, and indeed trapped ions and Rydberg atoms are
being used for quantum information processing owing to their long lifetimes and individual
addressability. This is also driving the miniaturisation and commercialisation of quantum
technologies.
Realistically, however, no quantum computer is likely to be made of cold atoms, at least
not anytime soon. Rather, these platforms hold the high ground because of their ability
to perform analogue quantum computation, i.e. quantum simulation. This renders them
ideal to implement, test and disprove theories from all fields of physics, from the latest
measurements of the electric dipole moment [45], to proposals to couple cold matter to
dynamical gauge fields [46], investigate AdS/CFT correspondance [47, 48] and detect
gravitational waves [49]. These might soon question the need for expensive and bulky
experiments such as particle accelerators.
Our work
We have implemented a novel geometry of optical lattices in order to study 1D and 2D
quasicrystalline phases of matter, which lack translational invariance and periodicity but
still possess long-range order.
The fractal and self-similar nature of quasicrystals results in non-trivial structure on all
lengthscales which adds complexity to the otherwise too simple and too clean optical
lattices.
Novel phases of matter are expected to emerge, from Anderson to many-body localised
states in both fermions and bosons (Bose glasses). Our system also allows investigation
of 4D physics, access to synthetic dimensions, and excitations of new quasiparticles such
as phasons.
The lack of a unit cell, the invalidity of Bloch’s theorem, and the non-separability of
Introduction 3
the 2D potentials discussed in this thesis make the study of quasicrystals a numerically
intractable problem. A quantum simulator is hence an ideal platform to investigate the
properties of this class of materials.
This thesis
This is one of the first theses produced by our group, and it is meant to be a manual for
all our future recruits. I tried to write the thesis that I would have wanted to read when I
started my own PhD, having transitioned from a different field and lacking the necessary
theoretical foundations and experimental knowledge.
The manuscript is divided into the following sections:
• Part I: This introduces statistical physics and the Bose-Einstein condensed phase
(chapter 1), and discusses cooling and trapping schemes for cold atom experiments
(chapter 2). A review of the current physical and mathematical understanding of
quasicrystals is presented in chapter 3.
• Part II: The experimental apparatus, including its hardware and software com-
ponents, is detailed in chapters 4 and 5. Chapter 6 summarises the imaging and
characterisation methods of atomic samples.
• Part III: Matter-wave diffraction experiments yielding sharp diffraction peaks and
establishing long-range order are presented in chapter 7, while chapter 8 discusses
the numerical and experimental investigation of an extended-to-localised phase
transition. Finally, other miscellaneous findings are presented in chapter 9.
Contributions
The experimental apparatus described in chapters 4 and 5 was jointly designed, assembled,
and characterised by (Dr.) Konrad Viebahn and myself, as we were the first PhD students
in the group. Major contributions were made by Dr. Stephen Topliss (group electronic
engineer), Hendrik von Raven, Oliver Brix, Edward Carter, Max Melchner, Michael Ho¨se,
and Jr-Chiun Yu.
Part I
Theoretical foundations
Modern Physics
Aristotle said a bunch of stuff that was wrong.
Galileo and Newton fixed things up. Then Einstein broke everything again.
Now, we’ve basically got it all worked out, except for small stuff, big stuff,
hot stuff, cold stuff, fast stuff, heavy stuff, dark stuff, turbulence, and the
concept of time.
Zach Weinersmith [50]
5
Do you guys just put the word quantum in front of everything?
Scott Lang, from Ant-Man and the Wasp [51]
1
Quantum, interacting gases
This chapter discusses the thermodynamic equilibrium distributions and the
phase transition to a Bose-Einstein condensate, introducing equations and no-
tions used later in the thesis such as phase space density and collisional dy-
namics.
1.1 Statistics
1.1.1 Indistinguishability
The Pauli exclusion principle
A wavefunction Ψ depending on two sets of variables {1, 2}, e.g. the positions of the two
electrons in a Helium atom, can be constructed in the absence of interactions from the
single-particle eigenstates ψ and ϕ:
Ψ(1, 2) = ψ(1)ϕ(2). (1.1)
The exchange operator Ex is defined to swap the labels 1↔ 2:
Ex [Ψ(1, 2)] = eΨ(2, 1), (1.2)
and upon its double action one would expect to return to the initial configuration:
Ex2 [Ψ(1, 2)] = e2 Ψ(2, 1) != Ψ(1, 2). (1.3)
The eigenvalues of Ex must therefore satisfy e2 = 1, and hence e = ±1: upon label ex-
change, the wavefunction may either acquire an overall minus sign (antisymmetric wave-
function) or remain exactly the same (symmetric wavefunction).
Quantum particles are identical and hence indistinguishable, a fact that needs to be
encoded in the wavefunction itself, as shown in eq. 1.4, to ensure the invariance of physics
7
8 Quantum, interacting gases
upon particle relabelling, for it was us the observers who had (arbitrarily) labelled them
in the first place:
Ψind(1, 2) =
1√
2
[ψ(1)ϕ(2)± ψ(2)ϕ(1) ] . (1.4)
As a consequence of eq. 1.4, an antisymmetric wavefunction with the same set of labels
(or quantum numbers) is equal to zero. Two identical particles can therefore not co-exist
at the same position, an effect known as the Pauli exclusion principle and observed by
fermions. A symmetric wavefunction, on the other hand, is displayed by bosons. The
particles’ positions can thus be affected in the absence of any external force, as illustrated
in fig. 1.1. This dramatically impacts the physical and chemical properties of macroscopic
materials, such as 3He and 4He which show distinct superconductivity mechanisms and
transition temperatures despite possessing the same electronic structure.
(a) symmetric:
|cos(x) cos(2y)+ cos(y) cos(2x)|2
(b) antisymmetric:
|cos(x) cos(2y)− cos(y) cos(2x)|2
(c) cut along y = 0
Figure 1.1 : Visualisation of the Pauli exclusion principle exhibited by two electrons in
the ground and first excited states of a square well. In (a), the symmetric combination
enables coexistence at x = y = 0 while in (b), the probability density ∀x = y is exactly
zero. The indistinguishability of quantum particles leads to bunching and anti-bunching
effects (c).
Topological argument
The following section is quite mathematical and is provided as a more rigorous proof for
the Pauli exclusion principle in all dimensions, in order to dot the i’s and cross the h¯’s.
It is not relevant for the rest of the thesis so it can be regretlessly skipped.
The argument presented in the section above is short, simple, and quite common in the
literature. It is however a bit too simple: why should Ex2 [Ψ(1, 2)] in eq. 1.3 be exactly
equal to Ψ(1, 2) and not just proportional to it? Quantum mechanics is defined over a
projective Hilbert space, so any phase factor would not affect the physical observables
either way.
Permutation of the particles’ labels is a continuous operation over a parameter space,
as illustrated in fig. 1.2, and it cannot reliably be characterised by the action of the
instantaneous operator Ex. The system will acquire a phase factor depending on the
specific loop that was traced out in this space, akin to the geometric and Berry phases
gained during parallel transport in general relativity and quantum mechanics [52].
1.1. Statistics 9
Figure 1.2 : The double action of the exchange operator Ex2 in eq. 1.3 is a continuous
transformation corresponding to a loop γ traced out by particles 1 and 2 in a pointed
topological space (X, a).
The description of loops and spaces requires basic topology. A topological space X is
a mathematical structure whose details are not needed here, while a pointed topological
space (X, a) is the same as X but with a specified point a ∈ X enabling definitions of
closed loops γ about a. The nth homotopy group consists of the topologically distinct
n-dimensional spheres Sn, i.e. that cannot be deformed into one another. In order to
classify the symmetries of the one-dimensional loops, we employ the first homotopy (or
fundamental) group pi1:
pi1(X, a) = {[γ] | γ : [0, 1]→ X, γ(0) = γ(1) = a}. (1.5)
The smooth and continuous label exchange operation defined in eq. 1.3 and shown in fig.
1.2 is a proper rotation in d dimensions, represented by the the Special Orthogonal group
SO(d):
SO(d) = {R being a d× d matrix : RT = R−1, detR = 1}. (1.6)
SO(d) is a Lie group, hence a differentiable manifold and a type of topological space: its
fundamental groups are listed in table 1.1.
dimension group fundamental group comments
d SO(d) pi1
1 SO(1) identity trivial, just one element
2 SO(2) Z winding number, countably infinite
> 3 SO(d > 3) Z2
only two topologically
distinct elements
Table 1.1 : Fundamental groups of SO(d).
In d = 2, it is impossible to avoid a crossing and hence there are countably infinite
topologically distinct ways of looping from a back to itself, each distinguished by the
winding number w as illustrated in fig. 1.3.
10 Quantum, interacting gases
(a) (b)
Figure 1.3 : In d = 2, rotations are represented by SO(2) and visualised as a circle S1.
Closed loops about a are topologically distinguished by the winding number w.
However, in d > 3, there exist only two topologically distinct permutations, as illustrated
in fig. 1.4. This is a consequence of the 2-sphere S2 being mapped onto SU(2), which is
the double cover of the rotation group SO(3): SO(3) ∼= SU(2)/Z2. The space of proper
rotations SO(3) is then confined to a hemisphere with opposite ends identified. This
redundancy is justified by noting that a rotation of θ about n is identical to one of pi − θ
about −n.
(a) (b) (c)
Figure 1.4 : The paths on the SO(3) hemisphere shown in (a) and (b) are topologically
distinct, as the former is contractible to a point while in the latter A and A′ are fixed
on the boundary. The path in (c) is topologically equivalent to (a), since deformation of
B and B′ allows bringing the loops above the boundary and hence shrinking them to a
point. In d = 3, therefore, there exist only two distinct loops.
The results in table 1.1 result from the existence of a double cover for any d > 3, which
guarantees only two topologically distinct loops.
Particle exchange can only amount to an overall phase factor θ in the wavefunction:
Ex[Ψ(2, 1)] = eiθΨ(1, 2), (1.7)
which can be acquired in pi1 distinct ways: 2pi/Z2 in d > 3 and 2pi/Z for d = 2. In 3D
space, the eigenvalues of the exchange operator are ±1 and correspond to bosons and
fermions, but in 2D they may be any rational divisor of 2pi, and are referred to as anyons.
1.1. Statistics 11
As shown in fig. 1.1, the symmetry of the overall wavefunction determines whether
particles can exist in the same quantum state. Hence, bosons and fermions are distin-
guished by the commutator (anti-commutator) of their creation and annihilation operators
a and a†:
for bosons:
[
a, a†
]
= 0,
for fermions:
{
a, a†
}
= 0.
(1.8)
Technical note
While quantum field theory is needed to relate particle statistics to spin, it is not required
here. The foundation of the proof lies in the existence of a double cover for SO(d > 3),
which is satisfied by both the Poincare´ P = SO(1, 3) n R1,3 and the Galilean group
G = (SO(3)nR3)× (R1 × R3) because SO(1, 3) 2:1←− SL(2, C) and SO(3) 2:1←− SU(2).
1.1.2 Classical, ideal gases
Classical and ideal are opposed to quantum and interacting1.
An atom in a classical gas follows a defined trajectory in real space (r) and in momentum
space (p), which together characterise its state s(r,p) in phase space.
The Hamiltonian for non-interacting particles in an external potential V (r) is given by:
H0(r,p) =
p2
2m + V (r). (1.9)
In the grand canonical ensemble, the system is in thermal and chemical equilibrium with
a large reservoir at temperature T and chemical potential µ. The occupation probability
of a state at energy E is given by:
P = 1
Z
e−β(E−µN) with Z =
∑
j
e−βE−µN , (1.10)
where Z is the grand partition function and β = 1/kBT . P describes a Boltzmann distri-
bution.
In the continuum, assuming the classical regime of dilute occupation of each state such
that E µ, one can write:
P (r,p) = 1
Z
e−H0(r,p)/kBT with Z =
∫
d3p d3r e−H0(r,p)/kBT , (1.11)
where P is now the probability density of finding a particle at position r and momentum
p. We then introduce the phase space density as the number of singly occupied points
per unit phase space volume, ρ(r,p) = N · P (r,p).
• The probability density of finding a particle at position r is n(r):
n(r) =
∫
d3p ρ(r,p) = N
∫
P (r,p) r =
ρ(0, 0) e−
V (r)
kBT
∫ ∞
0
e−
p2
2mkBT 4pip2dp
⇒ n(r) = n0 e−
V (r)
kBT ,
(1.12)
1However, even an ideal gas is weakly interacting, in order to be able to reach thermal equilibrium.
12 Quantum, interacting gases
where ρ(0, 0) and n0 are the central densities2 in phase space and in real space,
related by:
ρ(0, 0) = N
Z
= n0(2pimkBT )3/2
= n0λ
3
th
(2pih¯)3 =
D
(2pih¯)3 . (1.13)
λth ≡
√
2pih¯2/mkBT is the thermal de Broglie wavelength and D = n0λ3th ∝ N/Z
the degeneracy parameter. The regime where D ∼ 1 corresponds to one particle per
Planck volume, where the (classical) treatment of low state occupation is no longer
applicable thus signifying the onset of quantum degeneracy.
The number of particles Ntot can be expressed as:
Ntot =
∫
n(r)d3r = n0
∫
e−
V (r)
kBT d3r. (1.14)
The potentials V (r) usually employed to trap cold atom are power-laws of the form:
V (r) = V0r3/γ, (1.15)
where γ → 0 results in a spherical square well (the free system limit), γ > 3 in
dimple traps, and γ = 3/2 corresponds to the harmonic trap V0 = 1/2mω¯2r2 used
in our experiments.
For future use, it is useful to quote the following integral:∫
e−
V0r
3/γ
kBT d3r ∝
(
kBT
V0
)γ
. (1.16)
• The probability density of finding a particle at momentum p is n(p):
n(p) =
∫
d3r ρ(r,p) = Ntot
∫
P (r,p) r =
ρ(0, 0) e−p2/2mkBT
∫
e−
V (r)
kBT ,
⇒ n(p) = N
( 1
2pimkBT
)3/2
e−
p2
2mkBT︸ ︷︷ ︸
fM(p)
,
(1.17)
where the space integral was replaced by eq. 1.14.
fM(p) is the Maxwell-Boltzmann distribution, also expressed for the magnitude
p = |p| as:
fM(p) = 4pip2fM(p). (1.18)
The de Broglie wavelength λth = h/p controls the size of a wavepacket with momentum p.
Quantum degenerate effects become important when the wavepackets start overlapping
so that the wave nature of matter cannot be neglected, in the same way as geometric
optics is not applicable when the wavelength λ is comparable to the object size d.
This is equivalent to the regime where there is one particle per volume λ3th:
1 ∼ Ntot
V
λ3th = n0λ3th = D, (1.19)
2and usually also the peak densities, since most confining potentials used for trapping atoms have a
minimum at their centre.
1.1. Statistics 13
where Ntot is the number of particles, V is the volume, and D the degeneracy parameter.
Typical values of D are shown in table 1.2.
System T [K] D regime
Air 300 ∼ 10−10 Classical
Superfluid 4He 4 ∼ 1.5 Quantum
Valence electrons in Cu 300 ∼ 2, 000 Very quantum
Table 1.2 : The degeneracy parameter D determines when the wave nature of matter is
dominant hence requiring a quantum treatment.
1.1.3 Quantisation
Quantum distributions
When D ∼ 1 the assumption of sparse population of states breaks down and one needs
to account for particles in the same (quantum) state, which will be subject to the con-
sequences of indistinguishability discussed in section 1.1.1.
The Hamiltonian is replaced by a quantum operator with eigenstates |Ψi〉 and eigenvalues
Ei. In thermal equilibrium, a multitude of many-body energy eigenstates are occupied in
order to maximise the entropy, resulting in a thermal density matrix of the form:
e−β(H−µN) =
∑
i
e−β(Ei−µNi) |Ψi〉〈Ψi|. (1.20)
Every many-body state is then “Boltzmann” distributed as in eq. 1.10.
For non-interacting particles, the energy eigenstates are the single-particle states |nj〉,
yielding nj particles at energy Ej. The average number of particles per state, the occu-
pancy f , is defined as:
f = n¯ = 1
Z
∑
j
〈nj| e−(H−µN)/kBTn|nj〉 =
∑
j nj e−nj(Ej−µ)/kBT∑
j e−nj(Ej−µ)/kBT
. (1.21)
Expressing the mean as n¯ = ∑j njP (nj), we identify the probability density of finding n
particles in the same state of energy E to be:
P (n) = 1
Z
e−n(E−µ)/kBT with Z =
∑
n
e−n(E−µ)/kBT =
∑
n
(
e−r
)n
. (1.22)
For bosons, the commutation relation in eq. 1.8 imposes no restriction on the number of
particles in the same state. Hence Z = ∑∞n=0 rn = (1− r)−1 and:
n¯ =
∞∑
n=0
nP (n) = 1
Z
∞∑
n=0
n e−n(E−µ)/kBT = 1
Z
∞∑
n=0
n rns
⇒ n¯ = fBE(E) = 1e(E−µ)/kBT − 1 ,
(1.23)
14 Quantum, interacting gases
For fermions, on the other hand, only two particles of opposite spin can be accommodated
in each state. This restricts the sum Z = ∑1n=0 rn = 1 + r and results in:
n¯ =
1∑
n=0
nP (n) = 1
Z
e−(E−µ)/kBT = r1 + r
⇒ n¯ = fFD = 1e(E−µ)/kBT + 1 .
(1.24)
In the above, fBE(E) and fFD(E) are the Bose-Einstein and Fermi-Dirac equilibrium
distributions obeyed by quantum degenerate particles. Both of these taper off into the
classical, Boltzmann limit for E µ and high temperatures T , as shown in fig. 1.5.
(a) T1 (b) T2 (c) T3
Figure 1.5 : The occupancy f(E) determined by the three equilibrium distributions.
While they converge at large E, their behaviour is dramatically different for low temper-
atures T3 < T2 < T1.
Relation to spin
Section 1.1.1 required indistinguishability of quantum particles based on them being
identical. In the true sense of the word, identical particles transform in the same ir-
reducible representation of the Poincare´ group, uniquely identified by their mass and
spin. Proper treatment of spin requires quantum field theory, since quantum mechanics
is incompatible with special relativity as it does not obey causality.
For a theory to be causal, the time ordering of physical events affecting the system’s
evolution cannot be reversed. This is especially problematic for space-like separations
where a Lorentz boost may reverse the chronological order tfinal− tinitial < 0. For causality
to hold, any two space-like separated operators are required to commute:
[O1(x),O2(y)] = 0 if (x− y)2 < 0, gµν = (+,−,−,−), (1.25)
to ensure their time ordering is irrelevant and not resulting in any physical consequence.
The relativistic counterparts of the commutation relations in eq. 1.8 are:
bosons: [a(p, s), a†(p′, s′)] = 2Ep(2pi)3δ(3)(p− p′)δs,s′ ,
fermions: {a(p, s), a†(p′, s′)} = 2Ep(2pi)3δ(3)(p− p′)δs,s′ .
(1.26)
A general field operator φ for non-interacting particles A and B is given by:
φA,B(x) =
∫ d3p
(2pi)3
1
2Ep
∑
s
[
e−ipxfA,B(p, s)a(p, s) + eipxhA,B(p, s)a†(p, s)
]
p0=Ep
. (1.27)
1.2. Bose-Einstein condensation 15
Because operatorsO(x) are usually just a product of∏i φi(x), requiring [O1(x),O2(y)] = 0
is the same as requiring [φA(x), φB(y)] = 0.
This results in:
for bosons: [φA(x), φB(y)] ∝ 1− (−1)2j,
for fermions: [φA(x), φB(y)] ∝ 1 + (−1)2j,
(1.28)
where j, in the absence of orbital angular momentum, is the particle’s spin. The proof of
these relations is provided in [53], and is outlined in appendix D.1.
The ladder operators used in angular momentum algebra result in a quantised spectrum
bounded by j and −j, so that −j + n = j with n ∈ N. This requires j = n/2, i.e. either
an integer or a half integer.
Hence, in conclusion:
bosons⇔
[
a, a†
]
= 0⇔ integer spin,
fermions⇔
{
a, a†
}
= 0⇔ half-integer spin.
(1.29)
1.2 Bose-Einstein condensation
The theory presented here is thoroughly and rigorously discussed in [54] and [55].
1.2.1 A distinct phase of matter
There is a profound consequence of the Bose-Einstein distribution given in eq. 1.23. Not
only is the occupancy f(E) allowed to be larger than 1, meaning multiple bosons can exist
in the same quantum state, but it has to always be positive, requiring (Ej−µ)/kBT > 0 ∀j.
The ground state energy, chosen to be E0 = 0, sets an upper bound for the chemical
potential µ 6 0. The occupancy of any state j 6= 0 is therefore capped at:
max{fj(Ej)} = 1eEj/kBT − 1 . (1.30)
While still large for high temperatures, this starts becoming worryingly small as T → 0,
where the only state retaining infinite occupancy is the ground state with E0 = 0.
Decreasing the temperature T at constant particle number N , increasing N at constant
T , or a mixture of the two, eventually causes the excited states to reach their maximal
occupancies making only the ground state statistically accessible. Reaching the critical
number Nc or temperature Tc thus triggers a macroscopic occupation of the ground state
in a process known as Bose-Einstein condensation (BEC), as shown in fig. 1.6a.
16 Quantum, interacting gases
(a) Figure taken from [56] (b) Figure taken from [57]
Figure 1.6 : a) Bose-Einstein condensation can be interpreted as a saturation effect.
The solid lines show that at constant temperature, a cloud of non-interacting Ntot atoms
amass in the condensed fraction (N0) as opposed to the excited states N ′ when increased
above a critical value Nc. The presence of interactions (experimental data), changes the
critical point and the saturation behaviour. b) The visibility of the fringes in a matter-
wave interference pattern as a function of the spatial extent of the atomic cloud. For a
thermal state, this decays to zero within the thermal de Broglie wavelength λth, whereas
for a BEC it stays constant owing to the presence of off-diagonal long-range order.
The total number of particles Ntot can be written as:
Ntot = N0 +
∫ ∞
0
f(E)g(E) dE︸ ︷︷ ︸
Nex
, (1.31)
where the population of the ground (Bose-condensed) state N0 is separated because its
energy E0 = 0 is not taken into account by the integral of the occupancy f(E) and density
of states g(E), which therefore only results in the excited state population Nex.
The density of states g(E) is usually given by
g(E) ∝ Eα−1 with
{
α = d/2 for free particles in d dimensions
α = d for particles in a d−dimensional harmonic trap
The critical transition temperature Tc is reached when the total number of particles can
be just accommodated in the excited states:
Ntot = Nex =
∫ ∞
0
dE f(E) g(E) = c(α) · (kTc)α
⇒ kTc = N
1/α
tot
c(α) .
(1.32)
A BEC transition occurs if Tc exists and is non-zero, which is only allowed for α > 1 by
the convergence of the integrals in the constant c(α). Hence, a BEC can only be produced
in d > 3 dimensions, or in d > 2 if in the presence of a confining d > 2 harmonic potential.
The functional dependence of the density of states g(E) ∝ Eα−1 controls the value of Tc
as it determines whether or not particles can still be accommodated in the exicted states
as T → 0. This is also explained by the Mermin-Wagner theorem [58–60].
1.2. Bose-Einstein condensation 17
The condensed and excited (thermal) populations N0 and Nex can then be expressed as:
Nex = Ntot
(
T
Tc
)α
, N0 = Ntot
(
1− T
Tc
)α
. (1.33)
A BEC is described by a macroscopically occupied wavefunction, characterised by a single,
physical and non-gaugeable phase which results from breaking the U(1) symmetry cor-
responding to particle number conservatio [61]. The BEC is a coherent state |α〉 with
zero entropy, whereas a generic thermal ground state consists of the statistical mixture of
single-particle states that maximises entropy. Reaching a BEC therefore not only requires
loss of energy via cooling, but also, more importantly, the removal of entropy, as further
discussed in section 2.3.1.
The mechanism that guarantees stabilisation of the single phase in the BEC is constructive
interference. In the absence of interactions, the Hamiltonian commutes with the particle
number operator, resulting in the time evolution for single-particle eigenstates ψn of the
form ψn(x, t) = e−iEnt/h¯ψn(x).
A coherent state is defined as:
〈α|x〉 = φα(x) = e− 12 |α|2
∞∑
n=0
αn√
n!
ψn(x), (1.34)
and its time evolution is given by:
φα(x, t) = e−
1
2 |α|2
∞∑
n=0
αn√
n!
ψn(x, t) = e−
1
2 |α|2
∞∑
n=0
αn√
n!
ψn(x, 0)e−i
Ent
h¯ . (1.35)
In the absence of interactions, En = µ + ηnn, where µ = E0 is the ground state energy
and ηn is the energy of every particle in a level n, which then results in:
φα(x, t) = e−i
µt
h¯ φα(x, 0). (1.36)
The time evolution of a pure coherent state is solely controlled by the chemical potential,
which therefore guarantees its phase stability. In a thermal state with multiple occupied
single-particle eigenstates, the presence of a single phase is precluded by the destructive
interference between the different En phase factors.
The emergence of a single phase is confirmed by the asymptotic behaviour of the off-
diagonal one-body density matrix3. In the first quantisation formalism, this is defined as:
n(1)(r, r′) =
∑
i
ni ψ
∗
i (r)ψi(r′)
= n0 φ0(r)∗φ0(r′) +
∑
i 6=0
ni ψi(r)∗ψi(r)
= n0 φ0(r)∗φ0(r′) +
∑
i 6=0
ni e−
i
h¯
p·(r−r′),
(1.37)
where in the last term a special case of free particles was assumed, to express them as plane
waves. At increasing separation, this tends to a constant value because the contributions
of p 6= 0 average out:
lim
|r−r′|→∞
n(1)(r, r′)→ n0 6= 0. (1.38)
Experimental confirmation is provided in fig. 1.6b, where the BEC exhibits phase coher-
ence across its entire spatial extent.
3This is used in the Penrose-Onsager criterion [62] as a rigorous definition for a BEC.
18 Quantum, interacting gases
1.2.2 Useful formulae
For a confining potential V (r), the density of particles in the excited states can be ex-
pressed as:
nex(r) =
Nex
volume =
g3/2
(
e−V (r)/kBT
)
λ3th
, (1.39)
where Nex is the same as in eq. 1.31, and the polylogarithmic gγ and Riemann zeta ζ
functions are defined as:
gγ(z) =
∞∑
n=1
zn
nγ
, gγ(1) = ζ(γ). (1.40)
Most confining potentials in our experiments are three-dimensional harmonic traps of the
form:
V (r) = 12m(ω
2
xx
2 + ω2yy2 + ω2zz2) =
1
2mω¯
2r2, (1.41)
where ω¯ is the geometric mean of the trapping frequencies ωi, and m the mass of the
atom.
The critical temperature and atom number from eqs. 1.32 and 1.33 with α = 3 are:
Tc =
h¯ω¯
kB
(
N
ζ(3)
)1/3
= 0.94 h¯ω¯
kB
N1/3, (1.42)
N0
Ntot
= 1−
(
T
Tc
)3
. (1.43)
Defining the Thomas-Fermi radii Ri and their average R¯ as:
R2i =
2kBT
mω2i
, R¯2 = 2kBT
mω¯2
, (1.44)
the total number of particles Ntot in the absence of interactions is found using 1.14:
Ntot = n0
∫
d3r e−
V (r)
kBT = n0
∫
d3r e−
r2
R¯2 = n0pi3/2R¯−3
⇒ n0 = N
(
mω¯2
2pikBT
)3/2
.
(1.45)
The peak density in real space n0 allows expressing the degeneracy parameter D (eq.
1.19) in a 3D harmonic trap as:
D = n0λ3th = Ntot
(
h¯ω¯
kBT
)3
. (1.46)
In the presence of weak interactions the BEC can be described by the non-linear Gross-
Pitaevskii equation (GPE):(
− h¯
2
2m∇
2 + V (r) + g|ψ|2
)
ψ = µψ with g = 4pih¯
2a
m
, (1.47)
1.3. Interactions 19
where a is the scattering length quantifying the inter-particle interaction strength and is
discussed in section 1.3. The GPE is a mean-field approximation because each particle
(ψ) is made to experience the averaged interaction from the others (|ψ|2).
An approximate solution for the density n(r) of the many-body ground state is obtained
using the Thomas-Fermi approximation, where the cloud is assumed to be cold enough
to neglect the kinetic energy: (
V (r) + g|ψ|2
)
ψ = µψ
⇒ n(r) = |ψ(r)|2= µ− V (r)
g
.
(1.48)
The chemical potential µ = kBT controls the spatial extent of the cloud, from eq. 1.44:
R2i =
2µ
mω2i
. (1.49)
In a harmonic trapping potential, the density of the condensed fraction is therefore de-
scribed by the inverted paraboloid:
n(r) = n0
(
1− x
2
R2x
− y
2
R2y
− z
2
R2z
)
. (1.50)
Bose-Einstein Condensate vs Superfluid
Unlike many familiar phase transitions, the BEC is not driven by the interactions between
particles but is a natural consequence of their quantum statistics. The BEC is an equib-
librium phase of matter, in which one quantum state is macroscopically occupied4.
Superfluidity relates to the transport properties of the ultracold gas, introducing an
interaction-dependent critical velocity vc below which atoms do not experience viscos-
ity. In this context, a non-interacting BEC has vc = 0 whilst a weakly interacting BEC
has vc 6= 0 and may thus be referred to as a superfluid (the presence of interactions also
shifts the transition point [63]).
The two phase are, however, quite distinct.
Superfluidity, for example, does not necessarily interest only one quantum state (the
condensed fraction in liquid 4He is predicted to be ∼ 10% even at T = 0) and may arise
in systems where Bose-Einstein condensation is forbidden (such as in BKT transitions),
as further discussed in the introduction of [64].
Irregardless of these differences, the term “superfluid” is sometimes (inappropriately)
extended to also include the non-interacting BEC, especially in the context of the localised
phases of matter discussed in chapter 8.
1.3 Interactions
1.3.1 Collisions and Feshbach resonances
Collisions
The interaction between two atoms Vint(r) with separation r is usually modelled by the
Lennard-Jones potential shown in fig. 1.7a, which phenomenologically describes the Van
4Unless it is a fragmented BEC, see [62].
20 Quantum, interacting gases
der Waals interaction. The repulsion of the electronic clouds at short separation and the
attraction driven by the mutually induced electric dipoles give rise to an attractive well
of size r0 ∼ 10−9 − 10−10 m [65].
In cold and dilute gases, the inter-particle separation and the thermal de Broglie wavelength
are both much larger than r0. The atoms will hence not be able to resolve the details
of the interaction potential, which justifies its replacement with a spherically symmetric
pseudopotential resulting in the same behaviour at large distances.
One then expects an asymptotic scattering solution of the form:
ψk(r) ∼ eikz + f(k, θ)e
ikr
r
, (1.51)
where an incoming plane wave along the z axis is scattered into a spherical wave along r,
of amplitude f determined by the initial momentum k and the deflection angle θ.
The spherical symmetry allows use of the spherical Bessel functions j`(kr) and spherical
harmonics Y m` (θ, φ) = P`(cos θ), where the m dependence has been dropped owing to the
axial symmetry of the system. In this basis, the incident plane wave can be written:
eikz =
∞∑
`=0
(2`+ 1)i`j`(kr)P`(cos θ) with j`(kr) =
1
2ikr
[
(−1)`+1e−ikr + eikr
]
. (1.52)
The general solution ψk in eq. 1.51 can also be expressed in the spherical basis, identifying
e−ikr as the incoming waves and eikr as the outgoing waves. Since we expect the scattered
wave to retain the same functional form, the potential only amounts to a multiplicative
factor for the outgoing waves S:
ψk =
1
2ikr
∑
`
(2`+ 1)i`P`(cos θ)
[
(−1)`+1e−ikr + Seikr
]
. (1.53)
Upon comparing eq. 1.53 with eqs. 1.51 and 1.52, the scattering amplitude f is found to
be:
f =
∞∑
`=0
1
2ik (S − 1)(2`+ 1)i
`P` cos θ =
∞∑
`=0
(2`+ 1)i`P` cos θf`. (1.54)
If the flux is conserved, |S|= 1 and is hence a pure phase factor S = e2iδ` :
f` =
1
2ik
(
e2iδ` − 1
)
= 1
k cot δ` − ik . (1.55)
In [54] it is given that δ` ∝ k2`+1, which results in f` → 0 ∀` 6= 0. Physically, this
corresponds to atoms not possessing non-zero angular momenta because the kinetic energy
(∝ k2) at low temperatures is lower than the centrifugal potential barrier. The only non-
zero contributions to the scattering amplitude at low temperatures are from s-waves:
lim
k→0
f = lim
k→0
fs = − 11
a
+ ik +O(k2) = a, (1.56)
where a is the scattering length. The scattering cross section is given by σ =
∫ |f |2dΩ =
4pia2, which is the interaction term in eq. 1.47.
For bosons, low-temperature moderate interactions are fully determined by the s-wave
scattering length a which justifies treating them as contact interactions. For fermions,
1.3. Interactions 21
however, the anti-symmetry of the wavefunction precludes s-wave scattering and dramat-
ically reduces interactions at low temperatures.
Finally, a negative scattering length a may result in s-wave scattering also being absent,
which occurs at some k 6= 0 for which fs = 0 and is known as a Ramsauer-Townsend
minimum.
Feshbach resonances
For large separations, the energy associated with the internal atomic structure dominates
over the inter-atomic attractive potential. This gives rise to different potentials experi-
enced by distinct hyperfine states, and distinguishes the scattering channels between open
(energetically accessible) or closed, as shown in fig. 1.7a.
An external magnetic field B interacts with the spins of free atoms and may bring their
energy into resonance with a molecular bound state. This causes a drastic variation of
the scattering length with |B|= B [66]:
a(B) = abg
(
1− ∆
B −Bres
)
, (1.57)
where abg is the background scattering length in the absence of a magnetic field, and ∆
is the width of the Feshbach resonance about Bres.
Table 1.3 lists the resonances employed in our experiment to tune interactions and study
both many-body and non-interacting effects.
The strongly interacting regime where a→∞ is known as unitarity, where both two- and
three-body contact interactions are necessary to describe the ensuing phenomena [9].
A negative (positive) scattering length corresponds to attractive (repulsive) interactions,
as illustrated in fig. 1.7b where a square well was used as a pseudopotential.
(a) (b)
Figure 1.7 : a) Different hyperfine states experience different inter-particle potentials.
Free atoms with negligible kinetic energy may be brought into resonance with a molecular
bound state by tuning the external magnetic field. b) The scattering length a corresponds
to the shift of the scattered wave with respect to a free solution.
22 Quantum, interacting gases
nomenclature species resonant field width backgroundscattering length
zero-crossing
field
Bres/G ∆/G abg/a0 B0
inter-species 87Rb -39K 317.9 7.6 34 325.6
intra-species 39K -39K 402.70 −52 −29 350.5
Table 1.3 : The Feshbach resonances used in our experiments, for the scattering channels
|F = 1,mF = 1〉. Values taken from [9, 66].
1.3.2 Two- vs three-body losses
Elastic collisions result in energy redistribution and are essential for the success of the
evaporative cooling scheme discussed in section 2.3.3. Inelastic collisions, on the other
hand, may lead to particle loss as the released energy is usually larger than the trap depth.
For two atoms in their absolute ground state, there are no available final states after
the collision which prevents any interaction between them. Energy and momentum may
still be conserved, however, in three-body recombination events, where two of the atoms
form a molecule and release the associated binding energy. These phenomena are allowed
because the BEC is only a metastable state of matter, the true ground state being a solid,
and occur at a rate Γloss ∝ L3n2 where L3 ∝ a4 [67] and n is the atomic density.
To quantify the interplay between particle loss and strong interactions, the diluteness
parameter is defined as na3, such that:
Ekin
Eint
∝ 1(na3)1/3 ,
h¯Γloss
Eint
∝ na3, (1.58)
where
Ekin ∼ h¯
2n2/3
2m , Eint ∼
4pih¯2an
m
. (1.59)
In dilute and weakly interacting gases where na3 1, three-body recombinations are low
enough to guarantee a long lifetime of the BEC. The quantity
√
na3 also quantifies the
effects of quantum depletion [63]. Two-body losses may still arise due to the long-range
nature of the Lennard-Jones potentials. In the ground state, the inter-atomic separation
is much larger than the extent of the short-range (∝ 1/r6) interaction, hence not enough
kinetic energy is provided to escape the trap. However, if there are photons present, one of
the atoms may be promoted to the excited state which results in a long-range interaction
(∝ 1/r3) owing to the larger electric dipole. Two atoms can therefore participate in light
assisted collisions, shown in fig. 1.8, where the atoms may acquire enough kinetic energy
to escape the trap and result in a particle loss quantified by the two-body loss rate β [68].
1.3. Interactions 23
Figure 1.8 : In the presence of light, an atom in the ground state S may be promoted
to the excited state P , resulting in a long-range interaction capable of providing enough
kinetic energy ∆E to escape the trap. These light assisted collisions enable a two-atom
loss channel otherwise prevented by kinematic conservation laws.
By convention sweet is sweet, bitter is bitter, hot is hot, cold is cold,
colour is colour; but in truth there are only atoms and the void.
Democritus [69]
2
Cold atoms
This chapter reviews the scattering and confining effects arising from matter-
wave interactions, that are employed for cooling and trapping cold atoms. The
importance of removing both energy and entropy is emphasised, enabling the
increase of phase space density to reach quantum degeneracy. Optical lattices
are also introduced as a versatile and flexible platform to perform quantum
simulation of condensed matter systems.
2.1 Atom-light interactions
2.1.1 Semi-classical treatment
The atom will be treated as a quantum object with discrete energy levels, while the
electromagnetic (EM) field is assumed to contain a sufficiently large number of photons
not to require quantisation. This is a semi-classical approximation, and it does not apply
to spontaneous emission or to few-photon phenomena.
In the presence of an external EM potential Aµ = (φ/c, A), the Hamiltonian for the
atomic electron is:
H = (p + qA)
2
2m − qφ+ V (r), (2.1)
where V (r) is the Coulomb interaction with the nucleus. Choosing the Coulomb gauge
∇ ·A = 0 such that [p,A] = 0, this can be expanded:
H = p
2
2m + V (r)︸ ︷︷ ︸
H0
+ qA · p
m
− qφ+ q
2A2
2m︸ ︷︷ ︸
H1
, (2.2)
where H1 can be treated as a perturbation on the bare electronic Hamiltonian H0 provided
H0 H1.
25
26 Cold atoms
For an external static magnetic field B = ∇×A, the A · p and A2 terms are the linear
and quadratic Zeeman effects, discussed in section 4.3.1 and appendix A.
Here, a monochromatic plane wave is considered, with φ = 0, A = 2A0 sin (k · r− ωt) eˆ,
and a small enough amplitude to neglect the A2 term. H1 can then be written:
H1 = −iqA0
m
(
ei(k·r−ωt) − e−i(k·r−ωt)
)
eˆ · p, (2.3)
which is an example of a harmonic perturbation [70]. Acting on the stationary eigenstates
of H0 with phase factors eiω0t = eiE0t/h¯, the terms in eq. 2.3 change the energy to E0 →
E0 ± h¯ω and can thus be interpreted as absorption and stimulated emission.
The probability amplitude of a transition between an initial |i〉 and final |f〉 atomic state
is given by the matrix element 〈f |H1|i〉, where only the spatial dependence of H1 needs to
be considered. Detailed balance guarantees equal rates of the two processes, hence only
one needs to be analysed.
Redefining H1 as:
H1 = qA0ω eik·r eˆ · d, (2.4)
allows introducing the operator d:
d = −i
mω
〈f |eik·rp|i〉. (2.5)
A power expansion of the incident plane wave:
eik·r = 1 + ik · r + . . . (2.6)
then enables a perturbative expansion of H1 (derived in appendix D.2):
H1 = qA0ω eˆ · d =
= qA0ω
〈f |eˆ · r|i〉︸ ︷︷ ︸
electric dipole
+ 12mc(kˆ× eˆ) · 〈f |L + 2S|i〉︸ ︷︷ ︸
magnetic dipole
+ iω2c eˆ ·Q · kˆ︸ ︷︷ ︸
electric quadrupole
. . .
.
(2.7)
The names assigned to the first two terms stem from recognising qr as the electric dipole
moment, and 1/c (kˆ× eˆ) as the magnetic component of the plane wave.
For |i〉 6= |f〉, non-vanishing matrix elements result in the selection rules determining the
allowed transitions, discussed in the next section. On the other hand, the diagonal matrix
elements |i〉 = |f〉 cause energy shifts1 of the bare atomic states. For neutral atoms with
no permanent electric dipole moment and hence with an inversion symmetry r → −r,
energy corrections caused by the electric dipole term vanish and no first-order Stark shift
(∝ E) is expected. However, L + 2S is an axial vector and thus unchanged by a parity
transformation, resulting in a non-zero first-order Zeeman shift (∝ B).
1While transitions |i〉 6= |f〉 are represented by Feynman diagrams with one vertex only, diagonal
energy shifts |i〉 = |f〉 are visualised by Feynman diagrams with two or more vertices, and the same
incoming and outgoing states. This is further discussed in section 7.1.3.
2.1. Atom-light interactions 27
Selection rules
An atom can accommodate electrons in its discrete energy levels labelled by the principal
quantum number n and the orbital angular momentum `, defining electronic shells and
sub-shells. In the absence of an external magnetic field, the ` sub-shells are degenerate in
their 2`+ 1 states labelled by m`. This is known as the gross structure of the atom.
An electron possesses spin s = 1/2, which can be combined with ` to define the total
electronic angular momentum j = |` − s|, |` − s|+1, . . . , ` + s − 1, ` + s. When multiple
electrons occupy the same sub-shell, the same reasoning can still be applied by combining
the angular momenta of all shell occupants into L, S and J . J is only a good quantum
number within the LS coupling scheme, not valid in the presence of a strong interaction
with an external magnetic field, as discussed in appendix A. For the same n and L,
different J split the energy levels in the fine structure of the atom.
The electronic angular momentum also interacts with the nuclear angular momentum I.
Within the IJ coupling scheme, the total atomic angular momentum F = |J − I|, |J −
I|+1, . . . , J + I − 1, J + I is defined, leading to a further splitting of energy levels which
form the hyperfine structure. Each F is 2F + 1 degenerate in its mF sub-levels.
The lowest order transitions and their associated selection rules are listed in table 2.1.
When incident along the magnetic field, linearly polarised light (pi) drives ∆mF = 0
transitions, while circularly polarised light (σ±) drives ∆mF = ±1 transitions. The
spontaneous decay rate was estimated from analytical expressions for hydrogenic atoms.
The electric quadrupole transition is quoted for reference, but it does not affect electrons
in the s shell because of their isotropic charge distribution.
Order type symbol parity spontaneousdecay rate
selection
rule
forbidden
transitions
Dipole electric E1 odd 1 ∆F,∆mF F : 0 6↔ 0magnetic M1 even α−8 = 0,±1
Quadrupole electric E2 even α−2 ∆F,∆mF F : 0 6↔ 0, 1= 0,±1,±2 1/2 6↔ 1/2
Table 2.1 : Lowest order transitions and associated selection rules. The same rules are
valid beyond IJ coupling just by replacing F → J and mF → mJ . The spontaneous
decay rate is normalised to 1 = α5(mec2/h¯) as derived in appendix D.3.
2.1.2 AC Stark shift
Within the dipole approximation, only the lowest term in eq. 2.7 is retained, resulting in
the interaction Hamiltonian 〈f |HI|i〉 = q〈f |eˆ · r|i〉E0 cos(ωt). Defining the Rabi frequency
as Ω = qE0/h¯ 〈f |eˆ · r|i〉 allows writing:
HI = h¯Ω|e〉〈g|cos(ωt) + c.c., (2.8)
which only includes off-diagonal elements because of the inversion symmetry discussed
earlier.
28 Cold atoms
A two-level atom is considered for simplicity, consisting of a ground state |g〉 with energy
Eg = 0 and an excited state |e〉 with energy Ee = h¯ω0, detuned from the EM wave fre-
quency by ∆ = ω − ω0.
The evolution of the electronic wavefunction is encoded in the time dependence of its
bare, unperturbed stationary states:
|ψ〉t = cg(t)|g〉+ ce(t)|e〉e−iω0t. (2.9)
Solving the Schro¨dinger equation ih¯∂t|ψ〉 = HI|ψ〉 and neglecting the oscillatory terms
ω + ω0 as per the rotating wave approximation, results in:
ih¯
(
c˙g
c˙e
)
= h¯2
(
0 ei∆t Ω
e−i∆t Ω 0
)
︸ ︷︷ ︸
H(t)
(
cg
ce
)
,
⇒ |ce(t)|2= Ω
2
√
Ω2 + ∆2
sin2
(√
Ω2 + ∆2
2 t
)
,
(2.10)
where the population in the excited state |e〉 performs a Rabi oscillation.
It is beneficial to seek a solution where the Hamiltonian is not time-dependent as in eq.
2.10, hence we redefine the coefficient of the excited state in eq. 2.9 to c′e = ceei∆t:
|ψ〉t = cg(t)|g〉+ c′e(t)e−iωt|e〉,
change of basis : |g〉 → |g, h¯ω〉, e−iωt|e〉 → |e, 0〉,
⇒ |ψ〉t = cg(t)|g, h¯ω〉+ c′e(t)|e, 0〉.
(2.11)
The new dressed basis combines the bare electronic states with the external photon of fre-
quency ω, though the ground state energy is the same in both formalisms. The Schro¨dinger
equation now results in the time-independent Hamiltonian:
H = h¯2
(
0 Ω
Ω −2∆
)
. (2.12)
Diagonalisation of H yields the ground state energy Eg:
Eg =
h¯
2
(
−∆ +
√
Ω2 + ∆2
)
≈︸︷︷︸
∆Ω
± h¯Ω
2
4∆ , (2.13)
which is shifted from Eg = 0 depending on the detuning of the incident radiation, as
shown in the |∆| Ω regions of fig. 2.1. This shift is known as the AC Stark shift and
results in a repulsive interaction for blue detuned light (∆ > 0), and in an attractive
(confining) interaction for red detuned light (∆ < 0).
The functional form of the leading correction to Eg in eq. 2.13 is reminiscent of second-
order perturbation theory. This is in fact due to H(t) in eq. 2.10 only containing off-
diagonal elements:
H =
(
Eg a
a∗ Ee
)
=
(
Eg 0
0 Eg
)
︸ ︷︷ ︸
H0
+
(
0 a
a∗ 0
)
︸ ︷︷ ︸
V
, (2.14)
2.1. Atom-light interactions 29
whereH0 is the bare atomic Hamiltonian and V the perturbation which, because 〈k|V |k〉 =
0, results in no first-order corrections. In this context, the AC Stark shift can be inter-
preted as a virtual two-photon transition, quantitatively discussed in section 7.1.3.
Figure 2.1 : In the limit |∆| Ω, the perturbative effect of the external EM field results
in a shift of the bare ground state energy. Red detuned ∆ < 0 light (blue detuned ∆ > 0)
causes an effective attractive (repulsive) potential for the atom. In the region |∆|< Ω,
the presence of a coupling Ω opens a gap between the two diabatic states, enabling a
Landau-Zener transition between them.
Fig. 2.1 is also an example, in the |∆|< Ω region, of a minimal coupling Ω lifting the
degeneracy at ∆ = 0 and leading to level repulsion or an avoided crossing between the
new eigenstates {|−〉, |+〉}. In the experiment (section 5.3.3), this is employed to perform
a state transfer between two diabatic states |mF = −1〉 and |mF = 1〉 (|g, h¯ω〉 and |e, 0〉
in fig. 2.1) known as a Landau-Zener transition [71]. A coupling Ω between the two states
is introduced, and the detuning ∆ is ramped from −∞, where |−〉 = |mF = −1〉, to +∞,
where |−〉 = |mF = 1〉. The probability of tunnelling through the gap is proportional to
e−1/∆˙, hence in the limit ∆˙ → 0 the population is adiabatically transferred between the
two states.
2.1.3 Steady state forces
The unitary time evolution enforced by quantum mechanics prevents the introduction of a
loss term such as spontaneous emission. Hence, we extend the Hamiltonian H formalism
to the Liouville equation, which accommodates dissipation:
dρ
dt =
i
h¯
[ρ,H]−
( −Γρee Γ2ρge
Γ
2ρge Γρee
)
, (2.15)
whereH is from eq. 2.12, Γ is the spontaneous decay rate (assumed to be purely radiative),
and ρ the density matrix:
ρ =
(
ρee ρge
ρge ρee
)
with ρij = 〈i|ρ|j〉 = ci c∗j . (2.16)
30 Cold atoms
The above equation is valid under the rotating wave approximation and the constraints
ρee + ρgg = 1, ρge = ρ∗eg, and is usually cast as the Optical Bloch equations:
d
dt
uv
w
=
uv
w
×
Ω0
∆
−
Γ/2Γ/2
Γ
with
u = 1/2 (ρge + ρeg),
v = 1/2i (ρge − ρeg),
w = 1/2 (ρgg − ρee).
The state of the two-level atom is described by the Bloch vector (u, v, w) precessing about
(Ω, 0,∆) but decreasing in magnitude because of the damping terms Γ.
The steady-state solution is obtained by setting the time derivatives to 0, which results
in the following expression for ρeg:
ρeg = i
I/IS
2Ω
Γ2
2(Γ/2− i∆)(1 + s) , with s =
I/IS
1 + (2∆/Γ)2 , (2.17)
where I is the intensity and IS the saturation intensity, defined from:
I = 120cE
2
0 , IS =
pihcΓ
3λ3 , |〈e|qr|g〉|
2= 3pi0h¯c
3Γ
ω3
. (2.18)
The last term is needed to relate the electric field amplitude E0, the Rabi frequency
Ω = E0/h¯ 〈e|er|g〉 and the excited state lifetime Γ−1, and is obtained from the Wigner-
Weisskopf theory of spontaneous decay.
The force experienced from a two-level atom in the presence of an external potential U
is F = −∇U . Within the dipole approximation, the potential is due to the interaction of
the electron’s dipole moment qr with the external electric field E = E0 ei(k·r+ωt)eˆ. Hence:
F = q∇
(
r · eˆE0 eik·r
)
eiωt ∝ ∇(E0) + kE0, (2.19)
where the constant of proportionality is calculated from expressing r in density matrix
formalism:
r = 〈e|r|g〉ρeg + 〈g|r|e〉ρ∗eg. (2.20)
Using the steady-state value for ρeg in eq. 2.17, the force acting on the atom is given by:
F = − h¯∆2
∇( I
IS
)
1 + I
IS
+
(
2∆
Γ
)2
︸ ︷︷ ︸
Fdip
+ h¯k ·
Rscat︷ ︸︸ ︷
Γ
2
I
IS
1 + I
IS
+ (2∆Γ )2︸ ︷︷ ︸
Fscat
. (2.21)
The first term (Fdip) is the dipole force resulting from the spatially varying intensity of
the incoming electric field. For a large detuning ∆ Γ, Fdip ∝ ∇(Udip) = ∇(h¯Ω2/∆)
where the dipole potential Udip is the generalisation of the AC Stark shift of eq. 2.13. The
second term (Fscat) is the scattering force arising from the momentum imparted on the
atom by the absorbed (or emitted) photon.
We then identify two regimes of matter-wave interaction distinguished by the detuning
∆. For large detuning, the scattering contribution Fscat ∝ I/∆2 is negligible while the
dipole potential Udip ∝ I/∆ is dominant: this is the regime used for the optical trapping
techniques discussed in section 2.2.2. For small detuning, scattering dominates, which
underlies the cooling schemes presented in section 2.3.
2.2. Trapping 31
2.2 Trapping
2.2.1 Magnetic trapping
The Zeeman effect governs the interaction between an external magnetic field B and an
atom with magnetic moment µ, resulting in a potential U = −µ ·B. A trap can therefore
be engineered by a particular choice of a spatially varying magnetic landscape.
Earnshaw’s theorem
A static and source-less solution to Maxwell’s equations require∇×B = µ0J+µ00 ∂E∂t = 0,
where J is the current density and E the electric field. This results in:
((((
(((∇× (∇×B) = ∇(∇ ·B)−∇2B = 0 ⇒ ∇2Bx = ∇2By = ∇2Bz = 0. (2.22)
Extrema in the magnetic field occur when ∇|B|= ∇2B2 = 0, where the equality holds as
|B|=
√
B2 is a monotonic function in B2. Combined with eq. 2.22, this implies:
∇2B2 = 2(∇Bx)2 + 2Bx∇2Bx + 2(∇By)2 + 2By∇2By + 2(∇Bz)2 + 2Bz
∇2Bz ,
⇒ ∇2B2 > 0. (2.23)
The second derivative of B always being positive, a static magnetic field in vacuo never ex-
hibits maxima, which is known as Earnshaw’s theorem. Only the minima of the magnetic
field can thus be used to devise a trap.
The first-order energy shift of an atom in the presence of a magnetic field is given by the
Zeeman effect:
∆E = 〈mF |−µ ·B|mF 〉 = gFmFµBB, (2.24)
where the field was chosen to lie along the z axis in order to use the angular momentum
basis |F,mF 〉. gF is the Lande´ factor and µB the Bohr magneton.
States for which gFmF > 0 are termed low-field seekers as they increase their energy in the
presence of a magnetic field, and are suitable for a magnetic trap as they are energetically
localised near the field minimum. The states with gFmF < 0 are known as high-field
seekers as they lower their energy in the presence of a magnetic field. While this prevents
them from being trapped magnetically, in the optical traps discussed in section 2.2.2 these
states have a lower energy and atoms transition into them via spin-relaxation processes,
releasing energy and leading to particle loss.
Quadrupole field
A common magnetic field configuration is the quadrupole field, which is shown in fig. 2.2
and, near its centre, is given by:
B = b
xy
−2z
⇒ |B|= √x2 + y2 + 4z2 . (2.25)
The full expression depends on the geometry of the (electro)magnets generating it, and
is given in appendix B.2 for rectangular and circular coils.
32 Cold atoms
Figure 2.2 : The spatial configuration of the quadrupole magnetic field given in eq. 2.25.
At its centre, the field is exactly zero.
In order for atoms to remain trapped while moving through this spatially inhomogeneous
field, they need to adiabatically follow its changes and stay in the same mF state with
respect to the local field orientation. From dimensional analysis, this is only guaranteed
if:
v · ∇B
|B| ωL, (2.26)
where v is the velocity of the atom and ωL the Larmor frequency. In a region of space
where the field is too small or too rapidly varying, the adiabaticity condition is not satisfied
and atoms may flip to an untrapped mF state and hence be lost. These losses become
significant for cold clouds with an increasing population at the centre of the field, and
are known as Majorana spin flips [72]. They can be avoided by using different magnetic
field geometries (Ioffe-Pritchard [73] or QUIC [74] traps) or by superimposing an optical
repulsive potential (“plug”) at the trap minimum.
Optical pumping
In order to maximise the number of atoms captured by the magnetic trap, the sample
is spin-polarised into a low-field seeking mF state. Ideally, this is a state at the edge of
the manifold with the largest magnetic moment and the strongest interaction with the
magnetic field. This also suppresses spin-changing collisions by conservation of angular
momentum:
2× |F,mF 〉 → |F,mF − 1〉+ |F,mF + 1〉, (2.27)
where for a state with maximal mF there is no available mF + 1 state, hence reducing
particle loss.
Spin-polarisation may be achieved via optical pumping, where σ polarised light induces
successive transitions between neighbouring mF states, in order for the atoms to accumu-
late into a dark state, as shown in fig. 2.3.
2.2. Trapping 33
Figure 2.3 : Optical pumping of atoms into the |F = 2,mF = 2〉 state. σ+ radiation
causes ∆mF = 1 transitions to the excited manifold F ′, from which the atoms spontan-
eously decay back into F . Successive transitions eventually lead to the accumulation of
the atoms into the dark state, which does not couple to the due to the absence of an
mF ′ = 3 state.
2.2.2 Optical trapping
Polarisability
Optical trapping is enabled by the dipole potential Udip in eq. 2.21, which for large
detuning is equivalent to the AC Stark shift.
The full second-order perturbative treatment [75] results in an atomic energy shift ∆E:
∆E(r) = 3pic
2
2ω30
Γ
( 1
ω − ω0 +
1
ω + ω0
)
I(r), (2.28)
where ω0 is the transition frequency of the two-level atom, Γ the natural linewidth of the
excited state, ω the frequency of the incident radiation and I(r) its intensity distribution.
The rotating wave approximation was not used in order for this expression to be applicable
to arbitrarily large detuning, where the scattering force is suppressed.
The intensity profile of the incident radiation can be engineered to result in a spatially
varying potential enabling to trap the atoms. The dipole potential Udip(r) is usually
expressed as:
Udip(r) = α(ω)I(r), (2.29)
where α is the polarisability of the atom.
The species considered in this thesis, 87Rb and 39K, are both multilevel atoms with two
distinct transitions from the ground state, the D1 and the D2 lines discussed in section
4.3.1. Both need to be taken into account, with a relative strength determined by the
internal structure of the excited states [76]:
U(r) = pic
2
2
(
ΓD2
ω3D2
2
∆D2
+ ΓD1
ω3D1
1
∆D1
)
I(r), with 1∆D
= 1
ω − ωD +
1
ω + ωD
, (2.30)
where the pre-factor to the intensity I(r) is the scalar polarisability. Vector and tensor
contributions to the polarisability [77] depend on the internal states m and m2, and can
be ignored for far detuned light which cannot resolve them.
34 Cold atoms
For red detuned radiation, ∆ < 0 and hence U(r) < 0, resulting in an attractive potential.
Crossed-beam dipole trap
A common geometry for laser radiation is a Gaussian beam, the fundamental TEM00 mode
for the vacuum in the paraxial approximation. Its intensity profile I(r) when propagating
along y is given by:
I(r) = 2P
piwx(y)wz(y)
e−
2x2
w2x(y)
− 2z2
w2z(y) , wi(y) = w0,i
√√√√1 + ( y
yR
)2
, (2.31)
where wi are the waists (w0 at y = 0), yR the Rayleigh range and P the optical power. The
optical potential resulting from a single red detuned Gaussian beam is shown in fig. 2.4a,
and only allows for confinement in the transverse (x, z) directions. Hence, a crossed-beam
configuration comprising two orthogonal beams may be employed, shown in fig. 2.4b.
(a) single beam (b) crossed-beam
Figure 2.4 : Red detuned beams can be used to optically trap atoms.
At the centre of the trap, the intensity may be approximated as:
I(r) = 2P
piwhwv
(
1− 2x
2 + 2y2
w2h
− 4z
2
w2v
− . . .
)
, (2.32)
where wh = wx,0 = wy,0 and wh = wz,0 are the horizontal and vertical waists. The optical
potential is then given by:
U(r) = αI(r) = α 2P
piwhwv︸ ︷︷ ︸
U0
(
1− 2x
2 + 2y2
w2h
− 4z
2
w2v
+ . . .
)
, (2.33)
where U0 is the peak trap depth. This can be further simplified by approximating the
central region as a harmonic trap with trapping frequencies ωρ and ωz:
1
2mω
2
ρρ
2 = U0
2ρ2
w2h
⇒ ωρ =
√
4U0
mw2h
,
1
2mω
2
zz
2 = U0
4z2
w2v
⇒ ωz =
√
8U0
mw2v
.
(2.34)
2.3. Cooling 35
2.3 Cooling
2.3.1 Phase space density increase
As discussed in section 1.1.2, the onset of quantum degeneracy is enabled by the increase
of phase space density. In the following, we show that this can only be performed by
dissipative processes causing entropy loss.
Dissipative processes
Given the phase space density ρ(r,p, t) introduced in section 1.1.2, the total probability
of finding a particle is given by:
P =
∫
d3r d3p ρ(r,p, t). (2.35)
Conservation of probability requires:
dP
dt =
∂
∂t
∫
d3r d3p ρ(r,p, t) = −
∮
S
(r˙ · d2r + p˙ · d2p)ρ(r,p, t) =
−
∫
d3r d3p {∇r · [ρ(r,p, t)r˙] +∇p · [ρ(r,p, t)p˙]} = 0,
(2.36)
where Gauss’ theorem was used. Equating the integrands, as the integration can be
performed on an any arbitrarily chosen volume, yields the continuity equation:
∂ρ(r,p, t)
∂t
+∇r · (ρ(r,p, t)r˙) +∇p · (ρ(r,p, t)p˙) = 0. (2.37)
Using ∇q(ρq) = q˙ + ∇(ρ) + ρ∇ · q˙ and employing Hamilton’s equations of motion, we
obtain:
∇r · r˙ = ∂
∂r
(
∂r
∂t
)
= ∂
∂r
(
∂H
∂r
)
,
∇p · p˙ = ∂
∂p
(
∂p
∂t
)
= − ∂
∂p
(
∂H
∂p
)
,
⇒ r˙ · ∇r[ρ(r,p, t)] + p˙ · ∇p[ρ(r,p, t)] = 0,
(2.38)
which reduces eq. 2.37 to:
∂ρ(q, t)
∂t
+ q˙ · ∇q(ρ[q, t)] = dρ(q, t)dt = 0, (2.39)
where q = (r,p).
The phase space density of a system obeying Hamiltonian mechanics is constant, as guar-
anteed by Liouville theorem. The Hamiltonian formalism rests upon the conservation
of energy, enforced by position-dependent forces and conservative potentials. In order
to cause a change in phase space density, dissipative forces depending on velocity are
required, such as friction.
Entropy
Section 1.1.2 presented a thermodynamic reasoning to relate the phase space density to
the degeneracy parameter D. Here we show how this depends on the entropy of the
system S.
36 Cold atoms
Thermodynamics and statistical physics are related by the following expression involving
the Helmholtz free energy F = E − TS and the partition function Z:
F = −NkBT ln(Z) ⇒ Z = Ne−
F
kBT = N exp
(
− E
NkBT
+ S
NkB
)
, (2.40)
where N , E, T and S are the particle number, energy, temperature and entropy of the
thermodynamic system.
The degeneracy parameter introduced in eq. 1.13, D = n0λ3th ∝ NZ−1, is therefore given
by:
D ∝ exp
(
E
NkBT
− S
NkB
)
. (2.41)
The energy can be extracted from Z as:
E = NkBT 2
∂ lnZ
∂T
, (2.42)
where
Z ∝
∫
d3p e−
p2
2mkBT︸ ︷︷ ︸
∝T 3/2 from eq. 1.12
·
∫
d3r e−
V (r)
kBT︸ ︷︷ ︸
∝T γ from eq. 1.16
, (2.43)
which after some manipulation results in:
E ∝
(3
2 + γ
)
NkBT. (2.44)
The degeneracy parameter D can therefore be expressed as:
D = n0λ3th ∝ exp
(
γ − S
NkB
)
, (2.45)
where γ is the power law of the confining trapping potential.
An increase in the phase space density and therefore in D results from varying the trap
geometry, or from reducing the entropy S of the system. Hence, the laser and evapor-
ative cooling schemes presented in sections 2.3.2 and 2.3.3 make use of dissipative and
loss mechanisms in order to extract entropy from the atomic cloud and reach quantum
degeneracy.
2.3.2 Laser Cooling
Optical Molasses
Laser cooling relies on the momentum transfer between atoms and radiation enabled by
the scattering force Fscat in eq. 2.21. This depends on the ratio I/IS, IS being the
saturation intensity, above which the force saturates. For a high intensity resonant beam,
I IS and ∆ = 0, the maximum scattering force Fmax and acceleration amax exerted on
the atom are:
Fmax = h¯k
Γ
2 , amax =
h¯k
M
Γ
2 . (2.46)
When absorbing an incoming photon, an atom moving at speed v recoils in order to
conserve momentum, resulting in a decrease of its speed as it continues to propagate
along x:
dv
dt = v
dv
dx = −amax ⇒ v
2 = v20 − 2amaxx. (2.47)
2.3. Cooling 37
The atom returns to its ground state by spontaneously emitting a photon, but the recoil
associated with this event averages to zero because of the random direction of the scattered
radiation.
Photons absorbed from the coherent laser radiation have less entropy than the incoherent
spontaneously emitted photons, which results in the atomic cloud losing entropy.
Atoms in a thermal gas, however, require addressing velocities in arbitrary directions. This
is implemented by shining a pair of counter-propagating red detuned beams of frequency
ω < ω0 along each orthogonal direction. A stationary atom that has already been cooled
is not resonant with either beam, as shown in fig. 2.5a, so further light scattering is
avoided. However, atoms with a finite speed v experience Doppler-shifted laser frequencies
ω′ = ω ± kv which may be closer (farther) to (from) resonance, as shown in fig. 2.5b.
For a right-moving atom, the left-propagating beam is closer to resonance, increasing
its absorption rate and hence providing a net decrease of the atom speed, whereas the
right-moving beam is farther from resonance and results in fewer transitions.
The net force experienced by the atoms, Fmolasses, is shown in fig. 2.5c and is given by:
Fmolasses = Fscat[(ω − kv)− ω0]− Fscat[(ω + kv)− ω0] ' −2∂Fscat
∂ω
kv,
⇒ Fmolasses = −αv, with α = 2k∂F
∂ω
= 4h¯k2 I
IS
−2∆/Γ
1 +
(
2∆
Γ
)2 . (2.48)
Red detuned light with ∆ < 0 ensures α > 0 and results in the force Fmolasses ∝ −v being
dissipative, so as to reduce the kinetic energy of the atoms:
dE
dt =
d
dt
(1
2mv
2
i
)
= viFmolasses = −αv2i = −
2α
m
E. (2.49)
MOT
Optical molasses only decelerates the atomic sample but provides no confinement about a
point, preventing the increase of its spatial density. A Magneto-Optical Trap (MOT) com-
bines the velocity-dependent dissipative mechanism with a spatially-dependent restoring
force arising from a magnetic field modulating the scattering rate.
A pair of current-carrying coils in the anti-Helmholtz configuration generates the quad-
rupole field B which lifts the degeneracy among the mF states and shifts their energies
according to the Zeeman effect:
∆EZeeman = gF mF µBBz(z), (2.50)
where z is chosen as the direction of the local magnetic field defining the (local) quant-
isation axis, and the other symbols are the same as in eq. 2.24
The field is rather weak and does not provide confining by generating a magnetic trap,
but by spatially modulating the absorption rate of the incident radiation, as illustrated in
fig. 2.6. The counter-propagating beams are still red detuned but now also σ− polarised
(with respect to the local magnetic field) in order to drive ∆mF = −1 transitions: atoms
farther away from the trap centre will be closer to resonance with the incoming beam and
therefore scatter more photons.
38 Cold atoms
(a)
(b)
(c)
Figure 2.5 : Optical molasses consists of two red detuned counter-propagating beams.
While neither is resonant with stationary atoms (a), they induce a preferential absorption
of photons depending on the atom’s speed (b). This results in the atoms experiencing a
frictional force that slows them down (c).
2.3. Cooling 39
The total force on the atoms FMOT is given by:
FMOT = F→scat[
Doppler︷ ︸︸ ︷
(ω − kv)−
Zeeman︷ ︸︸ ︷
(ω0 + βz))]− F←scat[(ω + kv)− (ω0 − βz)], (2.51)
where
β = gFmFµB
h¯
dBz
dz z. (2.52)
This can be approximated as:
FMOT ' −2∂F
∂ω
kv + 2 ∂F
∂ω0
βz = −αv − αβ
k
z, (2.53)
which not only contains a frictional term (∝ −v) as in eq. 2.48, but also a restoring force
(∝ −z) providing confinement about the centre of the quadrupole field.
(a) (b)
Figure 2.6 : A Magneto-Optical trap provides both a dissipative force to slow down
the atoms, and a restoring force to confine them near the centre of the trap (a). The
quadrupole magnetic field is produced by a pair of coils in the anti-Helmholtz configuration
(b).
Optical molasses and MOTs rely on the atoms differentiating between the two incoming
beams by their Doppler shifted frequencies. When the frequency difference between the
two beams becomes comparable to the natural linewidth Γ of the atoms, cooling becomes
less efficient and eventually stops. The theoretical lower limit is the Doppler temperature
TD:
TD =
h¯Γ
2kB
, (2.54)
rigorously derived in [78], which is 146 µK for both 39K and 87Rb.
Polarisation gradient cooling
Sub-Doppler cooling schemes can further reduce the temperature by making use of the
multiplicity of the sub-levels in the atomic ground state. One such technique is polarisa-
tion gradient cooling, which combines the conservative potential resulting from the AC
Stark shift with the spontaneous emission needed to lose entropy.
40 Cold atoms
A common example is Sisyphus cooling [79], where two counter-propagating beams of
equal amplitude E0 but orthogonal linear polarisations (“lin ⊥ lin” configuration) interfere
to form a standing wave with a spatially varying polarisation:
E = E0 [(xˆ + yˆ) cos(ωt) cos(kz) + (xˆ− yˆ) sin(ωt) sin(kz)]. (2.55)
This results in a spatially varying AC Stark shift which distinguishes the internal states
mJ by their vector polarisability (section 2.2.2), as shown in fig. 2.7. If no coupling to
the excited state were present, the atoms would lose kinetic energy climbing the hill, to
be re-gained upon descending it, resulting in no energy loss. The spatial dependence of
the polarisation, however, allows atoms in one of the mJ sub-levels at the top of the hill
to transition to the excited states, from which it may spontaneously decay into the other
mJ sub-level. The short lifetime of the excited state (∼ 10 ns) guarantees the transitions
occur at the same position, and the atom decays to the bottom of the hill, thereby losing
a net amount of energy.
Figure 2.7 : In Sisyphus cooling, an atom in the |mJ = 1/2〉 state climbs the hill in the
spatially varying potential resulting from the AC Stark shift. At the top of the hill, σ−
polarised light enables a transition into an excited state which may then decay into the
|mJ = −1/2〉 state, at the bottom of the hill. This allows the atom to lose energy as it
propagates in space.
In our experiment, the same beams that are used in the MOT are also used to perform
polarisation gradient cooling. These are σ polarised but still allow sub-Doppler cooling,
though in the σ+ − σ− configuration which only results in a spatially rotating linear
polarisation:
E = 2E0 cos(ωt)[xˆ cos(kz) + yˆ sin(kz)]. (2.56)
As discussed in [80], this causes an imbalance of the radiation pressure exerted by the two
beams, because of the spatial dependence of the polarisation and hence of the transition
rate to the excited state.
2.3. Cooling 41
The ultimate limit of cooling schemes based on scattering of light is the recoil temperature
Trec:
Trec =
h¯2k2
2mkB
, (2.57)
where m is the mass of the atom, and corresponds to the energy acquired upon a single
spontaneous emission event. Polarisation gradient cooling is likely to become ineffective
even before this limit, as colder atoms increase their de Broglie wavelength λth and lose
the ability to resolve the details of the potential landscape.
The recoil temperatures for 87Rb and 39K are Trec ∼ 360 nK and ∼ 420 nK. Further
cooling is allowed by sub-recoil techniques such as evaporative cooling.
2.3.3 Evaporative cooling
The temperature of an atomic cloud can be reduced by adiabatically relaxing the trap
confining it, which however also decreases the spatial density and therefore prevents the
increase of phase space density. For example, in a harmonic trap of depth U0, the tem-
perature T ∝ h¯ω/kB and the number of particles N scale as:
T ∝ U1/20︸ ︷︷ ︸
from eq. 2.34
, N ∝ U3/20︸ ︷︷ ︸
from eq. 1.14
⇒ D = n0λ3th ∝ U−3/20 U3/20 ∼ 1, (2.58)
leaving the degeneracy parameter constant. This is classically explained in the context of
adiabatic invariants [81].
Once again, a dissipative process is needed in order to increase phase space density, such
as the particle loss used in evaporative cooling. This was first introduced in [82] and
is illustrated in fig. 2.8. The trap is gradually lowered to new depths U0 so as to lose
all atoms with energies E > U0 and allow the remaining cloud to thermalise to a lower
temperature.
In order for the evaporation to be successful, the elastic collision rate γel needs to be
larger than the inelastic collision rate, so that thermalisation of the remaining particles
in the trap occurs faster than particle loss mechanisms. The desired regime to operate
in is then the run-away regime, where γel increases with decreasing number of particles
and allows for a fast evaporative stage in order to minimise losses. A theoretical model
for evaporative cooling is presented in [55], and makes use of the truncation parameter
η = U0/(kBT ) to identify a suitable evaporation process, ensuring both a run-away regime
and phase space density increase.
42 Cold atoms
Figure 2.8 : Illustration of the working principle of evaporative cooling. The atoms in
the trap are in thermal equilibrium (a) and their energy distribution is given by eq. 1.18.
The trap depth is then lowered (b) to release highly energetic atoms, corresponding to
a truncation in the energy distribution. The remaining atoms thermalise (c) to a lower
temperature.
Evaporative cooling reaches its ultimate limit when the elastic and inelastic collision rates
become comparable, as they may be governed by different functional dependences at low
temperatures [79]. In experiments, however, a Bose-Einstein condensate usually emerges
before any of the assumptions break down.
2.4 Optical lattices
2.4.1 Quantum simulators
Optical lattices are generated from the interference of counter-propagating beams, which
gives rise to a standing wave pattern of intensity maxima and minima of variable geometry
and dimensionality, as shown in fig. 2.9.
In 1D (fig. 2.9a), the resulting intensity is:
I = 〈[E0 cos(kz − ωt) + E0 cos(−kz + ωt)]2〉 ∝ cos2(kz), (2.59)
with a separation between extrema of d = λ/2.
Because of the AC Stark shift (eq. 2.13), the atoms experience an attractive force towards
the intensity minima (maxima) of a blue (red) detuned optical lattice. Hence, they are
distributed according to the same geometry as the interference pattern generated from the
beams. The optical lattice determines the spatial arrangement of the atoms, analogous
to the role played by the ionic lattice for electrons in condensed matter. Because of the
qualitative similarity between the two systems, cold atoms in optical lattices are used as
quantum simulators for solid-state physics, as illustrated in fig. 2.10.
2.4. Optical lattices 43
(a) (b)
Figure 2.9 : A pair of counter-propagating beams gives rise to a 1D optical lattice (a)
in which each disc (“pancake”) is a 2D system. Interference of four beams at orthogonal
angles results in a 2D optical lattice with square geometry (b), where each tube is a 1D
system.
+ + +
J U
- --
J U U
J
+
--
Figure 2.10 : Electrons in ionic potentials in condensed matter systems are qualitatively
equivalent to neutral atoms in optical lattices, making the latter a platform to perform
quantum simulation of the former.
Quantum simulation was first introduced by Feynman [23] as a way to circumvent the
numerical intractability of large quantum systems by performing the investigation on
a different but analogous quantum system. For instance, electron transport in a real
crystalline solid cannot be reliably simulated on a classical computer given the exponential
growth of the Hilbert space governing the dynamics. However, an analogous system can
be engineered with cold atoms in an optical lattice in order to experimentally record the
atomic spatial distribution as a function of time.
An advantage of optical lattices over condensed matter systems is their much slower dy-
namics, which enables easier and better experimental investigations. A simple dimensional
analysis treatment of the Schro¨dinger equation results in the following relation between
the timescale T and lengthscale L of a system:
i
[
∂
∂t
]
[Ψ] = − h¯
2
2m
[
∇2
]
[Ψ] ⇒ i 1
T
∼ − h¯
2
2m
1
L2
. (2.60)
44 Cold atoms
The ratio of the timescales for electrons in solids and atoms in optical lattices is then:
Tatom
Te
∼ L
2
atom
L2e
matom
me
∼
(
10−7
10−11
)2 (1011
103
)
∼ 1015, (2.61)
where me ∼ 1 keV/c2, matom ∼ 100 GeV/c2, Le ∼ 0.1 A˚ as for the hydrogen atom, and
Latom ∼ 0.1 µm given by the minima spacing λ/2 with typical λ ∼ 1 µm.
While electronic interactions and chemical bonds formations occur on the attosecond
timescale [83], dynamical processes in optical lattices range from microseconds to milli-
seconds. This allows long exposure to incident radiation and thus high-resolution imaging
by a camera. Further differences between condensed matter systems and optical lattices
are summarised in table 2.2.
Interest in optical lattices sparked when the d-wave superconductivity exhibited in several
high-temperature superconductors was reproduced in the phase diagram of the Fermi-
Hubbard model [84]:
H = −J∑
〈i,j〉
(
c†i,σcj,σ + h.c.
)
+ U
∑
i
ni,↑ni,↓, (2.62)
where U is the on-site interaction, J the tunnelling between nearest neighbours, and c†, c
and n the creation, annihilation and number operators on lattice site i for a spin state
σ ∈ {↑, ↓}. This model is easily implemented by cold fermionic atoms in a square optical
lattice, which so far have allowed probing the anti-ferromagnetic area of the phase diagram
[85, 86] preceding the d-wave superconducting region.
Optical lattices cannot capture the full complexity of the materials displaying high-
temperature superconductivity, but they offer a platform to implement the toy model
that reproduces their key features. While real systems include defects and impurities
that undermine accurate investigation of the material, optical lattice solely rely on the
interference of light resulting in (essentially) perfect patterns, as displayed in fig. 2.11.
Figure 2.11 : The measured intensity pattern resulting from the interference of five slits
placed on the vertices of a pentagon. Optical lattices rely on interference to generate the
potentials for the atoms, and are therefore not subject to defects and impurities as in real
solids. Picture courtesy of Prof. Theodor Ha¨nsch.
2.4. Optical lattices 45
Cold atoms Condensed matter
Temperature 10−9 − 10−6 K 10−6 − 102 K
Confinement Optical/magnetic potentials positively chargedionic lattice
Density 1013 − 1015 cm−3 ∼ 1022 cm−3
number of
components Variable two (spin 1/2)
Particle statistics bosons or fermions fermions (electrons)
Geometry & dimensionality highly tuneable depends on chemicalcomposition, pressure ...
Interactions tuneable, also non-interacting Coulomb attraction
Disorder none but can be engineered unavoidable:defects, impurities
Table 2.2 : Comparison between cold atoms and condensed matter systems, based on
[87].
Why ultracold?
In a solid, the lattice spacing d is of order ∼ 1 A˚ while in an optical lattice it is controlled
by the wavelength of commercially available lasers, d = λ/2 ∼ 1 µm. In order to reproduce
the quantum regime, where the wave nature of matter is dominant, the temperature of
the atoms needs to be reduced so as to increase their de Broglie wavelength:
λth
d
1, λth ∝ T−1/2. (2.63)
2.4.2 Band structure
The potential experienced by atoms in the 1D standing wave of eq. 2.59 is given by
V = V0 cos2(kz) =
V0
2 [1 + cos(2kz)], (2.64)
where V0 is the lattice depth, often quoted in units of the recoil energy Erec = h¯2k2/2m.
The periodicity of the potential guarantees a periodic wavefunction as a solution, thanks
to Bloch’s theorem. An electron in a band n with quasimomentum2 q is therefore described
by the Bloch wave:
φn,q(x) = eiqxun,q(x), (2.65)
where the functions u have the same periodicity as the lattice, and are extended over all
space as shown in fig. 2.12a. Owing to their periodicity, the functions u can be expressed
as a discrete Fourier series:
φn,q(x) = eiqx
∑
`
cn,q ei2`kx. (2.66)
2Technically h¯q is the quasimomentum, though the term is often used for q itself.
46 Cold atoms
A Bloch wave is then composed of plane waves with wavectors q + 2`k, and in the limit
of a free particle this reduces to a single plane wave with momentum p = h¯q.
The Bloch wave can however also be constructed from a basis localised on each lattice
site xi. These are the Wannier functions wn(x−xi), the eigenstates of the band projected
position operator, and allow the following, equivalent expression:
φn,q(x) =
1√
N
∑
i
eiqxiwn(x− xi), (2.67)
whereN is the number of sites, needed for normalisation. The functional form of wn(x−xi)
is shown in fig. 2.12b, which illustrates how the confinement of the Wannier function to
a single site grows with lattice strength V0.
For deep lattices, the lattice sites that experience a significant contribution from the
ith Wannier state are only xi and its nearest neighbours. This is the tight-binding re-
gime, where the Hamiltonian only includes an on-site interaction energy U and a nearest-
neighbour tunnelling element J , as shown in fig. 2.10. For very deep lattices, the wave-
function is fully confined to a single potential minimum and transport is completely
eliminated. This then justifies treating each lattice site as an independent system, ap-
proximated by a harmonic oscillator as shown in fig. 2.12c.
(a)
(b)
(c)
Figure 2.12 : The Bloch wave φn,q can be expanded in the periodic and extended basis
un,q (a) as in eq. 2.65, or in the Wannier functions wn(x− xi), each confined to a lattice
site xi (b). Deep lattices cause the wavefunction to tightly localise on a single site, which
allows approximating it as an independent harmonic oscillator (c). 1D lattices were used
for these figures (shown in grey).
The strength of the lattice V0 determines the energies that the particles are allowed
to possess, which are clustered in bands. The band structure for a 1D and 2D simple
cubic lattice is shown in figs. 2.13 and 2.14, and was computed from diagonalising the
Hamiltonian constructed in the plane wave basis as described in section 7.1.4.
2.4. Optical lattices 47
(a) (b)
Figure 2.13 : Band structure for a 1D lattice. a) A free particle exhibits a parabolic
dispersion relationship (dashed line), which the presence of the lattice separates into finite
bands and gaps. The reduced band scheme is shown on the left. b) The width of the
bands depends on the lattice strength V0 and tends to a point in the deep lattice limit,
where a single harmonic oscillator can describe each potential minimum. Only the first
three bands are shown.
(c) (d)
Figure 2.14 : Band structure for a 2D square lattice. The bands are defined over a vector
quasimomentum q = (qx, qy) (a). Similarly to the 1D lattice in fig. 2.13, the widths of the
bands decrease with increasing lattice strength (b) but in this case the gaps only appear
at a finite depth V0. Only the first four bands are shown. In the deep lattice limit, the px
(blue) and py (green) bands become degenerate.
I wanted to persuade myself that these were only apparent flaws, that they were all part of a much
vaster regular structure, in which every asymmetry we thought we observed really corresponded
to a network of symmetries so complicated we couldn’t comprehend it...
Italo Calvino [88]
3
Quasicrystals
This chapter summarises the currently accepted description of regular crystals
and aperiodic structures, providing a historical introduction and establishing
the mathematical formalism necessary to appreciate the results in chapters 7
and 8. Quasicrystals display non-trivial structure on all lengthscales, and usual
investigation techniques such as spectral decompositions are not sufficient to
unequivocally discern order from disorder, placing aperiodic structures in an
interesting middle ground.
3.1 From periodic to quasiperiodic
The following review is largely based on [89–91].
This chapter is at times heavily theoretical and abstract, which is unavoidable given
the much vaster literature on aperiodic structures available in the field of mathematics.
Although I tried my best to provide consistent and rigorous definitions and proofs, this
chapter is not essential to the understanding of the rest of the thesis.
3.1.1 Regular crystals
Crystallographic restriction theorem
The elegant simplicity of symmetrical structures has always been a source of fascination
and inspiration for mankind. Ancient cultures used it for decorations, architectural plans,
ornative motifs, mosaics and painting proportions because of the apparent pleasing aes-
thetic of regular shapes. A quantitative investigation was performed by Johannes Kepler
in On the six-corner snowflake [92] where he explained the symmetrical pattern of snow-
flakes by means of close packing of spheres. This was the first inkling of the macroscopic
consequences of microscopic symmetries. In 1801 Rene´ Hau¨y published his four-volume
Traite´ de mine´ralogie [93], winning him the title of “father of modern crystallography”
49
50 Quasicrystals
and a name on the Eiffel tower. He specifically states that no fivefold symmetry is al-
lowed in crystals, which was the first connection between the ability to “pack spheres”,
or tile n-dimensional space via translations of the same microscopic constituent, and the
rotational symmetry of the resulting macroscopic structure. With the discovery of X-
rays and Bragg scattering at the beginning of the 20th century, the empirically-driven
and generally accepted consensus was that the ground state of matter always consisted
of a periodic arrangement of atoms and molecules, mathematically described by a space
group (230 in three dimensions) or a plane group (17 in two dimensions). It was how-
ever already known that, at finite temperatures, defects and impurities would spoil the
otherwise perfect symmetry. The rest of the discussion will focus on two dimensions (2D).
The plane group symmetry determines the microscopic arrangement of the constituents,
and its macroscopic consequence is the crystal’s morphology. The group symmetry implies
that the crystal is invariant under a certain combination of rigid (distance-preserving,
Euclidean) transformations:
r→ Rr + t, (3.1)
where rotations R and translations t constitute a plane group if the latter are generated
by a finite set of d vectors {ai}:
r =
d∑
j=1
nj aj, with n ∈ N. (3.2)
These vectors span the lattice structure, and underpin the translational invariance of the
crystal. Hence, the treatment of the solid can be simplified by only considering the unit
cell, the finite space from which every other point in the lattice is reached by translations.
The contents of the unit cell determine the structure of whole crystal.
The lattice is also invariant under rotations about a fixed point, i.e. point groups:
Rai =
∑
j
M(R)ji aj, (3.3)
where, having chosen the lattice vectors {ai} as the basis, M is an integer matrix. In
a different basis, e.g. orthogonal Cartesian (x, y) coordinates, M is the usual rotation
matrix:
M ′(R) = ±
(
cosφ − sinφ
sinφ cosφ
)
, (3.4)
where the minus denotes an improper rotation, i.e. a reflection.
Because M and M ′ are related by a similarity transformation M ′ = SMS−1, the trace is
invariant:
2 cosφ = n ∈ N crystallographic restriction theorem. (3.5)
The above equation is only satisfied for φ = 0, pi, pi/2, pi/3, 2pi/3, which restrict the
rotational symmetries of 2D lattices to the five 2D Bravais lattices exhibiting rectangular,
square, rhombic, oblique or hexagonal unit cells. A more detailed discussion is presented
in appendix D.5.
Tiling of the whole 2D plane is achieved by successive translations of the unit cell, with the
constraint imposed by eq. 3.5 that only allows a few rotational symmetries, establishing a
(one-way) relation between rotation and translation. A mathematical discussion of tilings
is provided in section 3.2.2.
3.1. From periodic to quasiperiodic 51
Reciprocal space
The symmetry of the atomic density ρ(r) also affects its Fourier transform ρˆ(k):
ρ(r) = ρ(Rr + t) ⇔ ρˆ(k) = ei(Rk·t) ρˆ(Rk), (3.6)
where R and t are elements of the plane group defined in eq. 3.1, and it is proved in
appendix D.4. This aids in structure determination as it provides extinction rules to
identify glide reflections, e.g. when k = Rk, the above relation implies either ρˆ(k) = 0 or
k · t = 0 mod 1. The space defined by k is termed momentum or reciprocal space.
Experimentally, one may probe the Fourier transform by bombarding a sample with X-
rays or neutrons, since the matrix element for scattering off a potential V (r) is given by:
〈kout|V (r)|kin〉 ∝
∫
ρ(r) e−ik·rd3r ≡ ρˆ(k), (3.7)
where k = kout−kin is the change of momentum between incident and detected particles.
In crystallography, ρˆ(k) is also referred to as the static structure factor, F (k):
F (k) =
∫
ρ(r) e−ik·r d3r =
N∑
n=1
fne−ik·r, (3.8)
where the expression was further simplified by noting that the density distribution of a
regular crystal is discrete. fn is the atomic scattering factor of the nth particle, and N is
the total number of constituents.
In real space, the position vector r can be expressed as an integer combination of lattice
vectors ai (eq. 3.2), and an analogous indexing scheme exists in reciprocal space:
K =
∑
i
hibi, (3.9)
where hi ∈ N are known as Miller indeces. The unit cell in reciprocal space is the Brillouin
zone.
The measured scattering intensity, ignoring diffuse and multiple scattering events, is
defined as:
I(k) ∝ |F (k)|2, (3.10)
from which it is evident that the presence of sharp peaks crucially depends on the con-
structive interference of the scattered radiation, enabled by the long-range order of the
structure. Diffraction experiments, such as the ones discussed in chapter 7, rely on eq.
3.8 and hence probe long-range order, whilst local ordering may be investigated with mi-
croscopes and in-situ techniques. Real systems, however, include impurities and defects:
this disorder increases the peaks’ size, reduces their heights by the Debye-Waller factor,
and adds a continuous diffuse background.
Fig. 3.1 shows examples of scattering intensities from a regular crystal (a), an amorphous
solid such as a glass (b), and a 1D quasicrystal. For regular crystals, the n > s part of
the sum in 3.8 is redundant, s being the number of particles in the unit cell, hence I(k)
consists of a repetitive pattern. The lack of translational order in a quasicrystal results in
the lack of a unit cell, hence the sum in eq. 3.8 extends to N →∞, uncovering non-trivial
structure on all lengthscales.
52 Quasicrystals
(a) Regular crystal (b) Amorphous solid (glass) (c) 1D Quasicrystal
Figure 3.1 : The unit cell uniquely determines the structure of a regular crystal, hence
the repeated, redundant pattern in its scattering intensity (a). The lack of a unit cell,
on the other hand, may have two distinct effects. In an amorphous solid (e.g. glass),
with particles at uncorrelated positions, F (k) = F (k) reduces to an average constant (b),
where residual short-range interactions may still give rise to limited non-trivial stucture.
A quasicrystal, however, provides an interesting middle ground between order and dis-
order, as it exhibits a dense set of sharp peaks (c) and hence non-trivial structure at all
lengthscales. The particular pattern shown in (c) is produced by a 1D Fibonacci chain,
rigorously discussed in [94].
The scattering intensity in eq. 3.10 depends on the absolute value of the structure factor,
which causes the loss of phase information. The structure is therefore not uniquely de-
termined by its diffraction pattern, and the symmetry of I(k) is larger than the one
of F (k), resulting in the existence of centrosymmetric points (k,−k) of equal intensity
termed Friedel’s pairs.
3.1.2 Aperiodic crystals
Paradigm shift
The conviction that the ground state of matter is always an ideal periodic crystal started
to weaken in the 1960s. It was already known that real crystals had defects and impurities,
and that these caused not only broadening of existing Bragg peaks, but also an additional
diffuse background. Furthermore, periodic variations such as spurious crystal growth
and twinning1 caused additional peaks to appear in the diffraction spectra, called lattice
ghosts or Gittergeister. All of the above had been modelled as mere perturbations on
regular crystal structures, though with increasing difficulty. For instance in [95] a material
was found where the magnetic properties occurred at a different period than the crystal
structure, though this was exquisitely solved by modelling it as a long-range modulation
with an incommesurate wavelength, the first instance of aperiodicity in condensed matter2.
This modulated structure approach, consisting of a periodic lattice with spacing a and an
incommensurate modulation λ such that a/λ 6∈ Q, quickly spread in popularity as it also
helped understanding previous results from alloys, such as [96], and provided a simpler,
more elegant framework.
From eq. 3.9, a modulation λ along an existing direction requires an additional Miller
index:
K = hb1 + `b2 = hb1 + (`′ + λ)b2 , (3.11)
1The intergrowth of two separate crystals that share some of the same lattice points.
2Two commensurate periods a and b will be periodic every a · b, but if one of them is an irrational
number, perfect recurrence will never happen.
3.1. From periodic to quasiperiodic 53
where b1 and b2 would be enough to span the 2D plane, but the structure requires the
use of n > 2 integers.
In 1972, a new description of the modulated phase method was proposed [97] that finally
managed to bring symmetry back in the picture: a 3D aperiodic structure can be thought
as a projection from a 4D regular lattice, where group symmetries reign supreme3. The
higher dimensional method is used to generate and simulate the results for the optical
quasicrystals presented in chapters 7 and 8, and is rigorously discussed in section 3.2.4.
Finally, in 1982, Dan Shechtman [99] found that electron diffraction from a particular alloy
of Al and Mn displayed sharp peaks, hinting at a high degree of coherence and long-range
order in the material, but arranged in the tenfold symmetric pattern of fig. 3.2. This was
in direct disagreement with the crystallographic restriction theorem, and hence paved the
way for the field of quasiperiodic crystals, or quasicrystals, which eventually won him the
2011 Nobel prize in Chemistry. In the publication they explicitly rule out the possibility
that the unexpected symmetry may be due to artefacts such as twinning. Unbeknownst
to Shechtman, another group had been working on the theory of crystalloids for years,
especially on the icosahedral phase shown in fig. 3.2, and published their results in [100].
Figure 3.2 : Selected-area electron diffraction patterns from an alloy of Al and Mn,
consistent with stereographic projections of an icosahedral phase (A5 × Z2). The tenfold
symmetry displayed by sharp Bragg peaks established the existence of quasicrystals, a
new form of matter between crystals and amorphous solids. Figure taken from [99].
Having established the existence of quasicrystals, a number of fundamental questions
arose:
1. What are the conditions needed for a material to solidify into a quasicrystalline
phase?
2. Can a real material display (or be made to display) a stable quasicrystalline phase?
3This idea stemmed from 4D space-time groups studied in the context of electro-magnetism [98].
54 Quasicrystals
3. What are the transport properties of quasicrystals?
4. Can crystals, quasicrystals and (structural4) glasses be unified under the same the-
oretical framework?
These are discussed in the remainder of the section.
Definitions
Aperiodic order has been drawing an increasing amount of attention and is now a topic
of study of pure mathematics, theoretical and experimental physics, material science and
metallurgy. Definitions are therefore not unique and not consistent, and I will here present
my favourite ones, taken verbatim from [90].
“ ”
1: Translationally ordered structure
A translationally ordered structure is a structure whose scattering amplitude is
given by a discrete sum of Bragg peaks.
“
”
2: Basis
A translationally ordered structure [...] with Bragg peaks centred at the discrete
set of reciprocal vectors {ki}, is said to have a basis if there exists a finite set of
vectors {bj}, such that each ki can be expressed as an integer linear combination
of [them].
“
”
3: Rank
A minimal set of vectors necessary to span a reciprocal lattice is a basis set of
reciprocal vectors, [...] the number of elements in this basis is the rank of the
reciprocal lattice.
“ ”
4: Crystal
A crystal in d dimensions is a translationally ordered structure with a basis whose
rank n is equal to d.
“ ”
5: Quasiperiodic structure
A quasiperiodic structure in d dimensions is a translationally ordered structure
with a finite basis whose rank n exceeds d.
The term “quasiperiodic” is used not only in contrast to “regular”, but also “chaotic”
and “disordered”. The generalisation of the reciprocal lattice K to aperiodic structures is
known as a Fourier module H of rank n.
Quasiperiodic structures can be further divided into two categories.
4Intended here as an amorphous solid with no long-range diagonal order, to be distinguished from the
Bose glass phase discussed in chapter 8, whose name refers to the lack of off-diagonal long range order
(phase coherence) and not necessarily to its microscopic structure.
3.1. From periodic to quasiperiodic 55
“ ”
6: Incommensurate crystal
An incommensurate crystal is a quasiperiodic structure with a crystallographic-
ally allowed orientational symmetry.
“ ”
7: Quasicrystal
A quasicrystal is a quasiperiodic structure with a crystallographically disallowed
orientational symmetry.
Crystals possess long-range translational order that restricts the rotational symmetry of
the unit cell, known as orientational symmetry, which leads to five 2D Bravais lattices as
per eq. 3.5. Translational order therefore implies orientational order, but the reverse is
not true, as shown by the Penrose and Ammann-Beenker tilings in fig. 3.5. Glasses and
amorphous solids possess neither long-range translational nor orientational order (apart
from the trivial isotropy)5.
Quasicrystals are a new and distinct state of matter as they display both long-range
translational order, and long-range orientational order with a crystallographically for-
bidden symmetry, distinguishing them from glasses, and from regular crystals. The in-
commensurate ratios present in the quasicrystal phases are different from the ones in the
modulated phases, as they are constrained by geometry (discussed in section 3.2.4) and are
hence insensitive to temperature and pressure. Modulated phases structures, on the other
hand, are observed to change the modulation period as a function of external parameters,
suggesting a distinct physical interpretation than in quasicrystals.
2D quasicrystals can still exist in 3D, but the aperiodicity is only noticeable in planes:
the rank of a stacked pentagonal quasicrystal is 4 > 3 but the redundancy in the linear
dependence is only among 2D vectors that span the same plane, which makes it an axial
quasicrystal [101], whereas the icosahedral structure in fig. 3.2 is a truly 3D quasicrystal.
All of the above only really apply in d > 2, since they rely on the presence of rotational
symmetries. It is therefore useful to quote the definition provided by the International
Union of Crystallography (IUCr) [102]:
“
”
8: IUCr definition
The term quasicrystal stems from the property of quasiperiodicity observed for
the first alloys found with forbidden symmetries. Therefore, an alternative definition
is: a quasicrystal is an aperiodic crystal with diffraction peaks that may be indexed
by n integral indeces, where n is a finite number, larger than the dimension of the
space (in general).
Not only does this, along with the above, establish the collapse of the old paradigm order
⇔ periodicity, but it requires quasiperiodic structures to (only) display sharp diffraction
peaks as a sign of long-range order, a criterion easily applicable to 1D structures as well.
5Technically, between glasses and crystals there are also paracrystals, defined to lack crystal-like long-
range order in at least one direction.
56 Quasicrystals
3.1.3 Physical properties and current research
The first question on this quasicrystalline phase of matter concerns its stability and oc-
currence. A huge variety of quasicrystals have been so far created in alloys and other
composite mixtures, though the only stable ones seem to be decagonal, dodecagonal and
icosahderal. A natural quasicrystal (the “second kind of impossible” [103]) was eventually
also discovered [104] in a meteorite fragment in Kamchatka6. The discovery faced fierce
scepticism because of the abnormal concentration of natural aluminium and the absence
of rust, which could only be the result of industrial treatment, at least on Earth. Indeed,
extreme conditions are needed for the formation of quasicrystals, such as those experi-
enced by a meteorite. The alloys discussed in the previous section are made by rapidly
cooling (quenching) the material, thereby preventing it from crystallising and freezing its
metastable quasiperiodic structure as in a glass. As a matter of fact, Shechtman in [99]
specifies how mean-field theory calculations had predicted the metastable phase, since
the diffusion velocity was slower than the crystallisation velocity but yet high enough to
allow for atomic rearrangements – essentially slow enough to allow its formation but rapid
enough to prevent it form crystallising into the true, regular ground state. The presence
of aperiodicity may be the result of structural deformations resulting in a lower energy,
of competition between mechanisms that favour different periodicities, or simply of the
structure stabilisation by the introduction of a distinct species. In fact, so far quasicrys-
talline order has always been observed in composite materials, and the existence of “pure”
stable quasicrystals remains an open question.
It is still undecided whether the formation of the phase is driven by energy or entropy
minimisation. The latter is favoured since aperiodic structures possess quasiparticle ex-
citations known as phasons (or simpletons), which may result in a rearrangement of the
atomic positions thereby increasing the entropy S and minimising the overall free energy7
F = U − TS. Phasons have been investigated in [105], observed in [106], and are further
discussed in section 3.2.4.
There is also active research in the spontaneous occurrence of quasicrystalline order. For
example, simulations in soft matter and colloids [107] showed that “a judicious design of
an isotropic pair potential” results in the spontaneous formation of clusters with forbidden
geometries. Self-emerging quasicrystalline order has also been proposed for cold atoms in
dipolar gases [108], in optical lattices [109], and in the time domain [110].
Quasicrystals possess promising properties for technological applications [111]: hardness,
low surface-energy, low friction, oxidisation resistance, hydrogen storage capacity and
a transition from brittle to ductile behaviour at high temperatures. Electrical, mag-
netic and thermal transport mechanisms are anisotropic [112, 113]: crystallographic-
ally non-equivalent directions exhibit ballistic, diffusive, and insulating behaviours, along
with thermoelectric power and Hall coefficients of opposite signs. Superconductivity in
quasicrystals has also been observed [114], though at a very low transition temperature
Tc ≈ 0.05 K.
The quenching processes utilised to create the alloys result in non-ideal quasicrystals, with
vacancies, dislocations, finite domains, chemical inhomogeneities, strains, and additional
disorder. Hence, a number of alternative, more controllable and cleaner experimental
platforms have been employed to reproduce and investigate quasicrystals. In material
6which I had only otherwise heard of from the boardgame Risk©.
7The importance of minimising the free energy is discussed in appendix D.6.
3.2. Mathematical treatment 57
science and condensed matter, these include layered semiconductors [115], liquid crystals
[116], polymers [117], colloids [118, 119], nanoparticles [120], and twisted bilayer graphene
[121]. Photonic structures have also been used [122], as well as optical diffraction patterns
[123, 124].
Experiments in cold atoms have so far only probed the dissipative regime [125, 126],
but there is a multitude of theoretical proposals – from transport properties [127], to
magnetism [128] and phason-stabilised order [129].
Our experiment is the first to probe degenerate quantum gases in a quasicrystalline po-
tential. As discussed in the next section, this results in non-trivial structure on all length-
scales and opens avenues to synthetic dimensions, so far only achieved via internal states
[37, 38].
The presence of non-trivial structure is of interest within computer science in the context
of search algorithms [130, 131], and has also been used in the proof of the undecidability
of the spectral gap theorem [132].
3.2 Mathematical treatment
The key to periodicity is the repetition of constituents at a fixed frequency. In order to
break this, various elements need to occur at incommensurate frequencies, easily achieved
by introducing an irrational number.
Quasicrystals may be generated by substitutional sequences, matching rules, and projec-
tions from higher dimensional spaces, as discussed in the remainder of this section.
3.2.1 1D: sequences
Fibonacci
The substitutional rule for the Fibonacci sequence is σF : L→ LS, S → L, or:
σF :
(
L
S
)
→
(
1 1
1 0
)
︸ ︷︷ ︸
=S
(
L
S
)
=
(
LS
L
)
. (3.12)
S, the substitution matrix, has eigenvalues 1 = ϕ and 2 = 1− ϕ = −1/ϕ, with ϕ being
the golden ratio, related to the Fibonacci numbers Fn:
lim
n→∞
Fn
Fn−1
= ϕ = 1 +
√
5
2 ≈ 1.618. (3.13)
One of the eigenvectors is (ϕ, 1), from which it is evident that a chain with L = ϕ, S = 1 is
self-similar and invariant under inflations (deflations) by ϕn, such as the one shown in fig.
3.10b. The eigenvalues of the substitution matrix therefore determine the self-similarity
of the structure.
Periodic lattices scale with integer factors, and so do the eigenvalues of their substitution
matrices S. In quasiperiodic lattices the eigenvalues of S are algebraic numbers8 or Pisot
numbers, which satisfy the Pisot-Vijayaraghavan (PV) property:
1 > 1, |i|< 1 ∀i > 1. (3.14)
8Actually integer algebraic numbers, that is they are solutions to polynomials with integer coefficients.
58 Quasicrystals
Octonacci
The substitutional rule for the Octonacci (or Pell) sequence is σO : L→ LLS, S → L, or:
σO :
(
L
S
)
→
(
2 1
1 0
)
︸ ︷︷ ︸
=S
(
L
S
)
=
(
LLS
L
)
. (3.15)
The eigenvalues of S are 1 = λ and 2 = −1/λ, where λ is the silver mean, related to the
Pell numbers Pn:
lim
n→∞
Pn
Pn−1
= λ = 1 +
√
2 ≈ 2.414. (3.16)
These eigenvalues also satisfy the PV condition, and the eigenvector (λ, 1) requires the
mapping L = 1+
√
2 and S = 1 in order for the structure to be self-similar upon inflations
(deflations) by λn.
However, it is useful for later considerations to find a basis of ratio L : S = 1 :
√
2 , such
as the sides of and . The nth inflation and deflation factor can be written in the
closed form:
(1 +
√
2 )n = Hn + Pn
√
2 , and (1− √2 )n = Hn − Pn
√
2 , (3.17)
where the Pell number Pn and its half-companion Hn satisfy the following recursion rela-
tion: (
Pn
Hn
)
=
(
1 1
2 1
)n
︸ ︷︷ ︸
S′
(
0
1
)
. (3.18)
The eigenvalues of S ′ are still λ and −1/λ, but the eigenvectors may be now mapped onto
L→ 1/√2 and S → 1.
Prime numbers
Substitution rules result in self-similarity and inflation rules, but are only a subset of the
deterministic recipe to construct a sequence based on the roots of a polynomial.
A famous polynomial is the Riemann zeta function ζ(s), whose non-trivial roots are given
by:
ζ(s) =
∞∑
n=1
1
ns
, ζ(sj) = 0 at sj =
1
2 + ikj, (3.19)
where the Riemann hypothesis was assumed. The 1D structure constructed from Im[sj]
is depicted in fig. 3.3, and its diffraction pattern is given by:
f(x) =
∣∣∣∣∣
∞∑
k=1
eikjx
∣∣∣∣∣
2
, (3.20)
which yields sharp peaks centred at prime numbers p and at integer powers of prime
numbers pn (n ∈ N) as shown in fig. 3.4.
Figure 3.3 : 1D structure made from the “matching rule” of knowing the zeros of the
Riemann zeta function.
3.2. Mathematical treatment 59
Figure 3.4 : The imaginary part of the zeroes of the Riemann zeta function, combined
in a 1D structure, yield a pure point diffraction spectrum centred at prime numbers p and
prime-powers pn, n ∈ N. As more roots are included, the peaks become infinitely sharp.
Despite my initial excitement and intrigue in the finding, this is actually a known result,
which pertains to the hypothesis that every 1D quasicrystal is uniquely identified by a
distinct algebraic number fulfilling the PV condition. Classifying 1D quasicrystals would
therefore also prove the Riemann hypothesis [133].
3.2.2 2D: tilings
Not just decorations
Euclidean or Achimedean tilings are coverings of the 2D plane by convex regular polygons,
and have been used since ancient times for art and decoration, such as in the mosaic of fig.
3.5a. While the Romans and Greeks made all of their tilings periodic, the Arabic and Per-
sian civilisations experimented with crystallographically forbidden rotational symmetries,
as shown in fig. 3.5b and 3.5c.
Tilings were formally classified by Johannes Kepler in his Harmonices Mundi [134], the
same book where he introduces the third law of planetary motion which he is, potentially,
better known for. Kepler did not only focus on periodic tilings, fig. 3.6a, but discovered by
trial and error that gapless tilings with various rotational symmetries could be created by
the careful use of distinct regular polygons, fig. 3.6b. It was not until Roger Penrose pub-
lished the celebrated Penrose tilings [135, 136] in 1974 that such coverings of the 2D plane
could be generated by just two distinct tiles, using Penrose’s matching rules, establishing
the order encoded in aperiodic structures. Penrose (fig. 3.6c) and Ammann-Beenker (fig.
3.6d) tilings both display crystallographically forbidden rotational symmetries, fivefold
and eightfold respectively.
60 Quasicrystals
(a) (b)
(c) (d)
Figure 3.5 : Examples of tessellation. a) Roman mosaic (IV century CE). b) Gunbad-i
Kabud tomb tower in Maragha (1197 CE), Iran (Harvard College Library). c) Alhambra,
Granada (XIII century CE). d) Circle Limit III, woodcut made in 1959 by Maurits Escher
showing tessellation in a non-Euclidean geometry, specifically an alternated octagonal
tiling of the hyperbolic plane.
3.2. Mathematical treatment 61
(a) (b)
(c) (d)
Figure 3.6 : a) and b): periodic and non-periodic tilings classified by Kepler in Har-
monices Mundi (1619) from [134]. c) and d): Penrose and Ammann-Beenker tilings,
showcasing crystallographically forbidden fivefold and eightfold rotational symmetries.
Ammann-Beenker tiling
Akin to the 1D substitution rules presented earlier, 2D tilings can be generated by the
inflation (deflation) rules of their elementary ingredients, the prototiles. Choosing these
to be and , with sides ratio 1 :
√
2 , the inflation of the tiling (area) follows from
the inflation of the side, given by Octonacci rule in eq. 3.18:
Σ = S ′2 =
(
1 1
2 1
)2
=
(
3 2
4 3
)
, (3.21)
where the matrix acts on the basis ( , ) as illustrated in fig. 3.7.
62 Quasicrystals
(a) (b)
(c) (d)
Figure 3.7 : The Ammann-Beenker tiling, with eightfold symmetry, can be generated
by the inflation rule (c) which, acting on (a), generates (b). The side of the tiling scales
with λ, the silver mean, according to the Octonacci sequence (eq. 3.18), while the area
scales with λ2.
3.2.3 Quasicrystals as the path to complexity
More advanced treatment of aperiodic order and the place of sequences is presented in
[137, 138].
Non-trivial structure at all lengthscales
The eigenvalues of a substitution or inflation matrix 1 > 2 are:
Eigs
[(
a b
c d
)]
⇒
1 =
a+ d+
√
(a− d)2 + 4bc
2 ,
2 =
a+ d−
√
(a− d)2 − 4bc
2 .
(3.22)
If the tiles are assumed to be indivisible, the entries of the matrix are ∈ Z, and the
characteristic polynomial is a monic (unit leading coefficient) polynomial with integer
coefficients, whose roots are known to be either irrational, or ∈ Z.
Additionally, it is trivial to show that:
1. 1 ∈ Z⇒ 2 ∈ Z,
2. 1 · 2 = ∆ = ad− cd.
Condition 1 implies that both eigenvalues belong in the same set of numbers, while condi-
tion 2 sets the relation between the two. For Z solutions, rational numbers only appear as
eigenvalues of the inverse matrix, since Eigs[A] = ⇒ Eigs[A−1] = −1, and so only when
performing a deflation operation on a previously inflated structure. On the other hand,
3.2. Mathematical treatment 63
substitution matrices obeying the PV condition have |2|< 1, which allows for non-trivial
deflation9.
When ∆ = 0, one of the eigenvalues is also 0: what does it mean to be self-similar when
scaled by 0, or even by 1/0? Maybe this the reason behind the peculiar diffraction spectra
of the Thue-Morse and Rudin-Shapiro sequences, discussed later.
Finally, if (a+ d) = (a− d)2 and bc = 1, the eigenvalues can be expanded as a continued
fraction with the same leading coefficient. For the Fibonacci and Octonacci sequences,
for instance, the golden and silver means can be written as:
ϕ = 1 +
√
5
2 = [1, 1, 1, 1, · · ·] = 1 +
1
1 +
1
1 + · · ·
,
λ = 1 +
√
2 = [2, 2, 2, 2, · · ·] = 2 + 1
2 +
1
2 + · · ·
,
(3.23)
and ϕ−1 = [0,−1,−1,−1,−1, · · ·], λ−1 = [0,−2,−2,−2,−2, · · ·]. This causes very slow
convergence and would hence require very large rational approximants to reliably describe
the ensuing pattern.
The non-trivial structure encoded in aperiodic tilings was already known to Erwin Schro¨dinger
when he proposed them as an explanation for how a fragile ensemble such as the DNA
could retain so much information while still obeying the second law of thermodynamics
[140, 141].
Spectra
The scattering intensity I in eq. 3.10, used to probe and classify the ordering of the
structure, is usually divided into a discrete, or pure point Ipp, and a continuous compon-
ent. The latter can be further divided into an absolutely continuous Iac and a singularly
continuous Isc contribution, for a grand total of:
I(H)total = Ipp(H) + Isc(H) + Iac(H). (3.24)
The character of the distinct spectral components is identified from the following defini-
tions [142]:
• A function f : [a, b]→ D is continuous at a point x0 ∈ [a, b] if ∀ > 0 ∃δ > 0 such
that ∀x ∈ [a, b]: |x − x0|< ⇒ |f(x) − f(x0)|< δ. Essentially, this means that the
function exhibits no discontinuous jumps of any size, which would otherwise result
in a lower bound for δ.
Continuity, however, only describes the behaviour of a function at a single point
x0. Analysis of its first derivative f ′ characterises the trend of the function in the
neighbourhood of x0 and allows to further classify it into absolutely and singularly
continuous.
A function f : [a, b]→ D is absolutely continuous if its derivative f ′, wherever it
exists, satisfies:
f(x) = f(x0) +
∫ x
x0
f ′(t)dt ∀x ∈ [a, b]. (3.25)
9At least when |∆|= 1, in accordance with the Pisot conjecture [139].
64 Quasicrystals
A function f : [a, b]→ D is singularly continuous if its derivative f ′, wherever it
exists, does not satisfy:
f(x) = f(x0) +
∫ x
x0
f ′(t)dt ∀x ∈ [a, b]. (3.26)
This means that f ′(t) is zero, i.e. the function f is composed of steps.
• A discrete function f : [a, b] → D does not satisfy continuity because there is a
lower bound on δ such that |x− x0|< 6⇒ |f(x)− f(x0)|< δmin.
The pure point contribution to the diffraction spectrum usually consists of the Bragg
peaks of a translationally invariant crystal, while the absolutely continuous component is
normally the diffuse background caused by points defects, impurities, dynamic excitations,
and structural disorder. However, [143] showed that even a deterministic structure may
exhibit only a continuous spectrum without any sharp peaks, suggesting that the presence
of disorder cannot be established solely from the absence a pure point spectral component.
An example of a singularly continuous function is the Cantor function, or “Devils’s stair-
case”, shown in fig. 3.8. This is defined as c : [0, 1]→ [0, 1], where c(x) is determined by
the following iterative procedure:
• Given a domain σ(i)n , starting from σ0 = [0, 1],
• divide σn into three equal segments σ(1)n+1, σ
(2)
n+1 and σ
(3)
n+1,
• assign the value 0.5 · n to the middle segment σ(2),
• and iterate for n.
This function satisfies the condition for continuity, since ∃n such that |c(x) − c(x0)|<
δ ∀x, x0 ∈ [0, 1], but it is not absolutely continuous because its derivative is zero wherever
it exists. The Cantor function is therefore singularly continuous, and it is of physical
relevance as it describes the energy spectrum of some 1D quasiperiodic Hamiltonians,
such as the ones described by the Aubry-Andre´ model [144].
Other 1D quasiperiodic models are also known to exhibit singularly continuous energy
spectra, such as the Fibonacci, the Thue-Morse and the Rudin-Shapiro chains [145–148].
However, the diffraction spectra of these structures are found to be pure point, singularly
continuous and absolutely continuous respectively, establishing that the diffraction spec-
trum is not sufficient to determine the ordering of the structure [143].
An example of a pure point diffraction spectrum from a quasiperiodic model is the
Riemann-based sequence presented in fig. 3.4, where the addition of more segments only
results in sharper peaks at pre-existing k points.
3.2. Mathematical treatment 65
(a) (b)
(c) (d)
Figure 3.8 : The Cantor function c defined in the text is a singularly continuous function.
Iterations n = 1, 2, 4 and 8 are shown in (a)-(d). Its fractal dimension is df = 0.61 6∈ N.
Fractals
Self-similarity and non-trivial structure on all lengthscales are properties characteristically
found in fractals. A fractal is a region of d-dimensional space whose filling capacity scales
with a fractal dimension df 6= d which may not be an integer. They are also generated
by a rule, substitutional (such as the Sierpinski gasket in fig. 3.9) or polynomial (the
Mandelbrot10 set). The Sierpinski gasket of fig. 3.9 is defined by subdividing an equilateral
triangle of side L into four equal equilateral triangles of side L/2, and removing the central
one. The ensuing structure is therefore self-similar only upon certain integer divisors of
L, and its area is known to scale as Ldf where df = 1.585 6∈ N. The discrete nature of
self-similarity is richer than scale invariance as it is non-trivial and leads to an increased
complexity.
Fractality is nature’s way to seek optimal area to volume ratio, enabling efficient capture
of sunlight by leaves or optimal exchange of oxygen in lungs. Applications of fractal
structures include antennae [149], biosensors and search and compression algorithms [150].
Synthetic fractal structures have been investigated experimentally so far only in solid-state
system [151] – our 2D quasiperiodic optical lattice provides a chance to implement this
kind of complexity to quantum simulation with ultracold atoms.
10The B in Benoˆıt B. Mandelbrot stands for Benoˆıt B. Mandelbrot.
66 Quasicrystals
(a) (b)
(c) (d)
Figure 3.9 : The Sierpinski gasket is generated from diving an equilateral triangle of
side L into four of side L/2 and removing the central one. Iterations 1, 2, 3 and 6 are
shown in (a)-(d). The ensuing structure is self-similar only for certain L/n(n ∈ N), and
the blue area possesses a fractal dimension of df = 1.585 6∈ N.
3.2.4 Higher dimensions
Method
An alternative, and probably more satisfying, procedure for generating aperiodic pat-
terns is the higher dimensional method, also known as cut and project or strip projection
method. The quasiperiodic sequence or tiling of interest is defined on an n-dimensional
physical space, be in 1D, 2D or 3D, which becomes the parallel space component V ‖ of a
regular and periodic m-dimensional hyperspace V :
V = V ‖ ⊕ V ⊥, (3.27)
where V ⊥ are the perpendicular or augmented dimensions. The m vectors defining
the Fourier module H (the generalised reciprocal lattice) are projections from the m-
dimensional regular lattice Σ:
H = P‖(Σ) and bi = P‖(di), (3.28)
where bi span the reciprocal lattice of the quasicrystal according to eq. 3.11, and {di}
is the rationally independent basis of Σ. Crystallographic symmetries, theorems and
techniques can be applied in the higher dimensional space owing to its periodicity.
For example, the Fibonacci sequence of eq. 3.12 comprises two distinct and incommen-
surate segments L and S, despite being a 1D structure. As shown in fig. 3.10, it can
3.2. Mathematical treatment 67
be generated by an irrational projection the 2D square lattice points that fall within a
specified projection window.
(a) tanα ∈ Q (b) tanα 6∈ Q
Figure 3.10 : The 2D square lattice points that are contained within a projection window
(grey) are projected onto a 1D line inclined at an angle α. This results in a periodic pattern
if the tanα is rational (a), and in a quasiperiodic structure if tanα is irrational (b). In
(b), tanα = ϕ (the golden ratio) generates the 1D Fibonacci chain specified by eq. 3.12.
The higher dimensional method is a technique common to other areas of physics, where an
internal state of the system is used to augment its physical space. For instance, protons
p and neutrons n are both composed of u and d quarks, which thus justify extending the
physical space by means of the isospin I3 = 1/2 (nu−nd), where ni is the number of quarks
of flavour i. This results in:
p =
(
1
0
)
, n =
(
0
1
)
⇒
(
p
n
)
SU(2)−−−→ exp
(
− i2θaσa
)(
p
n
)
, (3.29)
where σ are the Pauli matrices and θ are arbitrary parameters. In this higher dimensional
space, the nucleons are symmetric by forming an (approximate11) SU(2) doublet.
The 2D Ammann-Beenker tiling of fig. 3.6d can be projected from a 4D simple cubic
lattice. Due to the difficulty of printing 4D cubes on A4 paper, this procedure is divided
into projection from the 3D simple cubic lattice shown in fig. 3.11, stacking the resulting
2D patterns in the third dimension, and then projecting again resulting in fig. 3.12.
The projected structures bear a striking resemblance to the diffraction patterns produced
by real eightfold symmetric quasicrystals shown in fig. 7.15, and to the matter-wave
diffraction spectra from the 2D quasiperiodic optical lattice of fig. 7.14.
The (minimum) required dimension of the hyperspace is determined by the minimum
number of rationally linearly independent vectors required to span the Fourier module H.
In 2D, the basis vectors needed to span the diffraction pattern of a n-fold periodic struc-
ture are the nth roots of unity; the number of rationally independent nth roots of unity is
equal to the Euler totient function φ(n) [152], which counts the number of integers less
than n that are relatively prime to n. The dimension m of the regular lattices is related
to the n-fold rotational symmetry [153] as reported in table 3.1.
Not all quasicrystals can be obtained from higher dimensions, such as the pinwheel pat-
tern, and the projections are unlikely to be unique: for instance, fivefold symmetry can
be explained in 4D but also, more intuitively, in 5D.
11exact in the limit of equal masses mu = md.
68 Quasicrystals
Figure 3.11 : A 3D simple cubic lattice, projected by pi/4, results in a structure with
periodicity along one direction and quasiperiodicity along the other.
Figure 3.12 : The planes obtained in fig. 3.11 are stacked in z to mimic a 4D simple
cubic lattice, and then projected again by pi/4 to result in an eightfold symmetric pattern
which can be mapped onto the Ammann-Beenker tiling of fig. 3.6d.
3.2. Mathematical treatment 69
m-dimensional hyperspace n-fold symmetry in 2D
m n
0 1
1 2
2 3, 4, 6
4 5, 8, 10, 12
6 7, 9, 14, 15, 18, 20, 24, 30
8 16, 21, 28, 36, 40, 42, 60
10 11, 22, 35, 45, 48,56, 70, 72, 84, 90, 120
12 13, 26, 33, 44, 63, 66, 80,105, 126, 140, 168, 180, 210
14 39, 52, 55, 78, 88, 110, 112,132, 144, 240, 252, 280, 360, 420
16 17, 32, 34, 65, 77, 99, 104, 130, 154,156, 198, 220, 264, 315, 336, 504, 630, 840
18
19, 27, 38, 51, 54, 68, 91, 96,
102, 117, 176, 182, 195, 231, 234, 260, 312,
390, 396, 440, 462, 560, 660, 720, 1260
20
25, 50, 57, 76, 85, 108, 114,
136, 160, 170, 204, 273, 364, 385,
468, 495, 520, 528, 546, 616, 770,
780, 792, 924, 990, 1008, 1320, 1680, 2520
22
23, 46, 75, 95, 100, 119, 135,
143, 150, 152, 153, 190, 216, 224, 228, 238,
270, 286, 288, 306, 340, 408, 455, 480, 585,
624, 693, 728, 880, 910, 936, 1092, 1155,
1170, 1386, 1540, 1560, 1848, 1980, 2310
Table 3.1 : The m-dimensional regular hyperspace compatible with n-fold rotational
symmetry in 2D. The 120-fold symmetry in the m = 10 line is the icosahedral symmetry
(A5 × Z2) displayed by the quasicrystal in fig. 3.2. Table taken from [153].
70 Quasicrystals
Figure 3.13 : A 2pi/5 rotation in the physical 2D (parallel) space is equivalent to a 4pi/5
rotation in the augmented (perpendicular) dimension. Figure taken from [91].
Phasons
A fivefold symmetric pattern in 2D represented by (x, y) vectors on V can, as per table
3.1, be represented by a 4D basis on D: |1〉 = (1, 0, 0, 0), |2〉 = (0, 1, 0, 0), |3〉 = (0, 0, 1, 0),
|4〉 = (0, 0, 0, 1), and |5〉 = (−1,−1,−1,−1).
Rotation by 2pi/5 in V corresponds to a cyclic permutation of the vertices, and can
therefore be written in D as:
R =
0 0 0 −1
1 0 0 −1
0 1 0 −1
0 0 1 −1
D
, (3.30)
which can be block diagonalised12 to:
cos 2pi/5 − sin 2pi/5 0 0
sin 2pi/5 cos 2pi/5 0 0
0 0 cos 4pi/5 − sin 4pi/5
0 0 sin 4pi/5 cos 4pi/5
V
=
( R‖ 0
0 R⊥
)
. (3.31)
We identify this matrix to be constructed from a basis (x, y, x′, y′) corresponding to the
Cartesian coordinates of the parallel (physical) and perpendicular (augmented) spaces.
The rotation is depicted in fig. 3.13, which shows that the augmented dimension may
possess its own distinct symmetries. These may therefore result in new and distinct
excitations – phasons.
An excitation along the parallel space is a phonon, and one along the perpendicular space
is a phason. As discussed in [105], phononic u and phasonic w strains cause a phase φ:
φ = k‖ · u + k⊥ ·w. (3.32)
The phason can be visualised as originating from the modulation of the projection window,
as shown in fig. 3.14: for strong enough strain, more or fewer hyperspace points fall within
12Compute the complex eigenvalues of R, αj ± iβj , and its eigenvectors vj ± iwj , with j = 1, 2 and
αi, βi,vi,wi ∈ R. Construct a real basis by noting that Rvi = αivi−βiwi and Rwi = βivi +αiwi, such
that R
(
vj
wj
)
=
(
αj −βj
βj αj
)(
vj
wj
)
, i.e. the usual rotation matrix in Cartesian coordinates.
3.2. Mathematical treatment 71
the window and are therefore projected onto the physical space. As observed in [106], this
may lead to a non-local rearrangements of the atomic density.
(a) k⊥ = 0 (b) k⊥ 6= 0
Figure 3.14 : Phasonic strain is driven by modulating the projection window. In the in-
finite wavelength limit, this just corresponds to a translation in the augmented dimension
(a), but it is generally more complex (b).
It is known that there are as many phononic modes as the dimension d of the regular
crystal, though the transition from an amorphous to a crystalline solid actually breaks the
whole Euclidean group Ed = SO(d)n Rd, that is both rotations and translations. These
are continuous groups with d(d− 1)/2 and d generators respectively, hence the Goldstone
theorem would predict the presence of d(d+ 1)/2 massless modes.
To justify the absence of the expected additional modes, let us consider the following.
1. Invariance by a translation u:
r→ r + u ⇒ ψ(r)→ T (u)ψ(r) = e−iu·ph¯ ψ(r).
2. Invariance by a global, rigid rotation δφ about n:
r→ r + (n× r)δφ ⇒ ψ(r)→ R(δφ)ψ(r) = e−iδφn·Lh¯ ψ(r).
If the structure is a regular lattice with translational and rotational symmetries, then the
two must coincide for some u and n:
e−i
u·p
h¯ = e−iδφn·Lh¯ ⇒ u · p = (n× rδφ) · p, (3.33)
i.e. breaking translational invariance automatically breaks rotational invariance as well.
Additionally, δφ ∼ ∂ru, u ∼ 1/k and ∂r ∼ k, so even as k → 0, one finds that 〈|δφ(k)|2〉 6=
0: the rotational modes are gapped, and can therefore not be the results of a broken
continuous symmetry as required by Goldstone’s theorem.
Finally, the Mermin-Wagner theorem invoked in section 1.2 also applies to diagonal
(density-density) long-range order in crystalline structures. True long-range order in
d 6 2 at T 6= 0 is therefore forbidden and 2D crystals cannot strictly speaking exist.
However, the five 2D Bravais lattice can still be found in nearly 2D systems or in planar
cuts of 3D structures.
Part II
Experimental details
Physics and Engineering
Conceive of a sinner who is Catholic and devout. What complexity in his
feeling for the Church, what pities of observance live between his sins. He has
to make such intricate shows of concealments to his damned habits. Yet how
simple is the Church’s relation to him. Extreme Unction will deliver his soul
from a journey through hell.
So it is with physics and engineering. Physics is the Church, and engineering
the most devout sinner. Physics is the domain of beauty, law, order, awe,
and mystery of the pursue sort; engineering is partial observance of the laws,
and puttering with machines which never work quite as they should work:
engineering, like acts of sin, is the process of proceeding boldly into complex
and often forbidden matters about which one does not know enough - the law
remain to be elucidated - but the experience of the past and hunger for the
taste of the new experience attract one forward. So bridges were built long
before men could perform the mathematics of the bending moment.
Norman Mailer [154]
73
If it [engineering] was easy, everyone would do it.
Nigel Palfrey, Head of the Student Workshop until 2017
4
Design, assembly and control
This chapter provides an overview of the necessary equipment that enables
trapping and cooling of cold atoms, namely the vacuum, laser and software
infrastructure.
4.1 Design
4.1.1 Setup overview
The experimental apparatus is shown in figs. 4.1 and 4.2.
The atomic sources are heated to achieve a high vapour pressure in the 2D MOT, where
they are pre-cooled in large numbers. Both species are simultaneously loaded, cooled and
trapped in the 3D MOT and subsequently magnetically transported to the science cell.
Here, evaporative cooling is performed first in a magnetic trap and then in a crossed-beam
dipole trap, resulting in the creation of a Bose-Einstein condensate (BEC). The BEC is
then illuminated with the optical lattice to perform experiments.
Differential pumping sections between the various chambers guarantee a low pressure in
the science cell to ensure a long lifetime of the BEC, and are discussed in section 4.2.
The stages of the experimental sequence employed to produce a BEC are summarised in
table 4.1. A detailed explanation of each experimental stage is provided in chapter 5.
4.1.2 Starting a lab from an empty room
A good architecture allows postponing decision-making as long as possible.
In every design decision we made while setting up the lab from scratch, we tried to abide by
this rule and avoid accumulation of “technical debt”1. We strived to guarantee flexibility
and versatility of all aspects of the system to ease future upgrades and experiments, as
1Dr. Ulrich Schneider©.
75
76 Design, assembly and control
well as including plenty of diagnostics for the day-to-day maintenance and operation of
the machine.
Figure 4.1 : Schematic diagram of the experimental apparatus. Figure adapted from
[94].
Figure 4.2 : Assembled vacuum system, July 4th 2016.
4.1. Design 77
stage duration section
2D & 3D MOT 2.5 s 5.1
“temporal dark-spot” MOT 10 ms
polarisation gradient 4.5 ms
optical pumping 0.7 ms
magnetic transport 5 s 5.2
evaporation in
magnetic trap 7.5 s 5.3.2
state transfer 20 ms 5.3.3
evaporation in
optical trap 3 s
total ∼ 23 s
Table 4.1 : Overview of the experimental sequence.
The theme for our experiment had been established to be quantum simulation in a
quasicrystalline optical lattice, given the surprisingly little experimental research on qua-
siperiodic potentials with degenerate quantum gases.
Choice of atomic species
87Rb and 39, 40K atomic species were chosen because their cooling, trapping and manipu-
lation techniques had been thoroughly investigated and were thus well established. This
enabled performance of novel experiments just by applying the optical lattice tool-kit to
our new geometry.
The bosonic (A = 39) and fermionic (A = 40) isotopes of potassium possess experiment-
ally accessible Feshbach resonances, being at relatively low fields (< 600 G) and reasonably
broad (∼ 10s of G) to be stable against ambient magnetic field noise. These allow tuning
the inter-species scattering length thereby enabling investigation of single-particle (non-
interacting) and strongly interacting physics, and it is the main reason as to why K was
utilised despite the difficulties it presents compared to Rb.
The challenges and the solutions to producing large quantum degenerate samples of po-
tassium are listed as follows:
• Smaller concentrations. The fermionic isotope 40K has a low natural abundance of
0.012%. Our potassium source was enriched to increase the 40K concentration to
2.98%, and the pre-cooling stage in the 2D MOT enhances the number of atoms
loaded into the 3D MOT. Both isotopes, in addition, have low background vapour
pressures, which is compensated by constantly heating the samples to increase their
evaporation rate.
• A narrow hyperfine structure. The hyperfine splitting of the excited states of po-
tassium is of the same order as their natural linewidths (∼ 10 MHz), preventing
78 Design, assembly and control
individual addressability and hence hindering the efficiency of usual laser cooling
schemes. This results in the atoms reaching a higher final temperature at the end
of the laser cooling stage.
• A higher two-body loss parameter β (discussed in section 1.3.2) which limits the
size of the potassium MOT. For comparison, β for 39K is ∼ 2 · 10−10 cm3 s−1 [155],
10 times larger than for 87Rb, for which it is ∼ 2 · 10−11 cm3 s−1 [156].
This, combined with the higher final temperature of K atoms resulting from the
less efficient laser cooling, would require a larger particle loss during evaporation.
Hence, K is directly evaporated only in the final stage of the process, while relying
on sympathetic cooling with 87Rb for the most part.
• The negative background scattering length in 39K renders the BEC unstable and
leads to its collapse. This is avoided by an appropriate choice of scattering length,
by tuning the Feshbach resonance, to guarantee repulsive interactions.
• The negative scattering length in 39K also causes a Ramsauer-Townsend minimum
for the inter-species s-wave cross section at ∼ 400 µK [157], where other partial
waves’ contributions are negligible. This is circumvented by using a positive inter-
species scattering length, and also by using 87Rb as a sympathetic coolant for most
of the evaporation process. The use of 87Rb will also aid in the cooling of fermionic
40K, hindered by the lack of s-wave scattering channels.
All-optical cooling of 39K has been achieved [158] and would render the presence of 87Rb
redundant, were it not for the possibility of future experiments involving Bose-Bose and
Bose-Fermi mixtures.
At the time of writing this thesis, only a MOT of 40K has been obtained since we decided
to perform the first experiments with degenerate bosons.
Design guidelines and Munich model
The priorities in the design process were optimising the sequence duration and maximising
the optical access around the science cell. The former to aid in optimising, diagnosing
and obtaining results, and the latter to allow inclusion of future upgrades with minimal
disruption. Hence, all the laser light is prepared on different optical tables and routed to
the experiment chambers via optical fibres, which however cause power loss and increased
phase noise.
In addition, to guarantee stability and to preserve optical alignment, no moving parts are
present on the main experiment table, along with unnecessary sources of heat. Temper-
ature, humidity and airflow of the tables are all actively stabilised.
Our design was based on an existing experiment at the Ludwig-Maximiliens Universita¨t
in Munich, code named Fermi-II and described in the following theses, heavily used at
the beginning of my PhD: [159–163].
A number of upgrades were made on the Munich design:
• Rb and K were each given their own 2D MOT, so as to allow for independent
optimisation of parameters, from the magnetic field gradient to the vapour pressure.
• A longer differential pumping section between the 3D MOT and the science cell was
used, to allow for more space and hence more complex optical beam paths to be
4.2. Vacuum 79
implemented. The increase in length was accompanied by an increase in diameter
to maintain the same conductance (∝ diamter3/length).
• Our science cell is rectangular with parallel sides, which allows for closer experiment
coils and stronger magnetic fields, but results in back-reflection for normally incident
beams.
• The magnetic transport was re-designed to be more compact and to minimise heat-
ing of the atomic cloud.
4.2 Vacuum
Why a vacuum?
Collisions between trapped cold atoms (∼ 100 nK) and the room-temperature background
gas (∼ 300 K) would transfer enough energy to eject them from the trap, thereby causing
a significant loss. These are reduced at low pressures. Hence, while a high pressure region
is required at the location of the sources for efficient loading, low pressures are needed
in the science cell to ensure a long lifetime of the trapped atoms. This is achieved by
differential pumping.
Differential pumping
Steady-state pressure gradients are guaranteed by the differential pumping of the various
components, akin to voltage drops in electrical circuits. This analogy can be made quant-
itative in a regime where molecular flow dominates over viscous flow, i.e. where the mean
free path of the background particles λmfp is larger than the lengthscale of the system
D ∼ 1 m. The pressures in our experiment (10−7 − 10−11 mbar) satisfy this condition as
the Knudsen number Kn = λmfp/D 1, and λmfp is in the ∼ km range. The vacuum
system can be treated as an electronic circuit (fig. 4.3) with the following equivalences:
P ⇔ V, C ⇔ 1
R
, Q⇔ I, so V = IR ⇔ P = Q
C
, (4.1)
where P is the pressure (V the voltage), C is the conductance (R the resistance) and
Q = d(PV )/dt is the throughput (I the current).
The calculation is detailed in appendix B.1 and results in the following design pressure
gradients:
P1
P0
≈ 10−4, P2
P1
≈ 10−2. (4.2)
The atomic vapour pressure P0 for both species is kept at ∼ 10−7 mbar, hence the 3D
MOT pressure was expected to be P1 ∼ 10−11 mbar and the science cell pressure P2 =
P3 ∼ 10−13 mbar.
Experimentally, the 3D MOT pressure is found to be ∼ 7× 10−11 mbar and the science
cell ∼ 1.2× 10−11 mbar, the latter being higher than the target value but comparable to
other experiments, suggesting it is limited by the degassing rate of the inner surfaces.
80 Design, assembly and control
(a)
(b)
Figure 4.3 : Schematic diagrams of the vacuum system (a) and its electronic analogue
(b). The atomic sources act like batteries providing a constant evaporation rate, while
apertures and pipes hinder the flow. Pumps are modelled as having no pressure (groun-
ded), and with conductance equal to their effective pumping speed S∗.
Pumps and gauges
There are three types of vacuum pumps, distinguished by the underlying physical mech-
anisms and attainable pressures.
A positive displacement pump comprises a cavity and a seal: a crankshaft expands the
cavity and lets fluid in, then seals it gradually guiding it to the exhaust. The cycle repeats
until the pump reaches its ultimate pressure, governed by the tightness of the seal and
the performance of the mechanical components. We used a scroll pump, in which two
interleaving spiral scrolls, with low-degassing grease as a seal, rotate inside each other.
This allowed reaching 5× 10−2 − 1.2× 10−1 mbar base pressures.
A vacuum lower than 10 mbar was needed in order to operate the turbomolecular pump.
This is a type of momentum transfer pump, which relies on molecules impinging on rotat-
ing propellers that direct them towards the exhaust. This stage occurs at the boundary
between viscous and molecular flow, hence the pump’s operation is not just suction, but
akin to a “black hole”. The best pressures achieved were in the low 10−9 mbar.
The scroll and turbomolecular pumps were only temporarily connected to the system, via
vacuum hoses and corner valves, in order to reach a sufficiently low pressure to enable the
use of entrapment pumps, which are constantly operational. These are sealed containers
with a large surface area, covered in a getter material that background gas particles
chemically bond with. Our system comprises two Ion getter pumps and two Titanium
Sublimation Pumps (TSP). The former include a hot cathode and a strong magnet (800−
2000 G) in order to emit electrons and allowing them to spiral in the vacuum system,
ionising gas particles which are then attracted towards the pump inlet. Ion pumps are
needed as well as the other getter pumps, as noble gases would not be pumped unless
4.2. Vacuum 81
first ionised.
These pumps enable maintaining pressures of ∼ 10−11 mbar in the 3D MOT and in the
science cell which are measured with Bayard-Alpert gauges2.
Sources and materials
The base pressure inside the vacuum chambers is determined by the equilibrium between
the pumping speed of the entrapment pumps and the degassing rate of the inner surfaces,
due to evaporation, de-absorption or diffusion from within the material. Low degassing
materials such as specialised alloys, glues and glasses were hence selected, and thoroughly
cleaned following the procedure detailed in appendix B.1. The system was also heated
to high temperatures (“baked”) in order to accelerate the degassing and diffusion rates
and achieve lower pressures faster. The presence of alkali metals aids in attaining low
pressures, as their chemistry allows them to act like getters once they coat the chamber
walls. Their aggressive chemistry may unfortunately also result in the degradation and
eventual failure of non-alkali compatible materials, such as certain makes of viewports3.
The 87Rb and 39K, 40K atoms were contained in two hermetically sealed ampoules ∼ 7 cm
long (fig. 4.4) and have the following specifications:
• 200.0 mg of potassium: Z = 19, 39K at natural abundance (93%) and 40K enriched
to 2.98%. Chemical Form: potassium chloride.
• 1 g of rubidium: Z = 37, 87Rb at natural abundance (29%). Chemical form: rubid-
ium nitrate.
The ampoules were inserted in two bellows valved to the vacuum system prior to the
start of the pumps, and then broken when high vacuum was achieved. The bellows are
wrapped in aluminium foil and heating wire in order for the vapour pressure to be kept
at ∼ 10−6 − 10−7 mbar, as summarised in table 4.2.
Figure 4.4 : The sealed ampoule filled with argon, containing 200 mg of potassium, with
an enriched fermionic isotope concentration.
2Ionisation gauges use a hot cathode to create electrons and an anode to accelerete them. These create
positive ions in the gas, which are later collected by another cathode and whose current measurement
provides an estimate for the gas density. However, some electrons impinge on the anode and lead to X-ray
production, which generates an additional photocurrent in the collecting electrode thereby undermining
its accuracy [164]. The gauge next to the science cell includes enhanced features to force the X-ray barrier
below 5× 10−12 mbar, as opposed to 3× 10−11 mbar of the gauge next to the 3D MOT.
3Which we found out the hard way.
82 Design, assembly and control
species melting point pressure at 295 K pressure at 333 K
Rb 313 K 3.8× 10−7 mbar -
K 337 K 1.7× 10−8 mbar 9× 10−7 mbar
Table 4.2 : Vapour pressures for our Rb and K sources [165], the former kept at room
temperature and the latter heated to ∼ 60 ◦C.
4.3 Lasers and magnets
4.3.1 Transitions
Spectral structure
As discussed in section 2.3.2, increasing phase space density requires the absorption and
subsequent incoherent scattering of many photons so as to lose entropy. An effective
implementation would see cooling happening on a closed or cyclic transition, where selec-
tion rules permit the light to only couple two atomic states. The presence of other states,
however, results in transitions to dark states which do not participate in the cooling pro-
cess. While this is beneficial in the last stages of laser cooling, where further scattering
only increases temperature, these atoms generally need to be repumped back into a state
affected by the cycling transition.
Rubidium and potassium have the following electronic structures:
Rb 1s2 2s2 2p6 3s2 3p6 4s2 4p6 5s,
K 1s2 2s2 2p6 3s2 3p6 4s.
They both have only one electron in the outermost shell, which for both possesses orbital
angular momentum L = 0. All other shells are full, hence only the outermost electron
contributes to the angular momentum. The electron spin being s = 1/2, the ground state
of the manifold is then 2S1/2. The first excited state corresponds to the electron being in
the p shell with L = 1, leading to a fine structure splitting between J = 1/2 and J = 3/2,
written as 2P1/2 and 2P3/2. Transitions between the ground and the first excited states in
alkalis are called D1 and D2 lines.
The nuclear angular momentum is I = 3/2 for both 87Rb and 39K (I = 4 for 40K),
which results in the hyperfine structure splitting among different F states, calculated in
appendix A and shown in figs. 4.6 and 4.7. Each F state has 2F + 1 sub-states, labelled
by the magnetic quantum number mF , whose degeneracy is lifted by an external magnetic
field4, discussed in section 4.3.1.
Only the D2 lines are used in our cooling schemes, hence F and F ′ will denote the ground
(S1/2) and the excited (P1/2) states.
Laser cooling requires the following beams to illuminate the atoms:
• Cooling
The MOT cycling transition for 87Rb and 39K is |F = 2,mF = −2〉 → |F ′ = 3,m′F =
−3〉, driven by σ− polarised light.
4Interestingly, only an external magnetic field fully lifts mF degeneracy, as it violates time reversal
symmetry. The linear Stark shift, for instance, causes an energy splitting depending on |mF |.
4.3. Lasers and magnets 83
• Repump
The cooling beam may also cause the transition |F = 2,mF = −2〉 → |F ′ = 2,m′F =
−2〉5, from which atoms may decay to |F = 1,mF = −1〉. This is dark with respect
to the cycling transition, hence a repump beam is tuned to the |F = 1〉 → |F ′ = 2〉
transition in order to provide the atoms with another chance to decay into |F =
2,mF = −2〉. Because of the larger separation of the F ′ states in 87Rb than in 39K,
fewer atoms populate the dark state and hence lower repump optical powers are
required.
• Optical pumping
Atoms are spin-polarised by being pumped into the same mF = 2 or mF = −2, as
described in section 2.2.1. This is achieved by a σ+or− polarised beam tuned to the
|F = 2〉 → |F ′ = 2〉 transition.
• Push
In the 2D MOT, discussed in section 5.1.1, a repulsive blue detuned beam is used to
drive the atoms towards the 3D MOT, by causing a momentum kick in the opposite
direction to that received upon absorption of red detuned light.
• Imaging
Atoms are detected by recording their spatial position, as detailed in chapter 6. This
relies on them interacting strongly with the incident light, in order to maximise the
signal. Hence, the imaging beam addresses the cycling transition |F = 2,mF =
2〉 → |F ′ = 3,m′F = 3〉 and it is on resonance.
The lifetime of the excited states F ′ is ∼ 10 ns for both 87Rb and 39K, which corresponds
to a natural linewidth of Γ ∼ 2pi · 6 MHz. In 39K, the energy separation between the F ′
states are on the order of Γ, which prevents individual addressability of the transitions and
therefore limits the effectiveness of laser cooling by reaching a higher final temperature
than in 87Rb.
High fields
In the presence of an external magnetic field, the relevant contributions to the Hamiltonian
(appendix A) are:
H = Hhyp +HZeeman, (4.3)
where the hyperfine structure Hhyp is modified by the interaction HZeeman coupling the
atomic magnetic moment µ and the external magnetic field B, chosen to lie along the z
axis. Its first order energy shift ∆EZeeman is given by:
∆EZeeman = −〈µ ·B〉 =
gFmFµBB for weak B,gJmJµBB for strong B. (4.4)
For weak fields, Hhyp HZeeman and the energy shift is a perturbation on the total
angular momentum states |F,mF 〉. For strong fields, HZeeman Hhyp which causes J
and I to decouple and precess about the z axis, the hyperfine structure amounting to a
perturbation on the |mJ ,mI〉 basis.
For intermediate field strengths, the energy shifts are found by exact diagonalisation of the
Hamiltonian, and are shown in figs. 4.8-4.12. The low-field states |F,mF 〉 are divided into
5The magnetic field being inhomogeneous, the quantisation axis of the atoms vary with their positions
in the trap.
84 Design, assembly and control
high-field seekers (grey) and low-field seekers (blue), and are adiabatically connected to the
states |mJ ,mI〉. As discussed in chapter 6, at any finite field strength, the |mJ ,mI〉 states
are not pure and prevent a fully closed transition at high fields. Imaging was nonetheless
performed on the transition |mJ = −1/2,mI = 3/2〉 → |mJ ′ = −3/2,mI′ = 3/2〉, whose
predicted shift was found to agree with the experimental measurement shown in fig. 4.5.
Useful quantum numbers and parameters are provided in tables 4.3 and 4.4.
Parameter S1/2 P3/2
gF
−1/2 for F = 1
1/2 for F = 2 2/3
gJ 2.000 4/3
Table 4.3 : Lande´ factors for 39K and 87Rb. Values from [166, 167].
Dominant good as B→ 0 good as B→∞ not good† always good
Weak B Hhyp I · J, F2 Jz, Iz J2, I2, FzStrong B HZeeman Jz, Iz I · J, F2
Table 4.4 : Relevant quantum numbers for strong and weak magnetic fields, valid until
IJ coupling fails. †: but still apply for stretched states.
320 340 360 380 400
Magnetic field /G
−400
−350
−300
K
im
ag
in
g
lo
ck
fr
eq
u
en
cy
/
M
H
z
Figure 4.5 : The potassium imaging frequency for high magnetic fields was varied until
resonant with the atoms. A scan was performed to determine the frequency of maximum
signal, which corresponds to the black points. The solid line is the expected shift of the
|mJ = −1/2,mI = 3/2〉 → |mJ ′ = −3/2,mI′ = 3/2〉 transition from exact diagonalisation
of the Hamiltonian.
4.3. Lasers and magnets 85
Figure 4.6 : Hyperfine structure of 87Rb.
Figure 4.7 : Hyperfine structure of 39K.
86
D
esign,assem
bly
and
control
Figure 4.8 : Energy levels shifts caused by an external magnetic field in 87Rb S1/2. Feshbach resonances shown as red dashed rectangle.
4.3.
Lasers
and
m
agnets
87
Figure 4.9 : Energy levels shifts caused by an external magnetic field in 39K S1/2. Feshbach resonances shown as red dashed rectangle.
88
D
esign,assem
bly
and
control
Figure 4.10 : Energy levels shifts caused by an external magnetic field in 87Rb P3/2. Feshbach resonances shown as red dashed rectangle.
4.3.
Lasers
and
m
agnets
89
Figure 4.11 : Energy levels shifts caused by an external magnetic field in 39K P3/2. Feshbach resonances shown as red dashed rectangle.
90
D
esign,assem
bly
and
control
Figure 4.12 : Energy levels shifts caused by an external magnetic field in 39K P3/2, here plotted for low field strengths.
4.3. Lasers and magnets 91
4.3.2 Frequency Locking
Doppler free saturation spectroscopy
Laser cooling relies on the individual addressability of atomic transitions, which requires
the laser frequency to be stable to less than the natural atomic linewidth Γ ∼ 2pi · 6 MHz.
The frequency stability of a laser is undermined by thermal drifts and ambient noise,
and hence needs to be actively corrected (“locked”). To guarantee stability within Γ, a
PID loop responds to an error signal generated by the absorption lines of 87Rb and 39K
sources in a separate vapour cell using Doppler-free saturation spectroscopy [168, 169].
As illustrated in fig. 4.13, a portion of the laser radiation is split between a powerful pump
beam and a weak probe beam, both directed onto the vapour cell. The probe beam is
incident onto a photodiode measuring its transmission through the atomic cloud. As the
frequency of the laser is varied (“scanned”) a Doppler-broadened signal (∼ 1 GHz wide)
is recorded by the photodiode as none of the atoms are generally resonant with either
beam. A special case, however, are atoms at rest, for which both beams will become
resonant with the same transition: the pump beam will then saturate the transition,
resulting in reduced absorption of the probe beam and hence an increased transmission.
Another special case concerns atoms with a velocity such that the Doppler shifted beams
are resonant with distinct transitions, resulting in crossover peaks situated at the average
frequency of each pair of transitions. In 87Rb , the excited states |F ′〉 are well separated
and resolved, so each ground state |F = 1, 2〉 is coupled to them individually. At the
crossover, the pump beam will have already depleted the ground state, hence the probe
will again experience a reduced attenuation. In 39K, on the other hand, the |F ′〉 states
are too close to be individually resolved. At the crossover frequency, the pump and probe
beams then create a Λ system with the two ground states |F = 1, 2〉, pumping atoms
from one to the other, which results in an enhanced absorption of the probe beam. These
mechanisms are schematically shown in figs. 4.14 and 4.15, along with the experimental
signatures shown in figs. 4.16, 4.17, and 4.18.
Figure 4.13 : The Doppler-free saturation spectroscopy apparatus. The pump beam
ensures the ground state is depleted so as to induce a higher transmission of the probe
beam, measured by a photodiode.
92 Design, assembly and control
(a) v = 0
ω = ω1
(b) general v
general ω
(c) v = ±(ω1 − ω2)/2k
ω = (ω1 + ω2)/2
Figure 4.14 : In general, atoms at rest will be resonant with the both beams (a) while
the others experience a Doppler-broadened absorption (b). In 87Rb , the degeneracy in
the distinctly resolved excited states causes a reduced absorption of the probe beam at
the crossover peaks (c).
(a) v = 0
ω = ω2
(b) general v
general ω
(c) v = ±(ω1 − ω2)/2k
ω = (ω1 + ω2)/2
Figure 4.15 : In 39K, the degeneracy is in the ground states, which at the crossover
frequency form a Λ system causing an increased absorption of the probe beam (c).
Figure 4.16 : Doppler-broadened (red) and Doppler-free (blue) transmission signals of
the probe beam, for both isotopes of 85Rb and 87Rb present in the vapour cell.
4.3. Lasers and magnets 93
(a) Doppler broadened
(b) Doppler free
Figure 4.17 : Doppler-broadened (a) and Doppler-free (b) transmission signals of the
probe beam for the |F = 2〉 → F ′ transitions in 87Rb. Peaks 1, 3 and 6 correspond to the
transitions |F = 2〉 → |F ′ = 1, 2, 3〉, while 2, 4 and 5 are the crossover peaks. We lock on
peak 5.
(a) Doppler broadened
(b) Doppler free
Figure 4.18 : Doppler-broadened (a) and Doppler-free (b) transmission signals of the
probe beam for the |F = 1〉 → F ′ (peak 1) and |F = 2〉 → F ′ (peak 5) transitions in 39K.
Peak 2 is the crossover peak6 (which we lock to), and peaks 3, 4 and 6 are the transitions
and crossover peak for 41K, also present in the vapour cell.
PDH locking
A PID stabilisation circuit requires an antisymmetric signal in order to provide feedback.
The frequency may be locked to one of the slopes of the absorption peaks, side of fringe
locking, but this causes it to be highly sensitive to intensity fluctuations. Top of fringe
locking is therefore preferred as the peak’s position is fixed irregardless of the intensity of
the probe beam. Its symmetrical profile is circumvented by utilising its derivative as the
6Or, better, the resultant of the many crossover peaks caused by transitions among the internal |F 〉
and |F ′〉 states, as discussed in [94].
94 Design, assembly and control
error signal to feed into the PID electronics. The technique is known as Pound-Drever-
Hall (PDH) locking [170, 171] and it consists in modulating the laser carrier frequency ω
with a small frequency Ω in order to add sidebands at ω ± Ω:
E = E0 (1 + sin Ωt) e−iωt ∝ e−iωt + e−i(ω−Ω)t + e−i(ω+Ω)t, (4.5)
where amplitude modulation was used above, but phase modulation is also common. The
optical power P of the light can then be Taylor expanded to extract its derivative:
P (ω ± Ω) ≈ P (ω)± dPdωΩ + . . . . (4.6)
The modulation frequency Ω needs to be large enough for its higher harmonics to be
effectively filtered out, but also small enough not to interfere with the atomic transitions,
and was chosen to be between 20− 30 kHz.
In order to obtain the steepest slope and hence the best error signal, we locked on the
highest absorption peaks, i.e. the crossover lines in figs. 4.17 and 4.18. The linewidth of
the lasers is ∼ 100− 200 kHz, and the emitted frequency of 87Rb was shifted by 53 MHz
to aid in later manipulation.
Delay-line locking
With one master laser ωM locked to an atomic transition, the frequency of other diodes
ωD can be referenced to ωM by means of offset or delay-line locking. All other atomic
transitions are addressed using acousto-optical modulators (AOMs), where a laser beam
propagating through an RF-driven crystal absorbs a phonon and experiences a deflection
and a frequency shift. The optical power of this light may be later amplified by Tapered
Amplifiers (TAs), which owe their name to the tapered shape in order to avoid an increase
of the optical intensity which would lead to damage.
As illustrated in fig. 4.19, the two beams E1 and E2 are shone onto a photodiode, and
eventually split into two paths of unequal length (the delay-line) so as to acquire a net
phase difference φ(ωD − ωM = ∆ω), which yields the error signal:
V (tscan) = A(∆ω) E1 · E2 cos
(
ω0 −∆ω
c
∆x · tscan
)
, (4.7)
where A is the frequency dependence of the circuit, and ω0 is a local oscillator mixed with
the beat-note ∆ω. This allows remote change of the lock point (e.g. to vary the imaging
frequency for 39K in high magnetic fields) and results in a lower operational frequency,
reducing loss and attenuation. A typical error signal is shown in fig. 4.20.
4.3. Lasers and magnets 95
Figure 4.19 : Schematic diagram of a delay-line lock.
Figure 4.20 : The error signal for the Rb repump laser, locked at the marked position.
Note the symmetry about the point where ∆ω = 0. Figure adapted from [172].
Isotope switch
Appendix B.2 details the optical beam paths of all laser radiation.
The experiment was designed to address the energy manifolds of both isotopes 39K and
40K, which only differ by the isotope shift [78]. To implement this “isotope switch”, the
laser apparatus consists of double-pass AOMs that guarantee a fixed direction of the
outgoing beam irregardless of the drive frequency, so that it can inject the same fibre and
reach the experiment chamber. This is shown in fig. 4.21.
96 Design, assembly and control
Figure 4.21 : The optical switch used to address either 39K or 40K. The lens in the
double-pass AOM ensures that both the diffracted and undiffracted beams emerge along
the same direction.
4.4 Control
Probing of the BEC occurs at the end of the cooling sequence. In order to investigate its
evolution and its dependence on variable parameters, many identical sequences need to
be performed. These rely on the reproducibility and reliability of the programmed time
intervals, and is allowed by a centralised unit responsible for setting the absolute timing
which all hardware devices are synchronised to.
This unit needs to be independent from the operating system, which is designed to always
respond to tasks with higher priorities, from keyboard strokes to incoming network pack-
ages. Hence, the time keeping duties are delegated to external timing cards, that provide
independent reference clocks and counters along with programmable digital and analogue
output channels, schematically illustrated in fig. 4.23.
The software used for sequence programming and equipment control was written in python
and developed by reserach groups at LMU and MPQ in Munich. It was installed for us
by Hendrik von Raven. Having an open-source platform allowed developing and including
more functionalities, such as a genetic algorithm to aid in optomisation [173] and a large
number of devices such as cameras, the DDS7 and the signal generators creating arbitrary
optical lattice ramps. A short documentation for the signal generators programming is
provided in appendix D.9.
Software and hardware communicate over a private network (LAB network) in order to
reduce traffic and delays, and only require the building’s network (AMOP network) to
access the group’s shared drive and save the data. Network communication is handled by
Pyro, a python library consisting of a name server, listing the (logical) names of different
objects on the network, and an event server, publishing and subscribing to information
on the network.
Whilst the external cards ensure the timings of all controls are fixed relative to a master
clock, the mains synchroniser guarantees the sequence starts at the same time relative
to the 50 Hz mains frequency. Since mains noise cannot be completely eliminated, this
synchronisation guarantees that the effect is the same on each experimental run. Its
principle of operation is depicted in fig. 4.22, and its effect can be quantified by the
recovery of Rabi oscillations (eq. 2.10) shown in fig. 4.23.
7Direct Direct Synthesis, a programmable frequency generator developed by Hendrik von Raven [172].
4.4. Control 97
The machine also includes an alarm system, known as the Schutzschaltung thanks to (or
because of) the intrinsic German heritage of the group, and an independent monitoring
system that logs diagnostics online. CARBON is an open-source software that received
data over the network and stores it in order to be subsequently manipulated by various
web interfaces for monitoring, diagnostics, alarm raising etc. with API such as Icinga2,
Graphite and Grafana.
(a) Mains sync signals (b) Rabi oscillations
Figure 4.22 : a) The main synchroniser allows the sequence to start (Resync) only when
the previous sequence has finished (Output) and the mains frequency hits a rising edge.
b) A small magnetic field lifts the mF degeneracy in the 87Rb |F = 1〉 ground state.
An applied RF pulse of variable duration is expected to induce Rabi oscillations between
these states, but the ambient field noise does not allow a discernible signal to be identified
unless the mains synchroniser is operational.
98
D
esign,assem
bly
and
control
Hardware
Software
Master clock
Clock synchronisation
Mains synchroniser
Channels
Analogue
Digital
Analogue
isolation amplifiers
Digital
isolation amplifiers
coaxial cables
• AOM frequency ramps
• AOM amplitude ramps
• TA current ramps
• Coils' currents
• Camera triggers
• Shutters
• MOSFETs
saves hdf5 file with
sequence parameters,
script and images
physical connection
AMOP network
LAB network
QControl (python2)
Server: Timing server
running in background
Client: sequence & GUI
logical links
Sensors
Alarms
• AirCon temperatures, airflow, humidity
• MOT chamber and science cell pressures
• Temperatures of sources' ovens
• thermocouples
• leaks sensors
• flowmeters
• heartbeat
Schutzschaltung
turns off
software
Timing PC
Timing Cards
AMOP
network
LAB
network
Shared drive
CARBON server
twisted pairs
LAB
network
AMOP
network
LAB
network
LAB
network
Image analysis
Data analyser
Web interface
(Graphite, Graphana, Icinga2...)
• Arbitrary waveform generator
• DDS ramps & pulse shape
• Oscilloscope prep and data download
• Cameras setup and image download
• PC's with camera servers
AMOP
network
Figure 4.23 : Schematic diagram of the software and hardware control of the experiment. Dashed lines are yet to be implemented.
You can’t go back and change the beginning but
you can start where you are and change the ending.
C.S. Lewis
5
Reaching quantum degeneracy
This chapter details the components taking part in the cooling stages, provid-
ing quantitative information about their assembly, operation and performance.
5.1 MOT chambers
The left section of the apparatus shown in fig. 4.1 performs laser cooling using the optical
molasses, MOT and polarisation gradient techniques outlined in section 2.3.2.
5.1.1 2D MOT
The pressure in the 3D MOT is ∼ 10−11 mbar and it would take > 10 s to capture a large
amount of atoms from the background vapour. Hence, each atomic source is placed next
to a 2D MOT, filled with high-pressure atomic vapour (∼ 10−7 mbar) whose purpose is
to provide a stream of pre-cooled atoms to enable fast loading of the 3D MOT.
The 2D MOT is illustrated schematically in fig. 5.1, and shown in the experiment in fig.
5.4. This stage consists of the following simultaneous processes:
• a MOT along the two transverse directions x and y, giving it its name “2D” MOT;
• optical molasses along z provided by a linearly polarised axial cooling beam, whose
reflected intensity is attenuated by an absorptive filter;
• continuous loading of cold atoms along the central axis x = y = 0 into the 3D
MOT by a blue detuned, repulsive push beam propagating through the φ = 1.5 mm
differential pumping section.
The magnetic field is generated by two pairs of rectangular coils, arranged in the anti-
Helmholtz configuration, to result in a quadrupole field in the transverse xy plane which
is invariant along z, as shown in fig. 5.2. Table 5.1 summarises the magnetic field para-
99
100 Reaching quantum degeneracy
(a) 2D MOT (b) Differential pumping section
Figure 5.1 : Diagrams of the 2D MOT (a) and the differential pumping section (b). Two
σ polarised elliptical beams (1, 2) are shone from the sides and retro-reflected through
large λ/4 waveplates (3) to ensure maintenance of σ polarisation. A quadrupole magnetic
field is generated by coils wound around a holder (4) and water-cooled by a brass heat sink
(5). Optical molasses is performed along the z direction by the axial cooling beam (6),
retro-reflected by a mirror on the differential pumping section (8) and through a polariser
to tune its intensity. A blue detuned, repulsive push beam (7) continuously loads the cold
atoms on the central axis of the 2D MOT into the 3D MOT.
5.1. MOT chambers 101
meters, computed from the analytical solutions for rectangular coils quoted in appendix
B.2.
(a) Cross section (b) From above
Figure 5.2 : The magnetic field generated by the 2D MOT rectangular coils is a quad-
rupole field in xy (a) invariant along z (b).
parameter 87Rb 39K
quadrupole field
gradient @ 1 A (per pair) 2.4 G/cm
windings per coil 63
current per coil 8.5 A 9.0 A
field gradient 21 G/cm 22 G/cm
bias field
field @ 1 A (per pair) 1 G
windings per coil 8
current in x pair 0 A 2.5 A
current in y pair 2.5 A 0.6 A
Table 5.1 : Magnetic field parameters for the 2D MOTs of each atomic species. On top
of the quadrupole field coils connected in the anti-Helmholtz configuration, offset coils in
the Helmholtz configuration were also wound in order to cancel the strong fields caused
by the ion pumps and optical isolators.
The optical apparatus for the 2D MOT laser beams is shown in fig. 5.3 and summarised
in table 5.2.
The transverse beams are elliptical in order to address a larger volume of the vapour,
and their centres are shifted closer to the start of the differential pumping section as this
resulted in enhanced loading. While the push beam was essential in the 39K 2D MOT, it
was not used on the 87Rb side, where a power imbalance 60 : 40 between the incident and
reflected axial cooling beams was found to be sufficient.
Alignment of the axial and push beams were crucial for a successful, efficient and quick
loading of the 3D MOT.
102 Reaching quantum degeneracy
parameter 87Rb 39K
duration 2.5 s 1.0 s
cooling
detuning −19 MHz −28 MHz
power x : 230 mW, y : 320 mW x : 190 mW, y : 260 mW
single-waist 20 mm× 10 mm
repump
detuning 0 MHz −15 MHz
power x : 48 mW, y : 39 mW x : 230 mW, y : 170 mW
single-waist 20 mm× 10 mm
axial cooling
detuning −19 MHz −28 MHz
power 8.5 mW 1 mW
single-waist 2 mm
push
detuning 42 MHz
power 2 mW
single-waist 0.8 mm
Table 5.2 : The parameters for the lasers employed in the 2D MOT cooling stage.
Figure 5.3 : The optical setup that delivers light to the 2D MOT.
Assembly
The 2D MOTs were assembled in-house, selecting vacuum-compatible materials with sim-
ilar thermal expansion coefficients in order to allow baking.
A titanium (grade 21) frame and bespoke windows were ordered for the chamber to be
assembled in-situ. The components utilised in the assembly are listed in Table 3. These
all have similar thermal expansion coefficients, to avoid thermal shocks and cracks while
baking. The mirror, polariser and differential pumping sections are separated by titanium
spacers, which minimised the glued area and thus the potentially trapped air that could
undermine the integrity of the vacuum; this glue is thermally conductive to allow baking
of the components.
1The grade indicates which alloy and what concentrations are present. They differ in strength, ductility
and corrosion resistance, but should not be important for vacuum applications.
5.1. MOT chambers 103
Component Material Thermal expansion Thermal conductivity
10−6 K−1 10−3 W/(m ·K)
Frame Titanium, grade 2 9.2 20,800
Differential pumping section Titanium, grade 2 9.2 20,800
Windows BK7 glass 7.1 6.5
Mirror BK7 glass 7.1 6.5
Polariser N-BK7 glass 8.3 N/A
Spacers Titanium, unknown grade N/A N/A
Glue (windows 7→ frame) EPO-TEKr 353ND 54 N/A
Glue (mirror & polariser 7→ spacers) EPO-TEKr 920 24 970
Table 5.3 : Components used in the assembly of the 2D MOT.
In order to guarantee vacuum compatibility, the glue was degassed with a desiccator
pumped by the Scroll pump, reaching low 10−1 mbar.
A thin layer of Kapton film was added between the frame and the windows, so as for the
windows not to expel all the glue.
(a) (b)
Figure 5.4 : The assembled 2D MOT just after integration in the vacuum system (a)
and during the operation of the experiment. Once cured, the glue attaching the win-
dows turned orange. The yellow and blue wires are the quadrupole and the offset coils
respectively.
5.1.2 3D MOT
Simultaneous to the incoming flux of pre-cooled atoms from the 2D MOTs, another MOT
stage takes place for the combined species in the 3D MOT. This is exactly as outlined in
104 Reaching quantum degeneracy
section 2.3.2, with a pair of circular coils providing a quadrupole field and three pairs of
counter-propagating beams along each orthogonal direction.
The 3D MOT is shown in fig. 5.7 and its operating parameters are summarised in table
5.4. Cooling and repump light for both species are overlapped and distributed among the
6 fibres delivering them to the chamber by a commercial 2 7→ 6 fibre cluster.
The density and temperature of the atomic cloud is eventually limited by the re-scattering
of incoming photons, which results in additional effective forces preventing further increase
of phase space density. In the last 10 ms of the MOT stage, the 39K repump power was
lowered to 2% of its previous value, in order to perform a “temporal dark-spot MOT”
where the atoms in the F = 1 state would no longer participate in cycling transitions and
hence experience a reduced scattering rate.
The MOT is followed by a polarisation gradient cooling stage, performed by the same laser
beams and hence in the σ+−σ− configuration outlined in section 2.3.2, whose parameters
are summarised in table 5.5. This is very short because the magnetic field is turned off
and hence no restoring force counteracts the cloud’s expansion.
The laser cooling stage lasts 2.5 s resulting in ∼ 3 × 109 87Rb atoms at ∼ 30 µK and
∼ 2 × 108 39K atoms at ∼ 470 µK, while the temperatures before polarisation gradient
were ∼ 580 µK and ∼ 1.1 mK respectively. Both laser cooling stages are better performing
for 87Rb owing to its resolved excited state manifold. 39K is only loaded in the last 1 s
of the MOT stage, which was found to result in the maximum number of atoms for both
species.
A 87Rb MOT is shown in fig. 5.7, and the fluorescence curves recorded during the loading
stage are shown in fig. 5.6.
The size of the MOT is eventually limited by its high spatial density, which increases
collisions with the background gas and enhance light assisted collisions. High densities also
cause re-scattering of photons, which result in an equilibrium between a repulsive (from
atoms absorbing re-scattered photons) and an attractive (from the beams’ attenuation)
force [174, 175].
5.1.3 Optical pumping
The atoms need to be spin-polarised into a low-field seeking mF state in order to maximise
the capture efficiency of the magnetic trap, as described in section 2.2.1 and shown in fig.
2.3. The experimental parameters used are summarised in table 5.6.
A ∼ 10 G homogeneous field is present during this stage in order to provide a quantisation
axis and to lift the degeneracy among the mF sub-levels. For 87Rb, σ− light is used, and
repump is only active for the first half of the procedure, in order to accumulate atoms
in the |F = 1,mF = −1〉 state. For 39K, we did not manage to efficiently perform
optical pumping, hence no radiation addressing this species is used. The success of the
evaporative cooling stage established enough 39K was being transported to the science
cell, hence further investigations into the failure of optical pumping were not performed.
5.1. MOT chambers 105
(a) (b)
Figure 5.5 : The optical apparatus to shine laser light into the 3D MOT is schematically
shown in (a) and looks like (b) in real life. The brass circular structure houses the coils
generating the quadrupole magnetic field.
parameter 87Rb 39K
duration 2.5 s 1.0 s
cooling
detuning −23 MHz −52 MHz
power 95− 130 mW 40− 46 mW
single-waist 13 mm
repump
detuning 0 MHz −25 MHz
power 0.7− 1.0 mW 60− 87 mW
single-waist 13 mm
quadrupole field
z gradient @ 1 A 1.78 G/cm
current per coil 11 A
field gradient 20 G/cm
bias field
field @ 0.4 A (per pair) 1 G
windings per coil 6
current in x pair 1.5 A
current in y pair 1.5 A
current in z pair 0.3 A
Table 5.4 : Parameters for the dual-species 3D MOT stage.
106 Reaching quantum degeneracy
parameter 87Rb 39K
duration 4.5 ms
cooling detuning −63 MHz −4 MHzpower 26− 35 mW 16− 18 mW
repump detuning 0 MHz −45 MHzpower 0.7− 1.0 mW 1.9− 2.8 mW
Table 5.5 : Parameters for the polarisation gradient cooling stage that follows the 3D
MOT stage.
Figure 5.6 : Fluorescence curves measured by the appratus shown in fig. 6.1, when the
two species are loaded alone (opaque) or together (solid). These provide a measurement
for the loading time and final atom number of both species. Figure taken from [94].
Figure 5.7 : Picture of 87Rb in the 3D MOT chamber, without any 39K present. The in-
coming flux of atoms from the 2D MOT pierces the cloud and results in the tail appearing
in the top-left corner.
5.2. Magnetic transport 107
parameter 87Rb
pumping
duration 100 µs
detuning −3 MHz
power 1.5 mW
single-waist 3 mm
repump
duration 45 µs
detuning 0 MHz
power 0.4 mW
single-waist 3 mm
guide field
field @ 1 A (pair) 1 G/cm
windings per coil 15
field 10 G
Table 5.6 : Parameters for the optical pumping stage of 87Rb. No optical pumping was
performed for 39K.
5.2 Magnetic transport
After optical pumping, a quadrupole magnetic field in the 3D MOT chamber is ramped
to ∼ 100 G/cm to magnetically trap the atomic cloud. This is then transported to the
science cell via the series of overlapping coil pairs shown in fig. 5.8, each connected in
the anti-Helmholtz configuration and varying its current to allow the smooth transition
of the field minimum illustrated in fig. 5.9.
The atoms cover a total distance of ∼ 65 cm in 5 s. The magnetic field ramps were
designed to maintain the atomic cloud aspect ratio and minimise heating. Including a
smooth acceleration at the beginning and end of each current ramp proved to be crucial
to the success of magnetic transport. Non-magnetic materials were employed both in
the vacuum system and in the water-cooled coils holder in order to guarantee minimal
disruption to the simulated magnetic field configuration. Eddy currents were also reduced
by including slots in the metallic components, where possible.
The transport efficiencies for the two species are summarised in fig. 5.10. A 87Rb cloud at
∼ 50 µK at the start of the transport stage was found to have increased its temperature
to ∼ 200 µK when reaching the science cell.
Only 3 pairs of coils are active at any given moment during magnetic transport, enabling
the use of a limited number of current sources. These are switched between the coil pairs
via MOSFETs2, as shown in fig. 5.11a. The final coil pair, the “experiment coils”, was
manufactured in-house and is connected to an H-bridge (fig. 5.11b) to allow operation
both in the Helmholtz and anti-Helmholtz configuration, thus switching from the trapping
quadrupole field to a homogeneous Feshbach field. The technical details and magnetic
field parameters of the experiment coils are summarised in tables 5.7 and 5.8.
2The electronics for the magnetic transport was designed and manufactured by Dr. Stephen Topliss.
Field effect transistors (MOSFETs) are used instead of bipolar transistors (BJTs). In BJTs, the collector-
emitter current is related to the base-emitter current by the hFE parameter, such that a ∼ 100 A current in
the coils would require ∼ 100 mA to open the channel, large enough to require its own specialised current
source. In MOSFETs, on the other hand, the drain-source current is controlled by the gate-source voltage.
108 Reaching quantum degeneracy
2
3
5
1
4
Figure 5.8 : Magnetic transport is performed by a series of coils connected in the anti-
Helmholtz configuration, as shown in fig. 5.9. These include a “push” coil (1) which aids in
initiating the transfer, the coils used in the MOT stage (2), other transport coils (4, 5) and
finally the experiment coils, which also provide the Feshbach field. The coils are placed
in a water-cooled brass “cooling block” which guarantees stability and reproducibility of
the fields.
Figure 5.9 : In order to transfer the atomic cloud (green), the current in the next coil
pair is gradually increased while the one in the previous pair is reduced, resulting in a
smooth translation of the magnetic field minimum.
Figure 5.10 : The magnetic transport efficiency is measured by the cloud’s fluorescence
in the 3D MOT chamber. The return (left) and single-journey (right) efficiencies are
shown here. Figure adapted from [94].
5.2. Magnetic transport 109
(a) (b)
Figure 5.11 : a) Each power supply is connected to multiple MOSFETs, each enabling
current to reach a particular coil pair, in order to drive the 19 pairs of coils with only 3
current sources. The circuitry also includes a return path for the current (with a diode and
a varistor) when the MOSFET is open, in order to reduce the back-emf spike and avoid
damage. b) The upper coil in the final pair is connected to an H-bridge, which allows the
current direction to be reversed. This enables the “experiment coils” to generate either a
quadrupole or a homogeneous field.
thickness inner diameter outer diameter windings electrical resistance
h φin φout R
16.9− 17.1 mm 62− 63 mm 148 mm 75 45, 55 mΩ
Table 5.7 : The experiment coils were machined in-house, their parameters are reported
here.
configuration field @ 1 A
anti-Helmholtz 3.44 G/cm
Helmholtz 11 G
Table 5.8 : The H-bridge of fig. 5.11 allows the experiment coils to produce either a
quadrupole or a homogeneous magnetic field, when connected in the anti-Helmholtz or in
the Helmholtz configurations.
110 Reaching quantum degeneracy
5.3 Evaporative cooling
5.3.1 Science cell
Layout
The magnetic transport allows transferring the atoms to the science cell, shown in fig. 5.12,
where the evaporative cooling stage and all subsequent lattice experiments are performed.
Additional coils were wound around the science cell to cancel any ambient magnetic field
caused by the ion pumps or by the optical isolators, whose parameters are summarised in
table 5.9. In particular, the pair providing the vertical homogeneous field is driven by a
programmable current source, in order to shift the magnetic trap minimum to efficiently
load the optical trap.
Two breadboards are placed around the science cell, as shown in fig. 5.13 and schem-
atically illustrated in fig. 5.14. These provide the optical beam paths for imaging light,
the crossed-beam dipole trap and the optical lattices. The length of the beam paths was
minimised in order to increase stability and maximise optical access.
(a) (b)
Figure 5.12 : The science cell just after its inclusion in the vacuum system (a) and in
the operational experiment (b).
axis field @ 1 A (pair) windings current (pair)
D 0.6 G 6 4.5 A
T 0.6 G 6 1.5 A
Z 0.6 G 12 0.2 A
Table 5.9 : Three pairs of coils in the Helmholtz configuration are wound around the
science cell as shown in fig. 5.12b to cancel background magnetic fields.
5.3.2 Evaporation in a magnetic trap
The magnetic transport stage ends with the atomic cloud in the |F = 1,mF = −1〉
state, magnetically trapped in the science cell in a quadrupole trap of gradient 100 G/cm.
This has a large acceptance region and can therefore accommodate a larger number of
highly energetic (“hotter”) atoms, the majority of which would otherwise be lost if the
crossed-dipole optical were loaded directly.
5.3. Evaporative cooling 111
(a) Science breadboard (b) Knee breadboard
Figure 5.13 : A Hamiltonian in real life. The optical beams paths to probe and trap
the atoms are assembled on two breadboards, code-named “science” (a) and “knee” (b).
A first evaporative cooling stage is then performed in the magnetic trap. When compared
to the atomic cloud size, the extent of the magnetic trap is essentially infinite for any
field strength. Hence, the reduction of the trap depth explained in section 2.3.3 cannot
be performed by decreasing the magnetic field. Instead, a microwave (MW) radiation
generated by a waveguide is shone onto the atoms in order to cause a transition to the
magnetically untrapped state |F = 2,mF = −2〉, resulting in particle loss and enabling
thermalisation of the remaining atoms to a lower temperature. This is then referred to as
forced evaporation. The gradient of the magnetic trap is ramped to 300 G/cm to increase
the spatial density and enhance the elastic scattering and thermalisation rates.
Because of the smaller 39K cloud produced in the 3D MOT chamber, further loss of 39K
atoms in this stage is avoided by only evaporating 87Rb. The cold 87Rb cloud then sym-
pathetically cools 39K, whose temperature decreases while maintaining the same number
of atoms. The frequency of the MW radiation (often called a “knife”) is ramped from low
frequencies to high frequencies in order to first address the most energetic atoms, situated
far from the centre of the trap, as shown in fig. 5.15.
The stage consists of three linear ramps, each lasting 2.5 s, with edge values of 176 MHz,
190 MHz, 279 MHz and 327 MHz, which were mixed with a 6.8 GHz VCO in order to
address the 87Rb ground state hyperfine splitting. The precise values of the frequencies
were optimised by a genetic algorithm [173] programmed to maximise the phase space
density of the cloud loaded in the crossed-dipole trap, given by eq. 1.46.
112 Reaching quantum degeneracy
H]
Figure 5.14 : A schematic illustration of the laser beams illuminating the atoms in
the science cell. All radiation is collected in one of four axes X, Y, D (diagonal), and
T (transport). The dashed rectangle next to the knee is the fluorescence imaging setup
illustrated in fig. 6.1.
5.3. Evaporative cooling 113
Figure 5.15 : The microwave radiation ωMW results in atoms transitioning to the un-
trapped |F = 2,mF = −2〉 state and hence escape from the trap. The frequency is
increased in order to address less energetic atoms situated nearer the magnetic trap min-
imum.
This evaporation stage results in ∼ 3× 107 87Rb atoms at ∼ 10 µK and in ∼ 2× 106 39K
atoms at ∼ 7 µK. These temperatures were found to be sufficiently low to efficiently load
the crossed-beam dipole trap, which enables the use of a homogeneous magnetic field to
induce repulsive interactions in 39K, whose negative background scattering length would
otherwise lead to a collapse preventing the onset of a stable Bose-Einstein condensate.
Further cooling is hindered by the Majorana losses discussed in section 2.2.1, which be-
come more severe as the cloud increases its density near the centre of the trap. An optical
plug was originally superimposed with the magnetic trap minimum, which allowed the
evaporation to be performed for longer, reaching 2− 3 µK (quantitatively investigated in
[94]). However, the ∼ 10 µK clouds were found to already be efficiently loaded in the
dipole trap. Furthermore, the plug had a ∼ 20 µm single beam waist and was hence very
sensitive to misalignment. Its elimination from the apparatus thus caused more joy than
sorrow.
5.3.3 Evaporation in an optical trap
State transfer
The transfer from the magnetic trap to the crossed-dipole optical trap is illustrated in fig.
5.16. After the optical trap is ramped to its highest power, 7 W, the quadrupole field is
ramped to zero. A guide field of ∼ 2.8 G is produced by the z offset coils of the science
to provide a quantisation axis for the atoms not to lose their spin-polarisation.
A radio frequency (RF) wave generated by an antenna (whose design is detailed in [94])
above the science cell is shone onto the atoms for 20 ms, its frequency ramped between
1.95 MHz± 20 kHz in order to perform a Landau-Zener transition from the |mF = −1〉 to
the |mF = 1〉 state in both 87Rb and 39K, as explained in section 2.1.2. This state transfer
114 Reaching quantum degeneracy
allows the atomic cloud to now be in the absolute ground state of the energy manifold,
where the reduced loss channels guarantee a longer lifetime.
Figure 5.16 : The atoms are transferred from a magnetic to an optical dipole trap. A
guide field provides a quantisation axis in order for an RF wave (light red solid rectangle)
to perform a Landau-Zener transition from |mF = −1〉 to |mF = 1〉, the absolute ground
state. At the start of the second evaporative cooling stage, the experiment coils produce
a homogeneous magnetic field to access the Feshbach resonance.
axis wh /µm wv /µm
X 284 64
Y 280 47
Table 5.10 : Horizontal (h) and vertical (v) waists of the beams generating the dipole
trap.
Magnetic field calibration
The efficiency of the state transfer is also used to calibrate the magnetic field strength at
the atomic position. A Stern-Gerlach coil provides a magnetic field gradient in the vertical
(z) direction during the first 3 ms of time-of-flight. The interaction with the inhomogen-
eous field results in an mF dependent force which allows recording the population of the
individual mF sub-levels when imaging from an orthogonal direction (D), as shown in fig.
5.17. It is important for the Stern-Gerlach field to be in the same direction as the bias
field of the science cell in that direction: if not, at some time the two field will cancel,
resulting in no guide field for the atoms, which leads to a complete admixture of all mF
sub-levels.
The centre frequency of the RF sweep is varied until a complete state transfer is achieved.
The frequency at which this occurs provides a measurement for the magnetic field, using
the plots in section 4.3.1. The width of the sweep is reduced until it has no effect on the
measured linewidth of fig. 5.18a. This was found to be ∼ 20 kHz, and is due the ambient
magnetic field noise.
The magnetic field calibration is then used to locate the Feshbach resonance, as shown in
fig. 5.18b. Atoms are held in the dipole trap in the presence of a homogeneous magnetic
field which, as discussed in section 1.3.1, affects their scattering length a and increases
5.3. Evaporative cooling 115
Figure 5.17 : Imaging the cloud after time-of-flight yields no information on its mF
states (left), as they fall equally under gravity. A Stern-Gerlach coil produces a magnetic
field gradient and hence an mF dependent force. Without state transfer, the cloud is
in |mF = −1〉 (middle), while a successful state transfer results in the cloud being in
|mF = 1〉 (right). The spurious |mF = 0〉 are probably created by the Stern-Gerlach
pulse.
the three-body recombination losses (∝ a4). The Feshbach resonance Bres of eq. 1.57 is
then identified as the magnetic field resulting in the greatest loss.
Evaporation
The evaporative cooling stage in the crossed-beam dipole trap occurs exactly as described
in section 2.3.3. The intensity of the red detuned beams is decreased in order to reduce
the trap depth, inducing particle loss and thermalisation of the remaining atoms to a
lower temperature.
However, the dipole potential Udip is not the only interaction affecting the atoms. A
gravitational gradient mgz is also present and shifts the location of the trap minimum z0:
z0 ⇔ ddz (Udip +mgz) = 0, (5.1)
as shown in fig. 5.19. This results in a minimum value of the laser intensity below which
atoms cannot be trapped, referred to as “dipole bottom”, found to be ∼ 200 mW for 39K
and ∼ 400 mW for 87Rb. Because the gravitational sag depends on the atomic mass, it
causes a spatial separation between the two species, reducing the efficiency of sympathetic
cooling. Eventually, 87Rb is completely lost from the trap, leaving a pure 39K cloud.
In order to enhance the thermalisation efficiency, a homogeneous magnetic field is used to
tune the interaction strength of the atoms. As summarised in table 5.11, this is first tuned
near the inter-species 87Rb-39K Feshbach resonance, and, when 87Rb atoms are lost, near
the intra-species resonance in 39K to ensure a repulsive interaction and prevent collapse.
The trap depth is reduced until the onset of the Bose-Einstein condensate, as shown in
fig. 5.20.
The trapping frequencies of the crossed-dipole are measured by ramping the optical power
P to P+ and then instantaneously decreasing it to P . Because of the imperfect alignment
of the beams, this is sufficient to provide the atoms with a transverse velocity, resulting
in a simple harmonic motion in the trap. The trapping frequencies are shown in fig. 5.21,
and they exhibit the square root behaviour expected from eq. 2.34, but deviate from it
because of the presence of gravity.
116 Reaching quantum degeneracy
(a) (b)
Figure 5.18 : a) Magnetic field calibration via RF spectroscopy. The frequency of an
RF signal is varied until resonant with the Zeeman shift, enabling maximal state transfer
and hence providing a measurement of the field. b) The atoms are held in the dipole trap
at different homogeneous magnetic field strengths. Three-body recombinations increase
with scattering length (∝ a4) and hence cause maximal loss at the Feshbach resonance,
shown here for the inter-species interaction 87Rb-39K.
duration magnetic field scattering length
B a/a0
inter-species 87Rb-39K 1.7 s 315.5 G 141
intra-species 39K-39K 1.3 s 397.8 G 280
Table 5.11 : During the evaporation in the dipole trap, a homogeneous magnetic field
is used to access the Feshbach resonances for the |F = 1,mF = −1〉 scattering channels
given in table 1.3.
Figure 5.19 : Gravity introduces a gradient in the crossed-dipole trap potential and
shifts its minimum. This establishes a minimum of the intensity (“dipole bottom”) for
which the potential is confining, below which atoms are lost.
5.3. Evaporative cooling 117
(a) T > Tc (b) T < Tc (c) T Tc
Figure 5.20 : The atomic cloud measured in the direction perpendicular to the crossed-
beam dipole trap, at three different trap depths and hence temperatures. The decrease of
the temperature T below the critical value Tc triggers the macroscopic occupation of the
ground state associated with Bose-Einstein condensation. The thermal and condensed
fractions (grey solid lines) are individually fitted by the functions listed in section 6.2.
5.3.4 Quantum degeneracy
As explained in section 2.3.1, the removal of energy from the atomic cloud and the concom-
itant decrease in temperature are not sufficient to reach quantum degeneracy. The spatial
extent of the wavefunction also needs to increase, in order for the de Broglie wavelength
λth to become comparable to the inter-particle separation and ensure a dominant wave-
nature of matter. This requires an increase in the phase space density, quantified by the
degeneracy parameter D = n0λ3th, where n0 is the central spatial density.
A summary of the temperatures and atom numbers is provided in table 5.12, and the
associated increase in phase space density is shown in fig. 5.22.
118 Reaching quantum degeneracy
(a) Vertical
(b) Horizontal
Figure 5.21 : The vertical (a) and horizontal (b) trapping frequencies of the crossed-
beam dipole trap. The dashed lines are the expectations from eq. 2.34, whereas the solid
lines are the numerical solution accounting for the presence of gravity, with error bars
from the uncertainty on the waists reported in table 5.10. Figure taken from [94].
5.3. Evaporative cooling 119
stage Atom number87Rb
Atom number
39K
Temperature
87Rb
Temperature
39K
1 Backgroundvapour pressure 295 K 333 K
2 MOT 3× 109 5× 108 580 µK 1.1 mK
3 Polarisationgradient 30 µK 470 µK
4 Microwaveevaporation 3× 10
7 2× 106 10 µK 7 µK
5 Mid-dipoleevaporation 2× 10
7 1.2× 106 2 µK 2.6 µK
6 BEC ∼ 3× 105 ∼ 50 nK
Table 5.12 : Summary of the atoms numbers and temperatures after each cooling stage.
These not only result in the atomic cloud to be colder, but also to exhibit a macroscopic
de Broglie wavelength, quantified by the degeneracy parameter D ∼ 1 shown in fig. 5.22.
Figure 5.22 : The cooling stages of table 5.12 ultimately lead to a Bose-Einstein con-
densate of 39K (blue) when the degeneracy parameter D = n0λ3th is of the order ∼ 1.
87Rb, shown in red, is completely lost during the optical evaporation stage.
On the walls of the cave, only the shadows are the truth.
Plato, from “The Allegory of the Cave” [176]
6
Probing and characterisation
This chapter details the methods employed to image and characterise the
atomic cloud, and derives the functions used to extract the atom number and
the temperature.
6.1 Imaging
6.1.1 Fluorescence
In fluorescence imaging, the atomic cloud is illuminated with near resonant light which is
then scattered according to the second term in eq. 2.21. The number of photons collected
by a camera is related to the total scattering rate Rscat and hence to the atom number
Natoms. In our experiment, this method was used to estimate the number of atoms in the
3D MOT chamber, as shown in fig. 6.1.
Figure 6.1 : The optical apparatus to perform fluorescence imaging comprised both a
camera and two photodiodes, whose trace is shown in fig. 5.6. These provided an estimate
for the atom number loaded into the 3D MOT.
121
122 Probing and characterisation
The atom number Natoms is calculated as:
Natoms =
Nγ
ητ
∑
jWjRj
, (6.1)
and is derived in [177]. Nγ is the number of photons collected by a camera during an
exposure time τ and with a detection efficiency η, and R is the scattering rate of every
transition j driven by the incoming radiation, each weighted by the relevant Clebsch-
Gordan coefficients Wj. Accounting for multiple simultaneous transitions is especially
important for 39K whose F ′ states are not well resolved.
Scattered light was also collected by two photodiodes, each with an absorptive filter so as
to only record one species, providing a real-time diagnostic of the 3D MOT loading, as
shown in fig. 5.6. The number of atoms Natoms was extracted from the phenomenological
equation describing MOT loading [178]:
dNatoms
dt = L−
Natoms
ξ
− βN
2
atoms
V
, (6.2)
where L is the loading rate, ξ the lifetime of the atoms due to background gas collisions,
V the volume of the cloud and β the two-body loss parameter for light assisted collisions,
discussed in section 1.3.2. An analytical solution is allowed when assuming low densities
and hence eliminating the term with β, resulting in Natoms(t) = N0
(
1− e−t/τ
)
which was
used to fit the fluorescence traces of fig. 5.6. The use of this equation is also justified for
β 6= 0, since the functional form of the solution is qualitatively similar, as shown in fig.
6.2.
Figure 6.2 : Analytical (β = 0) and numerical (β 6= 0) solutions of eq. 6.2 describing
the number of atoms Natoms loaded in a 3D MOT. The two functions being qualitatively
similar, the use of the analytical solution also for β 6= 0 is justified.
This method only provided a lower estimate for the atom number in the MOT, limited
by re-scattering events when high densities are reached: an atom may scatter a photon
which, before being collected by the camera, may be absorbed and re-scattered by other
atoms.
6.1.2 Absorption
Formulae
The method employed to image the spatial distribution of the atoms in the science cell
was absorption imaging, which measures the transmission of a beam through the cloud.
6.1. Imaging 123
A laser beam with intensity I propagating along z through an atomic cloud of density n
is attenuated according to the Beer-Lambert law:
dI
dz = −σnI. (6.3)
σ is the absorption cross section defined from:
σI = h¯ωRscatt =
3λ2
2pi
1
1 + I
IS
+
(
2∆
Γ
)2 , (6.4)
where Rscatt is the scattering rate from eq. 2.21, and ω, λ and ∆ the frequency, wavelength
and detuning of the incident radiation, Γ the atomic natural linewidth and IS the satur-
ation intensity. For 87Rb and 39K, IS ∼ 2 mW/cm2.
Imaging was performed on resonance to maximise the signal, and with low intensities
I IS so as not to saturate the transition, which would cause an underestimation of the
atom number. For simplicity, in eq. 6.4 we assumed I = 0, since careful calibration of IS
was not performed, but is known to only result in a correction by a factor of 1.5−2 [179].
A beam with initial intensity I0(x, y) is incident on the camera with an attenuated in-
tensity I(x, y) given by:
I(x, y) = I0(x, y) exp
(
−σ
∫
n(x, y, z) dz
)
= I0(x, y) e−OD(x,y). (6.5)
The optical density OD is defined as:
OD(x, y) = σ
∫
n(x, y, z) dz︸ ︷︷ ︸
column density n(x,y)
= − ln
(
I(x, y)− Idark(x, y)
I0(x, y)− Idark(x, y)
)
, (6.6)
where Idark is the background intensity illuminating the camera even in the absence of
imaging light and/or atoms.
Three images are taken in a typical experimental run: an absorption image in the presence
of atoms to record I(x, y), a reference image in the absence of atoms to record I0(x, y),
and a dark image to measure Idark(x, y). These images are taken with the least possible
temporal separations in order to avoid intensity fluctuations of the light due to ambient
thermal drifts.
The number of atoms N is then calculated as:
N =
∫
n(x, y, z) d3r = 1
σ
∫
OD(x, y) d2r. (6.7)
For 87Rb and 39K, the imaging transition was chosen to be the cycling transition |F =
2,mF = 2〉 → |F ′ = 3,mF ′ = 3〉. The correct polarisation, σ+, needs to be ensured
in order for atoms not to accumulate in dark states and avoid detection. For atomic
clouds in the absolute ground state |F = 1,mF = 1〉, σ+ repump light was also shone in
order to induce the transition |F = 1,mF = 1〉 → |F ′ = 2,mF ′ = 2〉 from which atoms
spontaneously decay into |F = 2,mF = 2〉 and couple to the cycling transition. This
repump beam is referred to as “repump flash”.
124 Probing and characterisation
Camera
The choice of a suitable camera is motivated primarily by its cost and by our requirements.
In an ideal camera, each photon scattered by an atom excites an electron in the pixel,
which is converted into a signal and hence recorded. In reality, this is limited by quantum
efficiency (the number of electrons excited by incident photons), dark counts (electrons
excited in the absence of incident photons) and electrical read-out noise. Cooling the
chip reduces the extent of the latter two effects, and buying an expensive camera may
take care of the former. While low quantum efficiencies are detrimental to single-photon
measurements, they can easily be coped with in usual cold atom experiments.
We use a scientific CMOS (s-CMOS) camera, which has a lower quantum efficiency than
its CCD counterpart but a much faster read-out because the electrical charge on each
pixel is collected individually and at the same time. This allows to take the three images
(absorption, reference and dark) at ∼ 50 ms from each other.
Time-of-flight
Images are either taken in-situ, with the atoms sitting in the magnetic/optical traps, on
in time-of-flight.
This technique involves releasing the atomic cloud from any trapping potentials, and
allowing it to fall under gravity for a time ttof . To motion of each atom is then classically
described by:
r(ttof) = r0 + v0ttof − 12gt
2
tof . (6.8)
The initial velocity v0 is related to the initial momentum p0. Hence, a time-of-flight image
allows measuring the initial momentum distribution of the atomic cloud, as the different
velocities cause the atoms to spatially separate before being recorded.
For thermal clouds, the initial momentum distribution is determined by the temperat-
ure T , hence measurement of the width of cloud σ also provides a measurement of its
temperature:
σ ∝
√
〈p20〉t2tof
m2
, (6.9)
where m is the mass of the atom.
An example of time-of-flight imaging is shown in fig. 6.5.
Magnification
The optical apparatus between the atoms and the camera may include additional lenses to
increase (or decrease) the magnification of the image, as shown in fig. 6.3. This introduces
a factor M between the imaged and the real spatial distributions:
Image = M × Real, (6.10)
which, theoretically, is equal to the ratio of the eyepiece to the objective focal lengths
f2/f1.
For imaging along axes perpendicular to gravity (D,X,Y and T in fig. 5.14), M is easily
extracted from time-of-flight imaging of the central position of the cloud:
rimage(ttof) = M ·
(
r0 + v0ttof − 12gt
2
tof
)
. (6.11)
6.1. Imaging 125
For axes along gravity (Z), the magnification was calibrated by measuring distances of
known numerical value. These included cloud sizes previously measured by other, ortho-
gonal imaging axes, or the 2h¯k separation of the Kapitza-Dirac diffraction peaks of fig.
7.9. The imaging axes parameters for our experiment are summarised in table 6.1.
Figure 6.3 : The last lens f0 produces a collimated imaging beam that is shone onto the
atoms (green blob). The imaging beam is then made to propagate through a 4f setup
until incident onto the camera. The actual image x (green solid line) starts focussed at
the atomic position, and is focussed again onto the camera, albeit reduced/increased by
M , the magnification of the imaging system. Ideally, M = f2/f1.
Axis Angle from ↓ z measured M expected
M = f2/f1
pixel side
/ µm Camera type
X 0.7◦ 0.802 0.8 4.5 BeamCam
Y −0.5◦ 0.816 0.8 4.5 BeamCam
D 1.3◦ 3.07 3 5.86 Manta
T −1.12◦ 2.45 2.5 4.5 BeamCam
Z N/A 0.996 1 6.5 Andor Zyla
Table 6.1 : A list of the parameters pertaining to our imaging systems.
The lenses f0 and f1 in fig. 6.3 are also used by the dipole trap and optical lattice beams
(fig. 5.14) in order to be focussed onto the atomic position. The atoms, therefore, need
to be exactly at a distance f0 from the first lens, and at a distance f1 from the second.
The procedure to select the optimal position of the lenses is detailed as follows:
1. With the atoms in the magnetic trap, the position of the lens f1 is varied so as to
result in the smallest cloud size recorded on the camera. This corresponds to the f1
lens being exactly a distance f1 from the magnetic trap minimum;
2. Once the atoms are loaded into the optical crossed-beam dipole trap, the position of
the lens f0 is varied so as to result in the smallest cloud size with maximum signal
recorded on the camera. This corresponds to the dipole beam being focussed onto
the atoms and coinciding with the magnetic trap position.
Once the positions of the lenses are fixed, any other beam which is collimated when
incident on f0 is automatically focussed onto the atoms.
126 Probing and characterisation
High field imaging
For 39K, a repulsive interaction strength is needed in order to prevent its collapse when
reaching quantum degeneracy. This is guaranteed by a strong homogeneous magnetic
field, enabling access to its Feshbach resonance. The imaging of the cloud, therefore, also
needs to occur at high magnetic fields, for ramping them down to zero would cause the
BEC to implode into a bosenova.
As shown in fig. 6.4, in the presence of a (strong) magnetic field, the polarisation of the
imaging light is of crucial importance.
Figure 6.4 : ∆m ± 1 transitions are driven by σ polarised light. If incident along the
magnetic field (longitudinal), this is circularly polarised, but if incident perpendicular to
the field (transverse), it is linearly polarised. pi transitions causing ∆m = 0 are only
feasible when light is illuminated from the side.
For reliable imaging of the atoms, we seek a closed transition in the high-field region
where the atomic states are shown in figs 4.9 and 4.11. A suitable choice is the |mJ =
−1/2,mI = 3/2〉 → |mJ ′ = −3/2,mI′ = 3/2〉 transition driven by σ− light, with a
selection rule ∆mJ = −1 since the nuclear spin I can only be affected by magnetic
components of the radiation, which are ∝ 1/c reduced as per eq. 2.7. The ground state
|mJ = −1/2,mI = 3/2〉 is adiabatically connected to the low-field state |F = 1,mF = 1〉.
However, conservation of mF = mJ+mI is also obeyed by the low-field state |F = 2,mF =
1〉, adiabatically connected to the high-field state |mJ = 1/2,mI = 1/2〉.
6.2. Fit functions 127
The ground state in the high-field regime is therefore:
|ψ〉S1/2 =
√
1− 2 |mJ = −1/2,mI = 3/2〉+ |mJ = 1/2,mI = 1/2〉, (6.12)
with → 0 as the magnetic field strength |B|→ ∞.
This state is thus not pure and it prevents a fully closed transition, which undermines the
accuracy of the recorded atom number.
The 39K cloud also needed to be imaged from the side in order to measure its central
position to extract the dipole trapping frequencies shown in fig. 5.21. This was achieved by
illuminating the atoms with a linear (pi) polarised imaging beam parallel to the magnetic
field, with a frequency tuned to couple the |mJ = −1/2,mI = 3/2〉 → |mJ ′ = −1/2,mI′ =
1/2〉 transition, adiabatically connected to |F = 1,mF = 1〉 → |F ′ = 1,mF ′ = 0〉.
6.2 Fit functions
6.2.1 Atom numbers
Condensed fraction
The spatial density for a BEC was derived in section 1.2.2 and found to be the inverted
paraboloid |ψ|2 of eq. 1.50.
An absorption image along the z axis records the integrated column density n(x, y):
n(x, y) =
∫ ∞
∞
|ψ|2dz =
∫ ∞
∞
n0 ·max
[
0, 1− (x− x0)
2
R2x
− (y − y0)
2
R2y
− (z − z0)
2
R2z
]
dz =
⇒ n(x, y) = 43Rz︸ ︷︷ ︸
A0
·max
[
1− (x− x0)
2
R2x
− (y − y0)
2
R2y
]3/2
.
,
(6.13)
where Ri coincide with the Thomas-Fermi radii only when the cloud is imaged in-situ.
The number of atoms is therefore given by:
N = 1
σ
∫
n(x, y)d2r = 1
σ
∫
|ψ|2d3r =
1
σ
∫ ∞
∞
n0 max
0,(1− (x− x0)2
R2x
− (y − y0)
2
R2y
)3/2 dx dy =
⇒ N = A0
σ
· 2pi5 RxRy.
(6.14)
The BEC is a coherent state and hence not an eigenstate of the particle number operator.
The image measurement projects it onto a number-specific Fock state. Hence we expect
the atom number N to exhibit Poissonian fluctuations ∝
√
N which are however much
smaller than experimental uncertainties from other sources. Furthermore, the inverted
paraboloid of eq.1.50 assumes a sharp discontinuity of the cloud density profile at the
radii Ri. In reality, this would be smoothed by the healing length, ignored in the above
treatment.
128 Probing and characterisation
Deeply thermal fraction
For a fully thermal cloud T Tc, the momentum is Boltzmann distributed according to
eq. 1.10. The spatial density is then given by eq. 1.16 which, for a harmonic trapping
potential, is just a Gaussian with standard deviations σi. The number of atoms is then
given by:
Nex =
1
σ
∫ ∞
∞
dx
∫ ∞
∞
dy B0 exp
(
−(x− x0)
2
2σ2x
− (y − y0)
2
2σ2y
)
= B0
σ
· 2piσxσy. (6.15)
Partially condensed fraction
For a partially condensed cloud T . Tc, the Bose-Einstein distribution of the excited states
cannot be approximated by its classical, Boltzmann counterpart. The spatial density of
the thermal fraction is then given by eq. 1.39. Because the potential V (r) is harmonic,
the argument of the polylogarithmic function g is a Gaussian with standard deviations
σi.
The integrated density is then given by:
nex(x, y) =
∫ ∞
∞
|ψ|2dzB0 g3/2
[
exp
(
−(x− x0)
2
2σ2x
− (y − y0)
2
2σ2y
− (z − z0)
2
2σ2z
)]
=
⇒ nex(x, y) = B0 ·
√
2pi σz︸ ︷︷ ︸
B1
g2
[
exp
(
−(x− x0)
2
2σ2x
− (y − y0)
2
2σ2y
)]
.
(6.16)
The number of atoms is therefore:
Nex =
1
σ
∫ ∞
∞
dx
∫ ∞
∞
dy B1 g2
[
exp
(
−(x− x0)
2
2σ2x
− (y − y0)
2
2σ2y
)]
= B1
σ
·2piσxσy ζ(3), (6.17)
where ζ(3) = 1.202.
———-
After time-of-flight, the same fit functions are used but with a new cloud radius R(ttof) =
b(ttof)Ri, as per [180].
6.2.2 Temperature
The temperature of the atomic cloud is extracted from the time-of-flight expansion of its
size σ, which depends on the initial momentum distribution and hence on the temperature
T according to 6.9. The cloud size after time-of-flight is the results of a convolution
between the momentum p(t) and the initial spatial distribution r0.
For simplicity, the calculation is performed within the semi-classical approximation that
allows expression of the thermal atoms’ spatial distribution in a 3D harmonic trap with
frequencies ωi as a simple Gaussian, from eq. 1.16.
The initial phase space density of the cloud is given by:
ρ(r0,p) = N · P (r0,p) ∝ exp
(
−H0(r0,p)
kBT
)
∝ ∏
i=x,y,z
exp
(
− p
2
i
2mkBT
)
· exp
(
−mω
2
i r
2
0i
2kBT
)
,
(6.18)
6.2. Fit functions 129
where the overall proportionality factor was dropped.
The spatial density at time t is then given by the convolution:
nex(r, t) =
∫ ∫
d3r0 d3p ρ(r0,p) δ
(
r− r0 − pt
m
)
∝ ∏
i=x,y,z
∫ ∞
−∞
dpi exp
[
− p
2
i
2mkBT
]
exp
[
− 12kBT mω
2
i
(
xi − pit
m
)2]
,
(6.19)
which thanks to the symbolic integration performed by Mathematica can be expressed
as:
nex(r, t) ∝
∏
i=x,y,z
√
2pi√
1 + ω
2
i t
2
mkBT
exp
[
− x
2
i
2σi(t)2
]
. (6.20)
In the above,
σi(t)2 = σ2i0 +
kBT
m
t2tof , with σ2i0 =
kBT
mω2i
. (6.21)
Hence, the temperature T is related to the standard deviation σi of the Gaussian:
T = mσ
2
i
kB
ω2i
1 + ω2i t2tof
∼ mσ
2
i
kBt2tof
, (6.22)
which agrees with an estimate from the theorem of equipartition of energy:
1
2mv
2
i =
1
2kBT ====⇒vi=σi/t T =
mσ2i
kBt2tof
. (6.23)
The thermal atoms expand isotropically while the condensed fraction undergoes its char-
acteristic aspect-ratio inversion (driven by the greater interaction energy along the com-
pressed direction), as shown in fig. 6.5.
130
Probing
and
characterisation
Figure 6.5 : Time-of-flight images of a 87Rb BEC along the D axis, with ttof ranging from 2 ms to 15 ms. The thermal fraction expands
isotropically and can be used to extract the cloud’s temperature. The condensed fraction starts with an asymmetric profile dictated by the
dipole trap frequencies, and evolves into a profile with the inverted aspect ratio. The central position of the cloud was used in eq. 6.11 in
order to extract the magnification M for this imaging axis.
Part III
Results and findings
Order is for idiots, genius can handle chaos.
Albert Einstein
131
Ordeeeer! Ooordeeer! ORDER!
John Bercow, Speaker of the House of Commons, 2009-present.
7
Order: sharp diffraction peaks
This chapter provides a technical description of the eightfold optical lattice
apparatus, along with numerical considerations on the real space profile of the
potential and its energy spectrum. Analogously to X-ray crystallography, we
used matter-wave diffraction to investigate the 2D quasiperiodic optical lat-
tice, which yielded sharp peaks and hence confirmed the long-range order of
the structure. However, the seemingly pure point diffraction spectrum tends
to form a dense set, uncovering a singularly continuous component which pos-
sesses a fractal dimension.
7.1 Lattice
7.1.1 Implementation
Our experiment had the objective to investigate ultracold atoms in a quasicrystalline
potential, that is a lattice structure with a crystallographically forbidden rotational sym-
metry. The obvious choice would have then been a fivefold symmetric pattern, reminis-
cent of the Penrose tiling in fig. 3.6c. A minimalist implementation would have seen a
Penrose-like pattern projected onto the atoms with a DMD1, which would unfortunately
have required the use of a quantum gas microscope to achieve the required resolution, and
would suffer from imperfections in the potential landscape due to the presence of several
spatial frequencies.
An optical lattice was then a cleaner alternative, with exactly as many spatial modes as
laser beams. A set of five free running and mutually interfering beams would give rise to a
fivefold symmetric structure, but would require extremely accurate stabilisation of most,
if not all, of their phases in order to preserve the complex features of the interference
pattern. This technical limitation may be easily overcome by retro-reflecting the beams
and only allowing interference between each incident and counter-propagating pair. This
1Digital Mirror Device.
133
134 Order: sharp diffraction peaks
selective interference may be enabled by introducing a detuning between each pair of
beams such that its associated frequency scale is much larger than what the atoms could
respond to, and would result in an incoherent superposition of five distinct standing waves,
equivalent to ten pair-wise interfering lattice beams2.
Luckily my then colleague (and now Dr.) Konrad Viebahn realised that there is another
number, less than ten, that produces a crystallographically disallowed rotational sym-
metry: eight. We therefore opted for the eightfold symmetric pattern created by the
incoherent superposition of four standing waves shown in fig. 7.1.
Figure 7.1 : Our eightfold symmetric optical lattice is created by incoherently superim-
posing four stading waves, each separated by 45◦. Figure taken from [181].
The optical potential is described by:
V (r) = V0
4∑
i=1
cos2(ki · r + φi), (7.1)
where ki are the wavevectors of the lattice beams:
k1 = (1, 0), k2 = (1/
√
2 , 1/
√
2 ), k3 = (0, 1), k4 = (−1/
√
2 , 1/
√
2 ). (7.2)
The above expression assumes each beam is an ideal plane wave, though real laser beams
possess a Gaussian intensity profile, as further discussed in section 9.2. The four in-
dependent phases φi, each referenced to the position of the retro-reflecting mirror, are
passively stabilised by the mass of the mirror itself, and will be taken as 0 for the rest of
the thesis.
The laser providing the lattice radiation can be tuned over a large range of frequencies,
allowing the possibility of both a blue and a red detuned lattice (with respect to the D
lines of 87Rb and 39K). Blue detuning with a wavelength of λ = 725 nm was chosen, for
the following reasons:
• As further discussed in section 9.2, this option results in an overall additional an-
ticonfining potential, which is already mitigated by the red detuned dipole trap
beams. A red detuned lattice, on the other hand, would result in an overall confin-
ing potential;
2Three cross-interfering beams result in a global, superfluous phase factor and in two relative phases
whose noise only causes translations of the lattice’s centre of mass. The same happens for a square lattice
created by two standing waves. However, for three standing waves or four mutually interfering beams,
the number of phases exceeds the spanning set of the 2D plane. Phase noise will then not only translate
the lattice but also affect its shape and geometry.
7.1. Lattice 135
• Situated at the minima of the intensity, the atoms experience a reduced scattering
rate;
• Numerical investigations on both (red and blue detuned) lattice configurations
showed equally interesting and promising features.
7.1.2 Real space
Profile
The real space profile of the four-standing waves’ interference pattern is shown in fig. 7.2,
where it is compared to a scanning tunnelling microscope (STM) picture of an eightfold
symmetric solid-state quasicrystal.
Setting the phases φi in eq. 7.1 to zero results in a global eightfold point symmetry
about the centre of the lattice, as shown in fig. 7.3. A random choice of phases, on the
other hand, results in a pattern with only locally eightfold symmetric regions, as shown
in fig. 7.4. Ammann-Beenker tilings may be superimposed to the minima locations in
both configurations. Further details on the relation between the lattice potential and the
choice of phases are discussed in [94].
(a) in phase (b) random phases (c) STM, from [182]
Figure 7.2 : Four in-phase standing waves result in the centrally eightfold symmetric
interference pattern of (a). A random choice of phases (b) results in a distribution of
lattice minima locations (blue points) exhibiting the same local eightfold symmetry as
(c), a high-resolution image of the octagonal phase in Cr-Ni-Si obtained with a scanning
tunnelling microscope.
136 Order: sharp diffraction peaks
Figure 7.3 : Four in-phase standing waves result in a pattern exhibiting a global, central
eightfold symmetry. The minima of the potential are superimposed with an Ammann-
Beenker tiling. Figure taken from [94].
Figure 7.4 : Four standing waves with random relative phases result in a pattern which
exhibits eightfold symmetry only locally. The minima of the potential can still be super-
imposed with an Ammann-Beenker tiling. Figure taken from [94].
Lack of band structure
Investigation of the energy spectrum of the eightfold symmetric lattice is performed in
the deep lattice limit, discussed in section 2.4.2. Each potential minimum of the lattice
is approximated by a harmonic oscillator in order to compute the ground state energy:
E0 = Emin +
1
2 h¯(ωx + ωy), (7.3)
where Emin is the offset and ωi the on-site trapping frequencies, extracted according to
appendix D.10.
7.1. Lattice 137
Figure 7.5: The cumulative density of states f(E) defined in eq. 7.4 for different regions
of the same eightfold symmetric lattice, at single-beam lattice depth V0 = 40Erec. Each
curve is normalised to its maximum value. The energy spectrum is a singularly continuous
function, as per the definitions in section 3.2.3.
We define the cumulative density of states f(E) as:
f(E) =
∫ E
0
dE ′ g(E ′) =
∑
i
n(Ei), (7.4)
where g(E) is the density of states and n(E) the number of states at energy E.
The cumulative density of states f(E) is plotted in fig. 7.5 for different regions of the
same quasiperiodic potential. The resemblance to the Devil’s staircase of fig. 3.8 indicates
the lack of robust gaps and establishes the singularly continuous nature of the energy
spectrum, as per the definitions in section 3.2.3, in accordance with the results for other
quasiperiodic structures [145, 146, 183]. The shrinking of the gaps stems from the presence
of the irrational number
√
2 in the wavectors of eq. 7.2, allowing the minima of the
intensity to take any value in R.
7.1.3 Technical details
Alignment
In order for each beam to only interfere with its retro-reflected partner, their polarisation
was chosen to be out-of-plane and their frequencies offset by at least 10 MHz, larger than
any frequency the atoms could respond to. This frequency was also chosen so as not to
allow any Zeeman-assisted transitions at high fields as per the selection rules in table 2.1.
The waists of the beams are elliptical to increase the peak intensity and to guarantee
the lattice homogeneity over the atomic cloud. These are reported in table 7.1b along
with the angles between the beams, measured by fitting straight lines to the diffraction
patterns of 1D lattices, such as the one displayed in fig. 7.9. The maximum potential
depth achieved was 40Erec for each 1D lattice, for an optical power of ∼ 500 mW.
138 Order: sharp diffraction peaks
axis wh /µm wv /µm
D 158 73
Y 153 71
X 161 70
T 184 91
(a) waists
axis angle
TD 89.5◦
TX 45.6◦
TY 44.0◦
XY 91.6◦
XD 46.2◦
YD 45.4◦
(b) angles
Table 7.1 : The experimentally measured waists and angles of the lattice beams.
The procedure to align the lattice beams onto the atoms is detailed in [94], and sum-
marised as follows. A Mathematica simulation3 was used to model the beam propagation
on the experiment table until the last lens f0 before the science cell, which focusses it
onto the atoms’ trapped position. The pointing of each lattice beam was controlled by
a piezo-actuated mirror, and was optimised via the “lattice sliding method”: with the
retro-beam blocked, the anticonfining effect of the blue detuned travelling wave results in
a force shifting the atomic cloud sideways while at the same time broadening in, as shown
in fig. 7.7a. The beam was centred onto the atoms by maximising the cloud size whilst
causing minimal transverse displacement.
The retro-reflected beam was aligned by enhancing the ensuing matter-wave diffraction
pattern, with the logic that perfect retro-reflection would result in maximal peak visibility.
The retro-path included a lens and a mirror in the “cat’s eye” configuration, illustrated in
fig. 7.6, to ensure perfect beam-overlap even for Gaussian beams, and reduced sensitivity
to the inclination of the mirror. Alignment of the mirror had to be performed iteratively
with visual feedback from the diffraction patterns, for it to pass centrally through both
lenses thus imparting no net transverse momentum onto the atoms, as shown in fig. 7.7b.
Figure 7.6 : The retro-reflection of the lattice beam is performed within a “cat’s eye”
apparatus. A lens (f2) focusses the laser onto a mirror to maximise its overlap with the
counter-propagating beam and to reduce sensitivity on the mirror alignment.
Two-photon processes
The interaction between the lattice beams and the atoms is described by a general time-
dependent Hamiltonian H, resulting in transitions between an initial |i〉 and a final |f〉
atomic states. In the non-relativistic case, the transition rate Γi→f is determined by the
3using ABCD matrices in the paraxial approximation.
7.1. Lattice 139
(a) “Lattice sliding method” [94] (b) Retro-beam alignment
Figure 7.7 : The forward propagating beam is centred on the atoms by maximising its
anticonfining effects (a). Imperfect alignment of the retro-reflected beam results in an
unwanted transverse momentum imparted onto the atoms ∆k.
Fermi golden rule:
Γi→f =
2pi
h¯
|〈f |H|i〉|2ρ(Ef ), (7.5)
where ρ(Ef ) is the density of final states at energy Ef .
However, relativistic quantum mechanics provides corrections such that:
Γi→f ∝ |〈f |H|i〉+ higher order terms|2= |〈f |S|i〉|2, (7.6)
where S is the scattering matrix.
This is defined as S = limt→∞ U(t, t0), where U is the evolution or Dyson operator:
|f〉 = T exp
(
−
∫ t
t0
i
h¯
H(t′)dt′
)
|i〉 = T exp
(
−
∫ t
t0
i
h¯
HI(t′)dt′
)
︸ ︷︷ ︸
U(t,t0)
|i〉, (7.7)
with HI being the interaction component of H, as the kinetic energy is diagonal.
The transition rate is then given by Γi→f ∝ |limt→∞〈f |U(t, t0)|i〉|2 and is computed from
the Dyson series:
〈f |U(t, t0)|i〉 =
δfi − i
h¯
∫ t
t0
eiEf (t′−t0)/h¯〈f |HI|i〉e−iEi(t′−t0)/h¯dt′ +(−i
h¯
)2 ∫ t
t0
∫ t′
t0
∑
n
eiEf (t′−t0)/h¯〈f |HI|n〉 e−iEn(t′−t0)/h¯ eiEn(t′′−t0)/h¯〈n|HI|i〉 e−iEi(t′′−t0)/h¯dt′dt′′ . . . ,
(7.8)
140 Order: sharp diffraction peaks
which is derived in appendix D.7.
If HI is time independent, the integrals can be trivially computed in the limit t→∞. The
expansion can then be separated into the off-diagonal contributions to the Hamiltonian
|i〉 6= |f〉:
lim
t→∞〈f |U(t, t0)|i〉 =
+ 〈f |HI|i〉
Ei − Ef (i→ f direct scattering)
+
∑
n
〈f |HI|n〉〈n|HI|i〉
(En − Ef )(Ei − En) (i→ f scattering via staten)
+ . . . ,
(7.9)
and the diagonal terms |i〉 = |f〉:
lim
t→∞〈i|U(t, t0)|i〉 = 1−
i
h¯
〈i|HI|i〉t− i
h¯
∑
n6=i
〈i|HI|n〉〈n|HI|i〉
En − Ei t
2 . . . . (7.10)
We notice that the coefficient of the nth term in eq. 7.10 is the nth order perturbative
correction to HI. In the dipole approximation, the wavelength of the incident electric field
E(r)eˆ is much larger than the size of the atom, which then only experiences a spatially
homogeneous potential HI ∝ qr · eˆ. Hence the first order correction 〈i|HI|i〉 = 0 because
of the inversion symmetry in neutral atoms, due to the lack of a permanent electric dipole
moment. As a result, the leading term is the second-order perturbative correction. This is
the same conclusion drawn at the end of section 2.1.2, where the energy shift was termed
AC Stark shift. Here we confirm that this can be interpreted as a virtual two-photon
transition coupling |i〉 to itself via the excited states |n〉. For 87Rb and 39K, the D2 states
dominate the sum over |n〉, which leads to the expression found in eq. 2.30.
With this interpretation, we turn our attention to the transition terms appearing in eq.
7.9. The first order term is just a scattering event from |i〉 to |f〉, while the second
order term is again a two-photon transition coupling |i〉 and |f〉 via virtual excitations of
the states |n〉. The transition rate Γi→f also includes the density of states ρ(Ef ) which
enforces energy conservation. The frequency of the lattice beam ω being far detuned from
any atomic transition ωn, no direct scattering events are allowed. Only second or higher
(even) order terms in the perturbative expansion are permitted, and will be referred to
as two-photon processes in the remainder of the thesis.
The states |i〉 and |f〉 need to possess the same internal state to enforce energy conser-
vation, but may be different motional states owing to the momentum imparted by the
two-photon transition: |i〉 = |g〉 ⊗ |0h¯k〉 → |f〉 = |g〉 ⊗ |2h¯k〉, where |g〉 and |e〉 are the
ground and excited internal atomic states. In this context, two-photon processes may
be visualised by the physically incorrect picture of atoms absorbing one photon from the
forward beam and re-emitting it into the retro beam, depicted by the Feynman diagrams
of fig. 7.8.
In conclusion, the optical lattice beams result in two interactions for the atoms. Photons
propagating along the same direction mediate two-photon transitions coupling the same
ground state, a` la Raman, and cause an AC Stark shift which for a blue detuned lattice
results in an overall anticonfining potential, further discussed in section 9.2. Counter-
propagating photons can impart a net |2h¯k〉 momentum onto the atoms and therefore
7.1. Lattice 141
couple different motional states, a` la Bragg. This results in the Kapitza-Dirac scattering
discussed in the next section.
(c) (d)
Figure 7.8 : The transition terms in the perturbative Dyson expansion of eq. 7.9 can
be pictorially represented by Feynman diagrams. If the lattice beam were resonant, the
photon could be absorbed by an atom in the ground state |g〉 resulting in (a). For the far
detuned beams used in optical lattices, however, energy conservation causes all scattering
events to be mediated by “virtual states”4(b), ensuring the final internal state is still |g〉.
Kapitza-Dirac scattering
The two-photon processes described in the previous section result in the exchange of
momentum between light and atoms, which leads to the diffraction of matter-waves by
the optical lattice. This is the opposite of conventional X-ray crystallography, where
electromagnetic radiation is directed onto a solid-state sample and its diffraction pattern
recorded. The diffraction of matter from an optical lattice, a “crystal of light”, is known
as Kapitza-Dirac scattering. This was predicted in 1933 [184] and first realised with
electrons [185], and is quantitatively treated as follows.
A 1D optical lattice consists of two counter-propagating waves of amplitude E0 forming
a standing wave:
E = 2E0 cos(kz)eˆ. (7.11)
The single-photon Rabi frequency introduced in section 2.1.2 is then Ω ∝ E ∝ cos(kz).
For short and strong lattice pulses, the atom-light interaction energy is much larger than
the atomic kinetic energy, which can then be neglected. This is the Raman-Nath approx-
imation, where particles are assumed to be frozen in place and only acquire phase factors
from the lattice beams.
The relationship between the single-photon Rabi frequency Ω, the detuning ∆ and the
lattice depth in V0 is proved in [166] to be:
V0Erec =
h¯2Ω2
8∆ , (7.12)
where Erec is the recoil energy.
4In particle physics these would be termed “off-shell” and denoted by an asterisk.
142 Order: sharp diffraction peaks
Combining eq. 7.12 with the AC Stark shift of eq. 2.13 results in the following interaction
Hamiltonian:
HI(z, t) = −V0Erec2 cos(2kz). (7.13)
Time evolution leads to:
|ψt〉 = e− ih¯
∫
dt′HI(z,t′)|ψ0〉 ∝ ei
V0Erec
2h¯ cos(2kz)t|ψ0〉, (7.14)
where the constant of proportionality is only a pure phase factor.
Using eiα cos(β) = ∑∞n=−∞ inJn(α)einβ, where Jn is the nth Bessel function of the first kind:
|ψt〉 ∝
∞∑
n=−∞
inJn
(
V0Erec
2h¯ t
)
ei2nkz|ψ0〉
∝
∞∑
n=−∞
inJn
(
V0Erec
2h¯ t
)
|g〉 ⊗ |2nh¯k〉.
(7.15)
This confirms that upon interaction with the optical lattice, the atoms remain in the
ground state |g〉 but may gain a momentum 2nh¯k, n being the number of undergone
two-photon events and thus referred to as the diffraction order. The population of the
nth diffraction order is given by:
Pn = 〈ψn|ψn〉 = J2n
(
V0
2
t
τrec
)
, (7.16)
where τrec = h¯/Erec is the recoil time, ∼ 16 µs for 39K in a 725 nm optical lattice.
Spontaneous emission was ignored in this treatment, under the assumption that stim-
ulated emission in the counter-propagating beam would dominate owing to bosonic en-
hancement.
A typical Kapitza-Dirac diffraction pattern from a 1D optical lattice is shown in fig.
7.9. Upon interaction with the lattice, the atoms gain momenta in discrete units |2h¯k〉.
Imaging in time-of-flight along an axis perpendicular to the optical lattice allows the atoms
to spatially separate according to their speed. The resulting atomic spatial arrangement
provides a direct measurement of their momentum distribution. The n diffraction orders
are then said to be the basis states in momentum space, where they form a synthetic
lattice with spacing 2h¯k. All experiments presented in this thesis are performed on this
synthetic lattice.
Figure 7.9 : The two-photon transitions enabled by the optical lattice impart a 2h¯k
momentum to the atoms, which are then diffracted according to Kaptiza-Dirac scattering.
The atoms separate in n diffraction orders and form a synthetic lattice in momentum space.
7.1. Lattice 143
Calibration of lattice depths
The optical power in each lattice beam is set by the diffraction efficiency of an AOM,
controlled by a PID loop whose inverting and non-inverting inputs are connected to a
photodiode measuring the intensity close to the science cell, and to a signal generator
producing arbitrary pulse shapes. A signal generator was used as it ensures better DC
reproducibility, and allows simultaneous triggering of all four beams, avoiding a queue on
the analogue cards and therefore achieving sub-µs pulse synchronisation. The PID loops
were optimised for the specific pulse shape employed, and were made as stable as possible,
e.g. by minimising cable length and other sources of noise.
Various methods were employed to calibrate the voltage generated by the signal generator
to the lattice depth experienced by the atoms. Calibrations were performed for each
individual 1D lattice. An initial estimate for the required (optical) power was obtained
from eq. 2.30.
The first technique was lattice modulation spectroscopy.
Each 1D lattice was ramped to a depth V0, to allow the BEC to be adiabatically loaded
into its ground state, followed by an amplitude modulation about V0 of δV sin(ωmodt), to
drive inter-band excitations. Variation of the modulation frequency allowed observation of
the first→ second and first→ third band transitions5, which were quantified by measuring
the “temperature” of the central |0h¯k〉 peak in fig. 7.9. This method relied on the choice
of a suitable δV to cause an appreciable effect so as to be reliably captured by the fits,
and needed multiple depths to be sampled to allow extrapolation. Uncertainty in the
fit results was dominated by the asymmetry of the temperature profile, probably due to
quantum depletion and the presence of thermal atoms.
We then opted for taking a time-series of the Kapitza-Dirac scattering pattern (fig. 7.9),
and choosing the potential depth that best explained the populations of the peaks, as
shown in fig. 7.10. The Raman-Nath analytical solutions of eq. 7.16 were only reliable
at short pulse durations, hence the exact6 time evolution obtained from a dynamical
simulation (section 7.1.4) was used. This method allowed accurate depth calibrations to
within 0.1Erec, and were compatible with the modulation spectroscopy estimates.
Experimental sequence
The experimental sequence used to perform lattice experiments is summarised in fig. 7.11.
After creation of the BEC, the crossed dipole trap is ramped to a slightly higher power
in order to reduce the evaporation rate – usually from 190 mW, ωr = 2pi · 15 Hz and
ωz = 2pi · 75 Hz to 250 mW, ωr = 2pi · 18 Hz and ωz = 2pi · 115 Hz.
The magnetic field is ramped from the value used during evaporative cooling to a new one
determining the intra-species interaction strength during the lattice pulse – for most of
the results, this was chosen to be the zero-crossing in order to investigate single-particle
physics, with occasional exploration of the (repulsive) interaction regime.
The lattice is only shone when both the dipole trap and the magnetic field have reached
their desired values. Concomitant with the end of the lattice pulse, the dipole confinement
is turned off while the magnetic field is tuned back to its (repulsive) pre-lattice value in
5Parity conservation would only allow the latter [186] but dipole trap mixing also enables the former.
6though it still required to truncate the basis size and hence the Hamiltonian. The truncation was
chosen so as for the dynamics to affect only high and non-populated diffraction orders.
144 Order: sharp diffraction peaks
0
1
P
0
0
1
P
1
0 5 10 15 20 25 30
t/µs
0
1
P
2
Figure 7.10 : Depth calibration for a 15.7 Erec deep 1D optical lattice, such as the
one resulting in fig. 7.9. A rectangular lattice pulse illuminates the atoms for a variable
duration t. The numerical simulations for the time evolution (solid lines) and the Raman-
Nath analytical solutions (dashed) are fitted to the first three diffraction orders.
Figure 7.11 : Once a BEC is created, the lattice is shone for a time tlattice when both
the dipole trap and the magnetic field are tuned to the desired confining potential and
interaction strength. The atoms are then released and fall under gravity for ttof before
being imaged.
order to “puff up” the diffraction peaks, which would otherwise be optically too dense to
allow reliable extraction of the population7. The atoms are allowed to fall under gravity
for a time-of-flight ttof period of 33 ms, during which the diffraction orders caused by
two-photon processes separate in space to be recorded by a camera.
7.1.4 Simulation
As shown in fig. 7.9, two-photon transitions enable the optical lattice to impart 2h¯k
momentum to the atoms and results in a Kapitza-Dirac diffraction pattern. These points,
labelled by momentum, can be thought of forming a synthetic lattice where two-photon
transitions couple different synthetic states.
All experiments presented in this thesis are performed on the synthetic lattice generated
by Kaptiza-Dirac scattering. This synthetic lattice also underlies the analysis of the
7This technique was suggested to us by the Hadzibabic-Smith group in Cambridge (2017).
7.1. Lattice 145
vector 4D 3D 2D
|1〉 (1, 0, 0, 0) (1, 0, 0) (1, 0)
|2〉 (0, 1, 0, 0) (0, 1, 0) (cos pi/4, sin pi/4)
|3〉 (0, 0, 1, 0) (0, 0, 1) (0, 1)
|4〉 (0, 0, 0, 1) (− cos pi/4, sin pi/4)
Figure 7.12 : There are 4 basis states needed to span the eightfold symmetric reciprocal
lattice, corresponding to a higher dimensional lattice d = 4. These states can be written
in a 4D linearly independent basis, or in the 2D xy Cartesian representation. Any subset
d = 1, 2, 3 can be used to describe d-beam lattice configurations.
recorded images and the numerical simulations of the results.
Momentum space basis construction
As shown in fig. 7.9, a 1D optical lattice causes atoms to scatter into n diffraction
orders generating a synthetic lattice in momentum space, each labelled by |2nh¯k〉, n ∈ Z.
The optical lattice in our apparatus, shown schematically in fig. 7.1, consists of the
superposition of four independent standing waves.
A point in the full synthetic lattice can therefore be expressed as:
b = i|1〉+ j|2〉+ `|3〉+m|4〉 with (i, j, `,m) ∈ Z4, (7.17)
where the states |2nah¯ka〉 were renamed to na|a〉 to specify the ath 1D lattice causing na
two-photon transitions. {|a〉} is the basis set for the synthetic lattice, containing as many
vectors as active 1D lattices, as shown in fig. 7.12.
These can be expressed in 2D as per the wavectors in eq. 7.2. The presence of the
irrational number
√
2 is problematic for numerical simulations, as it prevents reliably
accounting of duplicates. Hence, the basis is defined over a 4D regular lattice, where
each vector is now linearly independent, as reported in fig. 7.12. This is also physically
justified by the higher dimensional method to produce quasicrystals described in section
3.2.4.
The generation n of a reciprocal vector b is the number of two-photon scattering events
it underwent from |0〉:
n =
d∑
j=1
abs[bj]. (7.18)
The number of new states appearing in a generation n (that were not in n − 1) is given
by:
N(d, n) =
d∑
k=0
2k
(
d
k
)(
n
k
)
, (7.19)
where d is the number of active lattice beams, determining the dimensionality of the
higher dimensional space. This formula is derived from the field of network theory, where
each generation of synthetic lattice states are given by the new vertices reached by 2d
edges starting from the original point.
146 Order: sharp diffraction peaks
1D Hamiltonian
The Hamiltonian for a particle in a 1D standing wave is:
H(x) = p
2
2m + V0 cos
2(kx), (7.20)
where
V0 cos2(kx) = V0
(
eikx + e−ikx
2
)2
= V02 +
V0
4
(
ei2kx + e−i2kx
)
. (7.21)
Because of the periodicity of the lattice, Bloch’s theorem can be used to write the wave-
functions of H as Bloch waves:
φn,q(x) = un,q(x) · eiqx, (7.22)
where q is the quasimomentum, conserved mod 2h¯k. The function u has the same
periodicity as the lattice, and can therefore be expanded as a discrete Fourier series in
the plane wave basis:
un,q(x) =
+∞∑
`=−∞
cn,`(q) ei2k`x. (7.23)
Inserting this into the Schro¨dinger equation Hφn,q = En(q)φn,q results in:
`=∞∑
`=∞
[(
− h¯
2
2m (2k`+ q)
2 + V02
)
ei2`k + V04
(
ei2(`+1)kx + ei2(`−1)kx
)]
cn,`(q) = En(q)
∞∑
`′=−∞
ei2`′kxcn,`′(q)
⇒
∞∑
`=−∞
H`,`′cn,`(q) = En(q)cn,`′(q).
(7.24)
Identifying ei2`kx = 〈x|2`k〉, |2`k〉 can be used as a suitable discrete basis to construct the
matrix Hamiltonian according to:
H`,`′ =
h¯2
2m(q + 2`k)
2 + V0/2 for ` = `′
V0/4 for ‖`− `′‖ = 1
0 otherwise.
(7.25)
Hamiltonian in arbitrary dimensions
We assume that the BEC, having free momentum k = 0, has a lattice quasimomentum
q = 0 as well. Different starting q 6= 0 can still be simulated and are discussed in section
8.2.5.
The kinetic energy is not just ∝ |k|2 like in eq. 7.25 as doing so would assume a physical
three- or four-dimensional space, whereas the matter-wave diffraction and the ensuing
synthetic lattice is restricted to a 2D plane. The kinetic energy associated with a d > 2
basis vector bi is then ∝ |P‖d ·bi|2, where P‖d is the projection operator defined in eq. 3.28.
These projection operators act on the augmented 3D/4D bases and return the physical
scattering vector in the 2D plane. From the bases listed in fig. 7.12, the 3D/4D projection
operators are required to take the following form:
P‖3 =
1 cos pi/4 0
0 sin pi/4 1
0 0 0
0 0 0
=
1
√
2/2 0
0
√
2/2 1
0 0 0
0 0 0
, (7.26)
7.2. Diffraction dynamics 147
P‖4 =
1 cos pi/4 − cos pi/4 0
0 sin pi/4 sin pi/4 1
0 0 0 0
0 0 0 0
=
1
√
2/2 −√2/2 0
0
√
2/2
√
2/2 1
0 0 0 0
0 0 0 0
. (7.27)
Eqs. 7.26 and 7.27 satisfy (P‖)2 = P‖ as expected for a projection operator.
The general d-dimensional Hamiltonian for d lattice beams is therefore:
Hdi,j(t) =
4Erec × |P‖d · bi|2+dV0(t)2 for i = j
V0(t)/4 for ‖i− j‖ = 1
0 otherwise,
(7.28)
The d V0/2 term added to the kinetic energy is a mere phase factor for time independent
Hamiltonians, but its diagonal and homogeneous nature ensures it also amounts to a
global phase factor for time dependent ones, as proven in appendix D.8.
An example for how the Hamiltonian looks like for a square 2D lattice is:
H2D(V0) =
〈0, 0| 〈1, 0| 〈0, 1| 〈1¯, 0| 〈0, 1¯|
0 + V0 V0/4 V0/4 V0/4 V0/4 |0, 0〉
V0/4 8 + V0 0 0 0 |1, 0〉
V0/4 0 8 + V0 0 0 |0, 1〉
V0/4 0 0 8 + V0 0 |1¯, 0〉
V0/4 0 0 0 8 + V0 |0, 1¯〉
which was truncated after the 1st generation, and 1¯ = −1.
Time evolution
Finally, the time dynamics is computed by direct integration of the Schro¨dinger equation,
cast for simplicity in dimensionless units:
H|ψ〉 = ih¯ ddt |ψ〉 ⇒ H
′|ψ〉 = i ddτrec |ψ〉 (7.29)
where H ′ = H/Erec, with Erec ≈ 9.7 kHz and τrec = h¯/Erec ≈ 16 µs for 39K in a 725 nm
lattice.
What underpins this elegantly simplistic treatment in the plane wave basis is the har-
monic nature of the optical lattice potential, which allows its decomposition into a finite,
discrete Fourier series. The above method cannot be employed to account for the effects of
interactions and of additional potentials (e.g. the dipole trap), which real space methods
such as the one presented later in section 8.1.3 are better suited to tackle.
7.2 Diffraction dynamics
The results in this section are contained in the publication [181], which was work primarily
produced by (now Dr.) Konrad Viebahn.
148 Order: sharp diffraction peaks
7.2.1 Multiple dimensions
The synthetic lattice generated up to the fifth generation is shown in fig. 7.13, while raw
images of the diffraction patterns obtained from a variable number of active lattice beams
are shown in fig. 7.14.
Figure 7.13 : The available synthetic lattice states reacheable by up to five two-photon
processes. Here Gi = 2h¯ki, and the solid lines are successive inflations by the silver mean
λ = 1 +
√
2 , emphasising the self-similarity of the diffraction pattern. Figure taken from
[181].
Figure 7.14 : Shining n lattice beams produces diffraction patterns compatible with pro-
jections from n-dimensional regular lattices, as explained in the text. The solid contours
in the final picture correspond to successive inflations by the silver mean λ = 1 +
√
2 .
Figure taken from [181].
The relationship between quasicrystals and higher dimensional regular lattices discussed
in section 3.2.4 is strikingly obvious in the three-beam (3D) and four-beam (4D) diffrac-
tion images of fig. 7.14, as these are exactly equal to the projections from the three-
and four-dimensional hypercubes of figs. 3.11 and 3.12. These are also very similar
to the electron-diffraction patterns recorded from solid-state quasicrystals with eightfold
rotational symmetry, shown in fig. 7.15.
7.2. Diffraction dynamics 149
Figure 7.15 : Electron-diffraction patterns from the octagonal phase of Cr-Ni-Si, which
bear outstanding resemblence to the 3D and 4D matter-wave interference patterns shown
in fig. 7.14. Figure taken from [182].
In addition, fig. 7.14 also shows the different structure uncovered by high diffraction or-
ders in the periodic (1D, 2D) and quasiperiodic (3D, 4D) lattices. For the former, high
diffraction orders always possess larger momenta, and thus extend the synthetic lattice
in the outward direction but does not increase its complexity owing to the periodic and
thus redundant features. In the 3D and 4D configurations, on the other hand, multiple
two-photon processes may culminate in high diffraction orders with a lower kinetic en-
ergy, causing a simultaneous inward and outward non-trivial and self-similar growth in
the structure. As discussed later, this allows the presence of a singularly continuous
component to the diffraction spectrum, which possesses a fractal dimension.
Multiple scattering vs Fourier transform
The atomic matter-wave diffraction patterns presented in fig. 7.14 are, though qualitat-
ively similar, fundamentally different from the scattering intensity of real quasicrystals
showed in fig. 7.15. Solid-state crystallographic techniques (described in chapter 3) only
assume single scattering events from the complex, ionic (Coulomb) potentials constituting
the crystals. Even when these potentials are periodic, they possess theoretically infinite
Fourier components, which allows the single-scattering diffraction pattern (eq. 3.10) to
result in a complex structure comprising a multitude of peaks.
Our optical lattices, on the other hand, are purely harmonic8 and their Fourier transforms
(i.e. the static structure factors, eq. 3.8) only comprise a finite set of elements. This is
further quantified by the following example for 1D lattices.
8for ideal plane waves.
150 Order: sharp diffraction peaks
The Fourier transform F of a 1D optical lattice consists of a finite set in k space:
F [−V0 cos2(pix/x0)] = −V0
√
pi
2
{1
2δ(k + k0) + δ(k) +
1
2δ(k − k0)
}
, (7.30)
as shown in fig. 7.16a.
By contrast, the Fourier transform of an infinite ionic potential comprises an infinite series
of k peaks:
F
[
−V0
∑
i
1
|x/x0 − i|
]
= 2
√
2pi
k2
V0
∑
j∈Z
δ(k − jk0), (7.31)
as shown in fig. 7.16b.
For simplicity, in the above we took x0 = λ/2 and k0 = 2pi. The equivalences in eqs. 7.30
and 7.31 are derived in appendix D.11.
(a)
......
(b)
Figure 7.16 : The Fourier transform F of the lattice, or static structure factor, indicates
the k states that can be reached via single-scattering events. While this is a finite set for
a harmonic optical lattice (a), it is infinite for a solid-state ionic potential (b).
The complexity of our matter-wave interference pattern is therefore due to the pres-
ence of multiple scattering events, that is multiple two-photon processes. The static
structure factor of our eightfold optical lattice, for instance, only comprises 9 elements
(k = |0〉,±|1〉,±|2〉,±|3〉,±|4〉) and is imaged by short lattice pulses that only allow single
two-photon transitions (second picture in fig. 7.17). Longer lattice pulses enable multiple
two-photon transitions and uncover new and non-trivial structures, and would correspond
to performing crystallography with thicker samples of real crystals.
Interestingly, however, the single scattering events from real, ionic potentials (in fig. 7.15)
and the multiple scattering events from our simpler, harmonic potentials result in qualit-
atively similar diffraction patterns.
7.2. Diffraction dynamics 151
7.2.2 Quantum walk
In the following results, a rectangular lattice pulse is used, as shown in fig. 7.11. This has
a steep rise time and short, variable duration (∼ 0−25 µs) and fixed depth V0 = 14.6Erec.
Because of the sudden presence of the lattice, the BEC |k = 0〉 is not loaded into its ground
state but is instead projected onto the many Bloch waves that are the eigenstates of the
potential. The diffraction dynamics, shown in fig. 7.17, constitutes a continuous-time
quantum walk on the synthetic lattice, where each hopping is a two-photon transition
with a detuning equal to the difference in kinetic energy between the initial and final
(reciprocal) states. This is quantitatively analysed in fig. 7.18, showcasing the difference
between the oscillatory dynamics in 2D and the ballistic spreading of the population in 4D,
where higher diffraction orders can possess lower kinetic energies and thus be energetically
favoured. The Raman-Nath approximation detailed in section 7.1.3 neglects the kinetic
energy and would correspond to a continuous-time quantum walk on a homogeneous
(synthetic) lattice.
Figure 7.17 : Raw absorption images displaying the diffraction dynamics at a fixed
lattice depth 14.6Erec for a variable pulse duration. The octagon has circumradius |2h¯k〉.
Figure taken from [181].
The transport properties on the synthetic lattice are quantified by the root-mean-square
momentum:
rms momentum =
√
p2 =
√√√√∑
i
pi(t)∑
j pj(t)
|bi|2 , (7.32)
where pi(t) is the population of the diffraction peak bi at any given time t. This is plotted
in fig. 7.19 for various lattice configurations and hence different (effective) dimensionalities
D, all exhibiting a ballistic expansion for short pulse durations. The extent of the linear
region depends on the lattice depth, and corresponds to the regime where the Raman-Nath
approximation is valid. Intuitively, at short pulse durations ∆t the Fourier broadened
frequency range ∆ω ∼ 1/∆t is large enough not to resolve the detuning between the
synthetic lattice points and hence enables a ballistic expansion9.
9The linear region can be extended by means of lattice sequence composed of multiple pulses, simulated
in Jonathan Mortlock’s Part III report [187].
152 Order: sharp diffraction peaks
Figure 7.18 : In a regular 2D lattice (square, here) the dynamics is oscillatory. On
the other hand, in the 4D quasiperiodic lattice some of the higher diffraction orders
possess lower kinetic energies and are thus energetically favored, which enables a ballistic
expansion shown on the right. The solid lines are obtained from simulations detailed in
section 7.1.4. Figure taken from [181].
ballistic
Figure 7.19 : The rms-momentum quantifies the transport of the atoms in the syn-
thetic lattice. For short times all lattice configurations exibit ballistic expansions, while
longer durations highlight the differences between the periodic and quasiperiodic ones,
the former showing revivals. The solid lines are numerical simulations (section 7.1.4)
for the experimental signal-to-noise ratio, while the dashed lines assume ideal detection
efficiency. Figure taken from [181].
7.2. Diffraction dynamics 153
The slopes collapse to the same line when rescaled by
√
D , which is explained by the
increased number of independent degrees of freedom contributing to the momentum p:
|p|= h¯
√√√√ D∑
i
k2i ∝
√
D
= h¯
√
k2x + k2y + k2u + k2v for the 4D case.
(7.33)
The
√
D scaling and the numerical prefactor vp are rigorously obtained in [94] and in the
supplemental material of [181].
At longer pulse durations, the distinction between periodic and quasiperiodic lattices
becomes evident. Periodic 1D and 2D lattices display an oscillatory behaviour, though
perfect revival (even in the simulation) is prevented from the system comprising more than
two levels. This can also be understood in terms of the Poincare´ recurrence theorem, which
states that for a bounded phase-space and a sufficiently long time, a dynamical system will
return to its original state, or at least very close to it. For two commensurate timescales
n and m, the Poincare´ recurrence time τP is n ·m. If m is irrational, it can be written as
a rational approximant m = p/q, with p, q ∈ Z, which only becomes exact in the limit of
large p, q →∞. The Poincare´ recurrence time n · q therefore tends to infinity τP →∞.
The quasiperiodic synthetic lattices take an infinite amount of time to return to the
original state because of the infinite number of reciprocal states at smaller momenta and
kinetic energies that become available at stronger and longer pulse durations. This is
illustrated in fig. 7.20, where regular 2D lattices always exhibit closed contours of equal
kinetic energy, thereby containing a finite number of states of the synthetic lattice. On the
other hand, fig. 7.21 shows that the equal-kinetic energy contours for the 3D lattice (or
equivalently a 3D hyperplane of the 4D lattice) are not bounded, but rather encompass
an infinite number of synthetic states, resulting in the infinitely slow revival dynamics
observed in fig. 7.19.
154 Order: sharp diffraction peaks
(a) Square lattice (b) “Slanted” lattice
(c)
Figure 7.20 : Regular 2D (synthetic) lattices created by two standing waves at 90◦ (a)
or 45◦ (b) angle between them, the latter referred to as “slanted lattice” giving rise to the
diffraction pattern in (c). The kinetic energy contours (bottom) are closed and therefore
contain a finite number of synthetic states.
Figure 7.21 : A 3D regular lattice where the z axis corresponds to the third beam
k3 = (1/√2 , 1/√2 ). The contours at constant kinetic energy are not closed and thus contain
an infinite number of synthetic states.
7.3. Diffraction spectrum 155
7.3 Diffraction spectrum
7.3.1 Singularly continuous component
In section 3.2.3, the scattering intensity (or diffraction spectrum) was divided into a pure
point, an absolutely continuous and a singularly continuous component:
I(k)total = Ipp(k) + Isc(k) + Iac(k). (7.34)
An image of the diffraction spectrum for the eightfold symmetric optical lattice is shown
in fig. 7.22. The pure point component Ipp is identified as the subset of the diffraction
peaks only caused by two-beam lattice configurations (resulting in crystallographically
allowed rotational symmetries), whereas the remaining peaks belong to the singularly
continuous contribution Isc, as shown in fig. 7.23. The distinction between the two spec-
tral components was based on the definitions presented in section 3.2.3. More intuitively
for the examples shown here, these amount to a closure-type argument of whether or not
another point belonging in the set could be found within a distance → 0 between two
other points in the same set. The spectral components for a three-beam lattice pattern
are shown in fig. 7.24.
Figure 7.22 : A non-linear colour bar allowed to visually emphasise up to the 9th dif-
fraction order, the central one being the 0th one.
156 Order: sharp diffraction peaks
(a) Full diffraction spectrum (b) Pure point contributions
Figure 7.23 : The full diffraction spectrum for the four-beam lattice (a) with its pure
point (blue) and singularly continuous (orange) contributions. The pure point component
consists of the diffraction peaks caused by all two-beam sub-lattices (b), which do not
violate the crystallographic restriction theorem. For the spatial extent of the spectrum
shown, the pure point component is complete while the singularly continuous one was
truncated.
(a) Full diffraction spectrum (b) Pure point contributions
Figure 7.24 : The full diffraction spectrum for a three-beam lattice (a) with its pure
point (blue) and singularly continuous (orange) contributions. The pure point component
consists of the sum of the diffraction peaks caused by all two-beam sub-lattices (b). For
the spatial extent of the spectrum shown, the pure point component is complete while
the singularly continuous one was truncated.
7.3. Diffraction spectrum 157
The absolutely continuous Iac contribution to the spectrum is not present in fig. 7.22
because the intra-species interaction strength was set to zero, preventing any scattering
between atoms. Using 87Rb, however, makes the scattering events noticeable as shown
in fig. 7.25. Besides reducing the visibility of higher diffraction orders and increasing
the peaks’ size, interactions add an absolutely continuous (in the limit of infinite particle
number) part to the spectrum. The scattering spheres are caused by the centre-of-mass
collisions of atoms with the same mass while they are separating inside the BEC, resulting
in the final states to lie on a sphere centred between the momentum states involved.
Figure 7.25 : Using 87Rb with its non-zero background scattering length allows inter-
atomic collisions, which result in scattering spheres, reduced visibility, increased peak size,
and an absolutely continuous contribution to the diffraction spectrum. Figure taken from
[94]
The size of the diffraction peaks is dominated by the near-field nature of the interference
pattern and its dependence on the initial BEC size. At longer times, the peaks’ size
would become infinitesimally small relative to their separations. Other contributions to
peak broadening are repulsive interactions (in 87Rb, fig. 7.25, they also cause the crescent
moon shape of the peaks10), and the finite size of the lattice beams and of the atomic
cloud.
7.3.2 Fractal dimension
When dealing with recursive and nested geometrical patterns, it is instructive to define
a measure of their space-filling capacity: if this does not scale as the dimensionality d of
the embedding space, it is referred to as fractal dimension, introduced in section 3.2.3.
The fractal dimension for the Penrose or Ammann-Beenker tilings (figs. 3.6c and 3.6d)
is trivially two, i.e. equal to the 2D plane they are drawn on, since the covering leaves
no gaps [188]. For the interference patterns shown in the previous section, however, the
discussion becomes more subtle since each diffraction order is a zero-dimensional point,
but their collectiveness tends to form a dense set which may tile the whole 2D plane.
In order to quantify the fractal dimension of the diffraction spectrum (S), we define the
box-counting dimension as the number of boxes N() of side required to fully cover the
set:
dimbox(S) = lim
→0
logN()
log(1/) . (7.35)
10which are present to some extent in 39K as well because of the “puff up” field in time-of-flight
158 Order: sharp diffraction peaks
For a finite set of n points, N() eventually becomes a constant, making the box-counting
dimension zero. This is the case for any pure point spectral contributions.
In general, the number of required boxes scales with the dimension of the set:
N() = k
(1
)d
⇒ logN() = log(k) + d log(1/), (7.36)
which can be used to extract d.
For a 2D square of normalised side 1, the above yields k = 1 and N = 1/2, and hence
results in d = 2. Similarly, a cube gives d = 3 and a line d = 1, all of which are equal to
the topological dimension of the set itself, hence trivial.
In order to find d for our diffraction spectra, we define as the smallest distance between
the diffraction orders in a finite section of radius `, and N() as the total number of states
present in the same section. This is shown in fig. 7.26, and the result of fitting eq. 7.36
is presented in fig. 7.27.
The “fractal” dimension obtained is d = 2, which is a trivial (non-integer) case but
still of relevance given that each individual diffraction order is a zero-dimensional point.
The ability of the diffraction peaks to cover the 2D plane is the result of the singularly
continuous component of the diffraction spectrum. For a three-beam lattice diffraction
spectrum, d = 1.
(a) n = 1 (b) n = 3 (c) n = 5
Figure 7.26 : Varying the number n of two-photon transitions events results in more
points N in a region of radius ` (red contour, here ` = 2h¯k) whilst at the same time
reducing the smallest separation between them.
7.3. Diffraction spectrum 159
0.0 0.5 1.0 1.5 2.0 2.5
log(1/)
1
2
3
4
5
6
7
lo
gN
(
)
` < 1h¯k, d = 2.1± 0.05
` < 2h¯k, d = 2.09± 0.11
Figure 7.27: The number of boxes N() of side required to fully account for the diffrac-
tion spectrum are related by eq. 7.36 which allows extraction of the fractal dimension d.
Each datum is computed from increasing the number of two-photon transitions, as shown
in fig. 7.26. The lower lying data points are caused by diffraction peaks on the rim ` and
are excluded from the linear fit.
———-
An interesting point that was raised in the VIVA (29th May 2019) was the following. A
true quasicrystal results in an infinite set of diffraction peaks, that would then pave the
whole 2D plane as proved in this section. Experimental images such as fig. 7.22, however,
only display a finite number of diffraction peaks, dictated by a finite number of atoms and
signal-to-noise ratio. One could then ask – what are the largest rational approximants
to
√
2 that would explain our quasiperiodic diffraction patterns? Would they depend on
the initial atom number N , and could one infer the limit for N →∞?
Chaos isn’t a pit. Chaos is a ladder.
Ser Petyr Baelish (Littlefinger), from Game of Thrones [189]
8
Disorder: localisation transition
In the following chapter we numerically and experimentally investigate the
ground state properties of bosons loaded in the 2D non-separable quasiperi-
odic potential. The quasi-disorder stemming from the lack of translational
symmetry results in a transition from an extended (“superfluid”) to a local-
ised phase at a critical lattice depth Vloc ≈ 1.78Erec in the absence of inter-
actions, akin to the mechanism described by the Aubry-Andre´ model. The
localised phase seems to be resilient against moderate repulsive interactions,
which would make this the first experimental realisation of a 2D Bose glass.
8.1 Theoretical aspects of localisation
8.1.1 Context
From Anderson to Many-Body Localisation
Disorder is ubiquitous in nature, and though generally treated as a mere perturbation,
often times it may lead to novel dynamics and phenomena. This was first discovered in
insulating materials: band and Mott insulators were already known and understood to
lack electrical and thermal conduction either due to the absence of available scattering
states or due to strong repulsive interactions. In 1958 P.W. Anderson [190] realised that
insulating behaviour may also stem from the presence of disorder, a phenomenon now
known as Anderson localisation. The disorder-induced (Anderson) insulator is a localised
phase of matter with an exponentially decaying correlation function [191] but, as opposed
to the Mott insulator, it is a single-particle phenomenon. Insulating behaviour is only a
mere consequence of localisation, for its defining feature is the lack of thermalisation and
its inability to equilibrate, as further discussed in the next section. The localisation of the
wavefunction can be understood to be a consequence of the destructive interference among
the (coherent) reflections from the random features of a disordered potential, as shown in
fig. 8.1. When the scattered amplitude is comparable to the incident one (quantified by
161
162 Disorder: localisation transition
the Ioffe–Regel criterion), destructive interference and a tight confining envelope ensue.
Figure 8.1 : Anderson localisation can be interpreted as the result of the destructive in-
terference of the multiple (coherent) reflections from the random features of the disordered
potential landscape. Figure taken from [30].
Anderson localisation has been experimentally observed in random [192, 193] and qua-
siperiodic [194] photonic lattices, in electronic systems [195–197], in optical fibres [198,
199], and in biological systems [200]. In cold atoms, this was first achieved for a BEC in
a 1D optical lattice illuminated with a disordered intensity pattern generated by a laser
propagating through a diffusing plate [30, 201], the results of which are shown in fig. 8.2.
In order for the exponential localisation to emerge, the thermal de Broglie wavelength
of the BEC needs to exceed the correlation length c of the disordered pattern, λth > c,
so that the atoms can react to the randomness of the potential instead of adiabatically
following it. The opposite limit would correspond to classical trapping of a particle of
kinetic energy lower than the potential step.
While Anderson formulated the theory for non-interacting particles, the presence of in-
teractions, providing an additional transport and delocalisation avenue, may not, in fact,
disrupt the insulating state, in what is known as Many-Body Localisation (MBL).
Anderson localisation in a random potential occurs for any arbitrarily small disorder1,
with stronger disorder only resulting in a tighter exponential envelope. However, when
the disorder is generated by an incommensurate modulation of an underlying regular
lattice, such as in the Aubry-Andre´ model, the localisation transition only occurs at a
finite critical value, when the reflected waves reach the same amplitude of the incident
one thereby resulting in destructive interference.
1Only in 1D and 2D. For d > 3, localisation only occurs above a critical disorder strength referred to
as the mobility edge. For any strength of the disorder there is a mobility edge at finite energy.
8.1. Theoretical aspects of localisation 163
a
b
z
(a)
-1
(b)
Figure 8.2 : A BEC in a 1D optical lattice with random disorder introduced by a diffusive
plate. In the presence of zero disorder (VR = 0) the atoms expand ballistically, whereas
for VR 6= 0 the BEC becomes exponentially localised and shows no transport (at least on
the experimental timescale). Figure taken from [201].
Importance of localised phase
A disorder-induced localised phase cannot and does not thermalise, hence it cannot be
described by any thermodynamic equilibrium ensemble, rendering common techniques of
statistical mechanics powerless – this is so intriguing that it deserves its own section in
this thesis, largely based on the review [202]. A more mathematical treatment is presented
in [203].
Classical and quantum statistical mechanics both assume that a general (non-integrable)
system will, at some point, thermalise. Thermalisation cannot be dependent on the
presence of an external reservoir, for an isolated quantum system would need to contain
such as a reservoir itself. Hence, the system needs to act as its own reservoir, with each
sub-system exchanging energy and particles with its surroundings. After thermalisation,
the properties of the system are not specified by any of the (local) initial conditions,
but by expectation values over the (global) thermodynamic equilibrium ensemble. If a
balloon is burst in vacuum, for example, the velocity of the air molecules will eventually
be described by the same Maxwell-Boltzmann distribution (eq. 1.18) no matter its initial
position or size. The starting conditions of the system cannot be simply erased by the
purely unitary evolution of a quantum system, however, but rather the information is
distributed over the whole system, encoded in the entanglement and hence recoverable
only by means of global operators.
Thermalisation may be prevented by the presence of additional conservation laws in in-
tegrable systems, though these can also be shown to thermalise in the presence of weak
perturbations [204].
A disorder-induced localised state is unable to act as its own reservoir, hence thermal-
isation and equilibrium are prevented; the final state, no matter the elapsed time, still
retains memory of the local initial details. This is a property of individual eigenstates,
and it is therefore not captured by equilibrium thermodynamics whose ensembles would
164 Disorder: localisation transition
average over many eigenstates. The BEC, crystals and structural glasses discussed in Part
I are all equilibrium (thermal) states, identified by distinct symmetries whose breaking
quantitatively characterises the phase transition. The localisation phase transition, on the
other hand, can only be probed “dynamically”2, by investigating the transport properties
and the memory of the initial conditions. While DC conductivity is strictly zero for local-
ised systems, it is not their defining feature, as there exist time-driven Floquet-engineered
localised states where DC observables would be ill-defined either way [206, 207]. Inter-
actions provide an additional avenue for particle transport and thermalisation, but MBL
states are found to be resilient against small but arbitrary perturbations. MBL states are
characterised by a set of localised conserved charges that are constants of motion for the
system, sometimes referred to as l-bits [202], and are hence (“locally”) integrable.
The Eigenstate Thermalisation Hypothesis [204] (ETH) has been found to serve as a
good discriminant between thermal and localised phases, in that it only holds for the
former. This essentially states that any initial wavefunction, provided the eigenstates
of the Hamiltonian are thermal, will evolve into a thermal state, i.e. dephasing will
eliminate the off-diagonal terms of the density matrix – this is for instance not true for
the Anderson insulator discussed in the previous section. Thermal states are also ergodic,
i.e. they explore all available phase-space so that their spatial and temporal averages look
the same and are independent of the initial state (except for conserved quantities such as
energy, in a closed system). Table 8.1 summarises the key differences between thermal,
single-particle (Anderson) and many-body localised states.
Thermal Single-particle localised Many-body localised
Memory of initial conditions
hidden in global operators
at long times
Some memory of local initial
conditions preserved in local
observables at long times
ETH true ETH false
DC conductivity = 0 or 6= 0 DC conductivity strictly 0
Continuous local spectrum Discrete local spectrum†
Eigenstates with
volume-law
entanglement
Eigenstates with
area-law entanglement†
Power-law spreading
of entanglement
from non-entangled
initial conditions
no spread of
entanglement
logarithmic spreading
of entanglement
from non-entangled
initial conditions†
Table 8.1 : Summary of the key differences between thermal and localised states. Table
adapted from [202]. †: these points are qualitatively discussed in appendix D.12.
2Which is not the same as the dynamical phase transitions in e.g. [205], where the free energy develops
a cusp as a function of time.
8.1. Theoretical aspects of localisation 165
Localisation so far
Cold atom experiments have hitherto introduced disorder either by generating a random
pattern from a diffusive plate, projecting it from a DMD, or by using bichromatic lattices
with incommensurate wavelengths resulting in a quasiperiodic modulation. A DMD has
recently also been used to project a 2D quasiperiodic pattern [208].
Non-localised states are usually referred to as extended or superfluid phases, the latter
term often being (inappropriately) used also for a non-interacting BEC before the Ander-
son localisation transition – see discussion at the end of section 1.2.2.
MBL has been studied mostly in the context of fermions in highly excited states, and it
has exhaustively been investigated in 1D [31, 209–212], while in 2D MBL has not always
been found to be stable against interactions [213, 214]. In addition, the 2D potentials
employed so far have always been separable.
The term Bose glass has been used for bosons in the ground state, and was theorised in
[215]. Bosons in the tight-binding regime described by the Bose-Hubbard model undergo
a transition from a compressible, gapless and long-range phase-ordered superfluid to a
gapped, incompressible and incoherent Mott insulator driven by the competition between
on-site interaction and nearest-neighbour tunnelling. In the presence of disorder, however,
the Bose glass phase appears, compressible and gapless but lacking off-diagonal long-range
order. Close to the transition, it is understood as an insulating phase surrounding and
enclosing small pockets of superfluid, known as “rare regions” and governed by Griffiths
effects, which grow in size and percolate at the transition. This is shown schematically in
fig. 8.3.
Bose glasses have been experimentally investigated in 1D [28, 29] but not yet in 2D. A
2D boson experiment in a disordered potential exhibiting MBL was carried out in [216],
but this involved quenches and therefore did not exclusively probe the ground state.
Figure 8.3 : Increasing the disorder strength ∆ in an otherwise regular lattice induces
a transition from a thermal (delocalised) to a localised (MBL, here) phase. The “rare
regions” near the critical disorder strength ∆c grow in size and percolate at the transition.
Figure taken from [214].
Several open questions remain in the field, two of my favourite ones being:
1. Can MBL happen in translationally invariant Hamiltonians, by encoding the dis-
order in the initial state?
Connections have been uncovered between the slow-relaxation mechanisms that oc-
cur in classical glasses and the non-ergodic behaviour of closed quantum systems
[217]. Glasses are classical examples that break ergodicity - are they just the classical
limit of MBL states, or a completely distinct phenomenon?
166 Disorder: localisation transition
2. Is there a difference between pure disorder and quasiperiodicity?
It has been argued [218] that these two systems belong to different universality
classes, and that MBL not only is expected to happen in quasiperiodic systems
[219], but that it would be more stable. However, other theoretical results are
conflicting, claiming that interactions in 2D quasiperiodic lattices lead to thermal
diffusion [220, 221]. Numerical simulations are not fully reliable because of the finite
size of the analysed systems, hence experimental platforms are the most promising
means to provide a definitive answer3.
Our experiments probe a system of bosons in their ground state. In the 2D non-separable
quasiperiodic potential, the observed behaviour is consistent with the bosons undergoing
an extended to localised phase transition at a critical lattice strength. The localised phase
seems to be resilient against moderate repulsive interactions, which would make this the
first experimental realisation of a 2D Bose glass.
8.1.2 Probing localisation in real and momentum space
Disorder-induced4 localised phases of matter have so far only been investigated in-situ,
studying the atomic density distributions in real space – e.g. with single-site resolution
in [216]. Typical studies usually involve imprinting a local density perturbation, either
removing part of the cloud or imposing a “density wave”, to determine how much of the
local information of the initial state is retained in its long term evolution. Localised phases
preserving local memory, they should always display traces of the original imbalance.
Investigations of the real space wavefunction in our experiment required construction
on a novel setup, plans for which had already started and are discussed in chapter 9.
However, we also wanted to test whether or not localised phases could be studied in
momentum space, by using the time-of-flight techniques already employed in chapter 7.
To our knowledge, disorder-induced localisation had never been investigated in momentum
space before.
In the rest of this short section we discuss how momentum space provides a comple-
mentary framework to establish the existence of a localised phase. The next section then
details numerical evidence confirming the presence of a disorder-induced phase transition
in our 2D quasiperiodic lattice, quantified by the Participation Ratio (PR) and Inverse
Participation Ratio (IPR).
A quantum state may be expressed in any complete orthonormal basis, for instance in
the Wannier |wi〉 or in the plane wave |φj〉 bases, corresponding to the eigenstates of the
(band-projected) position (x) and momentum (k) operators respectively:
|Ψ〉 = ∑
i
ψi|wi〉 =
∑
j
ψ˜j|φj〉. (8.1)
The (inverse) participation ratio PR (IPR) is defined in eq. 8.2 for both real (x) and
3To this date numerical simulations are limited to O(20) lattice sites, corresponding to at most 5× 5
sites in 2D. With & 100 µm× 100 µm lattice regions we can investigate regions that are 100− 1000 times
larger.
4Interaction-induced localised phases such as the Mott insulator, on the other hand, were originally
studied in momentum space: the absence of interference peaks after time-of-flight established the loss of
phase coherence associated with the insulating phase.
8.1. Theoretical aspects of localisation 167
momentum (k) space:
PRx =
1∑
i|ψi|4
, IPRx =
∑
i
|ψi|4,
PRk =
1∑
j|ψ˜j|4
, IPRk =
∑
j
|ψ˜j|4.
(8.2)
Normalisation imposes ∑i|ψi|2= 1, and hence ∑i|ψi|46 1, and similarly for ψ˜j. The
equality is only satisfied when the state is completely described by one basis function.
Hence, the PR and IPR quantify the “purity” (not in a density matrix sense) of a quantum
state in terms of its basis functions.
A fully localised state in real space (e.g. confined to one lattice site) is described by
a single Wannier function, which because of its narrow features would require infinite
Fourier components, meaning it would be fully delocalised in momentum space. From eq.
8.2, the localised phase therefore corresponds to IPRk → 0 in the thermodynamic limit.
On the other hand, in the absence of a lattice the state is fully delocalised in real space
and is thus described by the lowest plane wave with quasimomentum q = 0. As a result,
it would be fully localised in momentum space, being composed of only one Fourier
component, and corresponds to IPRk = 1. For shallow lattices below the localisation
transition, the state is still delocalised but its composition comprises higher harmonics
thus rendering IPRk = finite but not infinitesimal.
Momentum space thus provides a complementary and novel framework to study disorder-
induced localised phases, with the paradigm that localisation in real space corresponds to
delocalisation in momentum space, and vice versa. This is summarised in table 8.2.
definition delocalised(extended) localised
IPRx
∑
i|ψi|4 → 0in thermodynamic limit
finite
1 when fully localised
IPRk
∑
j|ψ˜j|4 finite1 for free system
→ 0
in thermodynamic limit
Table 8.2 : Inverse Participation Ratio (IPR) in real (x) and momentum (k) space.
8.1.3 Numerical evidence
In this section we present numerical evidence establishing the presence of a disorder-
induced localised phase in our eightfold symmetric quasicrystal. The studies are performed
both in momentum space (as will the experiments), and in real space for complementarity.
Momentum space
The single-particle Hamiltonian H was constructed in the plane wave basis |φj〉 for various
lattice configurations, but all involving beams of equal strength V0, as detailed in section
7.1.4. The ground state of H was computed by finding its lowest eigenvalue, and the
inverse participation ratio in momentum space (IPRk) was calculated. The results for
168 Disorder: localisation transition
the quasiperiodic eightfold symmetric lattice and for a regular square lattice are shown in
figs. 8.4 and 8.5 for an increasing number of basis states – the nth generation is reached
by n two-photon scattering events.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51.78
V0 /Erec
0.0
0.2
0.4
0.6
0.8
1.0
G
ro
u
n
d
st
at
e
IP
R
k
=
∑ j|ψ˜
j|4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
nu
m
b
er
of
ge
n
er
at
io
n
s
Figure 8.4 : The momentum space inverse participation ratio IPRk for the ground state
of the eightfold quasiperiodic lattice shows a sharp phase transition at V0 ≈ 1.78Erec =
Vloc. The red line is the limit for the non-truncated Hamiltonian (infinite generations).
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
V0 /Erec
0.0
0.2
0.4
0.6
0.8
1.0
G
ro
u
n
d
st
at
e
IP
R
k
=
∑ j|ψ˜
j|4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
nu
m
b
er
of
ge
n
er
at
io
n
s
Figure 8.5 : The momentum space inverse participation ratio IPRk for the ground state
of a 2D square lattice is always continuous, with no sign of any abrupt transitions. It is
always finite, and only tends to zero asymptotically.
All periodic lattices exhibit the same qualitative behaviour as fig. 8.5, for which it only
takes a few generations in order for the IPRk to converge to a smooth function of lattice
depth V0. By contrast, the eightfold quasicrystalline lattice exhibits a sharp transition
V0 ≈ 1.78Erec, hence identified as the critical strength for localisation Vloc. This value was
8.1. Theoretical aspects of localisation 169
obtained by plotting the IPRk of each lattice depth in fig. 8.4 against 1/n, n being the
number of generations. By fitting a straight line, its intercept allowed extrapolation to the
real, non-truncated system (n → ∞). Phase transitions for other lattice configurations
are discussed in appendix C.
When IPRk > 0, the state is localised in momentum space, meaning only a few plane
waves are occupied and hence the state is delocalised (extended) in real space – in the
limit of a free system, only one plane wave is occupied and IPRk = 1. However, for
V0 & 1.78Erec, the IPRk drops to an infinitesimally small value, just large enough to
ensure normalisation: all plane waves are occupied, which results in a localised real space
wavefunction with very narrow features.
The IPRk for the first few tens of low-lying eigenstates was also computed5 and is shown in
fig. 8.6. This was highly sensitive to both the size of the Hamiltonian and to the numerical
tolerance of the eigenvalue solver algorithm (because of degeneracies in the eigenenergies),
but it confirms that the first few excited states exhibit the same qualitative behaviour
as the ground state, which is good news for experiments since it would be challenging to
perfectly eliminate admixture from higher states.
Figure 8.6: The momentum space inverse participation ratio IPRk computed for the
first few tens of excited states of the 2D quasiperiodic Hamiltonian. Features past the
red dashed line had not yet converged. The calculation was performed with n = 25
generations.
Tight-binding
Confidence in the presence of a localisation transition had been provided by the 2016-2017
Part III student Attila Szabo´. The results and conclusions discussed in this section are
reproduced from his report [222], and will be the subject of a future publication.
Attila worked in the tight-binding regime modelling a 1D quasiperiodic potential as an
incommensurate modulation with period β = 1/
√
2 of amplitude λ on a regular lattice,
5though it needed to be multiplied by 8 in order to account for the eightfold degeneracy of states with
kinetic energy 6= 0.
170 Disorder: localisation transition
a.k.a. the 1D Aubry-Andre´ model:
H1D = −J
∑
n
(
a†nan+1 + c.c.
)
− λJ∑
n
cos (2piβn) a†nan, (8.3)
where a†n and an are the creation and annihilation operators for a lattice site n, and J is
the nearest-neighbour tunnelling element.
This was further extended to 2D with a modulation Λ in order to model the eightfold
quasiperiodic potential, treating two orthogonal beams as the regular (square) lattice and
the remaining two diagonal ones as perturbations:
H2D = −J
∑
nm
[
a†nm (an+1,m + an,m+1) + c.c.
]
−ΛJ∑
nm
[cos (2piβ(n+m)) + cos (2piβ(n−m))] a†nmanm.
(8.4)
As shown in fig. 7.5, it is not possible to construct a single-band tight-binding model with
four beams at equal strength, as used in the experiment. Hence the different geometry
investigated here.
The ground state participation ratio PRx6 was computed and it yielded a sharp phase
transition, as shown in fig. 8.7a. Besides the tight-binding model, Attila also constructed
a model for the 2D continuum case. Choosing to replace the irrational number by a
rational approximant
√
2 = N/M with N,M ∈ N (essentially constructing a unit cell of
size N ×M), the momentum space basis {eiki·r} was constructed. This method is well
suited for computing the ground state but not for performing dynamical simulations that
affect the excited states, for which the method outlined in the previous section is more
reliable. The ground state PRx for the 2D continuum was computed (blue curve, bottom
of fig. 8.7a) and is consistent the result of the previous section.
The analysis was extended to the excited states of H11D and H2D, highlighting the differ-
ences between the usual 1D Aubry-Andre´ model (fig. 8.7b top) and the 2D non-separable
model (fig. 8.7b bottom). In the former, localisation occurs at the same λ = 2 for all
states, whilst in the latter Λ alone is not sufficient to identify the phase. This is explained
by the diagonal perturbative beams↗ +↖= sin
[
(x− y)/√2
]
+ sin
[
(x+ y)/
√
2
]
being
written as 2 sin(x/
√
2 ) cos(y/
√
2 ), providing little to no disorder ∀y at x = 0 even for
strong Λ, and resulting in the extended states of fig. 8.8.
6Attila used the following convention. A delocalised wavefunction uniformly distributed over a lattice
of N sites has PRx = N , which is then normalised to N in order to be 1 in the thermal phase.
7Technically what is plotted here is a rescaled PR, which Attila used in order to investigate the
system-size dependence and to extract the critical exponents.
8.1. Theoretical aspects of localisation 171
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
Pa
rti
ci
pa
tio
n
ra
tio
Disorder
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5
Pa
rti
ci
pa
tio
n
ra
tio
Disorder
0.1
0.2
0.5
1.0
2.0
5.0
10
20
40
0 266 532 798
1064
1330
1596
Po
te
nt
ia
l t
o
ho
pp
in
g
ra
tio
λ
Eigenstate number
0.0
0.2
0.4
0.6
0.8
1.0
lo
g(
PR
) /
lo
g(
L)
0.1
0.2
0.5
1.0
2.0
5.0
10
20
40
0 1225
2450
3675
4900
6125
7350
8575
9800
Po
te
nt
ia
l t
o
ho
pp
in
g
ra
tio
Λ
Eigenstate number
0.0
0.5
1.0
1.5
2.0
lo
g(
PR
) /
lo
g(
L)
(a) (b)
Figure 8.7 : a) Ground state participation ratio (PR) of the continuum (blue) and the
tight-binding (red) quasiperiodic Hamiltonians, for 1D (top) and 2D (bottom). b) The
PR for the tight-binding excited states7 indicate that in 2D (bottom), as opposed to 1D
(top), the transition does not always occur at the same disorder strength. Figures from
[222].
172 Disorder: localisation transition
-288
-192
-96
0
96
192
288
-288 -192 -96 0 96 192 288
10-25
10-20
10-15
10-10
10-5
100
-288
-192
-96
0
96
192
288
-288 -192 -96 0 96 192 288
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Figure 8.8 : Real space profile of eigenstates of the 2D quasiperiodic tight-binding
Hamiltonian H2D. A low-lying eigenstate at Λ ∼ 3 (left) is localised near the origin,
whilst a mid-spectrum eigenstate at Λ ∼ 20 (right) can still be extended. Figures from
[222].
The novel and richer phenomena that result from the non-separability of the 2D qua-
siperiodic potential sets our optical lattice apart from any previous experimental platform.
Experiments, however, were not performed in the Aubry-Andre´ limit discussed here, but
rather with all beams of equal strength. This configuration exhibits equally promising and
interesting features, since Attila found the two limits to belong in the same universality
class8.
Real space
The 2D quasiperiodic potential was also numerically investigated in real space in collab-
oration with Dr. Alexander Gaunt from Microsoft Research.
The Gross-Pitaevskii equation (eq. 1.47) was solved using the split-step Fourier method
[223] in the presence of an external potential energy term:
ih¯∂ψ
∂t
= − h¯
2
2m∇
2
xψ + αV (x)Erecψ +
4pih¯2a
m
|ψ|2ψ, (8.5)
where Erec = h¯2k2lat/2m, α is the lattice depth and a the scattering length.
Introducing the dimensionless variables ξ = klatx, τ = (Erec/h¯)t, a′ = aklat and ψ →
ψ/k
3/2
lat , this becomes:
i∂ψ
∂τ
= −∇2ξψ + αV (ξ)ψ + 8pia′|ψ|2ψ, (8.6)
with V (ξ) = −∑i cos2 (Gi · ξ), G0 = (1, 0), G1 = (0, 1), G2 = 1/√2 (1, 1) and G3 =
1/
√
2 (−1, 1).
The simulation starts with a Gaussian wavefunction ψ(τ = 0) ∝ exp [−ξ2/(2σ2)], lets it
propagate in the 2D quasiperiodic potential according to eq. 8.6, and measures its spatial
extent 〈ψ|ρ2|ψ〉, with ρ2 = ξ2x + ξ2y , after some time τ , as illustrated in fig. 8.9.
8Which might even make sense within the PV conjecture discussed at the end of section 3.2.1, according
to which the irrational number
√
2 might be sufficient to classify and identify this type of quasicrystal.
8.1. Theoretical aspects of localisation 173
A specific radius
√
〈ρ2∗〉 = 35 reached at time τ∗ was chosen, so as to define a velocity
v =
√
〈ρ2∗〉 /τ∗ quantifying the real space transport properties in the potential. A sharp
transition into an insulating state (where τ∗ → ∞ and v = 0) is found to occur around
1.7 < α < 1.8, as shown in fig. 8.10.
Figure 8.9 : The spatial extent of a Gaussian wavefunction propagating in the quasiperi-
odic lattice is measured as a function of time, for different lattice depths α. The orange
line corresponds to the ballistic expansion in the absence of any potential, while the grey
area indicates where finite-size effects of the simulating region start becoming noticeable.
The insets show the spatial profiles of the wavefunction at τ = 90. Plot produced by Dr.
Alexander Gaunt.
Figure 8.10 : As the lattice depth α is increased, the time τ∗ required for the wave-
function to reach an arbitrarily chosen radius diverges. This results in its velocity v → 0,
indicating insulating behaviour and hence a transition into a localised phase, found to
occur around 1.7 < α < 1.8. Plot produced by Dr. Alexander Gaunt.
The critical lattice depth was found to be insensitive to the spatial extent σ of the starting
Gaussian wavefunction, and it is in agreement with the values found in both figs. 8.4
(plane wave basis) and 8.7 (tight-binding).
Unbeknownst to each other, Alex and I had been working on the same problem with two
174 Disorder: localisation transition
distinct but complementary approaches, and it was not until we casually ran into each
other near the Cambridge train station that we discovered our critical points agreed.
The GPE approach was used in section 8.2.4 to investigate the effects of interactions
on the extended-to-localised phase transition. The GPE is only valid within mean-field
theory, which is justified in our experiment because of the lack of strong confinement
along the z direction. This allows the z modes to be extended and to therefore carry
low kinetic energy (∝ k2), hence averaging over them is unlikely to result in significant
consequences. A strong confinement in z would result in Mott insulating physics, known
not to be captured by mean-field theory.
By evolving the system in imaginary time from an initially (spatially) uniform state, the
ground state of the 2D quasiperiodic potential was computed in real space. Lattice depths
greater than the critical value result in ground states exhibiting exponential localisation
akin to fig. 8.2b, as it is shown for a depth α = 2 in fig. 8.11.
0 20 40 60 80 100
ξx
10−18
10−14
10−10
10−6
10−2
|ψ
0
|2
(a) (b)
Figure 8.11 : The spatial distribution of the ground state ψ0 of the GPE of eq. 8.6,
for a = 0 (non-interacting). A cut through ξy = 0 (b) shows exponentially decaying
wings, in accordance with what is expected for a localised state. The region |ψ0|2. 10−13
is dominated by finite numerical accuracy. Plot produced from data provided by Dr.
Alexander Gaunt.
8.2. Experimental investigations of localisation 175
8.2 Experimental investigations of localisation
8.2.1 Technical details
Scheme
Disorder-induced localised phases are not thermal and thus cannot be detected by equi-
librium or steady-state analyses, but rather they need to be probed dynamically. The
lattice is then programmed to produce a triangular ramp from 0, to a final depth V0, and
then back to 0 in a total time 2τ , as illustrated in fig. 8.12a. The atoms start in a BEC,
hence in the zero-momentum plane wave |0〉 with amplitude ψ˜0 = 1. We measure the
fraction of the atoms recovered in |0〉 at the end of the ramp (after time-of-flight), as a
function of the maximum depth V0 and the total ramp duration 2τ .
Image here
2τ
V0
ttof
(a)
V0 = 0
V0 < Vloc
V0 > Vloc
|ψ0|2, V (x)
x
(b)
Figure 8.12 : Adiabatic following of the ground state from the extended to the localised
phase would require particle transport over an infinite spatial range (b) whose timescale
will therefore diverge. The divergence of said timescale provides a “dynamical” observable
that can identify either phase via triangular lattice ramps (a), an idea which stemmed
from [186]. In (b) the (real space) ground state wavefunctions (blue) are shown for lattice
(grey) depths V0/Erec = 0, 1 and 2.
For small ramp durations and hence abrupt lattice changes, the BEC is projected onto
the Bloch waves of the potential and the patterns of chapter 7 are recovered. When
176 Disorder: localisation transition
the lattice is slowly increased over a large interval τ , however, the atoms attempt to
adiabatically follow the ground state and would therefore return to the initial state at
the end of the triangular ramp. Longer pulse durations had already been established
to distinguish periodic from quasiperiodic lattice configurations in the synthetic lattice
transport measurements of fig. 7.19, and are thus expected to play a similar role here.
If the BEC were to be adiabatically loaded into the lattice ground state for every V0 in the
ramp, then it would smoothly turn into a delocalised Bloch wave for any finite V0 < Vloc,
but would become exponentially localised above a critical depth Vloc, as depicted in fig.
8.12b. Our quasiperiodic potential typically possesses a single absolute minimum, where
all four standing waves interfere destructively: the absolute ground state in the localised
phase would thus see all atoms tightly confined to this site, chosen to lie in the middle of
the x axis in fig. 8.12b for clarity.
As the lattice depth V0 is changed, the transition from the extended wavefunction for
V0 < Vloc to a localised state for V0 > Vloc would require particle transport over infinite
spatial separations, the time required for which diverges. In the framework of a synthetic
lattice in momentum space developed in chapter 7, this transition would require an infinite
number of two-photon transitions. Hence, adiabaticity is expected to break down at and
above the localisation transition Vloc.
The (predicted) lack of recovery of the initial BEC stems from the infinite number of
low-momenta states available in the quasiperiodic synthetic lattice. Instead of simply re-
turning to the central BEC peak at the end of the ramp, atoms may undergo multiple two-
photon transitions to terminate arbitrarily close it. The presence of these low-momenta
states, however, is independent of the lattice strength: so why is the transition driven by
a critical V0? While high-generation states may be energetically more favourable because
of their reduced kinetic energy ∆E, the (“Rabi”) coupling required to reach them is gov-
erned by the lattice strength V0. The competition between V0 and ∆E is what results in
a critical depth Vloc.
The fractal structure of the synthetic lattice is expected to yield non-trivial single-particle
physics, hence the majority of the numerical and experimental investigations was per-
formed in the absence of interactions.
Population extraction
The populations of the diffraction orders were extracted by fitting each peak independ-
ently. A fit was performed instead of summing the pixel counts because it allows for a
reliable characterisation of the peaks even in the presence of noise and (more importantly)
saturation. The 0th diffraction order (the BEC) is almost always severely saturated, as
shown in fig. 8.13. This is due to the high spatial density of the BEC, which causes the
imaging light to be nearly completely absorbed in its central region thus not reaching the
camera.
8.2. Experimental investigations of localisation 177
Figure 8.13 : To extract the population of the central peak (the BEC), the Thomas-
Fermi profile of eq. 6.14 (blue) was fitted to it. The BEC is heavily saturated due to its
high spatial density, thus all points above a certain cutoff are not considered in the fit.
According to eq. 6.6, this results in an optical density OD→∞ which is then discarded
in the image processing analysis: only points below a certain “cutoff” are considered in
the fit, chosen to be OD= 3.4. Only the wings of the atomic cloud provide data for the
fitting function, which was assumed to be the inverted parabola of eq. 6.14 obtained
within the weakly interacting Thomas-Fermi approximation.
As far as the other diffraction orders are concerned, the application of the fit functions is
generally reliable for the square lattice (e.g. fig. 8.14a) and for the short time dynamics
of the eightfold quasiperiodic one (e.g. fig. 8.19b) since the diffraction orders are well
separated. On the other hand, after longer and deeper quasiperiodic lattice ramps, atoms
may occupy up to the 10th diffraction order9 which renders the fitting procedure extremely
challenging, aggravated by the merging of neighbouring peaks (fig. 8.14b). This results
in the fit failing and the population not being recorded, causing an underestimation of
the total atom number.
The presence of moderate repulsive interactions during the lattice ramp, for many-body
physics investigations, causes the peaks to broaden enough to be reliably fitted even after
deep and long lattice ramps (fig. 8.14c).
8.2.2 2D periodic lattice
We started our investigations by probing a regular square lattice, obtained from the
superposition of two orthogonal standing waves.
Quantum adiabaticity
In the following we employ the quantum adiabatic theorem in order to estimate the
timescale required to adiabatically load a BEC in a regular square lattice, to be then
compared with what observed experimentally. The mathematical treatment here is based
on [224].
9As a comparison, the number of potentially occupied diffraction peaks in each image is 25 for the
2D regular lattice, and 8, 361 for the 4D quasiperiodic one. In theory, the atoms may occupy even higher
orders that are however not detected due to finite atoms numbers and signal-to-noise ratio. See fig. 8.23.
178 Disorder: localisation transition
(a) (b) (c)
Figure 8.14 : In the regular square lattice (a), atoms only occupy well separated peaks
which are thus easy to fit. In the deeply localised quasiperiodic lattices (b), some of the
peaks appear merged causing fits to fail and resulting in an underestimated total atom
number (here V0 = 2.4Erec, 2τ = 500 µs) – the red dots are the predicted locations of
the first 10 diffraction orders. The same parameters as in (b) but with the addition of
repulsive interactions (a = 40 a0) during the lattice ramp cause peak broadening and allow
for reliable fits (c).
The BEC having free space momentum k = 0, it only couples to bands at quasimomentum
q = 0 (conserved in the lattice), and in particular will only occupy the lowest band if the
ground state is to be followed adiabatically. Excitations into higher bands because of
imperfect adiabaticity can be estimated as follows.
A state in a band n at quasimomentum q is an eigenstate of the (instantaneous) Schro¨dinger
equation H(t)|φn,q(t)〉 = En(t)|φn,q(t)〉. The time-dependent wavefunction may be expan-
ded in the instantaneous basis: |ψt〉 = ∑n,q cn,q(t) exp [− ih¯ ∫ t0 En(t′) dt]|φn,q〉, which, when
substituted into the Schro¨inger equation, results in the following time dependence for the
coefficients cn,q:
c˙n,q = −
∑
m
〈φn,q|φ˙m,q〉 e−
i
h¯
∫ t
0 [Em(t
′)−En(t′)] dt′cn. (8.7)
This can be re-cast in a better form by first requiring the norm to be conserved:
d
dt〈φn,q|H(t)|φm,q〉 = 0 = (En − Em) 〈φn|φ˙m〉+ 〈φn|H˙|φm〉. (8.8)
For m = n, i.e. the diagonal phase, it gives rise to the pure phase factor known as the
Berry phase γn. For m 6= n, one can write:
〈φn|φ˙m〉 = 〈φn|H˙|φm〉(Em − En) . (8.9)
Hence, the time evolution of the coefficients may be written as:
c˙n,q = −iγncn −
∑
m 6=n
〈φn,q|H˙|φm,q〉
Em − En︸ ︷︷ ︸
non−adiabatic
e−
i
h¯
∫ t
0 [Em(t
′)−En(t′)] dt′︸ ︷︷ ︸
free evolution
cn. (8.10)
In order to adiabatically follow the ground state, the above time evolution should only
amount to a phase factor. For the second term, this requires the non-adiabatic term to
8.2. Experimental investigations of localisation 179
contribute less strongly than the free-evolution term:
∂
∂t
(non− adiabatic) ∂
∂t
(free evolution),
⇒ |〈φn,q|H˙|φm,q〉| (En − Em)
2
h¯
,
(8.11)
which is known as the adiabaticity condition. Technically it should really be [min(En −
Em)]2, since the gap may evolve in time or space as shown, for example, in fig. 2.1.
Writing H˙ = V˙ = ∆V0/τramp, where ∆V0 is the change in optical lattice potential strength,
and expressing the depths in terms of the recoil energy Erec, eq. 8.11 results in a lower
bound for the ramp duration:
τramp 1∆V0 τrec, (8.12)
where τrec = h¯/Erec is approximately 16 µs for 39K in a 725 nm lattice.
For atoms at q = 0 in a square lattice with the band structure of fig. 2.14, the minimum
relevant ∆V0 that may cause inter-band excitations is from the bottom of the first band
to the top of the second band, ∆V0 = 4Erec. This would require τramp 4 µs to ensure
the BEC is adiabatically loaded into the ground state of the square lattice.
Other time evolution contributions that were ignored here include interactions and the
dipole-trap harmonic confinement, though we work primarily in the non-interacting re-
gime which thus results in an infinitely slow interacting timescale. Our lattice ramps
are adiabatic with respect to the internal dynamics, as quantitatively discussed in this
section, but not with respect to the dipole trap. In fact, the harmonic confinement has a
maximum trapping frequency of ∼ 100 Hz (in z) and results in a timescale T/4 ∼ 2.5 ms,
much larger than the ∼ 100 µs lattice ramp durations employed in the experiment. The
ramps are fast (non-adiabatic) in order for the dipole trap not to affect the cloud dens-
ity and position. With respect to the dipole trap, the lattice ramp can be regarded as
instantaneous (∂/∂t dipole− trap = 0).
Experiment
The square lattice was ramped to depths V0 ranging from 0.6 to 4.6 Erec and then back to
0 in durations 2τ = 1 → 200 µs. The population of the 0th plane wave (|ψ˜0|2, the initial
BEC) was recorded as shown in fig. 8.15. Typical images are presented in fig. 8.16.
180 Disorder: localisation transition
Figure 8.15 : The recovery of the population of the initial BEC (|ψ˜0|2) as a function of
the total ramp duration 2τ . For long enough ramps, the atoms are adiabatically loaded
into the ground state of the square lattice for any depth V0, and they hence return to the
BEC at the conclusion of the ramp. The solid line is the numerical simulation described
in section 7.1.4.
(a) 2τ = 0µs (b) 2τ = 20µs (c) 2τ = 200µs
Figure 8.16 : Typical images taken at the end of the triangular ramps (and after time-
of-flight), for the V0 = 4.6Erec curve in fig. 8.15. The colour bar has been adjusted to
emphasise the higher orders.
The BEC population was always recovered with a slow enough ramp, in this case 2τ &
40 µs, which is compatible with the estimate 4 µs from the previous section, and agrees
with the numerical simulation described in section 7.1.4.
For short pulse durations, the lattice is abruptly increased and the BEC is hence projected
onto many Bloch waves, akin to the rectangular pulses of section 7.2.2. Multiple, smaller
dips in the BEC population are present even for 2τ > 40 µs, and are probably the result
of interference effects between the ramp time and the finite discrete set of frequencies in
the Hamiltonian.
The non-adiabaticity does not depend on the lattice depth V0 reached so much as it does
on the slope of the ramp, for the losses are always primarily caused by the same excitation
8.2. Experimental investigations of localisation 181
from the bottom of the first to the top of the second band discussed earlier. Hence, all the
depths in fig. 8.15 exhibit the same qualitative behaviour at equal ramp times. This is
further confirmed by the numerical simulation on the “slanted” lattice shown in fig. 8.17.
This lattice consists of two standing waves at 45◦ separation, where the difference between
the bottom of the first to the top of the second band is roughly half than for a regular
square lattice: this was observed to result in a doubled adiabatic recovery timescale, as
expected from eq. 8.12.
In addition, the fact that nearly total BEC population recovery is possible for up to
2τ = 200 µs sets a lower bound on the heating rates and hence on the lifetime of the
atoms in the square lattice. This is compatible with other groups routinely performing
experiments with lattice durations of up to ∼ 500 ms.
Figure 8.17 : The recovery of the BEC population is simulated for the regular square
lattice (blue) and for the slanted lattice (orange), with V0 = 7.5Erec. The inset shows
the width of the first two bands for both configurations. The minimum energy inter-
band excitation for atoms in q = 0 is 4Erec for the square and ∼ 2Erec for the slanted
lattice. The slanted lattice therefore exhibits an adiabatic recovery at a timescale roughly
∼ 4/2 = 2 times slower than for its square counterpart.
8.2.3 2D quasiperiodic lattice
Experiment
In the periodic lattices of the previous section it was always possible to fully recover the
BEC population, given slow enough ramps, even for peak depths V0 Vloc ≈ 1.78Erec.
This is very different from what observed in the 2D quasiperiodic lattice, shown in fig.
8.18 along with typical images presented in fig. 8.19. Two regimes are observed depending
on the lattice depth V0: some recovery curves seem to return to unity whilst others exhibit
the opposite behaviour and seem to tend to zero.
182 Disorder: localisation transition
Figure 8.18 : For the 2D quasiperiodic lattice, the recovery of the population of the
initial BEC (|ψ˜0|2) at large ramp durations 2τ is only observed for V0 6 1.2. The other
depths exhibit a gradually more significant negative slope (shown in fig. 8.20). The
solid line is the numerical simulation described in section 7.1.4. The dashed crimson line
denotes the maximum recovery of the population in the V0 = 2.4Erec curve.
(a) 2τ = 0µs (b) 2τ = 30µs (c) 2τ = 500µs
(d) 2τ = 0µs (e) 2τ = 30µs (f) 2τ = 500µs
Figure 8.19 : Typical images taken at the end of the triangular lattice ramps (and after
time-of-flight), for the V0 = 1.2Erec (top, a→c) and V0 = 2.4Erec (bottom, d→f) curves
in fig. 8.18. The colour bar has been adjusted to emphasise the higher orders.
The BEC population is observed to be recovered for V0 6 1.2Erec. The other curves
8.2. Experimental investigations of localisation 183
display a negative slope, which increases in magnitude with lattice depth V0, and are
further discussed in the following section in order to pinpoint the critical strength Vloc
triggering the localisation transition.
The apparent maximum reached by all curves around 2τ ∼ 50 µs is qualitatively similar
to what observed for the 2D regular lattice in fig. 8.15. At these short ramp durations, in
fact, the atoms have not yet had the time to explore enough phase space to display the
features that are unique to the quasiperiodic configuration. For the V0 = 2.4Erec curve,
a maximum recovery |ψ˜0|2= 0.9 is reached at 2τ = 55 µs, used later to discuss the effect
of interactions during the lattice ramp.
It is important to note that, in the localised regime V0 > Vloc ≈ 1.78Erec, the recovery
of the initial population becomes worse as slower ramps are employed. This may be
counter-intuitive at first, as it is the opposite of what would be expected from the Kibble-
Zurek mechanism10. However, this is explained by the many synthetic lattice points with
lower kinetic energies that become accessible at larger V0 and longer 2τ . The complexity
offered by the quasiperiodic lattice potential is what allows the atoms to populate an
increasing number of synthetic states via Kapitza-Dirac scattering, thus further depleting
the zero-momentum plane wave |0〉 (the initial BEC).
Since our investigation is limited to the 500 µs duration of the experimental run, we
in principle cannot exclude a BEC recovery occurring over much longer scales. The
observed behaviour is however consistent with the theoretical expectations from fig. 8.21a,
discussed later. For long and deep ramp durations, such as V0/Erec = 2.0 → 2.4 and
2τ & 250 µs, the BEC is not saturated and a pixel sum of the picture may be performed.
This yielded a qualitatively similar result to the fit analysis shown in fig. 8.18, though
significantly noisier.
Even in the localised phase (e.g. fig. 8.19f), the images still exhibit sharp and distinct
diffraction peaks, suggesting the atoms did not have enough time to lose phase coherence.
The timescale for this to happen is expected to be larger than what probed thus far, but
could be easily investigated by ramping and subsequently holding the lattice at V0 > Vloc
for a variable interval of time.
Critical strength for the localisation transition
The following describes the extraction of the critical lattice strength Vloc from the exper-
imental data, and its comparison to the theoretical expectation Vloc ≈ 1.78Erec obtained
from the numerical simulation of fig. 8.4.
A straight line was fitted to the 100 µs 6 2τ 6 200 µs region of fig. 8.18. This region was
chosen because, from the 2D regular lattice results of the previous section (fig. 8.15), other
sources of atomic loss were established to be negligible11. These are plotted in fig. 8.20,
and show that both theory and experiment already exhibit a negative slope for V0 < Vloc.
While this establishes that unit recovery is precluded on the 2τ = 500 µs timescale probed
in the experiment, it does not exclude that the V0 < Vloc curves may fully recover over
slower ramp durations. A simulation is then considered to explore longer dynamics.
The numerical simulation shown in fig. 8.18 is performed in the plane wave basis, as
10where essentially the slower the ramp, the more adiabatic the following.
11for the square lattice. For the 2D quasiperiodic, this is not necessarily also the case and would require
careful measurements.
184 Disorder: localisation transition
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
V0/Erec
−30
−25
−20
−15
−10
−5
0
slo
pe
/
10
4
×
µs
−1
Figure 8.20 : The downward slopes of the straight lines fitted to the population recovery
curves in fig. 8.18, along with the theoretical prediction (orange solid line). The predicted
critical localisation strength Vloc ≈ 1.78Erec (grey solid line) is obtained from fig. 8.4.
described in section 7.1.4. The momentum space basis is computationally intensive for
long ramp durations as the time dynamics involves multiplications of large12 Hamiltonian
matrices for each time step.
1.25 1.50 1.75
V0/Erec
0
25
50
75
100
2τ
@
∣ ∣ ∣ ∣ψ˜ 0
∣ ∣ ∣ ∣2 =
99
%
/
m
s
(a) (b)
Figure 8.21 : The real space simulation outlined in the last part of section 8.1.3 enables
numerical investigation of the BEC recovery fraction for much longer ramp durations (a),
establishing the different behaviour exhibited for lattices below and above the expected
critical lattice strength Vloc ≈ 1.78Erec. The adiabatic timescale is also computed and is
seen to diverge around the same lattice depth (b). Plots produced from data provided by
Dr. Alexander Gaunt.
Luckily, our collaborator Dr. Alexander Gaunt performed the same simulation in real
12For n = 25 generations, these are 283, 401× 283, 401 sparse matrices.
8.2. Experimental investigations of localisation 185
space (last part of section 8.1.3) and with more computing power, which enabled probing
much longer ramp durations, as shown in fig. 8.21a. The distinct recovery behaviour
exhibited by lattices above and below the expected localisation transition V0 ≈ 1.78Erec
is then very clear. Gentle downward slopes in the 100 µs 6 2τ 6 200 µs region, displayed
for V0 6 1.6Erec in the experiment, lead to a full recovery over much longer timescales.
On the other hand, the stronger negative slopes of V0 > 1.8Erec are seen to result in a
BEC recovery tending to zero (at least until the maximum simulated 2τ = 100 ms).
Hence, the experimental data presented in figs. 8.18 and 8.20 are consistent with the
localisation transition occurring between 1.7Erec and 1.8Erec, that is the blue and or-
ange curves in fig. 8.21a. A strong negative slope is thus taken to be a good indicator
of localisation, and will be further employed in the investigations involving interactions
presented in section 8.2.4.
It is also possible to quantitatively measure the breakdown of the adiabatic timescale,
predicted in section 8.2.1, by computing the minimum ramp duration required for all
subsequent points to exhibit a BEC recovery of > 99% (grey dashed line in fig. 8.21a).
This is shown in fig. 8.21b and is observed to diverge for a critical lattice depth Vloc/Erec =
1.80± 0.03, consistent with the prediction Vloc ≈ 1.78Erec from fig. 8.4.
The data in fig. 8.21b was fitted with:
y = a
(Vloc − V0)b
, with
value error
a 0.05 0.04
Vloc 1.80 0.03
b 3.15 0.80
(8.13)
which provided an estimate for Vloc/Erec = 1.80± 0.03, shown as the grey error bar.
Technical comments
We consider here a few technical aspects that may have affected the data.
The total atomic population in each image for the 2D quasiperiodic lattice experiments is
found to be roughly consistent, as shown in fig. 8.22a. This suggests that the fit analysis
to extract the peaks’ populations, discussed in section 8.2.1, is reliable. Deep and long
lattice ramps enable the atoms to occupy high diffraction orders and result in some of
the peaks to merge (fig. 8.14b). This causes fits to either be highly distorted or to fail,
resulting in increased noise or in an underestimation of the total population, respectively.
The lower recorded total populations for the 2.0 → 2.4Erec curves at 2τ & 300 µs in fig.
8.22 establish that the respective BEC recovery curves in fig. 8.18 should not be plateauing
but rather exhibit a stronger negative slope. For completeness, the total populations of
the 2D square lattice discussed in section 8.2.2 are shown in fig. 8.22b.
186 Disorder: localisation transition
(a) 2D quasiperiodic
(b) 2D square
Figure 8.22 : The fit analysis employed to extract the peaks’ populations is observed
to result in consistent total atom numbers. For the deeply localised regions of the 2D
quasiperiodic lattice (large V0 and 2τ in a), however, some of the diffraction peaks merge
causing fits to fail and the total population to be underestimated.
As explained at the end of section 7.1.3, repulsive interactions (a = 280 a0) are re-
introduced 2 ms into the ttof = 33 ms time-of-flight fall. This is done in order to broaden
(“puff up”) the diffraction peaks and allow the fits to reliably characterise them. The
2 ms time separation was intended to prevent scattering between particles of different
momenta which would otherwise lead to the scattering spheres of fig. 7.25. Although for
the arbitrarily closely spaced peaks in the 2D quasiperiodic lattice this might not be long
enough, the lack of scattering spheres in the images shown in fig. 8.19 suggests the “puff
up” does not cause any significant effects.
The experimental images for the 2D quasiperiodic lattice were observed to display up to
8.2. Experimental investigations of localisation 187
the 10th diffraction order. In order to verify that higher diffraction orders were indeed
expected not to be significantly populated, the image at V0 = 2.4Erec and 2τ = 200 µs
was compared to its simulated counterpart. This is shown in fig. 8.23, where orders
> 10th (red) are seen to account for extremely low populations, easily falling below the
detectable threshold of the experiment.
The peaks’ positions in the x− y plane of the camera are given by:
r/µm = 22 · (p/2h¯k) · (ttof/ms), (8.14)
so after ttof = 33 ms the 2h¯k peaks have covered a distance of 727 µm.
(a) (b)
Figure 8.23 : Comparison of the observed (a) and simulated (b) image for V0 = 2.4Erec,
2τ = 200 µs and ttof = 33 ms. The size of the simulated peaks are proportional to their
population. The > 10th diffraction orders (red) are not observed in the experiment but
contribute an insignificant fraction to the total population.
8.2.4 Interactions
2D Bose glass?
The lattice ramps were so far conducted at a magnetic field as close as possible to the
zero-crossing of the scattering length, in order to eliminate interactions and probe single-
particle physics.
In order to quantitatively investigate the effect of (repulsive) interactions on the extended-
to-localised transition in the 2D quasicrystalline lattice, the magnetic field during the
lattice ramp was varied to sample scattering lengths of up to 200 a0.
A priori, the effect of interactions could have been:
• non-existent;
• to fully destroy localisation for any a 6= 0, however small;
• to allow persistence of the localised phase, at least for moderate a.
188 Disorder: localisation transition
The result is shown in fig. 8.24, where the very low BEC recovery in a deeply localised
region (2τ = 500 µs and V0 = 2.4Erec) was observed to increase from ∼ 0.2 almost back
to unity with growing interaction strength. Typical images are shown in fig. 8.25.
Figure 8.24 : The BEC population in the 2D quasiperiodic potential at 2τ = 500 µs and
V0 = 2.4Erec is very low as it is deep in the localised phase. However, this is shown to
increase for a growing inter-species scattering length. The dashed line at |ψ˜0|2= 0.9 is the
maximum recovery experienced by the non-interacting 2.4Erec curve (inset). The error
bar corresponds to three repetitions of each datum.
(a) a = 0 a0 (b) a = 40 a0 (c) a = 200 a0
Figure 8.25 : Typical images taken at the end of the triangular ramps (and after time-
of-flight) for the V0 = 2.4Erec and 2τ = 500 µs curves in fig. 8.24. The colour bar has
been adjusted to emphasise the higher orders.
The increasing final BEC population at larger interaction strengths suggests the down-
ward slopes of the recovery curves (inset in fig. 8.24) decrease in magnitude and eventually
become positive for |ψ˜0|2> 0.9 (grey dashed line). 0.9 is the maximum recovered pop-
ulation occurring at 2τ = 55 µs, which is expected to be unaffected by the interaction
strengths probed here, as justified in the next section.
This establishes that interactions significantly affect the system’s dynamics, and suggests
that the localised phase may persist for a 6= 0, which is compatible with the theoretical
8.2. Experimental investigations of localisation 189
prediction of fig. 8.28 showing stronger interactions shift the transition to higher lattice
depths.
The interaction energy of the atomic cloud is quantified by the chemical potential µ ∝ a2/5.
When plotted against µ, the data asymptotically tends to a straight line, as shown in fig.
8.26. It would be interesting to perform further investigations to establish whether the
slope of this line depends on the specific ramp duration 2τ and lattice strength V0.
Figure 8.26 : The data in fig. 8.24, when plotted against the chemical potential µ ∝ a2/5
(quantifying the interaction energy), tends to a straight line (dashed) for large scattering
lengths a. The solid curve is a fit using eq. 8.15.
The fit function used in fig. 8.26 is:
y =
√
b2 + c2 · x2 , with
value error
b 0.19 0.08
c 0.12 0.01
, (8.15)
and the asymptotic straight line is given by y = cx.
In conclusion, figs. 8.24 and 8.26 show that the BEC population from a deeply localised
region may be (almost completely) recovered by increasing the interaction strength. This
suggests the localised phase may be resilient against moderate (repulsive) interactions,
potentially establishing this as a many-body localised phase. This would be the first
experimental realisation of a 2D Bose glass.
Negative slopes and interactions
The interpretation of fig. 8.24 heavily relies on the BEC recovery curves for V0 > Vloc to
exhibit a downward trend also in the presence of interactions. That is because a strong
negative slope had been established as a good indicator of localisation.
Here, we provide an argument for why the the maximum recovery |ψ˜0|2= 0.9 at 2τ = 55 µs
displayed by the V0 = 2.4Erec curve (inset of fig. 8.24) should remain constant for the
190 Disorder: localisation transition
Figure 8.27 : The recovery of the BEC population in the 2D square lattice (here at
V0 = 7Erec) shows no change for weak interactions, at least until 2τ = 150 µs. The solid
line is the (non-interacting) numerical simulation of section 7.1.4.
interaction strengths investigated here. The recovered populations at 2τ = 500 µs shown
in fig. 8.24 would then be enough to determine the slope of the whole curve13.
For small interactions (up to a = 20 a0), the BEC recovery in the 2D square lattice showed
no change around 2τ ∼ 55 µs, as illustrated in fig. 8.27. To extend the argument for up
to a = 200 a0, we consider the relevant timescale associated with the interactions, related
to the chemical potential µ.
The chemical potential µ of an atomic cloud in a harmonic trap is given by [54]:
µ = h¯ω¯2
(15Na
a¯ho
)2/5
, (8.16)
where a is the scattering length, ω¯ the geometric mean of the trap frequencies, N the
atom number and a¯ho the mean harmonic oscillator length, given by:
a¯ho =
√
h¯
mω¯
. (8.17)
During the lattice pulse, (ωx, ωy, ωz) = 2pi·(19.1, 17.5, 115.3) Hz and hence ω¯ = 2pi·33.8 Hz.
For our 39K BEC, a¯ho ≈ 2.8 µm. With typical atom numbers of 1.5 − 3.0 × 105, the
chemical potential of the cloud at the strongest interaction strength (a = 200 a0) is µ/h ≈
742± 84 Hz.
The (shortest) timescale τint associated with the interactions is then τint ∼ h/µ ∼ 1.5 ms
55 µs. It was argued in section 8.2.3 that the short-time (2τ . 55 µs) dynamics is approx-
imately the same for both the square the quasiperiodic lattices. Hence, we claim interac-
tions are unlikely to affect the V0 = 2.4Erec BEC recovery curves around 2τ ∼ 55 µs in
13The experimental apparatus has undergone a catastrophic failure in November 2018, from which
it has yet not recovered at the time of the final version of this thesis (7th October 2019). Hence the
unavailability of additional data, and the need to justify the negative slopes’ argument indirectly.
8.2. Experimental investigations of localisation 191
fig. 8.24. This would guarantee that the recovered population at 2τ ∼ 55 µs is constant
for the interaction strengths probed in the experiment, making the measured 2τ = 500 µs
points enough to determine the slope of the recovery curves.
Theoretical model
A real space simulation performed by Dr. Alexander Gaunt is shown in fig. 8.28. It
establishes that interactions require a larger critical lattice depth Vloc for localisation to
occur. Interactions thus shift the transition to higher lattice depths.
The simulation solved for the ground state of the lattice (eq. 8.6) subject to “twisted
boundary conditons” (based on [225]), whereby the phase of the wavefunction is required
to differ by a small angle θ between two boundaries of the simulation region. A wavefunc-
tion localised at the origin is unaffected by such twist, because of its vanishing magnitude
at the boundary: the ground state of the localised phase is hence the same in both
twisted and untwisted system. On the other hand, a delocalised wavefunction displays
a phase difference θ over the system size L, corresponding to a drift velocity v ∝ θ/L.
This provides the extended wavefunction in the twisted system with an additional kinetic
energy contribution, associated with the drift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51.78
V0/Erec
0.0
0.2
0.4
0.6
0.8
1.0
wa
ve
fu
nc
tio
n
st
iff
ne
ss
Vloc (a = 0) non-interacting
interacting
Figure 8.28 : Numerical simulations of interacting atoms in a 2D quasiperiodic potential
show that the critical point is shifted to larger depths14. The wavefunction stiffness is
related to the superfluid fraction and hence vanishes in the localised phase. Plot produced
with data provided by Dr. Alexander Gaunt.
In the simulation, the observable to distinguish between the localised and the extended
(“superfluid”) phase is the difference of the ground state energy between the untwisted
and twisted systems, related to the superfluid fraction and referred to as “wavefunction
stiffness”. This is plotted in fig. 8.28 against lattice depth: no energy difference in the
localised phase results in zero superfluid fraction (V0 > Vloc), while the entire wavefunction
moving at the drift velocity gives unit superfluid fraction (V0 = 0).
14The interaction strength is quantified by the 8pia′|ψ|2 term in eq. 8.6, where |ψ|2 (in dimensionful
units) is the atom number N . For the interacting (red) curve, a value 8pia′N = 345.2 was used.
192 Disorder: localisation transition
8.2.5 Limitations of simulation
The simulations discussed in this chapter (unless when otherwise specified) are performed
by me in the plane wave basis, as detailed in section 7.1.4.
Because of the finite RAM of the computers used (usually 8 GB), only a finite Hamiltonian
matrix could be loaded. Exact diagonalisation was numerically not allowed for & 12
generations, requiring the use of sparse matrices that allowed computing the ground state
for up to 25 generations used in the IPRk plot of fig. 8.4. The finite number of CPUs
(usually 4 logical cores) does not allow efficient parallelisation and hence results in long
computing times when simulating the time evolution of the state.
Hence, only relatively short 2τ ramp durations were simulated, and the plane wave basis
was truncated at some maximum generation nmax. The finite number of simulated dif-
fraction peaks then results in a finite number of basis states composing the synthetic
lattice. For the 2D quasiperiodic potential, the synthetic lattice is then a finite four-
dimensional hypercube. Long enough ramp durations therefore allow the population to
reach its boundaries and to be reflected from them, as shown in fig. 8.29.
Figure 8.29 : Simulated BEC recovery in the 2D quasiperiodic lattice, for depths V0 =
1Erec (blue) and V0 = 5Erec (orange) and maximum generation nmax = 5 (top) and
nmax = 15 (bottom). The finite size of the synthetic lattice allows for reflections from its
boundaries, whose effects is relegated to longer ramp durations 2τ for larger nmax.
All simulations presented so far used a Hamiltonian with quasimomentum q = 0, justified
by the BEC possessing free momentum k = 0. Experimentally, the BEC is not fully
in k = 0 because of quantum depletion and the finite size of the trap. In order to
quantitatively account for this, the Hamiltonian was computed for different quasimomenta
q 6= 0 and the BEC recovery simulations were performed. These are shown in figs. 8.30a
and 8.30b for the regular square and the 2D quasiperiodic lattices, respectively.
In both configurations, a higher q results in a less-adiabatic evolution, with the initial
population dip extending farther down and the population recovery occurring at larger
ramp durations 2τ . These may be simply explained by the higher initial kinetic energy
∝ q2 enabling to reach higher states and therefore further depopulating the initial BEC.
8.2. Experimental investigations of localisation 193
As a complementary explanation, a higher q reduces the energy gap to the next band,
thus increasing its coupling strength.
The slopes of the initial depopulation region are observed to be the same for all q, as they
are probably set by the quantum speed limit related to the “Rabi” coupling Ω ∝ V0. In
the square lattice case, the time dynamics for particles at the edge of the Brillouin zones
(q ∈ Z) exhibits periodicity and revival, justified by these points being degenerate with
points at −q. Because of this degeneracy, the band gap never vanishes and no dynamics
can ever be adiabatic with respect to it.
194 Disorder: localisation transition
0 25 50 75 100 125 150 175 200
2τ/µs
0.0
0.2
0.4
0.6
0.8
1.0
|ψ˜
0|2
q = 0
q = 0.25
q = 0.5
q = 0.75
q = 1
(a) 2D square
0 25 50 75 100 125 150 175 200
2τ/µs
0.0
0.2
0.4
0.6
0.8
1.0
|ψ˜
0|2
q = 0
q = 0.25
q = 0.5
q = 0.75
q = 1
(b) 2D quasiperiodic
Figure 8.30 : The Hamiltonian was computed for different quasimomenta q 6= 0 and
the BEC recovery simulations performed for the square (a) and quasiperiodic (b) lattices.
The lattice depth was set to V0 = 5Erec for all simulations.
Das ist nicht nur nicht richtig; es ist nicht einmal falsch.
Wolfgang Pauli
9
Plans for Z experiments
This chapter summarises miscellaneous results obtained while planning the
next upgrade to the experiment, a vertical (Z) apparatus enabling 2D confine-
ment, strong interactions and access to real space observables. A scheme to
compensate the anticonfining effects of the blue detuned quasiperiodic lattice
beams was devised, and the diffraction pattern of an axicon was numerically
and experimentally investigated.
9.1 Z setup
Originally, the localisation transition was going to be experimentally investigated in real
space and not in momentum space as presented in chapter 8.
Steps had already been taken to plan, design and characterise the apparatus required to
probe the atomic cloud in-situ. These were supposed to combine upgrades, diagnostics
and tools to perform a number of experiments from localisation transitions to topological
effects. This apparatus is code-named “Z setup” and is schematically shown in fig. 9.1.
Its components and the experiments they will allow to perform are described in the
following.
1. A novel imaging optical path including a high-magnification (×30) objective moun-
ted on a 5-axis stage to compensate for aberrations and misalignments. This will
allow high-resolution in-situ images of the atomic cloud;
2. A vertical optical lattice: “Z lattice”.
In the experiments described in chapters 7 and 8, the minima of the 2D quasiperiodic
optical lattice are actually 1D tubes, elongated along the vertical direction due to
the weak confinement of the optical trap1. The lack of confinement prevents strong
correlations and thus enables a mean-field treatment and use of the Gross-Pitaevskii
1The trapping frequency of the dipole trap along z for ∼ 0.25 W is about ωz ∼ 120 Hz, much lower
than the on-site trapping frequencies of the lattice minima ∼ 10 kHz.
195
196 Plans for Z experiments
equation. A vertical lattice will break the 1D tubes, thus enhancing correlations
and granting access to strongly interacting physics. This would allow production
of a Mott insulator in regular lattices, and the investigation of Mott physics in the
quasiperiodic ones;
3. A vertical attractive beam: “Z dipole”.
The in-plane blue detuned optical lattice beams result in an overall anticonfining
effect in the xy plane, which will be compensated by a vertical attractive beam thus
guaranteeing the dynamics of the atoms are controlled solely by the quasiperiodic
landscape. This is essential for the real space transport experiments that will be
conducted to probe the localisation transition by recording the expansion of the
atomic cloud in the 2D quasiperiodic lattice, akin to fig. 8.10. The anticonfining
potential and the Z dipole beam are numerically investigated in section 9.2;
4. A DMD will be used to superimpose arbitrary potential landscapes to the 2D qua-
siperiodic lattice, as shown in fig. 9.2. On the one hand, this will allow to include
transient local perturbations to the existing potential and probe their survival in
the long-term dynamics of the spatial density of the cloud, distinguishing thermal
from localised phases. On the other hand, the DMD will enable to create a sharp,
repulsive circular barrier around the quasiperiodic potential, in order to investigate
the topology of the edge states. To reduce the power loss in the generation of the
circular ring, an axicon will be employed to result in a more suitable beam profile.
This is numerically and experimentally studied in section 9.3.
5. To guarantee a large numerical aperture for the incoming DMD beam, the Stern-
Gerlach coils placed around the science cell had to be re-arranged. Their presence
is still essential as a diagnostic tool, and may be employed in future experiments to
provide transient gradients to the atoms, e.g. for quasiperiodic Bloch oscillations.
9.2 Anticonfining potential
In section 2.2.2 it was shown how a travelling wave with Gaussian intensity generates
an attractive or repulsive potential. The same effect is present for standing waves gener-
ated by Gaussian beams, and results in an additional overall confining or anti-confining
potential, for red or blue detuned lattices.
9.2.1 On-site trapping frequencies
A travelling wave E1 =
√
I0 sin kz is retro-reflected to create a standing wave:
Etot = E1 + E2 =
√
I0 sin(kz)−
√
I0 sin(−kz) = 2
√
I0 sin(kz). (9.1)
The intensity I and the potential V experienced by the atoms are then:
I = 4I0 sin2(kz), V = 4αI0︸ ︷︷ ︸
V0
sin2(kz), (9.2)
where α is the atomic polarisability.
The central intensity of a Gaussian beam is:
I0 =
2P
piwhwv
, (9.3)
9.2. Anticonfining potential 197
Figure 9.1 : The plan for the Z setup. This will include a high-magnification objective
to perform high-resolution in-situ imaging, a vertical lattice to enhance correlations, a
vertical attractive beam to compensate the anticonfining effects of the blue detuned optical
lattice beams, and a DMD to project arbitrary potential landscapes onto the atoms.
Figure 9.2 : With a DMD, the atoms (green) will not only be made to experimence
the 2D quasiperiodic potential (grey), but also sharp repulsive walls (yellow) and local
perturbations (red). Figure taken from [177].
198 Plans for Z experiments
and hence the central potential depth is:
V0 = 4αI0 = 4α
2P
piwhwv
, (9.4)
where wv and wh are the vertical and horizontal waists of the Gaussian beam.
In the deep lattice limit, the minima of the optical lattice potential can be approximated as
harmonic oscillators, as was shown in fig. 2.12c. Near each minimum, a Taylor expansion
can be performed:
V (z) = V0 sin2(kz) ≈ V0 k2z2, (9.5)
which can be then treated as a harmonic oscillator:
V0 k
2z2 = 12mω
2
on−sitez
2 ⇒ ωon−site =
√
2V0k2
m
, (9.6)
where ωon−site is the on-site trapping frequency.
In higher dimensions, the on-site trapping frequencies are obtained by:
V (x, y . . . )|min≈ V0 + 12m(ωxx
2 + ωyy2 . . . )
⇒ ωx =
√
1
m
∂2V
∂x2
, ωy =
√
1
m
∂2V
∂y2
, . . .
(9.7)
The ground state energy is then given as:
E0 = V0 +
1
2 h¯ (ωx + ωy + . . . ) , (9.8)
and depends both on the amplitude of the intensity V0 ∝ I0, and on its curvature ω ∝√
∂2I .
9.2.2 1D lattice
For red detuned beams, the polarisability is negative α < 0 and the atoms are attracted
to the maxima of the intensity pattern, which results in two effects:
1. an overall confining effect because of the Gaussian profile of the beam, with a
maximum intensity in the middle and hence the largest potential depth and offset
V0.
2. an overall anti-confining effect due to the decaying intensity profile reducing the
trapping frequency ω ∝
√
∂2I and hence the ground state energy away from the
centre of the beam. This is shown in fig. 9.3.
Both effects can also be characterised by a harmonic oscillator, by re-introducing the
9.2. Anticonfining potential 199
Gaussian envelope of the optical lattice beam V (z) = V0 sin2(kz) e
− 2x2
w20 :
Eg = −V0 + 12 h¯ωon site,
= −V0e
2x2
w20 + 12 h¯
√
V0k2
2m e
− x2
w20
≈ −V0
(
1− 2x
2
w20
)
+ 12 h¯
√
V0k2
2m
(
1− x
2
w20
)
. . .
= −V0 + 12 h¯
√
V0k2
2m +
1
2mω
2
confx
2 − 12mω
2
acx
2
∝ 12mωtotalx
2,
(9.9)
where
ω2total = ω2conf − ω2ac =
4V0
mw20
−
(
h
mw0λ
)√
V0
Erec
. (9.10)
Note that the waist of the anticonfining potential (wac) is a factor of
√
2 larger than that
of the lattice beam itself, wac =
√
2w0.
In a blue detuned optical lattice, the polarisability is positive α > 0 and the atoms are
placed at the intensity minima. The same derivation performed before applies, but the
offset V0 is negligible and only the anticonfining contribution prevails.
(a) (b)
(c)
Figure 9.3 : An optical lattice created by plane waves (a) does not result in additional
potentials. In optical lattices created by Gaussian beams (b), the intensity decrease in
the transverse direction (along the waist w0) reduces the trapping frequency (c) and thus
the ground state energy, resulting in an anticonfining effect.
200 Plans for Z experiments
Fig. 9.4 shows the numerical result for a 1D blue detuned optical lattice created by a
Gaussian beam. The anticonfining potential in this case is also exactly a Gaussian, with
a
√
2 larger waist, because of the separability between the propagation and transverse
direction enabled only in 1D.
Figure 9.4 : Difference in the on-site ground state energies between an optical lattice
created by plane waves and one created by Gaussian beams with waist w0 (black points).
Because of the separability of the problem in 1D, the anticonfining potential is analytically
a Gaussian (solid blue line), with waist wac =
√
2w0. Here, w0/λ = 275 and wac/λ = 389.
9.2.3 2D lattice
Method
2D optical lattices created by Gaussian beams are not separable in their variables, and
therefore require a fully numerical treatment. Blue detuned lattices are assumed in the
following, since they are used in the experiment.
The 2D quasiperiodic optical lattice potential generated by the four standing waves de-
scribed in section 7.1.1 is:
V (r) = V0
[
sin2(kx) + sin2(ky) + sin2
(
k√
2
(x+ y)
)
+ sin2
(
k√
2
(x− y)
)]
, (9.11)
where ideal plane wave were assumed.
With Gaussian beams, this becomes:
V (x, y) = V0 sin2(kx) e
− 2y2
w20 + V0 sin2(ky) e
− 2x2
w20 +
+V0 sin2
(
k√
2
(x+ y)
)
e
− (x−y)2
w20 + V0 sin2
(
k√
2
(x− y)
)
e
− (x+y)2
w20
. (9.12)
In order to approximate each potential minimum as a harmonic oscillator:
V (x, y) = 12mω
2
xx
2 + 12mω
2
yy
2, (9.13)
9.2. Anticonfining potential 201
we define the Hessian matrix H(x, y):
H(x, y) =
(
∂x,xV (x, y) ∂x,yV (x, y)
∂y,xV (x, y) ∂y,yV (x, y)
)
, (9.14)
and diagonalise it to yield: (
mω2x 0
0 mω2y
)
=
(
λ1 0
0 λ2
)
. (9.15)
The on-site ground state energies are then given by:
E0 =
1
2 h¯(ωx + ωy) =
1
2 h¯
√λ1
m
+
√
λ2
m
. (9.16)
The calculation was performed in dimensionless units as outlined in appendix D.10, and
was based on the method developed in [75].
Square lattice
We first consider a regular square lattice created by two orthogonal standing waves, with
a Gaussian intensity profile of waist w0:
V (x, y) = V0 sin2(kx) e
− 2y2
w20 + V0 sin2(ky) e
− 2x2
w20 . (9.17)
The on-site ground state energies are computed both for the potential of eq. 9.17 and
for its ideal, plane wave counterpart, their difference plotted in fig. 9.5. As for the 1D
case, lower energies are available away from the centre of the lattice, which results in an
anticonfining potential. This may be quantified by fitting a 2D Gaussian with waist wac:
Vac(x, y) =
√
V0
(
e−2
x2+y2
w2ac − 1
)
. (9.18)
However, the non-separability of the potential in eq. 9.17 forbids an analytical solution
for the anticonfining potential, rendering 2D Gaussians only an approximation. In fact,
different system sizes result in different anticonfining waists wac, as shown in fig. 9.6a.
This dependence can be reduced and eliminated by less stringent requirements on the
fitting function, for instance using a (truncated) power series:
V ′ac(x, y) =
√
V0
(
−2x
2 + y2
w2ac
+ 12x
2y2
w2ac
)
. (9.19)
This establishes that in 2D, the anticonfining potential is not described by an analytic
function, and can only be approximately compensated for a given system size. As in 1D,
fig. 9.6a shows that the waist of the anticonfining potential wac is larger that the waist
of each Gaussian lattice beam w0.
The potential being anticonfining for blue detuned lattices, the compensation is to be
performed by attractive (red detuned) beams. Experimentally, these could be the same
beams creating the optical crossed-beam dipole trap (section 5.3.3) which lie in the same
202 Plans for Z experiments
plane as the optical lattice. Alternatively, an ad-hoc vertical attractive beam can be
superimposed to the atoms, with waist and optical power optimised to provide the com-
pensation.
In order to quantify the effect of these two solutions, the quantity χ2 is defined as follows:
χ2 = 1
N
∑
j
(
∆Ej
Eidealj
)
, (9.20)
where Eideal is the on-site ground state energy in the lattice generated by plane waves, ∆E
the difference between the Eideal and the energy resulting from Gaussian beam lattices
and compensating potentials, and N the number of lattice sites the sum runs over. The
results for χ2 are plotted as a function of system size in fig. 9.6b, and show that a vertical
attractive beam, such as the Z dipole planned for in fig. 9.1, is the best option.
Quasicrystalline lattice
The on-site ground state energies of the 2D quasiperiodic lattice described by eq. 9.12
are computed, and their deviation from the ideal ones given by plane waves is shown in
fig. 9.7. As opposed to the regular lattices presented thus far, the lattice sites energies
of the quasicrystal are quasi-random due to the non-periodic geometry, so a fit can only
extract the overall trend. The results of the fit are shown in fig. 9.8a, and are performed
in the same way as previously described for the square lattice. In this case, however, the
waist wac of the anticonfining potential is smaller than the waist of each Gaussian lattice
beam w0. The compensation provided by the in-plane crossed dipole trap beams and by
a vertical beam is again quantified by χ2 and shown in fig. 9.8b.
Due to the scattered nature of fig. 9.7, the Gaussian fits quantifying the anticonfining
potential only provide an estimate for the average of all the lattice sites minima. The
fit is then performed on the ground state energies above or below a certain threshold, to
quantify their individual contribution to the anticonfining potential, as shown in fig. 9.9.
9.2. Anticonfining potential 203
Figure 9.5 : Difference between on-site ground state energies for the ideal (plane wave)
and real (Gaussian beams) square lattice. A Gaussian fit is performed in order to extract
the waist wac of the anticonfining potential.
(a)
10 20 30 40 50
system size / λ
10−7
10−5
10−3
10−1
101
χ
2
uncompensated
crossed dipole
Z dipole
(b)
Figure 9.6 : a) The Gaussian fit of fig. 9.5 is performed for different system sizes
2λ× 2λ, resulting in a variation of the extracted waist wac (blue points). This correlation
is explained by a Gaussian not being the analytical solution, and can be reduced by
performing the fit with a simple (truncated) power series (black points). b) The quantity
χ2 quantifies the deviation of the on-site ground state energies from their ideal value in
a lattice generated by plane waves. Both the in-plane crossed dipole trap beams and a
vertical beam reduce the anticonfining potential in the optical lattice, the latter providing
a better performance.
204 Plans for Z experiments
Figure 9.7 : Difference between on-site ground state energies for the ideal (plane wave)
and real (Gaussian beams) quasicrystalline lattice. A Gaussian fit is performed in order
to extract the waist wac of the anticonfining potential.
(a)
20 40 60 80 100
system size / λ
10−3
10−2
χ
2
uncompensated
crossed dipole
Z dipole
(b)
Figure 9.8 : a) The Gaussian fit of fig. 9.7 is performed for different system sizes 2λ×2λ,
resulting in a variation of the waist wac (blue points). This correlation is explained by
a Gaussian not being the analytical solution, and can be reduced by performing the fit
with a simple (truncated) power series (black points). b) The quantity χ2 quantifies the
deviation of the on-site ground state energies from their ideal value in a lattice generated
by plane waves. Both the in-plane crossed dipole trap beams and a vertical beam reduce
the anticonfining potential in the optical lattice, the latter providing a better performance.
9.2. Anticonfining potential 205
Figure 9.9 : The Gaussian fit of fig. 9.7 is performed for ground state energies above
(below) a certain threshold, given by the x coordinates, to quantify their contribution
to the overall anticonfining potential. This seems to suggest that the lowest energies
contribute the most to the anticonfining effect.
206 Plans for Z experiments
9.3 Axicon
The DMD will allow projection of a circular ring of blue detuned light, to provide sharp
repulsive boundaries to the quasiperiodic potential. Creation of a ring from an incident
Gaussian beam, however, requires the diversion of most of the incident power and hence
a dramatically low efficiency. To guarantee a higher efficiency, it would be beneficial
to illuminate the DMD with an intensity profile that already approximates a circular
ring. This is provided by the diffraction pattern of a system comprising an axicon and a
converging lens, shown in fig. 9.10 and investigated in the following section.
Figure 9.10 : The optical apparatus comprising an axicon (conical angle α) and a
converging lens (focal length f0) can be used to produce a circular ring as their diffraction
pattern (dashed line). The near-diffraction of the axicon itself results in a Bessel beam,
discussed later.
9.3.1 Mathematical derivations
The mathematical derivation reported here builds on the work presented in [226].
Incident Gaussian beam
Under the Helmholtz and paraxial approximations, a solution to Maxwell’s equations in
vacuo is a Gaussian beam propagating in z of the form:
Ein(ρ, z) =
w0
w(z) exp
[
− ρ
2
w2(z)
]
exp
[
i
(
kz − θ(z) + kρ
2
2Rc(z)
)]
. (9.21)
The waist w, radius of curvature Rc, and Gouy shift θ are given by:
w2(z) = w20
[
1 +
(
z
zR
)2]
,
Rc(z) = z
[
1 +
(
zR
z
)2]
,
θ(z) = arctan
(
z
zR
)
,
(9.22)
where zR is the Rayleigh range of the beam.
The above will be used as the incident beam onto the axicon in fig. 9.10.
9.3. Axicon 207
Thin lens and axicon
Within the thin lens approximation, the interaction between incident radiation and an
optical element only amounts to a phase correction to the wavefront, caused by the re-
fractive index affecting the propagation of the wave. Further interaction between the
dielectric material and the incoming electric field is neglected.
(a) Converging lens (b) Axicon
Figure 9.11 : The refractive index n of the optical element affects the propagation of an
incident EM wave, resulting in a phase correction dependent on the width of the material.
P (x) is the pupil function, limiting the (−∞,∞) integrations (discussed later) only to
the area covered by the aperture (the optical component).
An optical component with index of refraction n and central thickness ∆0 causes a ray
propagating through it to acquire a phase factor φ(x), depending on its transverse position
x, as shown in fig. 9.11:
φ(x) = kn∆(x) + k[∆0 −∆(x)]. (9.23)
For the converging lens in fig. 9.11a, the path length depends on the radii of curvature R
of the front (R1) and back (R2) surfaces:
∆(x) = ∆1 + ∆2 = ∆0 −R1
(
1−
√
1− x
2
R21
)
+R2
(
1−
√
1− x
2
R22
)
. (9.24)
Using the paraxial approximation such that x R and
√
1− x
2
R2
∼ 1− x22R2 , this can be
expressed as:
∆(x) = ∆0 − 12
( 1
R1
− 1
R2
)
, (9.25)
where the focal length f0 is defined as
1
f0
= (n− 1)
( 1
R1
− 1
R2
)
, (9.26)
leading to the a total phase factor:
φ(x) = kn∆(x) + k[∆0 −∆(x)] = kn∆0 − k2f0x
2. (9.27)
208 Plans for Z experiments
The non-trivial (only spatially dependent) correction to the wavefront of the propagating
wave by a thin lens is therefore:
τL(ρ) = e−i
k
2f0
ρ2
, (9.28)
while a similar calculation for the axicon in fig. 9.11b results in:
τA(ρ) = e−ibρ, with b =
2pi(n− 1)
λ
tanα, (9.29)
where α is the conical angle of the axicon.
Diffraction
The lens and the axicon are also finite apertures which will cause diffraction of the beam,
irregardless of the refractive index.
The resulting electric field distribution emerging from an aperture such as in fig. 9.12 is
given by the Fresnel-Kirchhoff diffraction formula [227]:
E(x, y, z) = −i
λ
∫ ∫
aperture
Ein(x′, y′, 0)
eik·R
R
(
1 + cos(R 6 zˆ)
2
)
dx′ dy′. (9.30)
Figure 9.12 : The electric field resulting from the diffraction through an aperture is
controlled by the incident radiation and by the distance to the screen. It is given by the
Fresnel-Kirchhoff diffraction formula in eq. 9.30.
Using the paraxial approximation, cos ∼ 1 and 1
R
∼ 1
z
, and focussing on the far-field
pattern where z2 (x− x′)2 + (y − y′)2, results in:
R = z
√
1 + (x− x
′)2 + (y − y′)2
z2
∼ z
(
1 + (x− x
′)2 + (y − y′)2)
2z2
)
, (9.31)
and hence:
E(x, y, z) ≈ −ie
ikzei k2z (x2+y2)
λz
∫ ∫
aperture
Ein(x′, y′, 0) ei
k
2z (x
′2+y′2)e−i kz (xx′+yy′)dx′ dy′. (9.32)
9.3. Axicon 209
In a system with cylindrical symmetry, where E(x′, y′, z = 0)→ E(ρ′, z = 0), x = ρ cosφ
and y = ρ sinφ, the integral can be written as:
E(ρ, z) = −ie
ikzei k2z ρ2
λz
∫
dφ′
∫
aperture
dρ′ρ′Ein(ρ′, 0) ei
k
2z ρ
′2e−i kz (ρρ′ cosφ cosφ′+ρρ′ sinφ sinφ′),
E(ρ, z) = −ie
ikzei k2z ρ2
λz
∫
dφ′
∫
aperture
dρ′ρ′Ein(ρ′, 0) ei
k
2z ρ
′2e−i kz (ρρ′ cos(φ−φ′).
(9.33)
The following identity involving the 0th Bessel function of the first kind J0:∫ 2pi
0
e−i kz (ρρ′ cos(φ−φ′)dφ′ = 2piJ0
(
k ρρ′
z
)
(9.34)
can be used to provide in the final expression for the far-field diffraction pattern in cyl-
indrical coordinates:
E(ρ, z) = 2pie
ikzei k2z ρ2
iλz
∫
aperture
dρ′ρ′Ein(ρ′, 0) ei
k
2z ρ
′2
J0
( 2pi
λz′
ρ′ρ
)
, (9.35)
where the integral is known as the Hankel transform of the incident field Ein(ρ′, 0).
In the apparatus shown in fig. 9.11, the resulting field distribution is given by the diffrac-
tion through both the lens and the axicon, weighted by the wavefront corrections they
impose on the propagating radiation:
E(ρ, z) = 2pie
ikz
iλz e
ipiρ2
λz
∫
aperture
τA(ρ′) τL(ρ′)Ein(ρ′, 0) e
ik
2z′ ρ
′2
J0
( 2pi
λz′
ρ′ρ
)
ρ′dρ′. (9.36)
For simplicity, the axicon and the lens are assumed to be at the same position, thereby
only requiring one integration over the element with the smallest diameter. All pure phase
factors in the integral can be discarded as the measured intensity only depends on the
squared modulus of the electric field.
9.3.2 Results
The near-field diffraction pattern of the axicon is known to be a Bessel beam, as shown in
fig. 9.13. This comprises a narrow region of high intensity which is exploited, for instance,
in laser drilling applications.
The far-field diffraction pattern of the axicon-lens compound of fig. 9.10 is shown in fig.
9.14. The incident Gaussian beam is transformed into a circular ring, which could then be
shone onto the DMD in order to provide a sharp boundary with far less incident optical
power. The simulated intensity pattern from eq. 9.36 is shown in fig. 9.14a, and the
recorded pattern in fig. 9.14b. The discrepancy between theory and experiment may be
explained by the finite distance between the axicon and the lens, which were assumed to be
at the same position when deriving eq. 9.36. This could easily be computed numerically
by allowing a region of free propagation of the beam between the two optical components.
Irregardless of these discrepancies, the measurement shows the intensity of the ring is
enhanced by a factor of ∼ 5 over the Gaussian amplitude. This confirms that the axicon
can be used in order for the DMD to generate a circular ring trap with far less optical
powers than if incident with a Gaussian beam.
210 Plans for Z experiments
(a) (b)
Figure 9.13 : The near-field diffraction pattern of an axicon results in a Bessel beam
with intensity I ∝ J0(r)2. A numerical simulation is shown in (a), while the measured
intensity pattern is recorded in (b).
Further numerical simulations are shown in figs. 9.16, in order to gain a qualitative
understanding on the effects of different parameters on the ring-shaped intensity pattern.
These were performed for an axicon of α = 2◦ and n = 1.5, a lens with f0 = 125 mm, an
incident Gaussian beam with single-waist w0 = 1.85 mm and wavelength λ = 532 nm.
(a) (b)
Figure 9.14 : A Gaussian beam (a) propagates through the axicon-lens compound shown
in fig. 9.11 and results in a ring-shaped intensity pattern (b).
9.3. Axicon 211
(a) theory (b) experiment
Figure 9.15 : The simulated intensity profile of the axicon-lens apparatus (a) is in
qualitative agreement with the recorded intensity pattern (b), for an axicon of α = 2◦
and n = 1.5, a lens with f0 = 125 mm, an incident Gaussian beam with single-waist
w0 = 1.85 mm and wavelength λ = 767 nm.
212 Plans for Z experiments
100 200 300 400
Focal length /mm
2
4
6
8
Ri
ng
ra
di
us
in
fo
ca
lp
la
ne
/m
m ring radius
0.025
0.050
0.075
0.100
0.125
0.150
FW
HM
/m
m
FWHM
(a)
1 2 3 4 5
Axicon angles /degrees
1
2
3
4
5
Ri
ng
ra
di
us
in
fo
ca
lp
la
ne
/m
m ring radius
0.01
0.02
0.03
0.04
0.05
FW
HM
/m
m
FWHM
(b)
1 2 3 4 5
Incident beam waist /mm
2.05
2.10
2.15
2.20
2.25
2.30
R
in
g
ra
d
iu
s
in
fo
ca
l
p
la
n
e
/m
m ring radius
0.02
0.03
0.04
0.05
F
W
H
M
/m
m
FWHM
(c)
Figure 9.16 : Further numerical simulations using eq. 9.36 were performed in order
to quantify the dependence of ring diameter and width on the lens’ focal length (a), the
axicon’s conical angle (b) and the incident beam waist (c).
Quaerite et invenietis; Seek and you will find.
Matthew 7:7
Conclusions and Outlook
In the 1980s, quasicrystals emerged as a class of materials that stands in an interesting
middle ground between order and disorder. While they lack translational invariance
and periodicity, they possess long-range order and display crystallographically forbidden
rotational symmetries. This distinguishes them from both crystalline and amorphous
solids, and sparked great interest in metallurgy and condensed matter physics. The
intrigue in quasiperiodic structures quickly spread to other fields of science, from search
algorithms and the Riemann hypothesis, to fractals and tilings. Computational models
are limited by the numerical intractability of large aperiodic systems, while the artificial
methods to synthesise real quasicrystals introduce unwanted impurities and defects.
Quantum simulation of quasicrystals with cold atoms in optical lattice provides a solu-
tion to all the limitations encountered thus far. It combines the flexible, versatile and
defect-free potential landscape resulting from the interference of light, with the highly
controllable parameters offered by cold atoms. The experiments presented in this thesis
are the first experimental realisation of degenerate quantum gases in a 2D, non-separable
quasiperiodic potential.
Short lattice pulses where first employed to study the synthetic lattice created in mo-
mentum space by the Kapitza-Dirac scattering of matter-waves from the optical quasicrys-
tal. This unveiled a singularly continuous diffraction spectrum, composed of a dense set of
sharp Bragg peaks displaying fractal-like self-similarity upon rescaling by the silver mean
λ = 1 +
√
2 . The measured diffraction spectra are compatible with projections from
higher dimensional regular crystals, which establishes the presence of long-range order
in the structure and enables ballistic transport on the synthetic lattice for short pulse
durations.
Longer, triangular lattice ramps were used to adiabatically load the BEC into the ground
state of the quasiperiodic lattice in order to probe its properties. The adiabatic timescale
was found to be finite for regular lattices, irregardless of their geometry and strength, but
was observed to diverge for quasiperiodic lattices above a critical lattice strength. This is
compatible with multiple distinct numerical simulations predicting a phase transition to
a disorder-induced localised state at a critical lattice depth Vloc ≈ 1.78Erec. The localised
phase of matter seems to be resilient against moderate interactions, which would make
this the first experimental realisation of a 2D Bose glass.
These first two experiments showcased the presence of both ordered and disordered fea-
tures in our system, and pave the way for many more future investigations.
The short term plans have already been briefly outlined in section 9.1. An in-situ imaging
apparatus is going to probe the real space transport properties of the quasiperiodic optical
lattice. A DMD will also result in the addition of sharp walls to probe topological edge
states, and the imprinting of local perturbations in the initial atomic density to further
quantify the onset of localisation. In addition, a vertical optical lattice will grant access to
213
214 Conclusions and Outlook
strongly interacting physics, enabling investigation of the Bose glass and Mott insulator
phases, and of their phase diagrams. The 2D disordered Bose-Hubbard model is predicted
to exhibit a re-entrant phase [228], so far only experimentally reproduced in a 1D Bose
glass [29].
A number of experiments for the long-term run of the machine are proposed as follows.
• The synthetic lattices in momentum space of chapter 7 may be revisited and probed
with pulsed sequences. These have been predicted [187] to enhance the effects of
Fourier broadening and therefore extend the linear region in fig. 7.19.
• The recovery curves of chapter 8 could be investigated further, theoretically and ex-
perimentally, in the presence of interactions in order to map the V0-a phase diagram
of the (potential) 2D Bose glass.
• A constant force may be exerted on the atoms in the optical lattice, which would
result in quasiperiodic Bloch oscillations.
• The retro-reflected mirrors could be controlled by a piezo-electric transducer in order
to enable dynamical control of the phase of each lattice beam. This can be used
to drive phasonic strain and shake the lattice to perform Hamiltonian engineering,
study Floquet physics and topological pumps such as in [229].
• The optics around the science cell is placed on a breadboard. This can be removed
and replaced with one enabling generation a fivefold optical lattice, to experiment-
ally prove the proposals in [230].
• The machine enables experiments on three distinct species, 87Rb, 39K and 40K, along
with Bose-Bose (87Rb-39K) and Bose-Fermi (40K-87Rb) mixtures. Mixtures would
allow experiments in self-generated disorder, where the lattice is deeper for one
species than for the other. They could allow experimental realisations of proposals
such as [46], where bosonic gauge fields are coupled to cold fermionic matter. In
addition, the 2D optical lattice proposed in [231] was shown to correspond to a
non-Abelian gauge theory, namely SU(2). Could the 4D (inherited) character of
our quasiperiodic optical lattice be a gateway to new non-Abelian gauge theories
such as SU(4)1 ?
1This is particularly interesting in the context of AdS/CFT correspondance. There is a duality between
Type IIB string theory on an AdS5 × S5 space and an N = 4 (super) Yang–Mills theory on the 4D
boundary of AdS5, as first discussed in [232–234]. This is just to emphasise the potentially interesting
physics that can only be investigated with our experiment, beyond MBL and other disorder-related
topics that have already been studied on other platforms. Together with the connections to cosmology
and high-energy physics discussed in the introduction, this shows how the field of cold atoms may rise
to be a cross-disciplinary, multi-purpose, cheap and compact experimental platform, which might one
day replace large and expensive structures such as telescopes, particle accelerators and (some) laser
interferometers.
Everybody brings joy to this office, some when they enter, most when they leave.
Ominous sign in the office of Gary Large, mechanical workshop manager
Acknowledgements
I finished a theoretical physics degree in 2015 from Imperial College London, with a final
year project on data analysis from the LHCb experiment at CERN1, which left me with
no hardware skills, no experimental intuition and no knowledge of what it really means
to carry out scientific research (best exemplified in the picture below).
Scientific research in films and outreach talks (left) vs day-to-day lab work (right).
Story of my life.
Multiple people have been directly or indirectly responsible for this masterpiece in prose,
and for the environment (physical and social) that enabled the research presented herein.
Convention dictates I now have to thank these people, or at least my favourite ones. This is
not a work of fiction. Names, characters, businesses, places, events, locales, and incidents
are neither the products of the author’s imagination nor used in a fictitious manner. Any
resemblance to actual persons, living or dead, or actual events is not coincidental.
I thank my supervisor Dr. Ulrich Schneider2 for recruiting an inexperienced theoretical
physicist in an initially (October 2015) fully German team3: Oliver Brix, Konrad Viebahn,
and Hendrik von Raven. Funnily enough now, at the end of my PhD, I am the most
German in #TeamQuasicrystal thanks to my maternal grandmother (...from germany to
gerfew?). Thank you Ulrich, for giving me the chance of building a lab and learning
so much. I appreciated, especially in the later years of my PhD, useful and insightful
discussions about experiments and theory, whence I greatly benefitted from your vast,
deep and intuitive understanding of the field.
(Dr.) Konrad Gilbert Heinrich Viebahn4, like all (future) PIs working on cold atoms in
optical lattices, came from the Bloch group at LMU Mu¨nchen, where he had already
done plenty of lasers, optics and theory - which made one of us, at least. Luckily for me,
1together with my valued undergraduate lab partner Hlynur Sveinbjo¨rnsson.
2Memorable quote: “Glimmer!”.
3Maybe he had not realised I was from the wrong side of the Su¨dtirol border.
4Memorable quote: “You’re not helping”.
215
216 Acknowledgments
he was not just a skilled source of knowledge and experience, but a respected leader, a
great mentor, and a clever guy, and I will forever be in his debt for the patient, diligent,
tenacious and never arrogant way he trained me. Luckily for both us, we shared the
same mentality and views on work ethics and lab routines: from the search for perfection,
elegance, simplicity and minimalism, to the efficient and effective use of time. Our PhD
was not done 9 to 5, Monday to Friday: I am most grateful for his presence and company
on the innumerable week-ends and late nights spent in the lab, and the constant exchange
of ideas via texts, emails, and audio messages. There are too many anecdotes to recite,
which I shall leave to my biographer and just mention here that one could tell who between
us left the lab last from the way the spanners had been put back in the drawer5. And
indeed his presence is something that I missed after he left: it was a funny feeling not
seeing him behind my computer screen, his presence reduced to a mere voice on the phone,
text, or email. American poet Maya Angelou said: “...people will forget what you said,
people will forget what you did, but people will never forget how you made them feel”.
And indeed, I will never forget.
Whilst I was not a fan of him never being hungry for lunch on time, having eaten a full
English breakfast beforehand, I am most grateful to the many formal dinners, breakfasts,
and feasts I got to enjoy at Trinity College6, and all the times Konrad took me out to
dinner and talked to me in my depressed moments. I learnt from the best, and I look
forward to a lifetime of friendship with my cherished colleague.
I also thank the rest of the quasi-scientists that joined #TeamQuasicrystal from our
second year onwards: Edward Carter, Jr-Chiun Yu and Shaurya Bhave - especially Ed
for his role of walking English dictionary and thesaurus, being the first and only fully
British member of the team. Thanks for helping running the lab, eventually taking over
and allowing us to remain in the office being useless doing theory and data analysis. I
also thank the Master’s students that joined our quest during my time here, first and
foremost Hendrik von Raven. Leaving aside the fact that for months he had a PCB
with bare 230 V worryingly within my arms’ reach, he installed, developed and served as
tech support for QControl, monitoring software and DDS. I congratulate Hendrik, along
with Konrad, for their (continued) patience dealing with my catastrophic ineptitude with
computing, software and coding. Oliver Brix was in charge of the magnetic transport and
experiment coils, and had the herculean task of designing and oversee the manufacturing
of the cooling block. Michael Ho¨se finalised the design of our reliable and cheap shutters
and worked with Konrad on the image analysis software. Max Melchner von Didiowa
designed and assembled the dipole trap, and I am especially grateful for his continued
help with Mathematica, and his Oreos.
Max enjoyed Cambridge so much that he joined the newly created (Summer 2017) Kagome
group, #TeamKagome, which soon after also recruited Daniel Reed, Dr. Tiffany Harte,
and Florian Sto¨rtz as a one-year Master’s student. Apologies (but not sorry) for my
occasional rants about borrowing and not returning components, for not replenishing stock
and for not maintaining tidiness and order in the quasicrystal lab. Let me also mention
the temporary, Summer and Part III students who worked with us during my time here:
Gregory Chant, Robin Croft, Elvinas Ribinskas, Luca Donini, Joannis Koepsell, Callum
Stevens, Attila Szabo´, Shreya Arya, Marius Weber, Pe´ter Juha´sz, Jonathan Mortlock,
and Jared Jeyaretnam.
5Konrad aligning them one by one next to each other, which I found to be an utter and complete
waste of time.
6Along with a roofs tour of questionable legality.
Acknowledgments 217
I am grateful to Max, Dan and Emily (Dan’s fiance´e) for our fortnightly breakfast at
Wetherspoons, for random meals here and there, and for our escapades using Dan’s car
around Cambridge and beyond (Heathrow airport). Max and I shared a tent both in
Snowdonia and on the top of Mt Kilimanjaro, during which I can barely imagine Max’s
pain in having to deal with the loquacious and talkative fellow that I am. Dan’s epic
improvement at table tennis is literally the most fascinating thing I have witnessed during
my PhD.
Special thanks to the Hadzibabic-Smith7 group at Cambridge, for constant collaboration,
lending of components, technical advice, and also social activities, sports, occasional office
meals and barbecues. I thank Zoran for his advice on Serbian tourist spots, careers and
physics questions, and I apologise for the time I stole from his PostDocs due to the lack of
such in my group. Indeed, I cannot thank enough the BEC I, II and III PostDocs that I
have bothered, hunted down, shadowed, interrupted, stalked and tormented on innumer-
able occasions: Dr. Raphael Lopes, Dr. Nir Navon8, and Dr. Julian Schmitt. Raphael
and Nir were there from day 1, and they spared neither mercy nor political correctness
in their opinions of my late-night questions on theory, experiment and software – I was
forged in fire. Thanks also to Dr. Richard Fletcher, Dr. Alexander Gaunt (especially for
the on-going collaboration, plots and support for chapter 8), Dr. Timon Hilker and Dr.
Jinyi Zhang, and the PhD/Master’s students Christoph Eigen, Jake Glidden, Jay Man,
Panos Christodoulou, Maciej Galka, Lena Bartha, Milan Krstajic, Maximilian Sohmen,
and Nathaniel Vilas.
I also thank other groups at the Cavendish for lending equipment and providing guidance,
such as Mete Atatu¨re’s group (especially Alejandro Rodr´ıguez-Pardo Montblanch, Lucio
Stefan and Dr. Matteo Barbone), experimental Astrophysics, SMF (especially Dr. David
Ward, Dr. Nadav Avidor and Fulden Eratam), TFM (especially Dr. Adrian Ionescu) and
SP (especially Dr. Joanna Waldie), along with Stephen Spurrier and Attila Szabo´ from
TCM for useful discussions.
Cambridge would not be able to maintain its worldwide academic reputation were it not
for its technical and administrative staff9. Peter Norman, and the maintenance team
headed by Alan Turner, whom I thank for providing assistance and intervention in tasks
outside their jurisdiction (e.g. bringing back the air-conditioning on more than one oc-
casion). The mechanical workshop, headed by Gary Large and before him Chris Sum-
merfield10, and the many times I liaised with Tom, Ollie, Matt, Jim, Kevin, Ashley and
Scott. The Cryogenics Facility Manager Dan Cross, the Electronics workshop and the IT
team, especially Alex Crook and Andrew Alderwick for the many questions they had the
patience to answer. The Rutherford hub but before them Pam Smith for her outstand-
ing efficiency in administrating AMOP. Marzena and the cleaning crew, stores, and the
kitchen staff, whose Thursdays’ curry will always have a place in my heart.
A special thank you goes to the AMOP’s electronic engineer Dr. Stephen Topliss, for
building, testing and closely working with students on several projects, especially the H-
bridge, the MOSFET boxes, and the DDS.
I also thank people from external companies who provided continued support throughout
my PhD, Jonathan Ramwell and Dariusz Swierad (Toptica), Ju¨rgen Borchers (Lens Op-
7Robert Smith moved to Oxford as an Associate Professor in the Summer of 2018
8Memorable quotes “It’s crazy”, “Spin-pol is a no-brainer”.
9Even though I may have unsubscribed from all HR-related mailing lists.
10Memorable quote: “Come back at 4”.
218 Acknowledgments
tics), and Radu and Ben from Thorlabs whom I called with the weirdest questions about
their parts and from whom I appreciate the honesty, sometimes against their commercial
interest.
I thank my college, Churchill, mainly for offering a free meal every once in a while. My
involvement in college was only through Dr. Lisa Jardine-Wright, IA & IB Physics DoS,
who gave me the opportunity to teach first year physics students. I particularly enjoyed
teaching and I thank Lisa and all of my students for the good questions that made me
grow personally and professionally.
Before starting my PhD, I spent two weeks in the Fermi-I group at LMU Mu¨nchen, under
the supervision of Henrik Lu¨schen and Pranjal Bordia whom I thank for their patience,
and for Pranjal’s advice not to do a PhD, which I sometimes wished I had followed. I
also thank the Fermi-II crew, i.e. the machine our lab is based on, for enduring my (our)
calls, emails and personal visits: Lucia Duca, Tracy Li, Martin Reitter and Jacob Na¨ger.
Finally, let me thank my housemates Jasleen11, Brian, Nick, Priscilla, Sho and Hannah,
along with all other friends and treasured acquaintances with whom I shared film nights,
lab breakfasts, formal dinners, travels, sports, and BBQs. Special mention to Alison Tully,
whom I met in my first year tutorial group at Imperial College, for our muffin tradition,
bouldering sessions, and for an exclusive CERN tour. I am also grateful to the Astronomy
boardgames crew who constituted the bulk of my social activities, the usual suspects
being Jasleen, Nick, Adam, Scottish Douglas, Shorts Douglas, Laura, Lukas, Rob, Sid,
and Sophie. The Madingley road Kebab van, for the occasional free drinks and desserts,
and for memorising my order. The Internet community of StackOverflow and Physics
StackExchange12. My one true love Aldi, and my newly found religion Mathematica. The
Cold Atom Laboratory group at JPL for making me press the button to make a BEC
on the International Space Station, which was quite cool. I am also grateful to EPSRC
for providing a stipend and the academic enrichment grant. Thanks to J.K. Rowling
for writing the Harry Potter series, which prompted me to learn English and led to me
enrolling at Imperial College. I tried to find suitable Harry Potter quotes to include in
the chapters, but unfortunately Hogwarts did not offer curricular activities on cold atoms,
quantum gases or quasicrystals. So I am going for “It is our choices, Harry, that show
what we truly are, far more than our abilities” [235], such as the choice of devoting four
years of my life to an underpaid PhD.
Last but not least, I thank my Mum Gabriella who, despite challenges of her own, always
found the time to listen, console, encourage, and comfort me.
11Memorable quotes: “Wonky”, “Innit bruv”, “Allow it”, “Bo’le of wa’er”.
12Few people know my username and please do keep it to yourself otherwise Imperial might ask for the
degree back, after seeing the sort of questions I asked...
Acknowledgments 219
(a) cold
(b) ultracold
Boson 1 and boson 2 (note the label swapping) at different cooling stages.
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Appendices
237
A
Hyperfine structure
A.1 Hamiltonian
The quantum numbers L, S, J, I and F are introduced in section 2.1.1.
The full Hamiltonian for an electron bound to an atom by the Coulomb interaction V (r)
and subject to an external electromagnetic field Aµ = (φ,A) is:
H = (p + qA)
2
2m − qφ+ V (r) +Hfine +Hhyp +HLamb, (A.1)
where Hfine ∝ L · S ∝ J2 is the fine structure and Hhyp ∝ I · J ∝ F2 is the hyperfine
structure. HLamb is the Lamb shift due to QED corrections and is ignored here. The
energy scales of these contributions are summarised in table A.1.
contribution energy scale
gross structure ∼ α2mc2
fine structure ∼ α4mc2
hyperfine structure ∼ α4m/mp
Lamb shift ∼ α5 ln(α)mc2
Table A.1 : Energy scales, m is the mass of the electron, mp of the proton, α the
fine-structure constant that sets the scale.
The total electronic J and atomic F angular momenta are defined as:
J = L + S, F = J + I. (A.2)
For a uniform magnetic field B in the Coulomb gauge ∇·A = 0, we can write A = 12B×r,
hence A · p = 12B · L and A2 = 12B2(x2 + y2).
239
240 Hyperfine structure
The Hamiltonian can therefore be expressed as:
H = p
2
2m + V (r) +Hfine +Hhyp −
q
2m(L ·B)︸ ︷︷ ︸
Linear Zeeman effect
+ q
2B2
8m (x
2 + y2)︸ ︷︷ ︸
Quadratic Zeeman effect
. (A.3)
At zero external field, the Coulomb and the fine structure contributions are dominant and
provide the basis to be used in the perturbative treatment of the hyperfine component
Hhyp.
The hyperfine structure takes into account all the aspects of the interaction between
the electron and nucleus that go beyond the simple point-charge treatment contained in
the gross structure V0 ∝ qeQnucleus. Analogously to eq. 2.7 , the full expression for the
Hamiltonian Hhyp can be broken into electric and magnetic multipole contributions:
Hhyp = AhfsI · J︸ ︷︷ ︸
magnetic dipole
+Bhfs
3(I · J)2 + 32(I · J)− I2J2
2I(2I − 1)J(2J − 1))︸ ︷︷ ︸
electric quadrupole
+ Chfs︸︷︷︸
magnetic octupole
+ . . . (A.4)
The constants are experimentally determined and are reported in table A.2, while the
vectors I and J are dimensionless and constructed as detailed below. The absence of
some terms in the multipole expansion is discussed later.
Construction of angular momentum matrices
For a generic angular momentum with associated quantum numbers s,ms, the (2s+ 1) ·
(2s+ 1) matrices are constructed in the |ms〉 basis according to:
sx =
1
2 (s+ + s−) , sy =
1
2i (s+ − s−) , sz = diag(s), (A.5)
with the following rules:
〈m′s|s+|ms〉 = δm′s,ms+1
√
s(s+ 1)−ms(ms + 1) ,
〈m′s|s−|ms〉 = δm′s,ms+1
√
s(s+ 1)−ms(ms + 1) ,
〈m′s|sz|ms〉 = δm′s,msms.
(A.6)
s+ and s− are the raising and lowering ladder operators.
For the atomic species used in this thesis, 87Rb and 39K, the matrices for the states
involved in the D2 transition were constructed in the |mI ,mJ〉 basis, where I = 3/2 and
J = 3/2:
Ix = sx(I)⊗ 1(J) = sx(3/2)⊗ 1(3/2)
. . .
I = (Ix, Iy, Iz).
(A.7)
Similarly for J, so as to define the total angular momentum F as:
F = I⊗ 1J + 1I ⊗ J. (A.8)
One could also employ the |mS,mL,mI〉 basis, which would result in both D1 and D2
states because of the different J = |L − S| ... L + S, though an appropriate L · S term
A.2. Higher multipole moments 241
needs to be added to the Hamiltonian in order to obtain the correct fine structure energy
splitting.
order term symbol 87Rb 39K
Magnetic dipole
2S1/2 Ahfs h · 3.417 GHz h · 230.86 MHz
2P1/2 Ahfs h · 408.328 MHz h · 27.77 MHz
2P3/2 Ahfs h · 84.72 MHz h · 6.093 MHz
Electric quadrupole 2P3/2 Bhfs h · 12.496 MHz h · 2.786 MHz
Table A.2 : Experimentally measured pre-factors for the multipole expansion of eq. A.4,
from [167, 236].
A.2 Higher multipole moments
The gross structure considers the nucleus and the electron as point charges, which is es-
sentially only taking the monopole moment in the multipole expansion of the electrostatic
interaction.
The full electrostatic Hamiltonian H is:
H = 14pi0
∫
re
∫
rn
ρe(re)ρn(rn)
|re − rn| d
3re d3rn, (A.9)
where r is the distance from the origin for both the electron (e) and the nucleus (n).
Assuming re > rn, one can perform the usual multipole expansion:
1
|re − rn| =
1√
r2e − 2re · rn + r2n
=
1
re
P0(cos θen) +
rn
r2e
P1(cos θen) +
r2n
r3e
P2(cos θen) + · · ·+ r
`
n
r`+1e
P`(cos θen),
(A.10)
where the Legendre polynomial is given by:
P`(cos θen) =
4pi
2`+ 1
∑`
m=−`
(−1)mY −m` (θn, φn)Y m` (θe, φe) (A.11)
with Y m` (θ, φ) = 〈θ, φ|`,m〉 being the spherical harmonics. These satisfy (Y m` )∗ (θ, φ) =
(−1)m Y −m` (θ, φ), L2Y m` = L2Y m` = `(` + 1)Y m` and LzY m` = m`Y m` . In general, L2, Lz
and ` can be any angular momentum quantum number among L, S, J, I and F .
The Hamiltonian can be therefore expressed as:
H =
∑
`
H` =
∑
`
1
4pi0
∫
re
∫
rn
ρe(re)ρn(rn)
re
(
rn
re
)`
P`(cos θen) d3re d3rn. (A.12)
With P0 = 1, P1 = x and P2 = 1/2 (3x2 − 1), this becomes:
H ∝
∫
ρ(rn) ρ(re)
1re︸︷︷︸
monopole
− rˆ · rn
r2e︸ ︷︷ ︸
dipole
+ 12r3e
(3(rˆ · rn)− r2n)︸ ︷︷ ︸
quadrupole
+ . . .
. (A.13)
242 Hyperfine structure
In general, the 2`-pole correction to the Hamiltonian is given by:
H` = Q` · (∇E)` =
∑`
m=−`
(−1)mQm` (∇E)m` , (A.14)
where we have separated the nuclear and electronic coordinates intoQ and∇E, as outlined
in [237]:
Qm` =
( 4pi
2`+ 1
)1/2 ∫
rn
ρ(rn)r`nY −m` (θn, φn)d3rn,
(∇E)m` =
1
4pi0
( 4pi
2`+ 1
)1/2 ∫
re
ρ(re)r−(`+1)e Y m` (θe, φe)d3re.
(A.15)
It is worthy of notice that ∇E = ∇ ⊗ E = ∂2V/(∂xi∂xj) quantifies the interaction
dependence on the electric field gradient rather than on its strength (captured by the
electric dipole term).
The ` = 0 term is the monopole term included in the gross strcture of the Hamiltonian.
Neutral atoms do not have a permanent electric dipole moment, and therefore display an
inversion symmetry re → −re which results in the contribution of odd powers r` to vanish.
The leading correction is therefore the electric quadrupole moment (` = 2), detailed below.
Note that these restrictions only apply to the diagonal corrections, i.e. energy shifts, of the
Hamiltonian, but not to off-diagonal elements corresponding to transitions: for example,
electric octupole transitions have been measured in trapped ions [238].
The above treatment only considered the electrostatic interaction. The magnetic interac-
tion can also be divided into a multipole expansion, though each therm possesses opposite
parity than its electric counterpart. Hence, the ` = 1 contribution is non-zero and is the
magnetic dipole term resulting in the I ·J interaction of eq. A.4. The next magnetic term
would be the magnetic octupole ` = 3 contribution, which has not yet been measured for
87Rb and 39K. There is no ` = 0 magnetic contribution as magnetic monopoles have not
yet been discovered at the time of writing this thesis.
Electric quadrupole
The following provides a derivation for the electric quadrupole contribution to eq. A.4.
Noting that re · rn = rern cos θen can be expressed as ∑i xnixei, the algebra detailed in
[239] shows that the electric quadrupole ` = 2 term can be cast in tensor notation:
H`=2 =
1
4pi0
∫
re
∫
rn
r2nρn(rn)
1
2(3 cos
2 θen − 1)ρe(re)
r3e
d3re d3rn
= 14pi0
∫
re
∫
rn
ρn(rn)
3
2
∑
ij
xnixnjxeixej − 12r
2
nr
2
e
ρe(re)
r55
d3re d3rn
⇒ H`=2 = −16
∑
ij
Qij(∇E)ij,
(A.16)
where
Qij =
∫
rn
ρn(rn)
(
3xnixnj + xnjxni2 − δijr
2
n
)
d3rn,
(∇E)ij = − 14pi0
∫
re
ρe(re)
r5e
(
3xeixej + xejxei2 − δijr
2
e
)
d3re,
(A.17)
A.2. Higher multipole moments 243
where the doubling in the x sum only serves to emphasise the symmetric nature of the
tensor, and normalisation constants have been dropped as they will be summarised in an
overall constant at the end of the calculation.
We now invoke the Wigner-Eckardt theorem, which essentially states that matrix elements
of tensor operators with respect to angular momentum eigenstates can be split into:
• a Clebsch-Gordan coefficient: this depends purely on geometry and is determined
by the orientation of the system with respect to the choice of the z axis, hence it is
only an angular contribution;
• a reduced matrix element: this summarises all non-angular contributions.
According to the above, a generic tensor operator Vi acting on a basis |α, `,m〉, where
`,m are angular momenta and α is a non-angular variable, can be decomposed as:
〈α, `,m|Vi|α, `,m〉 ∝ 〈α, `,m|L ·V|α, `,m〉 · 〈`,m|Li|`,m〉. (A.18)
In our specific case concerning the D2 line of 87Rb and 39K, the nuclear component Qij is
computed in the basis |I,m〉:
〈α, I,mI |Qij|α, I,mI〉 ∝ 〈α, I,mI |Ii ·Q · Ij|α, I,mI〉 · 〈I,mI |Ii Ij|I,mI〉, (A.19)
resulting in the simpler form:
Qij = B1
(
3IiIj + IiIj2 − δijI
2
)
, (A.20)
where B1 contains all non-angular dependencies.
Similarly, the electronic part with tensor operator ∇Eij is computed in the basis |J,mJ〉:
(∇E)ij = B2
(
3JiJj + JiJj2 − δijJ
2
)
. (A.21)
If the nuclear charge density is cylindrically symmetric about z, i.e. it is “squashed” along
z, then the dominant term is Qzz. The pre-factors B can be obtained from an integral in
the |I,mI〉 basis:
Q = 2〈I,mI |Qzz|I,mI〉 = 2
∫
rn
ρn
(
3z2n − r2n
)
d3rn =
= 2B1 〈I,mI |3I2z − I2|I,mI〉 = 2B1 (3I2 − I(I + 1)) = 2B1 I(2I − 1)
⇒ B1 = Q2I(2I − 1) ,
(A.22)
and similarly for the electronic component (the factor of 2 here is conventional) resulting
in:
Qij =
Q
2I(2I − 1)
(
3IiIj + IiIj2 − δijI
2
)
,
(∇E)ij = qJ
J(2J − 1)
(
3JiJj + JiJj2 − δijJ
2
)
,
(A.23)
where
qJ ∝
∫
re
ρ(re)
3z2e − r2e
r5e
d3re. (A.24)
244 Hyperfine structure
Note that for I or J = 1/2 the quadrupole correction vanishes, and thus does not affect the
S and P1/2 states. Intuitively, in the J = 1/2 state there is no orbital angular momentum
which prevents the electron from sampling the anisotropy of the electric field gradient.
Finally, tedious algebra covered in [239] leads to the final form the electric quadrupole
contribution to the Hamiltonian in eq. A.4:
H`=2 =
Bhfs
2I(2I − 1)J(2J − 1)
(
3(I · J)2 + 32(I · J)− I
2J2
)
, (A.25)
where Bhfs includes all other constants, and is experimentally measured as reported in
table A.2.
A.3 Magnetic fields
In the presence of magnetic fields, two more terms are to be included in the electronic
Hamiltonian of eq. A.3:
H = . . . q2m(L ·B)︸ ︷︷ ︸
Linear Zeeman effect
+ q
2B2
8m (x
2 + y2)︸ ︷︷ ︸
Quadratic Zeeman effect
. (A.26)
The quadratic Zeeman effect is usually ignored in our applications, as it only results in
sizeable contributions for very large fields (∝ µBB2(1−m2F )).
The linear effect, however, is the bread and butter of cold atoms experiments as it underlies
magnetic trapping and laser cooling schemes. It is the result of the interaction between
the external magnetic field and the magnetic dipole terms of the electron caused by its
angular momenta. These do not only include the orbital one L present in eq. A.26, but
also the spin S and the nuclear I contributions.
Depending on the basis used to construct the Hamiltonian, the linear Zeeman contribution
is given by:
HZeeman = µB(gSS + gLL + gII) ·B,
HZeeman = µB(gJJ + gII) ·B,
HZeeman = µBgFF ·B,
(A.27)
where all angular momenta are dimensionless. Usually, for a homogeneous field B = Bzzˆ,
the z axis is chosen as the axis of quantisation, simplifying the above expression to:
HZeeman = µB(gSmS + gLmL + gImI)Bz,
HZeeman = µB(gJmJ + gImI)Bz,
HZeeman = µBgFmFBz.
(A.28)
The Lande´ factors are reported in table A.3.
A.3. Magnetic fields 245
Parameter S1/2 P1/2 P3/2
gS 2.002
gL 0.999
gJ 2.002 2/3 4/3
gI
87Rb : −0.000995, 39K: −0.000142
gF
−1/2 for F = 1
1/2 for F = 2
−1/6 for F = 1
1/6 for F = 2 2/3
Table A.3 : Values from [167, 236].
The strength of the interaction with the external magnetic field determines the perturbat-
ive treatment of the atomic Hamiltonian. Namely, it determines which quantum numbers
and hence which basis is suitable for use. The different regimes are summarised in table
A.4. The plots in section 4.3.1 are computed from diagonalising the full Hamiltonian,
therefore not performing a perturbative expansion.
At strong enough fields, the coupling between I and J to form F fails, as they individually
precess about the magnetic field axis and the states are labelled by |mI ,mJ〉. This is the
Paschen-Back regime for IJ coupling.
The fine structure can also be included by adding an L · S term in the Hamiltonian.
When the energy from the interaction with the field is comparable to the fine structure,
LS coupling also fails and the states are labelled by |mL〉 and |mS〉. This is the Paschen-
Back regime for the LS coupling.
strength region comments
B
x
Landau limit
HZeeman H0 Landau tubes
HZeeman ≈ H0 n-mixing n not good
L-mixing L not good
HZeeman H0 Paschen-Back (LS) L and S good, J not good
Paschen-Back (IJ) I and J good, F not good
Zeeman effect F good
B = 0 Coulomb limit
Table A.4 : The relative strength of the Coulomb H0 and magnetic field interaction
HZeeman determines the regime governing the system.
B
More Experimental details
B.1 Vacuum
Differential pumping
A comprehensive treatment of the differential pumping formulae and vacuum calculus
behind the circuit in fig. 4.3 is presented here, based on the appendix of my first year
report [240] but with more revisions and fewer mistakes.
Conductances are in units of pumping speed, L s−1, and obey the following addition rules:
series: 1
Ctot
=
∑
i
1
Ci
, parallel: Ctot =
∑
i
Ci (B.1)
and for two components connected in parallel,
Ctot =
∑
i
Ci . (B.2)
The effective speed of a pump (S∗) is computed by combining its nominal pumping speed
(S) with the conductance of its connection to the system (CP ):
1
S∗
= 1
CP
+ 1
S
. (B.3)
The equations for the conductances of apertures and pipes are derived from statistical
physics, and are therefore only valid within the molecular flow regime where Knudsen
number Kn 1.
The conductance of a circular aperture of area A is:
Cap =
v
4A [L s
−1]. (B.4)
247
248 More Experimental details
This result can be derived from the flux of particles crossing a small patch of area dA at
an angle θ, in a cloud of density ρ:
flux = number of particlesdA dt =
ρ v cos θ dA dt
dA dt = ρ v cos θ → ρ cos θ vf(v) dv, (B.5)
where the last step generalised the expression to a a (classical) statistical ensemble via
v → vf(v) dv, where
f(v) dv = 4pi
(
m
2pikBT
)3/2
e−
mv2
kBT v2 dv sin θ dφ dθ (B.6)
is the Maxwell-Boltzmann distribution.
Integrating over the forward direction and defining the mean thermal velocity v¯ =
(
8kBT
pim
)1/2
,
the total flux J reads
J =
∫
ρ cos θ v f(v) dv = 4pi
(
m
2pikBT
)3/2∫ ∞
0
v3 e−
mv2
kBT dv
∫ pi
2
0
cos θ sin θ dθ
∫ 2pi
0
dφ
⇒ J · A = v¯A4 [L s
−1]
(B.7)
Similarly, the conductance of a pipe of diameter d and length ` can be found to be:
Cpipe =
pi v d3
12` . (B.8)
The total conductance CT of the circuit shown in fig.4.3 is:
CT =
1
S∗1 +
C2S
∗
2
C2 + S∗2
+
1
C1
−1
= CT =
C1
(
C2S
∗
2
C2 + S∗2
+ S∗1
)
C1 + S∗1 +
C2S
∗
2
C2 + S∗2
. (B.9)
The total “currents” (throughputs) flowing in the circuit are therefore
I0 = P0CT , I1 = P1 S∗1 , and I2 = P1
S∗2C2
S∗2 + C2
. (B.10)
Kirchhoff’s law requires:
I0 = I1 + I2. (B.11)
So:
P0CT = P1
(
S∗1 +
S∗2C2
S∗2 + C2
)
⇒ P1
P0
=
C1
C1 + S∗1 +
C2S
∗
2
C2 + S∗2
. (B.12)
Similarly,
P2
P0
=
C2
C2 + S∗2
· C1
C1 + S∗1 +
C2S
∗
2
C2 + S∗2
. (B.13)
All conductances above combine that of a pipe and of the aperture:
Cap+pipe =
(
1
Cap
+
1
Cpipe
)−1
=
pivd3
4(4d+ 3`). (B.14)
The TSP is modelled as a horizontal tube, ignoring its vertical dimension.
B.1. Vacuum 249
Part Symbol Length Diameter Aperture diameter Conductance
mm mm mm L s−1
2D MOT differential pumping section C1 54 1.5 1.5 4.6
3D MOT differential pumping section C2 198 10 16 367
Science cell tube C3 120 20 20 not needed
TSP MOT CP1 258 100 40 112
TSP Knee CP2 285 100 100 294
Ion pump MOT S1 75
Ion pump Knee S2 150
Effective Ion pump MOT S∗1 45
Effective Ion pump Knee S∗2 100
Table B.1 : Technical details of the vacuum components in fig. 4.3. All conductances
combine the contributions of the aperture and of the pipe For pumps, the effective pump-
ing speed S∗ combines the connection conductance (CP ) with the nominal pumping speed
(S).
Cleaning procedure
Our vacuum components were ordered from specialised companies and were therefore de-
livered clean, in sealed bags filled with inert gases (nitrogen or argon). Nonetheless, as
problems arose for some components, we cleaned them ourselves as well. Key protag-
onist in the cleaning procedure was an ultrasonic bath, which shook all surfaces at high
frequency causing dust and impurities to be detached.
The following steps were taken in cleaning vacuum equipment:
• ultrasonic bath in cleaning solution diluted in de-mineralised, warm, clean water for
∼ 40 minutes;
• ultrasonic bath in clean water for ∼ 20 minutes to remove the residual detergent
from the previous step;
• once dry, and immediately before placing the component under vacuum, they were
dusted with a pressurised-air container, and rinsed first with acetone and then with
IPA (Isopropyl alcohol) in order to remove all specs of dust.
Gaskets are made of copper, and I learnt the hard way that copper and acetone react to
form a blue-ish compound [241]. This caused the failure of the scroll pump, which had to
be disassembled and thoroughly cleaned to remove the blue residue which was preventing
it from descending below ∼ 12.5 mbar.
Bake-out
As discussed in the main text of section 4.2, the ultimate pressure in the system is de-
termined by the degassing rate of the internal surfaces. In order to speed this process
and attain low pressures faster, the apparatus was heated to high temperatures – it was
“baked”.
Two bakes were performed. The first one (“Pre-bake”) at maximum ∼ 400◦C, and only
included the differential pumping section, the knee and the science cell TSP (see fig. 4.1).
This was done to increase the diffusion of the hydrogen that accumulates in the steel
components during manufacturing.
250 More Experimental details
Stage Turbo MOT Science
Pre-
bake
1× 10−7 mbar start
6× 10−7 mbar day 15 (at 350 ◦C)
1× 10−9 mbar day 16 (at room temperature)
6× 10−11 mbar overnight, ion pump only
Main-
bake
2.5× 10−7 mbar 2× 10−5 mbar 3× 10−5 mbar start
∼ 8× 10−7 mbar ∼ 8× 10−7 mbar day 11 (at 130 ◦C)
∼ 3× 10−9 mbar ∼ 5× 10−9 mbar day 13 (at room temperature)
2.5× 10−9 mbar 4× 10−11 mbar 2.7× 10−11 mbar couple of days, ion pumps only
Atoms 7.92× 10−11 mbar 2.15× 10−11 mbar atomic sources open, sealed system
Table B.2 : Summary of the pressures reached in different parts of the apparatus.
The second bake (“Main bake”) was performed on the whole system, including the glass
parts, and was around 130◦C to allow evaporation of water vapour. The science cell was
heated via irradiation from a cylindrical aluminium case placed around it.
The bake procedure is detailed in the appendix of my first year report [240]. The temper-
atures were increased very slowly (∼ few degrees per hour) to allow all parts to expand
at the same rate, especially the metal to glass transitions.
Silver plated copper gaskets were used for the pre-bake components as they would not
oxidise, making it easier to remove them when assembling the rest of the apparatus.
Annealed copper gaskets were employed for viewport connections as they are more ductile
and less pressure was needed to tighten them.
A summary of the pressures reached during the bakes is reported in table B.2.
B.2 Laser tables
Lasers
The lasers are divided into two optical tables. The “Red” table, preparing the radiation
used for laser cooling and imaging, and the “High Power” (or “Harry Potter”) table with
the optical dipole trap and the lattice lasers. These are shown in figs B.2, B.3, and B.4.
All lasers use frequency selective components to tune the output frequency from the
broadband emission spectrum of the diode to a monochromatic radiation with linewidth
∼ 100 kHz, as shown in fig. B.1.
The lasers on the “Red” table are manufactured by Toptica, and use a piezo-actuated grat-
ing in the Littrow configuration and a controllable pump current for the diode to change
the effective length of the cavity and hence the frequency output. These are controlled by
a PID loop that stabilises the output frequency based on the error signal generated from
Doppler-free saturation spectroscopy, discussed in section 4.3.2.
The lattice laser (Matisse) uses several frequency selective components such as a birefrin-
gent filter, thin and thick etalons, and is locked to an external cavity (ECDL).
The laser for the optical dipole trap (Mephisto) is only passively stabilised by a sturdy
and rigid internal cavity, and is not locked.
B.2. Laser tables 251
Figure B.1 : Frequency selective components in the Toptica lasers on the “Red” table.
The black line is the emission frequency, corresponding to the maximum gain of the cavity.
Fibres and couplers
The 1064 nm light from the Mephisto is divided into ∼ 10 W per optical fibre, to be
transported to the experimental chambers. This is too much power for usual glass-based
optical fibres, hence large-mode photonic crystal fibres are used instead.
All other radiation, however, is coupled in glass-based optical fibres. In order to maximise
transmission, a collimated laser beam with diameter d1 is focussed onto the fibre core with
diameter d2 (also known as Mode Field Diameter, MFD) by a fibre coupler with a lens f :
d1 =
4fλ
pid2
. (B.15)
The MFD of the fibre is provided on the datasheet of the Thorlabs fibre used: P3-
630PM FC with core diameter d = 3.64 µm at λ = 630 nm, NA = 0.12 and cutoff
wavelength λc = 570 nm. The 630 nm fibre was chosen as it had a lower cutoff wavelength
so as to guarantee single-mode transmission of both 780 (Rb) and 767 nm (K) light.
To calculate the dependence of the MFD on the laser frequency, the following formulae
from [242] were used:
core = 22.4052pi ·
λc
NA , V =
pi
λ
· core · NA,
MFD = core ·
(
0.65 + 1.619
V 3/2
+ 2.879
V 6
)
,
(B.16)
resulting in the values reported in table B.3.
Wavelength V-number Mode field diameter
λ V MFD
nm µm
630 2.18 4.30
780 1.76 5.25
767 1.79 5.15
Table B.3 : Mode Field Diameter calculated for different frequencies coupled into a
P3-630PM FC Thorlabs optical fibre.
252 More Experimental details
22
2
2
2
2
2
2
2
4
4
2
2
2
2
4
2
Bl
ue
fib
re
clu
st
er
Re
d
fib
re
clu
st
er
Rb
C
oS
y
2
2
2
Rb
a
xi
al
Rb
2
D
tra
ns
v
co
ol
2
2
2D
R
b
re
pu
m
p
2
4
2
Rb
re
pu
m
p
flas
h
Rb
im
ag
in
g
pr
ep
Ka
go
m
e
re
pu
m
p
81.75 MHz
81.75 MHz
13
3.
5
M
Hz
-1
00
M
Hz
133.5 MHz
-8
0
M
Hz
4
-8
0
M
Hz
2
Rb
sp
in
-p
ol
4 175? MHz
2Rb
p
us
h
2
2
4
TA
p
ro
78
0
nm
Rb
m
as
te
r T
A
DL
p
ro
78
0
nm
Rb
re
pu
m
p
Bo
os
TA
78
0
nm
Rb
co
ol
in
g
M
OT
BoosTA780 nm
Rb loading MOT
Bo
os
TA
78
0
nm
Rb
7
80
TA
Bo
os
TA
76
7
nm
K
re
d
M
OT
Bo
os
TA
76
7
nm
K
bl
ue
M
OT
-1
25
M
Hz
11
0
M
Hz
2 2
4
-2
00
M
Hz
4
20
0
M
Hz
2
K
re
d
sw
itc
h
K
bl
ue
sw
itc
h
2
2
22
K
re
d
flas
h
2
4
-7
5
M
Hz
2
K
pu
sh
2
n
110 MHz
-110 MHz
2
110 MHz
2
2 K
bl
ue
flas
h
2
4
2
DL pro 767 nm
K imaging
2
2
K
2D
a
xi
al
2
2
22D
K
re
d
2D
K
b
lu
e
2
BoosTA 767 nm
K red loading
BoosTA 767 nm
K blue loading
n
2
80 MHz
-8
0
M
Hz
2
2
2 R
b
im
ag
in
g
pr
ep
2
K
sp
in
-p
ol
K
re
d
flas
h
4
-1
10
?
M
Hz
4
11
0?
M
Hz
K
re
d
flas
h
K
Co
Sy
-8
0
M
Hz
Im
ag
in
g
Z
2
2
Rb
re
pu
m
p
flas
h
2
Im
ag
in
g
D
Im
ag
in
g
T
2
Im
ag
in
g
sp
ar
e
2
n
n
200 MHz
n
TA
pr
o
76
7
nm
K
m
as
te
r T
A
n
n
λ/ 2
λ/ 2
λ/ 2
Rb
re
pu
m
p
off
se
t l
oc
k
K
im
ag
in
g
off
se
t l
oc
k
2
Figure B.2 : The “Red” table prepares the light for laser cooling and imaging. The
“CoSy” is the device performing the Doppler-free saturation spectroscopy in section 4.3.2.
B.2. Laser tables 253
2
2 2 2
2
2
Mephisto 1064 nm
Dipole trap
Cavity
Dipole X
Dipole Y
Dipole Z
Figure B.3 : The first half of the “High Power” (“Harry Potter”) table is dedicated to
preparing the optical dipole trap.
Millennia 532 nm
2
2
2
2
2
2
2
2
Ma�sse 730 nm
La�ce
Lattice
T axis
Lattice
D axis
Lattice
X axis
Lattice
Y axis
Figure B.4 : The second half of the “High Power” (“Harry Potter”) table is dedicated
to preparing the lattice light.
254 More Experimental details
Magnetic fields
Rectangular coils
Analytical expressions for the magnetic field generated by the rectangular coil shown in
fig. B.5 are derived in [243] and reported as follows.
Figure B.5 : The rectangular coil used in the magnetic field formulae reported in this
section.
Bz =
µ0I
4pi
4∑
a=1
[
(−1)αdα
rα(rα + (−1)α+1Cα) −
Cα
rα(rα + dα)
]
,
Bx =
µ0I
4pi
4∑
a=1
(−1)α+1z
rα(rα + dα)
,
By =
µ0I
4pi
4∑
a=1
(−1)α+1z
rα(rα + (−1)α+1Cα)
(B.17)
with
C1 = −C4 = a+ x, C2 = −C3 = a− x,
d1 = d2 = y + b, d3 = d4 = yb,
r1 =
√
(a+ x)2 + (y + b)2 + z2 , r2 =
√
(a− x)2 + (y + b)2 + z2 ,
r3 =
√
(a− x)2 + (y − b)2 + z2 , r4 =
√
(a+ x)2 + (y − b)2 + z2 .
(B.18)
Circular coils
The derivation of the field from a current loop is detailed in [244], and quoted as follows.
B.3. Plumbing 255
x
y
z
I
r
r'
r-r'
ρϕ
ϕ'
θ
ld
Figure B.6: Diagram for field generated from a current loop.
Bρ(ρ, z) =
µ0 I
2pi
z
ρ
√
(ρ+ a)2 + z2
[
a2 + ρ2 + z2
(a− ρ)2 + z2E(k)−K(k)
]
, (B.19)
Bφ(ρ, z) =
µ0 I
2pi
1√
(ρ+ a)2 + z2
[
a2 − ρ2 − z2
(a− ρ)2 + z2E(k) +K(k)
]
. (B.20)
These expressions can be shifted by replacing z → (z − L), so as to calculate the fields
from coils pairs in the Helmholtz and anti-Helmholtz configurations.
On the axis ρ = 0, the elliptical integrals are trivially equal to pi/2 and hence:
Bρ(ρ, z) = 0, and Bz(0, z) =
µ0 I
2pi
a2
(z2 + a2)3/2 , (B.21)
which is the known result for the field along the axis of a loop.
B.3 Plumbing
Calculation of the losses in the pressure and flow of the cooling water was performed in
order to determine the required specifications of the water supply. Suitable chillers were
purchased providing enough pressure to guarantee sufficient water flow to cope with the
thermal dissipation of the electromagnets. The calculation is detailed in the appendix of
my first year report [240] and reproduced here.
Coils producing the magnetic fields for 2D-MOTs and for the magnetic transport (and
Feshbach) generate a large quantity of dissipated heat. They therefore require to water-
cool the brass heat sinks which the coils are wound around.
256 More Experimental details
The ’cooling block’ (transport and experiment coils) and the 2D MOTs coils are connected
in parallel, as shown in fig. B.7. Considering this as an electronic circuit, parallel loads
need lower pressure, hence a cheaper chiller and a lower consumption.
Figure B.7 : 2D MOT coils and magnetic transport coils are connected in parallel.
Oliver Brix, the Master’s student that designed the magnetic transport system derived
the required chiller and derived that the pressure available at the connection (p2 − p1) is
0.18 mbar.
The continuity equation for an incompressible liquid requires the volumetric flow Q to be
constant throughout the circuit, so:
Q = uA = constant, (B.22)
where u is speed of fluid and A the cross section of the pipe.
The volumetric flow is fixed by the power dissipated (Pdissipated) by our electrical system
and by the desired temperature increase of the cooling liquid ∆T :
Qrequired =
Pdissipated
∆T ρC , (B.23)
ρ and C being the density and specific heat capacity of the cooling liquid.
A certain pressure is required to maintain a certain flow, due to the ’ideality’ of the
incompressible fluid being spoiled by the viscous forces with the pipe walls. Pressure
losses arising from the circuit are quantified with a head loss, which relates the kinetic
energy of an incompressible fluid to the potential energy stored static column of equivalent
volume, an analogy allowed by Bernouilli’s principle.
A hydraulic head is given by the elevation and the pressure heads. The elevation pressure
is required to pump the fluid from the ground to the level of the heat sinks (∼ 1 m).
Pressure heads are given by the pressure loss in pipes, due to friction, and in corners due
to direction change.
B.3. Plumbing 257
The friction in the pipe causes a pressure head give by the Darcy-Weisbach equation:
∆h = fD2g
u2L
d
, (B.24)
L and d being the length and diameter of the pipe, and fD the coefficient of friction given
by:
fD =
0.079
Re1/4
= 0.079
(
µ
ρ
)1/4 1
u1/4d1/4
, (B.25)
Re being the Reynolds number. This is an empirical expression that is valid for high
Reynolds numbers such as the ones encountered in a liquid.
The head loss associated with bends is given by:
∆h = ku
2
2g , (B.26)
k being a coefficient associated with the type of bend, e.g. for a square bend 90 degrees
is given by k1 = 1.3.
Sudden contraction and expansion also cause pressure loss, and have same expression as
above, with coefficients k2 = 0.44 and k3 =
(
1− A1
A2
)2
respectively. The flow being from
A1 to A2.
The pressure is derived from the head loss via
∆p = ρ g∆h. (B.27)
The chiller must then be able to supply a pressure pchiller:
pchiller︸ ︷︷ ︸
pressure required
= ∆pMOTs︸ ︷︷ ︸
2D MOTs
heat sink
+ ∆ppipe︸ ︷︷ ︸
pipe between
two 2D MOTs
+ ∆pbends︸ ︷︷ ︸
bends and
corners
+ ∆psudden︸ ︷︷ ︸
sudden expansions
and/or contractions
, (B.28)
pchiller︸ ︷︷ ︸
pressure
required
= fD
ρu2L
2d︸ ︷︷ ︸
2D MOTs
heat sink
+ N1k1
ρu2
2︸ ︷︷ ︸
pipe between
two 2D MOTs
+ f ′D
ρu2pLp
2dp︸ ︷︷ ︸
bends and
corners
+N2k2
ρu2
2︸ ︷︷ ︸
sudden
expansions
+N3k3
ρu2
2︸ ︷︷ ︸
sudden
contractions
, (B.29)
where the speed in the pipe up, from B.22:
up = u
(
d
dp
)2
. (B.30)
So:
1
2ρu
2
fDL
d
+ f ′D
Lp
dp
(
d
dp
)4
+N1k1 +N2k2 +N3k3
, (B.31)
fD = 0.079
(
µ
ρ
)1/4 1
u1/4d1/4
, (B.32)
258 More Experimental details
f ′D = 0.079
(
µ
ρ
)1/4 1
u
1/4
p d
1/4
p
= 0.079
(
µ
ρ
)1/4 d1/4p
u1/4d1/4
, (B.33)
k3 =
(
1− A1
A2
)2
=
(
1− d
2
1
d22
)2
. (B.34)
Combining B.22 with B.23 allows extraction of the necessary fluid speed u:
u = 4Pdissipated
piρC∆Td2 , (B.35)
and hence the expression for the pressure:
pchiller =
8P 2dissipated
pi2ρ∆T 2C2d4
0.079(µ
ρ
)0.25 (
piρ∆TC
4Pdissipated
)0.25 (
L
d3/4
+ Lp
d4
d
19/4
p
)
+
8P 2dissipated
pi2ρ∆T 2C2d4
N1k1 +N2k2 +N3k3
(
1− d
2
1
d22
)2 .
(B.36)
With pchiller equal to the pressure available between points 1 and 2 in fig. B.7, 0.18 mbar,
and the following data: Lp = 3 m, dp = 8 m, N1 = 27, N2 = 2, N3 = 2, we only need the
power dissipated by the coils to calculate the temperature increase of the coolant liquid
to make sure it is reasonable.
The power dissipated P by a current I in a conductor of electrical resistivity ρe, length L
and cross-sectional area A0 is:
P = I2 ρe
L
A0
. (B.37)
A single rectangular 2D-MOT coil, with maximum 10 A delivered from the power supplies
and 1.25 mm diameter coil, with L = 0.420 m of wire per turn with 70 turns in total (L ∼
30 m) dissipates ∼ 47 W. For eight coils, the total dissipated power is Pdissipated ∼ 380 W.
From this data, the temperature increase of the cooling liquid is ∆T = 2.7 ◦C, and the
water flow is Q ∼ 2 L s−1. This is the worst case scenario, with the coils assumed to be
always on and with maximum current. Radiative heating from the black-anodised coil
holder reduces the dissipated power the water needs to cool.
Compensation coils were not included in this calculation.
C
More localisation transitions
The same code used to produce the momentum space Inverse Participation Ratio (IPRk)
plots in section 8.1.3 was used to numerically probe localisation transitions in other lattice
configurations. As before, the size of the Hamiltonian was limited by the RAM of the
computers that performed the calculation.
Three beam case
The interference pattern for the three-beam lattice configuration is shown in fig. 7.14,
labelled as “3D”. This results in a regularly spaced array of 1D quasiperiodic systems.
The ground state IPR in momentum space for the three-beam lattice is shown in fig. C.1
and exhibits a transition between 2 < V0/Erec < 3, most likely around 2.75Erec.
Figure C.1 : The momentum space inverse participation ratio IPRk for the ground state
of a three-beam lattice, which consists of regularly spaced 1D quasiperiodic systems. The
red dashed line is computed for 50 generations.
259
260 More localisation transitions
Other four-dimensional cases
According to table 3.1, a four-dimensional regular lattice is not only compatible with
eightfold rotational symmetry, but also five-, ten- and twelvefold. The physical question
here is: can we probe the 4D Chern number in, for instance, a fivefold symmetric lattice?
While there is literature that claims no novel topological features are guaranteed by the
higher dimensional inheritance of quasicrystals [245], a quick investigation into other 4D
compatible rotational symmetries was performed.
In the following I quote the bases in 4D and 2D corresponding to the relevant n-fold
rotational symmetry, along with the projection matrices P‖ to obtain the latter from the
former. These are used to compute the kinetic energy contribution to the Hamiltonian
in eq. 7.28 while employing the same off-diagonal lattice (∝ V0) terms computed for the
eightfold case. The IPRk plots were all computed up to 25 generations, and essentially
all looked like fig. 8.4 and are hence are not shown.
The relation between the 4D hyperspace and the 2D bases reported below is justified
as the matrix representations of both obey the same groups operations. However, these
representations are not faithful, since the bases in 4D are not linearly independent. This
means that a particular point in momentum space cannot be uniquely identified by its
generation, as it was done for the eightfold case in section 7.1.4, which undermines the
confidence in the IPRk plots and in any dynamical simulation performed in this basis.
This is further confirmed by the projection matrices all giving (P‖)2 6= P‖, as opposed to
what found for eqs. 7.26 and 7.27.
In conclusion, I think that a reliable simulation for fivefold symmetry would require a 5D
parent regular lattice, where the 5 roots of unity can be mapped onto a set of 5 linearly
independent vectors. Similarly for the other rotational symmetries. Unfortunately this
would require even more RAM and CPUs than available at the time of these investigations.
Fivefold
vector 4D 2D
|1〉 (1, 0, 0, 0) (1, 0)
|2〉 (0, 1, 0, 0) (cos(2pi/5), sin(2pi/5))
|3〉 (0, 0, 1, 0) (− cos(pi/5), sin(pi/5))
|4〉 (0, 0, 0, 1) (− cos(pi/5),− sin(pi/5))
|5〉 (−1,−1,−1,−1) (cos(2pi/5),− sin(2pi/5))
261
P‖5 =
1 cos(2pi/5) − cos(pi/5) − cos(pi/5)
0 sin(2pi/5) sin(pi/5) − sin(pi/5)
0 0 0 0
0 0 0 0
. (C.1)
Notice how ∑i|G〉i = 0, so that we can always express one of the vectors as the sum over
the others: |5〉 = |1〉+ |2〉+ |3〉+ |4〉. This intuitively justifies why the four-dimensional
space still provides a representation for the fivefold rotational symmetry.
The ground state IPR in momentum space for the 4D basis showed a transition around
1.75 < V0/Erec < 1.80.
The Hamiltonian in the 5D basis was also computed, though only up to 20 generations
due to RAM limitations, showing a transition around 1.25 < V0/Erec < 1.35.
Tenfold
vector 4D 2D
|1〉 (1, 0, 0, 0) (1, 0)
|2〉 (0, 1, 0, 0) (cos(2pi/5), sin(2pi/5))
|3〉 (0, 0, 1, 0) (− cos(pi/5), sin(pi/5))
|4〉 (0, 0, 0, 1) (− cos(pi/5),− sin(pi/5))
|5〉 (−1,−1,−1,−1) (cos(2pi/5),− sin(2pi/5))
P‖10 =
1 cos(2pi/5) − cos(pi/5) − cos(pi/5)
0 sin(2pi/5) sin(pi/5) − sin(pi/5)
0 0 0 0
0 0 0 0
. (C.2)
The ground state IPR in momentum space for the 4D basis showed a transition around
1.75 < V0/Erec < 1.80. This is the same as for the fivefold symmetric case, as expected
because reflection symmetry makes the two synthetic lattice configurations equivalent.
262 More localisation transitions
Twelvefold
vector 4D 2D
|1〉 (1, 0, 0, 0) (1, 0)
|2〉 (0, 1, 0, 0) (cos(pi/6), sin(pi/6))
|3〉 (0, 0, 1, 0) (cos(pi/3), sin(pi/3))
|4〉 (0, 0, 0, 1) (0, 1)
|5〉 (−1, 0, 1, 0) (− cos(pi/3), sin(pi/3))
|6〉 (0,−1, 0, 1) (− cos(pi/6), sin(pi/6))
P‖12 =
1 cos(pi/6) cos(pi/3) 0
0 sin(pi/6) sin(pi/3) 1
0 0 0 0
0 0 0 0
. (C.3)
The ground state IPR in momentum space for the 4D basis showed a transition around
1.65 < V0/Erec < 1.75.
D
Miscellaneous
D.1 Spin-statistics theorem
The foundation of the proof leading to eq. 1.28 lies in the commutator [φA, φB], where φ
is defined in eq. 1.27. The full, rigorous and index-full proof is given in [53] in the section
about General causal fields. I am just going to sketch the proof here, as a guide for the
argument presented in the reference.
The commutator [φA, φB] depends on the commutators [fA, fB] and [hA, hB] of the func-
tions that define the field operator in eq. 1.27. Hence we will turn our attention to the
coefficients f and h.
The function f(p) for a generic momentum p can be generated from one at rest f(0) by
a Lorentz boost Λ: f(p) = D(Λ)f(0), with D being some irreducible representation of
the Lorentz group. The latter affects both space and time, and therefore obeys SO(4)
algebra – technically SO(1, 3) – which is isomorphic to SO(3) × SO(3). This can be
easily understood by considering the same isomorphism in one fewer dimension: SO(3)
is isomorphic to SO(2) × SO(2), because any position on a 3D sphere can be written
as a linear superposition of two orthogonal 2D circles. Hence, we can replace the four-
dimensional vector J ∈ SO(4) by two three-dimensional vectors J,K ∈ SO(3), with
quantum numbers j and k obeying the usual angular momentum algebra and hence 2j +
1, 2k + 1 ∈ N.
Luckily we already know that the spherical harmonics Y m` are the basis functions of
the irreducible 2` + 1 space, and they transform according to Y m` (−r) = (−1)`Y m` (r).
Extending this to the generic representation D(Λ), we expect a transformation D(Λ) →
(−1)j(−1)k = (−1)j+k.
Identical particles transform according to the same irreducible representation of the Poin-
care´ group, uniquely labelled by their mass m =
√
1/c2 pµpµ and spin. If A and B are
263
264 Miscellaneous
identical, that is with equal mass and spin, then:
[fA, fB] ∝ D(Λ)D(Λ) ∝ (−1)j+k(−1)j+k = (−1)2(j+k). (D.1)
Hence, the total angular momentum j+k determines the sign of the commutator and the
statistics obeyed by the particles, resulting in the expressions in eq. 1.28.
The treatment in this section was simplified by only considering real scalar fields, though it
can be easily extended to complex fields representing charged particles. More importantly,
the fields were assumed to be non-interacting and hence expressed in terms of single mode
p creation and annihilation operators. Proofs of the spin-statistics theorem for interacting
fields are much harder, and to be quite frank I did not understand them. However, it is
argued in [53] that the cluster decomposition theorem ensures interacting fields can always
be constructed from the non-interacting components considered here.
D.2 Derivation of transition matrix element
Here I report the tedious maths needed to result in the transition matrix element of eq.
2.7.
First of all, the d operator is defined as:
d = −i
mω
〈f |eik·rp|i〉, (D.2)
and the complex exponential is expanded as:
eik·r = 1 + ik · r + . . . . (D.3)
The dipole approximation assumes the electron’s orbit is much smaller than the wavelength
of the incoming radiation, k · r ∼ 0, resulting in the atom experiencing only a DC field.
Retaining only the first term of the expansion (1) and using:
[r, H0] =
ih¯p
m
, (D.4)
where H0 = p2/2m+ V (r), results in:
eˆ · d = −i
mω
eˆ · 〈f |eik·rp|i〉 ≈ 〈f |eˆ · r|i〉, (D.5)
which we identify as the electric dipole contribution to the transition element.
We now turn our attention to the leading correction in the above power expansion, ik · r.
The transition matrix element is given by:
eˆ · d = 1
mω
〈f |(k · r)(eˆ · p)|i〉 = 1
mc
〈f |(kˆ · r)(eˆ · p)|i〉. (D.6)
The rest of the section details suitable substitutions to bring the above expression to one
of greater physical interpretation.
First, we notice that defining nˆ = kˆ× eˆ enables writing:
nˆ · L = (kˆ× eˆ) · (r× p) = (kˆ · r)(eˆ · p)− (eˆ · r)(kˆ · p), (D.7)
D.3. Spontaneous decay rate 265
which is half of the expression in eq. D.6. We also take the liberty of adding 2S to L,
where S is the spin of the electron and 2 its (approximate) gyromagnetic ratio, motivated
by the experimental evidence that the two can be treated on equal footing.
The other half of eq. D.7 can be derived if we define a well-crafted concoction C:
C = im
h¯
[H0, (eˆ · r)(kˆ · p)], (D.8)
which can be expanded as:
C = im
h¯
eˆj kˆk[H, rj, rk] =
im
h¯
ej kˆk(rj[H, rk] + [H, rj]rk) (D.9)
Using the commutator of eq. D.4, this results in:
S = eˆj kˆk(rjpk + pjrk) = eˆj kˆk(rjpk + rkpj − ih¯δjk)
⇒ S = (eˆ · r)(kˆ · p) + (kˆ · r)(eˆ · p),
(D.10)
where eˆ · kˆ = 0 was assumed because we are considering a plane wave.
Eq. D.9 can also be used to directly compute the matrix element:
〈f |C|i〉 = im
h¯
eˆj kˆk〈f |(rjHrk − rjrkH +Hrjrk − rjHrk) |i〉 = im
h¯
eˆj kˆk〈f |rjrk|i〉, (D.11)
from which we define the electric quadrupole tensor as
Qjk = 〈f |rjrk − r2δjk/3|i〉, (D.12)
made traceless to guarantee the plane wave condition eˆ · kˆ = 0.
Combination of all of the above results in:
eˆ · d = 〈f |eˆ · r|i〉︸ ︷︷ ︸
electric dipole
+ 12mc(kˆ× eˆ) · 〈f |L + 2S|i〉︸ ︷︷ ︸
magnetic dipole
+ iω2c eˆ ·Q · kˆ︸ ︷︷ ︸
electric quadrupole
. . . (D.13)
D.3 Spontaneous decay rate
The spontaneous decay rate used in table 2.1 was estimated as follows. A hydrogen atom
was used, but the procedure can be extended to hydrogenic atoms by including Z, thereby
being applicable to the 87Rb and 39K atoms treated in this thesis.
From [246], the spontaneous decay rate wsp between a state |f〉 and |i〉 is given by:
wspf→i =
4αω3
3c2 |〈f |d|i〉|
2, (D.14)
where d is the transition operator and h¯ω the energy gap between the two states, α
the fine-structure constant and c the speed of light. In a hydrogen atom, we can take
h¯ω ∼ α2mc2 as per the gross structure, with m being the (reduced) mass of the electron.
For electric dipole transitions, d is the electric dipole operator and can be approximated
as ea0, where e is the charge of the electron and a0 = h¯/(mcα) is the Bohr radius. These
results allow for expressing the spontaneous decay rate for electric dipole transitions as:
wspelectric dipole ∼ α5
mc2
h¯
. (D.15)
266 Miscellaneous
For the magnetic dipole transitions, we need to identify a suitable d which from eq. D.13
may be taken as ∼ 1/mc (L + 2S). Estimating the angular momentum to be of order h¯,
and the ω between fine structure levels in the same n manifold (for parity rules in table
2.1) h¯ω ∼ α4mc2 (as per table A.1), we get:
wspmagnetic dipole ∼ α13
mc2
h¯
. (D.16)
For electric quadrupole transitions, we take d ∼ ω/cQ, and we estimate h¯ω ∼ α2mc2 as
for the electric dipole, and |Q|∼ a20 from eq. D.12. Leading to:
wspelectric quadrupole ∼ α7
mc2
h¯
. (D.17)
D.4 Proof of reciprocal space
I here provide a proof for eq. 3.6:
ρ(r) = ρ(Rr + t) ⇔ ρˆ(k) = ei(Rk·t) ρˆ(Rk). (D.18)
where ρˆ(k) is the Fourier transform (structure factor) of the density distribution ρ(r),
defined as:
ρˆ(k) =
∫
ρ(r)e−ik·r d3r. (D.19)
Under a plane group symmetry operation, can combine a translation t and a rotation R to
result in r′ = Rr + t, which upon being inverted yields r = R−1r′−R−1t. Because R−1 =
RT , then kR−1r⇒ (Rk)r, which in the exponent results in −k · r = −(Rk) · r′+ (Rk) · t.
Hence:
ρˆ(k) =
∫
ρ(r) e−ik·r d3r =
∫
ρ(r) e−i(Rk)·r′+(Rk)·t d3r = ei(Rk·t) ρˆ(Rk), (D.20)
as quoted in in eq. 3.6.
D.5 Groups and field extensions
Groups are the best way to mathematically classify and quantify symmetries. Based
on the relation between n-fold symmetry and m-dimensional regular lattices outlined in
section 3.2.4, one could speculate that if a point group possesses more than d d-dimensional
irreducible representations, it is not a crystallographic point group and it violates eq. 3.5.
The symmetry of regular n polygons is described by the dihedral group Dn. The number
of irreducible representations of Dn is (n − 1)/2 for odd n, and (n − 2)/2 for even n.
For d = 2, dihedral groups with n > 6 are then not crystallographic, with 5 and 6 being
undecided as the number of representations is exactly equal to d = 2.
We then look at the smallest splitting field for different Dn, listed in table D.1. This
can be thought of as the smallest extension to Z that contains all the coefficients of the
characteristic polynomial of the n-fold rotation matrix.
Table D.1 then confirms what already discussed about eq. 3.5. One-, two-, three-, four-
and sixfold symmetries are crystallographic because 2 cos(2pi/n) ∈ N, while fivefold and
eightfold symmetries require a field extension by an irrational number.
D.6. Free energy minimisation 267
Degree Order Dihedral group Number and dimensions Smallest
n 2n Dn of irreducible representations splitting field
1 2 D1 or Z2 1, 1 Q
2 4 D2 or V4 (Klein) 1,1,1,1 Q
3 6 D3 or S3 1,1,2 Q
4 8 D4 1,1,1,1,2 Q
5 10 D5 1,1,2,2 Q(
√
5 )
6 12 D6 1,1,1,1,2,2 Q
7 14 D7 1,1,2,2,2 Q(cos 2pi/7)
8 16 D8 1,1,1,1,2,2,2 Q(
√
2 )
Table D.1 : Summary of the properties of dihedral groups Dn.
Notice that threefold symmetry involves the angle φ = 2pi/3, such that cos(2pi/3) = 1/2 ∈
Q but sin(2pi/3) =
√
3 /2 6∈ Q. But threefold symmetry is still crystallographic because
Q(cos 2pi/3) = Q. Put simply, the n-fold symmetry will be non-crystallographic only if
cos(2pi/n) 6∈ N. This is justified by the argument leading to eq. 3.5, and also by quoting
a general non-orthogonal rotation matrix:
R =
(
0 −1
1 2 cos(2pik/n)
)
, (D.21)
with n, k ∈ N. This matrix is constructed with a non-orthogonal basis1 such as the unit
vectors in the directions of the vertices of an n polygon, and it still satisfies eq. 3.5.
D.6 Free energy minimisation
In thermodynamics, the equilibrium state is usually found by minimising the (Helmholtz)
free energy F = U − TS. But why?
An equilibrium macroscopic state, from the second law of thermodynamics, corresponds
to a state of maximal entropy S. The entropy is related to the statistical weight W
by Boltzmann’s relation S = kB ln Ω and therefore to the total number of microscopic
states N (Ω ∝ N). Equilibrium dictates maximising the entropy S, thus the number of
microstates states N .
In a closed system (the microcanonical ensemble), the number of microstates N at a fixed
energy U is obtained from the entropy:
S(U) ∝ lnN(U) ⇒ N ∝ eS(U), (D.22)
hence equilibrium is simply when the entropy is maximised.
If our system is put into contact with a reservoir at temperature T (the canonical en-
semble), equilibrium is attained when the total (reservoir+system) number of states Ntot
is maximised:
Ntot ∝ eS(U) · eSres(Utot−U) = eS(U)+Sres(Utot−U). (D.23)
1For the rotation matrix M in Cartesian coordinates of eq. 3.4, with φ = 2pi/n for an n-sided regular
polygon, the characteristic equation is M2 − tr(M)M + det(M) = 0. Applying to v1, with v2 = Mv1,
one gets Mv2 = −v1 + 2 cos (2pik/n) v2.
268 Miscellaneous
The entropy of the reservoir can be expanded as:
Sres(Utot − U) ≈ Sres(Utot)− U ∂S
∂U
= Sres(Utot)− U
T
, (D.24)
where we have used the thermodynamic relation ∂U/∂S = T .
Hence the total number of microstates is given by:
Ntot ∝ eS · e−U/T = e(TS−U)/T = e−F/T , (D.25)
where the (Helmholtz) free energy F was defined as F = U − TS.
Maximising the number of states corresponds to minimising the free energy F .
D.7 Derivation of Dyson expansion
This section provides a proof for the Dyson series of eq. 7.8.
In the Scho¨dinger picture, the time-evolution is carried by the states while the operators
are time-independent:
|ψt〉 = U(t, t0)|ψ0〉. (D.26)
In the interaction picture2, the time depence in encoded in the operators themselves,
requiring a shift U → U (int). In this picture, U (int) obeys the Schro¨dinger equation:
ih¯dU
(int)
dt = H
(int)
I U
(int), (D.27)
where HI is the interaction contribution to the Hamiltonian, which provides a non-trivial
evolution dynamics.
The equation may be solved iteratively with a boundary condition U (int)(t = t0) = 1 to
give:
U (int)(t, t0) = 1− i
h¯
∫ t
t0
H
(int)
I (t′) U (int)(t′)︸ ︷︷ ︸1− i
h¯
∫ t′
t0
H
(int)
I (t′′)U
(int)(t′′)︸ ︷︷ ︸
...
dt′′
dt′ (D.28)
which results in the expression:
U (int)(t, t0) = 1− i
h¯
∫ t
t0
dt′H(int)I (t′) +
(−i
h¯
)2 ∫ t
t0
dt′
∫ t′
t0
dt′′H(int)I (t′)H
(int)
I (t′′) . . . (D.29)
Finally, we return to the Schro¨dinger picture:
U (int) = eiH0t/h¯Ue−iH0t/h¯, H(int)I = eiH0t/h¯HIe−iH0t/h¯, (D.30)
where H0 is the free (non-interacting) component of the Hamiltonian.
Using the energy eigenbasis, eiH0t/h¯|i〉 = eiEit/h¯, we obtain the expression quoted in eq.
7.8.
Finally, for the integration:
2There are technical differences between the Heisenberg and interaction pictures that I will ignore.
D.8. Time-varying constant in the Hamiltonian 269
• Off-diagonal terms:∫
dt′ ei(Ef−Ei)(t′−t0) ∼ 1
Ef − Ei e
i(Ef−Ei)(t′−t0). (D.31)
The largest contribution form the pre-factor is when Ef ≈ Ei, and in the limit of
t → ∞, t0 → −∞ the phase factor becomes large and common to all perturbative
terms, and can hence be dropped.
• Diagonal terms: Ef = Ei hence the phase factor in the above integral is trivially
1, and all remaining phase factors are common to each perturbative terms and can
hence be dropped.
D.8 Time-varying constant in the Hamiltonian
Construction of the Hamiltonian detailed in 7.1.4 included diagonal terms d V0/2, where
d is the dimensionality of the simulated system and V0 the single-beam lattice depth.
While these terms only amount to phase factors for the time-independent lattices used
in chapter 7, it is not immediately obvious whether or not they can also be neglected
for the time-dependent V0(t) ramps presented in chapter 8. This is what I mean by the
oxymoron “time-varying constant”: a purely diagonal contribution to the Hamiltonian,
whose entries are all the same, but which happens to be time dependent.
In this case, the Schro¨diner equation can be written as:
id|ψ〉dt = (H
′(t) + Φ(t)) |ψ〉, (D.32)
with solution:
|ψt〉 = T exp
[
− i
h¯
∫ t
0
(H ′(t) + Φ(t)) dt
]
|ψ0〉, (D.33)
where Φ(t) = diag [f(t)] is the purely diagonal time dependent contribution to the
Hamiltonian, and T the time-ordering operator.
Using [H ′,Φ] = 0, we obtain:
|ψt〉 = T
[
exp−
( i
h¯
∫ t
0
H ′(t)dt
)
exp
(
− i
h¯
∫ t
0
Φ(t)dt
)]
|ψ0〉. (D.34)
Because Φ is diagonal,
exp
(
− i
h¯
∫ t
0
Φ(t)dt
)
= 1 exp
(
− i
h¯
∫ t
0
f(t)dt
)
. (D.35)
Also, since T (a · b) = θ(t1 − t2)(a · b) + θ(t2 − t1)(a · b) = T (a) · T (b), where θ is the
Heaviside step function:
T
[
1 exp
(
− i
h¯
∫ t
0
f(t)dt
)]
= exp
(
− i
h¯
∫ t
0
f(t)dt
)
, (D.36)
because the entries f(t) commute at unequal times ∀t.
Hence, the dynamical evolution of the state is given by:
|ψt〉 = exp
(
− i
h¯
∫ t
0
f(t)dt
)
T
[
exp
(
− i
h¯
∫ t
0
H ′(t)dt
)]
|ψ0〉. (D.37)
This proves that the time-dependence of a homogeneous and diagonal contribution to the
Hamiltonian only amounts to a phase factor.
270 Miscellaneous
D.9 ARB documentation
This section provides a short documentation for the class and methods used to generate the
ARBitrary lattice pulses of chapters 7 and 8. The signal generators used are the Keysight
33500 B series, with option 33512B. They have 16 MSa memory, and a maximum sample
rate of 250 MSa/s = 30 MHz.
A library was written to interface them with our timing software QControl, which is in
qcontrol\qcontrol\server\drivers\VisaDriver\CamWavGen.py. The main method
added to the signal generator class K33500B is:
s e l f . addChannels ( VisaChannel ( ’ Arbitrary ’ , None ,
( ’ arbname ’ , ’ 0 , . 1 , . 2 , . 3 , . 4 , . 5 , . 6 , . 7 , . 8 , . 9 ’ ,
’ arbname ’ , ’ 2 . 0 ’ , ’5 e 3 ’ ) ,
d e s c r i p t i o n = ’ Send a r b i t r a r y waveform ’ ,
hw hint = ’FUNC ARB; DATA:ARB %s , %s ; :FUNC:ARB %s ;
:VOLT %s ; :VOLT:OFFS: 0 ;
:FUNC:ARB:FILTER OFF; :FUNC:ARB:PER %s ’ ) )
where VisaChannel is the QControl class for all VISA (Virtual instrument software ar-
chitecture) devices. Its non-keyword arguments are the name and units (if any) of the
command, and a string with the details of the pulse sequence. The string in the example
above is the default value of the method, and will be replaced by a user-defined argument.
Each element in the string replaces the %s in the hardware hint hw_hint, which is the
actual string sent to the device. This employs the SCPI language, where commands are
separated by a semicolon (equivalent to a line break) and a colon specifies a sub-system.
FUNC ARB selects the arbitrary waveform functionality, DATA:ARB specifies that this will
be generated from a list of user-provided points, VOLT determines the maximum voltage
and the offset which all other points in the waveform are referenced to, and PER is the
duration of the waveform.
The methods defined above are then used in sequences such as in sequence\client\
latticepulse.py:
i f s e l f .P [ ’ do l a t t i c e ramp ho ld ’ ] :
arb = CamWavGen. Arb( s e l f .P [ ’ l a t t i c e r e s o l u t i o n ’ ] )
data = arb . compi le ( ’RAMPHOLD’ ,
arb . LinearRamp (0∗V, vo ltage , s e l f .P [ ’ l a t t i c e ramp dur ’ ] ) ,
arb . Hold ( vo l tage , s e l f .P [ ’ l a t t i c e h o l d d u r ’ ] + 10∗ms)
arb . ReturnZero (1∗ms)
where Arb is the class defining how to generate the points in the waveform. The output
data is an array of numbers (the steps in the output voltage), which the method compile
converts into strings with a waveform name that appears on the screen (in this case
RAMPHOLD), ready to be sent to the signal generator. Finally, lattice resolution is in
units of time, controls how many points the generator is going to output. We cannot
directly control the duration of the waveform, only the sample rate of the device. The
number of points in the waveform and the sample rate then determine the duration.
Understanding and implementing this is essential to guarantee the independent duration
of each separate part of the arbitrary waveform, and not cause an overall scaling. The
D.10. Dimensionless units for lattice potential 271
resolution needs to be increased for short pulses, and reduced for long ones in order not
to run out of memory.
Extensions
While I stood on the shoulders of giants when writing the above methods, I am not a
giant myself – literally and figuratively.
Hence my solution is certainly not definitive. Extensions and modifications that I would
have implemented with more time are reported below.
• Loading the arbitrary waveform to the signal generator during the upload phase of
the sequence, and not when the subclass is called. This would speed up the process
and prevents the GUI from freezing.
• Use the DATA:SEQ subsystem together with repeat mode. This allows to only specify
one voltage value that is repeated multiple times as opposed to waste memory on
generating a string of numbers with equal value.
D.10 Dimensionless units for lattice potential
Eq. 7.3 describes the ground state energy for atoms localised on each minimum of the
optical lattice potential. This assumes the deep lattice limit discussed in section 2.4.2,
where the minima are approximated as harmonic oscillators to result in a ground state
energy E0:
E0 = Emin +
h¯ωx + h¯ωy
2 , (D.38)
where Emin is the offset.
It is beneficial to compute this in dimensionless units, detailed as follows.
All the energies are defined in units of the recoil energy Erec = h¯k
2
2m , so the potential V is
expressed in terms of a dimensionless parameter V˜ : V = V˜ Erec. Likewise, positions are
in units of the wavelength λ: x = x˜λ and k = k˜λ, so that k˜ = 2pi.
The angular frequency ω is obtained via:
ω2 = 1
m
∂2V
∂x2
∣∣∣∣∣
min
= 1
mλ2
∂2V
∂x˜2
∣∣∣∣∣
min
= Er
mλ2
∂2V˜
∂x˜2
∣∣∣∣∣
min
= Erk
2
4pi2m
∂2V˜
∂x˜2
∣∣∣∣∣
min
= E
2
r
2pi2h¯2
∂2V˜
∂x˜2
∣∣∣∣∣
min
.
(D.39)
Now we define V˜ = αI where I = ∑ cos2 is the functional dependence of the intensity of
the optical lattice, and α essentially the atomic polarisability (section 2.2.2).
Hence:
ω2 = αE
2
rec
2pi2h¯2
∂2I
∂x˜2
∣∣∣∣∣
min
, ω =
√√√√αE2rec
2pi2h¯2
∂2I
∂x˜2
∣∣∣∣∣
min
,
h¯ω =
√√√√ α
2pi2
∂2I
∂x˜2
∣∣∣∣∣
min︸ ︷︷ ︸
Calculated from code, in units of Erec.
Erec,
E0 = αIErec +
1
2
√√√√ α
2pi2
∂2I
∂x˜2
∣∣∣∣∣
min
+
√√√√ α
2pi2
∂2I
∂y˜2
∣∣∣∣∣
min
Erec .
(D.40)
272 Miscellaneous
So, the energy E0 in units of Erec is:
E0/Erec = αI +
1
2
√√√√ α
2pi2
∂2I
∂x˜2
∣∣∣∣∣
min
+
√√√√ α
2pi2
∂2I
∂y˜2
∣∣∣∣∣
min
. (D.41)
D.11 Fourier transforms
The derivations of the equations quoted in section 7.2.1 are presented here.
A 1D harmonic optical lattice results in the following potential:
V (x) = −V0 cos2(pix/x0), (D.42)
where x0 can be taken as λ/2 in order for the lattice sites to occur ∀i ∈ Z. Its Fourier
transform is a finite set, as the potential can be expressed as a Fourier series:
F [V ] = F [−V0 cos2(pix/x0)] = −V0
√
pi
2
{1
2δ(k + k0) + δ(k) +
1
2δ(k − k0)
}
, (D.43)
where k0 = 2pi.
For a real ionic crystal, the potentials are given by the Coulomb interaction:
V = −V0
∑
i∈Z
1
|x/x0 − i| , (D.44)
which can be also written as a convolution (∗):
V = −V0
∑
i∈Z
δ(x/x0 − i) ∗ −1|x/x0|
(D.45)
so that its Fourier transform may be analytically obtained by means of the convolution
theorem:
F [V ] = F
−V0
∑
i∈Z
δ(x/x0 − i) ∗ −1|x/x0|
= F
−V0∑
i∈Z
δ(x/x0 − i)
· F [ −1|x/x0|
]
.
(D.46)
The Fourier transform of the Coulomb potential is known to be:
F
[ −1
|x/x0|
]
= 4pi
k2
, (D.47)
and that of an infinite Dirac comb is just another infinite Dirac comb:
F
∑
i∈Z
δ(x/x0 − i)
= 1√
2pi
∑
j∈Z
δ(k − jk0). (D.48)
Hence:
F
[
−V0
∑
i
1
|x/x0 − i|
]
= 2
√
2pi
k2
V0
∑
j∈Z
δ(k − jk0), (D.49)
i.e. an infinite series of Dirac peaks with an decaying envelope.
D.12. Properties of localised states 273
D.12 Properties of localised states
In this section we qualitatively discuss the last three points of table 8.1.
D.12.1 Local discrete spectrum
Imagine locally exciting a system, for example by kicking it at a fixed position x. The re-
sponse of the wavefunction can be inferred from the response of the eigenstates composing
it. A 1D optical lattice exhibits Bloch waves (given by eq. 2.65), which are an example of
delocalised, thermal states. As shown in fig. D.1a, there are infinitely many Bloch waves
(the first three of which are plotted) and hence a continuum of states affected by the local
excitation. On the other hand, the eigenstates of a disorder-induced localised state are
exponentially decaying wavefunctions centred on each lattice site; hence, as shown in fig.
D.1b, a local excitation only affects a finite number of them.
The local spectrum in the (disorder-induced) localised phase is hence discrete, as opposed
to a continuum spectrum for a thermal, delocalised phase.
(a) (b)
Figure D.1 : A local excitation (green bar) affects infinitely many Bloch waves in the
thermal phase (a) and hence results in a continuous local spectrum. On the other hand,
a local excitation of the localised phase (b) only affects a finite number of eigenstates and
thus results in a discrete local spectrum.
D.12.2 Volume- and area-law entanglement
As for the previous section, the wavefunction in a thermal, delocalised phase is composed
of thermal, delocalised eigenstates such as, for instance, the Bloch waves (eq. 2.65). The
wavefunction in the disorder-induced localised phase can be decomposed into exponen-
tially decaying functions centred on each lattice site. In a given “volume” of the system,
each eigenstate ψ (thermal or localised) contributes a weight ∝ |ψ|2.
While thermal states are delocalised and thus contribute more |ψ|2 for larger “volumes”,
localised states only contribute their |ψ|2 when the boundary (“area”) delimiting the
“volume” includes them, as shown schematically in fig. D.2. This qualitatively justifies
the volume-law growth of the entanglement in thermal states, and the area-law growth in
single-particle or many-body localised states.
274 Miscellaneous
(a) (b)
Figure D.2 : Thermal states (Bloch waves, here) are delocalised and their |ψ|2 weight
thus increases with an increasing “volume” of interest (a). Localised states (b) are tightly
confined on single lattice sites, and thus only contribute their weight when the boundary
(“area”) includes them.
D.12.3 Logarithmic spreading of the entanglement
In the delocalised phase, the spread of the entanglement is related to the transport prop-
erties of the material, which are characterised by a a power-law and thus categorised as
diffusive, sub-diffusive, ballistic etc.
In the localised phase, particle transport is prevented by the insulating nature of the
state. In the Anderson phase, the absence of interactions precludes the spreading of
entanglement over any distance L, or the change of its value should there be any to begin
with. On the other hand, in the many-body localised (MBL) phase, interactions allow for
a dynamic change of the entanglement.
Given the exponential confinement of the localised wavefunction, it is natural to assume
a similar expression for the tunnelling element J as a function of distance L:
J ∝ J0 e−
L
ξ , (D.50)
where ξ is the localisation length.
In the tight-binding model, the Hamiltonian H is directly proportional to the tunnelling
element J , H ∝ J . The Hamiltonian also controls time evolution by |ψt〉 = eiHt/h¯|ψ0〉. For
entanglement to spread over a region L, we define a time t such that Jt > 1. Re-arranging,
this results in L ∼ ξ ln(J0t) and hence in a logarithmic spread of the entanglement.
D.13. Comic relief 275
D.13 Comic relief
Not the absorption image we deserve, but the one we need right now. A silent guardian,
a watchful protector. A dark knight.
(a) The Sun (b) A spiral galaxy
The camera needs to be focussed at infinity, so we used astrophysical objects.
276 Miscellaneous
(a)
(b)
Konrad testing alternative ways to visualise projections from higher dimensional spaces.
Tilting the screen by 45 degrees (a) to serve as V ⊥ in fig. 3.13, and using two sets of 3D
glasses (b) to probe SO(4) ∼= SO(3)× SO(3).
(a) ♠ (b) ♣
Typical inexplicable bugs which would cripple our python code and make us reconsider
our life choices.
The search for truth is more valuable than its possession.
Gotthold Ephraim Lessing
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278 Miscellaneous