KONRAD G I L B ERT HE I N R I C H V I E B AHN
QUAS I C RY S TA L L I N E
OP T I C A L L AT T I C E S
FOR U LTRACO LD ATOMS
T H I S D I S S ERTAT I ON I S SUBM I T T ED FOR THE DEGREE OF DOCTOR OF PH I L O SOPHY.
T R I N I T Y CO L L EGE , OC TOBER 2 0 1 8
CAVEND I S H L A BORATORY, U N I V ERS I T Y O F CAMBR I DGE
Copyright © 2018 K.G.H. Viebahn
“ T H E MOS T E S S EN T I A L PART O F A L I V I NG CE L L – T H E CHROMOSOME F I -
B R E – MAY SU I TA B LY B E CA L L ED AN APER I OD I C CRYS TA L . I N P H Y S I C S
WE HAVE DEA LT H I T H ERTO ON LY W I T H P ER I OD I C CRYS TA L S . TO A HUM -
B L E PHYS I C I S T ’ S M I N D, T H E S E ARE VERY I N T ERES T I NG AND COMP L I -
C AT ED OB J EC T S ; T H E Y CONS T I T U T E ONE OF THE MOS T FA SC I N AT I NG
AND COMP L E X MAT ER I A L S TRUC TURES BY WH I CH I N AN I M AT E NATURE
PU Z Z L E S H I S W I T S . Y E T, COMPARED W I T H THE APER I OD I C CRYS TA L ,
T H E Y ARE RATHER P L A I N AND DU L L . T H E D I F F ERENCE I N S TRUC TURE
I S O F THE SAME K I N D A S THAT B E TWEEN AN ORD I N ARY WA L L PA P ER I N
WH I CH THE SAME PAT T ERN I S R E P E AT ED AGA I N AND AGA I N I N R EGU L AR
PER I OD I C I T Y AND A MAS T ERP I E C E O F EMBRO I D ERY, S AY A RAPHAE L
TA P E S TRY, WH I C H SHOWS NO DU L L R E P E T I T I O N , BU T AN E L A BORAT E ,
COHEREN T, M E AN I NG FU L DES I GN TRACED BY THE GREAT MAS T ER .”
E RW I N SCHRÖD I NGER , “WHAT I S L I F E ? ”

Declaration
This dissertation is the result of my own work and includes nothing which is the outcome of
work done in collaboration except as declared in the Preface and specified in the text.
It is not substantially the same as any that I have submitted, or, is being concurrently submit-
ted for a degree or diploma or other qualification at the University of Cambridge or any other
University or similar institution except as declared in the Preface and specified in the text. 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 or similar institution except as declared in the Preface and
specified in the text.
This thesis does not exceed the word limit of sixty thousand words, including abstract, tables,
footnotes and appendices, set out by the Faculty of Physics and Chemistry.

Abstract
Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth
of intriguing physical phenomena, such as the inheritance of topological properties from higher
dimensions, self-similarity, and the presence of non-trivial structure on all scales. The concept of
aperiodic order has been extensively studied in mathematics and geometry, exemplified by the
celebrated Penrose tiling. However, the understanding of physical quasicrystals (the vast major-
ity of them are intermetallic compounds) is still incomplete owing to their complexity, regarding
both growth processes and stability.
Ultracold atoms in optical lattices offer an ideal, yet untested environment for investigating
quasicrystals. Optical lattices, i.e. standing waves of light, allow the defect-free formation of a
large variety of potential landscapes, including quasiperiodic geometries. In recent years, optical
lattices have become one of the most successful tools in the large-scale quantum simulation of
condensed-matter problems.
This study presents the first experimental realisation of a two-dimensional quasicrystalline po-
tential for ultracold atoms, based on an eightfold symmetric optical lattice. It is aimed at bring-
ing together the fields of ultracold atoms and quasicrystals – and the more general concept of
aperiodic order. The first part of this thesis introduces the theoretical aspects of aperiodic order
and quasicrystalline structure. The second part comprises a detailed account of the newly de-
signed apparatus that has been used to produce quantum-degenerate gases in quasicrystalline
lattices. The third and final part summarises the matter-wave diffraction experiments that have
been performed in various lattice geometries. These include one- and two-dimensional simple
cubic lattices, one-dimensional quasiperiodic lattices, as well as two-dimensional quasicrys-
talline lattices. The striking self-similarity of this quasicrystalline structure has been directly
observed, in close analogy to Shechtman’s very first discovery of quasicrystals using electron
diffraction. In addition, an in-depth study of the diffraction dynamics reveals the fundamental
differences between periodic and quasicrystalline lattices, in excellent agreement with ab initio
theory. The diffraction dynamics on short timescales constitutes a continuous-time quantum
walk on a homogeneous four-dimensional tight-binding lattice.
On the one hand, these measurements establish a novel experimental platform for investigating
quasicrystals proper. On the other hand, ultracold atoms in quasicrystalline optical lattices are
worth studying in their own right: Possible avenues include the observation many-body local-
isation and Bose glasses, as well as the creation of topologically non-trivial systems in higher
dimensions.

Contents
Introduction: quasicrystalline potentials for ultracold atoms 11
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1 Quasicrystals and aperiodic order 15
1.1 What is aperiodic order? What is a quasicrystal? . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Diffraction patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Cut-and-project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Phasons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3 Quasicrystals in two dimensions, the octagonal tiling . . . . . . . . . . . . . . . . . . . 26
Crystallographic restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Cut-and-project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Phasons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Physical quasicrystals and current state of the art . . . . . . . . . . . . . . . . . . . . . 31
2 Design of the apparatus 35
2.1 Suitable constituents for creating quantum many-body systems . . . . . . . . . . . . . 35
2.2 Choice of atomic species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Overview of the experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Laser cooling & trapping of 87Rb & 39K 41
3.1 The alkalis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Rubidium-87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Potassium-39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 87Rb & 39K in magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Interlude: calibration of magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Spectroscopy and absolute frequency stability (laser locking) . . . . . . . . . . . . . . 44
3.4 Relative frequency stability (offset locking) . . . . . . . . . . . . . . . . . . . . . . . . . 46
Rubidium repump lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Potassium imaging lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Fluorescence signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Final MOT stage in potassium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8 Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
MOT compensation coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.9 Optical pumping (spin-polarisation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
10
3.10 Magnetic trapping, transport, and experiment coils . . . . . . . . . . . . . . . . . . . . 53
Technical implementation of transport and experiment coils . . . . . . . . . . . . 55
Compensation coils (offset fields) . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Microwave (MW) evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Majorana losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.12 Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
High-field imaging in 39K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Stern-Gerlach imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Quantum gases 61
4.1 Dipole trap: introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Choice of trapping geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Trapping potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Evaporation and sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Dipole trap laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Dipole loading, guide field, and RF state transfer . . . . . . . . . . . . . . . . . . . . . 68
RF antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Feshbach fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Quantum degeneracy and Bose-Einstein condensation . . . . . . . . . . . . . . . . . . 70
Atom number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Eightfold optical lattice 73
5.1 1D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Time-evolution and lattice depth calibration . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Eightfold optical lattice potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Incoherent superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Relation to octagonal tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Phase control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Fully-interfering axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Partly-interfering axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Setup and characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Control loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Matter-wave diffraction experiment 91
6.1 Basis in momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Method: Kapitza-Dirac diffration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Diffraction dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4 Quantum walks in high-dimensional homogeneous tight-binding lattices . . . . . . . 97
6.5 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Summary and outlook 101
Next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A List of components 105
Acknowledgements 109
Bibliography 113
Introduction: quasicrystalline
potentials for ultracold atoms
Quasicrystals are long-range ordered without being periodic [1–4].
Their order manifests itself in sharp diffraction peaks, exactly as
in their periodic counterparts. However, diffraction patterns from
quasicrystals often reveal rotational symmetries, most notably
fivefold, eightfold, and tenfold, that are incompatible1 with trans- 1 crystallographically ‘forbidden’
lational symmetry. Therefore it immediately follows that long-
range order in quasicrystals results not from a periodic arrange-
ment of unit cells but from a different paradigm. Quasicrystalline
order can be seen as an incommensurate projection of a higher-
dimensional periodic lattice; this enables investigation of physics
of higher dimensions, in particular in the context of topology [5–
7]. For instance, one-dimensional (1D) quasiperiodic models, such
as the Fibonacci chain and the Aubry-Andre model, are closely
connected to the celebrated two-dimensional Harper model [6].
An alternative approach to constructing quasicrystals was de-
scribed by Penrose [8] who discovered a pentagonal tiling. Using a
particular set of tiles and associated matching rules2 it is possible 2 imagine a puzzle game
to cover a plane with an infinite tiling, in which non-periodicity is
enforced by the matching rules [9]. The fivefold symmetric Pen-
rose tiling and the closely related eightfold symmetric Ammann-
Beenker tiling [10] (also called ‘octagonal tiling’ [9]) have become
paradigms of 2D quasicrystals [9, 11]. In addition to their dis
allowed rotational symmetries, these tilings have the remarkable
feature of being self-similar in both real and reciprocal space [3].
Self-similarity upon scaling in length by a certain factor (the silver
mean 1 +
√
2 in case of the octagonal tiling) implies that non-
trivial structure is present on arbitrarily large scales. Correspond-
ingly, diffraction patterns from quasicrystals display sharp peaks
at arbitrarily small momenta.
To date, quasicrystals have been extensively studied in material
science, condensed-matter, and crystallography [11, 12]. Various
quasicrystalline model systems have been created, most notably
in photonic systems [13] and optical lattices [14–17]. Optical lat-
tices are particularly adept for emulating quasicrystals since high
rotational symmetries can easily be imposed [17]. However, this
advantage has yet remained unexplored in the field of quantum
gases.
12
Ultracold gases in optical lattices
Quantum gases of neutral atoms are one of today’s most ex-
citing experimental frontiers in quantum many-body physics.
Sparked by the observation of a quantum phase transition us-
ing ultracold atoms in an optical lattice [18] there has been much
progress towards Feynman’s original proposal of a quantum sim-
ulator [19]. A quantum simulator is a machine that solves numer-
ically intractable quantum-mechanical problems3 by mapping3 such as the dynamics of the doped
fermionic Hubbard model them to another quantum mechanical system with experimen-
tally accessible observables. Quantum gases are ideally suited for
this application due to the exquisit control of external parame-
ters, such as dimensionality, type and strength of interaction, and
geometry.4 The current experimental realisations of ‘quantum4 both in lattices and in bulk systems
simulators’ [20, 21] have implemented a multitude of theoretically
relevant models, in which numerical simulations are either very
hard or impossible [22–25], particularly in dimensions higher than
one.
But there is more: quantum simulation is just one interpreta-
tion of quantum gas experiments. Ultracold atoms have recently
been used to create previously unseen phases of matter, such as
supersolids [26, 27] and unitary Bose gases [28]. Quantum gases in
lattices offer a controllable and clean environment to probe novel
systems, such as Floquet-engineered hamiltonians [29]. Therefore,
these novel quantum systems are worth pursuing in their own
right and it is expected that they play an important role in future
applications of quantum technology [21, 30].
Scope
The experimental apparatus described in this thesis is aimed
at bridging the two fields of quantum gases and quasicrystals,
thereby creating a new platform to study aperiodic order with ul-
tracold atoms. Drawing from established techniques and concepts
in both fields, there are numerous directions in which experimen-
tation can progress. In particular, the ability to create arbitrarily
complex quantum systems by employing aperiodic order [31], and
the possibility of inheriting topological properties from higher
dimensions [6] are most promising.
Outline
This thesis offers a comprehensive review of all experimental
methods that were used to create the first quasicrystalline optical
lattice for ultracold atoms. A very brief overview of the whole ex-
perimental apparatus is given in Chapter 2. The methods chapters
focus on initial cooling from room temperature to the µK regime
(Chapter 3), reaching quantum degeneracy in the optical dipole
trap (Chapter 4), and the setup of the eightfold symmetric optical
13
lattice (Chapter 5). Afterwards, the main experimental outcomes
of this work are discussed in Chapter 6, namely matter-wave ex-
periments. They highlight the self-similarity of our quasicrys-
talline optical lattice, and its ability to simulate quantum walks in
higher dimensions.
Initially, quasicrystals in one and two dimensions will be in-
troduced (Chapter 1), thereby summarising key concepts such as
aperiodicity, self-similarity, non-periodic diffraction patterns, the
cut-and-project scheme, phasonic excitations and the role of ro-
tational symmetries. The current state-of-the-art in quasicrystal
research is reviewed at the end of Chapter 1.
Contributions
The apparatus as it is described in this thesis was jointly designed,
implemented, and characterised by the author and Matteo Sbroscia,
under supervision of Dr. Ulrich Schneider. Major contributions
were made by Dr. Stephen Topliss (electronics engineer), Hendrik
von Raven, Oliver Brix, Edward Carter, Max Melchner, Michael Höse,
and Jr-Chiun Yu.

1
Quasicrystals and aperiodic order
The aim of this chapter is to provide an accessible introduction
to the topic of quasicrystals and the concept of aperiodic order. It
will form the theoretical basis for the experimental results (Chap-
ter 6). On the path towards understanding quasicrystals we will
follow these steps:
1.1 What is a quasicrystal? The definition of quasicrystals. The
notion of aperiodic order.
1.2 A pedagogical example of a one-dimensional quasicrystal: the
Fibonacci chain. In this section two important concepts will be
introduced: self-similarity, and the cut-and-project method.
1.3 Quasicrystals in two dimensions. Crystallographic restriction
and the role of rotational symmetries in quasicrystals. The
octagonal aperiodic tiling (or Ammann-Beenker tiling).
1.4 Quasicrystalline materials. What is known about physical
quasicrystals? Classification of quasicrystals and open ques-
tions in quasicrystal research. What could our experiment con-
tribute to these questions?
Figure 1.1: Is this a crystal? The image
shows a single grain of icosahedral
Zn-Mg-Dy. Courtesy M. Feuerbacher,
Forschungszentrum Jülich.
1.1 What is aperiodic order? What is a quasicrystal?
What is a black hole? What is a magnetic field? What is an atom?
Ask these questions to a group of physicists and you will get more
or less consistent answers.1 Of course, the depths of research are
1 If you do not have a group of physi-
cists at hand, Wikipedia will do.
always full of intricacies and, consequently, the answers to some of
these questions may depend on context. But overall, a quick look
into standard literature will bring clarity.
Defining quasicrystals is much trickier. First, this is due to some
initial confusion after the discovery of quasicrystals [12]. Second,
literature on quasicrystals is scattered over many different fields,2 2 see ref. [9] for a comprehensive list
including crystallography [11], condensed-matter physics [12],
and mathematics [9]. Third, even after 35 years of research on
quasicrystals and their descendants, there are still many open
questions [11]. Finally, there has been a paradigm shift in the
community from ‘quasicrystals’ to the more general concept of
‘aperiodic order’ [9, 12]. We will get back to this at the end of the
chapter.
16 quasicrystalline optical lattices for ultracold atoms
Figure 1.2: Crystallographic definition
of crystals and quasicrystals [32]. The
discovery of quasicrystals contributed
the distinction between periodic and
aperiodic crystals to this classification.
Up until then order had been thought
to be synonymous with periodicity.
periodic
disordered ordered (=  sharp diffraction peaks)
aperiodic
periodic crystals
amorphous
solids 
quasicrystals
aperiodic crystals
incommensurate crystals
modulated compound magnetic
Let us start by defining the terms ‘crystal’ and ‘aperiodic crys-
tal’ according to the International Union of Crystallography
(IUCr) [33, 34].
Definition. In the following by ‘crystal’ we mean any solid having an
essentially discrete diffraction diagram, and by ‘aperiodic crystal’ we
mean any crystal in which three-dimensional lattice periodicity can be
considered to be absent.
Although the term ‘essentially’ is deliberately vague,3 the over-3 “The word essentially means that most
of the intensity of the diffraction is con-
centrated in relatively sharp Bragg peaks,
besides the always present diffuse scatter-
ing.” [34, 35]
all message is clear: long-range order – in this case embodied by
crystalline solids – is defined by sharp diffraction peaks. The as-
sociation of ‘order’ with ‘sharp diffraction peaks’ was introduced
in the wake of the discovery of quasicrystals, which proved that
the previous paradigm (‘order⇔ periodicity’) had been false.
After some initial debate (see ref. [12] for a review) a paradigm
shift took place in the community of crystallography. Nowadays
there is broad consensus about the definition of crystals by the ex-
istence of sharp diffraction peaks. There can be no doubt that the
images shown in Fig. 1.3, which were obtained from a single grain
of Zn-Mg-Dy (Fig. 1.1), are indeed signatures of a type of crystal.
However, the image in Fig. 1.3C displays a tenfold rotational
symmetry which is forbidden for periodic crystals.4 Therefore this4 we will prove this later
material must be aperiodic and ordered. We will expand more on
the notion of aperiodic order in section 1.2.
What about quasicrystals then? Are they the same as aperiodic
ordered structures? The IUCr dictionary offers two different defi-
nitions of a quasicrystal, with a varying degree of overlap with the
class of aperiodic ordered materials [32].
Definition A. A quasicrystal is an aperiodic crystal that is not an in-
commensurate modulated crystal, nor an incommensurate composite
crystal. Often, quasicrystals have crystallographically ‘forbidden’ sym-
metries. These are rotations of order different from 2, 3, 4 and 6.
This definition explicitly sets quasicrystals apart from two other
classes of aperiodic crystals, namely ‘incommensurate modulated
crystals’ and ‘incommensurate composite crystals’, as shown in
Fig. 1.2. Incommensurate modulated crystals represent crystals
with reciprocal lattices that include at least two incommensu-
rate wave vectors [36]. Incommensurate composite crystals are
quasicrystals and aperiodic order 17
CBA
Figure 1.3: Electron diffraction images
of icosahedral Zn-Mg-Dy (Fig. 1.1)
along a twofold- (A), threefold- (B),
and fivefold- (C) rotationally sym-
metric axis. Note that the patterns in
(B) and (C) appear to be sixfold- and
tenfold-symmetric, respectively. This
is a consequence of the diffraction
intensity being proportional to the
square of the modulus of the structure
factor, making reflections with oppo-
site indices (h, k, l) and (h¯, k¯, l¯) appear
equally strong. This effect is known as
‘Friedel’s law’ [38]. Images: courtesy
M. Feuerbacher, Forschungszentrum
Jülich.
structures made of at least two subcrystals with incommensu-
rate lattice vectors [37]. According to this definition, the set of
quasicrystals and the set of incommensurate crystals (ICs) are
disjoint, both being subsets of the set of aperiodic crystals. There
are overlaps between the subsets within ICs (and a third kind of
ICs, namely ‘incommensurate magnetic structures’ in which the
size of the structural unit cell is incommensurate with the size of
the magnetic unit cell). However, the distinction between ICs and
quasicrystals in the one-dimensional case is non-trivial, as we will
see later by studying diffraction patterns of 1D aperiodic lattices.
Therefore the IUCr offers an alternative definition [32, 35]:
Definition B. The term quasicrystal stems from the property of quasiperi-
odicity observed for the first alloys found with forbidden symmetries.
Therefore, the alternative definition is: a quasicrystal is an aperiodic
crystal with diffraction peaks that may be indexed by n integral indices,
where n is a finite number, larger than the dimension of the space (in
general). This definition is similar to that of aperiodic crystal.
‘Indexing’ here means that the position of each diffraction peak
can be written as a unique set of integers5 times a set of recipro- 5 or a single integer in periodic 1D
latticescal lattice vectors. In other words, it means ‘count the minimum
number of integers you need to uniquely identify all the diffrac-
tion peaks of a D-dimensional crystal. If this number is finite and
larger than D, then the crystal is a quasicrystal.’ This classification
is summarised in table 1.1. Here there is essentially no distinction
between quasicrystals and aperiodic crystals because the index
counting method does not distinguish these cases. This definition
of quasicrystals is more mathematical and, consequently, slightly
# integers needed (n) class of material
n uncountably infinite → disordered, amorphous
n countably infinite → almost periodic crystal
n finite
n > D → aperiodic crystal
n = D → periodic crystal
Table 1.1: Alternative classification of
crystals according to the number n of
integers needed to index the diffrac-
tion peaks, or equivalently, the mini-
mum number of (Fourier-)components
needed to write the atomic density
distribution as a superposition of
plane waves. The class of ‘almost peri-
odic crystals’ is largely theoretical, in
analogy with the mathematical class
of ‘almost periodic functions’ [39],
since they have not been observed in
experiments [35].
18 quasicrystalline optical lattices for ultracold atoms
more tangible, compared to definition A. It has a certain appeal
since it lists periodic crystals as natural ‘extension’ of aperiodic
crystals in the case n = D (number of integers needed to index the
diffraction pattern equal to the spatial dimension of the crystal).
Note that there is no distinction between the terms ‘quasicrys-
tal’ and ‘quasiperiodic crystal’; the two can be used interchange-
ably [35, 40]. However, the adjective ‘quasiperiodic’ often refers to
a special class of mathematical functions called ‘almost periodic
functions’ [11, 39, 41].66 The definition of almost periodic
functions goes back the Danish mathe-
matician Harald Bohr, brother of Niels
Bohr, and winner of an olympic silver
medal in football (1908).
The question of rotational symmetries is a subtle one because
neither of the definitions list them as a necessary requirement.
However, due to crystallographic restriction (section 1.3) high ro-
tational symmetries in ordered structures are a sufficient criterion
for classification as quasicrystal. For simplicity, we can assume
two-dimensional quasicrystals to have ‘forbidden’ rotational sym-
metries in this work7 since the eightfold optical lattice obviously7 thereby ignoring the existence of
a few aperiodic structures without
forbidden symmetry [11, 12]
fulfils the criterion. Since the precise definition of quasicrystals
is tricky, we shall simply use the term ‘quasicrystal’ for ordered
structures with forbidden symmetries,8 whereas the broader con-
8 and also for 1D quasiperiodic struc-
tures, for historical reasons
cept of ‘aperiodic order’ shall include quasiperiodic structures
without forbidden symmetries, as well as quasicrystals.
In order to build some intuition on these abstract definitions of
quasicrystals, we will now take the Fibonacci chain as an exam-
ple. All of the concepts that are introduced in section 1.2 for one
dimension will be generalised to two dimensions in section 1.3.
1.2 Fibonacci chain
The Fibonacci chain is the archetypical manifestation of aperiodic
order9 in one dimension. We construct the Fibonacci chain by9 and quasicrystallinity
repeatedly acting with the following substitution rule
L → LS (1.1)
S → L
on a ‘seed’ letter L. Equation 1.1 means that one letter L is to be
replaced by a sequence of two letters (LS). An alternative formula-
tion of the substitution rule is(
L
S
)
→
(
1 1
1 0
)(
L
S
)
, (1.2)
to the same effect that the sum in the matrix multiplication is to be
understood as the concatenation of two letters (L + S → LS). This
procedure leads to an arbitrarily long sequence of letters
L→ LS→ LSL→ LSLLS→ LSLLSLSL→ LSLLSLSLLSLLS→ . . .
which is called a Fibonacci chain (or ‘Fibonacci word’). In a dia-
grammatic representation the letters L and S can be understood as
long and short line segments, respectively; in this representation
the vertices between line segments correspond to lattice points.
quasicrystals and aperiodic order 19
Table 1.2: Number of letters in the
Fibonacci word after m substitution
steps. This sequence of numbers has
been known for a very long time,
at least since the 13th century [42];
nowadays these numbers are known as
Fibonacci numbers.
m 1 2 3 4 5 6 7
NS(m) 0 1 1 2 3 5 8
NL(m) 1 1 2 3 5 8 13
The occurrence of S and L letters (NS and NL, respectively) after
m substitution steps is listed in table 1.2 from which we recognise
the famous Fibonacci numbers. We denote the mth Fibonacci num-
ber by Fm such that NS(m) = Fm−1 and NL(m) = Fm. Moreover,
the total number of letters is then Ntot = NS(m) + NL(m) = Fm+1.
In the limit of m → ∞ the ratio between two subsequent Fibonacci
numbers Fm and Fm+1 is [12]
lim
m→∞
Fm+1
Fm
= 1+
1
1+ 1
1+ 11+...
= τ , (1.3)
where τ is an irrational number called the ‘golden mean’. There-
fore we know that in the infinite Fibonacci word the ratio of the
overall occurrence of L letters compared to the total number of
letters (the ‘frequency’ of L letters NL/Ntot) is irrational.
Conversely, if the Fibonacci chain was periodic it would contain
(at least) one sub-word that is repeated infinitely often. Therefore
the frequency of letters in the limit m → ∞, given by the ratio
of letters in this sub-word, would be rational. Thus the Fibonacci
chain is aperiodic.10 10 This is true independent of the pre-
cise value of L/S (the ratio of lengths
in the diagrammatic representation), in
contrast to ref. [43]. The cases L/S = 1
and L/S → ∞ (L being finite) are a
trivially periodic inasmuch as they
contain just one non-zero spacing.
At the same time the Fibonacci word is not random, since it was
constructed using a deterministic rule (Eq. 1.1). In other words:
the Fibonacci chain is long-range ordered.11
11 this will be shown explicitly later by
its diffraction pattern
The corresponding ‘Fibonacci lattice’ can be constructed from
the chain by placing atoms at the vertices in between the line
segments. We can write down the position x of the nth atom in a
closed form [44]
xn = S
(
n +
1
ν
⌊
n + 1
ν
⌋)
, (1.4)
emphasising the presence of long-range order in the system. Here
b·c denotes the floor function and ν ≡ L/S the ratio of lengths of
the two line segments.
To summarise, the Fibonacci chain is both aperiodic and long-
range ordered.
It is intriguing that as early as 1944,
Schrödinger [45] introduced the con-
cept of aperiodic order as a possible
explanation for life on earth. At the
time, the microscopic function of genes
and the structure of DNA was un-
known, but the question was already
recognised on how such a fragile struc-
ture (then estimated at around one
million atoms) could robustly encode
enough information, lasting many
generations, to form a living organ-
ism and seemingly violate the second
law of thermodynamics. Schrödinger
proposed an aperiodic crystal as ‘code-
script’ in which all the information
was encoded by the exact configura-
tion of atoms. It is now known that
DNA does not represent a determin-
istic recipe as Schrödinger envisaged
but it functions as a resource that in-
teracts with its environment, thereby
upholding a constant non-equilibrium
of the organism in its surroundings
(‘life’) which does not violate the over-
all rise in entropy. However, the gist
of his original idea that information
is carried by aperiodic sequences (the
order in which the nucleotides A, C,
G, and T appear) has indeed proven
correct. Ref. [46] offers a very recent
introductory review on Schrödinger’s
book. See also ref. [12].
So far, we have considered arbitrary values for L and S, leading
to a deterministic and aperiodic arrangement of atoms in the
Fibonacci lattice. Next we consider how a particular choice of ν
can lead to self-similarity.
Self-similarity
Self-similarity means invariance under length scaling operations.
Therefore, in order to construct a self-similar Fibonacci lattice we
simply demand that replacing L with LS and S with L (Eq. 1.1)
corresponds to the same (relative) increase in length, namely
L + S
L
!
=
L
S
≡ ν (1.5)
20 quasicrystalline optical lattices for ultracold atoms
Figure 1.4: Self-similarity of the
‘golden’ Fibonacci chain (ν = τ). If
the upper Fibonacci chain is scaled
down by a factor τ, the resulting
structure is again a Fibonacci chain.
Figure taken and adapted from [47].
or, equivalently,
1+
1
ν
= ν (1.6)
ν2 − ν− 1 = 0 (1.7)
leading to
ν± = 1±
√
5
2
(1.8)
ν+ =
1+
√
5
2
= 2 cos
(
3pi
10
)
= 1.618033989 · · · ≡ τ (1.9)
The value τ is the aforementioned limit of two adjacant Fi-
bonacci numbers, the golden mean.12,13 A Fibonacci lattice with12 Another way to get the values ν± is
to simply diagonalise the substitution
matrix in Eq. 1.2.
13 It is called an ‘algebraic number’
because it is the solution to Eq. 1.7
which only has rational (in this case
integers) coefficients.
the self-similar property ν = τ is shown in Fig. 1.4. Here self-
similarity manifests itself in the appearance of structures on larger
and larger lengthscales.
Self-similarity is a characteristic feature of quasicrystals which
we are able to directly observe in the experiment (Chapter 6). It
constrasts the usual notion of coarse-graining in periodic lattices:
In the thermodynamic limit (system size V → ∞ and N → ∞,
while N/V = constant) the ratio of lattice spacing a to system
size tends to zero. Thermodynamic properties such as entropy
or magnetisation therefore do not depend on the value of a. In
self-similar systems such as quasicrystals this approach fails since,
even for arbitrarily big lengthscales, the system still resembles an
inflated version of the microscopic model.
Apart from being conceptually interesting, self-similarity
frequently appears in nature and society in the form of frac-
tal structures, for example in network traffic [48], biology and
medicine [49], and cosmology [50]. There are also well-known ex-
amples in various plants, such as romanesco broccoli, ferns, and
sunflowers.
The study of ultracold atoms in quasicrystalline lattices might
provide a novel approach to studying fractal structures in a well-
controlled environment. The use of massive, quantum degener-
ate particles – atoms – in this work represents a major advance
compared to previous studies of quasicrystals with massless pho-
tons [13] and classical particles in optical lattices [14–17].
Having covered the concept of self-similarity we can build some
physical intuition on long-range order in quasicrystals by looking
at their diffraction pattern, i.e. their structure in reciprocal space.
This is particularly relevant for the diffraction experiments that
were performed in this work using ultracold atoms. In addition,
quasicrystals and aperiodic order 21
diffraction patterns are the only way of fundamentally discrimi-
nating between quasicrystals and disordered aperiodic structures.
Diffraction pattern of the Fibonacci lattice
The density distribution of a 1D discrete lattice can be written as
ρ(x) =
1
N
N
∑
n=1
δ(x− xn) , (1.10)
where the sum goes over all lattice points, xn are the positions
of the atoms (taken from Eq. 1.4 in the case of a Fibonacci lat-
tice), and δ(x) represents the Dirac δ-distribution. The diffraction
pattern arising from ρ(x) is given by its structure factor which
corresponds to the modulus squared of its Fourier transform
F(q) ≡ |F [ρ(x)]|2 =
∣∣∣∣ 1√2pi
∫ +∞
−∞
ρ(x)eiqx dx
∣∣∣∣2 . (1.11)
Here q is the change in wave vector due to the diffraction.
Periodic lattices: periodic long-range order
0 1 2 3 4 5 6 7 8
q/(2pi)
0.00
0.05
0.10
0.15
F(
q)
0 1 2 3 4 5 6 7 8
q/(2pi)
0.00
0.05
0.10
F(
q)
A
B
0 1 2 3 4 5 6 7 8
q/(2pi)
0.00
0.05
0.10
F(
q)
C
Figure 1.5: Theoretical diffraction
patterns (Eq. 1.11) for three different
lattices. (A) Periodic lattice with
spacing S = 1. (B) Periodic lattice
of the type LSLSLS. . . with spacing
1 + τ (two atoms per unit cell). (C)
Fibonacci lattice with ν = L/S = τ.
The diffraction peaks in (A) and
(B) are perfectly regularly spaced
(see text), indicating the presence of
periodic long-range order. In contrast,
(C) consists of a dense set of peaks,
highlighting the aperiodic long-
range order of the Fibonacci lattice.
Although the envelope in (C) looks
very similar to that of (B), the main
peaks are not spaced regularly as can
be seen upon close inspection (see also
Fig. 1.7). All simulated lattices in this
figure consist of 21 atoms; the value of
τ (golden mean) is given by Eq. 1.9.
Before we come to the Fibonacci lattice, les us consider the diffrac-
tion pattern resulting from a periodic lattice with unity spacing
(S = 1). It is simply a comb of δ-peaks with regular spacing
q0 = 2pi/S = 2pi, as shown in Fig. 1.5A. There is a weak modu-
lation of the height of the peaks and on a much smaller scale than
q = 2pi there are ripples in the pattern which are not resolved on
the figure. Both of these effects arise from the finite size (21 atoms)
of the simulated lattice. This regularly-spaced diffraction pattern
is the signature of periodic long-range order.
Next, we consider another periodic lattice of the same size
(N = 21) but with two incommensurable spacings S = 1 and L =
τ, i.e. LSLSLS. . . . The diffraction pattern for this configuration is
plotted in Fig. 1.5B. Similar to Fig. 1.5A, the spacing between the
peaks is perfectly regular; in this case it is given by
q0 ≡ 2piS + L =
2pi
1+ τ
. (1.12)
As before, the regularly-spaced peaks herald periodic long-range
order. However, the height of the peaks is strongly modulated
which is a consequence of the presence of two incommensurable
lengths S and L. The overall structure is perfectly periodic.
Fibonacci lattice: aperiodic long-range order
The diffraction pattern of a 21-atom ‘golden’ Fibonacci lattice
(ν = τ)14 is plotted in Fig. 1.5C. At first sight it may look simi- 14 Note that any irrational value of ν
leads to a quasicrystal, not just ν = τ.
But we have chosen the special value
ν = τ because it nicely illustrates
the self-similarity of the ‘golden’
Fibonacci chain in reciprocal space.
This is already hinted at in Fig. 1.5B
which – although it is entirely periodic
in real space – shows signatures of
self-similarity.
lar to the periodic case but there are two important differences.
Firstly, the main peaks are not spaced periodically (in contrast to
Fig. 1.5B) but aperiodically. Secondly, there are many more peaks
in between the main peaks and there exists no single fundamental
distance q0. By indexing this diffraction pattern we can demon-
strate the aperiodic order of the Fibonacci chain.
22 quasicrystalline optical lattices for ultracold atoms
Indexing quasiperiodic diffraction patterns: aperiodic order
First, let us take a closer look at the diffraction pattern of the Fi-
bonacci lattice (Fig. 1.5C). A single reciprocal lattice vector, i.e. a
single index (h) where h ∈ Z, is not sufficient to uniquely identify
all the peaks since they are not spaced by equal distances. Simi-
larly, in the in-plane diffraction image of a physical quasicrystal
(Fig. 1.3C) two indices (h, k) are not sufficient to account for all
the peak positions since the distances in reciprocal space become
smaller and smaller. In fact, the peaks are densely spaced in these
cases due to the incommensurability of two (or more) lengths in
the quasicrystal. For example, in the ‘golden’ Fibonacci lattice
L = τ is incommensurable with S = 1. Consequently, there are as
many peaks as there are rational numbers: infinitely many. Does
this mean indexing the golden diffraction pattern (Fig. 1.5C) is a
lost cause?
1
1
1
2
1
3
1
4
1
5 · · ·
2
1
2
3
2
5
3
1
3
2
3
4
4
1
4
3
. . .
5
1
5
2
...
Figure 1.6: Cantor’s proof of Q+ being
a countable set. The sketch shows
the existence of a bijection from N to
Q+ by counting the rational numbers
diagonally [51].
Luckily it does not. An intuitive way of counting all positive ra-
tional numbers, introduced by Cantor in the 19th century, comes to
the rescue. We simply have to write the rational numbers down in
a two-dimensional array (Fig. 1.6), ordered by numerators (rows)
and denominators (columns). Subsequently we remove all dupli-
cates15 and then each rational number along the diagonal can be
15 e.g. 2/4 = 1/2, 2/2 = 1
identified with one non-negative integer, as shown in Fig. 1.6.
Inspired by the method of counting Q+ we can index all the
diffraction peaks of the 1D quasicrystal by using two integers
instead of one, namely
ql,m =
2pil
a
+
2pim
b
, where l, m ∈ Z and a, b ∈ R . (1.13)
At least one of the values a, b has to be irrational. In the case of
the golden Fibonacci lattice (ν = τ) the values for a and b can be
derived analytically [44] using Eq. 1.4. The result is
ql,m =
2pi
1+ 1
τ2
(
l
τ
+ m
)
. (1.14)
The positive indices for l, m ≤ 8 are shown in Fig. 1.7, together
with the simulated diffraction for the 21-atom golden Fibonacci
lattice (Fig. 1.5C).
How to tell the Fibonacci lattice apart from a random lattice?
The dense set of diffraction peaks resulting from the Fibonacci
lattice poses an interesting question: How can we be sure that the
Fibonacci chain is ordered and not random? On the one hand, the
deterministic construction rule should ensure perfect long-range
order. On the other hand, long-range order is really defined by the
existence of sharp diffraction peaks. Therefore it must be possible
to distinguish a random lattice from a Fibonacci lattice just by
observing their respective diffraction patterns.1616 In other words: The diffraction
process does not know about the rule
by which the lattice was created. It
only knows about the resulting pattern
of lattice sites.
For a random lattice, let us consider a non-deterministic con-
struction rule in which the length of each bond is randomly cho-
sen to be either S = 1 or L = τ. The probability for each line
quasicrystals and aperiodic order 23
0 1 2 3 4 5 6 7 8
q/(2p)
0.00
0.05
0.10
F(
q)
(0
,1
)
(1
,0
)
(1
,1
)
(1
,2
)
(1
,3
)
(2
,2
)
(2
,3
)
(2
,4
)
(3
,4
)
(3
,5
)
(3
,6
)
(4
,5
)
(4
,6
)
(4
,7
)
(4
,8
)
(5
,7
)
(5
,8
)
Figure 1.7: Two-dimensional indexing
for the 1D golden Fibonacci lattice.
The dashed lines are labelled accord-
ing to Eq. 1.14. The diffraction pattern
was obtained from a 21-atom Fibonacci
lattice with ν = τ.
0 1 2 3 4 5 6 7 8
q/(2pi)
0.00
0.05
0.10
F(
q)
(2
)
(3
)
(5
)
(8
)
(1
0)
(1
1)
(1
3)
(1
6)
(1
8)
(2
1)
0 1 2 3 4 5 6 7 8
q/(2pi)
0.00
0.05
0.10
0.15
F(
q)
/
0.
01
analytic
numerical
A
B
Figure 1.8: Diffraction patterns of 1D
random lattices with ν = L/S = τ (see
text). (A) Lattice length N = 21. The
most pronounced diffraction peaks
are spaced periodically in distances of
q0 = 2pi/(S + L) (c.f. Fig. 1.5B) which
are labelled according to their position.
These peaks arise from the frequently
occurring pattern LS. In between the
pronounced peaks there are many
more sharp peaks which are not
spaced periodically. These could be
(wrongly) interpreted as the signature
of aperiodic order. (B) N = 1000
(light blue). For a larger system size
the disorder becomes apparent: the
diffraction pattern becomes continuous
and smeared out. The numerical
curve is overlaid with the analytic
results from ref. [9] (black line) which
corresponds to an infinite random
chain. For even larger system sizes
the diffraction pattern would more
closely resemble that of the analytic
calculation. Some peaks remain sharp
due to a frequent occurrence of certain
patterns, the same as (A).
segment being L shall be 1/τ such that the overall occurrence
of L is roughly equal to that in the aperiodic ordered Fibonacci
chain.17 The diffraction pattern of such a random lattice (N = 21) 17 this random chain is discussed in
more detail in ref. [9]is shown in Fig. 1.8A. At first sight it does not differ greatly from
the diffraction pattern of the perfectly ordered Fibonacci lattice
(Fig. 1.7); the most pronounced peaks are spaced similarly but not
identically to the periodic case. For this random lattice a single
index is sufficient to label them, as demonstrated in Fig. 1.8A. The
background in the random case shows some noise which could be
interpreted as a dense set of peaks. In this system size (21 atoms)
there is no qualitative difference between a random lattice and an
24 quasicrystalline optical lattices for ultracold atoms
aperiodic ordered lattice. However, the randomness does show
up in a sufficiently large system (Fig. 1.8B). For N = 1000 the
diffraction pattern becomes smooth18 and tends towards the ana-18 although it retains some sharp
peaks which arise from the frequent
occurrence of certain combinations
such as LS and LLS
lytic solution for an infinite random chain [9]. From this we draw
the conclusion that system size is crucial for determining true
long-range order.19
19 As usual with quasicrystals, it is
not always as simple as that. A class
of pathological 1D quasicrystals was
constructed by Baake and Grimm [52]
whose diffraction patterns cannot
be distinguished from completely
disordered structures. Nevertheless,
for all practical purposes the sharp
peaks in the diffraction pattern are a
clear signature of order.
Projection from higher dimensions
The indexing method described above uses two indices to describe
the diffraction pattern of a 1D quasicrystal. On the one hand,
the requirement of two indices can simply be understood as the
consequence of the presence of two incommensurable lengths
in the system. On the other hand, the double-indexing naturally
leads to the notion that dimensions higher than the physical ones
are at play in quasicrystals. Therefore one can re-interpret the
diffraction pattern in Fig. 1.7 as the diffraction off a (fictitious)
two-dimensional periodic lattice, projected down to one dimen-
sion. This concept has proven extraordinarily powerful in the
determination of quasicrystalline structure and it is nowadays
routinely used to describe and classify quasicrystals [11, 53, 54]. It
extends naturally to physical dimensions higher than one. In gen-
eral, the number of extra dimensions needed to describe a given
quasicrystal is equal to the number of its linearly independent
lattice vectors over Q [55]. For example, the icosahedral symme-
try of the Zn-Mg-Dy quasicrystal (Fig. 1.1) requires six indepen-
dent lattice vectors to describe its diffraction pattern (Fig. 1.3)
and consequently the periodic ‘image’ lattice is six-dimensional.
Our experimental setting involves a planar, eightfold symmetric
optical lattice which can be uniquely described by four indepen-
dent lattice vectors. Consequently, the experiment is connected
to four-dimensional space; Chapter 6 shows the direct simula-
tion of a continuous-time quantum walk on a homogeneous 4D
tight-binding lattice using this notion.
The projection method is not only useful for crystallography
and indexing but it is also used to construct models of quasicrys-
tals in real space via the so-called ‘cut-and-project’ scheme.2020 also called ‘projection method’ or
‘strip-projection’
Figure 1.9: Schematic of the cut-and-
project scheme for a rational slope
(m = 2/3) leading to a periodic 1D
crystal. The width of the acceptance
window is given by the projection
of the 2D unit cell (black square) to
perpendicular space.
Cut-and-project
The cut-and-project scheme is widely used to contruct models of
quasicrystals, as well as study their physical properties. It is il-
lustrated in Figs. 1.9 and 1.10 for one-dimensional structures. The
construction involves defining a periodic lattice, for example a
simple cubic lattice (∼= Z2) in two dimensions (x, y). Rotation by
an angle α defines a new coordinate system (x′, y′) which can be
interpreted as parallel space (x′) and perpendicular space (y′). Par-
allel space corresponds to the new, physical 1D lattice. In perpen-
dicular space a certain ‘acceptance window’ (or ‘slice’) is chosen
and points within this window are then projected to parallel space
to form the new one-dimensional structure. In one dimension the
quasicrystals and aperiodic order 25
description can be written in terms of a slope m = tan α. The
width ∆ of the slice is given by the projection of the diagonal of
the 2D unit cell (black square in Fig. 1.9) to perpendicular space,
namely
∆ =
1+ m√
1+ m2
. (1.15)
Figure 1.10: Schematic of the cut-and-
project scheme for an irrational slope
(m = 1/τ) leading to an aperiodic 1D
crystals, namely the Fibonacci lattice.
The crucial parameter in this construction is the value of the
slope m. If it is chosen to be rational, the constructed lattice will be
periodic (Fig. 1.9). If it is chosen to be irrational, the constructed
lattice is aperiodic (Fig. 1.10). In particular, if we choose m = 1/τ
the resulting lattice is the golden Fibonacci chain.
Note that although the ‘window’ itself looks two-dimensional it
is actually just a 1D width in perpendicular space. All points that
fall into the width under perpendicular projection are ‘accepted’
and then projected to parallel (physical) space.
Phasons
Figure 1.11: Phasonic excitation of
the Fibonacci lattice. The acceptance
window has been displaced by 0.3
lattice spacings compared to Fig. 1.10
(the original Fibonacci lattice is also
shown). Small displacements of the
acceptance window in the perpen-
dicular direction can lead to sudden
displacement of some atomic posi-
tions. Sequences of LS turn into SL, a
process called ‘phason flip’.
Figure 1.12: Phasonic excitation with
non-zero wavelength (see text).
In periodic lattices there are two types of low-energy excitations,
namely electronic excitations and phonons (lattice vibrations).
Without going into too much detail, the cut-and-project scheme
can help us gain some intuition on additional excitations that
are unique to quasicrystals, called phasons [56]. Phasons are
markedly different from phonons which are characterised by wave
packets, propagating through the crystal.
In quasicrystals it can be energetically favourable for an atom
to perform a sudden ‘phason jump’ to a new equilibrium position,
rather than being excited around its original equilibrium posi-
tion. This process can be illustrated by phonons in perpendicular
space. In Fig. 1.11 the acceptance window in the cut-and-project
scheme has been displaced by a small amount (0.3 lattice spac-
ings). The resulting lattice is another golden Fibonacci lattice but
compared to Fig. 1.10 some atoms have jumped to a new position
by a distance 1− τ. These are just the simplest cases of phasonic
excitations, corresponding to a centre-of-mass movement of the
acceptance window (q = 0 phonon modes). Non-zero q phonon
modes, i.e. phonons with finite wavelength, introduce displace-
ments that can lead to the appearance of subsequences, such as
LSLSL or SLLLS, that do not appear in the Fibonacci lattice (see
Fig. 1.12). Moreover, a change in phase of the finite-wavelength
acceptance window can lead to the appearance of new spacings,
namely L − S and L + S in the resulting 1D lattice, as shown in
Fig. 1.13.
Figure 1.13: Phasonic excitation with
non-zero wavelength and different
phase compared to Fig. 1.12 above.
New spacings are introduced in this
case.
These phasonic excitations, in contrast to phonons, are not
propagative but diffusive, meaning that their ‘dispersion relation’
is purely imaginary [56]. Phasonic excitations can lead to broad-
ening of diffraction peaks and diffuse background scattering. It
is expected that phasons contribute considerable entropy to qua-
sicrystals, even at low temperatures. Phasons could potentially
stabilise them against energetically competing periodic configura-
26 quasicrystalline optical lattices for ultracold atoms
tions [53], i.e. minimise the free energy F = E− TS by increasing
the TS contribution (T being temperature, S being entropy, and E
being the internal energy).
Up to this point quasicrystals have been discussed in one di-
mension, thereby reviewing their most important characteristics,
namly quasiperiodicity, self-similarity, and the projection from
higher dimensions. However, one-dimensional space does not
support rotations. Therefore, 2D quasicrystals will be covered in
the following, beginning with the notion of ‘forbidden’ rotational
symmetries. After all, an unexpected tenfold symmetry in the
diffraction pattern of Al86Mn14 lead to the discovery of quasicrys-
tals [1] in the first place.
1.3 Quasicrystals in two dimensions, the octagonal tiling
Before we come to two-dimensional quasicrystals we need to take
a step back and consider two-dimensional periodic crystals.
A periodic lattice can be uniquely defined by its unit cell and a
set of lattice vectors. Translation of a given lattice point by any of
the lattice vectors results in an equivalent lattice point. The unit
cell describes the arrangement of atoms relative to a lattice point.
Note that in any physical system the lattice will be discrete, i.e. the
distance between any two lattice points is larger than zero. In the
following it will be shown that periodic lattices are incompatible
with rotational symmetries higher than sixfold rotational sym-
metry and also incompatible with fivefold rotational symmetry,
following an original proof of ref. [57].
Crystallographic restriction
Theorem. Periodic lattices are not compatible with any n-fold rotational
symmetry for n > 6 and n = 5.
P1 P2
P3
2�/8
A B
P0
Figure 1.14: Sketch of crystallographic
restriction for the case n = 8. (A) and
(B) represent the two contructions
of an eightfold symmetric polygon
that are used in the proof. An infinite
series of polygons can be constructed
by using the edges of (B) to construct
yet another polygon, and so on. The
fact that the perimeter of (B) is smaller
than that of (A) ensures that in the
limit of this series the value of the
perimeter vanishes, thus contradicting
the discreteness of the lattice.
Proof. Consider an n-fold rotational symmetry for n > 6 on a
discrete lattice (the case n = 5 will considered separately). We
start constructing a regular polygon ‘A’ from one lattice point
P1 by adding a lattice vector to reach another lattice point P2.
Subsequently we add a second lattice vector, rotated by 2pi/n,
to the point P2 to reach a third lattice point P3. We keep doing
this until we reach again P1, having constructed A in its entirety;
an example of this construction is shown in Fig. 1.14 for the case
n = 8. Let the perimeter of A be called sA.
Now, if we protract all the above vectors
−−→
P1P2,
−−→
P2P3, . . . from
one single lattice point P0 we have constructed another regular
polygon ‘B’ with smaller perimeter sB, shown in Fig. 1.14 (blue
line). The ratio between the two perimeters of A and B is only
dependent on n, namely
sB
sA
= 2 sin
(pi
n
)
< 1 . (1.16)
quasicrystals and aperiodic order 27
Taking the edges of the smaller polygon (B) as the new vectors
to construct yet another polygon C, and so on, will lead to an
infinite series of polygons. Since the ratio of perimeters of two
subsequent polygons in this series is strictly smaller than one
(Eq. 1.16) the edges of the polygons in the series converge to zero.
This contradicts the physical assumption of a discrete lattice.
The special case of n = 5 entails the identical construction of
the pentagon A but it necessitates a slightly more complicated
construction of the pentagon B (and the successive elements of
the series). In this case, the smaller pentagon (B) is constructed by
successively protracting the vectors
−−→
P5P4,
−−→
P1P5, . . . from the lattice
points P1, P2, . . . , rather than to one central point. The ratio of
perimeters is then
sB
sA
=
∣∣∣1− sin2 (pi
n
)∣∣∣ < 1 .
This construction can be used analogously to prove the theorem
for all cases including n = 5 and n > 6.
The proof of crystallographic restriction completes the discus-
sion of periodic lattices.
Since sharp diffraction patterns with tenfold rotational sym-
metry have been observed [1] (see also Fig. 1.3) it is immediately
clear that in an aperiodic lattice higher rotational symmetries must
be somehow possible. Furthermore, crystallographic restriction
implies that experimental evidence of sharp diffraction peaks
with, say, eightfold symmetry is a sufficient criterion to identify
a quasicrystal.21 In order to get a more microscopic picture of 21 However, one must avoid various
crystallographic pitfalls that can lead
to apparent rotational symmetries in
the diffraction pattern. For example, a
particular arrangement of four atoms
in a 2D simple cubic lattice produces
an approximately eightfold symmetric
diffraction pattern (see ref. [58], section
3, or ref. [9], Chapter 9). This effect is
very similar to the diffraction pattern
obtained from a 1D simple cubic
lattice with two atoms per site which
are arranged in a particular way
(Fig. 1.5B). This shows signatures of
self-similarity although the underlying
lattice is entirely periodic.
two-dimensional quasicrystal it is instructive to consider aperiodic
tilings, a subject within the mathematical field of geometry.22
22 Refs. [3] and [9] give an accessible in-
troduction to the topic. An exhaustive
treatment can be found in ref. [10].
Aperiodic ordered tilings
A tiling, or tessellation, represents a way to cover a plane using a
set of geometric shapes (‘tiles’), without any gaps or overlaps [10].
Until the early 60s it was assumed by Hao Wang [59, 60] and
others that all coverings of the plane using square tiles with deco-
rated edges would repeat themselves after a long distance, i.e. be
periodic. However, it was later shown that aperiodic coverings of
the plane do indeed exist [61]. Moreover, it was shown that, given
a certain set of tiles, it is generally impossible to decide whether
or not these tiles support an infinite covering of the plane; this is
known as the ‘undecidability of the domino problem’. This proof
is another representation of the undecidability of the Turing ma-
chine halting problem [62]. It has even been shown that these
‘Wang tiles’ can perform computations in the same way a Turing
machine does [63].
Very recently, the undecidability of the domino problem has
been mapped to a very general class of many-body hamiltoni-
ans, namely two-dimensional spin systems, with the result that
is impossible to predict whether these hamiltonians support
28 quasicrystalline optical lattices for ultracold atoms
gapless or gapped excitations [31]. It is therefore to be expected
that constructing physical systems of quantum particles in ape-
riodic potentials such as our quasicrystalline lattice will result in
many-body systems of unprecedented complexity. Maybe self-
assembly methods could be exploited similar to those that allow
DNA-strands to be used as Wang tiles in the construction of crys-
tals [64]. It is not unthinkable that one day atoms in quasicrys-
talline lattice might act as ‘quantum Turing machine’.
Figure 1.15: A patch of the octagonal
tiling.
The aperiodic tilings considered so far consist of square ‘Wang
tiles’ and hence they do not explain the existence of non-crystallographic
rotational symmetries. The first pentagonal tiling was found in
1974 by Penrose [8]:
I had often doodled by fitting together limited configurations of
pentagons and similar shapes but I had never found a good rule for
continuing such patterns indefinitely. However, recently I wanted
to design something interesting for someone who was in hostipal to
look at and I realised that there was a certain definite rule whereby
one could continue such a pattern to arbitrary size.
This pattern is nowadays widely known as the Penrose tiling.
Later, an eightfold analogue to the Penrose tiling was found, the
so-called Ammann-Beenker tiling [10] or ‘octagonal tiling’. A
patch of the octagonal tiling is shown in Fig. 1.15, consisting of
two types of tiles (a square and a rhombus) from which the entire
pattern can be pieced together indefinitely.
Construction rules
How is it possible to construct an aperiodic tiling, such as the
octagonal tiling? Similar to the Fibonacci chain there are construc-
tion rules that allow deterministic tiling of the plane using two
different shapes (in this case a square and a rhombus). In brief, the
construction relies on matching rules for adjacent tiles, similar to a
puzzle game. This process is illustrated in Fig. 1.16A by ‘decorat-
ing’ the vertices and the edges of the tiles with arrows. The arrows
on the vertices add local information to the tiles, thereby ensuring
‘validity’ of the infinite tiling, i.e. that there are no gaps. Non-
deterministic construction rules can be used to generate random
octagonal-like tilings with approximate eightfold symmetry [9].
Another way of constructing the octagonal tiling makes use of
its self-similarity.
Self-similarity
The octagonal tiling is self-similar under scaling the edges of the
tiles by the so-called silver mean,
λ = 1+
√
2 . (1.17)
Similar to the golden mean (Eq. 1.9) λ is an algebraic number,
obeying the equation
x2 − 2x− 1 = 0 , (1.18)
quasicrystals and aperiodic order 29
CBA
Figure 1.16: Octagonal tiling. (A) ‘Dec-
orated’ octagonal tiling, illustrating
the construction rules. (B) - (C) Self-
similarity of the octagonal tiling under
‘silver mean’ scaling. These images
were generated using the Mathematica
package ‘Aperiodic tilings’ [65], see
also ref. [47].
and it can be represented by a continued fraction analogous to
that in Eq. 1.3
λ = 2+
1
2+ 1
2+ 12+...
. (1.19)
However, the self-similar scaling operation of the octagonal tiling
is less trivial than its 1D counterpart, in which applying the scal-
ing operation to any patch of the Fibonacci chain results in an-
other valid Fibonacci chain (Fig. 1.4). In the two-dimensional case
of the octagonal tiling the scaling operation by a factor λ can nev-
ertheless be used if certain rules are obeyed which are shown
in Fig. 1.17. These allow self-similar scaling even if the position
of the origin is unknown. One step of self-similar scaling of the
octagonal tiling – also called ‘inflation’ – is shown in Fig. 1.16B-C.
Figure 1.17: Scaling by a factor λ
(Eq. 1.17) generates a self-similar copy
of the octagonal tiling (Fig. 1.16 B
and C).
The construction of the octagonal tiling based on local rules
(even in the case of self-similar inflation) is cumbersome because it
is an iterative process. Similar to the Fibonacci chain, the construc-
tion via cut-and-project is far easier and much more efficient. In
addition, it can provide intuition about phasonic excitations in this
lattice.
Cut-and-project
In the case of the octagonal tiling the cut-and-project scheme starts
with constructing a four-dimensional (4D) simple-cubic lattice
L ∼= Z4. The basis of L can be written as

1
0
0
0
 ,

0
1
0
0
 ,

0
0
1
0
 ,

0
0
0
1

 (1.20)
in canonical coordinates (x1, x2, x3, x4), xi ∈ Z. Similar to the
Fibonacci case we define a rotation [9]
R = 1
2

√
2 1 0 −1
0 1
√
2 1√
2 −1 0 1
0 1 −√2 1
 (1.21)
30 quasicrystalline optical lattices for ultracold atoms
Figure 1.18: Parallel (A) and perpen-
dicular (B) projection (‘shadow’) of a
hypercube (unit cell of 4D simple-cubic
lattice) using Eq. 1.22. The four basis
vectors (Eq. 1.20), i.e. edges of the unit
hypercube, are shown in blue (A) and
in red (B), respectively. The shadows
in both parallel and perpendicular
projection are identical. The hypercube
has an edge length of unity whereas
the shadows have edge lengths 1/
√
2.
(C) The acceptance window in per-
pendicular space (grey) for the 4D
cut-and-project scheme is the 2D
perimeter of the shadow in perpen-
dicular projection, i.e. outer green line
in (B). The points in (C) are the same
elements of Z4 (or, equivalently, the
4D simple-cubic lattice) that are also
shown in Fig. 1.19 in physical/parallel
space. (A-C) are drawn to the same
scale.
A B C
such that the new coordinates are
x
y
x′
y′
 = R

x1
x2
x3
x4
 . (1.22)
The first two coordinates (x, y) are parallel (physical) space and
the last two (x′, y′) are perpendicular space; a projection of (x1, x2, x3, x4)
to perpendicular space corresponds to multiplication by the first
two rows of R. The projection to parallel space, correspondingly,
is given by the multiplication by the last two rows of R.
Figure 1.19: Octagonal tiling generated
by projection from a Z4 using a 4D
hypercube with seven lattice sites per
edge and unity lattice spacing. The
bond lengths (black lines) resulting
from this projection have length 1/
√
2.
As before, the width of the acceptance window is given by the
projection of the 4D unit cell to perpendicular space, which means
that the acceptance window in this case is two-dimensional. The
projections of the 4D unit cell are shown in Fig. 1.18A-B; both
perpendicular and parallel projections (‘shadows’) have a 2D
octagonal perimeter of the same size. The acceptance window
is then chosen to be the perimeter of this octagon (green line in
Fig. 1.18B). The points that fall into this window under perpendic-
ular projection (Fig. 1.18C) form the octagonal tiling in physical
space, simply by performing a parallel projection (Fig. 1.19).
Phasons
Analogous to the Fibonacci case, phasonic excitations can be in-
troduced by shifting the acceptance window by a certain amount.
Figure 1.20 shows an example of a phasonic excitation by shifting
the acceptance window upwards by 0.2 lattice spacings.
Phasonic excitations of the octagonal tiling will play an im-
portant role in the description of ‘lattice sites’ in the eightfold
rotational optical potential (section 5.3).
Figure 1.20: One phasonic excitation
(red) introduced to the perfect octag-
onal tiling by shifting the acceptance
window (Fig. 1.18C) by 0.2 lattice
spacings in the vertical direction. The
original octagonal tiling (black) was
constructed by projection from Z4 us-
ing a hypercube of size 24 (three lattice
sites per edge, unit lattice spacing).
Physical diffraction of aperiodic models
The physical significance of aperiodic tilings was realised imme-
diately after the discovery of quasicrystals since the calculated
diffraction pattern of a Penrose tiling [66] closely resembled the
experimental images obtained by Shechtman et al. [1]. Nowa-
days aperiodic models such as the Penrose tiling and its three-
dimensional icosahedral counterparts [9, 11] are widely accepted
quasicrystals and aperiodic order 31
microscopic models for physical icosahedral and decagonal qua-
sicrystals whose crystallographic description is essentially com-
plete [11]. Although physical quasicrystals with eightfold sym-
metry are comparatively less common [54] the octagonal tiling
plays a central role in the modern understanding of quasicrystals,
particularly (but not only) from a geometric point of view [9, 67].
The diffraction pattern of the octagonal lattice can be calculated
analytically [9] and is shown in Fig. 1.21. It can be described by
four reciprocal lattice vectors23 {G1, G2, G3, G4} which point in 23 thus, a four-dimensional image
lattice is required to generate the
octagonal tiling via the cut-and-project
scheme
the same respective directions as the parallel projection of the 4D
basis vectors (red lines in Fig. 1.18). These lattice vectors are also
going to be used to describe the experimental diffraction patterns
in Chapter 6.
Figure 1.21: Theoretical diffraction
pattern of the octagonal tiling. The
image shows all peaks with at least
10−4 relative intensity (compared to
the q = (0, 0) peak), which leads to
the discrete appearance (the whole
diffraction pattern consists of a dense
set of peaks). The pattern is perfectly
eightfold symmetric and it shows
signatures of self-similarity, i.e. the
appearance of diffraction peaks at
powers of λ. Image taken from [9]
with kind permission of the authors.
The main concepts of quasicrystals, namely quasiperiodicity,
self-similarity and rotational symmetries have now been discussed
in one and two dimensions. In the final section of this chapter the
progress that has been made in research on quasicrystals in the
past 35 years will be reviewed.
1.4 Physical quasicrystals and current state of the art
To date, a huge variety of quasicrystals has been discovered [11,
54]. It is instructive to sort them according to their symmetry and
their chemical composition.
32 quasicrystalline optical lattices for ultracold atoms
Symmetry
The only thermodynamically stable quasicrystals which have been
found up to now exhibit either pentagonal (fivefold) or decago-
nal (tenfold) or icosahedral (Fig. 1.22) symmetry [11]. Icosahedral
quasicrystals are quasiperiodic along all three spacial directions.
Conversely, pentagonal and decagonal quasicrystals are quasiperi-
odic in the plane of the rotational symmetry but periodic along
the orthogonal direction. Therefore these quasicrystals are also
sometimes called ‘axial quasicrystals’ [54]. A large variety of
axial quasicrystals with varying degrees of structural integrity
have been found so far, including pentagonal, octagonal [54, 68],
decagonal, and dodecagonal (twelvefold) quasicrystals. However,
sevenfold symmetry or any other rotational symmetries not men-
tioned above have never been observed [11].
Figure 1.22: A regular convex icosa-
hedron. Each of the twenty faces is
an equilateral triangle. Around each
vertex there are five faces which form
a kind of pentagonal pyramid.
Chemical composition
All physical quasicrystals observed so far are binary or tertiary
compounds, the vast majority of them being intermetallic com-
pounds. For certain intermetallic materials the relative concentra-
tions of the constituents for which stable quasicrystals exist have
been mapped out [11]. However, it is currently an open question
whether quasicrystals made of a single chemical element can exist.
The stability of many quasicrystals seems to depend very strongly
on one chemical property, namely the valence electron concentra-
tion [69]. This suggests that electronic mechanisms may play an
role in stabilising the high rotational symmetries and aperiodic
order in general.
Transport
Electric and thermal transport properties of quasicrystals are gen-
erally anisotropic [70]. For example, a high-quality sample of
quasicrystalline material can exhibit metallic behaviour along one
direction and insulating behaviour along another, despite being
an intermetallic compound of aluminium and two other metals
(commonly). Similarly, the thermopower or the Hall coefficient
can change sign depending on the respective lattice axis it is mea-
sured on. These intriguing properties probably can be explained
by structural arguments, though it is currently unclear whether
local atomic order or long-range aperiodic order is responsible for
these unique features [70].
Open questions
Despite much effort to understand quasicrystals better there are
still many open questions, even on a fundamental level.
• Are physical quasicrystals energy- or entropy-stabilised? I.e.
is the Helmholtz free energy F = E − TS minimal for S be-
quasicrystals and aperiodic order 33
ing maximal, or for E being minimal? Is structural disorder
inherent in quasicrystals, or can an ‘ideal’ quasicrystal exist?
• How do quasicrystals form? In other words, quoting ref. [11],
How does the 1000th atom find its site in a giant unit cell with
thousands of atoms?
• Finally, why are certain rotational symmetries (fivefold, eight-
fold, tenfold) prevalent in quasicrystals, and why have others
(sevenfold, elevenfold, thirteenfold) never been observed?
Apart from the questions listed above which are primarily condensed-
matter- and crystallography-driven, research on quasicrystals also
touches on the topics of complexity, information science and biol-
ogy [71, 72].
Physical model systems
Curiously, the only evidence of naturally occurring quasicrystals
has been obtained from samples of meteorite material [73, 74].
All other known quasicrystalline materials have been synthesised
in the laboratory, some of them (Fig. 1.1) with similar structural
quality as best periodic crystals [11].
In order to tackle the questions mentioned above, a variety of
physical ‘model systems’ have been synthesised which aim to
emulate quasicrystals with a larger degree of experimental control.
Here is a list of physical settings in which quasicrystalline systems
have been studied experimentally
• aperiodically layered materials [12], e.g. semiconductors [75]
• liquid crystals [76] and polymers [77]
• self-assembled nanoparticles [78] and colloids [79]
• photonic waveguides [5, 13]. In these systems quasiperiodic
patterns can be imprinted by utilising a projected optical lat-
tice [80–82].
• laser-cooled atoms in dissipative optical lattices [14, 15]
• polystyrene spheres in optical lattices [16, 17]
• optical diffraction [66, 83]
• bilayered graphene [84]
Perspectives of ultracold gases in aperiodic lattices
This work establishes quasicrystalline optical lattices as a novel
platform for ultracold atoms. Several research avenues can be
imagined from here.
On the one hand, there are many open questions regarding
the physics of quasicrystals. It has been proposed by Macé et al.
[85] that ultracold atoms in a system very similar to ours (see sec-
tion 5.3) could help solve some of these long-standing mysteries.
34 quasicrystalline optical lattices for ultracold atoms
This would be particularly interesting from a chemical point of
view since up to now only binary and tertiary quasicrystalline
compounds have been discovered [11]. Using a single-component
quantum gas of potassium, for instance, would resemble this elu-
sive ‘unary’ quasicrystal.
On the other hand, ultracold atoms in quasicrystalline lattices
are interesting and rich systems in their own right. For example,
a topological charge pump has been demonstrated experimentally
using a 1D photonic quasicrystal [5]. Similary, ultracold atoms in
tilted superlattices have recently been shown to represent a 2D
topological charge pump, realising in effect a 4D quantum Hall
response [86]. These methods may in future be applied also to
our quasicrystalline potential thereby realising novel 4D model
systems with artificial gauge fields [87].
Another application of aperiodic lattices for ultracold atoms has
been the observation of many-body localisation [88]. In particular
the role of interactions in quasiperiodic media is poses fundamen-
tal questions in physics [89], and experimental results have only
recently become available [24, 88, 90, 91]. So far, all experiments
employed either 1D [88, 90, 91] or 2D separable [24] potentials.
However, more experimental evidence is needed, for instance, to
differentiate between two universality classes of localisation phe-
nomena, namely in random disorder and in aperiodic order [92].
For the latter class, our experiment is an ideal starting point since
the lattice potential closely resembles the octagonal tiling (see
section 5.3), a paradigmatic 2D quasicrystal. The eightfold lattice
described in this work is a highly flexible system, not only allow-
ing 1D quasiperiodic and 2D separable quasiperiodic potentials,
but also, for the first time, a 2D non-separable aperiodic lattice, i.e.
a 2D quasicrystal.
2
Design of the apparatus
This chapter offers a brief overview of the experimental appa-
ratus that is used to create quasicrystalline optical potentials
for ultracold atoms, before the apparatus is described in detail
in Chapters 3-5. First, the choice of atom (rubidium and potas-
sium) is motivated with a focus on the ability to perform quantum
many-body experiments in future. Then a brief overview of the
main vacuum apparatus is given, together with an outline of the
experimental sequence that is used to create samples of quantum
degenerate gases. For a detailed list of devices, part numbers and
components, the reader is referred to Appendix A.
2.1 Suitable constituents for creating quantum many-body sys-
tems
When setting up a new laboratory to study quantum many-body
physics, the choice of quantum constituents – neutral atoms, ions,
or molecules – is not an easy one.1 Often there is a trade-off be- 1 There are, of course, many more
possible experimental systems but
for simplicity only these three are
considered here.
tween the feasibility to prepare and manipulate ultracold samples
on the one hand and the capability to perform interesting, novel
experiments on the other.
Neutral atoms
First to mention are neutral atoms which can be broadly charac-
terised by four main properties: the atomic mass, the compound
particle flavour2 (boson, fermion), the feasibility to tune contact 2 If the number of neutrons is even,
the compound particle (atom) is
bosonic, otherwise it is fermionic.
Stable isotopes of the elements are
more commonly bosonic.
interactions [93], and the dominant type of interaction (short-
range or long-range). Alkali atoms are relatively easy to cool3 in
3 this will be discussed later in sec-
tion 3.1
large quantities but they lack intrinsic long-range interactions.
Ions
In contrast to neutral atoms, ions are subject to the Coulomb re-
pulsion and they offer long coherence times, suitable even for
quantum information processing [94]. However, they are hard to
cool en masse, the current record being around 200 ions [95].
36 quasicrystalline optical lattices for ultracold atoms
Polar molecules
Furthermore, polar molecules are very interesting for quantum
simulation applications since they feature long-range dipolar
interactions [96], but unfortunately their cooling process involves
serious technical challenges due to the plethora of rovibrational
sub-levels.
2.2 Choice of atomic species
To date, the most successful large-scale quantum many-body ex-
periments have been carried out using ultracold atoms in optical
lattices, the prime example here being the observation of a quan-
tum phase transition using 87Rb-atoms [18]. In the following short
review of atomic species we will therefore focus on neutral atoms.
The traditional atomic species that are best-suited for laser
cooling are 87Rb [97] and 23Na [98] which were the first to be
Bose-condensed.4 However, these species lack readily accessible4 together with 7Li [99]
Feshbach resonances [93] limiting their feasibility for studying
strongly-correlated systems.5 This limitation can be overcome by5 Long-range interactions can never-
theless be engineered in few-particle
systems by employing the Rydberg-
blockade, highlighted by recent ex-
periments on few-body quantum
systems [100].
using lithium [101] (6Li/7Li) or potassium [102, 103] (39K/40K)
which feature broad, accessible Feshbach resonances and are suit-
able for investigating strongly correlated physics. In addition,
both elements include a stable fermionic isotope (6Li and 40K),
a crucial ingredient to study electron-like many-body systems,
such as the Fermi-Hubbard model [104], as well as antiferromag-
netic quantum magnetism [105, 106]. Going beyond the ‘standard
repertoire’ in the alkalis there has been tremendous success in the
past decade in reaching quantum degeneracy of species with var-
ious special properties. These include elements with two valence
electrons, namely ytterbium [107, 108] and strontium [109–111],
which possess singlet ground states with with zero electronic an-
gular momentum, giving rise to intriguing SU(N)–symmetries
and extremely narrow optical transitions. Furthermore, there
are elements with very large magnetic moments, most notably
dysprosium [112, 113] and erbium [114, 115], which give rise
to dipolar interactions. Quantum degeneracy of these species
(ytterbium, strontium, dysprosium, and erbium) has been reached
with both their bosonic and their fermionic isotopes. However, the
advantages of these species often come with some technical cost,
for example in ytterbium and strontium: in order to perform laser
cooling on narrow optical transitions a laser with a very narrow
linewidth is required.
This experiment
The apparatus described in this thesis is designed to create a va-
riety of novel lattice geometries. This requires a large degree of
optical access, as well as reliable production of quantum degen-
erate samples without too much technical overhead. Therefore a
well-established ‘traditional’ combination of atomic species was
design of the apparatus 37
chosen, namely rubidium (87Rb) and potassium (39K/40K). The
setup employs sympathetic cooling of potassium using 87Rb as
coolant; with a suitable laser system it is therefore possible to
switch between 39K and 40K on a shot-to-shot basis using only
software commands. All the experiments carried out in this work
use 87Rb and 39K though we have also demonstrated a magneto-
optical trap (MOT) of 40K and the ability to easily switch between
the two potassium isotopes. For a more detailed review of the
atomic properties of 87Rb and 39K please refer to section 3.1.
2.3 Vacuum apparatus
The two most important design criteria for building a cold-atom
apparatus today are optical access and cycle time. On the one
hand, wide optical access allows flexibility of the type of exper-
iments performed. In our case, this is imperative as we require
four optical axes in one plane for the optical quasicrystalline lat-
tice. On the other hand, a short (< 1 min) cycle time is beneficial
for convenient experimentation and efficient optimisation of the
apparatus.6 6 and also for the sanity of the gradu-
ate students.The ‘Munich design’ pioneered by Greiner et al. [116] combines
the two requirements by separating a ‘high-pressure’ (∼ 1× 10−9 mbar)
MOT chamber with limited optical access from the science cham-
ber (∼ 1 × 10−11 mbar) with excellent optical access. The two
chambers are connected by an angled differential pumping section
through which the atoms are transported magnetically. This well-
established design is now being used in several laboratories, in-
cluding LMU Munich [116–120], ETH Zurich [121], Harvard [122],
and LPL (Paris) [123].
Our apparatus is shown schematically in Fig. 2.1. It is an adap-
tation of the latest iteration [119, 120] of the Munich design which
contains an additional high-pressure (∼ 1 × 10−6 mbar) cham-
ber for preparing a pre-cooled atomic beam (2D MOT). The main
differences between their design and ours can be summarised as
follows:
• We have two 2D MOT chambers, allowing independent control
of the background vapour pressure for rubidium and potas-
sium.
• The effective differential pumping section between the 2D
MOT chambers and the MOT chamber is probably shorter
than theirs.7 7 We ordered a titanium part with a
1.5mm clearance over 50mm length
but the part we received has a 1.5mm
clearence only over 20mm. The rest of
the differential pumping section has a
wider diameter (∼ 3 mm). However,
the pressure differential between 2D
MOT chambers (which are not actively
being pumped) and the MOT chamber
has never been a limiting factor in our
experiment and consequently a longer
clearence would have been of no avail.
• The differential pumping section (20 cm) between the MOT
chamber and the ‘knee’ is longer and its inner diameter (10mm)
is wider than theirs.
• Our science cell is a rectangular cuboid, allowing a more com-
pact geometry of experiment coils.
It should be noted that the large separation between the MOT
chamber (7.2× 10−11 mbar) and the science cell (1.2× 10−11 mbar)
38 quasicrystalline optical lattices for ultracold atoms
Ion pump
(150 l/s)
Ion pump
(75 l/s)
Knee
Science cell
Titanium sublimation
 pump (TSP)
Pressure
gauge
Pressure gauge
MOT chamber
Magnetic
transport
Experiment
coils
2D MOT chamber 
(39K/40K)
2D MOT chamber
(87Rb)
Rubidium ampoule
(1g, 27.8% 87Rb) 
Potassium ampoule
(100mg, 2.98 % 40K)
Valve
Differential
pumping section
Titanium sublimation
 pump (TSP)Differentialpumping section
Angled valve
Valve
Cooling block
Angled valve
Angled
valve
Angled valve
20 cm
Figure 2.1: Schematic of the main
experimental apparatus. For clarity,
only the lower half of the cooling
block and magnetic transport system
(brass-coloured) is shown. is now mostly practical and not physically required since the pres-
sure difference between the two chambers is only a factor of six.
Nevertheless we can maintain efficient MOT loading, facilitated
by a very large (∼ 104) pressure differential between the 2D MOT
chambers and the MOT chamber.
A more comprehensive treatment of the design, implementation
and characterisation of the vacuum apparatus can be found in
Matteo Sbroscia’s First-Year Report [124].
Table 2.1: Rough overview of the
experimental sequence.
stage duration section
MOT 2.5 s 3.5
final MOT 10ms 3.7
optical molasses 4.5ms 3.8
optical pumping 0.7ms 3.9
magnetic transport 5 s 3.10
MW evaporation 7.5 s 3.11
RF state transfer 20ms 4.5
dipole evaporation 3 s 4.3
total 23 s
2.4 Overview of the experimental sequence
Table 2.1 provides a rough overview of the experimental sequence
that is used to create quantum degenerate samples of 87Rb and
39K.
design of the apparatus 39
Figure 2.2: Completed vacuum setup
(4th July 2016). The photo was taken
after bake-out had been completed,
after the rubidium and potassium
ampoules had been broken, and
after the turbo-pump had been dis-
connected. The heavy cable of the
titanium-sublimation pump (TSP) is
still connected (top right).
Figure 2.3: Science cell and brass cool-
ing block seen from main breadboard
(see also Fig. 5.17), before any optics
have been placed on it (28th February
2017). The ribbon-wire can be seen
(black), connecting the upper and the
lower experiment coil. The compen-
sation coils are wound in red (T-axis),
black (D-axis), and blue wire (Z-axis)
on 3D-printed mounts.

3
Laser cooling & trapping of 87Rb & 39K
This chapter accompanies the atoms on their initial journey from
room temperature to the µK-scale before we later present the de-
tailed methods to reach quantum degeneracy in the optical dipole
trap (Chapter 4). First, each atomic species and their relevant en-
ergy scales are introduced. Then the experimental methods to cool
and trap these atoms are described. This chapter is focussed on
our experimental implementation1; the theoretical introduction 1 A detailed list of optical components
can be found in Appendix A.to each method is kept to a minimum. For a general introduction
to the topic, the reader is referred to the standard literature [125–
127].
3.1 The alkalis
Table 3.1: Vapour pressures for rubid-
ium and potassium for two different
temperatures [128].
T Rb K
295K 3.8× 10−7 mbar 1.7× 10−8 mbar
333K 1.6× 10−5 mbar 1.0× 10−6 mbar
The inevitable starting point for any experiment involving ul-
tracold gases is some kind of atom source at or above2 room-
2 In many cases atoms have to first
be heated to increase their vapour
pressure.
temperature. Thankfully, the vapour pressures of rubidium and
potassium are reasonable (see table 3.1) such that a small speci-
men (0.1 - 1.0g) of each species in a vacuum chamber is sufficient
to start the laser cooling cycle from background vapour.
Similar to atomic hydrogen, the ground level of the alkali atoms
is the 2S1/2-level of the valence electron. Since here the angular
momentum quantum number L is zero, there is no fine structure
splitting and consequently only the two hyperfine states F = 1, 2
are thermally populated at ambient temperature; The thermal en-
ergy required to significantly populate the excited electronic states
is much higher than room temperature, as shown in table 3.4.
The hyperfine structure of the alkalis is very favourable for
standard laser cooling techniques [126] due to the existence of
(almost) closed transitions on the D2-line (2S1/2 → 2P3/2).3 A 3 for a particular choice of F and mF
statestransition is said to be ‘closed’ if it can be treated as an effective
two-level system, neglecting decay into other (ground) states,
e.g. by spontaneous emission. Other advantages include reason-
able natural linewidths4 and the commercial availability of lasers 4 wide enough for the transitions to be
addressed with a conventional single-
mode diode laser and narrow enough
to have Doppler temperatures on the
order of 100 µK
in the relevant wavelength-range (though in past decades laser
technology has advanced considerably such that nowadays the
availability of lasers is not usually a limiting factor).
42 quasicrystalline optical lattices for ultracold atoms
Rubidium-87
-7
9.
9
+1
33
.3
F = 2
F = 1
5 2S1/2
5 2P3/2
F = 0
F = 3
C
o
o
li
n
g
R
ep
u
m
p
 l
o
ck
6.
83
4 
G
H
z
Im
ag
in
g
O
p
ti
ca
l 
p
u
m
p
in
g
 (
sp
in
-p
o
l)
72
.2
13
3.
3
F = 2
D
2-
sp
ec
tr
o
sc
o
p
y
 F
 =
 2
 t
o
 F
’ =
 2
,3
 c
ro
ss
o
v
er
 (
7
8
0
.2
4
1
 n
m
)
13
3.
3
15
6.
9
F = 1
Δcool
R
b
 m
as
te
r 
la
se
r 
em
is
si
o
n
53.4
gF = 1/2
gF = -1/2
220
Figure 3.1: Hyperfine structure of the
D2-line transition in 87Rb (not to scale)
 F‘
 =
 3
 F‘
 =
 2
 F‘
 =
 2,
3
 cr
os
so
ve
r
 F‘
 =
 1
 F 
= 
1
 F 
= 
2 3.036 GHz
85Rb 52S1/2
6.834 GHz 87Rb 52S1/2
87Rb 52P3/2
Figure 3.2: Rubidium D2-line Doppler-
free (blue) and Doppler-broadened
(red) spectroscopy. The 87Rb master
laser is locked 53.4MHz below (i.e. to
the red of) the F′ = 2, 3 crossover line
(see also Fig. 3.1).
In terms of aptitude for laser cooling 87Rb is one of the most
favourable atomic species (see also section 2.2) which is down
to three main factors: Firstly, the hyperfine splitting of both the
ground and the D2-excited state is much larger than the natural
linewidth (6MHz×2pi); therefore the individual hyperfine lev-
els (and their crossovers) are nicely resolved by our spectroscopy
(Fig. 3.2). Correspondingly the |F = 2, mF = 2〉 → |F = 3, mF = 3〉
(2P3/2) transition, driven by σ+-polarised light, is almost perfectly
closed and comparatively few repump photons (F = 1→ F’ = 2)
are required to transfer atoms back to the cycling transition. This
allows for a near-textbook-like implementation of magneto-optical
trapping and optical molasses. Secondly, the atomic mass is rela-
tively large, favouring thermalisation and reducing non-adiabatic
(‘Majorana’-)losses from magnetic traps. Thirdly, the background
scattering length is relatively large and positive (100 a0), favouring
fast thermalisation and BEC stability (section 4.6).
Potassium-39
In contrast to 87Rb, the hyperfine splitting of the 42P3/2 level in
39K is very narrow (Fig. 3.3, unresolved by our spectroscopy in
Fig. 3.4), precluding a ‘properly’ closed transition and necessi-
tating many more repump photons to combat this effect. Con-
sequently, molasses-cooling is much less efficient than for 87Rb.
Additionally, its background scattering length is negative which
means that an ultracold cloud of 39K can collapse under its own
inter-atomic attractive interactions [129]. However, 39K possesses
a readily accessible Feshbach resonance at 402.70(3) G [130] to
counteract the negative effects of attractive interactions. This Fesh-
bach resonance is indeed the main reason why 39K is widely used
for studying strongly interacting or non-interacting bosons [103,
131–136]. The well-established technique of sympathetic cooling
with 87Rb [103, 131, 132, 134] allows relatively straightforward
production of quantum degenerate samples of 39K. Although the
sympathetic production might be in future superseded by all-
optical production using the ‘grey molasses’ technique [129, 133,
137, 138] the presence of 87Rb proved to be useful in our lab. It
allowed us to benchmark the optical lattice in the early stages of
the experiment before we achieved condensation with 39K. More-
over, the experiment is designed to perform shot-to-shot switching
between 39K and fermionic 40K in future, aided by the abundant
87Rb-atoms for sympathetic cooling of either species.
So far the short review of atomic properties considered isolated
atoms; in the following the effects of external fields will be briefly
discussed.
laser cooling & trapping of 87rb & 39k 43
Laser frequencies
The relevant optical transitions for laser cooling of 87Rb and 39K
are shown in Fig. 3.1 and 3.3, respectively. In the following sec-
tions 3.5-3.9 the individual purpose of each laser frequency will
be discussed in detail. Note that in 87Rb the reference laser (green
line in Fig. 3.1) is emitting 53.4MHz to the red of the spectroscopy
line, whereas in 39K the reference laser is locked directly to the
spectroscopy signal.
14
.4
6.
7
F = 2
F = 1
4 2S1/2
4 2P3/2
F = 0
F = 3
C
o
o
li
n
g
R
ep
u
m
p
Im
ag
in
g
 (
lo
w
-f
ie
ld
)
O
p
ti
ca
l 
p
u
m
p
in
g
 (
sp
in
-p
o
l)
3.
3
F = 2
D
2-
li
n
e 
(7
66
.7
01
 n
m
)
9.
4
F = 1
K
39
m
a
st
er
 l
a
se
r 
em
is
si
o
n
, l
o
ck
ed
 t
o
 F
 =
 1
,2
 t
o
 F
’ c
ro
ss
o
v
er
 
21
.1
46
1.
7 2
26
.1
23
5.
6
53.0
gF = 1/2
gF = -1/2
Figure 3.3: Hyperfine structure of the
D2-line transition in 39K (not to scale).
Note the unequal distances between
the lock point on the two hyperfine
ground levels.
 F 
= 
1
 F 
= 
2
 F 
= 
1,2
 cr
os
so
ve
r
39K 42S1/2
39K D2-line
461.7 MHz
41K 42S1/2  
F =
 1
 F 
= 
2
226.1MHz 235.6MHz
Figure 3.4: Potassium D2-line Doppler-
free (blue) and Doppler-broadened
(red) spectroscopy. The 41K lines are
indicated in green. We lock on the
F = 1, 2 crossover line which has an
assymetric line shape.
3.2 87Rb & 39K in magnetic fields
In the presence of a magnetic field B the (2F + 1)-fold degeneracy
of mF states within a given hyperfine level F is lifted. In the case
of low field strengths the energy shift of an |F, mF〉 state is linear,
namely
∆E = gFmFµBB , (3.1)
called the anomalous Zeeman effect [127]. Values for gF can be found
in Fig. 3.1 and 3.3 and µB is Bohr’s magneton. For strong fields the
|F, mF〉 states cease to be good quantum numbers. In this case, the
levels of 87Rb or 39K with J = 1/2 (for example) are split into two
groups (mJ = ±1/2) of four, i.e. 2I + 1, where I = 3/2 (Paschen-
Back effect). The hyperfine and magnetic splittings are plotted in
Fig. 3.5 and 3.6 for 87Rb and 39K, respectively, in both the weak-
field and the strong-field regimes. For levels with L = 0 and S
= 1/2 the energies can be written in an analytic form called the
Breit-Rabi formula [127, 139, 140].
Interlude: calibration of magnetic coils
The splitting of hyperfine states in magnetic fields represents the
most accurate way of calibrating homogeneous magnetic fields
in our apparatus. It is a well-understood, single-particle effect
with no obvious systematic errors. The levels with J = 1/2 (e.g.
the 5 2S1/2 ground level of 87Rb) are particularly well suited for
magnetic field calibration as they are unaffected by the electric
quadrupole correction [139]. We can experimentally determine the
transition frequencies at high magnetic fields (several hundreds
of gauss) with a relative uncertainty of 10−4 using Landau-Zener
sweeps or Rabi oscillations. By comparing the experimental values
to the theoretical predictions (Fig. 3.5 B) we can quantify our mag-
netic field to similar precision. Table 3.2 shows some relevant cali-
brated field values using RF transitions from the |F = 1, mF = −1〉
to the |F = 1, mF = 0〉 state in the atomic ground level of 87Rb (ar-
row in Fig. 3.5 B). The calibration of our guide field coils can be
found in section 4.5 using the same method.
44 quasicrystalline optical lattices for ultracold atoms
3.3 Spectroscopy and absolute frequency stability (laser lock-
ing)
The production and manipulation of ultracold 87Rb & 39K requires
an absolute laser frequency stability (long-term and short-term) of
roughly 1MHz, corresponding to a relative stability on the order
of 10−9. This requirement means that the laser jitters and drifts
by much less than the natural linewidth (6MHz×2pi) of the rel-
evant optical transitions. In our case the short-term fluctuations
(linewidth) of the free-running reference laser is in the 100kHz-
range. However, the long-term (thermal) fluctuations are on the
order of 100MHz such that a stable reference and a frequency-
feedback-loop (‘laser lock’) are necessary. Absolute frequency sta-
bility for our 87Rb- and 39K-master lasers is achieved using stan-
dard saturated absorption spectroscopy and commercial Pound-
Drever-Hall locking electronics which will be briefly described in
the following. Relative frequency stability between separate laser
diodes and the master lasers is ensured using frequency-offset-
locks (see section 3.4 below).
Table 3.2: Magnetic field calibration
for our experiment coils using RF
transitions from the |F = 1, mF = −1〉
state to the |F = 1, mF = 0〉 state
in 87Rb. For each current value we
take an RF-resonance scan using a
Landau-Zener sweep with a half-range
of 50kHz and a duration of 20ms.
These frequencies are then converted
to magnetic field values using the
magnetic field hamiltonian (Fig. 3.5 B).
Current/A RF res./MHz B-field/G
29.257 228.825 317.68(9)
32.379 253.600 351.44(9)
36.600 287.213 397.19(9)
36.650 287.650 397.78(9)
Saturated absorption spectroscopy
The saturated absorption spectroscopy technique [141–144] over-
comes Doppler-broadening by employing the pump-probe beam
configuration. In our apparatus we have a commercial spec-
troscopy unit (one for each atomic species) which supplies a
fibre-coupled pump-probe setup and a vapour cell. We obtain a
spectroscopic absorption signal (y-axes in Figs. 3.2 and 3.4) by
scanning the laser piezo (x-axes) together with the diode current
(‘feed forward’) using a triangular wave.
Laser locking
Table 3.3: Spectroscopy, laser locking
parameters and lock performance.
The linewidths are measured with the
‘EagleEye’ reference cavity.
parameter 87Rb 39K
vapour cell temp. 50◦C 80◦C
optical power/mW ∼ 1 ∼1.4
modul. freq./kHz 28.00 20.00
modulation ampl. 0.012mA
laser linewidth/kHz 155 120
The master lasers are locked to spectroscopy lines using top-of-
fringe (Pound-Drever-Hall [145]) locking which is insensitive to
laser intensity fluctuations. The Pound-Drever-Hall technique
uses a weak (typically 5 × 10−4) modulation of the diode cur-
rent at a frequency around 20kHz to imprint sidebands onto
the laser frequency. The spectroscopy signal is mixed down
with the same 20kHz at a fixed phase delay to produce an er-
ror signal that represents the derivative of the spectroscopy
signal. This is then used to stabilise the laser frequency with a
proportional-integral-derivative (PID) loop. We choose the most
pronounced spectroscopy features as they promise the largest
feedback slopes and the best locking performance; in 87Rb we use
the F = 2 → F′ = 2, 3 crossover [127] line (Fig. 3.1), the 39K-laser
is locked to the F = 1, 2 → F′ crossover line, as shown in Fig. 3.3.
The parameters used for laser locking are itemised in table 3.3,
together with the lock performance.
laser cooling & trapping of 87rb & 39k 45
0 5000 10000
B-field/G
-20
-10
0
10
20
Fr
eq
ue
nc
y/
G
H
z
mJ = +12
mJ = -12
F = 2
F = 1
0 500
B-field/G
-400
-200
0
200
Fr
eq
ue
nc
y/
M
H
z
mF = 0
mF = +1
mF = -1
A B Figure 3.5: Hyperfine and magnetic
splitting of electronic levels in 87Rb.
(A) 5 2S1/2 level. In the presence of a
weak magnetic field the (2F + 1)-fold
degeneracy of each hyperfine level
F= 1, 2 is lifted. For strong fields
the lines are split into two groups
(mJ = ±1/2) of four (2I + 1, I = 3/2).
(B) 5 2S1/2 F = 1 (ground) level
in the low-field regime. The black
arrows indicate the magnetic dipole
transitions which can be stimulated
by RF-radiation in order to calibrate
magnetic fields (B), or perform MW
evaporation (A), as discussed in
section 3.11.
0 500
B-field/G
-1000
-500
0
500
1000
Fr
eq
ue
nc
y/
M
H
z
mJ = +12
mJ = -12
F = 2
F = 1
0 10 20
B-field/G
-10
0
10
Fr
eq
ue
nc
y/
M
H
z
mF = 0
mF = +1
mF = -1
A B
0 500
B-field/G
-2000
-1000
0
1000
2000
Fr
eq
ue
nc
y/
M
H
z
mJ = +32
mJ = +12
mJ = -12
mJ = -32
0 5 10 15 20
B-field/G
-40
0
40
Fr
eq
ue
nc
y/
M
H
z
F = 0
F = 3
C D
Figure 3.6: Hyperfine and magnetic
splitting of electronic levels in 39K.
(A) 4 2S1/2 level. (B) 4 2S1/2 F = 1
hyperfine (ground) level in the low-
field regime. (C) 4 2P3/2 (D2-excited)
level. There are 16 mF states which
are not resolved on this scale. In the
strong-field regime the states are
grouped by mJ, each containing four
mI states, i.e. 2I + 1, where I = 3/2.
(D) 4 2P3/2 (D2-excited) level (same
as C) at weak magnetic fields. Each
hyperfine level F = 0,1,2,3 is split into
mF states (2F+ 1).
hierarchy transition λ/nm frequency B-field/G temperature
hc/E E/h E/µB E/(kBT)
87Rb ‘gross’ 5 2S1/2 → 5 2P3/2 (D2) 780.241 210 384.230 484THz 3× 108 18 000K
fine 5 2P1/2 → 5 2P3/2 (D1-D2) 7.123THz 5× 106 342K
hyperfine 5 2S1/2 F = 1→ F′ = 2 6.834 682 611GHz 4900 0.3K
5 2P3/2 F = 2→ F′ = 3 266.650MHz 191 13mK
39K ‘gross’ 4 2S1/2 → 4 2P3/2 (D2) 766.700 921 391.016 170THz 3× 108 19 000K
fine 4 2P1/2 → 4 2P3/2 (D1-D2) 1.730THz 1× 106 83K
hyperfine 42S1/2 F = 1→ F′ = 2 461.720MHz 330 20mK
42P3/2 F = 2→ F′ = 3 21.1MHz 15 1mK
Table 3.4: Hierarchy of some rele-
vant energy scales (E) and energy-
equivalents in 87Rb [139] and 39K [140].
46 quasicrystalline optical lattices for ultracold atoms
3.4 Relative frequency stability (offset locking)
510 520 530 540
Rb repump beat note/MHz
-90
-60
-30
Si
gn
al
/d
Bm
Figure 3.7: Frequency scan of the
beat note (after mixing) between the
Rb repump and the Rb master laser.
A lorentzian fit to the peak gives
a centre frequency of 526.1MHz.
The measurement bandwidth of the
spectrum analyser was set to 1 kHz for
this measurement.
90 100 110 120 130
Rb repump DDS frequency/MHz
0.0
1.0
2.0
R
b
at
om
nu
m
be
r/
10
7
0.1ms
1.0ms
Figure 3.8: Frequency scan of Rb
repump flash (during imaging) for
two different pulse durations. Note
that the DDS unit drives an AOM in
double-pass configuration, i.e. the
laser frequency is changed by twice
the amount shown on the x-axis. The
atom numbers are 2d gaussian fits
to the atom cloud after loading the
dipole trap at a temperature of 6 µK in
12ms ToF. The repump flash is applied
1.5ms before the imaging pulse in
order to optically pump the atoms
from the |F = 1, mF = −1〉 state to
the F = 2 manifold. The lorentzian
fit (line) gives a centre frequency
of 110.1(1)MHz with a Γ-width of
6.5(1)MHz.
Frequency shifts on the order of 100MHz are routinely performed
by acousto-optic modulators (AOMs). Their radiofrequency (RF)
bandwidths of up to ± 20 % of the central frequency are sufficient
to tune the frequency by several natural linewidths in 87Rb or
39K (particularly if used in the convenient double-pass config-
uration). However, for the two situations described below we
require larger frequency shifts or wider tuning ranges, which can
be achieved using an ‘offset lock’ [146]. In this configuration the
laser that is to be stabilised is overlapped (on a non-polarising
beam-splitter) with a reference laser and the resulting beat note
is picked up on a high-bandwidth photodiode. The beat signal
can be mixed down with an external RF source, for example a
voltage-controlled oscillator (VCO) or a direct-digital-synthesis
(DDS) unit if frequency tuning is required. Afterwards the signal
is split into two paths with different cable lengths, then recom-
bined on a mixer and finally low-passed. The resulting DC voltage
can be approximated by a cosine [146], the argument of which
is given by the offset frequency times the constant time delay
(= length/speed of light) induced by the longer cable. Therefore
this setup is sometimes called ‘delay-line lock’. Laser locking is
then achieved by using a side-of-fringe lockpoint of the cosine as
the error signal for the PID-loop. In our experiment this technique
is used for two applications, namely the rubidium repump lock
and the potassium imaging lock.
Rubidium repump lock
The ground state splitting in 87Rb is 6.8GHz (table 3.4) and in
our experiment it is bridged by an offset lock. In this setup the
raw beat signal is first mixed down with a 7.5GHz reference
oscillator to make the RF signal easier to handle. We choose a
(constant) lockpoint 220MHz to the blue of the repump transition
F = 1 → F′ = 2 in 87Rb , frequency fine-tuning is then achieved
by a double-pass 110MHz AOM (-1st order, Fig. 3.8). Since our ru-
bidium master laser is locked 53.4MHz to the red of the crossover
line (Fig. 3.1) we should set our beat note (after mixing down with
the reference oscillator) to
7500.0− (6834.7− 133.3+ 53.4)− 220.0 = 525.2 (3.2)
(in MHz). In practice we have a beat note (after mixing) of 526.1MHz,
as shown in Fig. 3.7. In Fig. 3.8 we show a spectroscopic measure-
ment of the resulting repump laser frequency which is consistent
with the above considerations. This measurement was performed
using the ‘repump flash’ (see section 3.12) during imaging of the
|F = 1, mF = −1〉 state in the dipole trap. Since our main imaging
light is only resonant with atoms in the F = 2 level the measured
atom number is a good probe of the repump frequency, result-
ing from the optical pumping efficiency from |F = 1, mF = −1〉 to
laser cooling & trapping of 87rb & 39k 47
F = 2. For longer repump flash pulses we observe a broadening
effect similar to that of power broadening.
Potassium imaging lock
200 205 210 215
K master-imaging beat note/MHz
-80
-60
-40
-20
Si
gn
al
/d
Bm
Figure 3.9: Spectrum analyser trace of
the beat note between the 39K imag-
ing and the 39K master laser corre-
sponding to the highest data point
in Fig. 3.10 below. The measurement
bandwidth of the spectrum analyser
was set to 1 kHz for this measurement.
-220 -200 -180
K master-imaging beat/MHz
0
2
4
K
at
om
nu
m
be
r/
10
6
σ+ polarised
σ− polarised
Figure 3.10: Frequency scan of
39K imaging light at a guide field
of 2.8(1)G (offset coils at 4.5 A), using
σ+ and σ+ polarised light. Only the
σ+ polarised light results in a faithful
resonance signal at −208.6(1)MHz
(lorentzian fit) with a Γ-width of
7.6(5)MHz. The dashed line is a
lorentzian fit to the σ− data (wrong
polarisation) as a guide to the eye.
The frequency is red-detuned with
respect to the potassium master laser
(indicated by the negative values on
the x-axis). The atom numbers (y-
axis) are 2d gaussian fits to the atom
cloud after loading the dipole trap at a
temperature of 9 µK in 3ms ToF.
The ground level hyperfine splitting of 39K is only 462MHz (ta-
ble 3.4) and that of 40K is 1.3GHz which could, in principle, be
bridged by AOMs. However, we require a much larger degree of
flexibility than could be achieved by AOMs, since 39K is often im-
aged at magnetic fields close to the Feshbach resonance at 402.70G
(see section 4.6 on Feshbach resonances). In 39K the corresponding
frequency shifts are between 200 and 500MHz to the red of the
master laser; in 40K the shifts are even larger and of opposite sign
(to the blue, since the ground state hyperfine splitting is inverted).
Offset locks allow large tuning ranges of arbitrary sign and are
therefore ideally suited for this application.
In contrast to rubidium, we mix the beat signal with a DDS
signal (which in turn has been mixed with a VCO to cover higher
frequencies) such that we can vary the lockpoint in-sequence. Fig-
ure 3.9 shows the beat signal between the potassium master laser
and the imaging laser. By changing the DDS frequency we are
able to obtain a spectroscopic signal of the 39K imaging transition
F = 2→ F′ = 3 at low magnetic field, as shown in Fig. 3.10. It cru-
cial to use σ+ polarised imaging light in order to address the cor-
rect hyperfine state (Fig. 3.10 also shows a frequency scan using
the wrong polarisation, in this case the resonance is shifted in a
non-trivial manner). The expected imaging transition at 2.8G is at
155.5(2) MHz to the red of the D2-line (fine structure point). Tak-
ing into account the position of the 39K master lock point (Fig. 3.3)
we expect the imaging transition at
−155.5 MHz− 53.0 MHz = −208.5 MHz , (3.3)
in excellent agreement with the measured value of 208.6MHz
(Fig. 3.10).
Up to this point the stage is set for the atoms, and their jour-
ney towards quantum degeneracy begins. In the following the
individual cooling steps will briefly introduced and subsequently
described with the focus on our experimental implementation.
However, schematics of large-scale laser setups and lists of optical
components, part numbers, etc. have been omitted in favour of
readability.
48 quasicrystalline optical lattices for ultracold atoms
3.5 Magneto-optical trap (MOT)
Radiation pressure can be used to cool dilute vapours by means
of the Doppler effect. This was discovered as early as 1975 by
Hänsch and Schawlow [147]:
It is well known that light exerts a radiation pressure on any sub-
stance which reflects or scatters is. [. . . ] the radiation pressure of
laser light has been used to selectively deflect atoms of a chosen
isotopic species from a beam. We wish to point out that if the laser
radiation is essentially isotropic, but confined to frequencies on the
lower half of the Doppler-broadened absorption line of an atomic
vapour, the gas can be cooled. That is, the average translational
kinetic energy of the atoms can be reduced.
If the light is red-detuned with respect to the atomic transition,
only atoms moving against the direction of the laser beam will
predominantly experience photon momentum kicks and thereby
be slowed down. Adding a weak magnetic quadrupole field5 to5 weaker than would be necessary to
trap the atoms against gravity this situation leads to atoms that are further away from the field
minimum (i.e. the centre of the quadrupole field) reaching the
resonance condition more easily than atoms at the field mini-
mum. Therefore the atoms experience an effective restoring (i.e.
trapping) force towards the centre. This is the principle of the
magneto-optical trap (MOT) which nowadays forms the basis of
essentially all cold-atom experiments [125–127].
Table 3.5: MOT parameters. Laser
powers are given per beam. Detunings
are given in MHz from the respective
transitions (negative values mean red
detunings).
parameter 87Rb 39K
backgr. pressure 7× 10−11 mbar
loading duration 2.5 s 1.0 s
gradient (strong ax.) 20G/cm
coil current 11.0A
gradient @ 1A 1.78G/cm
cooling transition F = 2→ F′ = 3
repump transition F = 1→ F′ = 2
cooling power/mW 95-130 40-46
repump power/mW 0.7-1.0 60-87
cool. detuning/MHz -23 -52
rep. detuning/MHz 0 -25
beam waist 13 mm
All laser light required for MOT operation and the subsequent
near-resonant cooling stages is derived from the respective ref-
erence lasers (section 3.3). Frequency shifts are performed by
means of double-pass AOMs which are driven by home-built DDS
units and RF amplifiers. Subsequently the light is amplified us-
ing six separate tapered-amplifiers (TAs). The rubidium and the
potassium light is overlapped using ‘magic’ waveplates6 and then
6 which act as λ-plate at 767nm and as
λ/2-plate at 780nm
distributed via a commercial two-input-by-six-output setup called
‘fibre cluster’. Due to the nature of the polarising beam splitters in
Figure 3.11: 2D MOT vacuum cham-
ber. The frame is made of titanium
and the rectangular windows are made
of BK7 glass. The windows are glued
to the frame using high-temperature
exopy glue, with some kapton foil as
spacer material. Inside the chamber
(on the right) the mirror-polariser
setup with its small aperture (see text)
can be seen. All materials were cho-
sen such that their respective thermal
expansion coefficients match within
15% [124].
laser cooling & trapping of 87rb & 39k 49
the fibre cluster it is impossible to split the light from both input
ports into six equal parts. Therefore the waveplates in the fibre
cluster have been used to maintain pair-wise-balanced powers
(at least within 10%, typically better than 5%) for both inputs.
The MOT laser beams are large (table 3.5) and therefore relatively
insensitive in terms of alignment, as long as they are roughly cen-
tral, collimated and counter-propagating. We aligned one beam
of each counter-propagating pair by coupling the light into a fibre
plugged into the opposite fibre coupler. A list of relevant parame-
ters is given in table 3.5.
The weak magnetic field gradient is provided by the same
coils that are later used for magnetic transport (section 3.10). In
addition, there are three pairs of compensation coils (section 3.8)
which counteract stray magnetic fields.
3.6 2D MOT Figure 3.12: Schematic of the 2D MOT
chamber showing one of the four
coils. The brass block on the outside
contains a U-shaped hole which is
perfused by cooling water in order to
carry away the heat dissipated by the
coil. Figure taken and adapted from
ref. [124].
Figure 3.13: 2D MOT field configura-
tion seen from the axial direction. Fig-
ure taken and adapted from ref. [124].
Due to the low pressure in the MOT chamber (7× 10−11 mbar) we
observe no spontaneous MOT loading from background vapour.
Instead, the MOT is loaded using two 2D MOTs [148–152] which
are located on either side of the MOT chamber, each separated
by a differential pumping section that maintains a large pressure
differential (as described in section 2.3, see also Fig. 2.1). Separate
2D MOTs allow us to control the background vapour pressure for
each species independently, leading to improved loading rates
compared to refs. [119, 120]. These 2D MOTs are used to precool
the atoms and produce a collimated beam7 of 87Rb/39K-atoms
7 this method was also proposed in
the original paper by Hänsch and
Schawlow [147]
that can be directly captured by the main ‘3D MOT’. The basic
principle of the 2D MOT is the same as that of the 3D MOT, ex-
cept that there is one direction along which less cooling takes
place.8 In our 2D MOT design (Fig. 3.11) this manifests itself by an
8 Sometimes this is called a 2D+ MOT
since cooling also takes place along the
third direction. However, the notion of
cooling along the third axis depends
on the precise relative power levels
between the different axes. In this
work the term 2D MOT will be used
for simplicity.
elongated geometry9 and by an aperture in the axial retro-mirror
9 there is no magnetic field gradient
along this axis and hence no restoring
force
through which atoms leave the chamber. In addition, a polariser is
glued to the face of the retro-mirror (with some added spacers, as
shown in Fig. 3.11). The purpose of the axial beam is then twofold:
On the one hand, it provides some cooling in the axial direction
(though this contribution is limited due to the ring-shaped beam
profile coming back from the retro-mirror). On the other hand,
the central part of the axial beam serves to ‘push’ atoms towards
the 3D MOT. The mirror-polariser setup allows us to balance the
relative strength of cooling and pushing by changing the (linear)
polarisation of the incoming axial beam. For potassium we have
an additional laser beam along this axis that is blue-detuned in
order to push the atoms through the aperture. In contrast to the
3D MOT all beams are retro-reflected. A summary of relevant 2D
MOT parameters is given in table 3.6.
The 2D MOT vacuum chambers were assembled in-house,
closely following the Munich implementation [153, 154]. The
elongated geometry of the 2D MOT is intended to provide a large
50 quasicrystalline optical lattices for ultracold atoms
capture volume (though this effect cannot easily be quantified). A
detailed account of the design and setup of our 2D MOT chambers
can be found in Matteo Sbroscia’s First-year report [124].
Table 3.6: 2D MOT parameters. Laser
powers are given per beam. Detunings
are given in MHz from the respective
transitions (which are identical to
those in the MOT). Negative frequency
values mean red detunings. Here x
and y directions are parallel to the
transport axis and along gravity,
respectively.
parameter 87Rb 39K
vapour pr./mbar 4× 10−7 1× 10−6
diff. pumping  1.5mm
diff. pump. length 2 cm
duration 2.5 s 1.0 s
field gradient 21G/cm 22G/cm
coil current 8.5A 9.0A
gradient @ 1A 2.4G/cm
coil windings 63 per coil
heat dissipated 30W per coil
comp. windings 8 per coil
comp. current x 0.0A 2.5A
comp. current y 2.5A 0.6A
cooling detuning −19 MHz −28 MHz
repump detuning 0 MHz −15 MHz
push detuning - +42 MHz
cooling power x 230mW 190mW
cooling power y 320mW 260mW
repump power x 48mW 230mW
repump power y 39mW 170mW
axial cooling 8.5mW ∼1mW
push power - 2.0mW
beam waist (long) 20 mm
beam waist (short) 10 mm
axial beam waist 2mm
push beam waist - 0.8mm
The magnetic gradients in each 2D MOT are provided by four
rectangular coils, one on each face of the chamber, as shown in
Fig. 3.12. They produce a quadrupole-like field in the plane per-
pendicular to the axial direction (Fig. 3.13); field gradients in the
axial direction are negligible. Since considerable power is dissi-
pated by the 2D MOT coils (table 3.6) they are water-cooled in
parallel with the transport coils (section 3.10).
The alignment of the axial beam (and its counter-propagating
partner) is crucial for proper operation of the 2D MOT. A good
alignment technique is to first roughly maximise transmission
of the axial beam through the whole apparatus (coming out
of the opposite 2D MOT chamber). At this point the reflected,
ring-shaped beam should exit the entrance viewport which can
be made visible by holding a viewing card with a hole in place.
Probably the retro-reflected beam is then not exactly overlapped
with the incoming beam. This can be achieved by further beam-
walking, whilst keeping the ring-shaped profile approximately
constant. The alignment of the transverse beams is not crucial; it
has almost no effect on the MOT loading rate provided that the
beams are collimated and roughly counter-propagating.
Fluorescence signal
In order to characterise the loading of the MOT we monitor its
fluorescence. Scattered light is collected on a lens at a vacuum
viewport on the knee (Fig. 2.1), focussed, split into two parts, and
filtered by wavelength before it reaches a photodiode (one for each
species).Typical scope traces are shown in Figs. 3.14, 3.15, and 3.16.
0.0 1.0 2.0
Time/s
0
10
20
30
87
R
b
ph
ot
od
io
de
/m
V
87Rb alone
Figure 3.14: Fluorescence signal for
MOT loading without potassium
(rubidium only).
In potassium the optical powers for repump and cooling light
are comparable since the upper hyperfine levels are closely spaced
(Fig. 3.3) and there is no closed transition. The necessity for higher
laser powers leads to increased photon scattering and therefore
limits the achievable temperatures in a 39K-MOT to values well
above the Doppler-limit [127] of 145 µK. Additionally, we expect
light-assisted collisions to play a role in the comparatively modest
performance of the 39K-MOT in the presence of 87Rb [155, 156].
Compared to the 39K-only case (Fig. 3.15) we load roughly half as
many 39K-atoms if rubidium is present (Fig. 3.16). However, the
rubidium is almost unaffected by the potassium; this is impor-
tant because we rely on abundant 87Rb-atoms as coolant to reach
quantum degeneracy in 39K (section 4.3).
laser cooling & trapping of 87rb & 39k 51
3.7 Final MOT stage in potassium
0.0 1.0 2.0
Time/s
0
2
4
6
39
K
ph
ot
od
io
de
/m
V
39K alone
Figure 3.15: Fluorescence signal for
MOT loading without rubidium
(potassium only).
0.0 1.0 2.0
Time/s
0
2
4
6
39
K
ph
ot
od
io
de
/m
V
87Rb
39K
0
10
20
30
87
R
b
ph
ot
od
io
de
/m
V
Figure 3.16: Fluorescence signal
for MOT loading with both species
present. (oscilloscope on high-
resolution mode for all these mea-
surements).
Once loading of potassium is saturated (Fig. 3.16) we address the
problem of strong photon (re-)scattering (described above) by low-
ering the 39K-repump power in the final 10ms of the MOT. During
this ‘final MOT’ stage we linearly decrease the 39K-repump power
to 2% of its original value. Additionally, we decrease the detuning
of the 39K-cooling light to −28 MHz at the beginning of the final
MOT stage (the repump detuning remains essentially unchanged).
We observe a significant increase in 39K atom number using this
procedure. Explaining this effect in detail is non-trivial due to the
complexity of the process (dual species MOT, non-equilibrium
system, unresolved 39K-hyperfine structure). However, the most
plausible explanation seems that of a ‘temporal’ dark MOT [118,
157, 158] in which further heating due to photon (re-)scattering is
reduced by 39K-atoms accumulating in the dark F = 1 hyperfine
level.
Several experiments on 39K have found that increasing the
MOT gradient during the final MOT stage (‘compressed MOT’ or
cMOT [159]) has a positive effect on the MOT performance [120,
135, 160]. Other experiments (on lithium) have found the op-
posite to be true [161]. Yet different experiments (on rubidium)
have shown a somewhat chaotic behaviour to produce the best
results [162]; they employ an artificial neural network to perform
multiparameter optimisation of the final MOT stage. In our case
we observe strong fluctuations in the system when increasing the
MOT gradient and consequently we leave the MOT gradient un-
changed in this stage of the experiment. Similarly, there seems to
be no consensus as to whether the detuning of the cooling light
should be increased [120, 135] (i.e. shifted away resonance), de-
creased [160], or modulated in a non-trivial manner [162]. We find
decreasing the detuning of the cooling light (shifting its frequency
towards resonance) results in the best performance.
3.8 Optical molasses
Table 3.7: Molasses parameters. Laser
powers are given per beam. Detunings
are given in MHz from the respective
transitions, which are the same as for
the MOT (negative values mean red
detuning).
parameter 87Rb 39K
duration 4.5ms
cooling power 26-35mW 16-18mW
repump power 0.7-1.0mW 1.9-2.8mW
cooling detuning −63 MHz −4 MHz
repump detuning 0 MHz −45 MHz
A typical 87Rb-39K-MOT contains on the order of one billion
atoms of each species at temperatures in the millikelvin-regime.
The standard procedure to reach colder temperatures is the optical
molasses technique [163, 164]. Its setup is identical to that of the
MOT except that all magnetic field gradients are switched off. On
the one hand, this means there is no restoring force towards the
centre and consequently the atoms are not trapped any longer.
Consequently the duration of this cooling procedure is limited to
a couple of milliseconds. On the other hand, polarisation-gradient
cooling mechanisms [165] which were previously confined to the
trap centre are now happening across the entire atomic cloud. The
latter effect is so strong that within 4.5ms the temperature of the
cloud drops by almost a factor of ten. Our experimental parame-
ters are summarised in table 3.7.
52 quasicrystalline optical lattices for ultracold atoms
MOT compensation coilsTable 3.8: MOT compensation coil pa-
rameters. Here we use the convention
that the x-axis (yellow wire) coincides
with the direction of magnetic trans-
port, the y-axis (blue) points along
the axial direction of the 2D MOTs,
and the z-axis (grey) is aligned with
gravity. The coils for optical pump-
ing (spin-polarisation, s-p) are also
wound on the z-axis (black wire). The
number of windings is given per coil,
the conversion factor between field
and current in G/A, and the optimum
current values in A.
parameter x y z s-p
windings 6 5 34 6 15
conversion (G/A) 0.4 0.4 0.37 0.92
current/A 1.5 1.5 0.0 -
During optical molasses any residual magnetic field gradients can
lead to directional forces and thus inefficient cooling. Therefore
we use three pairs of ‘compensation coils’ to counteract stray mag-
netic fields caused by ion pumps, optical isolators, and Earth’s
magnetic field. The compensation coil parameters are listed in
table 3.8; the optimum current values were found by maximising
the number of atoms after transport. Changing the compensation
fields has negligible effect on the cloud temperature after trans-
port. The optical isolators for the lattice are the dominant source
of stray fields (by a factor of two) since placing them caused a
sizeable shift in the optimum currents.
3.9 Optical pumping (spin-polarisation)
Table 3.9: Optimum parameters
used for optical pumping of 87Rb.
Detunings are given in MHz from the
unshifted transitions (at zero field).
Negative values mean red detuning.
parameter value
total duration 0.7ms
pumping duration 100 µs
repump duration 45 µs
offset field 10G
pumping detuning −3 MHz
repump detuning +4 MHz
pumping power 1.5mW
repump power 0.4mW
beam waist 3mm
After optical molasses the atoms need to be magnetically trapped
and transported to the science cell for further (evaporative) cool-
ing. Only weak-field seeking mF states (i.e. states with a pos-
itive zeeman shift, gF × mF positive in Eq. 3.1) can be trapped
using magnetic forces.10 In the electronic ground levels of 87Rb
10 since local field maxima are pre-
cluded by Earnshaw’s theorem [124]
and 39K there are three trappable states: |F = 2, mF = 1, 2〉 and
|F = 1, mF = −1〉. After the optical molasses stage we optically
pump the 87Rb atoms into the |F = 1, mF = −1〉 state using a flash
of σ− polarised pumping (F = 2 → F’ = 2) and repumping light
(F = 1 → F’ = 2). Two frequencies are necessary to perform both
Zeeman pumping (mF-states) and hyperfine pumping (correct hy-
perfine level F = 1). The repump light is only on for the first half
of the pulse. An offset field is provided by dedicated coils that are
wound around the z-MOT-compensation coils (table 3.8). The cur-
rent for the offset field is produced by a capacitor discharge unit;
if needed the polarity of the coils can be switched using a current-
H-bridge. All relevant parameters can be found in table 3.9; they
were found by maximising the number of captured atoms in the
magnetic trap.
0 60 120 180
λ/4-waveplate/degrees
0
4
8
C
ap
tu
re
d
at
om
s/
10
8
data
no s-p
Figure 3.17: Optical pumping effi-
ciency. The number of captured atoms
is plotted as a function of pumping-
polarisation (λ/4-waveplate) in 87Rb.
Atom numbers are gauss sum fits to
fluorescence images in 5ms ToF after
release from the magnetic trap. The
minimum (maximum) atom numbers
correspond to almost σ+ (σ−) polar-
isation, probably with some residual
ellipticity.
In order to estimate our pumping efficiency we vary the polari-
sation using a λ/4-waveplate (Fig. 3.17). The maximum and min-
imum numbers of captured atoms in Fig. 3.17 can be interpreted
as most atoms being in |F = 1, mF = −1〉 and |F = 1, mF = 1〉
(un-trapped), respectively. Since these data points are roughly
symmetric around the situation without any optical pumping (‘no
s-p’) we can conclude that our Zeeman- and hyperfine-pumping
are both in principle working properly. The deviation from the
expected cosine-behaviour leads us to believe that our pumping
efficiency is limited by birefringent optical elements between the
waveplate and the position of the atoms (possibly the vacuum
windows). This birefringence may prohibit perfect σ+ and σ−
polarisation and therefore lead to a reduced optical pumping ef-
ficiency of roughly 75%. In principle, the birefringence could be
laser cooling & trapping of 87rb & 39k 53
accounted for by placing a λ/2-waveplate but this was not found
to be crucial for BEC production.
There is no general consensus as to which hyperfine state
should be used for transport. Some experiments [120, 160] opt for
the |F = 2, mF = 2〉 state because its trapping gradient is twice as
large compared to states with |mF| = 1. Other experiments [135]
report the |F = 1, mF = −1〉 to be beneficial. We use the latter state
since it gives the best results (condensate numbers). This could be
explained by the improved performance of the MW evaporation
(section 3.11) using the lower (F = 1) hyperfine manifold. For 39K
we have so far not managed to effectively pump atoms into the
|F = 1, mF = −1〉 state11 and consequently we do not perform any 11 however, pumping to the
|F = 2, mF = 2〉 state works efficientlyoptical pumping for this species. Potentially the relatively strong
offset field of 10G precludes any efficient Zeeman pumping since
at this field the upper hyperfine levels (Fig. 3.6 D) are split in a
nonlinear way. Moreover, the final MOT stage for 39K leads to a
large occupation of the F = 1 manifold which in our case suffices
to perform further cooling without any optical pumping.
3.10 Magnetic trapping, transport, and experiment coils
Magnetic quadrupole traps [127, 166] are used to capture the
atoms in the MOT chamber, transport them to the science cell,
and perform MW evaporation (section 3.11). Conceptually the
magnetic trap relies on a magnetic field minimum at which atoms
with positive Zeeman shift accumulate (gF ×mF has to be positive
in Eq. 3.1). The typical quadrupole coil configuration consists
of two axially aligned coils with counter-propagating currents;
this situation leads to a field minimum in between the two coils
that is at exactly zero field with linear gradients in all directions.
The main parameter characterising a magnetic trap, which for
all practical purposes has infinite trap depth, is its gradient. Due
to the second Maxwell equation ∇ · B = 0 the field gradient in
the axial direction is twice as big as the gradients in the planar
direction, leading to a 2:1-aspect ratio of the trapped atomic cloud.
In this work the trap gradient is always quoted along the ‘strong
axis’.
Table 3.10: Parameters used for mag-
netic trapping and transport.
parameter value
total distance 64.35 cm
total duration 4.99 s
acceleration 0.2m/s2
max. speed 0.5m/s
sampling rate 1kHz
MOT gradient @ 1A 1.78G/cm
MOT grad. (capture) 53G/cm
MOT grad. (transport) 100G/cm
exp. gradient @ 1A 3.5(1)G/cm
exp. grad. (transport) 100G/cm
exp. grad. (compressed) 300G/cm
A series of overlapping magnetic traps (Fig. 3.18) can be used
to transport atoms over relatively large distances (64 cm in our
case) in a ‘conveyor belt’ motion. By choosing appropriate cur-
rent ramps in the successive coil pairs the aspect ratio12 can be
12 between the longitudinal gradient
along the transport axis and the
transverse gradient
kept within reasonable bounds in order to avoid heating due to re-
peated compression and decompression of the cloud [117]. In our
configuration (Fig. 3.19) this aspect ratio varies from 1.0 (when the
cloud is at rest) to 2.4 (at the last coil pair before the atoms reach
the experiment coils). More information on the design of the coil
system can be found in the Master’s thesis of Oliver Brix [167].
54 quasicrystalline optical lattices for ultracold atoms
Figure 3.18: Schematic of magnetic
coils used for MOT & magnetic cap-
ture (2, green), transport (3/4, blue),
MW evaporation and Feshbach fields
(5, red). All coils except the experi-
ment coils (5) and the push coil (1) are
glued into a brass cooling block (only
the lower cooling block is shown for
clarity). Figure taken from [124].
2
3
5
1
4
Figure 3.19: Coil currents used for
magnetic transport. These were
determined by maintaining an ap-
proximately constant aspect ratio of
the atomic cloud over the travelled
distance [117, 167]. The colour code is
the same as in Fig. 3.18.
0 10 20 30 40 50 60
Distance/cm
0
20
40
60
80
100
C
ur
re
nt
/A
MOT push 1 - 16 17 exp
m
ot 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 ex
p
Coil index
0.0
0.5
1.0
Fr
ac
ti
on
(r
et
ur
n
jo
ur
ne
y)
87Rb alone
Figure 3.20: Fraction of rubidium
atoms recaptured after transport
(return journey).
The experiment coils themselves serve a further purpose, in
addition to providing a strong magnetic confinement for transport
and MW evaporation, namely to create homogeneous Feshbach
fields (section 4.6). They are designed such that if the polarity of
one experiment coil is flipped the resulting field at the position
of the atoms is as homogeneous as possible. In our case we ex-
pect residual curvatures on the order to 10−4 mG/µm2 at field
strengths of several hundred gauss [167]. If we had a uniform
optical trap for the atoms [168] these residual curvatures would
result in effective trap frequencies on the order of 1Hz.
m
ot 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 ex
p
Coil index
0.0
0.5
1.0
Fr
ac
ti
on
(r
et
ur
n
jo
ur
ne
y)
39K alone
39K with 87Rb
Figure 3.21: Fraction of potassium
atoms recaptured after transport
(return journey), with and without
rubidium present.
The sequence for magnetic transport starts with trapping the
atoms after optical pumping in the MOT chamber using a gradient
of 53G/cm. Subsequently the MOT field gradient is ramped to
100G/cm and the atoms are transported using the current values
shown in Fig. 3.19. In order to convert the distances to timestamps
in sequence we use a constant acceleration until we reach a certain
max. speed (and vice versa for decelerating). The atoms come to
a halt at the 45◦ bend in the transport path (‘knee’ in Fig. 2.1). All
relevant transport parameters are summarised in table 3.10.
The transport efficiency is characterised by performing a return-
journey to the experiment coils and back. Afterwards the number
of atoms that survived is counted (using fluorescence imaging in
the MOT chamber), as shown in Figs. 3.20 and 3.21. The result-
laser cooling & trapping of 87rb & 39k 55
ing transport efficiency is then given by the square root of the
fraction of atoms remaining after the return journey (table 3.11).
In general, the transport efficiency improves with higher gradi-
ents but we are limited to 100A by our power supplies. Another
way to enhance efficiency would be to transport the atoms in the
|F = 2, mF = 2〉 state which experiences a trap gradient that is
twice as large. However, the subsequent MW evaporation stage
works better in the |F = 1, mF = −1〉 state and therefore we trans-
port the atoms in this state. We observe that magnetic transport is
generally more efficient for 39K than for 87Rb (Fig. 3.21). We also
observe that lower acceleration improves the efficiency for 87Rb.
Table 3.11: Single-journey transport ef-
ficiencies for rubidium and potassium.
87Rb alone 64%
39K alone 80%
39K with 87Rb 66%
Technical implementation of transport and experiment coils
Table 3.12: Experiment coil parameters.
Note that the ribbon height has been
reduced by taking off < 0.5mm from
one side. The temperatures quoted
are steady-state values for a 50% duty
cycle at 100A; these are reached after
about 10minutes while the cooling
block is stabilised to 21.5◦C. The
temperature values are measured
using Pt100 sensors that are directly
glued to the coil using thermally
conductive epoxy glue.
parameter value
windings 75 per coil
ribbon height 16mm
ribbon width 0.5mm
resistance upper coil 66.2mΩ
resistance lower coil 66.2mΩ
temperature upper coil 45◦C
temperature lower coil 40◦C
All transport coils – except the pair of experiment coils – are glued
into a brass block (Fig. 3.18) that is watercooled and actively sta-
bilised to 21.5◦C (which is also the set temperature of the laser-
table air conditioning). More information on the design of the
cooling block can be found in ref. [167]. The experiment coils were
wound in-house using copper ribbon wire, kapton insulation be-
tween the windings, and thermally conductive but electrically
insulating epoxy glue. The side of each coil that faces the brass
block was milled down slightly (using a fly-cutter) and lapped in
order to ensure flatness and thereby improve thermal contact with
the brass block.13 Between the coil and the brass block there is an
13 despite this procedure there was
some residual unevenness, probably
due to the coil not being fully rigid in
itself
electrically insulating (but thermally conducting) sheet, as well
as heat compound paste that is directly applied to the coil face.
Carbon plates are screwed to the cooling block from both sides
in order to fix the experiment coils in place. The tightness of the
screws was carefully adjusted to optimise cooling performance
(table 3.12).
Unfortunately, there is a mechanical instability in the upper
experiment coil that makes fast switching impossible. If the upper
coil is switched off suddenly using a MOSFET switch the effective
field strength of the coil is reduced; consequently the position
of the magnetic trap moves upwards. Repeated pulses of this
kind lead to a degrading of field strength over time that takes
several weeks to relax. We observed shifts of several hundred
µm, corresponding to 1-2A of current ‘missing’ in the upper coil.
We attribute the effectively reduced current to electrical shorts
between the windings. These could be potentially introduced
by a jerk motion during the sudden switch-off of the coil (rough
estimates suggest an outwards force on the order to 10N within
one coil caused by the magnetic field of the other coil). The field
strength of the lower coil is insensitive to a sudden switch-off. We
have avoided fast switching for several months now and the cloud
position has been relatively stable since then.
56 quasicrystalline optical lattices for ultracold atoms
Compensation coils (offset fields)
20 40 60 80
Experiment coils current/A
-40
0
40
80
Ve
rt
ic
al
in
-s
it
u
po
si
ti
on
/µ
m 0.0
0.2
0.4
0.6
0.8
1.0
Figure 3.22: Optimisation of compen-
sation coil fields using the movement
of the quadrupole trap centre as
figure of merit. Data is shown for
six different current values in amps
(colour-coded in different shades of
blue) of the Z-compensation coils, the
optimum being around 0.2A (least
movement).
Similar to the MOT chamber we use three pairs of rectangular
compensation coils around the science cell to counteract any stray
magnetic fields. In an ideal quadrupole configuration the loca-
tion of the field minimum does not change with the trap gradient.
However, adding an offset field leads to a dependence of the trap
centre on its gradient. Consequently we can use the trap move-
ment to characterise our ambient magnetic fields and compensate
for them by minimising the gradient-dependence of the trap cen-
tre (Fig. 3.22).
After describing transport and the necessary magnetic hard-
ware we will now outline the evaporative cooling sequence in the
steep magnetic trap.
3.11 Microwave (MW) evaporation
The first stage of evaporative cooling in our experiment takes the
atoms from hundreds of µK down to the single-digit µK regime,
cold enough to be trapped by the optical dipole trap (Chapter 4).
In this technique the highest-lying atoms in the magnetic trap
(i.e. the hottest ones) are selectively removed via radio frequency
transitions to untrapped states. Evaporative cooling in magnetic
traps was first proposed for hydrogen [169] and then demon-
strated in 1995 [170, 171] in the wake of the first Bose-Einstein
condensates. It remains an important technique to this day due to
its conceptual simplicity, particularly in the context of atom-chip
microtraps [172]. Future large-scale cold-atom experiments will
probably move towards all-optical production of ultracold gases,
e.g. loading the dipole trap directly from grey molasses [133],
thereby avoiding the technical challenges posed by high-current
carrying coils (see above). However, in our setup the possibility of
selectively removing 87Rb-atoms (and not the 39K-atoms) is crucial
to our sympathetic cooling strategy.1414 For more information on sympa-
thetic cooling, please see section 4.3
For evaporation the |F = 1, mF = −1〉 → |F = 2, mF = −2〉
transition in the ground 52S1/2-level of 87Rb (arrow in Fig. 3.5A) is
addressed. After transport the 87Rb atoms have a temperature of
roughly 500µK in the steep magnetic trap (300G/cm). The initial
frequency for MW evaporation is chosen such that only the very
hottest atoms in the trap are removed at first. In our case we start
at a frequency of 6676MHz which addresses atoms situated at a
magnetic field around 76G (Fig. 3.5 A); this magnetic field cor-
responds to a thermal energy (kBT) of 5mK. In an ideal thermal
gas at 500µK only a tiny fraction (5× 10−5) of atoms have more
kinetic energy than this, which justifies our starting frequency.
Subsequently the MW frequency is lowered to 6827.2MHz in a
7.5 s ramp.
laser cooling & trapping of 87rb & 39k 57
Technical implementation
The MW radiation for evaporation is generated by mixing the
output of a home-built DDS unit [173] with a reference oscillator
at 6500.0MHz. The signal is then amplified several times (10W
max. power) and brought to the atoms by a waveguide and a
capped horn antenna which is situated around 13 cm from the
atoms in direct line of sight. The backreflection from the antenna
is dumped via a circulator into a high-power attenuator.
Majorana losses
The ‘kink’ at the centre of a magnetic quadrupole trap starts to
dominate the (quasi-)equilibrium15 in the trap for temperatures 15 Perfect equilibrium for a thermal gas
in our system can never be reached
since particle number is not conserved.
There is always an (exponentially
small) fraction of atoms at very high
energies that can escape the trap.
Subsequent re-equilibration leads to
an additional evaporation effect which
is negligible for our experimental
timescales.
in the µK regime. In this region, atoms passing very close to the
magnetic field minimum experience a sudden change in direction
of the magnetic field, leading to a non-adiabatic spin-flip [125]. In
our case this means that trapped atoms in the |F = 1, mF = −1〉
state become un-trapped (|F = 1, mF = 1〉) and lost from the cloud.
Even worse, since the atoms in the centre of the trap are the cold-
est ones, these atom losses lead to heating due to rethermalisation
of the remaining atoms. Therefore we terminate MW evaporation
at temperatures around 6.6µK before Majorana losses (Fig. 3.23)
and heating (Fig. 3.24) become dominant.
0.0 0.5 1.0 1.5 2.0
Hold time/s
0.0
0.5
1.0
N
or
m
al
is
ed
at
om
nu
m
be
r
6800.0 GHz
6827.2 GHz
6833.0 GHz
Figure 3.23: Majorana losses for 87Rb
(no 39K present) for three different fi-
nal MW frequencies. The hold time in
a steep magnetic trap (300G/cm) after
termination of the MW evaporation is
plotted on the x-axis. Temperatures
and atom numbers are measured after
release from the dipole trap in 33ms
ToF.
0.0 0.5 1.0 1.5 2.0
Hold time/s
4
6
8
10
12
Te
m
pe
ra
tu
re
/µ
K
6800.0 GHz
6827.2 GHz
6833.0 GHz
Figure 3.24: Majorana heating for
87Rb in the steep magnetic trap for
three different final MW frequencies.
Before we turn to the dipole trap and the physics of Bose-
Einstein condensation in Chapter 4 the experimental aspects of
absorption imaging are briefly discussed in the following. The
time-of-flight imaging technique is covered in more detail in sec-
tion 6.2 in the context of optical lattices.
3.12 Absorption imaging
The absorptive behaviour of a material in the presence light, char-
acterised by the Beer-Lambert law, has been known for more than
a century. In the case of an atomic vapour and laser light that
is resonant with a closed transition in the atom the laser inten-
sity is simply reduced exponentially with the distance travelled
through the vapour. Thus we can extract the density profile of an
atomic cloud by taking a sequence of two images of the laser beam
profile, one with the atomic cloud (seen as a shadow) and one
without [125]. The optical density of the cloud is then given by
OD(x, y) = σ ncolumn(x, y) = − ln
(
I(x, y)
I0(x, y)
)
(3.4)
Here σ is the absorption cross-section of a single atom, ncolumn is
the column density, I(x, y) is the ‘atom’ picture, and I0(x, y) is the
empty picture.16
16 in practice a third image is taken
without any laser light (background
image) an this image is subtracted
from both I(x, y) and I0(x, y)
All the measurements on Bose-Einstein condensates (with and
without lattices) were performed using this technique. Usually we
release the atoms from the dipole trap (see chapter 4 below) and
58 quasicrystalline optical lattices for ultracold atoms
perform absorption imaging in 33ms time-of-flight (ToF) on the Z-
axis (along gravity). There is resonant imaging and repump light
available for both species on all individual lattice axes (named X,
Y, D, and T, according to Fig. 5.17 in Chapter 5).
In some special cases it can be useful to image the cloud ‘in-
situ’, i.e. in the trap, for example to measure the position of the
magnetic trap (Fig. 3.22). Note that our capabilities of taking im-
ages in ToF from the magnetic trap are limited since we cannot
shut down the coil currents quickly enough (section 3.10). Short
imaging pulses of 100µs with low intensity17 (I  Isat) are typi-17 such that σ is only weakly depen-
dant on I cally used.
At low magnetic fields, both 87Rb and 39K are imaged on their
respective 2S1/2 |F = 2, mF = 2〉 → 2P3/2 |F = 3, mF = 3〉 transition
which comes closest to being a cycling transition. It is crucial for
the imaging light to have a definite polarisation with respect to the
quantisation axis of the atoms, given by a homogeneous magnetic
field. In particular, the low-field imaging scheme requires σ+
polarised light. A wrong polarisation can lead to an effective shift
in the imaging transition. Figure 3.10 nicely illustrates this effect
to the extent that σ− polarised light causes a shift by several MHz.
These imaging resonance scans were taken using a cold atomic
cloud released from the dipole trap.
Repump flash
In order to use the |F = 2, mF = 2〉 → |F = 3, mF = 3〉 cycling tran-
sition the atoms have to be prepared in the 2S1/2 |F = 2, mF = 2〉
state, or at least the F = 2 level within 2S1/2, prior to imaging. To
this end, a dedicated flash of repump light (see Figs. 3.1 and 3.3)
may be necessary, depending on the current stage of the experi-
mental sequence. Low-field imaging in 39K is almost never used
due to the slowness of our experiment coils (section 3.10), never-
theless, 39K repump light is available on two imaging axes in the
science cell. In the case of 87Rb, the repump flash is almost always
used.18 Figure 3.8 shows a resonance scan of the repump flash18 since the 87Rb atoms spend almost
all their lives in the ground F = 1
manifold
frequency (which can be tuned using an AOM in double-pass con-
figuration) using a cold atomic cloud. In order to maximise the
detected atom number, a relatively long flash of 1.0ms is used in
the experiment.
High-field imaging in 39K
Table 3.13: High-field imaging reso-
nances for 39K.
B-field/G Resonance/MHz
317.38 -301.4
351.44 -352.4
397.19 -419.0
In contrast to 87Rb, the potassium atoms are almost exclusively
imaged at strong magnetic fields (see section 4.6 below). The
atomic levels in 39K are detailed in Fig. 3.6 for both low and high
magnetic fields. In this case, the |F = 1, mF = 1〉 state connects to
the |mI = 3/2, mJ = −1/2〉 at high magnetic fields. Imaging is per-
formed on a transition from this state to the |mI = 3/2, mJ = −3/2〉
state in 2P3/2. Unfortunately, the mixing of hyperfine states at
high fields precludes the existence of a cycling transition and leads
a depumping effect [174] of approximately 3% per cycle. The neg-
laser cooling & trapping of 87rb & 39k 59
ative effects of depumping can be mitigated by performing an
independent measurement of atom number by means of locating
the BEC transition (section 4.7).
High-field imaging of 39K is routinely performed in our exper-
iment and some measured resonance frequencies are shown in
table 3.13. The experimental resonances agree excellently with
the theoretical prediction from Fig. 3.6, including the electric
quadrupole correction [140]. As with low-field imaging, the po-
larisation of the imaging light is crucial here. To this end, a setup
containing a λ/2 and a λ/4 waveplate is used to optimise the
imaging polarisation using atom number as figure of merit.
320 360 400
B-field/G
-300
-340
-380
-420Im
ag
in
g
re
so
na
nc
e/
M
H
z data
theory
Figure 3.25: 39Kimaging resonances at
strong magnetic fields. The agreement
between the experimental values (ta-
ble 3.13) and the theoretical prediction
(Fig. 3.6) is better than 0.6MHz.
Stern-Gerlach imaging
In order to discern the population of mF states of a given atomic
cloud, the Stern-Gerlach imaging technique is used. The cloud is
split according to the magnetic moment of each atom by subject-
ing the atoms to a magnetic field gradient while they are in free
fall (during time-of-flight).
In our apparatus a dedicated gradient coil has been wound in a
3D-printed mount that also contains the RF-antenna (section 4.5),
just above the science cell. It is driven by a home-built capacitor
discharge unit that is designed to provide a short pulse of high
current (table 3.14).
This coil provides a vertical field gradient of > 10 G/cm such
that the cloud is split vertically into the different mF components.
For 87Rb, a 2ms current pulse at the beginning of ToF (15ms)
leads to a vertical displacement of 246 µm between each mF com-
ponent.
Table 3.14: Parameters for Stern-
Gerlach imaging.
parameter value
coil windinds 360
peak current 12A
initial rise time 50 µs
pulse duration 87Rb 3ms
pulse duration 39K 1ms
It should be noted that neither the repump flash, nor the Stern-
Gerlach imaging technique has been achieved after the magnetic
trap but only in the dipole trap.
This discussion of absorption imaging, including specific ap-
plications thereof, such as high-field imaging in 39K and Stern-
Gerlach imaging, concludes the chapter on experimental methods
for laser cooling.

4
Quantum gases
Up to this point in the experimental sequence (table 2.1) the atoms
have been cooled, trapped, transported to the science cell, and
cooled down further to µK temperatures. During these stages
of the sequence the atom cloud represents a classical, thermal
gas. However, at even lower temperatures the collective prop-
erties of the gas start to be influenced by the wave-like nature
of particles, i.e. quantum mechanics. The first important mani-
festation of collective quantum-mechanical behaviour lies in the
collisional properties at low momenta. This will be briefly dis-
cussed in the context of Feshbach resonances (section 4.6).1 Once 1 Note that the bosonic or fermionic
particle symmetry already influences
thermodynamic properties, namely via
the type of prevalent low-energy col-
lision, long before quantum-statistical
considerations become relevant.
the thermal wavelength of the particles becomes comparable to the
inter-particle distance, the gas has to be described fully quantum-
statistically, reflecting the strikingly different low-temperature
phases of fermions and bosons (section 4.7).
The chapter begins with a brief introduction to optical dipole
traps. Afterwards our specific setup, namely a crossed-beam
dipole trap with elliptical waists, is characterised theoretically
and experimentally. Sympathetic cooling is discussed, together
with the general properties of Feshbach resonances in 39K. Finally
the concept of Bose-Einstein condensation is introduced.
4.1 Dipole trap: introduction
The interaction between atoms and monochromatic light can be
broadly divided into two separate regimes. On the one hand,
near-resonant light leads to dissipative behaviour and, conse-
quently, photon scattering. This effect is utilised by laser cooling
methods such as the molasses technique (section 3.8). On the
other hand, far-detuned light acts dispersively, leading to a con-
servative potential for neutral atoms which is often referred to
as the dipole potential. Relevant formulae can be derived both
semi-classically [127, 175] and purely quantum-mechanically [165,
175, 176]. The latter derivation is based on the Jaynes-Cummings
model, describing a two-level atom coupled to a quantised light-
field (a coherent state) resulting in a shift of the two atomic levels,
called light-shift or AC-Stark shift. Semi-classically, the light-shift
can be understood as the average energy of a dipole in the pres-
ence of an oscillating electric field, characterised by the real part
62 quasicrystalline optical lattices for ultracold atoms
of the polarisability of the atom. In this chapter we will not de-
rive the formulae from first principles but instead refer to Grimm
et al. [175]. In our experiment we use a standard red-detuned
dipole trap at 1064nm such that an intensity maximum results in
a trough in the potential. The photon scattering rates in the trap
centre are on the order of millihertz and can thus be neglected on
experimental timescales.
Choice of trapping geometry
Both the loading of the dipole trap and the subsequent evapora-
tion to quantum degeneracy contain several non-linearities, such
that the choice of optimal dipole beam parameters a subtle trade-
off between many different variables.
Several factors affect the loading efficiency of the dipole trap
from a magnetic quadrupole trap. Firstly, a high spatial overlap
between the two traps is required. The aspect ratio of an atomic
cloud confined to the centre of a radially symmetric magnetic
quadrupole field is always approximately 2 : 1 (as discussed in
section 3.10). This favours radially symmetric, oblate dipole traps
of similar aspect ratio. Secondly, the trapping volume should be
large, encouraging large beam waists, so as to transfer as many
atoms as possible to the dipole trap. But for the same reason the
trap should also be as deep as possible which, given a fixed avail-
able laser power, favours small waists.
The dipole trap geometry also crucially affects the final stages
of the experimental sequence, including optical evaporation and
loading into the optical lattice. Optical lattice experiments in two
dimensions profit from approximately homogeneous density
profiles, favouring oblate traps with high aspect ratios. In addi-
tion, anticonfining effects of blue-detuned lattice potentials can
be cancelled2 by suitably configured dipole traps. Moreover, the2 In the case of a one-dimensional,
blue-detuned, simple-cubic lattice
the resulting anticonfining poten-
tial is gaussian in both horizontal
and vertical directions which can be
compensated exactly by an attractive
dipole potential with a
√
2 times larger
waist. For two-dimensional lattices
the cancellation is approximate. Here,
instead of an in-plane crossed-beam
trap a single dipole beam in the axial
direction can be used to cancel the
quadratic terms in the anticonfining
potential.
geometric mean of the trapping frequencies ω¯ = 3
√
ωzω2r should
be kept as large as possible in order to avoid very high densities at
the centre of the trap which can lead to three-body losses. Finally,
the efficiency of sympathetic cooling using a coolant with a larger
atomic mass (e.g. cooling 39K with 87Rb) depends in a subtle way
on the gravitational sag in the dipole trap (as discussed in the fol-
lowing). On the one hand, maximising the spatial overlap between
the two species corresponds to having a small differential sag and
very small vertical waists. On the other hand, ideal sympathetic
cooling of 39K keeps all the potassium atoms in the trap, evap-
orating only 87Rb atoms. It turns out that the parameter region
of optical powers for which this is the case is larger for smaller
vertical waists.
Table 4.1: Dipole trap beam param-
eters at the position of the atoms,
as determined by measuring trap
frequencies.
axis w0h/µm w0v/µm max. pow.
x 285(5) 55(2) >7W
y 285(5) 55(2) >7W
In our experiment we decided to use a crossed-beam dipole
trap with elliptical beam profiles, summarised in table 4.1. The
two dipole beams propagate in the plane normal to gravity and
their foci intersect at an angle of just under 90◦. For each beam
the ratio of horizontal to vertical waist size is about five to one,
quantum gases 63
leading to a radially symmetric, oblate trapping geometry.
Trapping potential
The dipole potential for an idealised two-level (e-g) atom state can
be written as [175]
Ve-g(r) = −3pic
2
2ω3a
(
Γa
ωa −ω f +
Γa
ωa +ω f
)
I(r) , (4.1)
where ωa is the resonance frequency of the atom, ω f the fre-
quency of the light field, Γa the spontaneous scattering rate, and
I(r) the laser intensity profile. In a red-detuned dipole trap we
have ωa > ω f such that the dipole potential Ve-g is negative,
leading to an attractive force towards its centre. The relevant
transitions in the alkalis are the D-lines (section 3.1) which both
contribute to the total trapping potential, given by
Vdip(r) =
1
3
VD1e-g (r) +
2
3
VD2e-g (r) . (4.2)
Here we have assumed linearly polarised light for which the trap-
ping potential is independent of the given internal hyperfine state
of the atom.
The intensity profile of a focussed gaussian laser beam, as-
sumed to propagate along the z-axis is
I(r) =
2P
piwh(z)wv(z)
exp
(
− 2h
2
w2h(z)
− 2v
2
w2v(z)
)
. (4.3)
The paramaters h and v stand for the respective horizontal and
vertical coordinate and P is the optical power in watts. The beam
waists wi(z) are given by
wi(z) = w0i
√√√√1+( z
piw20i/λ
)2
, i ∈ {h, v} (4.4)
where w0i are the (single) waists of the elliptical gaussian beam in
h, v-direction and λ is the dipole trap wavelength.
Now we are going to change into lab coodinates. In the crossed-
beam configuration two dipole foci of equal waists are overlapped
at just under 90◦ which can be approximated by the sum of two
gaussian beams propagating along x and y, respectively. In the lab
frame z is chosen to be the direction of gravity and thus the axial
direction of the crossed-beam trap. Both beams are supposed to
have equal waists such that the trap is radially symmetric in the
xy-plane. The dominant trapping effect arises from the respective
transverse beam profile of each beam, whereas the beam profile
along the axes of propagation only add a weak confinement.
In the centre of the trap the potential is harmonic which can
be seen by Taylor-expanding Eq. 4.3. The characteristic trapping
64 quasicrystalline optical lattices for ultracold atoms
frequencies in the radial (ωr) and axial (ωz) directions for an atom
of mass m are then given by
ωr =
√
4V0
mw20h
(4.5)
ωz =
√
8V0
mw20v
, (4.6)
where V0 is the maximum dipole potential, i.e. Vdip in the centre
of the trap. Both dipole beams contribute to the vertical confine-
ment which leads to an increase in the vertical trapping frequency
by a factor of
√
2 compared to a single-beam trap.360 380 400
Dipole trap power/mW
0
20
40
R
b
BE
C
nu
m
be
r/
10
4
Figure 4.1: Minimal trapping power
for 87Rb in the dipole trap. Below a
certain threshold (385(5) mW in each
beam, marked by the shaded area) the
dipole trap cannot hold the rubidium
condensate against gravity.
180 190 200
Dipole trap power/mW
0
5
10
39
K
BE
C
nu
m
be
r/
10
4
Figure 4.2: Same as above, but for 39K.
Here the threshold is 190(2) mW.
Effects of gravity
The expressions 4.5 and 4.6 are good approximations as long as
gravity can be neglected. But as soon as the gravitational potential
becomes comparable to Vdip it has to tbe included, leading to a
total potential of the form
Vtot(r) = Vdip(r) + mgz . (4.7)
The maximum trap depth Vmax/kB expressed in µK is given by
the energy difference between the local minimum and the closest
local maximum of Eq. 4.7 which can be determined numerically.
In particular, the maximum trap depth vanishes at non-zero dipole
trap power, called ‘minimal trapping power’, causing the atoms to
spill out of the trap centre. The minimal trapping power is quan-
tified experimentally in Figs. 4.1 and 4.2 by measuring the con-
densate atom number for various dipole powers. Whereas for low
dipole trap powers the vertical trap depth is limiting, the high-
power maximum trap depth is limited in the horizontal direction
(Vhorz, dashed lines in Fig. 4.3A). Therefore the functional form of
Vmax (Fig. 4.3A) exhibits a kink for each atomic species, indicat-
ing the cross-over between the vertically and horizontally limited
regimes. This cross-over occurs at roughly 0.6W for 39K and 1.2W
for 87Rb. In our beam configuration the maximum trap depth for
Table 4.2: Measured dipole trap
frequencies for 87Rb. The frequencies
were experimentally determined by
slowly lifting a 87Rb-condensate by
means of increasing the dipole power
and then suddenly decreasing the
power to the precise values listed in
the table. This jerk in the position of
the dipole trap leads to oscillations
(along all axes, though primarily along
z) which are fitted to a cosine in order
to extract the frequency.
power 0.38W 0.40W 0.45W 0.50W 0.55W 0.70W 1.00W
ωr/(2pi)(x) 14.2(2) 15.9(2) 17.6(6) 19.2(5) 23.8(5) 28.4(5)
ωr/(2pi)(y) 12.2(9) 13.7(9) 14.7(5) 16.6(9) 20.6(6) 23.9(9)
ωz/(2pi) 90.8(3) 108.6(5) 120.9(8) 150.7(8) 183.1(7)
Table 4.3: Measured dipole trap
frequencies for 39K. The measurement
protocol is identical to table 4.2 above.
power 0.20W 0.25W 0.30W 0.50W 1.00W
ωr/(2pi)(x) 16.5(5) 19.1(5) 21.9(5) 29.06(2) 41.1(5)
ωr/(2pi)(y) 14.9(5) 17.5(5) 19.4(5) 25.49(2) 35.7(5)
ωz/(2pi) 74(1) 115.3(5) 135.9(5) 190(1) 280(5)
quantum gases 65
0.0 0.4 0.8 1.2
Dipole trap power/W
0
5
10
Tr
ap
d
ep
th
/
µ
K
Vmax (Rb)
Vhorz
Vmax (39K)
Vhorz
0.0 0.4 0.8 1.2
Dipole trap power/W
-20
-10
0
G
ra
vi
ta
ti
on
al
sa
g/
µ
m
Rb
39K
diff
A B Figure 4.3: (A) Maximum trap depth
(Vmax/kB) versus dipole trap power (in
each beam). For low powers the ver-
tical trap depth is limiting (the atoms
leave the trap from the centre along
gravity) whereas for higher powers
(above 1.2W) the horizontal depth is
limiting (the atoms leave the trap to
the side). (B) Calculated gravitational
sag for 87Rb (red), 39K (blue), and the
difference between the two (‘differ-
ential sag’, black dashed). The pale
blue (red) vertical bars are the mea-
sured minimal trapping powers from
Fig. 4.1 (4.2) which agree well with the
numerical results.
87Rb drops below that of 39K at about 1.0W. Fig. 4.3A also shows
the experimental values of minimal trapping power (vertical bars
of the same colour as in Figs. 4.1 and 4.2) which agree well with
the theoretical prediction. Sympathetic cooling works best if
the coolant (87Rb) is predominantly
evaporated, such that most of the
potassium atoms remain in the trap.
It has been argued by Campbell et
al. [131] that sympathetic cooling
in deep dipole traps does not work
well. However, we find decent cooling
results with this method despite
the limiting 39K trap depth. This
could be explained by a different
evaporation ramp, higher available
87Rb atom numers, or by the different
trap geometry
Gravity does not only affect the trap depth but also the vertical
position of the trap centre. This effect, also known as gravitational
sag,3 is summarised in Fig. 4.3B.
3 Efficient sympathetic cooling relies on
a good spatial overlap between the two
species.
4.2 Trap frequencies
The extent of a cloud of ultracold atoms in a dipole trap is typi-
cally much smaller than the beam waists and the expansion of the
potential around the trap centre can be restricted to include terms
up to quadratic order. For all practical purposes the dipole poten-
tial is therefore fully characterised by its vertical and horizontal
trap frequencies ωz and ωr = ωx = ωy, respectively. These can be
obtained numerically from fitting a parabola to the local minimum
of Eq. 4.7 (at the vertical position z given by the gravitational sag,
Fig. 4.3B). We find excellent agreement between our measurements
(values in tables 4.2 and 4.3) and the numerical predictions, shown
in Fig. 4.4. Note that the horizontal trap frequencies are measured
on the x/y-axes of the z-axis imaging camera and do not neces-
sarily correspond to ωx and ωy (the principal axes of the dipole
trap). The mismatch between the two measured horizontal fre-
quencies could be either explained by the not-perfectly-90◦ angle
between the two dipole axes4 or by slightly different horizontal 4 which should show up as beating in
the oscillation signal between the x-
and y-axes but was not resolved in our
experiments
beam waists (the latter seems more likely).
0.0 0.2 0.4 0.6 0.8 1.0
Dipole trap power/W
0
10
20
30
40
H
or
izo
nt
al
tra
p
fre
qs
./
H
z y (Rb)x
y (39K)
x
0.0 0.2 0.4 0.6 0.8 1.0
Dipole trap power/W
0
100
200
300
Ve
rt
ic
al
tr
ap
fr
eq
./
H
z
Rb
39K
A B Figure 4.4: Vertical (axial, A) and
horizontal (radial, B) trap frequencies
ω/(2pi) versus dipole trap power (per
beam). The solid lines and shaded
regions show the numerical prediction
with a 2.5 µm uncertainty of the
vertical beam waist. The dashed lines
represent Eqs. 4.5 and 4.6 (neglecting
gravity).
66 quasicrystalline optical lattices for ultracold atoms
4.3 Evaporation and sympathetic coolingTable 4.4: Evaporation parameters
used to reach quantum degeneracy
parameter 87Rb 39K
initial power 7W
initial V0/kB 43 µK 38 µK
final power/mW ∼400 ∼200
final V0/kB 3 µK 1 µK
exp. ramp dur. 3 s
exp. time const. 0.5 s
guide field 2.8(1) G
1st Fesh. (1.7 s)
0G 315.49(2) G
100 a0 141(1) a0 (inter)
2nd Fesh. (1.3 s)
0G 397.8(1) G
100 a0 280 a0 (intra)
Quantum degeneracy of 87Rb or 39K is achieved by forced evapo-
ration [177] in the dipole trap. In the case of 39K we start evapo-
ration at similar temperatures but with far fewer atoms compared
to 87Rb, necessitating a more efficient cooling method. We employ
the widely used sympathetic cooling technique [131, 178, 179]
which utilises the abundant 87Rb atoms as coolant for 39K. The
experimental parameters used for evaporation are summarised
in table 4.4. They are optimised to maximise condensate atom
number, given a fixed ramp duration of 3 s.
4.4 Dipole trap laser setup
This section is an interlude about the setup and characterisation of
the dipole trap.
All dipole beams are derived from a single-frequency solid state
laser5 at 1064nm. The two (x and y) dipole trapping beams are5 Mephisto MOPA 55W, Coherent Inc.
prepared on a separate laser table6 from which the laser power
6 ‘high-power’ or HP table, which also
hosts the lattice laser
is distributed via optical fibres to the experiment. The z-dipole
axis has been prepared but not implemented yet; it is designed to
provide additional radial confinement for the lattice potential once
the atoms habe been loaded into the quasicyrstalline potential.
Each beam passes through an AOM in order to shift its fre-
quency (with respect to the other beams, avoiding cross-interferences)
and to stabilise its intensity via PID feedback. After the initial
split, the x and y beams both carry 17W of power from which
max. 8W is measured after passing through the science cell. The
largest losses are caused by the AOM (15%), the DC fibre polar-
isation maintaining7 (∼25%), and the glass-cell/air interfaces7 The ‘cold’ fibre coupling efficiency
(AOM on pulsed mode with 1% duty
cycle) is about 95%. But in DC mode
the coupling efficiency is reduced to
∼ 75%, probably due to polarisation
drifts which manifest themselves
as power drifts after the Glan-Laser
cleaning prism.
(measured ∼6% each, 45◦ incidence). More information on the
preparatory stages of the dipole trap setup can be found in Max
Melchner’s Master’s thesis [180].
Table 4.5: Horizontal (w0h) and vertical
(w0v) dipole trap beam waists in µm.
‘Design’ refers to the values that were
aimed at, ‘theory’ are the expected
values using ABCD matrices, given the
approximate distances on the bread-
board and the measured divergence
from the fibre coupler, ‘exp(erimental)’
are the measured values using a cam-
era that is focussed to the position of
the atoms, and ‘freq(uency)’ refers
to inferred values by measuring trap
frequencies (table 4.1).
Dipole x Dipole y
waist w0h w0v w0h w0v
design 280 50 280 50
theory 284 53 279 52
exp 284 64 280 47
freq 285(5) 55(2) 285(5) 55(2)
After exiting the optical fibre on the experiment laser table each
beam passes through a Glan-Laser prim in order to clean the po-
larisation. The transmission through two subsequent bending mir-
rors is used as PID input signal (45◦ bend) and monitoring signal
(22.5◦ bend). At 1064nm, Silicon-based photodiodes are not suit-
able for high-bandwidth applications, such as the PID loop, due to
the slow diffusion time constant t2 in this wavelength range [181].
Therefore we use high-bandwidth amplified InGaAs photodiodes
for PID feedback. The feedback loop is otherwise identical to the
lattice intensity stabilisation 5.4. The optical power of each dipole
beam (for example, on the x-axis of Fig. 4.1) has been calibrated
using a power meter after the beam passed through the science
cell. Therefore the true optical power at the position of the atoms
is approximately 10% higher than the specified value. Although,
in principle, powers up to 8W have been measured after the sci-
ence cell, the maximum values used in the sequence are 7W in
order to leave some room for long-term drifts.
quantum gases 67
After the Glan-Laser prism, the beam first passes through a
cylindrical telescope and then a spherical telescope before it is
focussed to the atomic cloud via an achromatic lens (125mm fo-
cal length), as shown schematically in Fig. 5.17 in the following
Chapter. The gaussian beam propagation has been modelled8 for 8 using ABCD matrices
the actual physical setup on the experiment table, taking into ac-
count the measured beam divergence from the fibre collimation.
For elliptical beams, the horizontal and vertical waists propagate
independently from each other, making it (in general) impossi-
ble to focus both the vertical and the horizontal waist to the same
point. The horizontal waist of the dipole beam at the position of
the atoms is relatively large, corresponding to a small divergence,
whereas the vertical waist is strongly divergent. Therefore, all
remaining degrees of freedom9 were optimised to create an es- 9 distances between telescopes and
lenses (given the physical constraints
on the breadboard) and the available
focal lengths of lenses
sentially focussed beam at the atoms with specified design-waists.
The experimentally measured waists agree reasonably well with
the expected values from gaussian propagation (see table 4.5).
Note that the theoretical values resulting from the trap frequency
measurement assume a radially symmetric dipole trap. The ex-
perimental agreement could be improved by modelling the trap
frequencies in x and in y separately, accounting for the measured
mismatch between the two (Fig. 4.4B).
The final achromatic lens to focus each dipole beam to the
atoms is shared with the lattice setup by employing the grazing-
angle technique. The dipole beam is reflected off a D-shaped
mirror and for a short distance (5 cm) it propagates parallel to
the corresponding lattice beam. The lattice beam is incident on
the final achromat on its optical axis; the dipole beam is incident
roughly 5mm away from the optical axis. Since both beams im-
pinge on the lens as parallel rays they are focussed to the same
spot. Consequently, the dipole beams are overlapped at the po-
sition of the atoms under different angles than the lattice, in our
case just under 90◦. Due to the relatively large horizontal waists,
the trap geometry is insensitive to the exact angle under which
the dipole beams are overlapped. This technique avoids the use of
expensive dichroic mirrors while maintaining the necessary flexi-
bility in the alignment of both lattice and dipole trap. A schematic
of the optical setup around the science cell can be found in sec-
tion 5.4 below.
Alignment
All degrees of freedom of alignment, except the relative horizontal
angle between the dipole beams (as mentioned above), are impor-
tant for the resulting dipole trapping performance.
Firstly, the focus of the (vertical) dipole waists should coincide
with the atomic cloud, in order to ensure maximum trap depth.
This is achieved by first focussing a camera on each dipole axis
to the position of the atoms (using in-situ absorption imaging).
Subsequently, the last achromat is moved longitudinally (on a
68 quasicrystalline optical lattices for ultracold atoms
z-translation stage with µm screws) to focus the dipole beam.
Secondly, each dipole beam should propagate perpendicular to
gravity, thereby minimising gravitational sag. We achieve this by
beam walking using pinholes centred on the achromatic lenses,
thereby trusting that our optical table is also aligned perpendicu-
lar to gravity. Using the time-of-flight technique on the D-axis1010 one of the lattice axes which does
not have a dipole beam we measured the angle of the dipole trap beams with gravity to be
90.0(5)◦.
Thirdly and most importantly, the dipole beams have to inter-
sect at the same point in space (‘pointing’). This is necessary in
order to benefit from the crossed-beam configuration in which
the vertical trap depth of both beams add up. If the beams do
not intersect vertically, the trap depth is greatly reduced (in the
worst case by a factor of two). In addition, the vertical intersection
ensures the approximate radial symmetry of the trap (up to the
discrepancies in beam waist).
To this end, the last kinematic mirror before the final achromat
is piezo-driven. Rough pre-alignment is achieved by loading cold
atoms into a single-beam dipole trap and maximising the in-situ
optical density.11 Fine alignment is achieved by producing a con-11 An optical density of 1.8-2.0 for 87Rb
is a good benchmark here. densate and lowering the final evaporation depth until almost
no atoms are held against gravity. Then the vertical alignment
can be changed in small steps by the piezo-driven mirror to max-
imise atom number. This alignment technique can discriminate
the optimum position of the mirror to within 10 piezo steps; this
corresponds to a mirror-displacement of roughly 170nm and a
vertical displacement (at the atoms) of 1.6 µm.
The centre of the dipole trap is aligned 61 µm below centre of
the magnetic trap for optimum loading efficiency.
4.5 Dipole loading, guide field, and RF state transfer
Table 4.6: Magnetic field calibration of
the guide field. For each current value
we take an RF/MW-resonance scan
using a Landau-Zener sweep of 20ms
duration. In the case of the RF transfer
the transition |F = 1, mF = −1〉 →
|F = 1, mF = 0, 1〉 in the ground
level of 87Rb is used, in the case
of MW we use |F = 2, mF = 2〉 →
|F = 1, mF = 1〉. These frequencies are
then converted to magnetic field values
using the magnetic field hamiltonian
(section 3.2).
Current/A RF/MW resonance B-field/G
6.0 3.042(8) MHz 4.35(1)
6.0 6843.78(2) MHz 4.33(1)
4.5 1.95(5) MHz 2.8(1)
Before the magnetic trap is ramped down and during loading of
the dipole trap we apply a homogeneous magnetic field (‘guide
field’) using offset coils in the z-direction (Fig. 4.5). This serves
three purposes: Firstly, it ensures a definite quantisation axis after
the magnetic trap has been switched off and before the (strong)
Feshbach field comes on (in the case of 39K).12 Secondly, it com-
12 Though even without a quantisation
axis during the entire dipole evapo-
ration ramp we achieve condensation
with 87Rb.
pensates for long-term drifts in the position of the magnetic trap
(section 3.10), ensuring good spatial overlap between the magnetic
trap and the dipole trap. Our offset coils shift the vertical trap po-
sition a little more than 25 µm/A when applied to a high-gradient
magnetic trap of 300G/cm (strong axis). Thirdly, we perform a
radio-frequency (RF) state transfer, using the guide field to lift
the degeneracy of the internal states of 87Rb and 39K. The choice
of internal states during evaporative and sympathetic cooling is
of crucial importance to the success of these methods [182]. In-
elastic collisions between atoms in different internal states can
lead to heating and losses. Consequently it is desirable to trans-
fer the atoms to their respective absolute ground state, thereby
quantum gases 69
precluding inelastic processes. In our case this is achieved simul-
taneously for 87Rb and 39K using a Landau-Zener transition from
|F = 1, mF = −1〉 to |F = 1, mF = 1〉. The RF transfer is accom-
plished by a 20ms-long linear frequency sweep from 1.90MHz to
2.00MHz at a homogeneous magnetic field of 2.8(1) G.
-0.2 0.0 0.2 0.4
Time/s
0
3
6
Di
po
le
X
ph
ot
od
io
de
/V
dipole trap
exp. coil
guide field
RF transfer
-0.4
0.0
0.4
0.8
Cu
rre
nt
cla
m
p/
V
(1
00
A/
V)
Figure 4.5: Dipole trap loading pro-
cedure (for 39K). Before the magnetic
quadrupole field (dark blue) is ramped
down we apply a guide field (light
blue), as discussed in the text. The
polarity of the upper experiment coil
(dark blue) is then changed to provide
the inter-species Feshbach field during
evaporation. In case of 87Rb the dipole
loading procedure is identical except
that there is no Feshbach field.
Figure 4.5 shows a scope trace of our dipole loading procedure,
including a photodiode signal of the dipole trap, current clamp
measurements of the experiment coil and the offset-z coil, and
the timing of the RF transfer. The calibration of our guide field is
summarised in table 4.6.
RF antenna
The RF antenna that is used for the state transfer operates in the
near-field regime (λRF = 1.5 m) and consequently the radiation
consists mostly of an oscillating B-field. The antenna geometry
and its number of windings is therefore largely insignificant (see
Fig. 4.6). In addition, the antenna can be approximated by an
isotropic emitter, making the coupling efficiency of the radiation
independent of the precise alignment (angle) of the antenna with
respect to the atoms, as long as it is placed as close as possible (see
Fig. 4.6).
1 2 3 4
Distance from glass cell/cm
0
1
2
3
4
5
87
R
b
at
om
nu
m
be
r/
10
8 3
5
7
Figure 4.6: RF antenna coupling effi-
ciency. The data represents the surviv-
ing 87Rb-atoms after a 1s ramp from
30MHz to 5MHz in a steep-gradient
(300G/cm) magnetic trap, i.e. the
fewer atoms the larger the coupling
efficiency (or Rabi-frequency).
Our antenna has circular geometry, consisting of four windings
(radius 15mm) and it is wound on a 3D-printed plastic mount
that also houses the Stern-Gerlach coil, about 1 cm above the glass
cell. The antenna is not impedance-matched to any particular
frequency because it is designed to operate in a large range of
frequencies (1 - 400MHz).13
13 It was also designed to perform RF
evaporation on the mF-states in 87Rb.
4.6 Feshbach fields
Feshbach resonances [93, 183] are key to both the production of
ultracold atomic gases and the application thereof. The physi-
cal origin of this phenomenon lies in the quantum-mechanical
scattering process of two particles at low energies [184] which
can be characterised by a single parameter called the scattering
length.14 This parameter, usually quoted in units of the Bohr- 14 At low energies, s-wave scattering
dominates due to the centrifugal
barrier [183] which surpresses higher-
order scattering. The dominance of s-
wave scattering defines the ‘ultracold’
regime.
radius a0 = 5.29× 10−11 m, is a measure for the phase shift be-
tween the incoming and outgoing wave function of a scattered
particle. Its value depends on the respective energy landscape
of the inter-atomic molecular potential which, for a given atomic
species, is hard to predict. This combined quantum state of the
two scattering participants (in the limit of large separation) is
called a scattering channel. The energy of a scattering channel de-
pends on the composition of internal states and it can be shifted
according to the Zeeman effect by applying a magnetic field. If the
constituents of two scattering channels have a different composi-
tion of internal states then their respective molecular potentials
can be tuned relative to each other. In particular, the energy of
two free particles (an open scattering channel) can be brought
70 quasicrystalline optical lattices for ultracold atoms
into resonance with a bound state of a different channel (a closed
channel), leading to a maximum scattering phase shift of pi and
a diverging scattering length. This effect is called Feshbach reso-
nance.
310 314 318 320
B-field/G
0
1
2
39
K
co
nd
en
sa
te
nu
m
./
10
5 data
a(B)
-10
0
0
10
0
20
0
Sc
at
te
ri
ng
le
ng
th
/
a 0
Figure 4.7: Experimental optimisation
of the inter-species Feshbach field
for 39K - 87Rb. The optimum field
value (315.49G) produces the largest
39K-condensate. The data points are
averages over three identical realisa-
tions and the error bars are smaller
than the markers. The magnetic field
(x-axis) is calibrated using an RF trans-
fer in rubidium (table 3.2). The second
y-axis shows the scattering length a
(red line).
The scattering length (a) close to a Feshbach resonance is para
metrised by the following function of the magnetic field
a(B) = abg
(
1− ∆B
B− Bres
)
(4.8)
where the relevant values for abg, Bres, and ∆B are listed in ta-
ble 4.7.
When producing condensates of 39K we make use of both Fes-
hbach resonances during evaporation, firstly to sympathetically
cool 39K with 87Rb and secondly to reach quantum degeneracy
with 39K. The field is switched from inter-species to the intra-
species value at the point when all 87Rb has fallen out of the trap
due to gravity, as detailed in table 4.4. These parameters are op-
timised to produce the largest condensates, as shown in Figs. 4.7
and 4.8.
380 400 420
B-field/G
0
1
2
3
39
K
co
nd
en
sa
te
nu
m
./
10
5 data
a(B)
-20
0
0
20
0
40
0
Sc
at
te
ri
ng
le
ng
th
/
a 0
Figure 4.8: Experimental optimisation
of the intra-species Feshbach field for
39K (optimum value 397.8G) using
the 39K-condensate number as figure
of merit. The second y-axis shows the
scattering length a (red line).
4.7 Quantum degeneracy and Bose-Einstein condensation
The ultracold temperature regime is defined by the prevalence
of s-wave scattering for bosons [183] or p-wave scattering for
fermions, which are two-body collisional properties of the relevant
atomic species. Such a system may still be thermodynamically
described by classical Maxwell-Boltzmann statistics (although the
scattering has to be already described quantum-mechanically15).
15 The bosonic or fermionic particle
symmetry influences the collective
properties of the gas. This can, for
instance, show up in the dynamics
of the system, such as an anisotropic
expansion of a thermal Bose gas
after sudden release from the dipole
trap [185].
However, if the cloud is cooled down further, the density cru-
cially affects the thermodynamics of the system. The phase-space
density
n
λ3dB
where λdB =
h√
2pimkBT
(4.9)
replaces temperature as the relevant thermodynamic parameter
because it relates the average inter-particle distance n−1/3 to the
thermal de Broglie wavelength λdB. When the phase-space density
approaches unity [186] the classical Maxwell-Boltzmann descrip-
tion fails and must be replaced by the relevant quantum statistical
expression. In this regime the symmetry properties of the individ-
ual quantum particles dominate all thermodynamic quantities. In
particular, fermionic symmetry – the full many-body wavefunc-
tion is anti-symmetric under particle exchange – leads to Pauli
exclusion, i.e. the fermionic occupation of a single quantum state
Table 4.7: Feshbach resonance param-
eters. The values are taken from [93]
except the resonance position for
39K which has recently been measured
to a much greater precision [130].
species scattering channel Bres/G ∆B/G abg/a0
87Rb/39K 1×87Rb & 1×39K in |F = 1, mF = 1〉 317.9 7.6 34
39K 2×39K in |F = 1, mF = 1〉 402.70(3) -52 -29
quantum gases 71
kx
k y
T > Tc T << TcT < Tc
Figure 4.9: Bose-Einstein condensation
in a gas of 39K-atoms. The vertical
coordinate in the surface plot cor-
responds to the optical density. A
thermal gas (T > Tc) is well-described
by the Maxwell-Boltzmann distri-
bution, corresponding to a gaussian
profile in time-of-flight. Below the crit-
ical point, a sharp feature at k = (0, 0)
starts to appear which grows as the
gas is cooled down further. For tem-
peratures much lower than Tc, there is
no discernible thermal fraction.
is bounded from above by 1. Conversely, bosonic symmetry – the
many-body wavefunction is symmetric under particle exchange –
leads to particle-bunching which allows bosonic systems to lower
their energy by occupying a single quantum state. Statistically,
these effects are captured by the average occupation
〈ni〉 = 1e(ei−µ)/(kBT) ∓ 1 (4.10)
of a given state |i〉 with energy ei. The minus-sign corresponds to
bosonic, the plus sign to fermionic quantum statistics. The chem-
ical potential µ has been introduced to ensure particle number
conservation via
N =∑
i
〈ni〉 (4.11)
in the grand-canonical ensemble. In this work all lattice exper-
iments were carried out with bosons and the remainder of this
section will cover the relevant properties of ultracold bosons in a
harmonic trap.16 16 However, all necessary hardware has
been set up to produce a Fermi-gas of
40K.
Bose-Einstein condensation
As the temperature of the gas is reduced at constant density, the
chemical potential approaches the value of the lowest energy level
e0 =
0 in a uniform system3
2 h¯ω in an isotropic 3D harmonic trap
(4.12)
from below. The occupation of the excited states then tends to a
finite value, given by the number equation (Eq. 4.11).17 Therefore, 17 For all excited states (ei > e0)
the number equation can be solved
analytically by replacing the sum by an
integral.
all remaining particles (in the thermodynamic limit N → ∞) must
occupy the ground state e0. This effect is known as Bose-Einstein
condensation.
Alternatively, condensation can be achieved by increasing the
particle number N at fixed temperature. For an ideal Bose gas in a
harmonic trap the critical particle number Nc is given by [186]
Nc = 1.202
(
kBT
h¯ω¯
)3
. (4.13)
72 quasicrystalline optical lattices for ultracold atoms
Here ω¯ denotes the geometric mean of the trapping frequencies.
For N ≥ Nc (at fixed T) the excited motional states are ‘satu-
rated’ such that adding particles to the system solely increases the
ground state occupation.1818 In a realistic scenario, however, at
most half the added particles end up
in the condensate and the other half is
thermal [187].
Experimentally, the condensate can be detected in time-of-
flight (see sections 3.12 and 6.2). It appears as a sharply peaked
feature above the thermal background [97], as shown in Fig. 4.9.
The profile of the condensate peak, a paraboloid, is a consequence
of interatomic interactions and trapping geometry. Theoretically,
these can be taken into account by the Gross-Pitaevskii equation in
the Thomas-Fermi approximation [186].19 The resulting real-space19 by neglecting the kinetic energy term
− h¯22m∇2 profile in an anisotropic harmonic trap is then given by
n(r) = n0
(
1− x
2
R2x
− y
2
R2y
− z
2
R2z
)
, (4.14)
where Ri are the Thomas-Fermi radii in the three spatial directions
and n0 is the central density. Generally, the expansion of a trapped
condensate in time-of-flight is anisotropic [97, 186] in contrast to
the thermal gas which expands isotropically. Equation 4.14 gives
rise to a parabolic density profile of the atomic cloud in time-of-
flight which is used extensively in this work as a fit function to
measure atom numbers. Thereby the equations quoted in ref. [160]
are used.
Atom number calibration
The observation of a phase transition from a thermal to a Bose-
condensed state provides a measurement of atom number which
is independent of the specifics of the light-atom interactions of
the atomic species in question. For 39K in particular, a faithful
atom number measurement via summing the optical density (sec-
tion 3.12) is impossible due to the non-existence of a closed imag-
ing transition, even at high magnetic fields where no repumping
light is required (section 3.12).
5
Eightfold optical lattice
This chapter provides the theoretical and experimental ground-
work for the experiments carried out in this thesis (Chapter 6).
First, the basic equations for non-interacting particles in a 1D op-
tical lattice are derived (section 5.1). Second, the single-particle
dynamics following a sudden lattice quench is reviewed with a
focus on using this method for lattice depth calibration. Third and
last, the laser setup for the eightfold rotationally symmetric optical
lattice is described (section 5.4).
5.1 1D optical lattice
The matter-wave experiments described in Chapter 6 can be
treated in the usual Bloch-wave formalism. In this section the
matrix hamiltonian for a single atom in a one-dimensional optical
lattice is derived which forms the basis for all theoretical predic-
tions in Chapter 6.
Table 5.1: Relevant recoil energies
for 87Rb and 39K using the optical
wavelength λlat = 726 nm.
species Erec/h
87Rb 4.35kHz
39K 9.71kHz
The dipole potential (Eq. 4.2) of a retro-reflected laser beam can
be written as
V(x) = V0 cos2(klatx) =
V0
2
(
1+
ei2klatx + e−i2klatx
2
)
, (5.1)
where any inhomogeneities and (anti-)confining potentials have
been neglected.1 Further, it is assumed that the incoming and 1 This assumption is valid for our
experimental timescales since the
longest lattice pulses are t = 50 µs 
1
ωtrap
.
the retro-reflected beams are perfectly counter-propagating. The
lattice wavelength λlat = 2pi/klat = 726 nm is far-blue-detuned
with respect to the D-lines in 39K, ensuring that single-photon
processes are completely suppressed. The lattice depth2 (V0) can
2 In principle, the value of V0 could
be calculated directly from Eq. 4.2. In
practice, however, these estimations
suffer from systematic uncertainties
in the laser powers and beam waists.
Therefore the lattice dynamics that are
derived in the following will be used
to calibrated V0 since they provide a
much more precise measure of lattice
depth.
be expressed conveniently in units of photon recoil energy, i.e.
Erec =
h2
2mλ2lat
. (5.2)
Here m denotes the atomic mass and h Planck’s constant; the
relevant recoil energies for 87Rb and 39K are listed in table 5.1.
Due to the periodicity of the potential3 (Eq. 5.1) we can invoke
3 the lattice spacing a is given by
a =
λlat
2
=
2pi
2klat
(5.3)
Bloch’s theorem [188, 189] and introduce the quasimomentum q.
The quantity
q mod 2klat (5.4)
74 quasicrystalline optical lattices for ultracold atoms
-1 0 1
0
5
10
15
-1 0 1 -1 0 1 -1 0 1
Quasimomentum q/klat
En
er
gy
/E
re
c
V0 = 0Erec 2 Erec 20 Erec6 Erec
Figure 5.1: Band structure for non-
interacting particles in a homogeneous,
one-dimensional optical lattice. The
solutions to Eq. 5.9 for n ∈ {0, 1, 2, 3}
(deep blue to light blue) and V0/Erec ∈
{0, 2, 6, 20} (left to right) are shown.
For small values of V0 the optical
lattice causes only small perturbations
on top of the free-particle dispersion
p2/(2m). Conversely, in the limit of
deep lattices (V0 & 5Erec) the n = 0
Bloch waves become more peaked
around each lattice site, the bandwith
becomes small, and the dispersion
relation can be approximated by a
cosine (tight-binding limit).
is conserved in a periodic potential. In the Bloch formalism the
wave function takes the form
ψnq (x)︸ ︷︷ ︸
Bloch wave
= unq (x)︸ ︷︷ ︸
Bloch function
× eiqx︸︷︷︸
plane wave
, (5.5)
where the index n has been introduced to account for all possible
solutions (bands) of the energy eigenvalue equation Hψ = Eψ
(time-independent Schrödinger equation). The ansatz Eq. 5.5 can
be inserted to the eigenvalue equation to yield[
− h¯
2
2m
(
∂2
∂x2
+ q2
)
+
V0
2
(
1+
ei2klatx + e−i2klatx
2
)]
unq (x) = E
n
q u
n
q (x) .
(5.6)
Bloch’s theorem states that the functions unq (x) exhibit the same
periodicity as V(x), i.e. unq (x) = unq (x + a), and consequently they
can be decomposed in a discrete Fourier series:
unq (x) =
+∞
∑
l=−∞
cnl (q) e
i2lklatx . (5.7)
Inserting this expression into Eq. 5.6 results in
+∞
∑
l=−∞
[(
− h¯
2
2m
(
(2lklat)2 + q2
)
+
V0
2
)
ei2lklatx +
V0
4
(
ei2(l+1)klatx + ei2(l−1)klatx
)]
cnl (q) = E
n(q)
+∞
∑
l′=−∞
ei2l
′klatx cnl′(q) ,
(5.8)
which can be easily converted to the following matrix equation for
H
+lmax
∑
l=−lmax
Hl,l′cnl (q) = En(q)cnl′(q) . (5.9)
The simplicity of the potential V(x), which contains only two non-
trivial Fourier components, namely ±2klat, ensures the straightfor-
ward derivation of Eq. 5.8 (no double summation on the left-hand
eightfold optical lattice 75
side). Consequently, the matrix Hl,m is sparse, the only non-zero
elements being located on the diagonal and the first off-diagonal,
i.e.
Hl,l′ =

h¯2
2m (q + 2lklat)
2 + V02 for l = l
′
V0/4 for |l − l′| = 1
0 otherwise.
(5.10)
The truncation of the infinite basis of the coefficients cnl (q) at cer-
tain values ±lmax can be justified a posteriori since only a fraction
of bands get populated.4. 4 This is not the case for the eightfold
optical lattice and consequently the
calculation of a ground-state energy in
this formulation is usually impossible
in that case (section 6.1).
The Bloch waves are delocalised over the whole lattice. How-
ever, it is possible to recover localised wave functions for each
lattice site. These localised functions are called Wannier functions
wn(x− xi) = 1√
M
∑
q
e−iqxiψnq (x) (5.11)
for each lattice site i (M is some normalisation). Bloch waves
and Wannier functions are related via a discrete Fourier trans-
form. Therefore, a localised particle (i.e. unity occupation of a
single Wannier function) corresponds to a flat distribution of
Bloch waves (each Bloch wave is equally populated). The Wan-
nier functions form an alternative orthonormal basis to solve the
Schrödinger equation.
The derivation above can be generalised to more sophisticated
lattice geometries, as shown in section 6.1.
Band structure, tunnelling, and group velocity
In the case of periodic lattice geometries, for instance 1D or 2D
simple-cubic, or 2D honeycomb geometry, the hamiltonian matrix
in Eq. 5.10 can be diagonalised and the corresponding eigenvalues
and eigenstates are exact. The eigenenergies are the well-known
Bloch bands, depicted in Fig. 5.1 for different values of V0/Erec ∈
{0, 2, 6, 20}. Two regimes emerge here: On the one hand, shallow
lattices can be treated as a small perturbation on free-particle
behaviour. On the other hand, deep lattices (V0 & 5Erec) lead to
peaked Bloch functions around the lattice sites, corresponding to
the tight-binding limit.
-1 0 1
Quasimomentum q/pia
-2
0
2
D
is
pe
rs
io
n
E
(q
)/
J
E(q)
vgr
-2
0
2
G
ro
up
ve
lo
ci
ty
v g
r/
(
Ja h¯
)
Figure 5.2: Tight-binding dispersion
relation and its derivative, the group
velocity.
In the tight-binding limit the lowest band of the optical lattice
corresponds to a discrete lattice with a fixed tunnelling energy
between the sites. A measure for the tunnelling matrix element J
between neighbouring lattice sites is the bandwidth of the lowest
band, i.e.
J =
max(E0q)−min(E0q)
4
. (5.12)
The dispersion relation in this case (deep blue line in rightmost
plot in Fig. 5.1) is the characteristic cosine of the tight-binding
model [188]:
E(q) = −2J cos(qa) , (5.13)
76 quasicrystalline optical lattices for ultracold atoms
where a is the lattice spacing (Eq. 5.3). This dispersion relation is
plotted in Fig. 5.2 together with the corresponding group velocity
vgr =
1
h¯
∂E(q)
∂q
=
2Ja
h¯
sin(qa) . (5.14)
Higher dimensions and quantum walks
In the case of D-dimensional separable lattices Eq. 5.13 generalises
to
E(q) = −2J
D
∑
i=1
cos(aqi) , (5.15)
where q is now a D-component vector. Correspondingly, the indi-
vidual components of the group velocity are given by
vi(q) =
1
h¯
∂E(q)
∂qi
=
2Ja
h¯
sin(aqi) . (5.16)
For a given Wannier state that is initially localised to one lattice
site all Bloch waves are equally populated, leading to an average
root-mean-square group velocity
√
v2 =
2Ja
h¯
√√√√ D∑
i=1
sin2(aqi) =
2Ja
h¯
√
D
2
. (5.17)
The
√
D scaling will be recovered exactly in the experiment
(Chapter 6) for D ∈ {1, 2, 3, 4}.
5.2 Time-evolution and lattice depth calibration
Let us now consider the time-evolution of an optical lattice system
according to the time-dependent Schrödinger equation
ih¯
dψ
dt
= Hψ . (5.18)
As a first example the ballistic expansion of a particle initially
localised to one single lattice site will be shown.
Figure 5.3: (A) Quantum walk on a
1D homogeneous tight-binding lattice.
Initially the particle is localised to
one lattice site (deep blue, top). With
time, it spreads ballistically in both
directions, giving rise to a ‘light-cone’
with a constant group velocity given
by Eq. 5.17 for D = 1. Times are
given in units of tunnelling times
h¯/J. (B) Classical random walk
(schematic). Particles initially localised
to one lattice site are allowed to hop
to neighbouring sites with a certain
probability. Contrary to the quantum
case, the expansion is diffusive, i.e.
gaussian.
0 25 50
Lattice site
Pr
ob
ab
ili
ty
(a
.u
.)
0 25 50
Lattice site
A B
t = 0
t = 1
t = 2
t = 3
t = 4
t = 5
t = 6
eightfold optical lattice 77
Homogeneous tight-binding lattice
In the limit of deep lattices (V0/Erec  1) it is convenient to work
in the Wannier-basis and the system describes only the direct cou-
pling of nearest neighbours (homogeneous tight-binding model).
The hamiltonian reads
Hi,j =
J for |i− j′| = 10 otherwise. (5.19)
Note that we have changed basis compared to Eq. 5.10 such that
the lattice sites i denote the Wannier functions w0(x − xi). The
initial state ψ(t = 0) for time-evolution shall simply be unity
probability at one lattice site. The time-evolution ψ(t) according
to Equation 5.18 can readily be solved, for example using the
scipy.integrate package, leading to a linear expansion of the
wavepacket over the entire lattice. This light-cone-like behaviour
(depicted in Fig. 5.3A) has no classical analogon. Therefore this
effect has been coined ‘continuous quantum walk’ [22, 190, 191] in
order to differentiate it from a classical random walk (Fig. 5.3B).
Time-evolution in momentum space for square lattice pulses
Table 5.2: Scheme to group the basis
states for a 1D optical lattice.
order states
0th {|q = 0〉}
1st {|+2klat〉 , |−2klat〉}
2nd {|+4klat〉 , |−4klat〉}
nth {|+2nklat〉 , |−2nklat〉}
In our experiment we are primarily interested in the time-evolution
of a Bose-Einstein condensate initially at rest, i.e. localised in
momentum space. The time-evolution after a sudden switch-on
(quench) of the lattice potential can be calculated via integrat-
ing the time-dependent Schrödinger equation (Eq. 5.18), as be-
fore. In this case it will be beneficial to work in the original basis
{|q = 0〉 , |±2klat〉 , |±4klat〉 , . . . } and not in the Wannier basis. Al-
though the following considerations apply to reciprocal space, it is
instructive to keep the Wannier-picture at the back of one’s mind.
It provides a useful analogy in which momentum states sepa-
rated by 2klat correspond to ‘neighbouring lattice sites’, coupled
by a ‘tunnelling matrix element’ V0/4. This will become clearer in
Chapter 6.
The kinetic energy part on the diagonal of Hl,l′ (Eq. 5.10) now
crucially affects the dynamics of this system. It leads to an effec-
tive restoring force that causes the condensate to return to its orig-
inal state (|q = 0〉). In preparation for the quasicrystalline lattice
the basis shall be labelled according to the number of ±2klat steps
in momentum space from |q = 0〉, each step being one ‘diffrac-
tion order’, as shown in table 5.2. This is convenient because the
hamiltonian (Eq. 5.10) is symmetric in q ↔ −q such that all result-
ing states in the time-evolution obey the same symmetry.
In a one-dimensional periodic lattice5 the numerical solution 5 any periodic lattice, e.g. 2D and 3D
simple-cubic, or 2D triangularto Eq. 5.18 is exact, because only a limited number of orders get
populated. This result is shown in Fig. 5.4 (lines), together with
experimental data (points). The experimental points are the rela-
tive atomic population of each order of q-states.6 6 The measurement and data-analysis
procedures are explained in detail in
section 6.5 and 6.2, respectively.
Agreement between theory and experiment is very good, except
for short pulse durations for which a slight ‘lag’ of the experi-
78 quasicrystalline optical lattices for ultracold atoms
Figure 5.4: Time-evolution of a Bose-
Einstein condensate after a sudden
lattice quench (V0 = 15.7Erec) of the D
lattice axis (see below for naming con-
vention of axes). The plot shows the
time evolution grouped by diffraction
order; the 0th order corresponds to the
condensate, the 1st order to the satellite
peaks at ±2klat. For information on
image analysis the reader is referred to
section 6.5.
0 10 20 30 40 50
Lattice pulse duration/µs
0.0
0.5
1.0 0th
Po
pu
lat
io
n
1st
2nd
3rd
mental data can be observed. This is due to a finite ramping-up
time of the lattice pulse on the order of 1µs. This effect can be ac-
counted for by making the hamiltonian (Eq. 5.10) time-dependent
(see Chapter 6.5) but for now it will be neglected.
Lattice depth calibration
15.3 15.5 15.7 15.9
Lattice depth V0/Erec
0.0
0.2
0.4
R
es
id
ua
lχ
2
0th
1st
2nd
3rd
Figure 5.5: Lattice depth calibration
using time-evolution following a
sudden quench to a given lattice depth
V0. The y-axis shows the squared
and summed difference between the
experimental values (points in Fig. 5.4)
and the numerical calculation using
the depths shown on the x-axis. The
best fit in this case is obtained with a
lattice depth of V0 = 15.7Erec (lines in
Fig. 5.4).
We use the well-understood process described above to calibrate
precisely our lattice depth by comparing experiment and theory.
In fact, the lines in Fig. 5.4 were calculated using the calibration
method summarised in the following. For a given (unknown) ex-
perimental value of V0 a data trace of the type shown in Fig. 5.4
is taken by varying the duration of the (square) lattice pulse. Sub-
sequently, the exact time-evolution is calculated according to the
Schrödinger equation for a number of lattice depths and the best
fit to the experiment is found via the least-squares method (shown
in Fig. 5.5).
The lattice depths of all four lattice axes (see below) are cali-
brated in this way.
After discussing the properties and dynamics of 1D optical lat-
tice we turn to the description of the full, quasicrystalline lattice.
5.3 Eightfold optical lattice potential
The 2D quasicrystalline potential is created using a planar ar-
rangement of four mutually incoherent 1D optical lattices, each
formed by retro-reflection of a single-frequency laser beam, as
shown schematically in Fig. 5.6. The angle between two neigh-
bouring lattice axes is 45(1)◦, thereby imposing a global eightfold
eightfold optical lattice 79
Figure 5.6: Schematic of the eightfold
optical lattice. The four lattice axes
are mutually incoherent such that the
resulting potential is the sum of four
1D lattices. Each 1D lattice is formed
by retro-reflection, as shown in the
schematic. However, most of the optics
involved in the retro-reflection are not
shown. In the apparatus, a cat’s eye
configuration is used in which there
are two lenses in between the atoms
and the retro-mirror. The distance
from the atoms to the retro-mirror
should be exactly twice the sum of the
individual focal lengths of the lenses
in order to ensure approximately equal
beam waists at the position of the
atoms.
-4 -2 0 2 4
Position x/λlat
-2
0
2
Po
si
ti
on
y/
λ
la
t
0
4
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.7: Lattice potential resulting
from four non-interfering axes (blue-
detuned). The phases ϕi have been
chosen such that the symmetry point
is located at r = (0, 0) and coincides
with a potential maximum (maximally
constructive interference). There are
roughly as many maxima as there are
minima in this potential.
rotational symmetry in close analogy to the octagonal tiling (sec-
tion 1.3).
In the following section some basic properties of this eight-
fold lattice in real space are described. For the momentum-space
picture, the reader is referred to Chapter 6.
First, we consider the situation of four mutually incoherent lat-
tice beams, which also has been used for all experimental results
in Chapter 6. Afterwards, alternative schemes that make use of
inter-axis interference are considered.
-4 -2 0 2 4
Position x/λlat
0
2
4
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.8: Two cuts through the lattice
potential shown in Fig. 5.7. The curves
are color-coded by the horizontal
lines in Fig. 5.7. The distribution
of potential maxima and minima
is roughly symmetrical around the
average potential value (2 V0).
Incoherent superposition
The sum of four mutually incoherent7 lattice axes of equal depth 7 The terms ‘non-interfering’ and
‘incoherent’ only apply to the inter-
axis interference in this context. Of
course, the 1D lattices themselves are a
result of intra-axis interference.
V0 can be written as8
8 neglecting any anti-confinement
V(r) = V0
4
∑
i=1
cos2 (ki · r + ϕi) , (5.20)
80 quasicrystalline optical lattices for ultracold atoms
where the individual lattice vectors ki are given by
k1
klat
=
(
1
0
)
,
k2
klat
=
(
0
1
)
,
k3
klat
=
1√
2
(
1
1
)
,
k4
klat
=
1√
2
(
−1
1
)
(5.21)
and the individual phases ϕ1, ϕ2, ϕ3, ϕ4 are determined by the
longitudinal position of the retro-reflecting mirror.9 The potential9 The individual lattice axes are created
by retro-reflection, resulting in a
standing wave. Therefore the phase of
the incoming light is exactly cancelled.
Eq. 5.20 is shown in Fig. 5.7 for the case that all ϕi = 0. Since our
lattice is blue-detuned with respect to the D-lines in both 87Rb and
39K, an intensity maximum results in a potential maximum. In
this particular choice of phases (all ϕi = 0), the point r = (0, 0)
corresponds to the point of maximally constructive interference
(V(0, 0) = 4V0).10 In addition, the whole potential is eightfold10 The irrationality of cos 45◦ = 1/
√
2
ensures that this value of V(r) occurs
exactly once and consequently it is
the global maximum of the V(r).
However, there will always be values
of V(r) that are arbitrarily close to the
global maximum.
rotationally symmetric around this point.11 Two cuts through the
11 and only around this point
potential are shown in Fig. 5.8 which correspond to the horizontal
lines of the same colour in Fig. 5.7.
If we were to interpret each minimum of V(r) as a ‘lattice site’,
it is immediately clear from Fig. 5.8 that the vast majority of lattice
sites (or ‘wells’) is located at different energies. Moreover, the
ground level h¯ω of each well in harmonic approximation12 varies
12 given by the curvature of the well in
the x and y directions
strongly. Since deeper wells tend to have stronger curvatures
this effect leads to an even larger spread in energies. For these
reasons the potential V(r) cannot exhibit a band structure as in the
equivalent single-axis case (section 5.1). In addition, tunnelling is
inhibited by the energy offset between neighbouring sites.
Relation to octagonal tiling
It is interesting to compare the resulting potential V(r) to the oc-
tagonal tiling (Fig. 1.19). In the easiest case, we choose all phases
ϕi to be equal to pi/2 in which case the central point (0, 0) shows
maximally destructive interference (V(0, 0) = 0). This potential
bears a close resemblance to the ideal octagonal tiling (Fig. 1.19)
with a bond length of approximately 0.6λlat. Alternatively, the
actual minima of V(r) can be directly connected by lines to repro-
duce an approximate version of the octgonal tiling. This process is
shown in Fig. 5.9 for ϕi = pi/2.
The exact knowledge of all phases ϕi is unrealistic in a real ex-
perimental setting (see below). However, the octagonal tiling-effect
is relatively robust against a random choice of (constant) phases,
as shown in Fig. 5.10 in which all phases ϕi have been chosen at
random in between 0 and pi. The random choice of phases leads
to the appearance of ‘defects’ or closely-spaced ‘double wells’
which can be seen in Fig. 5.10 upon close inspection. These effects
can be interpreted as phasonic excitations, since the manner in
which they appear is identical to that of the ideal octagonal tiling
(Fig. 1.20). Despite the random choice of phases, the agreement
between the potential and the ideal octagonal tiling is very good.
Consider, for example, the length of the ‘bonds’ (black lines con-
necting the lattice points) and ‘gaps’ (the shorter diagonal of the
rhombus tile), the occurrence of which are plotted in Fig. 5.11 for
eightfold optical lattice 81
-4 -2 0 2 4
Position x/λlat
-2
0
2
Po
si
ti
on
y/
λ
la
t
0
4
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.9: Lattice potential resulting
from four non-interfering axes, over-
laid with the corresponding minima
(black points) and ‘bonds’ (lines).
Upon close inspection, the lattice
points and bonds can be seen not to
exactly represent the octagonal tiling.
The phases ϕi have been chosen such
that the symmetry point is located at
r = (0, 0) and coincides with a poten-
tial minimum (maximally destructive
interference). The bonds shown here
are not necessarily the physical bonds
(shortest distance or largest tunnelling
matrix element) between lattice sites
but they nicely illustrate the similarity
to the octagonal tiling.
the random-phases potential. The ratio between the mean values
of the two is remarkably close to the ideal value of the octagonal
tiling:
0.478(6)
0.60(2)
= 0.80(4) ' 2 sin
(pi
8
)
= 0.765 . . . . (5.22)
Sampling from a larger window than Fig. 5.10 would result in an
even better agreement.
-4 -2 0 2 4
Position x/λlat
-2
0
2
Po
si
ti
on
y/
λ
la
t
0
4
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.10: Lattice potential resulting
from four non-interfering axes. The
phases ϕi have been randomly chosen
between 0 and 2pi.
0.45 0.55 0.65
Bond length/λlat
0
100
200
300
O
cc
ur
re
nc
e
Figure 5.11: Histogram of bond lengths
(blue) and gap widths (red) for the
random-phases lattice patch shown
in Fig. 5.10. The gaps are the short
diagonals of the rhombus-like tiles.
The mean (standard deviation) of
this distribution is 0.60(2) λlat and
0.478(6) λlat, respectively.
So far we have considered static phases ϕi and the resulting
potentials V(r) whose overall structure closely resembles the
octagonal tiling in introduced in Chapter 1. Further studies are
necessary to decide how exactly the eightfold lattice potential can
be used to faithfully model the octagonal tiling. Some of these
points will be addressed in the Outlook. Experimentally, one
very important question is whether or not phase changes during
82 quasicrystalline optical lattices for ultracold atoms
the course of the sequence will result in strong heating in this
eightfold lattice, as discussed in the following.
Phase control
The phase ϕ of a 1D lattice potential that is created by retro-
reflection is effectively determined by the path length from the
atoms to the retro-reflecting mirror. If this path length varies with
time the phase ϕ changes accordingly.
Alternatively, the phase of a lattice that is created by the in-
terference of running waves can be controlled by changing the
incoming phase of the laser light, e.g. by a phase-locked-loop or
by deliberately changing the laser frequency.
Control over the phase13 of optical lattices have become a cor-13 achived either by stabilisation or
deliberate phase changes nerstone in quantum simulation experiments, for example in
splitting and combining double-well potentials [192]. However, the
application of phase control to quantum gas experiments some-
times shows unexpected subtleties. For instance, the powerful
method of engineering a variety of hamiltonians by periodically
modulating the lattice phases (Floquet engineering [29, 193]) has
often been limited to short measurement times and weakly in-
teracting gases [194] due to heating. Understanding the role of
interactions, in particular, in the heating processes of periodically
modulated systems is a field of active research [195–197]; the pre-
cise control over the bandwidths of the higher-lying Bloch bands
might be a way to mitigate heating [25].
In our experimental setting, the question of phase control is
particularly subtle. The fundamental reason for this is precisely
the aperiodicity of our lattice potential. In contrast to periodic
lattices, in which up to two relative phases can be absorbed into
the global centre-of-mass motion of the atoms, in our case there
are more than two such phases (ϕi).
The non-local effect of changing any of the phases ϕi can be
seen by equating an arbitrary-phase potential V(r) with a fixed-
phase potential that is displaced by r0 = (x0, y0), i.e.
4
∑
i=1
cos2 (ki · r + ϕi) =
4
∑
i=1
cos2 [ki · (r− r0)]
which leads to
ϕ1 = klatx0
ϕ2 = klaty0
ϕ3 = − klat√
2
(y0 + x0)
ϕ4 = − klat√
2
(y0 − x0) .
Combining these equations yields
ϕ3 = − 1√
2
(ϕ2 + ϕ1) (5.23)
ϕ4 = − 1√
2
(ϕ2 − ϕ1) , (5.24)
eightfold optical lattice 83
-4 -2 0 2 4
Position x/λlat
-2
0
2
Po
si
ti
on
y/
λ
la
t
0
16
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.12: Lattice potential resulting
from four fully-interfering axes (blue-
detuned). The phases ϕi have been
chosen such that the symmetry point is
located at r = (0, 0) and coincides with
a potential maximum (maximally con-
structive interference). Compared to
the non-interfering case, the potential
maximum is a factor of four higher in
energy.
which defines a plane in R4 on which the vector (ϕ1, ϕ2, ϕ3, ϕ4)
has to move in order to maintain a simple centre-of-mass motion
of the quasicrystalline lattice potential. Any phase changes outside
this plane will result in a physical change of the lattice potential.
In particular, changing any single phase ϕi without changing the
others accordingly, will result in a non-local potential deformation;
some sites will disappear and others will appear from nowhere, in
close analogy with phasonic excitations mentioned in Chapter 1
that are characteristic for quasicrystals. Therefore it seems likely
that the mechanical stability of the retro-reflecting beam path
will play a crucial role in the performance of the eightfold optical
lattice, particularly once atoms have been loaded into the lattice
potential. It would then be very interesting to deliberately excite
phasonic modes by slightly displacing one (or several) of the retro-
mirrors.
Before we turn to the experimental results in momentum space,
the situation of fully-interfering lattice beams will be discussed
briefly in the following.
Fully-interfering axes
-4 -2 0 2 4
Position x/λlat
0
4
16
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.13: Two cuts through the
interfering lattice potential shown
in Fig. 5.12 color-coded accordingly.
In contrast to the non-interfering
case, potential maxima are steep and
isolated.The lattice potential from four interfering lattice axes can be writ-
ten as14 14 although the potential should,
strictly speaking, include eight in-
dividual phases, four incoming and
four retro-reflected phases. However,
for practical purposes in this work it
will be sufficient to consider just four
independent phases.
V(r) = V0
[
4
∑
i=1
cos (ki · r + ϕi)
]2
. (5.25)
Experimentally, this can be achieved by simply removing the
frequency offsets between each lattice beam. If two or more lattice
axes interfere with each other, their respective polarisations also
crucially affect the resulting potential. Equation 5.25 only covers
the situation of out-of-plane polarisation, i.e. the polarisation of
each lattice beam being at right angles to the lattice plane. The
opposite case (in-plane polarisation) leads to a modification of the
84 quasicrystalline optical lattices for ultracold atoms
potential but the overall structure remains similar to the out-of-
plane configuration.
In comparison to the non-interfering case discussed above
(Figs. 5.7 and 5.8), constructive interference between four axes
occurs much less frequently. Consequently, the resulting poten-
tial landscape (Eq. 5.25) is dominated by steep, isolated max-
ima, which are surrounded by a ‘floor’ of minima, as shown in
Fig. 5.12. This effect becomes even more apparent when con-
sidering cuts through the potential (Fig. 5.13), emphasising the
importance of interference in the eighfold lattice (c.f. Fig. 5.8).
0 2 4
Energy of extremum/V0
0
10
20
30
O
cc
ur
re
nc
e
Figure 5.14: Histogram of minima
(blue) and maxima (red) of the exact
potential shown in Fig. 5.10 (random
phases, non-interfering, blue-detuned
lattice). The non-interfering case
produces roughly equal amounds of
minima and maxima, their distribution
is approximately symmetric around
2V0.
0 4 8 12 16
Energy of extremum/V0
0
10
20
O
cc
ur
re
nc
e
Figure 5.15: Histogram of maxima
(red) of the exact potential shown in
Fig. 5.12 (ϕi = 0, fully-interfering,
blue-detuned lattice). The minima
would all cluster around zero and
have thus been omitted from this plot.
In comparison the non-interfering
case (Fig. 5.14 above) the maxima
are spread out more. The eightfold
rotational symmetry is reflected in
the histogram by the appearance of
maxima in groups of eight.
Similar to the non-interfering case, the potential in Eq. 5.25
bears strong similarities with the octagonal tiling, a fact that has
been discussed extensively by Jagannathan et al. [85, 198–200].
However, in contrast to the non-interfering case, the octagonal
tiling only applies to points of maximally constructive interfer-
ence (the isolated maxima discussed above). This means that in
order to model the octagonal tiling physically with cold atoms we
would have to change the detuning of our lattice from blue- to
red-detuned in order to convert the intensity maxima to potential
minima. Conversely, the tunnelling between these lattice points
(as discussed in ref. [198]) might more closely resemble physical
‘tight-binding’ bonds between lattice sites, compared to the non-
interfering case in which the bonds between lattice sites dictated
by the octagonal tiling may or may not correspond to the physi-
cally most relevant tunnelling matrix elements. Another drawback
of the interfering case is that there are numerous shallow ‘local’
intensity maxima surrounding the strong ‘octagonal’ lattice points
and it is unclear whether these affect the equilibrium behaviour
of a quantum gas in such a lattice. Regarding the ‘bands’ of en-
ergy minima, the non-interfering case comes closer to forming
a well-defined band of allowed energies (Fig. 5.14), whereas the
fully-interfering case shows maxima distributed over the entire
range of the potential (Fig. 5.15).
Partly-interfering axes
The case of partly-interfering lattice axes is particularly interesting
since it allows the creation of an separable 2D quasiperiodic po-
tential. If two orthogonal lattice axes are chosen to interfere with
each other, whereas the other two axes shall be non-interfering,
the potential can be written as
V(r) = V0
2
∑
i=1
cos2 (ki · r + ϕi) +V0
[
4
∑
i=3
cos (ki · r + ϕi)
]2
.
(5.26)
A potential of this kind is shown in Fig. 5.16, which is quasiperi-
odic in both x and y direction, but without global eightfold ro-
tational symmetry. This potential landscape is similar to the 2D
bichromatic lattice employed in ref. [24]. It highlights the flexi-
bility of our eightfold lattice setup, being able to produce both
eightfold optical lattice 85
-4 -2 0 2 4
Position x/λlat
-2
0
2
Po
si
ti
on
y/
λ
la
t
0
6
La
tt
ic
e
po
te
nt
ia
lV
(r
)/
V 0
Figure 5.16: Lattice potential re-
sulting from two fully interfering
axes and two non-interfering axes
(Eq. 5.26. The phases ϕi are chosen to
be (0, 0,pi/2,pi/2).
separable and non-separable quasiperiodic potentials in two di-
mensions.
This completes the discussion of attainable real-space poten-
tials. Next, the setup and characterisation of the eightfold lattice
(non-interfering case) will be summarised.
5.4 Setup and characterisation
The lattice setup is depicted schematically in Fig. 5.17. The axes
have been labelled X, Y, D(iagonal), and T(transport). Only the last
achromat before the atoms is shown in the schematic.
Laser setup
The lattice beams are derived from one single-frequency Ti:Sa
laser15 at λlat = 726 nm which is far-blue-detuned with respect 15 Matisse, Sirah Lasertechnik GmbH,
max. 7W with RefCell locking. The
Ti:Sa crystal is pumped by a Millennia
laser (Spectra Physics) producing 24W
of power at 532nm.
to the D-lines in 39K, ensuring suppression of single-photon scat-
tering. Cross-interferences between lattice axes are avoided by
offsetting their frequencies by more than 10 MHz from each other.
Therefore, the corresponding beat notes between the axes oscillate
much faster than the atomic kinetic energy scale, given by Erec/h
(table 5.1).
Beam waists
The choice of lattice beam waists is governed by essentially the
same trade-off as for the dipole beam waists.
• Similar to the dipole trap waists, small lattice waists lead to the
highest peak intensity and, consequently, the deepest lattice
potential. Therefore, the lattice beam waists should be chosen
as small as possible, given a certain maximum optical power
(on the ordon the order off 200mW per beam).
86 quasicrystalline optical lattices for ultracold atoms
Figure 5.17: Schematic of the science
cell optics (lattice in blue and dipole
in red). The axes have been labelled
X, Y, D(iagonal), and T(transport).
The last achromats before the atoms
are shown as black double-headed
arrows, with their corresponding focal
lengths (except for the T-axis for which
only one of them is shown). The glass
cell is shown in light blue (actual
dimensions).
f = 125mm
f = 125mm
f = 125mm
f = 125mm
f = 125mm
f = 100mmf = 100mm
D-axis
T-axis
Y-a
xis
X-axis
LatticeDipoleDip
ole
Lat
tice
25.0mm
• Conversely, large in-plane lattice waists (compared to the extent
of the atomic cloud) are beneficial in order to ensure homo-
geneity of the lattice potential.1616 An overall homogeneous potential
is desirable to compare cold-atom
quantum simulation experiments to
condensed-matter theory, which is
often performed in homogeneous
lattices. In most lattice experiments the
potential envelope is gaussian which
necessitates a density-dependent
theoretical description of the system.
A well-known example is the bosonic
or fermionic Mott-insulator. The
characteristic ‘wedding-cake’ density
profile can be described by the local-
density approximation.
Elliptical beam profiles are a good compromise between the two.
The in-plane waists are chosen as large as possible, whereas the
vertical waists can be reduced in order to increase the maximum
achievable lattice depth. In addition, the retro-reflecting configu-
ration increases the maximum depth (given a certain maximum
available beam power) by a factor of two.
In the fully-interfering case, the maximum lattice depth is a
factor of four higher in the fully-interfering case (see Fig. 5.12,
compared to Fig. 5.7), which could support even larger lattice
waists (given a maximum power).
Blue-detuned lattices lead to an overall anti-confinement re-
sulting from the spatially-varying on-site trapping frequencies
ωon-site(r). The ground state energy in each well is given by the
lowest harmonic oscillator energy E(r) = h¯ωon-site(r) in the two-
dimensional case (two lattice beams overlapped in-plane at 90◦).
The in-plane envelope E(r) is also approximately gaussian.1717 In a 1D lattice the situation is sep-
arable and thus the gaussian anti-
confinement is exact.
Consequently the anti-confining effect can compensated by weak
co-propagating red-detuned dipole beams with larger waists by
a factor of
√
2. Alternatively, a single red-detuned laser beam in
the vertical direction (also with a factor of
√
2 larger waist) can be
used to the same effect.
eightfold optical lattice 87
In the eightfold optical lattice these considerations are tricky
since the lattice potential is not periodic and it is unclear which
of the ‘lattice sites’ – which are all at different energies, even in
the homogeneous case – should be taken into account for the
anti-confinement. As a guideline, we aimed at a ratio of in-plane
waists between lattice and dipole trap of roughly
√
2. However,
spatial and physical constraints (gaussian beam-propagation with
certain minimal distances in between the components and limited
choices of focal lengths) led to slightly smaller waists than initially
intended (see table 5.3). In addition, preliminary calculations
taking into account the entire spread of ‘lattice sites’ show that
the eightfold case leads to an anti-confining waist that is smaller
than the lattice waists by a factor of 21/4. We plan to counteract for
anti-confining effects using a dedicated dipole trap beam along the
z-axis.
Table 5.3: Experimental values for the
in-plane (wr) and vertical (wz) lattce
beam waists (single waists), measured
on cameras that are focussed to the
position of the atomic cloud. The T-
waists are approximately 15% larger
than the other waists. Probably this is
due to the different optical path length
on the incoming beam side (40 cm
distance between last lens and atoms).
axis wr/ µm wz/ µm
D 158 73
Y 153 71
X 161 70
T 184 91
The vertical beam waists (table 5.3) are chosen such that the
lattice intensity varies as little as possible across the atomic cloud.
Using our maximum power (several hundred mW per beam)
we can achieve single-axis lattice depths of at least 40 Erec.
Lattice angles
One of the limiting factors in observing very high momentum
states is the angle between lattice beams. By fitting a straight
line to diffraction peaks in time-of-flight the angle of each lattice
beam with respect to the absorption image axes is determined.
From these measurements (table 5.4) we conclude that the relative
angles are 45(1)◦.
Table 5.4: Relative lattice beam angles
used in this work.
axes angle
TD 89.5◦
TX 45.6◦
TY 44.0◦
XY 91.6◦
XD 46.2◦
YD 45.4◦
Control loops
The intensity of each lattice beam is actively stabilised using a
PID feedback loop. Directly after the fibre and the optical isolator
(which also acts as a polarisation-cleaning cube) a small (' 4%)
proportion of the beam is directed to the ‘PID-photodiode’ using
a wedged glass plate. This constitutes the feedback signal for
the control loop, the active stabilisation being performed by the
RF-power given to the AOM (before the fibre). The lattice pulse
signals are created by waveform generators in order to have sub-
µs time-resolution and high reproducibility of the DC voltages.
In order to maximise the bandwidth of the control loops all cable
lengths have been chosen to be as short as possible; the largest
contribution comes from the cable connecting the PID-photodiode
with the input of the PID-unit (12m).
The active control loops have bandwidths on the order of sev-
eral hundred kHz and consequently µs-risetimes of lattice pulses
can be expected. The PID-parameters have been optimised for
a given lattice depth (15Erec) such that a square pulse produces
a maximal rise time (0% to 90% set voltage) of 1.1µs, thereby
minimising any overshoot. For different pulse shapes and lattice
depths the parameters have to be re-optimised for best perfor-
88 quasicrystalline optical lattices for ultracold atoms
mance; as usual, there is a trade-off between short rise times and
minimal overshoot.
-40
-20
0
20
40
In
-s
it
u
po
si
ti
on
/µ
m
-600 -300 0 300 600
70
80
90
100
In
-s
it
u
cl
ou
d
si
ze
/µ
m
Figure 5.18: Lattice sliding method.
The upper plot shows the in-situ
position of the condensate on the
Z-camera, at right angles to the lat-
tice beam direction. The dashed line
corresponds to the unshifted con-
densate position. The lower plot
shows the same scan, but using the
Thomas-Fermi radius of the cloud as
observable. This data was taken using
the horizontal knob of the D-axis piezo
mirror.
Setup and alignment
The precise alignment of the lattice beams is similarly critical as
that of the dipole trap.18 In brief, the following alignment steps
18 We showed in section 4.4 that the
vertical position of the dipole beam
can be discrimiated to within 2um.
should be observed.
• The focus of the incoming beam (beam waist) has to be over-
lapping with the atomic cloud. Similar to the dipole trap, the
final achromat before the atoms is translated longitudinally (z-
translation stage with µm-skrews) to minimise the beam waist.
The beam waist can be measured with a dedicated camera on
each lattice axis that has been focussed to the position of the
atoms via in-situe absorption imaging. In the case of the X and
Y axes, which are shared with the dipole trap, this alignment
step is omitted since we trust the achromat to be already well-
aligned having minimised the dipole trap waists. This assumes
firstly that the focal shift due to the different wavelengths is
minimal, and secondly, that the expected beam propagation of
the lattice beam (using ABCD-matrices) is faithful. The latter
point is assured by measuring the actual beam divergence after
the fibre coupler for each individual lattice axis.
• The pointing of the lattice beam can be aligned via the so-called
‘lattice sliding method’. It entails blocking the retro-path and
shining a relatively weak lattice potential onto an atom cloud
which is imaged in-situ along an axis that is at right angles to
the lattice beam. Since our lattice is blue-detuned, the beam
leads to a repulsive potential. If the lattice beam is off-centre
in one direction, the atomic cloud is displaced towards the
opposite direction. By taking a scan of vertical and horizontal
displacement using a piezo-driven mirror, as shown in Fig. 5.18,
the lattice beam can be centrally aligned to within a few µm.
Note that due to the anti-symmetric response of the cloud, it
is crucial to take an entire scan and find the symmetry point
which corresponds to the optimum, i.e. unshifted position.
In addition, the repulsive potential leads to a broadening of
the atom cloud, providing an additional signature of good
alignment (lower half of Fig. 5.18).
• On the retro-path, the transverse position of both lenses in the
cats-eye configuration [201] is important in order to ensure
good directional overlap between incoming and retro-reflected
beam. This can be achieved by placing a pinhole on each lens
mount and maximising the transmission. In addition, the lon-
gitudial position of the retro-mirror and the lenses is crucial
to ensure approximately equal19 beam waist of incoming and19 The beam emerging from the fi-
bre coupler (beam diameter roughly
1.2mm) can never be perfectly col-
limated. Hence a 1:1 reproduction
of beam waists at the position of the
atoms is precluded by the laws of
gaussian propagation.
retro-reflected lattice beams. It should be noted that for ellipti-
cal beams, there are, in principle, no degrees of freedom in the
longitudinal position of both the lenses and the retro-mirror,
eightfold optical lattice 89
because the cat’s eye requires the distance between atoms and
retro-mirror to be exactly twice the sum of the two focal lengths.
The achromat on the atoms’ side is assumed to be aligned to
the in-situ position of the atoms via absorption imaging. The
second cat’s eye lens is placed in good faith by measuring the
distances with a ruler.
• Finally, the directional pointing of the retro-reflected beam at
the position of the atoms is relatively insensitive to the angle of
the retro-mirror due to the cat’s eye configuration. In particular,
the diffraction angle (Kapitza-Dirac diffraction [202]) at the
position of the atoms does not depend on the angle of the retro-
mirror. Nevertheless, the angle can be optimised by maximising
the effect of 1D Kapitza-Dirac diffraction.

6
Matter-wave diffraction experiment
This chapter presents the main experimental results of this thesis,
namely the matter-wave diffraction experiments. Short lattice
pulses (Kapitza-Dirac diffraction [202, 203]) are used to probe the
2D quasicrystalline structure in reciprocal space.
First, the basis of momentum in space of the 2D quasicrystal
is described in section 6.1, building on the 1D results from sec-
tion 5.1. Section 6.2 covers the experimental methods that are used
(short lattice pulses). Afterwards, both the static and the dynamic
diffraction patterns are described in theory and experiment. The
diffraction dynamics, for very short pulses, are shown to represent
quantum walks. Finally, our methods of data-analysis are detailed
in section 6.5.
6.1 Basis in momentum space
Similar to the indexing method for diffraction pattern of 1D qua-
sicrystals (section 1.2) we are going to write down a method to
identify all basis states of the 2D quasicrystalline potential in mo-
mentum space using four integers. The basis states in momentum
space correspond to the matter-wave diffraction peaks that are
going to be observed in the experiment.
As before, the real-space potential of our optical lattice can be
written as
VD(r) = V0
D
∑
i=1
cos2
(
Gi
2
· r
)
, (6.1)
where D = 1, 2, 3, or 4 is the number of mutually incoherent lattice
beams and V0 = 14.6(2) Erec is the individual lattice depth.1 The 1 The lattice wavelength λlat =
2pi/klat = 726 nm is far detuned
with respect to the D-lines in 39K,
ensuring that single-photon processes
are completely suppressed. The value
of Erec is listed in table 5.1.
reciprocal lattice vectors Gi are defined as
G1 =
(
1
0
)
, G2 =
(
0
1
)
, G3 =
1√
2
(
1
1
)
, G4 =
1√
2
(
−1
1
)
(6.2)
where we have switched to dimensionless units in which 2klat = 1
as shown in Fig. 6.1 (right inset). In contrast to a periodic lattice
the combination of several Gi here may give rise to new, smaller
momentum scales, as shown the left inset of Fig. 6.1; for example,
the combination −G1 + G3 − G4 results in a new k-vector (red
arrow) that is shorter than the original G1 by a factor of 1 +
√
2
(the silver mean). This process can be repeated ad infinitum and
92 quasicrystalline optical lattices for ultracold atoms
results in a self-similar structure containing arbitrarily small k-
vectors, as indicated by the sequence of octagons in Fig. 6.1. Con-
sequently, it is impossible to assign a maximum characteristic
length to this quasicrystal, heralding the presence of structure
on all scales. Furthermore, the momenta that are reachable from
k0 = (0, 0) form a dense set in the kx, ky-plane and any element G
of this set is determined by four integers (i, j, l, n) ∈ Z4 as
G = iG1 + jG2 + lG3 + nG4 . (6.3)
While physical momentum remains two-dimensional, all four
integers are nonetheless required to describe a given G, since
cos(45◦) = sin(45◦) = 1/
√
2 is irrational and hence incommensu-
rable with unity. In fact, Fig. 6.1 can be viewed as an incommensu-
rate projection of a 4D simple-cubic ‘image’ lattice to the 2D plane,
similar to the ‘cut-and-project’ scheme for constructing the octag-
onal tiling (section 1.3). By using fewer than four lattice beams we
can control the dimensionality of the image lattice and Z4 reduces
to ZD with D ∈ {1, 2, 3, 4}.
-6 -4 -2 0 2 4 6 8
kx/klat
0
2
4
k
y
/k
la
t
0th
1st
2nd
3rd
≥ 4
Gˆ1
Gˆ2 Gˆ3
Gˆ4
−Gˆ1
+Gˆ3 −Gˆ4
Figure 6.1: Momentum space struc-
ture. The first 15 orders of possible
diffraction peaks are shown. They are
constructed by iteratively adding or
subtracting one of the four reciprocal
lattice vectors Gi (inset on the right) to
the peaks in the previous order, start-
ing with k = (0, 0). This results in a
fractal structure, whose self-similarity
is illustrated by a sequence of oc-
tagons, which are each scaled by the
silver mean 1+
√
2 relative to the next.
The left inset shows one inflation step
(see text).
Basis in momentum space
We can use Eq. 6.3 to express any accessible k-state as an integer-
valued vector bi in ZD, where D is the number of active lattice
beams. As described for the 1D case in section 5.1 the nth order of
this basis is defined as the set of all elements {bi} with
D
∑
j=1
abs
(
[bi]j
)
= n (6.4)
where [a]j denotes the jth component of a vector a and abs()
denotes the absolute value. The total number of basis states N up
to and including the nth order for D = 4 is listed in table 6.1; it can
matter-wave diffraction experiment 93
be written in a closed form as
N(n) =
min(4,n)
∑
k=0
2k
(
4
k
)(
n
k
)
. (6.5)
Table 6.1: Number of basis states up to
and including the nth order in the 4D
case (eightfold lattice).
n N(n)
0 1
1 9
2 41
3 129
4 321
5 681
. . .
11 11969
The in table 6.1 are also known to be
the coefficients of the Taylor expansion
of the following function [204]
(1+ x)4
(1− x)5 (6.6)
around x = 0.
We will later truncate ZD at the nth order, meaning we only
take into account states that can be reached by at most n two-
photon scattering events.
Projection
For D = 3 and D = 4 the projection matrices for the states bi onto
the kx, ky–plane are given by
M3 =
 1 0
1√
2
0 1 1√
2
0 0 0
 (6.7)
and
M4 =

1 0 1√
2
− 1√
2
0 1 1√
2
1√
2
0 0 0 0
0 0 0 0
 , (6.8)
respectively. The third and fourth dimensions are projected to the
in-plane diagonals with respect to the first two dimensions. The
first two rows of M3 and M4 are simply given by the Gi defined in
Eq. 6.2. Note the similarity of M4 to the rotation matrix for the 4D
cut-and-project scheme (Eq. 1.21 in section 1.3).
Fig. 6.1 shows elements of the basis of order 15 for D = 4,
projected onto the kx, ky–plane using the matrix M4. For D = 1(2)
the matrices M1(M2) are trivial because here the dimension of bi
is the same as the physical dimension.
Hamiltonian in momentum space
Having constructed the basis we can write down the hamiltonian
HD in momentum space
HDi,j =

V0/4 for |bi − bj| = 1
4Erec × |MD · bi|2 +V0/2 for i = j
0 otherwise
(6.9)
Here the norm of a D-dimensional vector a is given by
|a| =
√√√√ D∑
j=1
(
[a]j
)2
. (6.10)
The non-zero off-diagonal elements correspond to transition ele-
ments for stimulated two-photon scattering events, where atoms
scatter photons from one lattice beam into its counterpropagating
partner. These transitions connect discrete momentum states sep-
arated by ±Gi and effectively realise a tight-binding hamiltonian
94 quasicrystalline optical lattices for ultracold atoms
1D 2D 3D 4D
Op
tic
al 
de
ns
ity
0.0
0.3
Figure 6.2: Raw time-of-flight images
of 39K resulting from four different lat-
tice configurations at fixed lattice pulse
duration (t = 3.5 µs). Using just one of
the lattice axes results in a regular 1D
simple-cubic lattice characterized by
G1; adding the perpendicular lattice
creates a regular 2D square lattice
with G1 and G2. By adding the first
diagonal lattice we obtain a lattice of
1D quasiperiodic potentials with dense
sets of momentum states along G3
whereas the direction perpendicular
to G3 remains regular (labelled 3D).
Finally, using all four axes we create
the 2D quasicrystal (labelled 4D); its
self-similarity is illustrated by the
octagons.
in momentum space [205]. The matrix elements on the diagonal
are given by the kinetic energy term, where the prefactor 22Erec
arises from the momentum scale 2h¯klat of the individual lattices
and a constant offset of V0/2 which arises from the k = 0 Fourier
component of VD(r) (it has no physical significance since it corre-
sponds to a global shift in energy).
Truncation of basis
As in the 1D case (Eq. 5.10), this hamiltonian is infinite-dimensional
and, consequently, we need to make it numerically tractable by
truncating it. Since the experiment starts with a pure condensate
in the |h¯k = 0〉 state, and we apply only short pulses of lattice
light, it is sensible to work with a basis of order n = 11. This can
be justified a posteriori since even for the longest applied lattice
pulses our simulation (using a basis of order 15) shows that the
orders n > 11 get populated by less than 15 per cent. As before,
the time-evolution can be calculated by numerically integrating
the time-dependent Schrödinger equation. This is achieved by
writing the hamiltonian matrix HDi,j in the truncated momentum
basis; integration is performed using the scipy.integrate library.
6.2 Method: Kapitza-Dirac diffration
The experimental sequence starts with the preparation of an al-
most pure Bose-Einstein condensate of 39K atoms in a crossed-
beam dipole trap, as described in Chapters 3 and 4. Using the
Feshbach resonance centred at 402.70(3)G (section 4.6) we tune
the contact interaction to zero just before we release the conden-
sate from the trap. Then we immediately expose it to the optical
lattice for a rectangular pulse of duration t. During this pulse,
atoms in the condensate can undergo several stimulated two-
photon scattering events (Kapitza-Dirac scattering [206]), which
scatter photons from one lattice beam into its counterpropagating
partner and transfer quantized momenta of ±2h¯klat, where h¯klat is
the momentum of a lattice photon and |Gi| = 2klat. Throughout
matter-wave diffraction experiment 95
1D 2D 3D 4D
Op
tic
al 
de
ns
ity
0.00
0.25
Figure 6.3: Raw time-of-flight images
of 87Rb resulting from four different
lattice configurations at fixed lattice
pulse duration (t = 5.0 µs). Compared
to Fig. 6.2, which was taken with
39K, the resolution in momentum
space is worse. In addition, strongly
pronounced ‘scattering shells’ (see
text) appear for 87Rb (scattering length
a = 100 a0) which lead to a diffuse
background in the 4D situation.
this chapter, the lattice depth of each individual axis is 14.6(2)Erec.
Finally, we record the momentum distribution of the atomic cloud
by taking an absorption image after tToF = 33 ms (time-of-flight).
In a first experiment we fix the lattice pulse duration at t = 3.5 µs
and vary the number of lattice beams, as shown in Fig. 6.2. Start-
ing from the single-axis (1D) case, we subsequently add lattice
axes, finally completing the eightfold symmetric case (4D), rep-
resenting the quasicrystalline structure with its striking self-
similarity under (1+
√
2) scaling.
Interaction effects
For diffraction experiments that are carried out with finite contact
interactions (e.g. using 87Rb, Fig. 6.3) one finds that the time-of-
flight images feature pronounced ‘scattering shells’ [207, 208]
connecting the discrete momentum peaks. These shells appear as
characteristic rings on the absorption images and arise from two-
body s-wave collisions between parts of the atomic cloud which
are moving with respect to each other. In order to eliminate this
effect we perform all experiments with 39K and tune the contact
interaction to zero by ramping the magnetic field to a value of
351.5(1)G, just before the optical lattice pulse is applied. How-
ever, atomic clouds of 39K at vanishing interactions are optically
dense enough to absorb essentially all imaging light, preventing
any faithful atom number measurement. Therefore we turn inter-
actions back on (back to the previous value of roughly 280 a0) once
the diffraction orders have separated from each other, such that
the individual peaks expand and reveal their atom populations.
6.3 Diffraction dynamics
The diffraction dynamics offers one of the most striking signa-
ture of the aperiodicity of the eightfold optical lattice. During the
lattice pulse the condensate explores reciprocal space in discrete
steps of ±Gi, leading to profoundly distinct behaviours in the
periodic (2D) and in the quasicrystalline case (4D). Fig. 6.4 shows
absorption images for four different values of pulse duration t in
96 quasicrystalline optical lattices for ultracold atoms
t = 0.0μs t = 1.0μs t = 3.0μs t = 6.0μs
Op
tic
al 
de
ns
ity
0.0
0.3
Figure 6.4: Dynamics of Kapitza-Dirac
diffraction in the quasicrystalline
optical lattice. The figure shows raw
absorption images for four different
lattice pulse durations. After 1µs,
only the first diffraction order has
been populated, while longer pulses
lead to populations in successively
higher orders as the atoms perform a
quantum walk in momentum space.
Each octagon has a circumradius of
|Gi | = 2klat, illustrating the fun-
damental momentum scale due to
two-photon processes.
the latter configuration, illustrating the occupation of more and
more closely spaced momenta. Using individual fits we extract
the number of atoms in every k-state up to the seventh diffrac-
tion order, i.e. those momenta reachable by seven or fewer two-
photon scattering events. In the 2D simple cubic lattice (Fig. 6.5
on the left) only a few momentum states are accessible, since ki-
netic energy acts as a harmonic confinement in momentum space.
Therefore the dynamics is oscillatory and reminiscent of a sim-
ple harmonic oscillator. In the quasicrystalline case, in contrast,
the diffraction dynamics is aperiodic (4D, right of Fig. 6.5), since
large parts of the population propagate ballistically to progres-
sively higher orders, as illustrated by the light blue ‘wave front’.
Due to the self-similar momentum space structure, the atoms can
access momentum states in successively higher diffraction orders
Figure 6.5: Kapitza-Dirac diffrac-
tion dynamics in a periodic (2D) and
quasicrystalline (4D) lattice. The nor-
malized populations (coloured dots)
of the condensate (0th order) and the
first seven diffraction orders are plot-
ted against pulse duration, together
with the numerical solution to the
Schrödinger equation (lines). The peri-
odic case (2D) is oscillatory as kinetic
energy limits the accessible momenta.
In contrast, the quasicrystalline lattice
(4D) contains a dense set of k-states,
c.f. Fig.6.1, enabling the population of
higher and higher orders. Correspond-
ingly, the expansion carries on linearly,
indicated by the light blue ‘wave front’
as a guide to the eye. Error bars denote
the standard deviations from five re-
alisations of the experiment, and are
typically smaller than symbol size.
0.0
0.5
1.0
0 5 10 15 20
Lattice pulse duration (µs)
0 5 10 15 20 25
Lattice pulse duration (µs)
2D 4D
1st
2nd
3rd
4th
5th
6th
0th
1st
2nd
3rd
4th
5th
6th
0th
7th7thP
op
ul
at
io
n
matter-wave diffraction experiment 97
0 1 2 3
Lattice pulse duration (µs)
0.0
0.5
1.0
1.5
2.0
rm
s
m
om
en
tu
m
(2
h¯
k
la
t)
1D
2D
3D
4D
0 3
0.0
1.0
rm
s/
√ D
(2
h¯
k
la
t)
Figure 6.6: Continuous-time quantum
walk in momentum space in D dimen-
sions, where D is controlled by the
number of lattice beams. Dots repre-
sent the measured root-mean-square
momentum (see text), while lines rep-
resent numerical solutions to the full
Schrödinger equation. The inset shows
the same data, but scaled by
√
D. Here
the dashed line represents the expan-
sion dynamics of a continuous-time
quantum walk on a homogeneous
D-dimensional tight-binding lattice.
The
√
D scaling (Eq. 5.17) is a direct
consequence of the separability of hy-
percubic lattices. Deviations from the
linear behaviour at later times are due
to kinetic energy, and the lines would
differ from each other at long times.
Error bars denote standard deviations
from five identical realisations of the
experiment.
that correspond to ever smaller momenta. Our data agrees ex-
cellently with exact numerical solutions (lines in Fig. 6.5) of the
single-particle time-dependent Schrödinger equation in momen-
tum basis.
6.4 Quantum walks in high-dimensional homogeneous tight-
binding lattices
0 2 4 6 8 10 12 14
Lattice pulse duration (µs)
0.0
0.5
1.0
1.5
2.0
rm
s
m
om
en
tu
m
/√
D
(2
h¯
k
la
t)
vp × t
1D
2D
3D
4D
Figure 6.7: Root-mean-square mo-
mentum for longer times, scaled by
a factor
√
D as in Fig. 6.6 (inset) in
the main text. The dashed black line
is the analytic result (Eq. 5.17) in a
D-dimensional homogeneous tight-
binding model. The dashed blue lines
are the solutions to the time-dependent
Schrödinger equation, using all states
with n ≤ 11. If we take into account
that we only detect atoms up the sev-
enth diffraction order, the expansion is
reduced to slightly lower momentum
values (solid lines).
In the regime of very short pulses kinetic energy can be neglected
for all dimensions and the discrete momentum space structure
can be mapped onto a homogeneous tight-binding lattice [205]. A
hopping event in this effective lattice corresponds to a two-photon
scattering event and connects momenta differing by ±h¯Gi. In this
mapping, the diffraction dynamics is equivalent to the expansion
of initially localised particles and gives rise to a continuous-time
quantum walk with its characteristic light-cone-like propaga-
tion [22, 190, 191]. For a hypercubic lattice in D dimensions, the
separability of the tight-binding dispersion relation (section 5.1)
leads to an average group velocity proportional to
√
D (Eq. 5.17).
Due to the correspondence between the number of lattice beams
and the dimension of the tight-binding hamiltonian, we are able to
extend the dynamics to up to four dimensions. Using the appro-
priate form of Eq. 6.3 in ZD, we extract the effective root-mean-
square momentum in D dimensions, e.g.
√〈i2 + j2〉 in the 2D
case and
√〈i2 + j2 + l2 + n2〉 in the 4D case, from the individual
populations of all diffraction peaks, and find excellent agreement
between the measurements and the analytic result vp ∝
√
D, as
shown in Fig. 6.6. The departure from linear behaviour at longer
times is due to kinetic energy and is captured well by the exact
numerical solution to the Schrödinger equation (solid lines in
98 quasicrystalline optical lattices for ultracold atoms
Fig. 6.6). The extent of the linear region is controlled by the lattice
depth. For even longer times, kinetic energy enforces fundamen-
tally different behaviours for periodic and quasicrystalline lattices,
as shown in Fig. 6.5 (and in Fig. 6.7).
6.5 Data analysis
The data analysis procedure for determining the population in
each diffraction peak is described in the following. First, we deter-
mine the position of the condensate, find the angle of one lattice
axis relative to the camera axes, and calibrate the magnification us-
ing reference images showing only zeroth and first order diffrac-
tion peaks. With this information we can calculate the expected
position of each momentum peak. Around each calculated peak
position we perform an individual fit to a 2D Thomas-Fermi pro-
file (Eq. 4.14) in a square bin of 28 × 28 pixels (56 × 56 for the
central condensate). In order to mitigate effects of imaging satu-
ration, the fit ignores pixels with optical densities above 2.0. The
corresponding atom population of each basis element pi(t) is pro-
portional to the integrated Thomas-Fermi profile, the result of
which is in Fig. 6.8 as an example. If the population pi(t) is below
0.04% of the total population we ignore it in order to avoid count-
ing spurious populations in high diffraction orders, which would
otherwise dominate the rms. For Fig. 6.5 we sum all populations
in one diffraction order.
Figure 6.8: Simplified example of
the population count. (A) The raw
absorption image (D = 4, t = 3µs).
At each calculated peak position with
n ≤ 7 we perform an individual
fit to a 2D Thomas-Fermi profile.
The resulting populations pi(t) are
depicted in (B) by the area of the
circles.
A B
Root-mean-square extraction
We calculate the root-mean-square momentum in D dimensions as
a function of time as √
∑
i
pi(t)
∑j pj(t)
|bi|2 , (6.11)
where pi(t) are the populations in each diffraction peak bi at a
given time t. The sums go over all elements of a basis.
matter-wave diffraction experiment 99
0 5 10 15 20 25
Lattice pulse duration [µs]
0
200
400
600
800
1000
1200
1400
1600
1800
S
um
of
po
pu
la
tio
n
in
al
lp
ea
ks
(a
.u
.)
0 5 10 15 20 25
Lattice pulse duration [µs]
0
200
400
600
800
1000
1200
1400
1600
1800
S
um
of
po
pu
la
tio
n
in
al
lp
ea
ks
(a
.u
.)
2D 4D
Figure 6.9: Total detected population
(a.u.) in all diffraction peaks for
the 2D and 4D situations. The solid
black lines correspond to the total
population and the dashed lines are
the simulated populations in peaks
up to seventh order and additionally
account for peaks that fall below the
cutoff. When single peaks (such as
the central condensate) are strongly
populated we systematically detect too
few atoms due to imaging saturation
and the finite signal-to-noise ratio of
the camera. This effect is reflected
by the apparent rise in total detected
population during the first few µs. It
also explains the subsequent ‘dips’ in
atom number in the 2D case.
Total population
Figure 6.9 shows the total detected population ∑j pj(t) summed
over all diffraction peaks for the cases D = 2, 4. For D > 2, the
detected population is reduced for longer lattice pulses since more
and more peaks are weakly populated and fall below the cut-
off. In addition, we underestimate the population of very highly
populated peaks, such as the initial condensate, due to the finite
signal-to-noise of the camera.
Conclusion
In conclusion, a quasicrystalline potential for ultracold atoms
was realised, which can facilitate the creation of ever more com-
plex many-body systems [31]. By observing the occupation of
successively closer-spaced momenta, we were able to confirm its
self-similar structure in momentum space. In addition, the fun-
damentally different diffraction dynamics between periodic and
quasicrystalline potentials have been verified, in excellent agree-
ment with theory. Finally, we demonstrated the ability to simulate
tight-binding models in one to four dimensions, by observing the
ballistic spreading of particles in reciprocal space.

Summary and outlook
In summary, this thesis establishes quasicrystalline potentials as
a novel platform for ultracold atoms. The eightfold optical lattice
is a highly versatile system, allowing investigation of a large vari-
ety of lattice potentials. These include several 1D and 2D periodic
geometries, 1D quasiperiodic lattices, and 2D quasiperiodic po-
tentials, both separable and non-separable. This work represents
the first demonstration of a non-separable, 2D quasicrystalline
lattice for ultracold atoms, thereby bringing closer the fields of
quasicrystals and ultracold atoms.
The focus of this thesis has been on the experimental realisa-
tion of above mentioned platform. Experimental details have been
given on initial laser cooling (Chapter 3), the production of quan-
tum gases (Chapter 4), and the setup and characterisation of the
eightfold optical lattice (Chapter 5).
In a series of proof-of-principle experiments, two important
properties of 2D quasicrystals, namely self-similarity and the
connection to higher dimensions, have been demonstrated with
ultracold atoms for the first time. These include the direct obser-
vation of a quantum walk on a homogeneous tight-binding model
in four dimensions.
Next steps
Having established the eightfold lattice potential, there are many
interesting steps to be taken from here.
Adiabatic loading
0 10 20 30
Time/µs
0
1
2
3
La
tt
ic
e
de
pt
h/
V 0
Figure 6.10: Triangular lattice pulse
shape, duration 20 µs, lattice depth
2.5 Erec.
So far, all the experiments described in this thesis have been prob-
ing the quasicrystal in momentum space, using transient pulses
of laser light. The next obvious step would be to study the sys-
tem in equilibrium (if this is possible) and load atoms into the
potential for longer times. Here, one of the main challenges will
be understanding adiabaticity in this aperiodic potential. Prelim-
inary studies show that, for small lattice depths (V0 . 1.5Erec),
adiabatic loading of the lattice should indeed be possible. This
can be seen by considering triangular lattice pulses as shown in
Fig. 6.10. After ramping up the eightfold lattice to a certain value
V0 and subsequently ramping it back down, the fraction of atoms
remaining in the condensate (0th diffraction order) is a signature
102 quasicrystalline optical lattices for ultracold atoms
Figure 6.11: Theoretical final popu-
lation in 0th diffraction order after
triangular pulse of duration t (x-axis)
for various lattice depths V0 in the
eightfold optical lattice. After an initial
dip, the condensate fraction (0th order)
recovers to almost unity for shallow
lattice depths (light blues). For higher
depths (deep blues), the long-term
behaviour shows no recovery of the
0th order, which can be interpreted as
non-adiabaticity on these timescales.
For t > 100 µs the curves have been
fitted with linear functions, the slopes
of which are shown in Fig. 6.12. 0 100 200
Triangular pulse duration/µs
0.0
0.5
1.0
Po
pu
la
ti
on
0t
h
or
de
r
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Figure 6.12: Slopes from Fig. 6.11,
plotted against lattice depth. Non-zero
slopes start to appear above 1.5 Erec
heralding non-adiabatic behaviour.
1.0 2.0 3.0
Lattice depth V0/Erec
0
10
20
N
on
-a
di
ab
at
ic
sl
op
e
×1
04
/(
−
0t
h
po
p/
µs
)
of adiabaticity [209].
This has been theoretically investigated in Fig. 6.11 for the full
eightfold lattice potential (solutions to time-dependent Schrödinger
equation). For shallow lattices (faint blues) adiabaticy can be
achieved, even in the aperiodic lattice. However, lattice depths
higher than ' 1.5Erec (deep blues) show a non-zero slope for
longer lattice pulse durations, heralding non-adiabatic behaviour.
The slopes have been fitted for t > 100 µs as a quantitative esti-
mate of adiabaticity and they show a cross-over behaviour from
adiabadic to non-adiabatic (Fig. 6.12).
Experimentally, this behaviour will be very interesting to inves-
tigate, in particular as a function of interaction strength.
Topology
One of the great achievements in the field of quantum gases has
been the experimental realisation of topologically non-trivial sys-
tems [210–212]. In future, aperiodic lattices may find an appli-
cation in this context. For example, a topological charge pump
has been demonstrated experimentally using a 1D photonic qua-
sicrystal [5]. It would be worth pursuing further studies along this
matter-wave diffraction experiment 103
direction in quantum gas systems, particularly in the light of the
recent observation of a 4D quantum Hall response in tilted optical
superlattices [86].
Many-body localisation
Another application of aperiodic lattices for ultracold atoms has
been the observation of many-body localisation [88]. In particular
the role of interactions in quasiperiodic media is still poses fun-
damental open questions in physics [89], and experimental results
have only recently become available [24, 88, 90, 91]. So far, all ex-
periments employed either 1D [88, 90, 91] or 2D separable [24]
potentials. However, more experimental evidence is needed, for
instance, to differentiate between two universality classes of local-
isation phenomena, namely in random disorder and in aperiodic
order [92]. For the latter class, our experiment is an ideal starting
point since the lattice potential closely resembles the octagonal
tiling (see section 5.3), a paradigmatic 2D quasicrystal.
Moreover, our system will be able to investigate the fate of a
Mott-insulator in the presence of a weak aperiodic perturbation in
two dimensions. In one dimension, there has been evidence of a
Bose-glass phase [213, 214], but the two-dimensional case has not
been experimentally investigated so far.
More ideas
In the four-beam lattice the total number of basis states grows
very fast with the diffraction order (table 6.1). For example, a basis
of order eleven contains almost 12000 states. Therefore it would
be interesting to reduce the atom number in the diffraction experi-
ment by some orders of magnitude until it becomes comparable to
the number of basis states. Will the discreteness of the population
then show in the diffraction dynamics?

A
List of components
Near-resonant laser system (‘red table’)
identifier description part number manufacturer
Lasers K master TA 766.7nm, 2.0W, 760-775nm coarse tuning, 50GHz mode-hop-free TA pro 765-3V0 Toptica Photonics AG
Rb master TA 780.24nm, 3.0W, 770-795nm coarse tuning, 50GHz mode-hop-free TA pro 780-3V0 Toptica Photonics AG
K imaging 766.7nm, 80mW DL pro Toptica Photonics AG
Rb repump 780.24nm, 100mW, 765-805nm coarse tuning, 30GHz mode-hop-free DL pro 780-3V0 Toptica Photonics AG
K red loading 767nm, 2.0W BoosTA pro 765-1V0 Toptica Photonics AG
K blue loading 767nm, 2.0W BoosTA pro 765-1V0 Toptica Photonics AG
K red MOT 767nm, 2.0W BoosTA pro 765-1V0 Toptica Photonics AG
K blue MOT 767nm, 2.0W BoosTA pro 765-1V0 Toptica Photonics AG
Rb loading MOT 780nm, 3.0W BoosTA pro 780-1V0 Toptica Photonics AG
Rb cooling 780nm, 3.0W BoosTA pro 780-1V0 Toptica Photonics AG
Rb 780/repump 780nm, 3.0W BoosTA pro 780-1V0 Toptica Photonics AG
Optics M45 Laser mirror HR 630-900 nm/0-45◦ s+p, R>99.6% HR 630-900nm Lens-Optics GmbH
W2 λ/2 0th order wave plate W2Z15-773 Lens-Optics GmbH
W4 λ/4 0th order wave plate W4Z15-773 Lens-Optics GmbH
magic-W λ/2 at 780nm and λ at 767nm 0th order wave plate W2Z15-L/2@780&L@767 Lens-Optics GmbH
PBS 1/2 inch polarising beam splitter PBC12-600/900 Lens-Optics GmbH
BS50 Non-polarising beam splitter 50:50 BS005 Thorlabs, Inc
F15 Fibre coupler, f = 15 mm APC, 600-1050nm (used for TA beams) 60FC-4-M15-02 Schäfter-Kirchhoff GmbH
F15-A Fibre coupler, f = 15 mm APC, 750-1550nm (used for TA beams) 60FC-4-M15-37 Schäfter-Kirchhoff GmbH
F8 Fibre coupler, f = 8.1 mm APC, 630-980nm (for all other beams) 60FC-4-M8-10 Schäfter-Kirchhoff GmbH
WW30 Glass plate beam spliter, fused silica, 30min of arc, λ/10 quality WW1Q Lens-Optics GmbH
AOMs 80 AOM 80(30) MHz, 700-1100nm, 1×2mm2 active aperture, >2W RF MT80-B30A1-IR AA opto-electronic
110 AOM 110(50) MHz, 700-1100nm, 1×2mm2 active aperture, >2W RF MT110-B50A1-IR AA opto-electronic
200 AOM 200(50) MHz, 700-900nm, 1×2mm2 active aperture, >2W RF MT200-B50A1-800 AA opto-electronic
Other Optical fibres Single-mode polarisation-maintaining fibres P3-630PM-FC Thorlabs, Inc
CoSy K Fibre-coupled spectroscopy CoSy TEM Messtechnik AG
CoSy Rb Fibre-coupled spectroscopy CoSy TEM Messtechnik AG
106 quasicrystalline optical lattices for ultracold atoms
Vacuum
part description part number manufacturer
Vacuum chamber Custom, mostly 316LN (MOT, 20 cm diff. pumping section  10mm, 2 TSPs, knee) - PiNK GmbH
Small ion pump Vacion plus 75 l/s starcell pump 9191440 Agilent Technologies LDA
Large ion pump Vacion plus 150 l/s starcell pump with installed heaters 220V 9191542 Agilent Technologies LDA
Titanium sublimation pump TSP filament cartridge 9160050 Agilent Technologies LDA
DN16CF bellow Bellow between 2D MOT and 3D MOT MEW0750251C1 Kurt J. Lesker
DN16CF straight valve Valve between 2D MOT and 3D MOT 48124-CE01-0001 VAT Vacuum Products Ltd
DN16CF angled valve Valve for potassium and rubidium ampoules 54124-GE02-0001 VAT Vacuum Products Ltd
DN40CF angled valve (high-T) Angled valve for high-temperature bakeout section (near science cell) 57132-GE02-0005 VAT Vacuum Products Ltd
DN40CF angled valve (low-T) Angled valve for low-temperature bakeout (all others) 54132-GE02-0001 VAT Vacuum Products Ltd
Lowest-pressure gauge Pressure gauge for science cell section Barion-extended Vacom GmbH
Standard gauge Pressure gauge for MOT chamber and turbopump BAT40C Vacom GmbH
Non-magnetic viewport DN40CF 316LN steel, viewport used near science cell, BBAR coating 550-1100nm XTEMP-FT Kurt J. Lesker
Standard viewport DN40CF, BBAR coated (two parts broke under vacuum in 2D MOT) VPZL-275BBAR Kurt J. Lesker
Non-magnetic viewport DN40CF BBAR coated, replacement for above VPCF40DUVQ-L-BBAR-316L Vacom GmbH
MOT setup
part description part number manufacturer
2D MOT Vacuum chamber Titanium (grade 2) frame and diff. pumping section (custom made, partly faulty) S2-CF-5045/5044 Vaqtex S.r.l.
Window BK7 optical window, 85mm×50mm, thickness 10mm, AR 760-790nm PL.OF.85.50.760 Pegasus Lasersysteme GmbH
Polariser 25.0mm  linear glass polariser 510-800nm LPVISC100 Thorlabs, Inc
Large λ/4 Mico retarder λ/4 wave plate, 70mm×35mm, 780/767nm RQM7035-C/RQM7036-C Knight optical Ltd
MOT Fibre cluster 2 to 6 port fibre cluster, 767nm and 780nm, incl. single-mode fibres and couplers - Schäfter + Kirchhoff GmbH
λ/4 wave plate Polymer λ/4 wave plate (custom made) WPQ10E-773-SP Thorlabs, Inc
Fluorescence setup Rb filter Laser line filter for 780nm FL05780-10 Thorlabs, Inc
K filter Laser line filter for 766nm 65-116 Edmund optics
list of components 107
Dipole trap laser system
part description part number manufacturer
Laser 1064nm, 55W, master diode and four integrated amplifiers Mephisto MOPA 55W Coherent Inc
Water-water chiller 1000W at 20◦C (operating) and 15◦C (house), 5.6 l/min P201-16675-WW-S Termotek AG
Coolant Cooling mixture P/N 1607-0546L Nalco® CCL100 Nalco
Optical isolator 1064nm free-space isolator IO-5-1064-HP Thorlabs, Inc
Mirror (M45-1064) High-power mirror, 45◦ s+p R> 99.6%, 500kW/cm2 M1064 Lens-Optics GmbH
Wave plate (W2-1064) λ/2 0th order wave plate W2Z15-1064 Lens-Optics GmbH
Cube (PBS-1064) 1/2 inch polarising beam splitter PBC12-1064 HP Lens-Optics GmbH
AOM 80MHz, 1064nm, 1×2mm active aperture, >2W RF MT80-A1-1064 AA opto-electronic
Fibre Photonic crystal fibre, SMA connector, 5m, E324-443-500 aeroGUIDE POWER-15-PM NKT photonics A/S
Fibre coupling lens f = 18.40 mm, mounted geltech aspheric lens, AR 1050-1700nm C280TMD-C Thorlabs, Inc
Polariser Glan-Laser polariser, 10mm diameter CA, 1064nm V-coating GL10-C26 Thorlabs, Inc
Photodiode InGaAs amplified detector 900-1700nm PDA10CS-EC Thorlabs, Inc
Cylindrical lens f = −30 mm, C-coating 1050-1700nm LK1982L1-C Thorlabs, Inc
Cylindrical lens f = 150 mm, C-coating 1050-1700nm LJ1934L1-C Thorlabs, Inc
Lens f = −100 mm, C-coating 1050-1700nm LC1120-C Thorlabs, Inc
Lens f = 125 mm, C-coating 1050-1700nm LA1986-C Thorlabs, Inc
Piezo mirror mount 1 inch, Picomotor™ piezo clear edge center mount, 2-axis 8821 Newport Spectra-Physics Ltd
Lattice laser system
part description part number manufacturer
Pump laser 532nm, 25W, frequency-doubled laser Millennia EV20S Spectra-Physics (Newport)
Ti:Sapph laser 700-790nm, 5.5-8.0W (in this wavelength range, at 25W pump) Matisse 2 TS Sirah Lasertechnik GmbH
Water/air chiller Chiller used for Millennia and Ti:Sapph crystal, typ. 20.0◦C, 20psi 6000 Series PolyScience
Coolant Cooling mixture P/N 1607-0546L Nalco® CCL100 Nalco
Mirror (M45) Laser mirror HR 630-900 nm/0-45◦ s+p, R>99.6% HR 630-900nm Lens-Optics GmbH
Wave plate (W2-730) λ/2 0th order wave plate W2Z15-730/W2Z24-730 Lens-Optics GmbH
Cube (PBS HP) 1/2 inch polarising beam splitter PBC12-730-HP Lens-Optics GmbH
Optical isolator Linos Faraday isolator FI-730-5SV Qioptiq Photonics GmbH & Co KG
AOM 110MHz, 700-1100nm, 1×2mm2 active aperture, >2W RF MT110-A1-IR AA opto-electronic
Fibre FC/APC to FC/PC connector (custom-made) TS1754306-PM630-HP Thorlabs, Inc
Fibre coupler (F8) f = 8.1 mm APC, 630-980nm (fibre in-coupling) 60FC-4-M8-10 Schäfter-Kirchhoff GmbH
Fibre coupler (FP5) f = 5.1 mm PC, 630-980nm (fibre out-coupling DYX axes) 60FC-0-M5-10 Schäfter-Kirchhoff GmbH
Fibre coupler (FP15) f = 15 mm PC, 600-1050nm (fibre out-coupling T axis) 60FC-0-M15-02 Schäfter-Kirchhoff GmbH
Photodiode Si amplified detector, 350-1100nm PDA36A-EC Thorlabs, Inc
Cylindrical lens (L -40 cyl) f = −40 mm, N-BK7 B-coated cylindrical lens LK1283L1-B Thorlabs, Inc
Cylindrical lens (L 100 cyl) f = 100 mm, N-BK7 B-coated cylindrical lens LJ1567L1-B Thorlabs, Inc
Achromat (LA100) 1 inch, f = 100 mm, B-coated achromatic lens (DT axes) AC254-100-B Thorlabs, Inc
Achromat (LA125) 1 inch, f = 125 mm, B-coated achromatic lens (YX axes, shared with dipole) AC254-125-B Thorlabs, Inc
Achromat (LA400) 1 inch, f = 400 mm, B-coated achromatic lens (T axis) AC254-400-B Thorlabs, Inc
Achromats (LA XX) 1 inch, f =XX mm, B-coated achromatic lens (telescopes) AC254-XX-B Thorlabs, Inc
Piezo mirror mount 1 inch, Picomotor™ piezo clear edge center mount, 2-axis 8821 Newport Spectra-Physics Ltd.
Dichroic mirror (M-DICH) 40mm , HR725-780nm HT1064nm/45◦ (used on YX axes to split off dipole) B-14749/B-14687 S-00929 Laseroptik GmbH (Garbsen)
108 quasicrystalline optical lattices for ultracold atoms
Miscellaneous
part description part number manufacturer
Cameras Andor Zyla Z axis camera, sCMOS 6.5 µm pixel size, 0.5◦ wedged window, 780nm ZYLA-5.5-USB3-9WT Andor Technologies
Manta Imaging on D axis, CMOS 5.86 µm pixel size, Gigabit Ethernet AV Manta G-235B POE RCG Allied Vision/Stemmer Imaging
IDS uEye Imaging on YXT axes, MOT fluorescence, CMOS 4.5 µm pixel size, USB2.0, global shutter UI-1252LE-M IDS Imaging Development Systems GmbH
Beam cam Beam profiling applications, CMOS 4.5 µm pixel size, USB2.0, rolling shutter UI-1542-LE-M IDS Imaging Development Systems GmbH
Timing PCIe cards 4× X series multifunction DAQ card, 32 AI, 48 DIO, 4 AO, 1.25MS/s NI PCIe-6353 National Instruments
PC Black 4U case, 250GB SSD, Intel Core i5-6500 (3.6GHz), 8GB DDR4, Asus intel LGA1151 various various/World of Computers Ltd
Offset lock Photodiode High-bandwidth MSM photodetector G4176-03 Hamamatsu
Reference oscillator 7.500GHz, 50Ω SMA, custom made KU LO 7500 RO-587 Kuhne electronic GmbH
Amplifier with Bias-T 23dB 1-7000MHz, 50Ω SMA (rubidium offset, used directly after PD) KU LNA BB 0180 A-SMA-BT Kuhne electronic GmbH
Amplifier 23dB 1-7000MHz, 50Ω SMA (rubidium offset) KU LNA BB 0180 A-SMA Kuhne electronic GmbH
Amplifier 20dB 5-1500MHz, 50Ω SMA (rubidium offset, used after mixer) KU LNA BB 0515 A-SMA Kuhne electronic GmbH
MW setup Power amplifier 6400-7200MHz, 40dBm max. output power KU PA 640720-10 A Kuhne electronic GmbH
Reference oscillator Programmable oscillator unit MKU LO 8-13 PLL Kuhne electronic GmbH
Circulator 4.5-9.0GHz, 3-port circulator CS-20 MCLI MW communications laboratories, Inc
Antenna Horn-antenna WG14(WG137), standard gain 14240-20 UAR70 Flann Microwave Ltd
Elements Rubidium Rubidium, 99.75% (metals basis) 044214 Alfa Aesar
Potassium Potassium-40 enriched 2.98% from potassium chloride, 200mg - -
Acknowledgements
‘Experimental science is teamwork.’ I cannot imagine a project for
which this statement could be more appropriate than ours. In the
past three years I have had the great pleasure and honour to work
in a team of clever, supportive, creative, curious, and challenging1 1 ‘But why?’
minds.
This team was brought together and led by our supervisor
Ulrich Schneider, who managed to convince everyone of us to em-
bark on a long journey into the unknown and build a quasicrystal
with ultracold atoms. For me, his deep knowledge of physics has
been a never-ending source of inspiration and information. I am
very grateful for his continuous support and constant availability
to answer questions, even in busy weeks, be it in the office or on
the phone, at any time of day or night.
The first members of the team were Matteo Sbroscia, Oliver Brix,
Hendrik von Raven, and myself.
Matteo has been the central figure during my PhD. It is not
easy to find words that adequately express my feelings towards
him, except simply to state that I am extraordinarily grateful for
his presence, both as friend and colleague. I could fill pages with
anecdotes but still fail to do justice to the role he played in the
past three years. Therefore I am going to borrow some phrases
from a famous Cavendish alumnus who advised, ‘Work with a
teammate who is your intellectual equal’ and, ‘[. . . ] bring together
the complementary talents of those involved.’ While I seriously
doubt that I have been, or ever will be, Matteo’s intellectual equal,
I think we did an O.K. job of profiting from each other’s talents.2 2 his certainly being the more glam-
orousOliver was part of the duo, with Hendrik, that had joined us
from LMU for a year’s Master’s project. He won the black belt in
the disciplines of Autodesk Inventor and Mathematica, developing
our magnetic transport system and designing the cooling block.
Scientists often dismiss supernatural abilities and black magic
for not being compatible with the ‘laws of nature’,3 but Hendrik’s 3 if there is such a thing
coding abilities certainly fall into these categories. He single-
handedly built our timing system while at the same time plan-
ning, developing, and testing one of the most-relied-on compo-
nents in the lab, the DDS board.
Ed Carter joined within the first year of this project, building
a phase-stable interferometer for his Part III project. His arrival
marked a new era, him being the first truly English person in
the group which consisted, at the time, of four Germans and one
110 quasicrystalline optical lattices for ultracold atoms
Italian.4 Ed introduced wit and good humour, which, had it not4 and, rumour has it, two British
passports been for Matteo, would have come as a complete novelty to us.
Later, having started his PhD in the group, he quickly advanced to
playing a vital role in the lab, leading the day-to-day activities and
taking the next generation of students under his wing. Among
many other accomplishments, Ed perfected the art of writing
sequences and lab book entries.5 I am also grateful for him taking5 which sometimes read like crime
novels over my position on the grad student committee.
In the year 2016 we were joined by Max Melchner (from ETH)
and Michael Höse (from LMU) for half-year internships. Max
designed our dipole trap laser system and performed useful nu-
merical simulations of the adiabatic loading process of the dipole
trap. I am grateful for his mathematica scripts, including the trap
frequency calculations. Michael, who – by complete coincidence
– moved into the same flat share that Oliver had lived in, devel-
oped several useful GUIs, completed and characterised our beam
shutter design, and implemented the microwave setup.
Jr-Chiun Yu started his PhD one year ago and he quickly ab-
sorbed a huge amount of information on how to run the machine.
His contributions to the experiments are now indispensable and
I am confident Jr-Chiun is going to ensure a bright future of the
quasicrystal lab.
In the course of the experiment, we were joined by a num-
ber of hard-working project students, namely Callum Stevens,
Joannis Koepsell, Attila Szabó, Robin Croft,6 Shreya Arya, Marius6 who implemented an evolutionary
algorithm that we use to optimise RF
evaporation sequences
Weber, and Elvinas Ribinskas.
One year ago, the optical lattice group expanded and qua-
sicrystal’s sibling, the Kagome lab, was born. Kagome boasts a
fantastic team with Tiffany Harte, Max Melchner, Dan Reed, and
Florian Störtz, as well as their former project students Péter
Juhász, Greg Chant, and Luca Donini. Now the older sibling is
constantly fretting that the younger one will do everything much
better and more efficient than herself. Somehow it feels like ‘Team
Kagome’ has been around since day one, and I am grateful for all
their contributions to the group, regarding both physics and life
outside the lab.
The optical lattice group is embedded in a wider Cavendish-
cold-atom community with a strongly collaborative atmosphere.
This is thanks to Zoran Hadzibabic and his team, namely
Rob Smith, Christoph Eigen, Jay Man, Jake Glidden, Jinyi Zhang,
Panos Christodoulou, Julian Schmitt, Lena Bartha, Timon Hilker,
Maciej Galka, Milan Kristajic, and Nathaniel Vilas. I am particu-
larly indebted to two former postdocs,7 Nir Navon and7 the lack of postdocs in the quasicrys-
tal lab was partly compensated by an
abundant supply in Zoran’s group
Raphael Lopes, as well as Zoran & Rob, for their continuous sup-
port and advice while setting up the apparatus. I owe Zoran a
great deal, both professionally and personally, and I am grateful
for having been infected by his natural enthusiasm for science.
Besides, I believe he is one of the select few8 who truly share my8 together with Nir, and Richard
Fletcher limitless appreciation for Trinity’s Full English.
Next, I would like to express thanks to Andreas Nunnenkamp,
acknowledgements 111
Tijmen Euser, and Mete Atatüre and his team for their support
during my time at the Cavendish.
I am also grateful for the invaluable contributions to our
lab by the Cavendish support staff, including Stephen Topliss,
Dan Cross, Marzena Kulma, Graham Cox, Alan Turner, Dave
Smith, Huw Prytherch, Nigel Palfrey, Kevin Mott, Gary Large,
Chris Summerfield, and others. In addition, I would like to thank
Pam Smith and the Rutherford Hub for doing a great job in ad-
ministration.
Let’s face it, setting up a new laboratory is a mixed bag. It can
be an overwhelming, enjoyable, sometimes frustrating, sometimes
rewarding, daunting, both stressful and exhilerating9 task, but, 9 The emotions while seeing the first
MOT and the first BEC in the lab could
be described as a mixture of childish
joy and what I imagine zero-g must
feel like.
more than anything else, it is time-consuming. I am deeply grate-
ful to Matteo and our colleagues for sharing so many hours at
the lab.10 Despite those long hours, life outside the lab did exist.
10 The total number of hours spent
together is comparable to the duration
of a typical mission to the ISS.
Unfortunately, naming all those wonderful people who made my
life worth living (and will hopefully continue to do so) would
probably lead to a very long and still incomplete list, so I am not
going to attempt it. There must be a few exceptions, though. First
and most important, I would like to thank my family, my parents
Victoria and Christoph, my brothers Johannes and Emanuel, as
well as Emanuel’s wife Mai and their boys Samuel and Adrian.
Emanuel’s lot spent half a year in Cambridge, six wonderful
months that I will cherish for the rest of my life.
Figure A.1: Typical diffraction-limited
image of two single bosons in their
excited state.
112 quasicrystalline optical lattices for ultracold atoms
Then, had Mirjam not saved me, I probably would have de-
volved during my PhD to an undernourished eydeet,11 who uses11 a kind of cave man
all his energy for pondering physics problems and forgets the an-
imate world around him. Thank you, Mirjam, for being there and
being you.
Another two consistent figures shall not go unnamed here:
Daniel Malz and Angela Wittmann. Apart from being brilliant
physicists, Daniel and Angela share my passion for music. And
while I am happy that all three of us completed our PhDs,12 I will12 within weeks of each other
sorely miss their frequent company in the future.
The same goes for my dear friends at 4 Portugal St, Pearce A,
Burrell’s E, as well as the Barnwell lerds, EBC rowing, CUCO
and CUMSSO, cold-atom football, chamber music sessions in
the Frazer room, breakfast at Trinity’s, snooker at WT’s, the BA
society, Pembroke Choir, CMS lunches, lab breakfasts, Trinity
evensongs, astronomy board games, and the dihedral group.
I acknowledge financial support from the Cambridge Common-
wealth, European & International Trust, as well as the EPSRC.
This document was typeset using the tufte-book template1313 tufte-latex.github.io/tufte-
latex/ which is licensed under the Apache License, Version 2.0. I would
like to thank Michael Feuerbacher, as well as Michael Baake and
Uwe Grimm, for providing figures. I am grateful to Anika Bayer
for her Adobe Illustrator skills and her advice on formatting. Fi-
nally, I would like to thank Ulrich, Matteo, Ed, Angela, Daniel,
and Mirjam for proofreading.

114 quasicrystalline optical lattices for ultracold atoms
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