Excitations in Superfluids of Atoms
and Polaritons
Florian Pinsker
St. Edmund’s College, University of Cambridge
This dissertation is submitted for the degree of Doctor of Philosophy
June, 2014
Contents
1 Bose-Einstein condensation theory 14
1.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.1 Particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.2 Macroscopic occupation of the lowest energy level . . . . . . . . . 16
1.1.3 BEC within the formalisms of quantum mechanics . . . . . . . . . 18
1.2 Equilibrium condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.1 Tuneable scattering length a . . . . . . . . . . . . . . . . . . . . . 23
1.2.2 State equations for atomic BEC . . . . . . . . . . . . . . . . . . . 26
1.2.3 Superfluid velocity and streamlines . . . . . . . . . . . . . . . . . 28
1.2.4 State equations for condensates in motion . . . . . . . . . . . . . 29
1.2.5 Critical velocity and Quantum Hydrodynamics . . . . . . . . . . . 31
1.2.6 BEC in lower dimensions . . . . . . . . . . . . . . . . . . . . . . . 33
1.2.7 Excitations in BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.2.8 Energy and mass conservation in GP theory . . . . . . . . . . . . 39
1.2.9 Rigorous mathematical treatment of BEC . . . . . . . . . . . . . 40
1.3 Condensates out of equilibrium . . . . . . . . . . . . . . . . . . . . . . . 44
1.3.1 Exciton-polariton condensates . . . . . . . . . . . . . . . . . . . . 44
1.3.2 A non-equilibrium complex nonlinear Schro¨dinger-type model . . 52
1.4 Spinor condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.4.1 Multicomponent atomic condensates . . . . . . . . . . . . . . . . 56
1.4.2 Spinor polariton condensates . . . . . . . . . . . . . . . . . . . . . 57
2 Analytical properties of a rapidly rotating Bose-Einstein condensate in a
homogeneous trap 62
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.1.1 Rescaled Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3.1 Estimates on the Ginzburg-Landau Functional for 1/ε Ω . . . . 76
2.4 Lower bound for the kinetic energy . . . . . . . . . . . . . . . . . . . . . 84
3 A Nonlinear quantum piston for the controlled generation of vortex rings
and soliton trains 88
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Physical idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2
3.4 Emergence of soliton trains in quasi-one dimensional single component
BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Smooth vs. abrupt change in self-interactions . . . . . . . . . . . 97
3.4.3 Properties of soliton trains . . . . . . . . . . . . . . . . . . . . . . 100
3.4.4 Analytical approximations to the soliton train profiles . . . . . . . 100
3.5 Emergence of soliton trains in quasi-one dimensional two-component BECs103
3.6 Controlled generation of vortex rings and soliton trains in 3D . . . . . . 108
3.6.1 Dynamics of single component BEC . . . . . . . . . . . . . . . . . 109
3.6.2 The two component case . . . . . . . . . . . . . . . . . . . . . . . 111
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.8 Stationary solutions to GPE with step-like coupling parameter . . . . . . 114
3.9 Determining self-interaction strength g of the condensate via the form of
the soliton train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.10 Dark-bright soliton train solutions for two component BEC . . . . . . . . 117
4 Transitions and excitations in a superfluid stream passing small impurities 119
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 1. Asymptotic expansion for a flow around a disk below the criticality . . 121
4.2.1 Critical velocity of nucleation . . . . . . . . . . . . . . . . . . . . 125
4.3 2. Nucleation of excitations: vortices and rarefaction pulses . . . . . . . . 126
4.4 3. Superfluid regimes in the lattice of impurities . . . . . . . . . . . . . . 131
4.4.1 Impurities arranged into regular lattices . . . . . . . . . . . . . . 132
4.4.2 Uniformly distributed impurities . . . . . . . . . . . . . . . . . . . 135
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5 On-demand dark soliton train manipulation in a spinor polariton condensate144
5.1 Main text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.1.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.3 Soliton train generation . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.4 Optical control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.5 Electric control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.6 Polarization control . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.2.1 Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.2.2 Dependencies of the train frequency . . . . . . . . . . . . . . . . . 152
5.2.3 Critical density modulation . . . . . . . . . . . . . . . . . . . . . 152
6 Coupled counterrotating polariton condensates in optically defined annular
potentials 157
6.1 Main Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1.1 Author contributions . . . . . . . . . . . . . . . . . . . . . . . . . 157
3
6.1.2 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.4 Petal-condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.1.5 Power dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2 Mode selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2.1 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2.2 Condensate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2.3 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . 168
6.2.4 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . 168
6.3 Supporting information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3.1 Macroscopic coherence . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3.2 Laguerre-Gauss modes . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3.3 Optical pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3.4 Stability with respect to defects and disorder . . . . . . . . . . . . 172
6.3.5 Power dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3.6 Simulations of the complex Ginzburg-Landau equation . . . . . . 177
6.3.7 Analytic estimate of the relation between number of lobes and
condensate radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.3.8 Polariton energies at different positions . . . . . . . . . . . . . . . 182
6.3.9 Condensate energy and condensation threshold vs radius . . . . . 183
6.3.10 Flexibility of excitation method and variations of the double-ring
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.3.11 Video 1: Sample disorder and lobe orientation . . . . . . . . . . . 188
6.3.12 Video 2: Simulation of the dynamics of a locally disturbed con-
densate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4
Well, I’m beginning to see the light..., The Velvet Underground
5
Declaration
This dissertation describes work undertaken at the Department of Applied Mathematics
and Theoretical Physics at the University of Cambridge. Some of its content has already
been submitted to peer reviewed journals or published. Except where stated otherwise,
this dissertation is the result of my own work and as such contains nothing which is the
outcome of work done in collaboration. This dissertation has not been submitted, in
whole or in part, for any degree or diploma other than that of Doctor of Philosophy at
the University of Cambridge. The dissertation does not exceed the word limit for the
Degree Committee.
Florian Pinsker
Summer 2014
6
Acknowledgements
I have been fortunate to get a scholarship for doing my PhD study at DAMTP starting
in 2010 partly funded by EPSRC and by a KAUST grant. After the first year of courses
on different mathematical subjects within a doctoral training site at DAMTP I was able
to directly start my PhD research. I had the opportunity to work the following three
years on further developing the theory of superfluids with Natasha Berloff as supervisor.
Pursuing my research on Bose-Einstein condensation has been a persistent pleasure and
in particular collaborating with Natasha Berloff, Hugo Flayac, Alex Dreismann, Peter
Cristofolini, Hamid Ohadi, Jesus Sierra, Jakob Yngvason and Jeremy J. Baumberg. The
inspiration I have gained from working with those researchers has essentially impacted
my understanding of the subject under consideration here and as such a great deal
of this thesis has to be credited to them. During my time as a PhD student I was
visiting the recently founded KAUST University in the holy land of Saudi Arabia. For
this possibility I am grateful and for the funding of my research trip by KAUST. The
openness, friendliness and interest I have encountered there was leading to new research
projects and a throughout wonderful visit. I am in particular deeply honored to have
become part of a permanent exhibition in a tea room in Jeddah.
7
Prologue
Our ability to identify entities of our experience with abstract mathematical structures
has been key for the development of ourselves and our modern societies. A physicist
models his/her observations mathematically and by doing so s/he admits that the rules
of his/her logical picture are mathematical axioms, which are consistent with experience
or experiment and universal in the sense that other human beings might share their
meaning. For example when analysing the motion of a planet around a star in the way
Newton suggested it, we are guided to think of a four dimensional continuous space
in such a way that each object within this space has the property of a position at a
given point in time and can be associated with a planet, a star or any other object.
This allows us to relate the planet and the star via their positions in this abstract
space. Furthermore, invoking the concept/axiom of force between objects enables us
to make statements about this relation over periods of times. Thus our mathematical
consideration of reality allows us to extrapolate a statement about reality. If it fits
with experience our mathematical consideration accurately describes the logical form of
reality. Indeed we observe planets to be moving around stars and changing their position
according to the gravitational force acting upon them and in many cases we are able
to predict future behaviors. In general a physical theory might be regarded as settled
once it has shown its reliability and appropriateness in describing aspects of reality and
dated if a more general and/or accurate framework has been developed. The truth
of a physical statement or picture of the world [1] is only determined by its relation to
nature and not by its number of citations [2], the status of the person proposing it or that
one just believes in it. The possibility to extrapolate within a physical theory led to the
observation of a variety of new physical phenomena such as the Aharonov-Bohm effect in
quantum mechanics, where electron beams are affected by an applied magnetic field far
from them and so without any counterpart within classical mechanics or entanglement
of particles. More generally speaking, extrapolation within physical theories led to the
implementations and observations of novel physical states in the laboratory not seen
before on our planet.
Over hundreds of years accessing reality by the means of partial differential equations
(PDEs) has been proven to be remarkably successful. Newton’s classical mechanics,
the second law of thermodynamics or the Maxwell equations are ingenious examples
in physics with tremendous applications in all technical disciplines and implications on
the state of our society. With the rise of wave mechanics in the beginning of the 20th
century new kinds of PDEs emerged such as Schro¨dinger’s equation (SE), which gives
an accurate picture of e.g. the hydrogen atom and the dynamics of quantum mechanical
single-particle processes. Some systems of many particles, however, are very hard to
be computed by the means of Schro¨dinger’s many-body quantum mechanics. Instead
8
it is often more practical (and indeed more accurate when particle numbers vary) to
invoke the general framework of quantum field theory. For fixed particle numbers both
frameworks are identical and it turns out that under certain physical circumstances
effective equations of state functions for particular many-body systems are similar to
Schro¨dinger’s equation describing the state of a single-particle. In contrast to SE these
effective equations in many-body systems now include mathematical adaptions such as
nonlinearities accounting for interactions with other particles or growth and decay terms
that vary the mass of the probability amplitude.
In the course of this thesis we are considering Schro¨dinger-type equations arising as
effective theories in the context of the quantum many-body problem. These are en-
sembles of particles such as atoms or excitations in semiconductors for which quantum
effects are to a certain degree non-negligible, thus traces of the quantum world remain
within the effective state equation describing them. In general these non relativistic
PDEs have first order in time and second order spatial derivatives, while some adap-
tions include first order spatial derivatives or fractional derivatives. Solutions to these
differential equations describe certain kinds of waves within ensembles of particles. Al-
though mathematically similar in structure they can be identified with ensembles of
entities/particles that physically have very different constituents. It turns out that the
microscopic structure of an ensemble often plays a secondary role for the form of the
wave - a phenomenon encountered in many classes of complex systems and referred to
as universality [3]. Quantum particles such as atoms and electrons behave themselves as
waves, each obeying some wavelength. So in the study of waves in many-body systems
the concept of coherence has been widely used describing their temporal and spatial in-
terference. Important examples of coherent waves in physics are lasers or Bose-Einstein
condensates (BEC) of atoms and molecules.
Atomic BEC were first realized in experiment unambiguously in dilute and weakly
interacting Bose gases [4, 5, 6]. The concept of BEC carries at its heart the idea of
coherence as it is indeed a macroscopically occupied quantum state of a single particle
mode, i.e., each particle within the condensate is described by the same spatially ex-
tended wave function and so the ensemble of particles forms a giant matter wave out
of the constituents of the ensemble. By the indistinguishability of its components the
individual meaning of a single particle vanishes in the collective form of the condensate.
Bosons can share the same quantum state and in principle can form such a collective.
In contrast, fermions are prevented to occupy the same quantum state due to Pauli’s
exclusion principle. Some of the species of molecules used in experiments to form BEC,
however, indeed are fermions - by forming quasiparticles due to interactions between
them, some fermions can arrange as new entities that behave as bosons and as such can
condense into the same single-particle quantum mode and thus induce a giant matter
wave in the many-body system. In a proper regime the giant matter wave function
satisfies a nonlinear Schro¨dinger-type equation (NLSE). In the context of Bose-Einstein
condensate theory this wave equation was derived first by E.P. Gross and L.P. Pitaevskii
in order to describe quantum vortices in liquid Helium [7, 8] and now is referred to as
Gross-Pitaevskii equation (GPE). Solutions to the NLSE or GPE include such phe-
nomena as solitons, quantum vortices or quantum vortex rings. These elementary and
9
topologically stable excitations are spatially localized entities that evolve without chang-
ing their form over time. Excitations within the condensate wave function are caused
by an access of energy of the BEC and in conservative systems they persist in time due
to the missing mechanism to release the energy.
The attempt of generating a single occupied mode in a many-body system might be
regarded as being a further logical step in the long tradition of the quest of to purify
elements, which led to such developments as extracting aluminum into its elemental
state in the beginning of the 19th century [9]. Atomic BEC, besides often consisting of a
single kind of atoms, is a purication regarding the states these atoms obey, and as such
is very articial and not encountered on earth apart from laboratories starting in 1995
[4, 5, 6].
Investigations on BEC have been key in understanding quantum matter at low temper-
ature with its distinctive properties [10, 11]. BEC is a macroscopic quantum phenomenon
often involving a great and seemingly unbounded number of interacting coherent quan-
tum particles [11]. A major route of research on BEC has been on applying the concept
of macroscopically occupied many-body states [12, 13] to a great variety of different ma-
terials such as solid-state light-matter systems, atoms and molecules of different species,
photons or classical waves [14, 15, 16, 17]. Although different in nature, once conden-
sation is established these many-body systems often obey similar coherent matter prop-
erties. Important examples are superfluidity, i.e., frictionless flow [18] below a critical
velocity [19] or the unique response to motion via elementary excitations such as quan-
tum vortex rings, quantum vortices and dark or bright solitons [20, 21, 22, 23, 24, 25, 26].
On the other hand do various BECs show specific properties depending on the nature of
their carrying physical system, such as variable inter-particle interactions due to Fesh-
bach resonances within certain external magnetic fields [27] or local particle sources and
sinks in non-equilibrium systems.
An outstanding example are exciton-polaritons, quasiparticles which in some param-
eter regime show Bose-Einstein statistics and for which recently condensation has been
established experimentally [16]. Excitons are coupled electron-hole pairs of oppositely
charged spin-half particles in a semiconductor held together by an effective Coulomb
force between them [28] as the energy to form a pair is lower than a free electron and
a free hole. They interact with light fields [29] and can form quasiparticles themselves,
so called exciton-polaritons (polaritons), in the strong coupling regime when confined to
a micro-cavity [30]. As they possess integer spin they can condense in dilute systems,
despite the fact that some of the elementary constituents are fermions. In particular, as
exciton-polaritons are 109 times lighter than rubidium atoms [16], condensation could be
observed at the Kelvin range in CdTe/CdMgTe micro-cavities or more recently even at
room temperatures in flexible polymer-filled micro-cavities [31], which clearly sets them
apart from atomic Bose-Einstein condensates observed below 200 nano Kelvin [4, 5, 6].
As a consequence for exciton-polariton quasiparticle systems much less technological
effort is required to cool the system below condensation threshold in order to achieve
BEC [16, 31]. The requirement for high vacua and the necessity of very good insulation
are major obstacles for technological applications of atomic BEC, which stands in stark
contrast to the requirements for exciton-polariton condensation and clearly favors the
10
latter for technological devices feasible outside the laboratory.
Another key property of exciton-polaritons is that they are quasiparticles which can be
generated at will by pumping the system with a laser beam (see e.g. [32]). Once created
they obey a finite lifetime of about 1-100ps [28, 33]. This finite lifetime is a consequence
of the leakage of photons through the cavity mirrors covering the semiconductor [28] and
non-radiative dissipative processes [15]. In this sense these BEC are non-equilibrium
systems, which besides being feasible at room temperatures allow very high control over
the experimentally generated condensate wave function [32]. Taking this into account
it is clear that exciton-polariton condensates are excellent candidates for a variety of
technical devices as the plethora of recent proposals shows [34, 35, 36, 37, 38, 39, 40, 41]
of which some already have been implemented in experiments [42, 43, 44]. Here one
very remarkable example of an application for exciton-polariton condensation is that
for room-temperature polariton lasers based on GaN micro-cavities [45, 46]. These
lasers emit coherent and monochromatic light without population inversion - a major
distinction to conventional lasers [46, 15].
From a theoretical viewpoint a successful mean-field model for exciton-polariton con-
densates is a complex Ginzburg Landau-type equation, which induces the time evolution
of the order parameter (condensate wave function) of the system [47, 48, 26, 32] and
which in some scenarios is coupled to a reservoir of non-condensed polaritons [49, 32].
Several recent papers show that these systems of PDE’s provide good results in modeling
exciton-polariton condensates [49, 32], but often a more thorough analysis indicates the
necessity to improve our current models by including more physics, such as the disorder
due to the sample imperfections [50].
Now, the purpose of this thesis is to introduce the reader to the theories of equilibrium
and non-equilibrium BEC described in terms of NLSEs and to present new results within
those frameworks in the context of condensed matter physics. It is clear however that
the results can be relevant for investigations in analogous systems within different fields
due to universality and might have a possible impact for applications and on related
areas as well. My thesis is devided in chapters as follows.
1. In the first chapter I introduce the concept of Bose-Einstein condensation as a
macroscopically occupied state due to Einstein. In the following I present the
Gross-Pitaevskii theory, the basics framework of Bose-Einstein condensation, for
atoms as it is derived from a second quantisation scheme and correspondingly
mathematically rigorous theorems that show the interconnection with the many-
body Schro¨dinger equation. I discuss state-of-the-art theoretical concepts within
the theory of equilibrium condensates such as elementary excitations and the dis-
persion law due to Bogoluibov. Subsequently I explain the meaning of exciton-
polaritons in semi conductor micro cavities in terms of a second quantization model
and their condensation. These quasiparticles when condensed are a prime exam-
ple for non equilibrium BEC. An analog to the atomic Gross-Pitaevskii theory
is presented including growth and decay processes, which is widely used for the
description of exciton-polariton condensates. Finally, I discuss spinor condensates
both for the equilibrium and non equilibrium case.
11
2. I first present a novel rigorous energy theorem within the 2d Gross-Pitaevskii
theory of tightly trapped atoms under rapid rotation based on a collaborative
effort with Jakob Yngvason. The main tools for the mathematical proof are:
• finding a proper wave function minimizing the convex energy functionals
• splitting up the GP energy functional additively into two parts called re-
duced GP functional and a Ginzburg-Landau-type energy functional due to
the latter formal analogy to the theory of Type-I semiconductors
• establishing statements on the wave function via the direct use of the Gross-
Pitaevskii equation, i.e. the equation describing the minimizer. This allows
us to relate the minimizer of the GP equation to the corresponding Thomas
Fermi (TF) density.
• employing a lower bound for the Ginzburg-Landau-type functional developed
in [112] by appropriate rescaling of the functional.
Secondly I present a so far unpublished proof for a novel lower bound to the kinetic
energy of the reduced Gross-Pitaevskii energy functional. It uses results from the
previous proof relating the reduced GP density to the TF density. Furthermore I
apply partial integration in order to express the lower bound to the kinetic energy
solely in terms of the TF density, which is an explicitly given function, hence
yielding the asymptotic lower bound.
3. In chapter III I present results made in an collaborative effort together with
Natasha Berloff and Victor Pe`rez-Garc`ıa for a quantum machine enabling the
generation of solitons and quantum vortex rings in highly elongated harmonically
trapped atomic Bose-Einstein condensates. This proposal is based on the pos-
sibility of locally changing interactions within atomic condensates via applying
a magnetic field utilizing the phenomenon of Feshbach resonances. In addition
we present numerical results as well as novel mathematical expressions for soliton
trains within two component/spinor condensates. Another particular outcome of
this investigation is numerically generated evidence for skyrmions/core vortices
within two component BEC.
4. In chapter IV I present a study of superfluid flow of an atomic condensate passing
small impurities and arrays thereof, which has been contrived in joint work with
Natasha Berloff. In the beginning of this chapter the BEC model is introduced in-
cluding an impurity in relative motion. Then an asymptotic analysis is developed
in the stationary frame yielding the critical velocities for the first generation of vor-
tices in BEC when streaming around the small impurity. Subsequently we present
numerical results supporting the analytical considerations. Finally we extend the
setup of a single impurity in the superfluid’s stream to arrays of impurities. This
generalization gives rise to new phenomena such as different phases of structurally
different excitation generation within such lattices depending in particular on the
lattice constant and the superfluid velocity.
12
5. This chapter is devoted to present results for an effective soliton train generator
within exciton-polariton condensates developed collaboratively with Hugo Flayac.
The main idea is that pumping on a potential step induced via an external electric
field will induce excitations within the condensate due to the induced flow caused
by the difference in potential energy and the incoming stream of pumped particles.
As increased pumping also increases the density of the condensate and, therefore,
the local speed of sound, less solitons are generated within the stream. We show
the very high control over the number of generated solitons via this effect over time
with varying parameters. The excitations are generated as the nonlinearity of the
superfluid’s state equation cannot compensate the local access of energy at the step
and the phase of the wave function starts winding up. Electric control over soliton
generation via varying step function is also demonstrated. Finally we show that
via TM-TE splitting we can generate half -soliton trains in both components which
separate due to the repulsive interactions between the solitons. We believe this
proposal has the potential to be at the heart of polariton semiconductor devices.
6. In the final chapter of this thesis I present the results of a collaboration with ex-
perimenters Alexander Dreismann, Peter Cristofolini, Ryan Balili, Gabriel Christ-
mann, Zaharias Hatzopoulos, Pavlos G. Savvidis, and Jeremy J. Baumberg and
the theoretician Natasha Berloff on ring like exciton-polariton condensates. In an
appropriate parameter regime symmetry breaking within the condensate occurs.
This patter formation has been shown to be feasible within experiments and the
complex GP (cGP) model and the interconnection between both are extensively
discussed. In particular the cGP model predicts the same radius dependence of
the pattern structure (number of lobes) as the experimentally generated conden-
sate does, thereby demonstrating the strong interconnection between theory and
experiment. Furthermore dynamical considerations are made by disturbing the
lobes pattern with a laser beam, which is shown to be in good agreement with the
model.
13
1 Bose-Einstein condensation theory
1.1 Fundamental concepts
1.1.1 Particle statistics
An anomaly at low energy levels
The dawn of research on Bose-Einstein condensation was set with S.N. Bose’s discovery
of a new method for counting occupation numbers of possible states of photons in phase
space, which he developed to derive Planck’s law for the spectral distribution in a cavity
at thermal equilibrium without the need of any reference to classical theory [51, 52].
Subsequently Einstein applied this remarkable way of counting occupation numbers to a
case of massive particles. More specifically he considered the atomic ideal gas [12] made
of entities now classified as bosons, i.e. particles having integer spin. By considering
all possible distributions for these particles in phase space with respect to their defining
properties of indistinguishability and the possibility to occupy the same state more than
once, one gets a formula for the mean number of atoms with an energy Es = cs
2/3 of a
phase space cell s,
ns =
1
eA+
Es
κT − 1
, (1.1.1)
where A, κ, c are constants and T is the temperature of the system. Here the constraint
of fixed particle number n =
∑
s ns = const has to be satisfied as well as the mean
energy has some fixed value due to the condition of thermal equilibrium of the many-
atom system.
For the possible number of particles within a fixed volume V and at a temperature
T he found that particles have to occupy or ”condense” into the lowest energy state
to conserve the total particle number a phenomenon now referred to as Bose-Einstein
condensation. We write
n = n0(T ) + nc(T ), (1.1.2)
and denote the number of atoms in the ground state (GS) by n0. The second term on
the r.h.s. in (1.1.2) must satisfy nc = (cT )
2/3 → 0 as T → 0 [12]. Underlying this
observation was that he presumed all particles can in principle share a common ground
state - a key aspect of bosons. The complementary class of particles that make up our
visible world, fermions, would fill up every free energy cell depending on their energy
just once due to Pauli’s exclusion principle. This introduces a different particle statistic
in phase space and thus a different formula for distributions of particles along energy
levels.
14
Comparison of boson, fermion and Maxwell-Boltzmann statistics
We consider an ensemble of N noninteracting particles of energies εi with i ∈
{1, 2, 3, . . .}, introduce the chemical potential µ of the reservoir with which the sys-
tem is in thermal equilibrium and write β = 1/(kBT ) with kB denoting Boltzmann’s
constant. For bosons in a state i being indistinguishable and able to occupy same en-
ergies the mean occupation number for particles of energy εi follows from the grand
canonical partition function of statistical mechanics. It is given by [10, 11]
〈ni〉bosons = 1
eβ(εi−µ) − 1 , (1.1.3)
where the chemical potential is fixed by
∑
i〈ni〉bosons(µ, β) = N . Similarly for fermions
that are indistinguishable but cannot share the same energy state one obtains
〈ni〉fermions = 1
eβ(εi−µ) + 1
(1.1.4)
and classically (Maxwell-Boltzmann) for distinguishable particles that can occupy the
same energy spot εi we have
〈ni〉classical = 1
eβ(εi−µ)
. (1.1.5)
In Fig. 1.1 we show a comparison of the graphs for the different particle distributions.
In all three cases µ is fixed by the normalization condition due to the fixed number of
particles in closed systems. In particular we see that energy levels for fermions can only
be singly occupied while as εi → µ that for bosons diverges. If the chemical potential
µ were bigger than the lowest energy eigenvalue ε0, the mean occupation number would
become negative for the ground state. Hence, by reductio ad absurdum this case can be
considered unphysical. Once the atoms in excited states are less than the total number
of particles N occupation of the lowest energy state has to be the consequence [53] and
as the temperature decreases lower energy levels are increasingly occupied.
(a)MB
Fermions
Bosons
〈ni〉
εi
(b)
MB
Fermions
Bosons
〈ni〉
εi
Figure 1.1: Occupation numbers vs energy. Parameters are β = 1 in picture (a) β = 3
in picture (b) while µ = 1 in both. As εi → µ with εi > µ the boson
distribution diverges.
15
To distinguish the macroscopically occupied state from the other states one separates
it out of the total particle number and defines
N =
∞∑
i=0
〈ni〉bosons = N0 +NT , (1.1.6)
where we introduced
N0 =
1
eβ(ε0−µ) − 1 (1.1.7)
and NT denotes the remainder, i.e. the thermal component of excited bosons [57] [62].
From (1.1.7) it follows straightforwardly
ε0 − µ = kBT ln
(
1 +
1
N0
)
' kBT
N0
(1.1.8)
relating the key parameters to the ground state.
1.1.2 Macroscopic occupation of the lowest energy level
Critical temperature in 3 spatial dimensions (3d)
Expression (1.1.8) shows that for Bose gases the lowest energy level ε0 becomes increas-
ingly occupied, once the chemical potential µ→ µc = ε0. This is enhanced when the gas
is cooled down below a certain temperature - inevitably a fraction of bosons will occupy
the lowest energy level [12, 54, 11]. Now, for an ideal 3d Bose gas of particles with mass
m confined to a box of volume V the critical temperature is easily obtained by replacing
the sum in (1.1.6) with an integral and by supposing that the thermal particles at the
critical temperature satisfy
NT (Tc, µ = ε0) = N, (1.1.9)
by which we presume that N0 → 0 at the transition temperature Tc and that the excited
states εi are well populated close to the ε0 energy level [53]. One assumes µ = ε0, since
then the number of particles in excited states close to the GS is maximized, which we
set equal to N . Thus it follows [11, 53] that
kBTc =
2pi~2
m
( n
2.612
)2/3
(1.1.10)
with n = N/V and ~ denoting Plank’s constant devided by 2pi. Below that the number
of particles in the GS with energy ε0 is [11]
N0(T ) = N
(
1−
(
T
Tc
)3/2)
. (1.1.11)
Applying the same setup of particles in a box leading to a critical temperature for 1d
or 2d would fail, since for that case Tc would have to be zero [53]. However, trapping
16
Figure 1.2: BEC at different stages/temperatures of the cooling process [54].
of bosons in a harmonic trap allows a finite critical temperature to occur for 1d and 2d
systems as well as discussed in more detail later. To get a rough idea of the particle
numbers within different samples of BEC we note that they can vary significantly from
N ' 103 − 1010 in dilute systems [10, 4, 5, 6] to N ' 1023 in chemical samples up to
N ≥ 1045 in hypothetical boson stars [55]. Among the bosons for which BEC has been
unambiguously demonstrated are 87Rb, 85Rb, 23Na, 7Li [10] and 6Li [56] or quasiparticles
in semiconductor micro cavities such as polaritons [16] or photons [14] or classical waves
[17].
Figure 1.3: (a) shows the velocity distribution above the condensation threshold, (b) just
below the condensation threshold and (c) showing an even cooler BEC [5].
17
Formation of a giant matter wave
The thermal de Broglie wavelength, i.e. the mean wavelength of a particle for a free
ideal gas in equilibrium with no internal degrees of freedom at some temperature, say
T , is given by [28]
λdeBroglie =
2pi~√
2pimkBT
, (1.1.12)
with m being the mass of the particle. It increases as the temperature T → 0 in principle
to infinity. As the wave functions of each boson of the gas start to overlap, however,
interactions between particles lead to synchronization of waves and the creation of the
giant matter wave, so the ideal gas approximation becomes inaccurate and a microscopic
theory including interactions becomes relevant. Usually the giant matter waves formed
are confined to a finite area in space, due to magnetic trapping for atomic BEC or the
finite sized semiconductor micro cavities in which polariton condensates exist, thereby
imposing a constraint on the maximal wavelength. Fig. 1.2 illustrates the stages of
the formation of a giant matter wave as the condensate cools down below the critical
temperature. As the wave function extends spatially its momentum p distribution of all
particles narrows due to de-Broglies postulate λ = h/p for matter waves. In Fig. 1.3
experimental data for the transition to BEC is shown in terms of a narrowing of the
momentum distribution. The momentum of the measured particles is tightly located at
very low temperatures as expected for the formation of BEC and proofing both spatial
coherence of the particle waves and macroscopic occupation of a single particle mode
[5], both properties now regarded as key requirements for BEC [10].
1.1.3 BEC within the formalisms of quantum mechanics
After reviewing the basics for BEC we introduce the reader in this section to the the-
oretical formalism of many-body quantum field theory to describe in those terms the
phenomenon of macroscopically occupied states and to enable us to discuss interact-
ing systems. A key concept is the field operator of the quantum many-body system.
Knowing its form exactly corresponds to having the most complete knowledge about the
many-body system.
Annihilation and creation operators
We consider N identical separable quantum particles each described by a wave function
ψi ∈ H1 with i ∈ {1, . . . , N}. N -particle states are vectors in a Hilbert space
HN = H1 ⊗H1 ⊗ . . .⊗H1︸ ︷︷ ︸
n times
, (1.1.13)
which is spanned by states |ψ1, ψ2, . . . , ψN〉 with ψi ∈ H1 where i ∈ {1, 2, . . . , N}. To
include the possibility of particle creation and annihilation the Fock space is introduced
as
F(H1) = C⊕
⊕
N≥1
HN(R3n, d~x1, . . . , d~xN), (1.1.14)
18
where the so called Fock vacuum is given by Ω = (1, 0, 0, . . .). Generally, vectors of
F(H1) are sequences
Ψ = {ψi}i≥0 (1.1.15)
with ψi ∈ Hi and must satisfy
‖Ψ‖2 =
∑
n∈N0
‖ψn‖2 <∞. (1.1.16)
The inner product of states in F(H1) is defined as
〈Ψ,Φ〉 =
∑
i∈N0
〈ψi, φi〉, (1.1.17)
so the Fock space has the structure of a Hilbert space. In this work we consider
|ψ1, . . . , ψN〉 ∈ HN = L2(~x1, . . . , ~xN), i.e., many-body states dependent on positions
~x1, . . . ∈ Rd and in chapter 8 include the spin degree of freedom |ψ1, . . . , ψN〉 ∈ HN =
L2(~x1, s1, . . . , ~xN , sN). The Hilbert space HN can be divided into a symmetric part HsN
and an asymmetric part HaN with respect to permutations of the single particle states.
Correspondingly the Fock space F(H1) can be divided into F(Hs1) and F(Ha1). As we
consider bosons throughout this work, we restrict our consideration to the symmetric
part HsN . We write for multiple occupied single-particle states
|n1, . . . , n′N〉 = |ψ1, . . . , ψ1, ψ2, ψ2, . . . , ψN ′〉. (1.1.18)
In this terminology one defines the creation operator for an additional boson as [57],
ψiaˆ
†
i |n0, n1, . . . , ni, . . .〉 =
√
ni + 1|n0, n1, . . . , ni + 1, . . .〉 (1.1.19)
mapping from HN → HN+1 and the annihilation operator as
ψiaˆi|n0, n1, . . . , ni, . . .〉 = √ni|n0, n1, . . . , ni − 1, . . .〉 (1.1.20)
mapping HN → HN−1. Here ni are the eigenvalues of nˆi = |ψi|2aˆ†i aˆi that give the
numbers of bosons in state i. We write synonymously ψiaˆ
†
i = aˆ
†
i (ψi). An important
feature is that on a symmetric Fock space the commutation relations of the creation and
annihilation operators satisfy canonical commutation relations (CCR) [11],
[aˆi, aˆ
†
j] = δij, [aˆi, aˆj] = 0 = [aˆ
†
i , aˆ
†
j]. (1.1.21)
On the other hand Fermions do satisfy canonical anti-commutation relations. Employing
those terms we introduce the field operator as a superposition of states of different
particle numbers [10, 11]
Ψˆ†N =
N∑
i
ψiaˆ
†
i . (1.1.22)
These field operators act on the Fock vacuum by
Ψˆ†N |Ω〉 = |ψ1, . . . , ψN〉 (1.1.23)
19
The creation operators within the field operators represent a linear combination of single
particle states corresponding to an ONB in HsN ⊆ F(Hs1), i.e.
aˆ†i |Ω〉 = (0, 0, . . . , 0, 1︸︷︷︸
i’s position
, 0, . . .), (1.1.24)
so that
Ψˆ†N |Ω〉 = (ψ1, 0, 0 . . .) + (0, ψ2, 0 . . .) = |ψ1, . . . , ψN〉. (1.1.25)
In this sense ψiaˆ
†
i (ψiaˆi) describe the creation (annihilation) operators of the single
particle mode ψi at position i in a Fock space vector. It is clear that strictly speaking
an infinite sequence in Fock space is not equal to a vector in Hilbert space, but from a
physicists view the content of information about the physical system is equivalent and
we simply write (. . . , ψN) = | . . . , ψN〉.
In its most general form the field operators Ψˆ†N acting on the vacuum |Ω〉 gives the N
particle state. For fixed particle numbers it is sometimes more convenient to consider
the wave function (Schro¨dinger field) ψ(~x1, . . . , ~xN) for an N -particle state instead of
a field operator acting on Fock space. This is possible as in the case of fixed particle
numbers both frameworks are equivalent and the distinction is only due to the different
notations. Here the Schro¨dinger field gives the probability amplitude |ψ|2 of an ensemble
being at some point in R3N phase space and ψ ∈ HN . In the course of this thesis the
reader will encounter both notions used when appropriate.
Macroscopic occupation of a single mode
Multiple occupation of a single particle state, say M particles are in a state f and the
remainder in states ψi, means that our field operator can be written as
Ψˆ†N =
N−M−1∑
i=0
ψiaˆ
†
i +
N∑
j=N−M
faˆ†j. (1.1.26)
Acting on the Fock vacuum we have
Ψˆ†N |Ω〉 = |ψ1, . . . , ψN−M−1, f, . . . , f〉 (1.1.27)
So essentially the Hilbert space
⊗M H1 of multiple occupied states is equivalent to a
single particle Hilbert space H1, since each entry of the M dimensional vector is the
same.
In terms of the above quantum field operators one defines the one-body density matrix
n(1)(~r, ~r∗) = 〈Ψˆ†N(~r)ΨˆN(~r∗)〉. (1.1.28)
This definition implies the special case n(~r) = 〈Ψˆ†N(~r)ΨˆN(~r)〉, which is referred to as
the diagonal density of the system [11] or the one-particle density. Furthermore the
corresponding eigenvalue equation of the one-body density matrix is∫
d~r∗n(1)(~r, ~r∗)ψi(~r∗) = niψi(~r) (1.1.29)
20
and its solutions ψi(~r
′) are called the single particle modes of the condensate and provide
an ONB in the corresponding Hilbert space [11]. Consequently in its diagonalized form
the one-body density matrix can be written as
n(1)(~r, ~r∗) =
∑
i
niψi(~r∗)ψi(~r), (1.1.30)
where ni denote the occupation numbers of the single particle modes ψi(~r). These
occupation numbers correspond to field operators given by
Ψˆ†N =
n0−1∑
i=0
ψ0aˆ
†
i +
∑
j=1,N ′
nj+1−1∑
i=n0
ψj aˆ
†
i . (1.1.31)
We assume aˆ†i (ψi) to provide an ONB in HsN ⊆ F(Hs1). The quantum field operator
(1.1.31) acts on the Fock vacuum by
Ψˆ†N |Ω〉 = |ψ0, . . . , ψ0, ψ1, . . .〉 = |n0, n1, . . . , nN ′〉 (1.1.32)
with N ′ ≤ N . So different modes j are occupied with multiplicity nj.
Complete Bose-Einstein condensation
Now in terms of (1.1.30) complete (100%) Bose-Einstein condensation is achieved in a
many-body system, iff
n(1)(~r, ~r∗) = n0ψ0(~r∗)ψ0(~r), (1.1.33)
i.e. n0 = N or likewise in the Fock space vector notation
Ψˆ†N |Ω〉 = |ψ0, ψ0, . . . , ψ0〉 = |N〉. (1.1.34)
Here ψ0(~r) is referred to as condensate wave function [11, 58], which inherits the complete
information about the many-body system. It shows the huge reduction of mathematical
complexity of the multidimensional system of e.g. N ∼ 5 × 105 particles in dilute
gaseous BEC [5], which effectively can be regarded a one-dimensional system described
by a single wave function ψ0 in H1. Generally, if there is a c > 0, so
n0
N
≥ c (1.1.35)
for all large N one has BEC [13], i.e. even when other states are occupied too.
According to C.N. Yang [59, 60, 10] an additional necessary criterion for BEC is the
need to have off-diagonal long-range order (ODLRO), where the one-body density matrix
does not vanish at large distances, i.e. when
lim
|~r|→∞
n(1)(~r, 0) = lim
N→∞
n0
V
> 0. (1.1.36)
21
For example for the simple condensate wave function ψ0 = c with c being a constant
one has ODLRO [59], which corresponds to the picture of an spatially highly extended
condensate wave function.
Once we have macroscopic occupation of a single mode ψ0, i.e. when n0  1 one can
write for the occupation number operator
〈ψ|aˆ†(ψ0)aˆ(ψ0)|ψ〉 = n0〈ψ|ψ〉 = 〈ψ|aˆ(ψ0)aˆ†(ψ0)− 1|ψ〉 ' 〈ψ|aˆ(ψ0)aˆ†(ψ0)|ψ〉. (1.1.37)
Here the non-commutativity has been neglected as an approximation in the final step,
which is fairly reasonable for N ≥ 103 particles in the 0 mode. So for a macroscopic
population of a single particle mode one can introduce the Bogoliubov approximation
[11] and we write for the creation operator of a BEC mode aˆ†(ψ0) =
√
n0ψ0 = aˆ(ψ0)
corresponding to neglecting the CCR satisfied by the creation and annihilation operators.
By such an approximation fluctuations of quantum numbers of the different modes are
assumed to be irrelevant for the description of the condensate.
22
1.2 Equilibrium condensates
Bose-Einstein condensates in equilibrium have a fixed number of particles. Such systems
include condensed rubidium atoms (87Rb) trapped in an electric field [4], where the gain
and drain of particles both are zero. When confined to a finite volume most bosons
interact with each other and alter their state functions. For separable bosons there exist
a great variety of interaction processes and potentials describing their nature [61, 11, 53].
Once particles have interacted and moved far away from each other, however, often the
details of the interaction can be neglected - a simplification widely used in scattering
theory, where only relations between incoming states and outcoming states are made.
In dilute quantum many-body systems at low temperature particles are in slow motion
and only occasionally interact with each other. So for short ranged interactions between
particles details of the interactions indeed can be neglected to understand the whole
system to a satisfying degree. Indeed in cold atomic vapors the mean particle separation
is of order 102nm while the length scales of atom-atom interactions is one order of
magnitude smaller [53]. Simplified interaction potentials, i.e., without considering all
details of the scattering process, are a first step towards understanding the complete
interacting many-body system. A key concept when these details are less important is
that of the two-particle scattering length [63, 61, 62], which we introduce in the beginning
of this section.
1.2.1 Tuneable scattering length a
The concept of scattering length a
We consider two slowly and freely moving quantum particles, which scatter due to a
short ranged scattering potential, say v. Furthermore they scatter elastically, i.e. have
unchanged internal degrees after the collision. As a simple model we take the two-body
Schro¨dinger equation, which imposes the time evolution of the wave function [61],
i~∂tψ(~r1, ~r2, t) = Hˆψ(~r1, ~r2, t), (1.2.1)
where ψ(~r1, ~r2, t) is the two-body wave function. We assume ~ri ∈ R3 with i ∈ {1, 2}.
The Hamiltonian for two colliding particles is
Hˆ =
~p2
2M
+ V (~r), (1.2.2)
where we introduce the relative coordinate ~r = ~r1 − ~r2 and ~p = ~p1 − ~p2, and the
reduced mass M = m1m2/(m1 + m2). Long time after the collision of the particles
one will get time independent states. These correspond to wave functions of the form
ψ(~r, t) = e−itEn/~ψ(~r), so (1.2.1) becomes
Hˆψ(~r) = Enψ(~r), (1.2.3)
where En is the corresponding nth eigenvalue of an eigenfunction ψ. Freely moving
particles which move along z-direction are of the form ψ(z) = eikzz with ~r = (x, z).
23
The scattered particle is asymptotically for large distances described by a wave function
including a term with some information of the scattering process [61, 53]
ψ ' eikzz + f(θ)e
i~k~r
r
, (1.2.4)
where f(θ) is the so called scattering amplitude for which one assumes spherically sym-
metric interactions between atoms, hence only dependence on the scattering angle θ.
The scattering amplitude for spherically symmetric potentials, implying axial symmetry
with respect to the direction of the incident particle, can be expanded as [53]
f(r, θ) =
∞∑
l=0
AlRkl(r)Pl(cos(θ)) ' 1
i2k
∞∑
l=0
(2l + 1)(ei2δl − 1)Pl(cos(θ)) = f(θ) (1.2.5)
with l ∈ {0, 1, 2, . . .} corresponding to the s, p, d, . . . partial waves of the scattering
amplitude. The approximation in the second step is due to an asymptotic approximation
of the radial profile, see [53] for details. Here Pl are Legendre polynomials and we have
introduced the phase shifts δl. As a measure of the strength of a scattering process one
introduces the scattering cross section, which under the made assumptions depends only
on the magnitude of ~k [53],
σ(k) = 2pi
∫ ∣∣f(θ)∣∣2 sin(θ)dθ. (1.2.6)
For (isotropic) s-wave scattering, i.e., at low energies, the cross section for indistinguish-
able particles is simply given by [53, 61, 63]
σ(k) = 8pi|f |2 = 8pi
k2
sin2 δ0 (1.2.7)
and in the small k limit on gets
lim
k→0
σ(k) = 8pia2. (1.2.8)
Here we define the scattering length as
a = − lim
k→0
tan δ0(k)
k
. (1.2.9)
In this sense the scattering length a does give the complete information of the elastic s-
wave scattering process, which clearly is an approximation of the interaction process due
to the original potential V (r). On the other hand it can be shown that slowly moving
particles, which scatter, have in the first Born approximation a scattering length given
by [53]
a =
M
2pi~2
∫
d~rV (r), (1.2.10)
24
where V (r) denotes the spherical symmetric interaction potential between the scatterers.
In general a different potential implies a different scattering length, however, apparently
equivalence classes of different potentials with same scattering length are possible. For
two particles with the same mass this implies an effective potential [53]∫
d~rVeff(r) =
4pi~2a
m
= V0, (1.2.11)
that could w.l.o.g. be of the form Veff(r) = V0δ(r), which indeed is a good approx-
imation for short ranged interaction processes [11]. For the scalar scattering length
one can distinguish two cases, that for positive scatterers, a > 0, which corresponds to
predominantly repulsive interactions V and negative scattering lengths corresponding
to predominantly attractive interactions. For a meaningful treatment of the scattering
length in many-body systems it is important to assume a condition for diluteness as then
the exact scattering potentials become less relevant for the many-body processes. An
appropriate conditions for repulsive scattering potentials is that the mean inter-particle
distance has to be much larger than the scattering length a [62]. With attractive scat-
terers many-body systems might collapse, if there is no process forcing the particles
to separate. The main contributor to the effective scattering length in atomic BEC is
the repulsive hard core and the van der Waals interaction, which is due to the electric
dipole-dipole interactions between atoms [53]. Typical scattering lengths are 102 times
the atomic length scale [53].
Feshbach resonances
Scattering processes and their effective interactions can in certain cases be significantly
altered by external parameters, e.g. due to magnetically induced Feshbach resonances
[53]. In such systems the scattering length becomes a function of an external magnetic
field [64, 65, 11, 53]
a(B) = a∞
(
1− ∆B
B −Bres
)
. (1.2.12)
Here B denotes the magnetic field strength, ∆B is the width of the resonance and Bres
its position and a∞ denotes the scattering length far away from the resonance. Feshbach
resonances allow the spatial and temporal control over self-interactions, which as we will
see later will turn out to be employable to generate flows within atomic Bose-Einstein
condensates. Furthermore, the sign of effective atom-atom interactions can be changed
[53], so it allows us to externally switch from condensates with attractive interactions
to a condensate with repulsive interactions and vice versa.
25
1.2.2 State equations for atomic BEC
Time dependent Gross-Pitaevskii equation
In the general framework of second quantization the time evolution of the time dependent
quantum field operators Ψˆ†N(~r, t) is governed by the Heisenberg equation [11]
i~∂tΨˆN(~r, t) =
[
ΨˆN(~r, t), Hˆ(~r, t)
]
(1.2.13)
with the Hamiltonian given by
Hˆ =
∫ (
~2
2m
∇Ψˆ†N∇ΨˆN
)
+
1
2
∫
Ψˆ†N(~r)Ψˆ
†
N(~r∗)V (~r∗ − ~r)ΨˆN(~r)ΨˆN(~r∗)d~r∗d~r. (1.2.14)
To simplify the interaction potential in (1.2.14) one assumes effective highly local inter-
actions between particles within the condensate, so the interaction potential is approxi-
mated by
V (~r − ~r∗) = gδ(~r − ~r∗), (1.2.15)
with interaction strength g = 4pi~
2a
m
and a the scattering length associated with the
condensed particle scattering as derived in the previous section. Using (4.3.1), (1.2.14)
and (1.2.13) we get
i~∂tΨˆN(~r, t) =
[
ΨˆN(~r, t), Hˆ(~r, t)
]
=
=
(
−~
2~∇2
2m
+ Vext(~r, t) +
∫
d~r∗ΨˆN(~r∗)V (~r∗ − ~r)ΨˆN(~r)
)
ΨˆN =
=
(
−~
2~∇2
2m
+ Vext(~r, t) + g|ΨˆN |2
)
ΨˆN . (1.2.16)
By assuming the field operator to be a c-number ΨˆN =
√
n0ψ0 and setting ψ0 =
1√
n0
ψ
the time evolution simplifies to the so called Gross-Pitaevskii equation [8, 7]
i~∂tψ(~r, t) =
(
−~
2~∇2
2m
+ Vext(~r, t) + g|ψ|2
)
ψ(~r, t). (1.2.17)
Here m is the mass of a single particle and g|ψ|2 represents the self-interactions of the
particles with a magnitude g within our condensate. The order parameter (condensate
wave function) is a complex function ψ(~r, t) : R3+1 → C and inherits the density distri-
bution of the condensed particles within the condensate by the expression |ψ|2, similarly
in meaning to usual single particle quantum mechanics. We note that the correspon-
dence to the many-body density matrix will be discussed further in a following section.
A crucial difference to Schro¨dinger’s single-body quantum theory is the factor for the
energy density due to self-interactions, which is
U =
4pi~2a
m
|ψ|2. (1.2.18)
26
It gives rise to nonlinear phenomena unique to BEC. Mathematically one can classify
(1.2.17) as NLSE for a classical field [66]. The energy of the BEC is analogously obtained
as in quantum mechanics by finding the minimizer for the energy functional associated
with (1.2.17),
EGP[ψ] =
∫ (
~2
2m
∣∣∣~∇ψ∣∣∣2 + Vext(~r, t)|ψ|2 + g
2
|ψ|4
)
d~r. (1.2.19)
As we consider here a system of fixed particle number, the minimizing function has
to satisfy the mass constraint ‖ψ‖L2(R3) = 1 in rescaled units. In contrast to linear
quantum mechanics an energy shift is induced via (1.2.18). The PDE (1.2.17) is the
Euler-Lagrange equation for (1.2.19), when including a Lagrange multiplier to respect
the conservation of mass.
Galilean invariance of the GPE
For any solution to (1.2.17) a new solution can be obtained by replacing ~r → ~r+~vt and
by multiplying the wave function with a factor e−i
m
~ ~v(~r+~vt/2), i.e.
ψ(~r, t)→ ψ(~r + ~vt, t)e−im~ ~v(~r+~vt/2). (1.2.20)
This transformation property is analogous to a Galilean boost in linear quantum mechan-
ics and it corresponds to switching the consideration to another relatively moving frame
of reference as will be discussed later in more detail. Indeed the additional nonlinear
term (1.2.18) does not interfere this interpretation.
Time independent GPE
The equation for stationary states can be obtained by restricting the condensate wave
function to be of the form
ψ(~r, t) = φ(~r) exp(−iµt/~), (1.2.21)
with a constant µ associated as chemical potential. By inserting a stationary state into
the time dependent GPE (1.2.17), which implies assuming the external potential to be
time independent as well, Vext(~r, t) = Vext(~r), one gets the stationary GPE
µφ(~r) =
(
−~
2~∇2
2m
+ Vext(~r) + g|φ|2
)
φ(~r). (1.2.22)
Generally the chemical potential µ is fixed by the normalization condition ‖φ‖L2(R3) =
1. This thermodynamical quantity measures the cost of a particle to be added to a
thermally isolated system and the value of µ satisfies the thermodynamical relation
µ = ∂EGP/∂N [11] and as such depends on the total particle number N . In order to
obtain the stationary GPE (1.2.17) via the energy functional (1.2.19) one employs the
technique of Lagrangian multipliers where one sets
E ′GP[φ] = EGP[φ]− µN. (1.2.23)
27
The global minimum is obtained for a vanishing variational derivative
δE ′GP[φ]
δφ∗
= 0, (1.2.24)
and yields (1.2.22) [67]. Finally we note that for the uniform BEC φ =
√
n0 = const.
(1.2.22) simplifies to
µ = gn0. (1.2.25)
Small variations thereof which include the slowly varying potential are referred to as
Thomas-Fermi approximation borrowing the term from the density functional theory of
electrons due to formal similarity.
1.2.3 Superfluid velocity and streamlines
When we apply a Galilean transformation to the condensate wave function in order to
boost into the frame moving with velocity −~v0 and coordinates ~r∗ = ~r+~v0 · t and t′ = t,
we get another order parameter
ψ′(~r∗, t) = ψ(~r∗ − ~v0 · t, t)e
(
im~
(
~v0~r∗−~v
2
0t
2
))
. (1.2.26)
For the simple homogeneous ground state, i.e. when we set ψ =
√
n0 = const. then by
(1.2.26) we have
ψ′(~r∗, t) =
√
n0 exp
(
i
m
~
(
~v0~r∗ − ~v
2
0t
2
))
=
√
n0 exp (iφ0) . (1.2.27)
In the last step we have defined a new function φ0 that is unique up to a multiple of
2pi due to different sheets of the complex exponential. Consequently one obtains the
following relation for the velocity of the condensate
~v0 =
~
m
~∇φ0(~r) (1.2.28)
Now the superfluid velocity ~v is identified by generalizing the relation with the velocity
of uniform flow ~v0 (1.2.28) by writing
~v(~r, t) =
~
m
~∇θ(~r, t), (1.2.29)
with θ : R3+1 → R denoting the phase of an inhomogeneous BEC. According to the
Helmholtz decomposition a general vector field can be decomposed in a sum of an ir-
rotational and a divergence free vector field, i.e., a vector valued function, say ~f , on a
bounded domain Ω ⊂ R3 can be written as ~f = −∇k + ~∇× ~A for appropriately chosen
functions ~A and k [68]. By the definition (1.2.29) the superfluid velocity only obeys the
first part of that sum, which will turn out to have crucial restrictions on its properties.
28
Also note that by using (1.2.29) and (1.2.17) one can write the superfluid velocity in
terms of the condensate wave function
~v(~r, t) = − i~
2m|ψ|2
(
ψ∗(~r, t)~∇ψ(~r, t)− ψ(~r, t)~∇ψ∗(~r, t)
)
. (1.2.30)
As we are going to see in a later section the definitions (1.2.29) and (1.2.30) connect
with a conservation law, the continuity equation for the NLSE.
Furthermore we remark that an alternative interpretation of quantum mechanics,
Bohmian mechanics, derives its law of motion for position-trajectories in phase space
from the corresponding velocity field. It is given by the so called guiding equation
∂tXt(~x) = ~v(t,Xt(~x)) with initial condition X(0, ~x) = ~x ∈ R3. (1.2.31)
Here Xt(~x) denotes a possible trajectory of a quantum particle and the velocity field ~v
is dependent on the wave function ψ of the linear SE in the same way as in (1.2.29) and
(1.2.30). The nonlinearity introduced in (1.2.17), however, would modify the trajectories
to linear quantum mechanics and so would represent streamlines of the possible mean
particle positions.
1.2.4 State equations for condensates in motion
Linear motion
Let us write the time dependent GPE (1.2.17) as
i~∂tψ(~r, t) = O[ψ] · ψ(~r, t) (1.2.32)
and introduce the nonlinear operator
O[ψ] =
(
−~
2~∇2
2m
+ Vext(~r, t) + g|ψ|2
)
. (1.2.33)
We say the condensate is at rest in a frame K, has energy E0 specified by (1.2.19) and
momentum defined as
~p
~
=
∫
n~∇φd~r = m
~
∫
n~vd~r. (1.2.34)
When put in motion with velocity ~v0 in the static frame the energy and momentum of
the moving fluid are simply obtained via a Galilean transformation of the observables
and given by [11, 67]
~p′ = ~p−M~v0 (1.2.35)
E ′ =
|~p′|2
2M
= E0 − ~p · ~v0 + M
2
|~v0|2. (1.2.36)
M = m, as we are working with rescaled units in N , denotes the total mass of the
fluid. On the other hand we can transform the condensate state equation (1.2.32) to the
moving frame by simply using (1.2.26) and obtain
i~
(
∂t +
m
~
~v0 · ~∇′
)
ψ(~r′, t) = O′[ψ′]ψ(~r′, t), (1.2.37)
29
with
O′[ψ′] =
(
−~
2 ~∇′2
2m
+ Vext(~r
′, t) + g|ψ|2
)
. (1.2.38)
Let us define the momentum operator pˆ = −i~~∇, so we can transform (3.8.13) into
(i∂t +
1
2
mv20)ψ(~r
′, t) =
1
2m
(pˆ′ −m~v0)2ψ(~r′, t) (1.2.39)
for a gauge field Aµ = m(−1
2
v20, ~v0) of coupling strength m. In this sense a velocity might
introduce comparable effects as an electromagnetic field would, when introduced to the
system of quantum mechanical equations and vice versa a gauge field of this form will
induce linear motion to the system. We note that the form (3.8.13) of the GP equation
corresponds to the transformation O′(~r′, pˆ′) = O(~r′, pˆ′) + ~v0 · pˆ′.
General rotation
We have seen in the derivation of the GP equation (1.2.17) that an external potential
can depend on time, i.e. Vext(~r, t) with (~r, t) ∈ R3+1. An important special case of
a time-dependent potential is that of a rotating trap. For this case one can remove
time dependence from the potential via transforming to a non inertial coordinate frame
r˜ = r˜(~r, t) with t˜ = t via
r˜i =
∑
j
Tijrj (1.2.40)
for a time-dependent, real and unitary matrix Tij [67] that can be derived from
∂tr˜ = −Ω · (~n× r˜). (1.2.41)
Ω denotes the rotation frequency, ~n the unit vector along the axis of rotation [67]. The
transformation to the rotating frame implies ∆ = ∆˜ and
∂t = ∂˜t + ∂tr˜ · ∂r˜ = ∂t˜ − Ω(~n× r˜)∂r˜. (1.2.42)
Dropping the r˜ and instead writing ~r the GP equation in the rotating frame becomes
i~∂tψ(~r, t) =
(
−~
2~∇2
2m
+ Vext(~r) + g|ψ|2 + iΩ(~n× ~r) · ∇
)
ψ(~r, t). (1.2.43)
This form of the GP equation can analogously be obtained by the transformation
H ′(~r′, ~p′) = H(~r′, ~p′) − ~Ω · ~L(~r′, ~p′) with ~L = (~r′ × ~p′) [61, 69]. Consequently the en-
ergy functional (1.2.19) becomes
EGP[ψ] =
∫ (
~2
2m
∣∣∣~∇ψ∣∣∣2 + Vext(~r)|ψ|2 + g
2
|ψ|4 − ψ∗Ω(~n× ~r) · ∇ψ
)
d~r (1.2.44)
and all variables are defined in the rotating frame [69]. By defining the vector potential
~A =
m
~
Ω(~n× ~r) (1.2.45)
30
and the effective potential that includes the centrifugal potential
Veff (~r) = Vext(~r)− Ω
2r2⊥
2γ
(1.2.46)
with r⊥ denoting the distance to the rotation axis one can rewrite (1.2.44) as
EGP[ψ] =
∫ (
~2
2m
∣∣∣(~∇− i ~A)ψ∣∣∣2 + Veff (~r)|ψ|2 + g
2
|ψ|4
)
d~r. (1.2.47)
There is an inevitable loss of conservation laws for the rotating case, but one retains U(1)
symmetry corresponding to the conservation of mass [67]. Corresponding to (1.2.47) the
Gross-Pitaevskii equation in the rotating frame is
i~∂tψ(~r, t) =
(
− ~
2
2m
(
~∇2 − i ~A
)
+ Veff (~r) + g|ψ|2
)
ψ(~r, t). (1.2.48)
Rotation around an axis nˆ induces a particle flow away from the axis of rotation. So in
order to allow stationary states one has to introduce an appropriate trap Vext(~r) confining
the condensate by opposing the centrifugal force due to rotation. This turns out to be
a key prerequisite for the results presented in chapter 4. We conclude that the effect of
motion on the state equation of the condensate is via additional complex-valued terms
within the nonlinear Hamiltonian, which to some extent resembles effects encountered
in the study of non equilibrium condensates discussed later.
1.2.5 Critical velocity and Quantum Hydrodynamics
Landau’s criterion
In a moving superfluid within a capillary excitations, i.e., deviations from the ground
state, may occur. An excitation with momentum ~p present in the fluid has an energy ε(~p)
and the total energy of the fluid at rest carrying such an excitation is given by E0 +ε(~p).
Let us consider a capillary frame K ′ where the fluid moves with ~v0, or likewise K ′
moves with −~v0 relative to the fluid. In K ′ the energy including an excitation and the
momentum are given by
E ′ = E0 + ε(~p) + ~p · ~v0 + M
2
|v0|2 (1.2.49)
~p∗ = ~p+M~v0. (1.2.50)
The elementary excitation spectrum in the frame of the capillary K ′ has the form [15]
ε′(~p) = ε(~p) + ~p · ~v0. (1.2.51)
Consequently elementary excitations in K ′ are energetically favorable, iff
ε′(~p) < 0, (1.2.52)
31
so that it is thermodynamically favorable to create excitations [15] and superfluidity
vanishes. By the means of this condition one defines the critical velocity
vc = min
~p,∀~p
ε′(~p)
|~p| , (1.2.53)
as a threshold below which no excitations are energetically feasible. The quantity ε
′(~p)
|~p| is
non zero as we will see later, hence establishing a regime of superfluidity. It is important
to note that the self-interactions are essential for the appearance of a superfluid velocity
vc > 0, as the excitation spectrum ε
′(~p) essentially depends on self-interactions. Similarly
to (1.2.53) one defines
Ωc = min
~l,∀~l
ε(~l)
|~l| , (1.2.54)
with ε(~l) being the energy of elementary excitations in the frame of the capillary as
function of the angular momentum of the elementary excitation ~l. For large ~l inducing
rapid variations of the density profile (1.2.54) is expected to fail and the necessity for a
microscopic theory arises.
Quantum Hydrodynamics
We now turn to an equivalent treatment of the condensate wave function, which was in-
troduced in a similar context by Madelung [70, 71] as an analog to the linear Schro¨dinger
equation. The motivation for writing down an alternative set of equations was to pro-
vide an intuitively more easily accessible framework for quantum mechanics at that
time. This set of equations obey a similar form as classical hydrodynamical equations.
So this system is usually referred to as quantum hydrodynamical system (QHD) and
it can be easily extended to nonlinear Schro¨dinger-type equations such as the GPE. To
do this one writes the condensate wave function in polar coordinates ψ =
√
n · eiφ with
φ,
√
n : (~r, t) ∈ R3+1 → R and defines the superfluid velocity as (1.2.29). By inserting
this ansatz in (1.2.17) and separating imaginary from real parts a continuity equation
n˙+ ~∇ ·~j = 0 (1.2.55)
and a Hamilton-Jacobi-type equation
mv˙ + ~∇
(
Vext(~r) + α1n− ~
2
2m
√
n
~∇2√n+ mv
2
2
)
= 0 (1.2.56)
is obtained. Here we have introduced the matter flux
~j =
~
m
(
ψ~∇ψ∗ − ψ∗~∇ψ
)
=
~
m
|ψ|2~∇φ. (1.2.57)
It is connected to the superfluid velocity (1.2.29) via
~j = |ψ|2~v. (1.2.58)
32
Through the hydrodynamical relation (1.2.58) one can alternatively identify the super-
fluid velocity, which complements the method of generalization previously leading to its
definition (1.2.29). Furthermore it is worth mentioning that the continuity equation has
a unique solution with respect to the Cauchy problem [72]. Generally it represents the
conservation of mass for the equilibrium condensate. This holds true as well when a vec-
tor potential is included into the GPE (1.2.17) as for the rotating condensate (1.2.48).
For this case, however, the superfluid velocity has to be extended to
~v =
~
m
(~∇φ− ~A) (1.2.59)
to retain (1.2.55) [67]. Now, the so called quantum pressure term in the Hamilton-
Jacobi-type equation (1.2.56),
~2
2m
√
n
~∇2√n, (1.2.60)
distinguishes QHD from classical hydrodynamics (derived from Newtonian mechanics).
This can be seen by neglecting quantum pressure (via ~→ 0), which yields an Euler-type
equation for a potential flow and zero viscosity fluid,
mv˙ + ~∇
(
Vext(~r) + α1n+
mv2
2
)
= 0 (1.2.61)
for functions satisfying sufficient regularity conditions [74].
It is worth noticing that the nonlinear QHD system is not equivalent to NLSE in the
sense that solutions to the nonlinear QHD do not necessarily provide solutions to the
corresponding NLSE. To gain equivalence the change in phase around any closed contour
has to satisfy ∮
L
∇φ · dl = 2pin (1.2.62)
with n being an integer [75]. This condition corresponds to the single-valuedness of the
condensate wave functions, which is necessary under elementary assumptions [76] and
allows such effects as quantum vortices as discussed subsequently.
1.2.6 BEC in lower dimensions
The concept of Bose-Einstein condensation depends significantly on the spatial dimen-
sions (due to the different energy dependence of the density of states) and is feasible
in quasi 2d and 1d systems, when taking certain adaptions to the setup into account
[77]. It can be shown that for infinitely large uniform Bose Gases in 1d and 2d no Bose-
Einstein condensation can occur at non-zero temperatures. This changes for harmoni-
cally trapped condensates. Applying a semiclassical approximation the total number of
particles of the Bose gas can be written as
N = N0 +
∫ ∞
0
dEρ(E)N
(
E − µ
T
)
(1.2.63)
33
with the density of states given by ρ(E) ∼ Ed−1 due to harmonic confinement and
N(z) = 1/(exp(z) − 1)[77]. After a short calculation one obtains for the number of
bosons in the ground state for quasi 2 spatial dimensions
N2d0 ' N
(
1−
(
T
Tc
)2)
, (1.2.64)
with Tc =
√
6N
pi2
~ω, while for quasi 1d one gets [77]
N1d0 ' N −
(
T
~ω
)
ln
(
T
~ω
)
. (1.2.65)
Both formulas for the occupation number of the lowest energy state show the feasibility
of macroscopic occupation for small enough temperatures. It turns out that the Gross-
Pitaevskii equation in appropriate parameter regimes remains an accurate model, how-
ever changes to the interaction parameter g have to be made. I refer to [77, 62, 11, 80, 78]
for in depth derivations and explicit formulas for the self interaction strength g and mo-
tivate the simple argument as follows.
Gross-Pitaevskii equation in quasi 1d.- One starts with a cylindrically-symmetric trap
geometry V (~r) = (m/2)(ω2zz
2 + ω2rr
2) with the trapping frequencies ωr and ωz and
assumes tight confinement in r direction ωz  ωr to provide a quasi 1d regime. Under
this assumption the radial component of the condensate wave function approaches a
static harmonic oscillator state and can be separated out ψ3d(z, r) = φ(r)·ψ(z). Inserting
in (1.2.19) one obtains
EGP1d =
∫ (
~2
2m
|∂zψ|2 + ~ωr|ψ|2 + Vext(z, t)|ψ|2 + g
4pil2r
|ψ|4
)
dz, (1.2.66)
with lr =
√
~/mωr, which includes an energy shift ~ωr/2 from each r component of
the transverse harmonic oscillator [78]. The minimizer of (1.2.66) satisfies again 1d
GP equation with a modified chemical potential µ1d = µ − ~ωr and can be obtained
analogously as in 3d. The GP equation is the Euler-Lagrange equation of (1.2.66) and
given by
i~∂tψ(z, t) =
(
−~
2∂2z
2m
+ Vext(z, t) +
g
4pil2r
|ψ|2
)
ψ(z, t). (1.2.67)
Depending on the context we will write g → g
4pil2r
for the sake of simplicity. Finally we
note that analogous considerations apply for a 2d description of the condensate wave
function. There have been a variety of successful experimental implementations for lower
dimensional condensates and we refer to [79, 16, 78] for details.
1.2.7 Excitations in BEC
Solutions to the Gross-Pitaevskii equation (1.2.17) are nonlinear waves, i.e., linear com-
binations of solutions cannot be solutions as well. This holds even though the initial
34
many-body problem with which the GPE can be associated is linear. The adaption
introduces certain kinds of new solutions with properties only possible by the presence
of the nonlinearity. Depending on the sign of the cubic nonlinearity in (1.2.17) the
corresponding additional term can balance other terms and thus certain kinds of ener-
getically stable excitations arise such as solitary waves or vortices [80, 67]. However, a
consequence is that a threshold of disturbance is necessary in order to generate these
nonlinear excitations which we discuss now as follows.
Bogoliubov energy spectrum
Previously we have introduced the notion of a critical velocity (1.2.53) below which
no excitations may occur for appropriate dispersion curves of the possible excitations
in the fluid. This phase is referred to as superfluidity and depends significantly on
the nonlinearity within the GPE. To see this we turn to the Gross-Pitaevskii equation
(1.2.17) without external trapping, but including an explicit chemical potential through
ψ → ψ exp(iµ/~t) with µ = n0g, i.e,
i
∂ψ
∂t
= − ~
2
2m
∆ψ + g|ψ|2ψ − µψ. (1.2.68)
For the small amplitude harmonic modes we make the ansatz of the form ψ = φ0 + δψ,
where φ0 represents the unperturbed part solving the GPE and δψ a small perturba-
tion. By inserting this ansatz in (1.2.68) and dropping terms of order δψ2 we get the
Bogoliubov equation for the perturbation
i
∂δψ
∂t
= − ~
2
2m
∆δψ + gn0(δψ + δψ
∗). (1.2.69)
where n0 = |ψ0|2. Assuming δψ = Aei(~k~r−ωt) + B∗e−i(~k~r−ωt) with A,B ∈ C one gets the
Bogoliubov law
ω(k)2 =
~2
2m
n0gk
2 +
(
~2
2m
)2
k4
4
. (1.2.70)
In Fig. 1.4 we plot the dispersion relation (1.2.70) of the excitation spectrum. The
sound-like part of the dispersion (linear dispersion) corresponds to the speed of sound
v2s = ~2/(2m)n0g and implies a finite and nonzero critical velocity (1.2.53), which is
equal the speed of sound.
Solitons, dark or bright
Solitons are localized waves moving without changing their form and obeying elastic
scattering properties [81]. Historically such waves were first observed in a narrow chan-
nel of water, however with less strict assumptions on their behavior when scattering
[82]. On the other hand in the strict sense, none of those excitations discussed in this
thesis will satisfy all the properties to classify them as solitons, but we use the term
‘soliton’ simply because it is done so in most literature. More recently the study of
35
Figure 1.4: The continuous line represents the Bogoliubov dispersion law in rescaled
units, k2 + 2k4, the dotted line the 2k4 contribution corresponding to a free
particle-like dispersion and the dashed line the sound-like dispersion k2.
solitons has reached areas from nonlinear fibre optics [83] to Bose-Einstein condensates
[84]. These physical systems provide exceptional playgrounds for the study of the prop-
erties of nonlinear waves in NLSE systems. For the GPE we distinguish between bright
and dark solitons that are excitations in quantum matter consisting of particles waves
with effectively repulsive or attractive interactions. Usually solitons are stable in highly
elongated geometries, i.e. quasi 1d systems and the (1.2.67) indeed admits solutions,
which can be identified as solitons.
Dark solitons,- Consider the case of repulsive interaction a > 0 and a chemical poten-
tial µ = n0g (as follows from (4.2.26)) in (1.2.67). Then setting ξ = z−vt and supposing
ψ(z, t) = ψ(ξ) a solution for a moving dark soliton is given by [11]
ψds(ξ) =
√
n0
(
i
v
c
+
√
1− v
2
c2
tanh
(√
1− v
2
c2
ξ
l0
))
. (1.2.71)
To evaluate the energy of a moving dark soliton one calculates the difference between
energy without excitation to the energy with excitation to provide convergence of the
energy functional (1.2.66) over an infinite domain [11]. The energy of the dark soliton
per unit surface is then given by [11]
EGPds =
4
3
~cn
(
1−
(v
c
)2)3/2
. (1.2.72)
Writing the order parameter (1.2.71) in polar form ψrmds(z − vt) =
√
ρ(z) · exp(iφ(z))
one sees that the phase has a phase jump of
∆φ = 2 arccos
(v
c
)
(1.2.73)
as z varies from −∞ to ∞.
36
Bright solitons,- For attractive self-interactions a < 0 the GP equation (1.2.67) admits
bright soliton solutions. In the stationary case these are explicitly given by [11]
ψbs(z) = ψ0(0) sech
(
z√
2l0
)
e−
iµt
~ (1.2.74)
with the healing length l0 = ~/
√
2m|g|n0, the density at the maximum n0 = |ψ0(0)|2
and the chemical potential given by µ = gn0/2. The generalization for a moving bright
soliton is given by
ψbs(z − vt) = √n0
(
i
v
c
+
√
1− v
2
c2
sech
(√
1− v
2
c2
(z − vt)
l0
))
, (1.2.75)
where v corresponds to the velocity of the bright solitary wave and c denotes the speed
of sound. Apparently the form of the wave (1.2.75) does not change for z − vt = const.
but it moves along the z-direction. For similar expressions of bright solitons I refer to
[85, 81].
Quantized vortices in 2d
Quantum vortices.- First predicted for superconductors, quantized vortices are a basic
topological excitation in 2d superfluids [86, 87]. Compared to the non-excited con-
densate wave function, which in the simplest case is constant and associated with a
homogeneously distributed condensate, an adaption thereof by quantized vortices is the
superfluids response to an intake of energy, which can be the superfluids response e.g.
to rotation [69] or obstacles within the superfluids stream [88]. One defines the math-
ematically necessary condition of a quantum vortex as follows. The condensate wave
function ψ = |ψ|eiθ : R2 → C has a quantized vortex with a winding number n ∈ N at
a single point ~r ∈ R2 if the wave function |ψ| = 0 and θ increases by an amount of 2pin
along any trivial curve enclosing the singularity at ~r. We note that a requirement for the
condensate wave function is to be single valued and to be continuously differentiable.
Now, the possibility of such a quantum vortex is due to the structure of the condensate
wave function imposed by (1.2.17), which leads to a unique definition of the superfluid
velocity (1.2.29). As the superfluid velocity is defined as a gradient of a potential field
(1.2.29) it is irrotational, i.e.,
~∇× ~v = ~∇×∇θ = 0. (1.2.76)
Around nodal points where ~j = 0 and |ψ| = 0 the condensate wave function becomes
indeterminate and so any relation defined upon meaningless, because the phase θ and
derivatives thereof are indeterminate.
Generally the principal value of the phase is not unique, since
exp(iθ) = exp(iθ + i2pin). (1.2.77)
37
Therefore, using the definition for (1.2.29) and supposing that the path of integration
does not pass any singularities we get∫ ~r2
~r1
~v · d~r =
∫ ~r2
~r1
~∇θ · d~r = (θ(~r2)− θ(~r1)) + 2pin. (1.2.78)
For any path not passing a nodal point the superfluid velocity ∇θ is well defined. Clearly
the line integral does not depend on the path of integration and closed curves give a zero
integral or discrete positive values due to the non-uniqueness in (1.2.77) as can be seen
in the formula (1.2.78). Generally the value of a line integral for closed loops depends
on the homotopy class, ∮
~v · d~r =
∮
~∇θ · d~r = 2pin, (1.2.79)
i.e. no intersection with a nodal point where ψ = 0 is associated with integral zero,
and the more nodal points are enclosed by the integration path the higher the winding
number n. Thus, the circulation is quantized and the formal definition of a quantum
vortex fulfilled. Condition (1.2.79) is referred to as the Onsager-Feynman quantization
condition [15]. It is due to this quantization that vortices remain stable as no contin-
uous transition between condensates with vortex and without exists, i.e. vortices are
topologically stable excitations for the NLSE.
Mathematical form of quantum vortices.- To determine the adaption of the condensate
wave function due to quantum vortices we consider the NLSE in rescaled units (~r → l0~r,
ψ → √n0ψ, so that ψ → 1 at ∞), i.e.
0 = −1
2
∆ψ − (1− |ψ|2)ψ (1.2.80)
in two spatial dimensions. Let us make an ansatz of the form
ψ(x) = U(r)einθ. (1.2.81)
Then inserting in (1.2.80) the modulus satisfies
U ′′ +
1
r
U ′ − n
2
r2
U + (1− U2)U = 0 (1.2.82)
which is well posed for U(0) = 0 and U(∞) = 1 [80]. For r → 0 one obtains by inserting
into (1.2.82) the asymptotic solution
Un = a1r
|n| + a2r|n|+2 + . . . (1.2.83)
and for r →∞ we obtain
Un = 1− b1
r
− b2
r2
− . . . (1.2.84)
with b1 = 0 and b2 = n
2/2 [80, 11]. Alternatively one obtains for a singly charged vortex
the approximate vortex solution for (1.2.17) in 2d given by [73]
U1(r) =
√
n0
r/l0√
(r/l0)2 + 2
(1.2.85)
with n0 denoting the density as r →∞.
38
Figure 1.5: Vortex Ring of radius R.
Vortex rings in 3d with radius R
Roughly speaking, the 3d analog to the 2d quantum vortex are quantized vortex rings.
These are the stable elementary excitations of the atomic BEC within 3d. Considering
the profile of a vortex ring corresponds to considering two oppositely winded vortices as
illustrated in Fig. 1.5. The energy of a vortex ring depends in particular on its radius
and we refer to [11] for a more detailed analysis of the form of these 3d excitations.
However, for big radii compared to the healing length, R  l0, one obtains the simple
expression for the energy of the vortex ring [11],
E(R) = 2pi2R
~2
m
n ln
(
1.59R
l0
)
(1.2.86)
where n denotes the density of the superfluid. Consequently its velocity of a large vortex
ring is given by
v =
dE(R)
dp
=
dE/dR
dp/dR
=
~
2mR
ln
(
1.59R
l0
)
. (1.2.87)
1.2.8 Energy and mass conservation in GP theory
We now derive fundamental properties of the GP theory of equilibrium condensates and
in particular in the beginning show its formal connection to classical mechanics.
Classical Hamilton equations
In classical mechanics a many body state is determined by its pairs of conjugated vari-
ables {qj, pj} with j = 1, 2, 3, . . .. The Hamilton equations of classical mechanics induce
trajectories for {qj, pj} in phase space for the conjugated variables Γ ≡ R3Nq ×R3Np over
time t ∈ R and are given by
∂tqj =
∂H
∂pj
(1.2.88)
39
∂tpj = −∂H
∂qj
, (1.2.89)
where we introduced the Hamilton function H[qi, pj, t], which corresponds to the energy
as time dependent function of phase space. Defining the complex canonical variable
aj = αqj + iβpj, where α, β ∈ C we can rewrite the system of Hamilton equations
(1.2.88) and (1.2.89) as
iλ∂taj =
∂H
∂a∗j
(1.2.90)
with λ = 1
αβ∗+α∗β [67]. Furthermore, the global U(1) symmetry, i.e. aj → ajeiθ with
θ ∈ R implies that N = ∑j |aj|2 is a constant of motion, ∂tN = 0.
We now show that the Gross-Pitaevskii equation derived from a second quantization
scheme (1.2.17) can analogously be considered as a complex order parameter satisfying
a Hamilton equation. This will conveniently allow us to discuss the conserved quantities
within GP theory.
The condensate wave function in a classical field framework and its properties
An important sub-class of Hamiltonian classical field theories is that for which the rela-
tion
iψt(~r) =
δE [ψ]
δψ∗(~r)
(1.2.91)
holds, where E [ψ] is a real-valued energy functional of the field ψ [67]. A particular
example of such an energy functional is the Gross-Pitaevskii functional (1.2.19). Putting
(1.2.19) into (1.2.91) we obtain the Gross-Pitaevskii equation (1.2.17) describing the
state evolution of the condensate wave function.
Energy.- By using (1.2.91) one sees that the energy is conserved over time, i.e.
∂EGP[ψ]
∂t
=
∫ (
∂EGP[ψ]
∂ψ
∂ψ
∂t
+
∂EGP[ψ]
∂ψ∗
∂ψ∗
∂t
)
d~r = 0. (1.2.92)
Mass conservation.- As for the classical field the U(1) symmetry of the functional EGP[ψ]
is defined as invariance with respect to the transformation ψ(~r)→ ψ(~r)eiθ, where θ ∈ R.
From this follows conservation of mass
0 =
∂EGP[ψ]
∂θ
=
∫ (
∂EGP[ψ]
∂ψ
iψ − ∂E
GP[ψ]
∂ψ∗
iψ∗
)
d~r =
=
∫ (
ψψ˙∗ + ψ˙ψ∗
)
d~r = N˙ (1.2.93)
1.2.9 Rigorous mathematical treatment of BEC
Quantum many-body problem
An alternative and equivalent viewpoint on BEC for fixed particle numbers can be
taken by starting from Schro¨dinger’s equation for N -particles, which resembles the
40
second quantization terminology when considering fixed particle numbers. Instead of
considering operators on a Fock space we now work with the concept of N -particle
wave functions ψN(~x1, . . . , ~xN) ∈ H(R3N). For an ensemble of bosons they satisfy
ψN(~x1, . . . , ~xN) = ψN(~xpi1 , . . . , ~xpiN ) for any permutation pi corresponding to the CCR of
the annihilation and creation operators within the Fock space formalism. Many-body
sates consisting of fermions would be multiplied by sign(pi) under the action of a permu-
tation. The permutation properties of bosons can be taken into account by restricting
the many-particle Hilbert space to its symmetric subspace. Usually one assumes the
state or wave function ψ ∈ L2(R3N+1) (including time) and this is a good choice in
particular for spatially extended quantum gases.
Now, the many-body Schro¨dinger equation imposes the time evolution of the wave
function,
i~∂tψ(C, t) = Hˆψ(C, t), (1.2.94)
with some Hamilton operator of the form [62],
Hˆ =
(
− ~
2
2m
N∑
n=1
∆i + Vext(C, t) +
∑
1≤i<j≤N
v(~xi − ~xj)
)
. (1.2.95)
Eq. (1.2.94) is a linear partial differential equation, which obeys a unique solution [55].
Due to linearity superpositions of many body quantum states are solutions as well. We
write for particle positions C = (~x1 = ~x, ~x2 . . . , ~xN), Vext(C, t) denotes a time dependent
external (electric field) trap and v(~xi−~xj) two-body interactions between particles ~xi and
~xj. This Hamilton operator e.g. pre-excludes the existence of three body interactions,
which for dilute ensembles are quite rare events compared to two body interactions.
Hence the model provides a reasonable approximation for this case. The (complex
valued) quantum states satisfy Born’s rule: - |ψ( ~X, t)|2 is the probability density for a
particle configuration at ( ~X, t) ∈ R3N+1 and the states are normalized ‖ψ‖2L2 = 1. As we
will see later non equilibrium systems do not necessarily have a conserved mass, since
particles can be created or destroyed, but only in stationary situations where the influx
equals the drain of particles. For this special case one has a finite and time independent
norm as well, however with a slightly different interpretation as according to Born’s rule.
In (1.2.95) we assume an electric field confinement of the Bosons, but the absence of
a magnetic field. We do so as this is a usual setting for experimentally realized BEC. In
the special case of non-interacting particles one can simply write the many body wave
function as a plain product
ψ(C, t) =
N∏
i=1
ψi(xi, t). (1.2.96)
In a previous section we have defined BEC as a macroscopic occupation of a single
mode. Hence a product of some single particle functions ψi(xi) could be envisaged as
a candidate wave function for a BEC state. Indeed for noninteracting systems we can
write
ψ(C, t) =
N∏
i=1
ψ0(xi, t), (1.2.97)
41
which corresponds to a special case, i.e., without external trap V = 0, conceived by
Einstein [12] in the paper where he introduced BEC. For interacting particles, however,
such a factorization cannot be expected in a strict sense of a simple product of math-
ematical functions, but as we will see later, in a modified way. An example for the
non appropriateness of product states in this formalism is the scenario of particles with
delta function point interactions. Here one has infinite energy at the collision points, so
excluding those solutions. In the proofs on BEC discussed in [62], they therefore intro-
duce a function which smoothens the wave function, once particles collide, i.e. employ
an ansatz ψ(C) = φ0(~x1) · · ·φ0(~xN) · F (~x1, . . . , ~xN), where F vanishes for ~xi = ~xj with
i = j, but is 1 otherwise.
The Schro¨dinger equation (1.2.94) gives the dynamics of the many body system. On
the other hand time independent states correspond to wave functions of the form ψ =
e−itEn/~ψ, so it becomes
Hˆψn(C) = Enψn(C), (1.2.98)
where En is the corresponding nth eigenvalue of a solution. Bose-Einstein condensation
for atoms is achieved for very cold atoms and therefore for energy levels close to or at
the absolute ground state. In a rigorous treatment often condensation in the ground
state with energy E0 is considered as it was envisaged by Einstein initially.
Gross-Pitaevskii Energy functional.- The stationary quantum many-body problem
(1.2.98) can be significantly simplified by effective theories in certain parameter regimes
[62, 55, 11]. For dilute and weakly interacting Bose gases in the ground state, which can
be formalized by the parameter limit referred to as Gross-Pitaevskii limit, where N →∞
and g = 4piNa fixed [62], the Gross-Pitaevskii equation (1.2.17) provides a convenient
tool to understand BEC even in a mathematically rigorous sense. Corresponding to the
eigenvalue problem (1.2.98) one defines the GP energy functional (1.2.19). The energy
of the system is given by
inf EGP[φ] = EGP, (1.2.99)
which is provided by inserting a solution to (1.2.17). Now, it has been shown that in the
GP-limit, N → ∞ while N · a = fixed, energy and density of the GP theory converge
to the corresponding quantities of the quantum many body problem as stated in the
following rigorous theorem.
Theorem 1.2.1 (Lieb-Seiringer-Yngvason) If N →∞ while N · a = fixed , then
lim
N→∞
E0
N
= EGP (1.2.100)
and
lim
N→∞
n(~r) = |φ|2 (1.2.101)
in the weak L1-sense.
42
Bose-Einstein condensation in the Gross-Pitaevskii limit
A key concept for BEC in the second quantization framework is that of the one-particle
density matrix (4.4.2). In the many body Schro¨dinger equation framework the reduced
one-particle density matrix for the ground state ψ0 is defined similarly as
γ(~x, ~x′) = N ·
∫
ψ0(~x,X) · ψ0(~x′, X)dX, (1.2.102)
with X = (~x2, . . . , ~xN) and dX =
∏N
j=2 d~xj. It implies
∫
γ(~x, ~x) = tr[γ] = N and BEC
means that the operator has an eigenvalue of order N in the thermodynamical limit [62].
In the GP limit the reduced one-particle density matrix factorises, so that γ/N be-
comes a simple product f(~r)f~(r′). The function f(~r) is then called the condensate wave
function. This is analog to (1.1.33). Further it could be proved that in this limit the
condensate wave function is the solution to the Gross-Pitaevkii equation φGP as stated
in the following theorem.
Theorem 1.2.2 (Lieb-Seiringer) For each fixed g
lim
N→∞
1
N
γ(~r, ~r′) = φGP(~r)φGP(~r′). (1.2.103)
Convergence is in the senses that Trace
∣∣ 1
N
γ − PGP∣∣ → 0 and∫ (
1
N
γ(~r, ~r′)− φGP(~r)φGP(~r′))2 d~rd~r′ → 0.
We note that this convergence is not pointwise. From a physicists view this theorem
indeed proves rigorously that the GP equation can correctly describe the condensate
wavefunction. Furthermore one can deduce for the interpretation of the GP density that
the one-particle density associated with the many-body wave function gives the same
predictions as the GP density. Because the one-particle density is the mean-value of
particles at a given point the GP density is.
43
1.3 Condensates out of equilibrium
Open particle systems allow gain of energy and particles/quasiparticles, which sets them
apart from the conserved atomic systems in thermal equilibrium introduced in the pre-
vious section. In an open system the notion of BEC is complicated in particular as
particle distributions do not satisfy the same simple relation for the occupation numbers
of states as the ideal Bose gas in thermal equilibrium does (1.1.3), while weakly interact-
ing and closed systems relate to this distribution to a higher degree. The importance of
the distribution of particles on energy levels is that for (noninteracting and conserved)
bosons there is a statistical pressure for particles to occupy the lowest mode signifi-
cantly, when compared to the higher modes and once a critical temperature has been
surpassed. On the other hand BEC is by definition a macroscopically occupied single
particle mode with very high spatial coherence and often when formed in appropriate
physical setups coherent particles could be added to the condensate or removed from
it without compromising the concept. Indeed BEC is proven recently to be feasible in
non-equilibrium systems of exciton-polaritons within semiconductor micro cavities [16].
Besides this outstanding example there are a variety of other feasible systems constitut-
ing non-equilibrium BEC such as bulk nonlinear crystals, atomic clouds embedded in
optical fibers and cavities, photonic crystal cavities and superconducting quantum cir-
cuits based on Josephson junctions [15]. In this thesis, however, our emphasis is on the
non equilibrium exciton-polariton (polariton) condensates. Polaritons are quasiparticles
formed within a semiconductor micro cavity and constitute themselves as superpositions
between a light field and bound electron-hole pairs, the so called excitons [29]. These
quasiparticles can be generated at will by external excitation through a laser field, while
on the other hand they have a lifetime from a few up to hundreds of ps [33], controlled
primarily by the leakage of photons from the microcavity [16, 47, 28, 15], which is small
in comparison with the time scale of polariton condensates that persists over minutes
[32]. Hence, polariton condensates are a true manifestation of a non-equilibrium conden-
sate that can be analyzed experimentally with relative ease compared to atomic BEC
[15, 16], due to the lack of complicated cooling devices, which indeed are necessary for
atomic bose gas condensation.
In the beginning of this section I introduce the basic second quantization formalism
of polariton condensates. Thereafter I will introduce the phenomenological model for
exciton-polariton condensates and its various extensions. Then I discuss elementary
properties of the non-equilibrium system.
1.3.1 Exciton-polariton condensates
Foundations of exciton-polaritons
Photons.- Photons can be trapped inside a cavity. We suppose the trapping geometry to
restrict each photon mode to a finite interval in z-direction. This spatial confinement of
photons is achieved by metallic or planar dielectric mirrors [15]. The dielectric crystalline
mirrors consist of alternating layers of two different dielectric constants. The reflectivity
44
increases with the number of layers [28, 49]. On the top and bottom two such distributed
Bragg reflectors (DBR) sandwich the cavity spacer of length l0 and refraction index n0.
Fig. 1.6 shows an illustration of such a setup. It is important to remarks that although
the reflectivity of the DBR’s is very high some photons will leak out of the system, which
can be used to analyze the physics within experimentally. Due to this confinement the
Figure 1.6: Schematic illustration of a micro cavity made of DBR’s enclosing a cavity
spacer. The illustration hash been taken from [87].
wave function of the photon is of almost finite extend and thus quantized along z-
direction, while we assume the photons to propagate freely in the (x, y) plane. We split
the wave vector of each photon ~k ∈ R3 into the in plane component ~k|| = ~kx + ~ky and
the component along cavity axis that is quantized ~kz = piM/l0 with quantum number
M . Consequently the dispersion relation of the photon within the cavity becomes
Eφ(~k||) =
~ck
n0
=
√
E2z +
~2c2k2||
n20
. (1.3.1)
Here c is the speed of light and Ez = ~c|~kz|/n0 is the minimum energy of a cavity
eigenmode [15, 49]. Applying a first order Taylor expansion we get
Eφ(~k||) ' ~c
n0
kz +
1
2
~ck2||
n0kz
, (1.3.2)
which is a good approximation of (1.3.1) for small k||. For (1.3.2) the first term can be
associated with a rest energy of the cavity photon, while the second term represents the
kinetic energy of the photon for in plane motion. Consequently one can associate an
effective mass to confined photons in analogy to the kinetic energy of intrinsic massive
bodies, i.e. E = ~
2k2
2m
, by writing
mφ =
~kz
c
. (1.3.3)
45
We want to point out that this is only an effective mass and not a fundamental property
of the cavity photon itself.
Excitons.- We now turn to a quantized quasi particle called exciton within a quantum
well. A quantum well consists of a thin semiconductor layer in which the quasiparticles
are formed and which is embedded in different semiconductors that provide a barrier
for the quasiparticles and its constituents [15]. The geometry is chosen such that we
have confinement to the same plane (x, y) as the photon modes are confined to. These
quantum wells are placed within the cavity spacer as Fig. 1.6 illustrates.
When in a semiconductor an electron is excited into the conduction band it leaves
behind a hole in the valence band. This hole can be regarded to carry a charge +e
opposite that of the free electron. An exciton is a quasiparticle consisting of an electron
bound to a hole by the Coulomb interaction between them. Forming a composite boson
it is effectively a neutral quasiparticle. Such a coupled state of an electron with mass
me and a hole with effective mass mh can be written in the center of mass frame as
eˆ ~K,n =
∑
~k,~k′
δ ~K,~k+~k′fn
(
mh~k −me~k′
me +mh
)
aˆ†~kbˆ
†
~k′
(1.3.4)
with K being the center of mass momentum and where a†~k denotes the creation operator
of the electron and bˆ†~k′ the creation operator of the hole, while fn(
~k) is the wave function
of the electron-hole pair [28]. It follows that excitons are bosons for large interparticle
spacings in terms of its Bohr radius [28]. Furthermore does the exciton form a dipole
that indeed interacts with a light field [29] and as such can form new quasiparticles itself.
Exciton-Polaritons.- Let us now introduce the concept of exciton-polariton quasipar-
ticles in semiconductor microcavities as illustrated in Fig. 1.6. To do so we employ a 2d
coupled harmonic oscillator model, which takes into account the physics of the polari-
tons as quasiparticles made of photons and excitons. The basic Hamiltonian describing
the interaction between laser and excitons is [15, 28, 87]
Hˆpol = Hˆcav + Hˆexc + HˆI =
∑
~k||
Eφ(~k||)φ
†
~k||
φ~k|| +
∑
~k||
Eξ(~k||)ξ
†
~k||
ξ~k||+
+ ~ΩR
∑
~k||
Eφ(~k||)
(
φ†~k||
ξ~k|| + ξ
†
~k||
φ~k||
)
. (1.3.5)
Here φ†~k||
is the creation and φ~k|| the annihilation operator of a photon, while ξ
†
~k||
and ξ~k||
are the corresponding operators for the exciton. ΩR is the Rabi frequency, its amount
corresponds to the coupling-strength between both fields. For the sake of simplicity we
introduce the abbreviation g0 = ~ΩR. The coupling constant is the amount of light
matter interaction and can be determined by [89]
g0 = e~
√
fosc/S
8εdme
∫
ψ∗exc(~r)ψcav(~r)d~r. (1.3.6)
46
Here φexc is the quantum-well (QW) exciton wave function and φcav the cavity pho-
ton wave function. e denotes the elementary charge, fosc/S the QW-exciton oscillator
strength per surface area S, ε the dielectric constant, d the quantum well thickness and
me the electron mass [90].
Figure 1.7: Panel (a) shows the concept of the exciton-polariton in a semiconductor
micro cavity. Panel (b) shows the schematic illustration of the upper and
lower energy dispersion curve of the polariton. In particular comparisons
with the photon and exciton curves are given. The figure has been taken
from the original work [16].
To write down (1.3.5) we have assumed that excitons and light interact, which however
is only true for certain spin states of the excitons (bright states) as we will discuss in
more detail in a later section. One diagonalizes (1.3.5) by finding the eigenvalues and
eigenvectors of
A =
(
Eφ(~k||) g0
g0 Eξ(~k||)
)
(1.3.7)
yielding the two eigenvalues
EL,U(~k||) =
1
2
(
Eφ(~k||) + Eξ(~k||)∓
√
(Eξ(~k||)− Eφ(~k||))2 + 4g20
)
(1.3.8)
The energy eigenvalues EL,U(~k||) are referred to as the upper and lower (polariton)
branch. An illustration of them is given in Fig. 1.7. These two dispersions correspond
47
to the basis vectors
ψL(~k||) =
(
h2,~k||
−h1,~k||
)
, ψU(~k||) =
(
h1,~k||
h2,~k||
)
, (1.3.9)
where the so called Hopfield coefficients are given by [28, 90]
h1,~k|| =
√√√√√1
2
1− δ(~k||)√
δ(~k||)2 + 4g20
 (1.3.10)
and
h2,~k|| =
√√√√√1
2
1 + δ(~k||)√
δ(~k||)2 + 4g20
. (1.3.11)
In addition we have introduced the micro cavity’s detuning [87, 90]
δ(~k||) = Eξ(~k||)− Eφ(~k||). (1.3.12)
If δ(~k||) = 0 the Hopfield coefficients satisfy |h1,~k|||2 = |h2,~k|| |2 = 12 and the polaritons are
exactly half exciton and half photon quasiparticles. In any case the real-valued Hopfield
coefficients satisfy the normalization condition h2
1,~k||
+ h2
2,~k||
= 1 and they quantify the
excitonic and photonic content of the polariton; |h1,~k|||2 and |h2,~k|| |2 correspond to the
fraction of photon and exciton within a given exciton-polariton. By employing the
Hopfield coefficients one obtains the diagonalized Hamiltonian [15, 28]
Hˆpol =
∑
~k||
EL(~k||)p
†
~k||
p~k|| +
∑
~k||
EU(~k||)q
†
~k||
q~k|| . (1.3.13)
Here p†~k||
are the creation and p~k|| annihilation operators of a lower branch mode, while
q†~k||
and q~k|| are the corresponding quantities for the upper polariton branch. We note
that (1.3.13) has a translational symmetry along the cavity plane [15]. The dispersion
curve of the lower polariton branch are illustrated in Fig. 1.7 and for low |~k||| resembles
that of the cavity photon (1.3.2) and for higher |~k||| that of the exciton. The difference
between EU(0) − EL(0) gives the Rabi splitting, i.e. the coupling strength between
exciton and photon, if the detuning is zero at ~k|| = 0.
Condensation
Polaritons are quasiparticles within a semiconductor micro cavity obeying two different
energy dispersions, the lower and the upper polariton branch. The effective mass of the
48
exciton-polaritons is determined by the curvature of the dispersion relations for each
branch and given by
mU,L = ~2
(
∂2EU,L
∂k2
)−1
. (1.3.14)
Consequently the effective mass of the polaritons when |~k||| ∼ 0 is [28]
mL(|~k||| ∼ 0) ' mφ|h2,~k|| |2
∼ 10−4mexc, mU(|~k||| ∼ 0) ' mφ|h1,~k|||2
, (1.3.15)
with mexc = mh + me denoting the effective exciton mass. The relatively light mass of
polaritons in comparison to atoms or excitons implies a much higher critical tempera-
ture at which the transition to a condensed state happens. That polariton’s/exciton’s
can condense is due to the fact that they have effectively integer spin. But due to the
polariton’s composite nature an upper bound on the possible polariton density has to
be imposed. Indeed polaritons lose their bosonic features at roughly nB = 1/a
2
B, where
aB denotes the Bohr radius of the exciton in the given material. To allow polariton con-
densation, this condition has to be taken into account [49]. Analogue arguments apply
for composite atomic condensates, where atoms considered at their own are fermions,
but the composite particles obey bosonic statistics. A comparison of key parameters
between atoms, excitons and polaritons is shown in Fig. 1.8; it particular shows quan-
titativ values for the critical temperature depending on the species. Note that excitons
themselves are bosons and thus can in principle condense to a single mode as well.
Figure 1.8: Comparison of key parameters for various candidates eligible for the BEC
transition as taken from [28].
The weak and the strong coupling regime
Regarding the interaction between the excitons and the photons at least two di?erent
regimes can be distinguished, that of strong coupling and that of weak coupling, al-
though clearly a continuum exists in-between [47]. Generally speaking weak coupling
between two fields accounts for a perturbation of each field by the other without a signif-
icant adaption of their properties. This regime is often utilized for semiconductor-based
49
emitters like VCSELs (vertical cavity surface emitting lasers) or quantum-dot polariton
based single-photon sources. Strong coupling implies new quantum mechanical eigen-
states for the system and a splitting between the exciton and cavity modes appears
[90].
To illustrate the differences between those two regimes we now take into account the
finite lifetime of the photon and exciton that make up a polariton. We introduce the
cavity photon decay rate γcav and γexc, which takes non radiative exciton decay into
account [28]. For uncoupled decaying exciton-photon states one writes
H~k|| =
(
Eφ(~k||)− i~γcav g0
g0 Eξ(~k||)− i~γexc
)
(1.3.16)
with new energy eigenvalues given by
EL,U(~k||) =
1
2
(
Eφ(~k||) + Eξ(~k||)− i~(γexc + γcav)
)
∓
√
g20 +
1
4
(
δ(~k||)− i~(γexc − γcav)
)
). (1.3.17)
The energy dispersion of the upper and lower dispersion branch is modified by the decay
represented by γcav and γexc and the energy difference between the branches becomes
~Ω′ =
√
4g20 − ~2(γexc − γcav)2. (1.3.18)
Now, strong light matter coupling means that (1.3.18) is real-valued, which corresponds
to the regime where the coupling dominates dissipative processes [15, 28].
From the decay rates one in particular deduces the lifetime of the polaritons [49, 90],
which for the lower branch is [28]
1
τLp (k)
=
|h1,~k|| |2
τcav
+
|h2,~k|||2
τexc
(1.3.19)
with γcav = 1/τcav, γexc = 1/τexc and for the upper branch
1
τUp (k)
=
|h1,~k|| |2
τcav
+
|h2,~k|| |2
τexc
. (1.3.20)
In state-of-the-art semiconductor micro cavities one has γcav ' 1−100ps and γexc ' 1ns
and thus the main contribution for the finite lifetime is due to the leakage of cavity
photons and the induced decay of polaritons [28, 33]. The lifetime significantly depends
on the number of DBRs forming the micro cavity.
Measuring the polariton condensate
The emission of leaking photons from the microcavity either stems from the cavity
photon or usually in polariton experiments the lower dispersion polaritons depending on
50
the coupling strength between the exciton and photon modes. In this sense the system
shows either regular lasing or polariton condensation. Furthermore it shows a cross over
between weak coupling at higher temperatures and high pumping strengths to strong
coupling at lower temperatures and pumping [47], while here we turn our focus to the
latter case. The decay of polaritons is mainly due to the leakage of photons of energy
EL,P = ~ω out of the cavity and has the same in plane wave vector ~k|| as the polariton
it has been emitted from. The decay rate depends on the quality of the cavity mirrors,
which can be increased or decreased in a simple fashion e.g. by adding additional layers
to the DBRs [28, 49]. In measurements the in-plane ~k|| vector of the leaking cavity
photons directly corresponds to the angle of emission out of the microcavity and the
energy allowing to measure directly the full exciton-polariton wave vectors [28]. This is
illustrated in Fig. 1.6 and Fig. 1.7.
Self-interactions of lower branch polaritons
Until now we have neglected in our discussion interactions between the condensed quasi-
particles. Such interactions are present and indeed stem from the exciton quasiparticles,
which interact with each other via the exchange of electrons [15, 47]. The effective
Coulomb interactions between electron and hole pairs with each other can be simplified
by contact two-body interactions between excitons given by [15, 87],
Hˆξξ =
1
2
∑
~k||,~k′||,
V ξξ~q ξ
†
~k||+~q
ξ†~k′||−~q
ξ~k||ξ~k′||
(1.3.21)
where ξ~k|| (ξ
†
~k||
) are the exciton annihilation (creation) operators. In the polariton basis
these are taken into account in the Hamiltonian [15, 87] with an additional term
HˆPP =
1
2
∑
~k||,~k′||,~q
V PP
~q,~k||,~k′||
p†~k||+~q
p†~k′||−~q
p~k||p~k′||
, (1.3.22)
with
V PP
~q,~k||,~k′||
= V ξξ~q ξL(
~k|| + ~q)ξL(~k′|| − ~q)ξL(~k||)ξL(~k′||) (1.3.23)
and ξL(~k
′
||) (ξ
†
L(
~k′||)) denoting the lower exciton branch annihilation (creation) operators,
which follow from the eigen vectors (1.3.9) [87]. So the combined Hamiltonian of (1.3.13)
and (1.3.22) is then given by
Hˆpol = HˆP + HˆPP . (1.3.24)
As in the derivation of the Gross-Pitaevskii equation for atomic condensates (1.2.17)
delta function self-interactions between the field operators can be assumed, if the con-
ditions for diluteness and relative shortness of the scattering length are satisfied for the
system.
51
1.3.2 A non-equilibrium complex nonlinear Schro¨dinger-type model
The Superfluid State equation
A key property of polaritons in polariton condensates is that they interact with each
other inducing enriched dynamics of the coherent matter waves they form, such as a
nonzero finite speed of sound below which there is a superfluid phase [15]. In contrast
does the linear single-body Schro¨dinger equation obey excitations, the so called Reigh-
leight waves, which occur once only a small modulation of the condensate is induced,
while the nonlinear counterpart lacks such excitations [15]. In contrast to the linear
equation the nonlinearity can compensate a sufficiently small local gain of energy and
therefore prevent the phase of the superfluid to wind up, and so the emergence of excita-
tions such as quantized vortices. Similarly to the atomic case we formulate an alternative
description to the second quantization framework [15, 47, 28] for the condensate wave
function of the lower polariton branch in terms of a GP-like equation,
i~∂tψ(~r, t) =
(
EˆL + Vext(~r, t) + α|ψ|2
)
ψ(~r, t), (1.3.25)
where the lower branch dispersion EˆL is often approximated as −~2∇22m∗ with m∗ denot-
ing the effective mass of the polariton. This approximation corresponds in ~k||-space
to setting the dispersion relation (1.3.8) for low momentum polaritons (small ~k||) to
EˆL(~k||) ∼ −~k2||. ψ(~r, t) : R2+1 → C is the order parameter of the many-body system,
i.e. the condensate wave function. Further (1.3.25) includes the self-interactions be-
tween polaritons of strength α that stem to the leading order from the electron-electron
exchange interactions when two excitons swap their electrons [47]. The magnitude of
self-interactions is given through [87, 15]
α =
6EbX
2
exca
2
B
S
, (1.3.26)
where aB is the 2d Bohr radius, Xexc the excitonic fraction and S is a normalization and
Eb is the excitonic binding energy.
Gain and Loss
We present two alternatives of modelling growth and decay. The nonuniqness of the
theoretical model comes from the phenomenological nature of the model. So far a
unique model has not been completely derived from a second quantization scheme, but
only motived by physical arguments. Now, introduced in [48] gain and loss of polaritons
can be modeled by extending the condensate dynamics via
∂tψ =
(
P (~r)− γ|ψ|2)ψ, (1.3.27)
where P (~r) is the pumping distribution and γ the factor for the density dependent
decay rate. On the other hand according to [91, 15] the time dynamics of (1.3.25) in an
52
incoherently pumped mean field polariton condensate are enhanced by
∂tψ =
(
P
γR +R|ψ|2 − γL
)
ψ, (1.3.28)
where the first term on the r.h.s. in (1.3.28) is a saturating nonlinearity, which for small
amplitudes becomes (1.3.27) as can be seen by applying a first order Taylor expansion.
Here γR represents the reservoir decay rate, γL the condensate decay rate and R the
scattering rate from the reservoir into the condensate. Multiplying (1.3.28) by ψ∗ and
adding the hermitian conjugate we get for the condensate particle magnitude
∂tn = 2
(
P
γR +Rn
− γL
)
n. (1.3.29)
The dynamics imposed by (1.3.29) implies that the higher the magnitude n(~r) is at a
given point the lower the gain for the condensate wave function there and vice versa. If
the left hand side in (1.3.29) is negative a decrease of particles follows while when it is
positive the condensate gains particles/mass at a given point ~r. Now, adding (1.3.28)
to (1.3.25) we obtain a non equilibrium order parameter equation for the polariton
condensate
i~∂tψ(~r, t) =
(
−~
2∇2
2m
+ Vext(~r, t) + α|ψ|2
)
ψ(~r, t)+
+ i
(
P
γR +R|ψ|2 − γL
)
ψ(~r, t). (1.3.30)
We note that the complex terms due to pumping and decay resemble to some extend
those induced by motion (3.8.13) and (1.2.48) or an electromagnetic field. Furthermore
for stationary condensates the U(1) symmetry is preserved. Generally energy relaxation
can be taken into account by introducing additional decay of particles as a function
of energy. The most straightforward way to phenomenologically model the process of
energy relaxation is via a factor ∂t → (1+ iµ)∂t introduced to the nonlinear hamiltonian
in (1.3.30) [47].
Dynamical reservoir
In addition to the equation for the condensate field ψ, one can introduce a dynamical
reservoir population equation for the wave function in correspondence with the form in
(1.3.30). The latter is indeed an approximating of the reservoir when ∂nR
∂t
' 0 applies
and can be extended to include time dependence explicitly [91] by
∂nR
∂t
= P − γRnR −R |ψ|2 nR. (1.3.31)
P denotes the spatial and time dependent pumping geometry associated with the applied
laser beam that corresponds as coherent photon source. γR represents the reservoir decay
rate and the third term on the r.h.s. represents the loss due to stimulated scattering
from the reservoir into the polariton condensate. As γL  γR the reservoir dynamics is
faster than that of the condensate and one can indeed assume ∂nR
∂t
' 0 as only the time
averaged mean value of the reservoir is felt by the condensate.
53
Figure 1.9: Measured far-field emission (momentum distribution) at 5K as condensation
within a 2d polariton condensate takes place, showing the typical narrowing
at k|| = 0 for increased power. (a) is 0.55Pthr, (b) is Pthr and (c) is 1.14Pthr,
where the power Pthr = 1.67kWcm
−2 and corresponds to the condensation
threshold. [16]
Incoherent pumping
With the mean field theory presented by (1.3.30) we have introduced a model describing
the dynamics induced through an incoherent pumping scheme. This scheme pre supposes
a physical situation where a significant density of incoherent polaritons are located in
the so called bottleneck region in k||-space, that is closely around the inflection point
of the lower polariton dispersion. Then further relaxation to the bottom of the lower
polariton branch via phonon-polariton scattering is decreasing due the reduced density
of states in the final states. However, once the polariton density is above some threshold
polariton-polariton scattering provides another energy relaxation process as they scatter
into the bottom of the lower branch, while the other polaritons scatters into the excited
excitonic region. Due to the statistical pressure by the bosonic nature of polaritons
once the density of polaritons at the bottom of the lower branch is above order one,
increase of particles is stimulated. When the gain of polaritons there exceeds the losses
caused by decay a coherent polariton condensate forms at |~k||| ∼ 0 [15]. Fig. 1.9 shows
the condensation of polaritons in momentum space around the threshold as measured
in experiments. Fig. 1.7 illustrates the form of the lower polariton branch and at
which position on the dispersion the condensate forms on the lower energy branch. As
mentioned earlier, the momentum vector of the polariton is experimentally determined
by the angle and magnitude of outgoing photons.
54
Bogoliubov excitation spectrum for the lower polariton branch
To quantify the effects of pumping and decay on the elementary excitation spectrum
of the modified GPE (1.3.25) for a homogeneous ground state ψL =
√
nL exp(−iωt)
one considers the linearized Bogoliubov equations for the perturbation δψL(~r) and its
complex conjugate,
i∂t
(
δψL(~r)
δψ∗L(~r)
)
=
=
(
ωL(~k||) + 2αnL − ω − iΓ/2 αnL − iΓ/2
−αnL − iΓ/2 −ωL(~k||)− 2αnL + ω − iΓ/2
)(
δψL(~r)
δψ∗L(~r)
)
, (1.3.32)
with ωL(~k||) = ω0L − ~~∇2/2mL [15]. ω0L is the frequency of the bottom of the lower
polariton branch and gL the magnitude of polariton self-interactions. In addition we
have introduced an effective damping rate [15]
Γ =
PRnL
(γR +RnL)2
. (1.3.33)
Then by considering the small amplitude harmonic modes as in the corresponding section
for equilibrium condensates we obtain
ωBog(~k||) = ±
√√√√~2~k2||
2mL
(
~2~k2||
2mL
+ αnL
)
− Γ
2
4
− iΓ
2
. (1.3.34)
Here P is the constant pumping [15]. In contrast to equilibrium BEC additional terms
due to the gain and loss have been introduced to the dispersion curve in (1.3.34). The non
equilibrium dispersion resembles the equilibrium case once the limit to the conservative
system Γ→ 0 is applied.
In the equilibrium case we have seen that there are a variety of elementary excitations,
which depend on the dimensionality of the condensate. Analog excitations indeed do
exist within polariton condensates [15]. However, it has been shown numerically that
due to the pumping and decay terms these excitations do not persist for infinite times
but disperse [92].
55
1.4 Spinor condensates
By definition BEC is a many-body phenomenon of macroscopically occupied states of
very high coherence. While we so far have considered single mode condensates, we
now turn to condensates of two components. So the operator of the many-body system
(1.1.22) becomes
Ψˆ†N =
N−M−1∑
i=0
f1aˆ
†
i +
N∑
j=N−M
f2aˆ
†
j. (1.4.1)
Applying the Bogoliubov c-number approximation one can simplify the operator to be
effectively described by two classical condensate wave functions f1 and f2. From a theo-
retical point of view it is clear that several components of BEC can in principle coexists,
but depending on internal parameters communicate their presence to each other. Ex-
amples of several components of condensates are systems including different spin states
of the condensed atoms that induce a hyperfine splitting between the components [11]
or one can envisage different species of atoms to condense into the same trap, which due
to their different mass, and scattering length would impose enhanced dynamics on the
system of condensates. First we shortly review the two component formalism for atomic
condensates and secondly that for polariton condensates which stems from the pseudo
spin of the polariton quasiparticles.
1.4.1 Multicomponent atomic condensates
Although a second quantization approach applies for the treatment of multicomponent
condensates we will shortcut the arguments leading to an effective mean-field model,
but note that similar considerations as for the single component case apply for the
derivation. So to capture the two component atomic BEC (occupying the hyperfine
states |F = 1,mF = −1〉 and |F = 2,mF = 1〉) we simply introduce the generalized GP
energy functional [11]
EGP2 [ψ1, ψ2] =
∫ (
γ
2
∣∣∣~∇ψ1∣∣∣2 + γ
2
∣∣∣~∇ψ2∣∣∣2 + V (1)ext (~r)n1+
+ (V
(2)
ext (~r) + Vhf )n2 +
1
2
g1n
2
1 +
1
2
g2n
2
2 + g12n1n2
)
d~r. (1.4.2)
In contrast to the convex single component functional (1.2.19), EGP2 [ψ1, ψ2] is an ex-
pression of the wave function of the condensate component ψ1 and of component ψ2
and its minimizer, say Ψ = (ψ1, ψ2), provides the energy of the BEC system. Further
introduced symbols in (1.4.2) are given as follows: gi denotes the self-interactions of
component i, while g12 isotropic cross-interactions between components and ni denotes
the magnitude of the component i. Here Vhf is the hyperfine splitting between two
atomic states due to the relative energy shift between components [11]. V
(i)
ext(~r) are the
trapping potentials felt by component 1 and 2, respectively. Analogously to the single
56
component theory the minimizing functions are then obtained by the variational ap-
proach i~∂ψi/∂tψi = δE/δψ∗i with i = 1, 2. Applying the variational approach under
the assumption of fixed mass (included by a Lagrangian multiplier) one gets a system of
two coupled Gross-Pitaevskii equations, corresponding to Euler-Lagrange equations for
the spinor field Ψ = (ψ1, ψ2),
i~∂tψ1 =
(
− ~
2
2m
~∇2 + V (1)ext + g1|ψ1|2 + g12|ψ2|2
)
ψ1 (1.4.3)
and
i~∂tψ2 =
(
− ~
2
2m
~∇2 + V (2)ext + Vhf + g2|ψ2|2 + g12|ψ1|2
)
ψ2 (1.4.4)
with self-interactions explicitly given by g1 = 4pi~2a1/m and g2 = 4pi~2a2/m and cross-
interaction given by g12 = 4pi~2a12/m. Here we denote the scattering length between
particles of state 1 and 2 by a12. Atom numbers are assumed to be conserved in this
model, so neglecting the scattering from one component into the other [11]. Remarkably
mixtures of dilute 87Rb atom gases in the condensed phase in two different hyperfine
states have been realized in experiment and are well described by this model [11]. More
recently different species consisting of e.g. a 87Rb atom condensate and a 85Rb atom
condensate were realized [93]. Now as the above system of coupled GPE suggests, one
can distinguish different regimes depending on the relation of self- and cross-interactions.
To induce a separation of phases the condition for the condensate to be immiscible
g12 >
√
g1g2 has to be satisfied, as the cross talk between components will force them
apart [11], which has been shown particularly in [93]. As soon as g12 ≤ √g1g2 condensates
become miscible and only fluctuations between components might appear through the
relatively small g12. Also in two component systems Feshbach resonaces allow the control
over the scattering length and therefore the self-interaction parameter of each component
separately [93]. This insight will be utilized in one of the following chapters, where an
effective mechanism to generate skyrmions/core vortices will be presented.
1.4.2 Spinor polariton condensates
Similarily to atoms, polaritons obey two component properties due to an intrinsic spin,
which becomes relevant once decoherence between the components is induced. It turns
out that this spin allows the separation of spin modes e.g. by applying an external
magnetic field on the polariton condensate or excitation by oppositely polarized light and
thus two component spinor states formally similar to their atomic counterpart become
feasible.
Let us introduce first the concept of spin in polaritons and then present a theoreti-
cal framework for polariton condensates which takes their spin degree of freedom into
account in a circularly polarization basis.
57
Pseudospin model for polaritons
As the polariton is a quasiparticle the arrangement of the angular momentum of its
components is key for the manifestation of the polariton’s quasi spin. In standard
quantum wells, the electron in the conduction band has a S-symmetry projection of
its angular momentum Jez = S
e
z = ±1/2 [15, 87]. A hole with P -symmetry in the
valence band has possible angular momenta Jhz = S
h
z +M
h
z = ±1/2,±3/2. This includes
the case of a hole with spin projection antiparallel to mechanical momentum Mhz that
implies Jhz = ±1/2, but otherwise one has angular momentum Jhz = ±3/2 [87]. So
the total angular momentum of excitions can be J = ±1,±2. It turns out that only
the J = ±1 states can be excited optically, which is essential to form polaritons in the
semiconductor micro cavity [29, 15], i.e. excitons which interact with light have ±1 spin
projections. Due to this interaction one can polarize exciton-polaritons by illuminating
the excitons with a polarized light source. Circularily polarized light yields J = 1,−1
and linearly polarized light a superposition of J = +1 and J = −1, with absolute zero
spin projection. In Fig. 1.10 we show an analog to the so called Stokes vector on the
Poincare sphere representing the polarization state of the polariton stemming from the
illumination with the polarized light field. Taken over from the polarization of the light
each polarization of the polariton can be associated with a point on this sphere. The
north and south poles of the sphere are the circular polarizations (due polarization with
circularly polarized light) while the equatorial plane corresponds to linear circulation
(stemming from linearly polarized light) [49, 87]. The polariton has spin projection
Figure 1.10: Schematic illustration of the pseudo spin representation, where S denotes
the stokes vector. The image has been taken from [87].
J = ±1 that couples to light and separates from the other component J ± 2 in a way
that the latter is negligible for the J = ±1 condensate dynamics. A separation of spin
components J = 1 and J = −1 can be by achieved by excitation through σ+ and
σ− circulated light [94]. Formally the spin structure of polaritons is analog to that of
58
electrons and can be represented by the 2x2 pseudo-spin density matrix
ρ =
N
2
I + ~S · ~σ, (1.4.5)
where I is the 2x2 unit matrix, N the total number of particles, ~S = (Sx, Sy, Sz) the
pseudo-spin vector and ~σ the vector of Pauli matrices [94, 87]. The pseudo-spin’s orien-
tation along the z-axis is associated with circular polarized polaritons, while when the
pseudo-spin is within the (x, y)-plane one has linear polarized polaritons. For circularly
polarized polaritons (1.4.5) reduces to
ρ =
N
2
I + Sz · σz. (1.4.6)
TM-TE splitting
Polaritons with ~k|| 6= 0 can be classified as states of dipole momentum orientated along
and perpendicular to the wave vector as they in particular obey different energy levels;
excitons with dipole moment parallel to the wave vector are called transverse magnetic
(TM) and excitons with dipole momentum perpendicular to the wave vector transverse
electric (TE). The long-range electron-hole interaction imposes a longitudinal-transverse
splitting (LT) of the two types of excitons referred to as TE-TM splitting [94]. This split-
ting further translates to polaritons with k|| 6= 0 and accounts for the energy difference
between the TM and TE states [49]. In [90] a pseudo spin representation modelling the
TE-TM splitting has been introduced and can be described as follows. The propaga-
tion of a single free polariton is governed by the effective Hamiltonian extended by an
effective magnetic-type field acting upon it [90], i.e.
Hˆ(~k||) =
~2~k2||
2m∗
+ µBg(~σ ~Heff). (1.4.7)
Here ~σ is the Pauli matrix vector, m∗ the effective mass of the polariton and we have
introduced the effective field
~Heff =
~
µBg
Ω~k|| (1.4.8)
where Ω~~k|| has the components
Ωx =
Ω
~k2||
(k2x − k2y), Ωy = 2
Ω
~k2||
kxky, (1.4.9)
with ~k|| = (kx, ky), Ω = ∆LT/~ and ∆LT is the longitudinal-transverse splitting. Conse-
quently the strength of the energy splitting of the two species of polaritons is by (1.4.7)
due to the magnitude of the LT splitting. We refer to [90] for details on the orientation
of the effective field in terms of the in-plane wave vector ~k||.
The density matrix for the spin without interactions evolves in time by the Liouville-
von Neumann equation [87],
i~∂tρ = [Hˆ(~k||), ρ], (1.4.10)
59
with the Hamiltonian given by (1.4.7). Note that the general form of (1.4.10) can be
derived from the Schro¨dinger equation.
TE-TM splitting in the circularly polarized spinor polariton condensate
Previous considerations on the pseudo-spin dynamics for polaritons apply for many co-
herent polaritons without interactions as well. When interactions between polaritons
play a crucial role, however, the model needs to be extended. In the circularly polar-
ized basis the Hamiltonian respectively Gross-Pitaevskii-type energy functional for the
condensate spinor wave function Ψ = (ψ+, ψ−) with interactions is becomes [94, 87],
E =
∫ (
Ψ∗AΨ +
α1
2
(|ψ+|4 + |ψ−|4)+ α2|ψ+|2|ψ−|2) d~r. (1.4.11)
α1 denotes the self-interaction strength within each spin component of the spinor and
α2 accounts for the cross-interactions between each spin component. The kinetic energy
operator of (1.4.11) is given by
A =
( − ~2∆
2m∗ β(∂y − i∂x)2
β(∂y + i∂x)
2 − ~2∆
2m∗
)
. (1.4.12)
The off-diagonal elements correspond to the TM-TE splitting and impose a splitting of
spin components, while it is the simplest form with which they can be associated with
the effective magnetic field (1.4.9) [94, 87]. The diagonal terms are the conventional
kinetic energy terms of the lower polariton branch. We have used the effective mass m∗
of the polariton, which can be written in terms of the effective masses of the TM (mTM
) and TE (mTE) energy branches,
m∗ =
mTMmTE
mTM +mTE
. (1.4.13)
The factor β = ~2/4 · (1/mTM − 1/mTE) in (1.4.12) denotes the magnitude of the TE-
TM splitting between the components. Applying the Lagrangian equations as in the
previous section on (1.4.11) yields the state equations for the minimizers ψ+ and ψ− of
the energy functional (1.4.11). These are given by
i~
∂ψ+
∂t
= − ~
2
2m∗
∆ψ+ + α1|ψ+|2ψ+ + α1|ψ−|2ψ+ + β(∂y − i∂x)2ψ− (1.4.14)
i~
∂ψ−
∂t
= − ~
2
2m∗
∆ψ− + α1|ψ−|2ψ− + α1|ψ+|2ψ− + β(∂y + i∂x)2ψ+. (1.4.15)
Besides self-interaction and between components one has the additional TE-TM splitting
terms modifying the evolution of the spinor polariton condensate Ψ = (ψ+, ψ−). The
latter terms induce in particular an exchange between particles of each spin component.
We point out that the mass of particles is not conserved for a polariton system, but
varies with time, when no stationary state is accessible. We note that another form
of splitting between spin components in polariton condensates is due to the intrinsic
Zeemann splitting, we refer to [87] for details.
60
TM-TE splitting model in 1d
In one spatial dimensions the spinor polariton field ψ = (ψ+, ψ−)T due to TM-TE
splitting evolves according to the simplified system of cGLE [26]
i~
∂ψ+
∂t
=
[
−~
2∆
2m
+ α1
(|ψ+|2 + nR)+ α2|ψ−|2]ψ+
+
[
U − i~
2
(γd − γpnR)
]
ψ+ − Hx
2
ψ− (1.4.16)
i~
∂ψ−
∂t
=
[
−~
2∆
2m
+ α1
(|ψ−|2 + nR)+ α2|ψ+|2]ψ−
+
[
U − i~
2
(γd − γpnR)
]
ψ− − Hx
2
ψ+ (1.4.17)
(1.4.18)
Here we have included pumping and decay within the set of equations and a reservoir
nR, which can be dynamical or statical as discussed in a previous section. In contrast to
the 2d case (1.4.12) the TM-TE splitting can be assumed to be constant along the 1d
geometry; in addition to the case without spin we now have an effective magnetic-type
field with amplitude Hx = 0.01 meV induced by the energy splitting between TE and
TM eigenmodes and that couples both spin components. γp denotes a pump rate and γd
the linear decay of particles. A crucial point is that the polariton-polariton interactions
are spin-anisotropic and that the interactions within a spin component are much bigger
than between the spin polariton components, i.e. α2 = −0.1α1, and vary with the
number of quantum wells, their separation within the cavity spacer and the detuning
between the light and exciton fields [87].
61
2 Analytical properties of a rapidly
rotating Bose-Einstein condensate in
a homogeneous trap
Florian Pinsker and Jakob Yngvason, published in a joint work with Michele Correggi
and Nicolas Rougerie as part of the paper J. Math. Phys. 53, 095203 (2012);
We present an asymptotic analysis of the effects of rapid rotation on the ground state
properties of a superfluid confined in a two-dimensional trap. The trapping potential
is assumed to be radial and homogeneous of degree larger than two in addition to a
quadratic term. The energy asymptotics are found between the two critical rotational
velocities marking respectively the vortex lattice regime with the creation of a hole of
low density within the vortex lattice and the giant vortex phase. These phenomena have
previously been established rigorously for a flat trap with fixed boundary but the soft traps
considered in the present paper exhibit some significant differences.
2.1 Introduction
Since their first experimental realization in 1995, atomic Bose-Einstein condensates
(BECs) have become a subject of tremendous interest, stimulating intense theoretical
activity. The BE-condensed alkali vapors that are nowadays produced in many labora-
tories offer a spectacular level of tunability for several experimental parameters, making
them a favorite testing ground for many intriguing quantum phenomena. Among these
is superfluidity, i.e., the occurrence of frictionless flow, previously observed in liquid he-
lium. Atomic BECs offer a valuable alternative to the latter system, since experiments
with dilute gases of ultracold atoms allow to test theories of superfluidity in much more
detail.
From a theoretical point of view an appealing aspect of the field is its sound mathe-
matical foundation. The most commonly used model for the description of BECs, the
so-called Gross-Pitaevskii (GP) theory, is now supported both by extensive comparisons
with experiments [95, 182] and a solid mathematical basis [97]. Indeed, substantial ad-
vances on the connection between GP theory and many-body quantum physics have been
made in recent years, leading to a rigorous derivation of both the stationary [98, 99, 100]
and dynamic aspects [101, 102, 103] of the theory.
One of the most striking features of superfluids is their response to rotation of the
confining trap. Typical experiments (see [182] for further references) start by cooling
62
a Bose gas in a magneto-optical trap below the critical temperature for Bose-Einstein
condensation. If the trap is set in rotational motion the gas stays to begin with at rest
in the inertial frame, but if the rotation speed exceeds some critical value, vortices are
nucleated and remain stable over a long time span. This behavior is in strong contrast
with that of a classical fluid, which in the stationary state rotates like a rigid body and
thus remains at rest in the rotating frame.
In a rotationally symmetric trap, the main two experimentally tunable parameters are
the rotation speed and the strength of interparticle interactions, written as 1/ε2 with
ε > 0 in the sequel. The ground state of the system strongly depends on the relation
between these two parameters.
In this paper we study the ground state of a rotating Bose gas in the framework of
the two-dimensional GP theory1. We consider the minimization, under a unit mass
constraint, of the GP functional2
EGPphys[Ψ] =
∫
R2
d~r
(
1
2
∣∣∣(∇− i ~Arot)Ψ∣∣∣2 + (V (r)− 1
2
Ω2rotr
2
)
|Ψ|2 + |Ψ|
4
ε2
)
(2.1.1)
~Arot = ~Ωrot ∧ ~r = Ωrotr~eθ, (2.1.2)
where Ωrot is the rotational velocity, r = |r| with r ∈ R2 is the distance from the rotation
axis, and eϑ stands for the unit vector in the direction transverse to r. The confinement
is provided by a potential of the form
V (r) := krs + 1
2
Ω2oscr
2 (2.1.3)
with k > 0. We restrict to the case 2 < s <∞ (anharmonic, ‘soft’ potentials) and shall
mainly study the Thomas-Fermi (TF) limit ε → 0. The case s = 2 is special in many
respects because the potential −Ω2rotr2 due to the centrifugal force is also a quadratic
function of r. As a consequence an upper bound is set to the allowed values of the
rotational speed and different physics is expected in the regime where the centrifugal
force nearly compensates the trapping force (see [95] and references therein, [104] and
[105]).
The parameter regime considered here concerns the rotational velocities where a vor-
tex lattice wavefunction is an appropriate trial function for the minimizer of the GP
energy functional. This vortex lattice solution then gives the next to the leading or-
der contribution of the energy functional, which matches the contribution of the lower
bound. We introduce now additional functionals in which terms we will in particular
express the GP energy functional.
1Such a description is justified if the trap almost confines the gas on a plain orthogonal to the axis of
rotation or, on the contrary, the trap is very elongated along the axis, in which case the behavior
can be expected to be essentially independent of the coordinate in that direction.
2The subscript ‘phys’ stands for physical, i.e., this is the functional in the original physical variables,
as opposed to the rescaled functionals introduced in Sect. 1.1 below. Units have been chosen such
that h¯ as well as the mass are equal to 1.
63
2.1.1 Rescaled Functionals
In this paper we focus on the analysis of the minimization of the GP energy functional
(2.1.1) under the mass constraint
‖Ψ‖22 :=
∫
R2
d~r |Ψ|2 = 1.
The trapping is given by the potential (2.1.3) with Ωosc < Ωrot, and 0 < ε 1, i.e., we
study the TF limit of the model. The power s in (2.1.3) characterizes the homogeneous
trap and we assume that 2 < s <∞. We also introduce the parameter
Ωeff :=
√
Ω2rot − Ω2osc, (2.1.4)
so that the effective potential in (2.1.1) can be written in the form
V (r)− 1
2
Ω2rotr
2 = krs − 1
2
Ωeffr
2. (2.1.5)
Since we are interested in exploring the rotation regime Ωrot →∞ but want also to keep
track of the effect of the quadratic term in the expression above we shall assume that
Ω2eff = γΩ
2
rot (2.1.6)
for some given 0 < γ ≤ 1.
In this excerpt of the paper [23], we shall consider rotational velocities that satisfy
Ωrot & ε−
s−2
s+2 . (2.1.7)
Then a natural scaling parameter is given by the position Rm of the unique minimum
point of the effective potential (2.1.5), which is explicitly given by
Rm :=
(
γΩ2rot
sk
) 1
s−2
= O
(
Ω
2
s−2
rot
)
. (2.1.8)
Rescaling
~r = Rm~x, Ψ(~r) = R
−1
m ψ(~r), Ωrot = R
−2
m Ω, ~AΩ = Ωx~eθ (2.1.9)
the GP functional becomes
EGP[Ψ] =
∫
R2
(
|(~∇+ iΩx~eθ)Ψ|2 + γ2Ω2
(
2
s
xs − x2
)
|Ψ|2 + ε−2|Ψ|4
)
d~x. (2.1.10)
In the course of this work we will consider a similar GP functional as well, which is
associated with (2.1.10),
EˆGP[ψ] =
∫
R2
(
|~∇ψ|2 + γ2Ω2
(
2
s
xs − x2
)
|ψ|2 + ε−2|ψ|4
)
d~x. (2.1.11)
64
Both functionals map functions ψ ∈ H1(R2) with ‖ψ‖2 = 1. For simplicity we introduce
the notation
V (x) ≡ γ2Ω2
(
2
s
xs − x2
)
(2.1.12)
for the potential term in above energy functionals and a simple expansion at the poten-
tials’ minimum x = 1 yields
V (x) ' γ2Ω2
((
2
s
− 1
)
+ (x− 1)22 (s− 2)
)
. (2.1.13)
Now, the unique minimizer of (2.1.11) under the mass constraint denoted by g satisfies
the variational equation
−∆g + V (x)g + 2
ε2
g3 = µg, (2.1.14)
where µ is a Lagrangian multiplier given by
EˆGP + ε−2‖g‖44 = µ. (2.1.15)
Thomas Fermi (TF) energy functional and rough estimates
Besides the GP functional we also consider the TF functional given by
ETF[ρ = ψ2] =
∫
R2
(
γ2Ω2
(
2
s
xs − x2
)
|ψ|2 + ε−2|ψ|4
)
d~x (2.1.16)
on the domain {
ρ ∈ L2(R2)|ρ ≥ 0, rsρ ∈ L1(R2)} (2.1.17)
with the mass constraint ‖ρ‖1 = 1. Its unique minimizer, the TF density, is
ρTF(x) =
ε2
2
(
µTF − γ2Ω2
(
2
s
xs − x2
))
+
'
' γ
2ε2Ω2
2
(
2
s
(xsin − 1) + (1− x2in)− (x− 1)22(s− 2)
)
+O(ω2|1− x|3). (2.1.18)
Multiplying (2.1.18) with ρTF, integrating and the normalization ‖ρTF‖1 = 1 yields
µTF = ETF +
1
ε2
‖ρTF‖22. (2.1.19)
We refer to [106] for the analysis of the support of the TF density. In that work they
find that there is an inner radius, which we denote by xin and an outer radius denoted
by xout at which the TF density vanishes and between which it is positive. Beyond those
65
radii the density is zero. By the normalization of the TF density ‖ρTF‖1 = 1 and a
Taylor expansion we have
µTF
(
x2out − x2in
2
)
−
∫ xout
xin
dxxV (x) =
=
(
µTF − γ2Ω2
(
2
s
− 1
))(
x2out − x2in
2
)
− γ2Ω2
∫ xout
xin
dxx(x− 1)22 (s− 2) =
=
1
piε2
. (2.1.20)
The condition ρTF(xin) = 0 and a Taylor expansion yields for the chemical potential
µTF = γ2Ω2
(
2
s
xsin − x2in
)
=
= µTF(xin) ' γ2Ω2
((
2
s
− 1
)
+ (xin − 1)22 (s− 2)
)
. (2.1.21)
This combined with (2.1.20) yields the width of the support given by
l ≡ xout − xin ∼ (γω)− 23 ·
(
1
pi4(s− 2)
)1/3
(1− o(1)). (2.1.22)
Taking the formula for the TF density and the estimate on the width implies
ρTF ≤ Cω2/3. (2.1.23)
Ginzburg-Landau type Functional
A key ingredient of our proof of the energy asymptotics is the energy functional
Fg[u] =
∫
R2
{∣∣∣(~∇− i ~A)u∣∣∣2 g2 + 1
ε2
g4|(1− |u|2)|2
}
(2.1.24)
with u(~r) being a complex function for a given real function g. It will be used to addi-
tively decouple the GP energy functional. It also provides the vortex lattice contribution
to the energy asymptotics.
2.2 Main result
The main result of this work concerns the energy asymptotics of the rotating BEC. It
is stated in particular in terms of the TF energy, which gives the leading order in this
parameter regime. The subsequent term is due to the kinetic energy of the vortex lattice.
Theorem 2.2.1 (Energy Asymptotics to EGP)
For ε sufficiently small, and if Ω satisfies ε−1  Ω ε−4
EGP = ETF +
Ω| log (ε4Ω) |
6
(1 + o(1)). (2.2.1)
66
2.3 Proofs
To proof the above theorem several particular problems need to be solved. First we
prove a decomposition of the GP functional (2.1.10), which enables us to estimate two
energy functionals seperately.
Proposition 2.3.1 (Energy Decoupling)
For any Ψ ∈ DGP satisfying the normalization condition ‖Ψ‖2 = 1 we define the decom-
position of the wave function by Ψ(~r) ≡ g(r)·u(~r), with g denoting the positive minimizer
of (2.1.11), and u(~r) beeing complex-valued. Thus, we have the energy decoupling
EGP[Ψ] = EˆGP + Fg[u]. (2.3.1)
with
Fg[u] =
∫
R2
{∣∣∣(~∇− i ~A)u∣∣∣2 g2 + 1
ε2
g4|(1− |u|2)|2
}
(2.3.2)
Proof: We write Ψ as Ψ = ϕ ·u with ϕ real-valued and positive and u complex-valued.
We use the identity ∫
|~∇Ψ|2 = −
∫
|u|2ϕ∆ϕ+
∫
ϕ2|~∇u|2 (2.3.3)
to rewrite the kinetic energy term in (2.1.10) as∫ ∣∣∣(~∇− i ~A)Ψ∣∣∣2 = −∫ |u|2ϕ∆ϕ+ ∫ ϕ2 ∣∣∣(~∇− i ~A)u∣∣∣2 , (2.3.4)
which enables us to write the GP functional as
EGP[Ψ] =
∫
R2
{
−|u|2ϕ∆ϕ+ ϕ2
∣∣∣(~∇− i ~A)u∣∣∣2 + V (x)|Ψ|2 + ε−2|Ψ|4} . (2.3.5)
We now choose ϕ = g and insert the corresponding variational equation (2.1.14) in
(2.3.5) and afterwards use (2.1.15) and the mass constraint ‖Ψ‖2 = 1 to obtain
EGP[ψ] =
∫
R2
{
|Ψ|2
(
− 2
ε2
g2 − V (x) + µ
)
+g2
∣∣∣(~∇− i ~A)u∣∣∣2 +V (x)|Ψ|2 +ε−2|Ψ|4} =
= EˆGP +
∫
R2
{∣∣∣(~∇− i ~A)u∣∣∣2 g2 + 1
ε2
g4|(1− |u|2)|2
}
+ EˆGP + Fg[u]. (2.3.6)

Next we find an upper bound for the minimum of (2.1.11). This together with an
approveriate lower bound allows to state precise asymptotics of the condensate GP
energy. A trivial lower bound for the EˆGP is given by ETF and sufficient to proof the
main statement [106]. In a subsequent section, however, we will present a refinement of
the lower bound.
67
Proposition 2.3.2 (Upper Bound to EˆGP)
For ε sufficiently small, and if Ω satisfies ε−1  Ω ε−4
EˆGP ≤ ETF + Cω4/3| log ε|. (2.3.7)
Proof: To prove an upper bound to the reduced GP energy we use as trial function a
regularization of the TF density defined by
ρ(x) =

ρTF(x) xin +  ≤ x ≤ xout − 
ρTF(xin + )
(x−xin)2
2
xout −  ≤ x ≤ xout
ρTF(xout − ) (x−xout)22 xin ≤ x ≤ xin + 
0 otherwise.
(2.3.8)
The norm of the trial function satisfies
‖ρ‖1 ≥ 1− CρTF(xin + ). (2.3.9)
Now, a Taylor expansion of the TF density yields
ρTF(x) = Cγ2
(
(εΩ)2/3 − (εΩ)2(x− 1)2) (1 + o(1)) =
= Cγ2(εΩ)2/3(1− t2)(1 + o(1)) (2.3.10)
with t = (εΩ)2/3|x− 1|. Hence, we have∣∣~∇ρTF∣∣ = Cγ2(εΩ)2|x− 1|(1 + o(1)) = Cγ2(εΩ)4/3|t|(1 + o(1)). (2.3.11)
This together with (2.1.20) implies
|~∇ρTF| ≤ C(εΩ)4/3 (2.3.12)
and therwith we have
ρTF(xin + ) ≤ C2(εΩ)4/3. (2.3.13)
So we find for the norm of our trial function
‖ρ‖1 ≥ 1− C2(εΩ)4/3. (2.3.14)
Note that in comparison the TF energy is
ETF ∼ −Ω2 + (εΩ)
2/3
ε2
. (2.3.15)
By the definition of the TF functional the potential error term to the potential TF
energy is smaller than Ω22(εΩ)4/3 and this term satisfies
Ω22(εΩ)4/3  (εΩ)
2/3
ε2
(2.3.16)
68
as long as   (εΩ)−4/3. Defining ζ ≡ (εΩ)2/3 this condition can be written as ζ 
(εΩ)−2/3. The upper bound to the potential error term succeeds the upper bound to the
interaction error term given by (εΩ)
2/3
ε2
2(εΩ)4/3, i.e.,
(εΩ)2/3
ε2
 Ω2, (2.3.17)
if Ω 1/ε, which is an assumption in our proof. Now, defining 1− t2 = u we get∫
t∈C·[0,1−]
|~∇ρTF|2
ρTF
= C(εΩ)4/3
∫
t∈C·[0,1−]
t3
1− t2dt ≤
≤ C(εΩ)4/3
∫
u∈C·[,1]
1− u
u
du ≤ C(εΩ)4/3| log |. (2.3.18)
We have as condition for the upper bound to the kinetic energy term to be smaller than
the lower order term of the TF energy,
(εΩ)4/3| log |  (εΩ)
2/3
ε2
, (2.3.19)
which is true as long as Ω ε−4| log ˜|−1 for some appropriate adaption of  denoted by
˜. More important the upper bound to the kinetic energy succeeds the error term
Ω22(εΩ)4/3  (εΩ)4/3| log | (2.3.20)
for an appropriate . Summarizing above estimates we get
EˆGP ≤ ETF + C(γω) 43 | log |+O (Ω22(εΩ)4/3)+O((εΩ)2/3
ε2
2(εΩ)
)
=
= ETF +O ((εΩ)4/3| log ζ(εΩ)−2/3|)+O (Ω2ζ2)+O((εΩ)2/3
ε2
ζ2
)
. (2.3.21)
Minimizing the r.h.s. of (2.3.21) in terms of ζ yields
ζ =
(εΩ)2/3
Ω
 (εΩ)−2/3 (2.3.22)
as long as Ω  ε−4. Hence, we have proved all terms beside the TF energy terms are
smaller then those of the TF energy for the regime ε−1  Ω  ε−4| log ˜|−1 and the
main contribution beyond the TF energy is estimated in (2.3.18).

69
Useful estimates on and properties of g2
Proposition 2.3.3 (The Minimizer Achieves a Maximum at a Unique Radius.)
The minimizer achieves a maximum at the unique radius denoted by RGP.
Proof: We rewrite (2.1.11) by a variable transformation r2 → k, such that the functional
EˆGP[g] is
pi
∫ ∞
0
k
{
|~∇g|2 + 1
ε2
g4 + V ′(k)g2
}
dk, (2.3.23)
and the normalization condition is ∫ 1
0
g2dk = 1/pi. (2.3.24)
The variational equation (2.1.14) implies that g is not constant on any open interval
(otherwise g ≡ 0 that contradicts the mass constraint ‖g‖22 = 1). This is a consequence
of the fact that g satisfies a Lipschitz condition.
Suppose g(x) has more than one local maximum. Then it has a minimum at some
point k = k2 with 0 < k2 < ∞, positioned on the right side of a maximum at k = k1,
i.e., k1 < k2. For 0 < ε < g
2(k1)− g2(k2) we consider the set
Iε = {k < k2 : g2(k1)− ε ≤ g2(k) ≤ g2(k1)}. (2.3.25)
Since g2 is continuous, the function
F (ε) ≡
∫
Iε
g2dk (2.3.26)
is strictly positive and F (ε)→ 0 for ε→ 0. Likewise, for a κ > 0 we consider
Jκ = {k > k1 : g2(k2) ≤ g2(k) ≤ g2(k2) + κ}. (2.3.27)
So, the function
G(κ) ≡
∫
Jκ
g2dk (2.3.28)
has the same properties as F . There is at least another maximum at k = k3. For
0 < δ < g2(k3)− g2(k2) we consider the set
Kδ = {k > k2 : g2(k3)− δ ≤ g2(k) ≤ g2(k3)}. (2.3.29)
Again since g2 is continuous and strictly positive, the function
H(ε) ≡
∫
Kδ
g2dk (2.3.30)
is strictly positive as well and F (ε)→ 0 for ε→ 0.
70
Since g2 is a continuous function there always exist δ, ε, κ > 0 with g2(k2) + κ <
g2(k1) − ε and g2(k2) + κ < g2(k3) − δ. This implies that Iε, Jκ and Kδ are disjoint
sets. Since the integrand of F , G and H is positiv and continuous, G is continuous in
κ as well as F is continuous in ε etc., we are able to specify those parameters by the
condition
F (ε) = G(κ) = H(δ). (2.3.31)
Given any potential V with a unique minimum. Then we have V (k1) > V (k2) or
V (k3) > V (k2). If the first case applies mass can be rearranged from k1 to k2, i.e., we
consider the function
g21(k) =

g(k1)
2 − ε if k ∈ Iε
g(k2)
2 + κ if k ∈ Jκ
g(k)2 otherwise
(2.3.32)
The energy due to the potential term V in (2.3.23) evaluated for (2.3.32) is smaller than
it is for g2. Also the mass constraint of this function equals the normalization condition
(2.3.24): What we subtract from g2 in Iε equals what we add to g2 in Jκ, or rather
F (ε) = G(κ). Furthermore, the kinetic energy of g2 vanishes on the sets Iε and Jκ, but
doesn’t differ from g2 elsewhere. Therefore it is smaller than the kinetic energy of g2.
By the definition of g2, mass is rearranged from Iε to Jκ, where the density is lower,
such that ∫
g4 <
∫
g4. (2.3.33)
If we consider V (k3) > V (k2) we have to put mass from k3 to k2 to lower the energy,
i.e., we define a function
g22(k) =

g(k2)
2 + κ if k ∈ Jκ
g(k2)
2 − δ if k ∈ Kδ
g(k)2 otherwise
(2.3.34)
We find that the energy due to the potential term V in (2.3.23) evaluated for (2.3.34) is
smaller than it is for g2. The arguementation in the previous case for the kinetic energy
and the interaction term can be applied straightforewardly to this case, which leads us
to the following conclution.
The functional evaluated for g2? (? = 1, 2) is strictly smaller. This contradicts the
assumption that g2 is the minimizer. Therefore by the construction g2? the minimizer g
has only one maximum.

Proposition 2.3.4 (Difference between the chemical potentials µ and µTF)
The difference between the chemical potentials can be estimated by
|µ− µTF| ≤ CΩ| log ε|1/2 + Cω 43 | log ε|. (2.3.35)
71
Proof: For the sake of brevity we define A ≡ Cω 43 | log ε|. By the definition of the TF
functional (2.1.16), the TF density (2.1.18), the chemical potential µTF (2.1.19) and the
upper bound for EˆGP we find the estimate∫
R2
d~xx(g2 − ρTF)2 = ‖g‖44 + ‖ρTF‖22 − 2
∫
R2
d~xg2ρTF ≤
‖g‖44 − (γεΩ)2
∫
R2
d~xx2g2 +
∫
R2
d~x · 2
s
xsg2 + ‖ρTF‖22 − ε2µTF ≤
≤ ε2 (ETF[g2]− ETF) ≤ ε2 (EˆGP − ETF) ≡ ε2A, (2.3.36)
where we used an abberivation in the last step. Using Cauchy-Schwarz inequality, the
trivial bound ‖ρTF‖22 ≤ ‖ρTF‖∞ = ρTF(1) yields
‖g‖44 − ‖ρTF‖22 = 2
∫
R2
d~xρTF(g2 − ρTF) +
∫
R2
d~x(g2 − ρTF)2 ≤ Cε (ρTF(1)A)1/2 + ε2A.
(2.3.37)
As a consequence we get an estimate for the difference between the chemical potentials
µTF and µ,
ε2
∣∣µ− µTF∣∣ = ε2 (EˆGP − ETF)+ ‖g‖44 − ‖ρTF‖22 ≤ Cε (ρTF(1)A)1/2 + ε2A. (2.3.38)
Using ρTF(1) ≤ Cω 23 yields the result.

Proposition 2.3.5 (Pointwise Estimate for g(x)2 on ATF)
Suppose ε→ 0 then for
~x ∈ ATF ≡ {~x : x ∈ [xin + max[CΩ−1/2, ε1/3ω−5/9], xout −max[CΩ−1/2, ε1/3ω−5/9]]}
(2.3.39)
we have
|g(x)2 − ρTF(x)| ≤ ρTF(x)o(1). (2.3.40)
Proof: At first we rewrite the variational equation (2.1.14) for the minimizer g as
−∆g = 2
ε2
[
ρ˜(x)− g2] g (2.3.41)
with the density
ρ˜(x) ≡ ε
2
2
(
µ− Ω2
(
2xs
s
− x2
))
. (2.3.42)
An important fact is that above function differs from the TF density only by its chemical
potential.
Now we determine the set where ρ˜ = ρTF(1± o(1)): One finds by the estimate to the
difference between the chemical potentials (2.3.35)
‖ρ˜(x)− ρTF‖∞ ≤ Cε2Ω| log ε|1/2, (2.3.43)
72
By a simple Taylor approximation of the density we determine the set of ~x such that the
difference of the chemical potentials is beyond the first order by x ∈ [xin +CΩ−1/2, xout−
CΩ−1/2] and denote it as A. Moreover, we split this set into the part defined by the
property x ≤ 1 denoted by A< and the complement A> with the property x > 1.
The following strategy in this proof is to find pointwise estimates to g2 on A by
providing suitable super- and subsolutions inside a local interval x ∈ [x0 − κ, x0 + κ]
where x˜in + κ < x0 < x˜out − κ with κ ≥ 0 and x˜in denoting the inner radius of the set
A and x˜out the outer radius. We refer to [107] chapter 9.3 for the justification of the
method of sub- and supersolutions.
As candidate for the supersolution on A< we consider a function of the form
W (x) =
=
√
ρ˜(x0 + κ) coth
[
coth−1
(√
ρ˜(1)
ρ˜(x0 + κ)
)
+
κ2 − |x− x0|2
3κε
√
2ρ˜(x0 + κ)
]
. (2.3.44)
As it was shown in [108] one has
−∆W ≥ 2
ε2
(
ρ˜(x0 + κ)−W 2
)
W ≥ 2
ε2
(
ρ˜(x0)−W 2
)
W, (2.3.45)
for any x ∈ [x0 − κ, x0 + κ], because ρ˜(x) is an increasing function in x for x ≤ 1. Note
that the proof for the domain A> is very similar, with the important distinction that
ρ˜(x) is an decreasing function in x. Thus, the candidate for the subsolution (2.3.44)
has to be adapted by κ → −κ. For the sake of brevity we turn again to the case A<
and make notes on the other case when it is necessary. At the boundary of the interval
the function (2.3.44) coincides with the density ρ˜(x), i.e, W (x0 − κ) = W (x0 + κ) =√
ρ˜(1) ≥√ρ˜(RGP/R = xGP), which is larger than g(xGP). Hence W is a supersolution
to g and by the maximum principle we have
g(x0) ≤ W (x0) ≤
√
ρ˜(x0 + κ) coth
[ κ
3ε
√
2ρ˜(x0 + κ)
]
, (2.3.46)
where we have used the fact that coth(x) is a nonincreasing function. Above inequality
can be written as
g(x0) ≤ W (x0) ≤
√
ρ˜(x0)
(
1 +
|ρ˜(x0 + κ)− ρ˜(x0)|
ρ˜(x0)
)
coth
[ κ
3ε
√
2ρ˜(x0 + κ)
]
. (2.3.47)
When the argument in coth(x) tends to infinity one has
coth(x) =
1 + e−2x
1− e−2x ≤ (1 + Ce
−2x) (2.3.48)
The argument of coth in our formula (2.3.47) tends to infinity if the condition
κ
ε
√
ρ˜(x0 + κ) 1 (2.3.49)
73
is satisfied. Assuming that is the case we have
g(x0) ≤
√
ρ˜(x0)
(
1 +
|ρ˜(x0 + κ)− ρ˜(x0)|
ρ˜(x0)
)
(1 + o(1)) . (2.3.50)
To estimate the second term in the first brackets of the r.h.s. of (2.3.50) we use
|(ρ˜(x+ κ)− ρ˜(x))| = Cκω4/3 (2.3.51)
This together with (2.3.43) implies
|(ρ˜(x+ κ)− ρ˜(x))|
ρ˜(x)
≤ Cκω
4/3
ρ˜(x)
 1 (2.3.52)
To fulfill both conditions ρ˜  (εω4/3)2/3. We summarize our finding by stating the
result
g(x) ≤
√
ρ˜(x) (1 + o(1)) . (2.3.53)
for any x ∈ [x0 − κ, x0 + κ]. To cover the whole set A we extend the estimate to
x˜in ≤ x ≤ x˜out with the help of the formula (2.3.52).
As before we fix some x0 in a interval x˜in + δ ≤ x0 ≤ x˜out − δ. Since ρ˜(x) is an
increasing function for x ≤ 1 and g is positive,
−∆g ≥ 2
ε2
[
ρ˜(x0 − δ)− g2
]
g. (2.3.54)
for any x ∈ [x0 − δ, x0 + δ]. We find a subsolution by imposing a Dirichlet boundary
condition to the same problem on the boundary ∂B1. In [109] it was proven that there
is a unique positive function h satisfying{
−∆h = 1/ε˜2[1− h2]h in Ω
h = 0 on ∂Ω
(2.3.55)
as ε˜→ 0 and it is stated that h ≤ 1. In particular, if we consider the domain Ω = B1, it
follows from the uniqueness of the positive solution h, that it is radially symmetric. In
[110] it was proven that there is a lower bound to h, namely
1− C exp
{
−dist(r, ∂Ω)
2ε˜
}
≤ h. (2.3.56)
This implies for our problem with Ω = B1
1− C exp
{
−1− x
2
2ε˜
}
≤ h(x) ≤ 1. (2.3.57)
If we now define
h˜(x) ≡
√
ρ˜(x0 − δ)h
(
x− x0
δ
)
(2.3.58)
74
and
ε˜ ≡ ε
δ
√
2ρ˜(x0 − δ)
, (2.3.59)
then for any ~x ∈ B(x0 − δ, x0 + δ) the function h˜ solves
−∆h˜ = 2
ε2
[
ρ˜(x0 − δ)− h˜2
]
h˜, (2.3.60)
with Dirichlet conditions at the boundary x = x0 ± δ. Therefore, h˜ is a subsolution for
the problem (2.3.41) and the maximum principle states g(x) ≥ h˜(x): By (2.3.52) with a
similar condition as on κ but now for δ we have
g(x0) ≥ h˜(x0) ≥
√
ρ˜(x0)
[
1− C exp
(
− 1
2ε˜
)]
(1 + o(1)) (2.3.61)
for any x0 such that x˜in + δ ≤ x0 ≤ 1− δ and find the condition
ε˜ 1, (2.3.62)
which is the analogon to (2.3.49) in the proof of the upper bound to get
g(x0) ≥
√
ρ˜(x0) (1 + o(1)) . (2.3.63)
The proof for the case A> can be done analogously but with δ → −δ. To extend (2.3.63)
to the boundaries we use (2.3.52).

Proposition 2.3.6 (Exponential Smallness of the Minimizer)
Suppose that ε→ 0 and ε−1  Ω, then on the set
T ≡ {~x ∈ R2 : x2 ≤ x2in − Ω−1/2ε1/2| log ε|}, (2.3.64)
the minimizer g satisfies
g2(x) ≤ Cg2(xGP) exp
{
− x
2
in − x2
Ω−1/2ε| log ε|
}
. (2.3.65)
Proof: As starting point we take the inequality −∆(g2) ≤ −2g∆g, and insert the
variational equation for g (2.1.14) to obtain
−∆(g2) ≤ 2
ε2
(
ε2
2
(
µ− Ω2
(
2xs
s
− x2
))
− g2
)
g2. (2.3.66)
Now we rewrite this inequality as
−∆(g2) ≤ 2
ε2
(
ρTF(x) + ε2
(
µ− µTF)− g2) g2 (2.3.67)
75
In the following we consider the set defined by
T ≡ {~x ∈ R21 : x2 ≤ x2in − z2}, (2.3.68)
and find on this set by the estimate of the difference between the chemical potentials
(2.3.35)
ρTF(x) + ε2
(
µ− µTF) ≤ −Cω2 · z2 + Cε2Ω| log ε|1/2 ≤ −Cω2 · z2 (2.3.69)
for z2  Ω−1| log ε|1/2. Combining (2.3.67) and (2.3.69) we find the following differential
equation
−∆(g2) + C
ε2
ω2z2g2 ≤ 0. (2.3.70)
Now we consider the function
W (x) = exp
{
−x
2
in − x2
f
}
, (2.3.71)
For z2  f the function W is small on the boundary ∂T . Exponential smallness requires
that
z2
f
≡
(
1
ε
)α
(2.3.72)
with α > 0. For any x ≤ xin the function (2.3.71) satisfies
−∆W + C
ε2
ω2z2 ·W ≥ W
(
−C
(
4x2in
f 2
+
2
f
)
+
C
ε2
ω2z2
)
≥ 0 (2.3.73)
for any
z  C
fΩ
(2.3.74)
We choose z = Ω−1/4ε1/4| log ε|1/2 . If we multiply W by g(x)2 we obtain a supersolution
to the solution of (2.3.70), i.e.,
g(x)2 ≤ Cg(xGP)2W (x), (2.3.75)
for any x2 ≤ x2in − z2.

2.3.1 Estimates on the Ginzburg-Landau Functional for 1/ε Ω
Theorem 2.3.1 (Lower Bound to the GL functional)
For ε−1  Ω ε−4 as ε→ 0 we have
Fg[u] ≥ (1− o(1)) Ω| log (ε
2Ω/ρ) |
2
(2.3.76)
with ρ ∼ (εΩ)2/3.
76
Proof: To deal with the dependence of (2.3.2) on g2(x) we decompose the integration
domain into small cells: Let Lˆ be the lattice
Lˆ =
{
~xi = (mlˆ, nlˆ),m, n ∈ Z|Qi ⊂ ATF
}
(2.3.77)
where Qi denotes the lattice cell centered at ~xi ∈ Lˆ and the lattice constant satisfies√
| log ε|
Ω
 lˆ Cω−1/3. (2.3.78)
By reducing the integration domain to ATF and with the help of (2.3.40) we obtain
Fg[u] ≥
∫
ATF
d~x
{
1
ε2
g4
(
1− |u|2)2 + ∣∣∣(~∇− i ~A)u∣∣∣2 g2} =
=
∑
~xi∈Lˆ
∫
Qi
d~xρTF(x)
{
1
ε2
g2
(
1− |u|2)2 + ∣∣∣(~∇− i ~A)u∣∣∣2} (1− o(1)) =
=
∑
~xi∈Lˆ
ρTF(xi)E ′(i)[u](1− o(1)). (2.3.79)
In the last step we have defined the functional
E ′(i)[u] ≡
∫
Qi
d~x
{
1
ε2
g2
(
1− |u|2)2 + ∣∣∣(~∇− i ~A)u∣∣∣2} . (2.3.80)
As above we can exchange g2 with ρTF and a remainder by the pointwise bound (2.3.40),
E ′(i)[u] ≡
∫
Qi
d~x
{
1
ε2
ρTF
(
1− |u|2)2 + ∣∣∣(~∇− i ~A)u∣∣∣2} (1− o(1)) ≡
≡ E (i)[u](1− o(1)). (2.3.81)
Proposition 2.3.7 (Lower Bound inside cells)
If ε−1  Ω ε−4 as ε→ 0 , it is possible to find a lˆ satisfying (2.3.78) such that
E (i)[u] ≥ (1− o(1)) Ωlˆ
2| log (ε2Ω/ρ) |
2
(2.3.82)
with ρ ∼ (εΩ)2/3.
Proof: As in [111] the strategy is to rescale the cells Qi in such a way that we consider
the minimization of a GL functional in a different regime. For this purpose we set
~s ≡ lˆ−1(~x− ~xi),
u˜(~s) ≡ u(~xi + lˆ~s) ~B(~s) ≡ lˆ ~A′(~xi + lˆ~s) (2.3.83)
In those coordinates (2.3.81) is
E (i)[u] = E˜ (i)[u˜] ≡
∫
Q1
d~s
{
1
ε2
ρTF(xi)lˆ
2
(
1− |u˜|2)2 + ∣∣∣(~∇− i ~B) u˜∣∣∣2} . (2.3.84)
77
with Q1 being an unitary square centered at the origin. Note that the rescaled vector
potential ~B is explicitly given by
~B(~s) =
Ωlˆ~ez ∧ ~xi
2
+
Ωlˆ2~ez ∧ ~s
2
, (2.3.85)
and the corresponding magnetic field is
h˜ ≡ curl ~B = Ωlˆ2. (2.3.86)
Our concern is the minimization of (2.3.84). By gauge invariance one can simplify the
functional to
inf
u˜∈H1(Q1)
E˜ (i)[u˜] ≥ inf
u˜∈H1(Q1)
∫
Q1
d~s
{
1
ε2
ρTF(xi)lˆ
2(1− |u˜|2)2 +
∣∣∣(~∇− ilˆ2 ~A) u˜∣∣∣2} . (2.3.87)
We now introduce a new infinitesimal parameter  defined as
 ≡ ε
lˆ
√
ρTF(xi)
 1. (2.3.88)
It follows that
E (i)[u] ≥ inf
u˜∈H1(Q1)
∫
Q1
d~s
{∣∣∣∣(∇− ih˜ex~ez ∧ ~s2
)
u˜
∣∣∣∣2 + 12 (1− |u˜|2)2
}
, (2.3.89)
for a magnetic field h˜ex satisfying the conditions
| log |  h˜ex ≡ Ωlˆ2  1
2
≡ ρ
TFlˆ2
ε2
. (2.3.90)
The upper bound on the r.h.s. of (2.3.90) implies the condition
ρTF  ε2Ω. (2.3.91)
Additionally, to ensure that the cell size is much smaller than the width of the annulus
the condition
lˆ g−2 ∼ ω−2/3 (2.3.92)
has to be satisfied. The upper bound of the first condition,i.e.,
| log |  Ωl2  −2 ≡ l
2g2
ε2
(2.3.93)
is satisfied as long as
Ω ε−4 (2.3.94)
The lower bound is simultaneously fulfilled as long as  → 0. The second upper bound
(2.3.92) enforces as estimate for the quantity Ωl2
Ωl2 . Ω
ω4/3
(ε4Ω)α (2.3.95)
78
for some α ≥ 0 and with the help of (2.3.94). The lower bound in (2.3.90) can be
satisfied simultaneously as long as
| log(ε4Ω)|  (ε4Ω)α− 13 , (2.3.96)
where we assumed | log | ∼ log(ε4Ω)|. This is always fulfilled as long as 0 < α < 1/3,
because
| log x|  x−α. (2.3.97)
For our proof of the lower bound we need that the condition
lˆ
√
ρTF(xi) 1, (2.3.98)
holds, but this is satisfied as a consequence of (2.3.92). (Note that we have 1/
√
ρTF ≥
Cω−1/3). Hence
Ωlˆ2 ≥ | log ε| ≥ | log |, (2.3.99)
because 0 ≥ log  = log ε − log
(
lˆ
√
ρTF(xi)
)
, which implies log  ≥ log ε and | log | ≤
| log ε|.
The functional on the right hand side of (2.3.89) is precisely the GL functional on Q1
with external magnetic field h˜ex~ez and parameter , i.e.,
E˜GL
[
u˜, ~A
]
≡
∫
Q1
d~s
{∣∣∣(∇− i ~A) u˜∣∣∣2 + ∣∣∣curl ~A′ − h˜ex~ez∣∣∣2 + −2 (1− |u˜|2)2} , (2.3.100)
evaluated on the configuration(
u˜ , ~A′
)
=
(
u , h˜ex()~ez ∧ ~s/2
)
(2.3.101)
and (2.3.90) corresponds to the GL regime where the external magnetic field is between
the first and the second critical fields. We can thus apply the lower bound for the GL
functional proven in [112], Theorem 1.1 (note that in the definition of the GL functional
given in [112] there is overall factor 1/2), to get
E (i)[u] ≥ (1− o(1))h˜ex log 1

√
h˜ex
= (1− o(1))Ωlˆ
2
2
log
ρTF(xi)
ε2Ω
≥
≥ (1− o(1)) Ωlˆ
2| log (ε2Ω/ρ) |
2
(2.3.102)
with ρ ∼ (εΩ)2/3 beeing the order of magnitude of the TF density ρTF.

We note that the normalization of ρTF over the set where ρTF(xi)  ε2Ω holds is
1− o(1) if Ω ε−4.
79
Collecting the lower bounds inside all cells proven in the proposition above, we have
E˜GP [u] ≥ (1− o(1)) Ωlˆ
2| log (ε2Ω/ρ) |
2
∑
~xi∈L
ρTF(xi)(1− o(1)). (2.3.103)
To replace the Riemann sum by the integral we use the symmetry of the lattice cell and
the L1−normalization of ρTF,∑
~xi∈L
ρTF(xi) ≥ 1
lˆ2
(∫
∪iQi
d~x ρTF(x)− C max[lˆ, ωlˆ]
)
≥ (1− o(1))
lˆ2
. (2.3.104)

Theorem 2.3.2 (Upper Bound to the GL functional)
For ε−1  Ω ε−4 as ε→ 0 we have
Fg[u] ≤ (1 + o(1)) Ω| log (ε
2Ω/ρ) |
2
. (2.3.105)
with ρ ∼ ω2/3.
Proof: We start with the trial function for the rescaled GL functional Fg[u]: Similarily
to [111] we decompose the support of our trial function R2 into cells Qi whose centers
~xi are arranged in a regular lattice denoted by L. The lattice constant l is choosen such
that each cells area is
|Qi| = 2pi · Ω−1. (2.3.106)
and the lattice constant is given by
l = (const.)Ω−1/2. (2.3.107)
Hence, as in [111] the total number of cells in the unit disc B1 is
N =
Ω
2
(1 + o(1))) . (2.3.108)
Now, since the area of support for the TF density is ∼ ω 23 the total number of lattice
points inside supp
(
ρTF
)
is given by
NTF = N · ω− 23 . (2.3.109)
The position vector ~x ∈ R2 can also be considered as a complex number ζ = x+ iy ∈ C.
Hence, we define a phase function by
v(~x) ≡
∏
ζi∈L
ζ − ζi
|ζ − ζi| . (2.3.110)
80
To avoid singularities at the lattice points in the trial function we multiply v with a
function defined as
ξ(~x) ≡
{
1 if |ζ − ζi| > t for all i
t−1|ζ − ζi| if |ζ − ζi| ≤ t
(2.3.111)
where t is a variational parameter. Thus, we choose our trial function for the upper
bound to (2.3.2) to be
h = ξv. (2.3.112)
Proposition 2.3.8 (Vortex Kinetic Energy)
For ε→ 0 and Ω ε−4 we have∫
R2
g2|(~∇− i ~A)u|2 ≤ (1 + o(1)) Ω| log (t
2Ω) |
2
(2.3.113)
Proof: Since v is a phase and ξ real-valued we obtain∫
R2
g2|(~∇− i ~A)vξ|2 =
∑
i
∫
|ζ−ζi|≤t
g2|~∇(ξ)v|2 +
∫
R2
g2|iξ ~∇(v)− i ~Aξv|2 =
=
∑
i
∫
|ζ−ζi|≤t
g2|~∇ξ|2 +
∫
R2
g2|ξ ~E|2. (2.3.114)
We turn to the first term in (2.3.114). By definition we have |~∇ξ| = t−1 inside each
vortex disc Bit and the gradient is zero in the complement. Inserting the estimate g2(x) ≤
ρTF(x)(1 + o(1)) inside ATF, we obtain for the first term on the r.h.s. in (2.3.114) the
estimate
‖g~∇ξ‖22 ≤ C/t2
∑
i
∫
Bit∩ATF
ρTF(x) (1 + o(1)) + remainder. (2.3.115)
The remainder in (2.3.115) is inside T exponential small, because g2 satisfies on this set
(2.3.65). The area of the remaining set R2\{ATF ∪ T } ≡ W is
AW ≤ Cε1/3 (2.3.116)
and the number of vortices in W satisfies
NW ≤ CN · ε1/3. (2.3.117)
Hence, we find for the remainder in (2.3.115) by Cauchy-Schwarz inequality and using
g2 ≤ Cω2/3
remainder ≤ Cω2/3Ω1/2ε1/6. (2.3.118)
By the number of vortices in the support of ρTF, i.e., NTF and ‖ρTF‖L1(ATF) = 1− o(1)
we find argueing as we did for the remainder
C
t2
∑
i
∫
Bit∩ATF
ρTF(x) (1 + o(1)) ≤ Cω1/3Ω1/2. (2.3.119)
81
Next, we turn to the second term in (2.3.114). We use the property that g2 is close to
ρTF in ATF, i.e., g2/ρTF ≤ (1 + o(1)) justified by the pointwise bound (2.3.40) to find∫
R2\ATF
d~xg2(x)ξ2(x)| ~E(~x)|2 +
∫
ATF
d~x
g2
ρTF
(x) · ρTF(x)ξ2(x)| ~E(~x)|2 ≤
≤
∫
R2\ATF
d~xg2(x)ξ2(x)| ~E(~x)|2 +
∫
ATF
d~xρTF(x)ξ2(x)| ~E(~x)|2 · (1 + o(1)) ≤
≤
∫
R2\ATF
d~xg2(x)ξ2(x)| ~E(~x)|2+
+
(
1 +O((t2Ω)1/2))∑
i
sup
~r∈Qi
ρTF(~x)
(
pi
∣∣log (t2Ω)∣∣+O(1)) · (1 + o(1)) . (2.3.120)
To get the second term in the last line of above inequality we use a calculation already
made in the paper [111]. Recall from [111] the facts
| ~E(~x)|2 ≤ | ~Ei(~x)|2 + const.
(
Ω1/2
|~x− ~xi| + Ω
)
(2.3.121)
and
| ~Ei(~x)|2 ≤ 1|~x− ~xi|2 . (2.3.122)
Hence they find∫
Qi\Bit
d~x | ~E(~x)|2 −
∫
Qi\Bit
d~x | ~Ei(~x)|2 ≤
≤ const.
∫ C Ω′−1/2
t
dx x
(
Ω1/2x−1 + Ω
)
= O(1) (2.3.123)
while∫
Bit
d~x ξ(~x)2| ~E(~x)|2 −
∫
Bit
d~x ξ(~x)2| ~Ei(~x)|2 ≤
≤ const.
∫ t
0
dx x (x/t)2(Ω1/2x−1 + Ω) = O((t2Ω)1/2). (2.3.124)
On the other hand, since ~Ei(~x) ≤ |~x− ~xi|−1,∫
Qi\Bit
d~x | ~Ei(~x)|2 ≤ 2pi
∫ C Ω−1/2
t
dx x x−2 = pi| log(t2Ω)|+O(1) (2.3.125)
and ∫
Bit
d~x ξ(~x)2| ~Ei(~x)|2 ≤ 2pi
∫ t
0
dx x (x/t)2x−2 = O(1). (2.3.126)
82
We now estimate the Riemann approximation error
R ≡ |Q0|
∑
i ~x∈Qi
ρTF(~x)−
∫
ATF
d~xρTF(~x) ≤
≤ |Q0|
∑
i
{
sup
~x∈Qi
ρTF(~x)− inf
~x∈Qi
ρTF(~x)
}
. (2.3.127)
Multiplying the area of a cell times the value of the lattice constant l and ‖dρTF/dx‖∞ ≤
Cω4/3 and the number of vortices inside supp
(
ρTF
)
is
R ≤ C|Qi| · l · ω4/3 ·NTF  1 (2.3.128)
as long as Ω ε−4. Hence,∫
R2
d~xg2(x)ξ2(x)| ~E(~x)|2 ≤
∫
R2\ATF
d~x · ξ2(x)g2(x)| ~E(~x)|2+
+
(
1 +O((t2Ω)1/2)) |Q0|−1 (pi ∣∣log (t2Ω)∣∣+O(1)) · (1 + o(1)) . (2.3.129)
It remains to estimate the first term on the r.h.s. of (2.3.129). By the definition of ξ
and since | ~Ei(~x)|2 ≤ 1|~x−~xi|2 we have∫
W
d~x ·ξ2(x)g2(x)| ~E(~x)|2 ≤
∫
W
d~xg2(x)
ξ2(x)
|~x− ~xi|2 +C
∫
W
d~xg2(x)
Ω1/2
|~x− ~xi|+CΩ ·o(1) ≤
≤
∫
W
d~xg2(x)
ξ2(x)
|~x− ~xi|2 + CΩ (2.3.130)
For the first term on the r.h.s. of (2.3.130) we find with NW ≤ C max[Ω−1/2, ε1/3ω−5/9]
(see pointwise estimate and exponential smallness)∫
W
d~xg2(x)
ξ2(x)
|~x− ~xi|2  CΩ ·
(
1 +O((t2Ω)1/2)) (pi ∣∣log (t2Ω)∣∣+O(1)) (2.3.131)
as long as Ω ε−4.

Proposition 2.3.9 (Vortex Interaction Energy)
For ε→ 0 and ε−1  Ω . ε−4 we have∫
R2
d~x
{
g4
ε2
(1− |u|2)2
}
≤ CΩ. (2.3.132)
83
Proof: First we consider the contribution to the vortex interaction energy due to W .
Since g2 ≤ CΩ−1/2ω4/3 and with NW ≤ C max[Ω−1/2, ε1/3ω−5/9] (see pointwise estimate
and exponential smallness) we have
∑
i
1
ε2
∫
Bit∩W
g4
(
1−
(x
t
)2)2
(1 + o(1)) ≤
≤ Cω
4/3
ε2
∑
i
∫
Bit∩W
(
1−
(x
t
)2)2
(1 + o(1)) ≤
≤ Cω4/3ε−2Ω−1/2Ω · t2 + Cω4/3ε−2ε1/3ω−5/9Ω · t2 ≤ CΩ. (2.3.133)
Using ρTF ≤ Cω2/3 we find for the contribution due to ATF
∑
i
1
ε2
∫
Bit∩ATF
(ρTF)2
(
1−
(x
t
)2)2
(1 + o(1)) ≤
≤ Cω
2/3
ε2
∑
i
sup
~r∈Qi
ρTF(xi)
∫
Bit
(
1−
(x
t
)2)2
(1 + o(1)) ≤
≤ Ct
2ω2/3
ε2
∑
i
sup
~x∈Qi
ρTF(xi)(1 + o(1)) ≤ Ct
2Ωω2/3
ε2
(1 + o(1)). (2.3.134)
Choosing t2 = ε2/ρ with ρ = ω2/3 yields the result.


2.4 Lower bound for the kinetic energy
This section is devoted to present a novel lower bound to the kinetic energy term of the
reduced GP energy functional.
Theorem 2.4.1 (Lower Bound to reduced kinetic energy)
For ε sufficiently small, and if Ω satisfies ε−1  Ω ε−4
∫
R2
|~∇g|2 ≥ (1 + o(1))
∫
ATF+
(
~∇ρTF
)2
ρTF
. (2.4.1)
with ATF+ = ATF ∩ {x : x ≤ 1}.
Proof: The starting point for our proof is the definition of the spherically symmetric
rescaled GP density in terms of the corresponding variational equation
ρGP(x) ≡ g2(x) = ε
2
2
(
µ− µTF + 2
ε2
ρTF +
∆g
g
)
(x) (2.4.2)
84
and let us by the way recall the explicitly given rescaled TF density
ρTF(x) =
ε2
2
(
µTF − Ω2
(
2xs
s
− x2
))
. (2.4.3)
A crucial point in our argumentation is proposition 4.3.5, i.e.,
ρGP(x) = ρTF (1 + o(1)) (x) (2.4.4)
holds for some specified set ATF ⊆ R2. On the other hand this equation determines the
asymptotically small function o(1)(x) in terms of the introduced densities. Clearly, by
the radial symmetry of the densities o(1) is radial symmetric too. But be aware that in
the following the symbol o(1) denotes different functions as well, but they all share the
property of being o(1) for all x ∈ ATF and having their origin in above function o(1) are
radially symmetric. Now, in particular (2.4.4) says considering the definition of the GP
density (or rather the variational equation) (2.4.2)
∆g
g
(x) =
o(1)
ε2
ρTF(x) for x ∈ ATF ⊆ R2, (2.4.5)
which is a useful insight in the following argumentation. For the sake of analogy to
previous estimates [111] on the kinetic energy for the TF density we rewrite the kinetic
energy of the reduced GP functional due to its minimizer g in terms of its GP density
(2.4.2) yielding
∫
R2
|~∇g|2 =
∫
R2
|~∇ρGP|2
ρGP
=
∫
R2
(
~∇ρTF + ε2
2
~∇
(
∆g
g
))2
ρGP
. (2.4.6)
In difference to the kinetic energy from the TF density (2.4.6) has two additional terms
involving the GP minimizer and the divisor is now the GP density. However, we show
that the leading order kinetic energy is exactly the kinetic energy due to the TF density.
To do so we exploit the (pointwise) positivity of the integrand to reduce the integration
domain and by using (2.4.4) on this subdomain we get an estimate to (2.4.6) from below:
∫
R2
|~∇g|2 ≥
∫
ATF+
(
~∇ρTF + ε2
2
~∇
(
∆g
g
))2
ρGP
≥
≥
∫
ATF+
(
~∇ρTF + ε2
2
~∇
(
∆g
g
))2
ρTF
(1 + o(1))(r) ≥
≥ inf
ζ∈ATF+
(1 + o(1))(ζ)
∫
ATF+
(
~∇ρTF + ε2
2
~∇
(
∆g
g
))2
ρTF
(2.4.7)
85
Here the domain ATF+ ⊂ ATF ⊆ R2 is specified later. Furthermore, we suppress the
quadratic and therefore positive term due to the minimizer g within the integrand to
get ∫
R2
|~∇g|2 ≥ (1 + o(1))
∫
ATF+
(~∇ρTF)2 + ~∇ρTF · ε2~∇
(
∆g
g
)
ρTF
. (2.4.8)
Our knowledge about the (to us) not explicitly known term ~∇
(
∆g
g
)
is quite limited,
but having (2.4.5) in our pocket we proceed by using partial integration to disburden
this term from the gradient, where in addition we exploit the rotational symmetry of
the problem:
2pi
∫ xo
xi
(
d
dx
ρTF
)2
+ d
dx
ρTF · ε2 d
dx
(
∆g
g
)
ρTF
=
= −2pi
∫ xo
xi
d
dx
(
d
dx
ρTF
ρTF
)
ρTF − 2pi
∫ xo
xi
d
dx
(
d
dx
ρTF
ρTF
)(
∆g
g
)
ε2+
+ 2pi
(
d
dx
ρTF
ρTF
)
ρTF
∣∣∣∣xo
xi
+ 2pi
(
d
dx
ρTF
ρTF
)(
∆g
g
)
ε2
∣∣∣∣xo
xi
= ? (2.4.9)
Here xi denotes the inner radius of ATF+ and xo its outer radius . We get using (2.4.5)
? = −2pi
∫ xo
xi
d
dx
(
d
dx
ρTF
ρTF
)
ρTF(1 + o(1)) + 2pi
(
d
dx
ρTF
ρTF
)
ρTF
∣∣∣∣xo
xi
(1 + o(1)). (2.4.10)
Now let us discuss the sign’s of both terms from above by calculating these functions of
ρTF. The derivative of the TF density is
d
dx
ρTF(x) = ω2
(
x− xs−1) (2.4.11)
being positive on ATF as long as x ≤ 1. Moreover, we have
d
dx
(
d
dx
ρTF
ρTF
)
=
= −2Ω2 s
[
µTFs ((s− 1)xs − x2) + Ω2 {s2xs+2 + s(x4 − 5xs+2) + 2(xs + x2)xs}]
x2 (µTFs− 2Ω2xs + Ω2sx2)2
(2.4.12)
which is negative making by the way the first term in (2.4.10) positive as long as
1− (s− 1)xs−2
µTF − Ω2 (2xs
s
− x2) + 2Ω2(x− xs−1)(xs−1 − x)(µTF − Ω2 (2xs
s
− x2))2 ≤ 0. (2.4.13)
86
Here both divisors are positive on the domain of interest, because both are modulo some
Cε the TF density or the TF density squared. The second term in (2.4.13) is negative as
long as x ≤ 1, but the first one isn’t in general for x ≤ 1. However, aiming to consider
the behavior as ε → 0 we rewrite the condition (2.4.13) using the explicit formula for
ρTF
1− (s− 2)xs−2 + ω
2(x− xs−1)(xs−1 − x)
ρTF
≤ 0 (2.4.14)
or
(x− xs−1)(xs−1 − x) ≤ − (1− (s− 2)xs−2) ρTF
ω2
. (2.4.15)
Using the fact ρTF ≤ Cω2/3 we find the stronger condition
(x− xs−1)(xs−1 − x) ≤ − (1− (s− 2)xs−2) C
ω4/3
ω→∞−−−→ 0 (2.4.16)
Since the l.h.s. is negative for x ≤ 1 on ATF we conclude that both terms in (2.4.10) are
positive as long as x ≤ 1. As a consequence we define ATF+ = ATF ∩ {x : x ≤ 1}. Let
us complete these explicit considerations by a simple ceck: Using a Taylor expansion we
get for the TF density
ρTF ' Cγ2 ((εΩ)3/2 − (εΩ)2(x− 1)2) (2.4.17)
with the derivative being
d
dx
ρTF ∼ Cε(1− x) ≥ 0 (2.4.18)
for x ≤ 1 on ATF and in addition one has
d
dx
(
d
dx
ρTF
ρTF
)
· ρTF ≤ 0. (2.4.19)
We find using (2.4.10) and the positivity of the involved terms as lower bound for the
kinetic energy∫
R2
|~∇g|2 ≥ inf
ξ∈ATF+
(1 + o(1))(ξ)
(
− 2pi
∫ 1
xi
d
dx
(
d
dx
ρTF
ρTF
)
ρTF + 2pi
(
d
dx
ρTF
ρTF
)
ρTF
∣∣∣∣1
xi
)
(2.4.20)
and reverse the partial integration to get
∫
R2
|~∇g|2 ≥ (1 + o(1))
∫
ATF+
(
~∇ρTF
)2
ρTF
. (2.4.21)

87
3 A Nonlinear quantum piston for the
controlled generation of vortex rings
and soliton trains
Florian Pinsker, Natalia G. Berloff and Vı´ctor M. Pe´rez-Garc´ıa, Physical Review A 87,
053624 (2013)
We propose a simple way to generate nonlinear excitations in a controllable way by
managing interactions in Bose-Einstein condensates. Under the action of a quantum
analogue of a classical piston the condensed atoms are pushed through the trap generat-
ing vortex rings in a fully three-dimensional condensates or soliton trains in quasi-one
dimensional scenarios. The vortex rings form due to transverse instability of the shock
wave train enhanced and supported by the energy transfer between waves. We elucidate
in which sense the self-interactions within the atom cloud define the properties of gen-
erated vortex rings and soliton trains. Based on the quantum piston scheme we study
the behavior of two component Bose-Einstein condensates and analyze how the presence
of an additional superfluid influences the generation of vortex rings or solitons in the
other component and vice versa. Finally, we show the dynamical emergence of skyrmions
within two component systems in the immiscible regime.
3.1 Introduction
One of the most remarkable achievements in quantum physics in the last decade was that
of Bose-Einstein condensation (BEC) in ultra-cold alkaline atomic gases. These physical
systems have a high potential for supporting quantum nonlinear coherent excitations
and many types of nonlinear waves have been experimentally observed [114, 115, 116,
117, 118, 119, 120, 121, 122, 123, 124, 125] or theoretically proposed to exist (see e.g.
the reviews [126, 127]) in ultra-cold quantum degenerate gases. The list includes dark
solitons [113], bright solitons [114], bubbles [128] and gap solitons [115], vector solitons
[116], vortices, vortex lattices and giant vortices [21, 22, 23], vortex rings [120, 121, 122],
dark ring-shaped waves [123], shock waves [121, 124], collapsing waves [125] and many
others. In this way Bose-Einstein condensates (BECs) are, apart from their fundamental
role in quantum physics, exceptional physical systems for the manifestation and study of
nonlinear phenomena and in particular due to their rather simple theoretical description.
Among the various nonlinear excitations the vortex ring occupies a special place. Vor-
tex rings are essentially three-dimensional topological nontrivial structures appearing in
88
either classical [129] or quantum [130] fluids. They are able to propagate in cylindri-
cally trapped BECs as stable objects [131], similarly to classical fluids [132]. This is
an essential difference to most other solitonic structures that become unstable when
passing to a fully three-dimensional setting, e.g. one dimensional bright solitons that
are unstable to blow-up [125] or dark solitons, that are unstable to the snake instability
[134, 133, 120]. This leaves the vortex ring as the only dynamically nontrivial nonlinear
excitation observed in fully three-dimensional BECs.
Vortex rings were first observed in BECs as the outcome of the decay of dark soli-
tons [120] and as a result of the decay of quantum shock waves [121]. More recently
they have been observed to appear during complex oscillations in soliton-vortex ring
structures [122] and during the merging of BEC condensate fragments [135]. How-
ever, generating vortex rings involved complicated nonlinear phenomena and in general
a simple mechanism allowing the controlled generation of a prescribed finite number
of vortex rings is still missing. The main purpose of this paper is to propose such a
mechanism allowing the generation of a few vortex rings in a highly controllable way.
The same method can be used to create soliton trains of certain frequencies within a
one-dimensional model. We will also extend the concepts to coupled BECs showing how
skyrmions can be generated using similar techniques.
The plan of this paper is as follows. First in Section 3.2 we introduce the main phys-
ical idea of a nonlinear quantum piston. In Section 3.3 we introduce the mathematical
equations and nondimensionalisations used throughout the paper. Next we discuss the
nonlinear excitations in the form of dark and bright soliton trains for one dimensional
problems for single (Sec. 3.4) and two-component (Sec. 3.5) condensates. The con-
trolled generation of vortex rings and skyrmions is discussed in Section 3.6. Finally we
summarize our conclusions in Section 3.7.
3.2 Physical idea
The process of vortex ring generation in classical fluids has received a substantial treat-
ment in the literature. One of the most standard ways to obtain vortex rings in classical
fluids involves moving a piston through a tube, resulting in a vortex ring being generated
at the tube exit. A standard generation geometry consists of the tube exit mounted flush
with a wall with the piston stroke ending at the tube exit [136].
In this paper we will use something conceptually much simpler utilizing the possi-
bilities opened by space-dependent Feschbach resonance management in a BEC. Since
the first achievements in scattering length control in BECs [27], the technique of Fes-
chbach resonance management has been improved and used in many different applica-
tions. Presently, the level of control of the scattering length allows for its very pre-
cise tuning [137] and nothing prevents an extended control of the interactions leading
to a space dependent scattering length. A large number of theoretical papers have
studied nonlinear phenomena in systems with managed interactions (see e.g. Refs.
[126, 138, 139, 140, 146, 149, 150] and references therein).
The physical idea is illustrated in Fig. 3.1. Starting from a single equilibrium BEC
89
Figure 3.1: Schematic diagram of the quantum piston idea, where the ellipse symbolizes
the trapped BEC. The trap is supposed to be radially symmetric and elon-
gated along the z-axis. The area of change in self-interactions is illustrated
by the hatched lines. The small circles represent the cross sections of possi-
ble vortex rings obtained as a result of the flow induced by the asymmetric
interactions playing the role of a quantum piston.
90
in a trap with a given value of the scattering length a > 0, we propose modifying
interactions in half of the space (say the left hand side z < 0) to a larger value aL > a =
aR instantaneously. This change would affect the initial configuration by inducing the
transverse expansion of the atomic cloud for z < 0 generating a flow towards the region
with a smaller interaction value at z > 0. This process is analogous to the piston-driven
flow through an aperture used to generate vortex rings in classical fluids. These effects
are achieved simultaneously with a single action on the interactions without restoring to
complicated external potentials. We also consider a smooth change of scattering length
that would be more realistic in experiments. Modifying interactions to be attractive
a < 0 on one side would allow the atom cloud with repulsive self-interaction at z > 0 to
expand towards this region. The nature of nonlinear excitations generated in this way
would be different to the repulsive case due to the difference in interatomic relations [11,
151]. Generating a controlled flow within a Bose gas with entirely attractive interactions
by imposing a change in interactions would cause similar nonlinear excitations.
Applying the concept of a quantum piston to a two component BEC enables new
physics to come into play. One scenario we will consider in this paper is the case in
which the interactions are changed only in one of the components that would generate
a counterflow between two components thus creating excitations involving the other
component, for example generating skyrmions that are stable vortices with the second
component filling in the core.
3.3 Mathematical models
We consider a single component BEC modeled in the mean field limit by the nonlinear
Gross-Pitaevskii equation [7, 8]
i~
∂ψ
∂t
= − ~
2
2m
∆ψ + V natext (r, z)ψ + g
nat(z)|ψ|2ψ, (3.3.1)
for particles of mass m with self-interactions defined by gnat(z) = 4pi~2as(z)/m, where
as(z) is the s-wave scattering length. The condensate wave function ψ describing a
condensate of N bosons located in Ω ⊂ R3 satisfies the mass constraint ‖ψ‖2L2(Ω) =
N . Confinement is due to an external elongated axisymmetric trap V natext = (mω
2r2 +
mω2zz
2)/2, where the frequencies satisfy ω  ωz. We use the rescaled GPE given by
i
∂ψ
∂t
= −∆ψ + Vext(r, z)ψ + g(z)|ψ|2ψ. (3.3.2)
Here the spatial coordinates are measured in the healing length of the transverse ground
state a0 = (~/mω
√
2)1/2 and time t in
√
2/ω, respectively, while the energies and frequen-
cies are measured in units of ~ω/
√
2 and ω/
√
2 respectively, and Vext = (λ
2r2 +λ2zz
2)/2.
We rescale the wave function such that it is normalized to 1, i.e., ‖ψ‖2L2(Ω) = 1. Then
g(z) = 4pias(z)N/a0, which is proportional to the local value of the s-wave scattering
length as(z) and will be taken, starting from t = 0
+, to be the step-like function defined
91
by
g(z) = gL +
gR − gL
1 + e−2kz
, (3.3.3)
where the constituents of the local self-interaction are gR and gL and the ‘smoothness’
parameter is taken to be k > 0. Finite k accounts for a gradual change in scattering
length across z = 0. In the limiting case k → ∞ the coupling parameter g(z) becomes
a step function
g(z)
k→∞−→

gR, z > 0
(gR + gL)/2, z = 0
gL, z < 0.
(3.3.4)
In addition we will study two component Bose-Einstein condensates within a similar
quantum piston setting. The wave functions of component A and B will be assumed to
be governed by a system of coupled nonlinear Schro¨dinger-type equations [11, 77].
i~
∂ψA
∂t
=
(
− ~
2
2mA
∆ + V natext,A(r, z) + g
nat
A |ψA|2 + gnatAB|ψB|2
)
ψA, (3.3.5a)
i~
∂ψB
∂t
=
(
− ~
2
2mB
∆ + V natext,B(r, z) + g
nat
B |ψB|2 + gnatBA|ψA|2
)
ψB. (3.3.5b)
Here the mass of a boson from component i ∈ {A,B} is denoted mi, the symbol
gnati refers to the corresponding self-interactions and we consider a scaling for which
the normalization is ‖ψi‖2L2(Ω) = Ni, where Ni denotes the number of atoms of species i.
Depending on the context self-interactions gnati can be thought of either as being constant
or step functions. Those step-functions are formally defined as in the single component
case. The cross-interaction strengths are given by gnatij = 2pi~2aij/mij with i 6= j and
the reduced mass given by mij = mimj/(mi + mj). In this paper we will take aij = aji
and m = mA = mB. The harmonic potentials are given by Vext,i = mi(ωr
2 + ω2zz
2)/2,
where the ω’s denote the corresponding trapping frequencies.
The non-dimensionless form is obtained via the transformation x→ a0x, t→ t
√
2/ω
and ψi → ψi
(
1
Li
)1/2
/a0 with
1
Li
= gi~2/(2mgnati ) where the nondimensionalized self-
interaction strength gi has been introduced. In those terms our coupled system is given
by
i
∂ψA
∂t
=
(−∆ + Vext,A + gA|ψA|2 + gAB|ψB|2)ψA (3.3.6a)
i
∂ψB
∂t
=
(−∆ + Vext,B + gAB|ψA|2 + gB|ψB|2)ψB (3.3.6b)
with Vext,i = λ
2
i r
2 + λ2z,iz
2 and gij = gjg
nat
ij /g
nat
j . We choose parameters such that the
mass constraints ‖ψi‖2L2(Ω) = 1, Ω ⊂ R3.
To follow the time evolution of the fields ψi numerically we have used a fourth order
finite difference scheme in space together with a fourth order Runge-Kutta discretization
in time.
92
In the next section we consider the generation of nonlinear excitations by changing
the interaction strength on one half of the domain for quasi-one dimensional BECs.
3.4 Emergence of soliton trains in quasi-one dimensional
single component BECs
The starting point of our investigation of the controlled generation of nonlinear exci-
tations is a strongly cigar shaped Bose-Einstein condensate, i.e., ω  ωz. Neglecting
trapping in z by setting ωz = 0, the evolution equation (4.2.26) for the wave function
becomes [152]
i
∂
∂t
ψ1D = − ∂
2
∂z2
ψ1D + g(z)|ψ1D|2ψ1D − µψ1D, (3.4.1)
where we have introduced a chemical potential via ψ1D = ψ1De
−iµt, dropped the mass
constraint as we consider an infinitely spread BEC here and rescaled time. Initially,
i.e., for t < 0 the single coherent condensate is uniformly distributed and lies at rest.
By n0 = µ/g we denote the associated constant equilibrium density distribution and
the corresponding self-interaction strength by g. To see the effect of changing self in-
teractions at t = 0 instantaneously, i.e., g → g(z), on the initial state (such that we
have step-like self interactions g(z) for all times t > 0) we have simulated the dynamical
behavior of ψ governed by (3.4.1) for different parameter combinations of g(z) [207].
3.4.1 Results
Starting with a uniformly distributed Bose gas with repulsive self-interactions set to
g = 1 we observe that for a moderate increase in the interaction strength on the left-
hand side the outflow does not produce any solitary trains. When the spatial change
in self-interactions is sufficiently large, specifically gL/gR > 2.2 the transport of atoms
from the region of higher interactions on the left to the one of lower interactions on the
right is accompanied by the emergence of a dark soliton train.
The wave generated when the interactions are increased on the left half of the cloud
(z < 0) via g → g(z) leads to a formation of dispersive shock that propagates on the
background density that sets the reference sound speed. As shown by many authors
[141, 142, 143, 144, 145] the 1D shock profile can be found by matching the high- and
low-intensity boundaries. In such a shock the inner (slow) nonlinear part of the front
is a train of dark or gray solitons, while the outer (fast) part is a low-intensity region
with oscillations that are effectively sound-like [141, 142, 143, 144, 145]. As the fast
outer part propagates further into the less interactive region, the inner part adopts more
and more pulses in the solitary train, which is clearly seen on Fig. 3.2. We note that
besides the characteristic density depletion the phase of these arrays of solitons show the
characteristic phase jumps at each soliton by pi. Subsequently the flow of the condensate
manifests itself as a regular array of density depletions moving at a constant speed while
keeping the shape over time. Results in this regime also agree well with simulations of
93
Figure 3.2: Pseudo-color density plot ρ(z) = |ψ1D(z, t)|2 of a uniformly distributed Bose
gas with constant interaction g = 1 = gR at t = 0 evolving in time as a
change of gL/gR = 3.4 has been implemented for t > 0. The change in
interactions is sharp at z = 0, i.e., k → ∞ in (3.3.3). Here luminosity is
proportional to density. The dimensionless units are used as specified in the
main text .
94
Figure 3.3: Pseudo-color density plot ρ(z) = |ψ1D(z, t)|2 of a uniformly distributed Bose
gas with constant interaction g = 1 = gR at t = 0 evolving in time as a
change of gL/gR = −1 has been implemented for t > 0. The change in
interactions is sharp at z = 0, i.e., k → ∞ in (3.3.3). Here luminosity is
proportional to density. The dimensionless units are used as specified in the
main text.
the nonpolinomial Schro¨dinger equation in trapped systems reported in Ref. [146]. In
addition we note that in a different context soliton patterns arise due to a mechanism
where two spatially distinct condensates collide within a harmonic trap [147] and thereby
generate nonlinear excitations. The two condensates in this case have different global
phases, so joining the two condensate together is analogous to phase imprinting in a
single condensate [148]. In our case we have a condensate with the same global phase
where vortices and solitary trains are formed dynamically.
In Fig. 3.3 we show the evolution for a BEC between t = 0 and t = 160 in the
case where we have changed self-interactions to negative (attractive) values at the l.h.s.
In this particular example parameters have been chosen to be g = 1 = gR and gL =
−1. The emerging structure can be identified as a bright soliton train appearing in a
similar process of formation of individual solitons in a localized reservoir. [149, 150].
Furthermore it can be seen in Fig. 3.3 that bright solitons remain approximately at the
same position. Similarly to the formation of a dark soliton train as an initial shock wave
propagates a bright soliton train is generated. In contrast to the dark soliton train which
mainly is due to the reallocation of mass in a directed manner, the bright soliton train
forms due to the perturbation at the breaking point such that due to its attractiveness
the condensate collapses locally to form bright solitons.
Starting with a system of attractively interacting atoms the introduction of even small
95
Figure 3.4: Pseudo-color density plot ρ(z) = |ψ1D(z, t)|2 of a uniformly distributed Bose
gas with constant attractive interaction g = −1 = gR at t = 0 evolving in
time as a change of gL/gR = 0.99 has been implemented for t > 0. The
change in interactions is sharp at z = 0, i.e., k → ∞ in (3.3.3). Here
luminosity is proportional to density. The dimensionless units are used as
specified in the main text.
96
ρ(z)
z
Figure 3.5: Density plot ρ(z) = |ψ1D(z, t)|2 of numerically computed density profiles at
t = 510 for gL/gR = 4 and gR = 1 and k → ∞ (solid line) and k = 1.5
(dashed line). The dimensionless units are used as specified in the main text
change in interactions leads to the generation of a bright soliton train as Fig. 3.4 il-
lustrates for the case of g = −1 = gR and gL/gR = 0.99. Unlike the cases involving
repulsive interactions this bright soliton train slowly expands in both directions. How-
ever, as gL/gR → 0 bright solitons can only be observed for z > 0.
3.4.2 Smooth vs. abrupt change in self-interactions
In real experiments one would expect a more gradual change of the interaction strength
across z = 0. Thus to describe more realistic situations a finite (though maybe small) k
has to be considered within Eq. (3.3.3).
In several series of simulations for attractive as well as repulsive condensates we have
observed the same qualitative dynamics as for the step-function case, the structure of
the solitary wave train and the threshold for its appearance has been very similar even
for small k, i.e., a very smooth step. In Fig. 3.5 we present an example where k = 1.5
showing the very small deviation in emerging soliton trains due to sharp and gradual
changes in self-interactions. Hence, for simplicity we will turn our attention on the limit
k →∞ in what follows.
97
ρ(z)
z
Figure 3.6: Density plots ρ(z) = |ψ1D(z, t)|2 of numerically computed density profiles
with step function interactions (k → ∞) at time t = 500 of an initially
uniformly distributed Bose gas with g = 1 = gR for different self interac-
tion strenghts ratios gL/gR = 2.5 (solid line), gL/gR = 4 (dashed line) and
gL/gR = 6 (dotted line). The dimensionless units are used as specified in the
main text
98
Figure 3.7: A schematic diagram of the relationship between the frequencies f of the
soliton trains and the change in interactions s = gL/gR.The dimensionless
units are used as specified in the main text
99
3.4.3 Properties of soliton trains
For repulsive interactions one would expect that an increase in the change of interactions,
gL/gR, produces larger flow, so leads to an increase in the spatial frequency, i.e., the
number of solitons within some fixed space interval. An example of this behavior is
shown in Fig. 3.6 where an initial distribution specified by g = 1 = gR changes its
profile when the interactions are set to gL/gR = 2.5, gL/gR = 4 and gL/gR = 6. The final
profiles for t = 500 are shown. Fig. 3.6 shows that larger asymmetries in the interaction
lead to higher spatial frequencies. In addition we observe that for large asymmetries in
interactions the wavefunction profiles resemble the square of a sinus function while for
smaller asymmetries the profiles of the density depletions resemble an array of squares
of hyperbolic tangents near their minima, i.e., there is a qualitative difference in the
form of the density profile depending on the imposed change in interactions. We have
also studied the velocities of generated dark soliton trains and found them to be almost
independent on the change in interaction strength.
The numerically obtained relationship between the frequencies of soliton trains and
the change in interactions gL/gR ∈ [−2, 8] is given in Fig. 3.7 for g = 1. Here r denotes
entirely repulsive BEC and ra a condensate where we switched to attractive values on
one side. For entirely attractive interactions one finds that changing self-interactions
to different values leads to bright soliton trains with in general different frequencies on
both sides.
3.4.4 Analytical approximations to the soliton train profiles
Eq. (3.4.1) with spatially dependent interactions does not admit in general exact analyt-
ical solutions. While we will be constructing solutions of the full equation in Appendix
3.8 (for step-function like self-interactions in the static case), for now we will restrict
our attention to the construction of phenomenological solutions fitting the dynamics for
uniform g that can be expected to approximate the solutions far from z = 0 (the point
where the nonlinearity has the transition from gL to gR). First we consider the repulsive
and then the attractive case. The differential equation (3.4.1) for constant g is integrable
[154] and the solution representing a single dark soliton [155] for g > 0 can be written
as
ψg>0(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
· tanh
(√
1−
(v
c
)2
·
√
n0g
2
· (z − vt)
)]
. (3.4.2)
Here v is the velocity of the soliton, c =
√
2n0g is the Bogoliubov speed of sound, n0
denotes the equilibrium one particle density distribution and one sets µ = n0g. In order
to get a periodic solution describing a dark soliton train we interchange the hyperbolic
tangens in (3.4.2) with a Jacobi elliptic function by letting tanh→ sn,
ψdt(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
· sn
(√
1−
(v
c
)2
·
√
n0g
2p2
· (z − vt)
∣∣∣∣p2
)]
. (3.4.3)
100
A straightforward calculation shows that it is approximately solving (3.4.1), if µ is
chosen to be µ = gn0
(
(1+p2)
(2p2)
(1− v2
2n0g
) + v
2
2n0g
)
. Indeed, the above approximate solution
becomes exact either if p→ 1, where sn(z|1) = tanh(z), or if v → 0, so it generalizes the
single soliton expression (3.4.2). It is known that Jacobi elliptic functions are periodic
solutions of the GP equation with repulsive and attractive interactions [156, 157, 158,
159, 160]. The sinus amplitudinis interpolates between a trigonometric and a hyperbolic
function and its dependency on each is controlled by the real-valued elliptic modulus
p ∈ [0, 1].
One property of the analytic dark soliton train (3.4.3) is that its density profile fits with
the computationally obtained profile, iff the self-interaction strength g of the analytical
solution equals the one used for generating the numerical solution. This in turn enables
us to deduce the effective interaction strength g between the condensed atoms from the
form of the numerically generated dark soliton train, i.e., from the density profile of the
atom cloud by means of (3.4.3). For details we refer to the Appendix 3.9.
We have compared the numerical generated soliton trains with the analytical periodic
soliton trains due to the condensate wave functions (3.4.3). A typical example is pre-
sented in Fig. 3.8 where parameters for the analytical solution where chosen to be as
follows. v = 0.1018, p = 0.9978, n0 = 0.1566 and g = 1 while the numerical solution
is considered on a space interval where self-interactions have been g = 1 for t ≤ 0 and
gL = 3 for t > 0.
A single bright soliton solution to (3.4.1) with constant attractive interactions is given
by replacing tanh(z)→ 1/ cosh(z) in (3.4.2) and setting g → |g|. The transition to the
bright soliton train is obtained by tanh→ cn, i.e.,
ψbt(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
· cn
(√
1−
(v
c
)2
·
√
n0|g|
2p2
· (z − vt)
∣∣∣∣p2
)]
. (3.4.4)
Again a straightforward calculation shows that it is approximately solving (3.4.1), if µ
is chosen to be µ = |g|n0
(
(1−2p2)
(2p2)
(1− v2
2n0g
) + v
2
2n0g
)
. Note that the chemical potential
is negative for 1 > p2  0 and for v → 0 and √p → 1 converges to µ = −|g|n0/2.
Furthermore, this solution becomes exact either if
√
p → 1, where cn(z|1) = 1
cosh(z)
,
or if v → 0, thereby generalizing the single bright soliton expression. We note that
similar considerations on the density profile like those made above for dark soliton train
solutions apply to bright soliton train solutions as well. However, as the bright soliton
train in Fig. 3.3 is not freely expanding the limit v → 0 gives the best approximation to
our numerics.
The densities corresponding to the dark soliton train solution (3.4.3) and the bright
soliton train solution (3.4.4) are related via
|ψdt(z, t)|2 + |ψbt(z, t)|2(
1 + v
2
c2
) = n0. (3.4.5)
101
ρ(z)
z
Figure 3.8: Details of analytical (dashed line) and numerically computed (solid line) den-
sity profiles |ψ1D(z, t)|2 of dark soliton trains. At time t > 0 the interactions
of the condensate on the left are set to gL = 3 on the left-hand side of the do-
main in the numerical solution. The dimensionless units are used as specified
in the main text.
102
3.5 Emergence of soliton trains in quasi-one dimensional
two-component BECs
In the previous sections we have found that the emergence and properties of soliton trains
depend on the magnitude of change in self-interaction strength. Next we discuss how the
state of a one-dimensional condensate of component A ψA1D is affected by the presence
of a second component B represented by ψB1D. Supposing ω  ωz, rescaling time and
neglecting trapping in z-direction the wave functions are governed by the system
i
∂
∂t
ψA1D =
(
− ∂
2
∂z2
+ gA|ψA1D|2 + gAB|ψB1D|2
)
ψA1D, (3.5.1)
i
∂
∂t
ψB1D =
(
− ∂
2
∂z2
+ gB|ψB1D|2 + gAB|ψA1D|2
)
ψB1D. (3.5.2)
Here the self-interactions gA, gB and cross-interactions gAB are either constants or step
functions. The dynamical stability of the mixture depends on the criterion gAgB > g
2
AB,
therefore, by changing the interaction strength on one part of the cloud it is possible to
have a miscible regime on one half and the phase separation regime on the other half of
the domain.
First, we assume that initially (t = 0) condensates A and B are spatially homogeneous
with uniform and repulsive self- and cross-interactions. Then (t = 0+), self-interactions
are changed in component A, i.e., formally gA → gA(z) leading to the generation of
dark solitons and the appearance of complex dynamics for t > 0. An example of the
dynamics is shown in Fig. 3.9 (with gLA/g
R
A = 3 and gAB = g
R
A = gB = 1). In that case,
the soliton train in component A is raised by the presence of the second condensate.
Dark soliton trains generated in the presence of a second repulsive condensate are not
as stable as single component condensates - solitons decay faster. However, we have
observed dark soliton trains to appear at slightly lower interaction ratios than in single
component condensates, which depends in particular on cross-interaction strength.
Next we consider a two component condensate where one component is attractive
and the other component is repulsive. Initially both components are mixed and uni-
formly distributed. In Fig. 3.10 snapshots of a two component condensate with initial
parameters gA = 1, gB = −1 are shown (dotted line). After changing self-interactions of
component A to gLA/g
R
A = 2.1 a dark soliton train is generated. Fig. 3.11 and Fig. 3.12
illustrate the spectrum of the time evolution for each component. The dark soliton train
in component A represents the part of the effective potential for the other component
B and, therefore, induces excitations in condensate B producing a bright soliton train
(dashed line). As it can be seen in Fig. 3.10 the density depletions of one component
are at the maxima of the other and vice versa. In particular the frequency of the dark
soliton train in A is correlated with that in component B. As we showed above a change
of self-interactions in an attractive single component condensate leads to a soliton train
expanding in both directions (see Fig. 3.4). Hence, as the dark soliton train in A for
z > 0 induces a bright soliton train in the other component B, which expands in both
directions, this density depletion itself induces a dark soliton train expanding towards
103
ρ(z)
z
Figure 3.9: Density plot ρ(z) = |ψ1D(z, t)|2 of component A of a coupled BEC (solid
line), component B (dashed line) at time t = 390 and a single component
condensate (dotted line). Initial state has gA = gB = gAB = 1. At time t = 0
the interactions of the condensate A are set to 3 on the left-hand side of the
domain. The dimensionless units are used as specified in the main text.
104
ρ(z)
z
Figure 3.10: Snapshots of density plots ρ(z) = |ψ1d(z, t)|2 of a two component BEC at
t = 0 (dotted line) and t = 300 - component A (solid line) and B (dashed
line). Initial state has gA = 1, gB = −1, gAB = 1. At time t = 0 the
interactions of the condensate A are set to 2.1 on the left-hand side of the
domain. The dimensionless units are used as specified in the main text.
105
Figure 3.11: Pseudo-color density plot ρ(z) = |ψ1D(z, t)|2 of component A of a uniformly
distributed Bose gas with constant interaction gA = gAB = 1 = g
R and
gB = −1 at t = 0 evolving in time as a change of gL/gR = 2.1 has been
implemented in A for t > 0. The change in interactions is sharp at z = 0,
i.e., k → ∞ in (3.3.3). Here luminosity is proportional to density. The
dimensionless units are used as specified in the main text.
106
Figure 3.12: Pseudo-color density plot ρ(z) = |ψ1D(z, t)|2 of component B of a uniformly
distributed Bose gas with constant interaction gB = −1 and gA = gAB =
1 = gR at t = 0 evolving in time as a change of gL/gR = 2.1 has been
implemented in A for t > 0. The change in interactions is sharp at z = 0,
i.e., k → ∞ in (3.3.3). Here luminosity is proportional to density. The
dimensionless units are used as specified in the main text.
107
z < 0 in component A (Fig. 3.10.) Starting from the same initial distribution and
changing interactions in the attractive condensate has a comparable effect, i.e., soliton
trains in both components on the whole line are created. In any case only a very small
change in self-interactions is sufficient to start this process, which is comparable to the
behavior of the attractive single component condensate and due to its instability.
General analytical solutions to two component condensates, where one component is
in a state corresponding to a dark soliton train while the other component represents a
bright soliton train can be constructed using the previous expressions (3.4.3) and (3.4.4).
Thus, a dark soliton train of the form
ψA(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
·
· sn
(√
1−
(v
c
)2
·
√
n0(gA − gAB)
2p2
· (z − vt)
∣∣∣∣p2
)]
, (3.5.3)
can be coupled to a bright soliton train of the form
ψB(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
·
· cn
(√
1−
(v
c
)2
·
√
n0(gB − gBA)
2p2
· (z − vt)
∣∣∣∣p2
)]
, (3.5.4)
where one has to introduce appropriate chemical potentials in (3.5.1) and (3.5.2) and
both trains have the same periodicity. We refer to appendix 3.10 for a short outline.
3.6 Controlled generation of vortex rings and soliton
trains in 3D
Let us now remove the constraint of one-dimensional geometries, but we still consider
cigar-shaped traps. Due to the phenomenon of snake instability in dimensions higher
than one, dark solitons in repulsive condensates decay into more stable excitations such
as vortices [162, 163] or vortex rings. Thus we would expect that once dark solitons are
generated, they would decay into vortex rings in three-dimensions and the threshold in
the self-interaction imbalance for the generation of these excitations would be close to
the one obtained in the quasi-one dimensional system discussed earlier.
The scenario of vortex rings nucleation is very similar to that of vortex rings formation
after a cavity collapse [133]. When the train of shock waves/dark solitons is formed, some
part of the front breaks into vortex rings with an extra energy necessary to drive such
transition provided by the part of the train traveling behind [165]. The energy transfer
also counteracts the effect of the friction allowing the ring to travel a long distance before
breaking apart.
108
(a)
(b)
(c)
(d)
Figure 3.13: (Color online) Same as in Fig. 3.13 but for gL/gR = 2.1 and times (a)
t = 0.75, (b) t = 4.5, (c) t = 7.5, (d) t = 11.25. The spatial region
shown corresponds to z ∈ [−20, 20], r ∈ [0, 8]. Black corresponds to low
atom densities and yellow to high ones. The dimensionless units are used
as specified in the main text.
The presence of the solitary wave train enhances the instability leading to the forma-
tion of vortex rings in comparison with the instability of a single grey soliton. The faster
the soliton moves the more stable it becomes. To overcome this stability there has to
exist the supply of energy which is provided by the waves traveling behind.
3.6.1 Dynamics of single component BEC
We have numerically simulated Eq. (4.2.26) for various parameter combinations [214] and
will describe the typical outcome for a specific example corresponding to a large repulsive
BEC with g = 105, with λx = λy = 1, λz = 0.05 (i.e. soft longitudinal trapping). Our
initial configuration is a ground state BEC corresponding to g = gL = gR. At time t = 0
we suddenly raise interactions strength gL for z < 0 and then observe the subsequent
109
(a)
(b)
(c)
(d)
Figure 3.14: (Color online) Pseudocolor plot of atom density |ψ(r, z, t)|2 snapshots for
different values of time: (a) t = 0.75, (b) t = 4.5, (c) t = 8.625, (d)
t = 13.875 and (e) t = 18.375. The values of the interactions gL/gR = 2.3
and the spatial region shown corresponds to z ∈ [−30, 30], r ∈ [0, 8] in
subplots (a-d) and to z ∈ [−10, 50] in subplot (e). Black corresponds to low
atom densities and yellow to high ones. The dimensionless units are used
as specified in the main text.
110
evolution of the condensate.
Once the non-equilibrium situation is generated there is a flow of atoms from z < 0
to z > 0 with a flow intensity depending on the ratio gL/gR. When a critical value
gL/gR ' 2 is surpassed the vortex rings are generated as seen on Fig. 3.13. It shows
density plots with density depletions that characterize vortex rings, i.e. zero density
around a closed curve in 3d. In addition these density depletions are accompanied by
the 2pi phase jumps not shown in the pictures. In Fig. 3.13 we present the stages of
vortex rings formation for gL/gR = 2.1. The change in interaction strength leads to a
generation of a train of dark solitons, see Fig. 3.13(a), that evolve into vortex rings which
enter the condensate around z = 0 coming from the low density region, see Fig. 3.13(b),
and move slowly through the condensate remaining stable for long times [Fig. 3.13(c,d)].
Another vortex ring with a large radius seems to be present in the lower density regions
where it would be experimentally difficult to detect.
Increasing the interactions even further to gL/gR = 2.3 leads to a richer dynamics as
summarized in Fig. 3.14. The short-time dynamics is analogous to the previous cases
[Fig. 3.14(a)] but then a complex transient appears where several vortex rings enter the
condensate; also rarefaction pulses are clearly identified [see Fig. 3.14(b)]. After that,
some of those vortices counter-flow and disappear and a much more regular picture arises
with several vortex rings moving to the right in a very clear way. Fig. 3.14(c) shows
few vortex rings slowly moving through the condensate for t = 8.625 and one being
generated around z = 0. Fig. 3.14(d) shows a later stage of the evolution where three
long-lived vortex rings travel smoothly through the condensate although their relative
positions changes due to differences in their speeds (notice the small differences in their
radii) and their interaction with sound waves originated after the reflection of the shock
wave in the condensate boundary [see Fig. 3.14(e)].
Fig. 3.15 shows that changing interactions to attractive, i.e., a change of gL/gR =
−10−3 for z < 0 causes the formation of a shock wave there and the formation of
solitonic waves, similar to the formation of bright soliton trains in quasi-one dimensional
BECs. Due to the small attractive force between particles and the tight potential in
the transverse direction the condensate gently collapses into a quasi-one dimensional
setup, thereby carrying stable solitons in its attractive part. Increasing the attractive
force to more negative values of scattering length leads to a blowup of the condensate
wave function, while lowering implies more stable solitary waves even for BEC without
interaction (gL = 0). Lowering scattering lengths on the l.h.s. to positive values leads
to vortex ring generation once a threshold is surpassed and to solitary waves as gL tends
to become smaller.
3.6.2 The two component case
We now turn to two component systems of clearly distinct states ψA and ψB with
interactions to be chosen within the phase separation regime, g2AB > gAgB, i.e. cross-
interactions between both components are dominating. The harmonic trapping potential
for the repulsive two component BEC has been specified by λx = λy = 1, λz = 0.05
(i.e. soft longitudinal trapping) for both components. To generate a quantum piston
111
(a)
(b)
Figure 3.15: (Color online) Pseudocolor plot of atom density |ψ(r, z, t)|2 snapshots for
different values of time: (a) t = 0.75 and (b) t = 3.75. The values of the
interactions are gL/gR = −10−3. Black corresponds to low atom densities
and yellow to high ones. The spatial region shown corresponds to z ∈
[−20, 20], r ∈ [0, 6] in subplot (a) and to z ∈ [−30, 10], r ∈ [0, 4] in subplot
(b). The dimensionless units are used as specified in the main text.
induced evolution of the condensate wave functions containing skyrmions we tested
various different initial conditions. The initial ground state at t = 0 on which we
apply the quantum piston scheme consists of one component surrounded by the second
component. The initial states are naturally generated by putting both components
(each localized regarding a harmonic trap specified by λx = λy = 1, λz = 0.05 but one
translated along the z-axis to the left and the other to the right) into the single trap
without any overlapping of the atom clouds and by evolving the corresponding state in
imaginary time until the new common ground state is reached at t = 0.
In Fig. 3.16 we show an example of density profiles of a two component BEC in
such a ground state at t = 0, which is specified by its self-interactions gA = 6005,
gB = 2650 and cross-interactions gAB = 6000. After a change in self-interactions by a
factor gLA/g
R
A = 2.26 on the l.h.s. in component A has been implemented vortices are
generated. In particular we observe the emergence of a vortex ring in component A
that is filled with mass of component B, which can be identified as a skyrmion - the
corresponding area within the density distributions in Fig. 3.16 is encircled.
3.7 Conclusions
In this paper we have studied several examples of how the spatial and temporal control of
the self-interactions in an atomic BEC leads to the formation of nonlinear excitation such
as dark/bright solitons/ solitary trains and solitary waves using a nonlinear “quantum
piston” concept. In an axisymmetric elongated condensate vortex rings form as the
interaction strength on one half of the condensate changes by a factor exceeding 2.
This mechanism can be used to controllably generate and study such excitations. Our
112
(a)A
(b)B
(a)
(b)
Figure 3.16: (Color online) Pseudocolor plot of atom density |ψ(r, z, t)|2 snapshots of
component A and component B in the initial state at t = 0. The spatial
region shown corresponds to z ∈ [−120, 120] and r ∈ [0, 10]. Subplots
(a),(b) at t = 35.25 show details for a region z ∈ [30, 60], r ∈ [0, 7]. The
circle marks the position of a “skyrmion” – vortex in the first component
with the density maximum of the second component. The dimensionless
units are used as specified in the main text.
113
proposal, in addition to being conceptually simple and accessible to present experimental
techniques improves essentially currently used methods to produce nonlinear excitations.
Two component Bose-Einstein condensates could be used to amplify the generation of
vortex rings or solitons and give rise to another set of excitations such as skyrmions.
The number of vortex rings/skyrmions generated can be controlled by changing the
transverse confinement ω/ωz. For weak transverse confinement a moving solitary wave
is subject to snake instability leading to the formation of vortex rings. For a sufficiently
tight transverse confinement the solitary wave becomes stable to the snake instability
and vortices will not form.
The faster a solitary wave moving the more stable it becomes to the snake instability.
In the periodic train the stability is reduced because of the energy transfer between the
parts of the train [165]. The vortex generation in the proposed method, therefore, is
the result of an intricate interplay between shock wave train generation and the snake
instability enhanced by the energy transfer between the parts of the train.
3.8 Stationary solutions to GPE with step-like coupling
parameter
We will describe here a procedure to construct time independent solutions of the GPE
(3.4.1) with step function coupling parameters. To introduce the method we will consider
the second derivative of a ‘two branches of the real line’ ansatz defined as
u(z) = tanh(g+(z)) + tanh(g−(z)). (3.8.1)
Here the basic idea is that one branch (denoted by the subscript −) takes into account
properties of the solution for z < 0 and the other branch (denoted by the subscript +)
properties relevant in z > 0. The arguments of the hyperbolic tangent, g+ and g−, are
explicitly given by
g+(z) = lim
k→∞
log
(
e2kz + 1
)
2k
=
{
|z| z > 0
0 otherwise
(3.8.2)
and
g−(z) = lim
k→∞
− log (e−2kz + 1)
2k
=
{
0 z ≥ 0
−|z| otherwise (3.8.3)
Consequently the derivatives of these functions are
g′+(z) = lim
k→∞
1
1 + e−2kz
=

1 z > 0
1/2 z = 0
0 otherwise
(3.8.4)
and
g′−(z) = lim
k→∞
1
1 + e2kz
=

1 z < 0
1/2 z = 0
0 otherwise
(3.8.5)
114
and the second derivatives satisfy
g′′+(z) =
{
0 z 6= 0
∞ z = 0 (3.8.6)
and
g′′−(z) =
{
0 z 6= 0
−∞ z = 0 . (3.8.7)
These functions obey the property
−g′′+(0) = g′′−(0) = lim
k→∞
(
−k
2
+ k
)
=∞. (3.8.8)
Now consider the second derivative of our ansatz, i.e.,
∂2zu(z) = sech
2(g+)g
′′
+ + sech
2(g−)g′′−−
− 2sech2(g+) tanh(g+)g′2+ − 2sech2(g−) tanh(g−)g′2−, (3.8.9)
at z = 0, i.e., where by (3.8.8) and the fact that g+(0) = g−(0) = 0 one gets
sech2(g+)g
′′
+ + sech
2(g−)g′′− = (g
′′
+ + g
′′
−) = 0. (3.8.10)
Hence the second derivative of (3.8.1) satisfies
∂2zu(z = 0) = 2
(
tanh(g+)
2 − 1) tanh(g+)g′2++
+ 2
(
tanh(g−)2 − 1
)
tanh(g−)g′2− = 0, (3.8.11)
and
∂2zu(z 6= 0) = 2
(
tanh(g±)2 − 1
)
tanh(g±)g′2±, (3.8.12)
where the subindexes + and − correspond to z > 0 and z < 0 respectively
Let us go a step further and rescale our ansatz in order to get a solution for
∂2zu = θ(z)|u|2u− µu (3.8.13)
with
θ(z) =

g1 if z < 0
c if z = 0
g2 if z > 0.
(3.8.14)
The rescaled two branches ansatz is given by
u ≡ u+ + u− ≡
√
µ
g2
tanh
[
±
√
µ
2
g˜+(z)
]
+ √
µ
g1
tanh
[
±
√
µ
2
g˜−(z)
]
(3.8.15)
115
ρ(z)
z
Figure 3.17: Density plot |ψ1d(z, t)|2 of an example of an exact two branch real line
solution to the GPE with constant self-interactions.
with phase functions defined by
g˜+(z) = lim
k→∞
log
(
e2
√
g1kz + 1
)
2
√
g1k
=
{
|z| z > 0
0 otherwise
(3.8.16)
and
g˜−(z) = − lim
k→∞
log
(
e−2
√
g2kz + 1
)
2
√
g2k
=
{
0 z ≥ 0
−|z| otherwise (3.8.17)
To verify that (3.8.15) has the desired property we consider its second derivative
∂2z (u+ + u−) =
µ√
2g2
sech2(g+)g
′′
+ +
µ√
2g1
sech2(g−)g′′−−
− µ3/2
(
1√
g2
sech2(g+) tanh(g+)g
′2
+−
− 1√
g1
sech2(g−) tanh(g−)g′2−
)
= g2u
3
+ − µu+ + g1u3− − µu−. (3.8.18)
Note that u+ is nonzero, iff z > 0, as well as u− is nonzero, iff z < 0.
116
Furthermore one can interchange one branch or both by a Jacobi elliptic type function
of a similar form as the hyperbolic tangent branches, which is a solution to the same
differential equation (3.8.13). In Fig. 3.17 one finds an example of such a solution.
On the left hand side one finds the Jacobi elliptic part while on the r.h.s. the density
corresponds to a hyperbolic tangent wave function.
3.9 Determining self-interaction strength g of the
condensate via the form of the soliton train
The ‘free’ parameters of the condensate wave function (3.4.3) are {n0, v, p, g} and the
form of the analytical dark soliton train solution depends on the parameter p in (3.4.3)
- if p is close to 1 the contribution of the hyperbolic tangens is dominating, while for
smaller p the solution resembles properties of a squared sinus. Hence, one selects the
elliptic modulus p by comparing the form of the numerically generated profile with the
form of the analytical expression. The amplitude of the solution and the depth of each
soliton are fixed by the requirement that at a maximum of the density graph we have
max
∣∣ψdt∣∣2 = n0 = c1, (3.9.1)
and at a minimum
min
∣∣ψdt∣∣2 = ( v√
2g
)2
= c2, (3.9.2)
where c1 and c2 are fixed numbers. The periodicity is fixed as well, i.e.,√
1−
(v
c
)2
·
√
n0g
2p2
= c3, (3.9.3)
where c3 again is a constant. Inserting (3.9.1) and (3.9.2) in (3.9.3) determines g. Hence
we can determine the effective interactions between atoms and the velocity of the dark
soliton train from the density profile of the condensate at a particular instant in time.
3.10 Dark-bright soliton train solutions for two
component BEC
We now show that there exist analytical expressions for a dark soliton train in component
A coupled to a bright soliton train in component B. We recognize that
ψA(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
·
· sn
(√
1−
(v
c
)2
·
√
n0(gA − gX)
2p2
· (z − vt)
∣∣∣∣p2
)]
, (3.10.1)
117
satisfies the equation
i∂tψA =
(−∂2z + (gA|ψA|2 − gX |ψA|2 − µ))ψA. (3.10.2)
We rewrite some terms
− gX |ψA|2 − µ = −gXn0
(
v2
c2
+
(
1− v
2
c2
)
sn2
)
− µ =
= gXn0
(
v2
c2
+
(
1− v
2
c2
)
cn2
)
− µ˜ (3.10.3)
with µ˜ = 2gXn0
v2
c2
+ gXn0
(
1− v2
c2
)
+ µ. Hence, by setting gX → gAB and defining
ψB(z, t) =
√
n0
[
i
v
c
+
√
1−
(v
c
)2
·
· cn
(√
1−
(v
c
)2
·
√
n0(gB − gY )
2p2
· (z − vt)
∣∣∣∣p2
)]
, (3.10.4)
with gB − gY = c = gA − gAB > 0 (3.10.1) satisfies
i∂tψA =
(−∂2z + (gA|ψA|2 + gAB|ψB|2 − µ˜))ψA. (3.10.5)
On the other hand (3.10.4) satisfies
i∂tψB =
(−∂2z + ((gB − gY )|ψB|2 − µ′))ψB =
=
(−∂2z + (gB|ψB|2 + gAB|ψA|2 − µ′′))ψB, (3.10.6)
by setting gY → gAB and for an appropriately chosen µ′′.
118
4 Transitions and excitations in a
superfluid stream passing small
impurities
Florian Pinsker and Natasha G. Berloff, Phys. Rev. A 89, 053605 (2014).
We analyse asymptotically and numerically the motion around a single impurity and
a network of impurities inserted in a two-dimensional superfluid. The criticality for the
break down of superfluidity is shown to occur when it becomes energetically favourable to
create a doublet – the limiting case between a vortex pair and a rarefaction pulse on the
surface of the impurity. Depending on the characteristics of the potential representing
the impurity different excitation scenarios are shown to exist for a single impurity as
well as for a lattice of impurities. Depending on the lattice characteristics it is shown
that several regimes are possible: dissipationless flow, excitations emitted by the lattice
boundary, excitations created in the bulk and the formation of large scale structures.
4.1 Introduction
Superfluidity is the property of extraordinary low viscosity in a fluid for which evidence
was first found in liquid helium II, i.e., He-4 below the λ-point at 2.17K [166, 167]. Later
it was proposed that liquid helium II can be regarded as a degenerated Bose-Einstein
gas in the lowest energy mode [18] and the hydrodynamical picture was completed by
arguing that the condensed fraction [12] of the superfluid does not take part in the
dissipation of momentum [168]; it is due to the non-condensed atoms/molecules or quasi
particles that viscosity occurs in superfluids. In 1995 condensation to the lowest energy
state was achieved experimentally for weakly interacting dilute Bose-gases [4, 5, 6] and
subsequently research on hydrodynamic properties has been presented; superfluidity
could be confirmed by moving laser beams of different shapes through the condensate,
which showed a drag force only above some critical velocity and dissipationless flow
below [169, 170, 19, 171]. Experimental investigations have been accompanied by a great
advancement in our theoretical understanding of this matter; a variety of scenarios [8],
like superfluid flow around obstacles of different shapes at sonic or supersonic speed
[172, 173, 174, 175, 176, 180, 181, 177, 178, 179], solitary waves due to inhomogeneities
[25, 26, 88] or the transitions emerging in rotating Bose gases [182, 183, 184, 21, 22]
have been considered. In more recent years superfluids made out of quasiparticles such
as gaseous coupled fermions (Cooper pairs) [185, 186], spinor condensates [187, 188,
119
189], exciton-polaritons [190, 191, 16, 192, 193] or classical waves [194, 195, 17] have
received much attention [15, 10]. New aspects emerge for investigation and the quest of
elucidating defining properties of these novel superfluids goes on [196].
To study the nature of a superfluid, in particular the key property of zero viscosity at
zero temperature T = 0, a well-established scenario to consider is an obstacle in relative
motion to the fluid [169, 170, 19, 172, 173, 174, 175, 176, 180, 181, 177, 178, 179]. It
was noted as early as in 1768 by d’Alembert [197] that an incompressible and inviscid
potential flow past an obstacle encounters vanishing resistance, i.e., such a fluid is in a
state of superfluidity. In this paper, however, we shall consider a compressible superfluid
with zero viscosity obeying a finite speed of sound, which is to be described by a nonlinear
Schro¨dinger-type equation. Among such superfluids governed by nonlinear Schro¨dinger-
type equations are dilute and weakly interacting Bose-gases [11], condensates of classical
waves [17] or exciton-polariton condensates close to equilibrium [193]. Due to the finite
speed of sound the fluid obeys a critical velocity vcr above which excitations occur
within the condensate; superfluidity starts to dissipate, nodal points in the condensate
wave function (with zero absolute value and discontinuities in the phase) might emerge
forming quantized vortices [198]. In the presence of an obstacle this criterion needs to
be modified as the full depletion of the condensate on the surface means that the local
speed of sound is zero, this however does not lead to the excitation formation at low
velocities. It has been shown by numerical simulations that this transition takes place
when it becomes energetically favourable for a vortex to appear inside the healing layer
on the surface of the obstacle [199]. By generating such elementary excitations as vortices
or rarefaction pulses (for smaller perturbations) the system limits the superfluid flow
velocity by the onset of a drag force on the moving obstacle due to those excitations.
Emerging rarefaction pulses in the wake of the obstacle may exchange energy due to
sound waves propagating through the condensate or via contact interaction as they
encounter other excitations. When acquiring energy rarefaction pulses may transform
into vortex pairs (or vortex rings in 3d) or by radiating energy vortex pairs collapse to
excitations of lower energy [202].
In this paper we consider a condensate in 2 spatial dimensions (2d) passing finite size
obstacles that are about the size of the superfluid’s healing length. Initially the obstacles
(which are assumed to be repulsive) either induce vortex pairs or rarefaction pulses into
the superfluid’s stream; weaker impurities imposing just a slight dip of small radius
on the superfluid density support the generation of rarefaction pulses while stronger
interacting impurities favor the appearance of a vortex pair. Experimentally it has
been shown in [203] that vortex pairs are stable excitations in oblate and effectively 2d
Bose-Einstein condensates.
Using a species-selective dipole potential the localized impurities were created in ex-
periment [204]. To elucidate what kind of flows can exist in such systems we consider
many impurities arranged on specific lattices and demonstrate that various regimes are
possible depending on the network configuration. The motion of generated excitations
within the lattice is strongly affected by their attraction to the impurities and interac-
tions between excitations.
Our paper is organized as follows. In Section 1 we use asymptotic expansions to
120
estimate the density, velocity and energy of the condensate moving past a fixed obstacle
and determine analytically the speed at which the vortex nucleation takes place as a
function of the radius of the obstacle. In Section 2 we analyze how the form of the
potential modelling the impurity affects the excitations nucleated. In Section 3 we
study various regimes in superfluid flow passing impurity networks. We conclude by
summarizing our findings.
4.2 1. Asymptotic expansion for a flow around a disk
below the criticality
In this section we extend the analysis done in [177] for the flow around a two-dimensional
disk of a radius large compared to the healing length to obtain the corrections due to the
finite disk size. The condensate order parameter satisfies the Gross-Pitaevskii equation
[11] in the reference frame moving with velocity ~v, oriented along the positive x-direction:
i~∂tψ = − ~
2
2M
∆ψ + V natψ − (E + 1
2
Mv2 − g|ψ|2)ψ. (4.2.1)
The expression E + 1
2
Mv2 is the energy in the moving reference frame at which the
impurity is at rest. In this model M is mass of a boson, g > 0 is the strength of the
repulsive self-interactions within the fluid and dependent on the number of its atoms
[11], V nat denotes the potential modeling the impurities, E the single-particle energy
in the laboratory frame. A natural scale in our discussion is the healing length, ξ =
~/(2ME)1/2.
We develop the asymptotics of the order parameter ψ =
√
ρ exp(iφ). We use the
nonlinear Schro¨dinger equation (4.2.1) where, for simplicity, we drop the potential V nat
in favor of boundary conditions on the order parameter for which the potential is an
impenetrable barrier of radius b, so ψ(r < b, t) = 0, where r2 = |~x|2 = x2 + y2. We write
Eq. (4.2.1) in hydrodynamical form using the Madelung transformation
ψ = ReiS, (4.2.2)
so that
ρ = MR2, φ = (~/M)S, (4.2.3)
and rescale the resulting equations using ~x → b~x, t → (ξbM/~)t, v → (~/ξM)U and
ψ → ψ∞ψ, where ψ∞ = (E/g)1/2, and using the dimensionless parameter ε = ξ/b. The
resulting system of equations becomes
ε2∇2R−R(∇S)2 = (R2 − 1− U2)R (4.2.4)
R∇2S + 2∇R · ∇S = 0, (4.2.5)
subject to the boundary conditions R = 0 at r = 1, S → −Ux and R → 1 as r → ∞.
We shall assume that both ε and U are small and consider an asymptotic expansion of
the solution to Eq. (4.2.4) and (4.2.5).
121
Boundary layer.- At the boundary layer the quantum pressure contribution plays a
crucial role. We introduce r = 1 + εχ and expand R and S as
R(χ, θ) = R̂0(χ, θ) + εR̂1(χ, θ) + ε
2R̂2(χ, θ) + . . . (4.2.6)
S(χ, θ) = Ŝ0(χ, θ) + εŜ1(χ, θ) + ε
2Ŝ2(χ, θ) + . . . (4.2.7)
The solutions to O(ε2) in Sˆ and the leading order for R were found in [177] for a spherical
object, for the disk these become
R̂0 = g(θ) tanh
(
g(θ)χ/
√
2
)
, Ŝ0 = Ŝ0(θ), Ŝ1 = Ŝ1(θ) (4.2.8)
Ŝ2 = − ∂
∂θ
(
h(χ, θ)
d
dθ
Ŝ0θ
)
+ ζ2(θ), (4.2.9)
where
h(χ, θ) =
∫ χ
0
dχ′
R̂20(χ
′, θ)
∫ χ′
0
R̂20(χ
′′, θ)dχ′′
=
1
2
χ2 −
√
2χ
g(θ)
coth
(g(θ)χ√
2
)
, (4.2.10)
and Sˆ0(θ), Sˆ1(θ) and ζ2(θ) are functions that will be determined by matching to the
mainstream and
g(θ) =
√(
1 + U2 −
(
Sˆ ′0(θ)
)2)
. (4.2.11)
Mainstream.- To leading order, the mainstream flow is governed by
R2 = 1 + U2 − (∇S)2 (4.2.12)
R2∇2S +∇R2∇S = 0 (4.2.13)
that can be combined to a single equation on S
(1 + U2 − 3(∇S)2)∇2S = 0. (4.2.14)
We expand S in powers of ε as in [177]
S(r, θ) = S0(r, θ) + εS1(r, θ) + εS2(r, θ) + · · · (4.2.15)
where S0, S1 etc. are expanded in powers of U as
S0 = US11(r) cos θ + U
3(S31(r) cos θ + S33(r) cos 3θ) + · · ·, (4.2.16)
where we assumed that θ = 0 is parallel to ~v. The solutions for the mainstream we find
up to O(U11); the first few are
122
S11 = −r
2 + 1
r
, S31 =
c1
r
+
6r2 − 1
6r5
, S33 =
c2
r3
+
1
2r
,
S51 =
c1
2r5
− 2c1
r3
+
c2
2r7
− c2
r5
+
c3
r
− 7
30r9
,
+
19
12r7
− 8
3r5
+
3
2r3
S53 = − c1
2r
+
3c2
5r7
− 3c2
r5
+
c4
r3
− 1
36r9
+
11
30r7
,
− 2
r5
+
1
2r
S55 = −3c2
2r3
+
c5
r5
− 3
4r3
− 1
4r
. (4.2.17)
To carry out the asymptotic matching, we substitute r = 1 + εξ into the mainstream
functions, expand the solution (4.2.16) in powers of ε and match it to the boundary
layer solution. To this order it is the same as to request that S ′ij(r) = 0 at r = 1, so on
the boundary of the disk. Thus we found the solution S0 to O(U
11). The first few terms
are (correcting the expression given in [177])
S0(r, θ) = −U (r
2 + 1)
r
cos θ + U3
[(
6r2 − 1
6r5
− 13
6r
)
cos θ +
(
1
2r
− 1
6r3
)
cos 3θ
]
+U5
[(
− 7
30r9
+
3
2r7
− 43
12r5
+
35
6r3
− 479
60r
)
cos(θ)+
(
− 1
36r9
+
4
15r7
− 3
2r5
+
43
30r3
+
19
12r
)
cos 3θ
+
(
7
20r5
− 1
2r3
− 1
4r
)
cos 5θ
]
+ · · ·. (4.2.18)
The boundary layer function becomes Ŝ0 = S0(1, θ) and the maximum flow velocity is
when cos θ = 1 (θ = pi/2). The corresponding maximum velocity is
u0max =
1
r
∂S0(r, θ)
∂θ
∣∣∣∣
r=1,θ=pi
2
=
= 2U +
7
3
U3 +
176
15
U5 +
1511639
18900
U7 +
5084105183
7938000
U9 +
311688814107079
55010340000
U11 + · · ·
(4.2.19)
This coincides with the expression for the velocity (in terms of Mach number) obtained
in [205] via a Janzen-Rayleigh expansion applied to the classical problem of the flow of
a compressible fluid passing around a solid disk.
The equation (4.2.14) becomes hyperbolic beyond a critical velocity. It first happens
at umax such that
1 + U2 = 3u2max. (4.2.20)
123
Solving this equation for umax as shown in Eq. (4.2.19) gives U = 0.263, or in dimen-
sionless units v = 0.37c. By considering the terms of the mainstream expansion up to
O(U40) we recover v = 0.36969(7)c in agreement with [205].
To get the analytical expression for the critical velocity for a finite size of the disk
we need to consider the O(ε) contribution to the mainstream solution that satisfies the
equations
R0R1 = −∇S0 · ∇S1 (4.2.21)[
1 + U2 − 3(∇S0)2
]∇2S1 = 6∇S0 · ∇S1∇2S0. (4.2.22)
For S1 we employ the expansion similar to Eq. (4.2.16), solve the ordinary differential
equations for Sij to get
S1(r, θ) =
d1
r
U cos θ + U3
[
cos θ
(
d1
2r5
− 2d1
r3
+
d2
r
)
+ cos 3θ
(
d3
r3
− d1
2r
)]
· ··, (4.2.23)
where the constants of integration di are found by matching the boundary-layer solution
to (6.5). We substitute r = 1 + εχ in (6.5), expand the solution in powers of ε and
match to the dominant linear in χ term in (4.2.9). The corresponding term in Ŝ2 is
expanded in powers of v and in trigonometric functions. The resulting expression for
the mainstream becomes
S1(r, θ)√
2
= −U 2
r
cos θ + U3
[(
5
3r3
+
1
r
)
cos 3θ +
(
− 1
r5
+
4
r3
− 31
3r
)
cos θ
]
+
+ U5
[(
− 17
20r5
− 3
r3
− 5
2r3
− 1
2r
)
cos 5θ
+
(
− 5
18r9
+
53
15r7
− 16
r5
+
1753
60r3
+
1
r
+
31
6r
)
cos 3θ
+
(
− 7
3r9
+
37
3r7
+
5
6r7
− 65
3r5
− 31
6r5
+
106
3r3
− 1161
20r
)
cos θ
]
+ . . . (4.2.24)
124
The ε term in the expansion for the maximum value of the velocity on the disk is
u1 max =
1
r
∂S1(r, θ)
∂θ
∣∣∣∣
r=1,θ=pi
2
(4.2.25)
=
√
2
[
2U +
46U3
3
+
8453U5
60
+
5525323U7
3780
]
+ · · ·
It is clear from this expression that the maximum velocity on the surface of the obstacle
is growing as the radius of the obstacle decreases for constant velocity of the main-
stream. Therefore, the nucleation of the excitations on the surface of the object is not
directly relevant to the maximum velocity for the objects of a finite radius as numerical
simulations show (see Section 2). As we show in the next section the vortices and other
excitations appear when it becomes energetically possible to create a doublet on the
surface. The asymptotics for ψ developed in this section allows us to get estimates of
the energy of the system.
4.2.1 Critical velocity of nucleation
The effect of the finite size of an obstacle on the critical velocity of vortex nucleation
has been studied numerically [199]. It was demonstrated that the nucleation takes place
when it becomes energetically favourable to create a vortex on the surface of the obstacle.
Here we use this criterion to obtain the critical velocity of nucleation by analytical means.
It is convenient to consider a different rescaling of (4.2.1) in units of ~x → ξ~x, t →
(ξ2M/~)t, v → (~/ξM)U and ψ → (ψ∞e−iUx)ψ such that ψ → 1 as |~x| → ∞. The
Eq. (4.2.1) becomes
2i∂tψ = −∆ψ + V ψ + (|ψ|2 − 1)ψ + i2U∂xψ. (4.2.26)
V denotes the rescaled potential modeling the fixed impurities inserted into the fluid’s
flow. The energy of the system (4.2.26) is [206]
E =
∫
|∇ψ|2 + (1− V − |ψ|2)2d~x. (4.2.27)
The solitary wave solutions such as vortex pairs and rarefaction pulses were analysed
numerically in [206] and asymptotically in [202]. The lowest energy of vortical solutions
is for the limiting case between a vortex pair and a rarefaction pulse: a doublet – a
single nodal point of ψ when two vortices of opposite circulation collide. The doublet is
moving through the uniform superfluid with velocity U ≈ 0.45. Its explicit form can be
approximated by adapting the Pade´ approximations considered in [202]:
ud = 1 +
a10x
2 + a01y
2 − 1
1 + c10x2 + c01y2 + c20x4 + c11x2y2 + c02y4
,
vd =
x(b00 + b10x
2 + b01y
2)
1 + c10x2 + c01y2 + c20x4 + c11x2y2 + c02y4
, (4.2.28)
125
where ud(x, y) = Re(ψ) and vd(x, y) = Im(ψ) for the doublet. The far field expansions
for |~x| → ∞ were considered in [206]:
ud ≈ 1 +m(2U −m)x
2 − 2U(1− 2U2)y2
2(x2 + (1− 2U2)y2)2 ,
vd ≈ − mx
x2 + (1− 2U2)y2 . (4.2.29)
To match it with (4.2.28) we set
a10 =
1
2
c20m(2U −m), a01 = −c20m(1− 2U2)U,
b10 = −mc20, b01 = −mc20(1− 2U2), (4.2.30)
c11 = 2c20(1− 2U2), c02 = c20(1− 2U2)2
and determine b00, c20, c10, c01 by expanding the stationary Eq. (4.2.26) with (4.2.28)
around zero and setting the constant term, the terms at x2 and y2 in the real part of
(4.2.26) as well as the term at x in the imaginary part of (4.2.26) to zero. The known
value of m = 3.32 [206] for U = 0.45 complete the determination of unknowns in (4.2.28).
We approximate the wave function of the doublet sitting on the surface of the disk of
the radius b by
ψd = ud(x, y − b) + ivd(x, y − b) (4.2.31)
and obtain the energy from (4.2.27) where the integration is for r > b. The critical veloc-
ity of the vortex nucleation from the surface of the moving disk is then associated with
the disk velocity at which the asymptotic solution (4.2.2) with S given by Eqs. (4.2.15),
(4.2.18) and (6.15) and R given by Eq. (4.2.12) reaches the energy of the doublet sitting
on the surface of a stationary disk. Figure 4.1 summarizes our findings and compares the
resulting critical velocities with the numerical solutions. We found that the procedure
for determining the criticality gives a good approximation for 20 > b/ξ > 2, for large
obstacles the criterion of the velocity exceeding the local speed of sound becomes more
accurate, whereas for the obstacle sizes of the order of the healing length the asymptotic
expansion of the solution breaks down.
In the next section we show that the shape of the potential modelling the impurity
has a profound effect on the type of excitation created.
4.3 2. Nucleation of excitations: vortices and
rarefaction pulses
The nonlinear Schro¨dinger equation (4.2.1) possesses elementary excitations in the form
of solitary waves: vortex pairs and rarefaction pulses – finite amplitude sound waves
[206]. In 2d rarefaction pulses have lower energy and momentum than vortex pairs,
so one may expect that narrow impurities, with radii smaller than healing length, will
generate rarefaction pulses rather than vortex pairs [180]. It is also clear from the
126
vcr/c
b/ξ
Figure 4.1: Critical velocity of vortex nucleation as a function of obstacle radius b/ξ.
Solid line – numerically determined critical velocity. The critical velocity
vcr = 0.36970c for an infinite obstacle is given by the thin red dashed line.
Thick green dashed line – analytically determined critical velocity as ex-
plained in the text.
127
ææ
æ
æ
æ
æ
æ
æ
0 1 2 3 4
0.5
0.6
0.7
0.8
0.9
1.0
à
à
à
à
à
à
à
à
à
à
à
à
à
0.0
0.2
0.4
0.6
0.8
ò
ò
ò
ò
vcr/c
V0
Figure 4.2: Critical velocity at which excitations are generated as function of obstacle
height V0 for fixed b = ξ. Dots represent the generation of vortex pairs, while
triangles represent rarefaction pulses. Squares represent the depletion of the
condensate (uniform density minus the minimum of the density at the centre
of the potential) due to the repulsive interactions with the obstacle [207].
topology of the system that if the obstacle does not bring about the zero of the wave
function of the condensate through the repulsive interactions the formation of a vortex
pair always starts from a finite amplitude sound wave. Therefore, one can envision that
depending on the properties of the repulsive interaction induced by the impurity on the
condensate different excitations are generated.
We start by modelling the repulsive interactions between the obstacle at positions
(xi, yi) and the condensate by a potential
V = V0
(
1− tanh [(x− xi)2 + (y − yi)2 − b2]) . (4.3.1)
Here V0 > 0 is the repulsive interaction strength between the impurity and the conden-
sate and b the impurity radius.
Fig. 4.2 shows the dependence of the critical velocity on the height V0 of the impurity
potential. In particular we have numerically found that for a weaker potential (i.e., small
V0) rarefaction pulses are generated rather than vortex pairs. In order to distinguishing
the generation of vortex pairs from the generation of the rarefaction pulses we have
128
throughout this work analyzed the excitations by evaluating if both the real and the
imaginary part of the wave function are zero, when passing a radius of two healing lengths
measured from the center of the impurity; setting a fixed radius is an unambiguous way
to make an identification as for example a rarefaction pulse might gain energy when
leaving the obstacle due to sound waves present in the condensate [202] and evolve into
a vortex pair. We have observed that the smaller is the strength V0 the greater is the
critical velocity of nucleation; for zero depletion of the condensate the criticality agrees
with the asymptotics considered in the previous section.
(a) (b) (c)
Figure 4.3: V ortex pair generation due to stronger BEC-impurity interaction.- Pseudo-
color density plots of superfluid flow around an obstacle specified by V0 = 0.7,
b = 1 (measured in healing lengths) at t = 60; (a), t = 84; (b) and t = 90;
(c) with velocity v = 0.753c. The more luminous the picture the higher the
density. Each picture shown corresponds to an area of 15 × 15 [208]. The
dimensionless units are used as specified in the main text.
As mentioned above if the obstacle does not completely deplete the condensate density
at its centre, then it is topologically impossible to create a vortex pair on non-zero
background. Instead a finite amplitude sound wave is created at the condensate density
minimum which can evolve into either a vortex pair or a rarefaction pulse as the wave
separates from the obstacle and gains energy entering the bulk. The outcome in this
case depends on the energetics created by the obstacle: the larger V0 the more energetic
solution emerges. This is illustrated in Figs. 4.3 and 4.4. In Fig. 4.3 we show time
snapshots illustrating the emergence of a vortex pair for a stronger barrier. A finite
amplitude sound wave formed at the impurity atom (a) evolves into a pair of vortices
(b) leaving towards the direction of the stream of the superfluid (c). Fig. 4.4 illustrates
the formation of a rarefaction pulse at a weaker obstacle; here after a finite amplitude
sound wave is formed at the obstacle it evolves into a rarefaction pulse that is carried
away by the stream. In Fig. 4.5 we present the full wave functions of a rarefaction pulse
and a vortex pair.
129
(a) (b) (c)
Figure 4.4: Rarefaction pulse generation due to weaker potential.- Pseudo-color density
plots of superfluid flow around an obstacle specified by V0 = 0.3, b = 1 at
t = 150; (a), t = 175.5; (b) and t = 186; (c) with flow velocity v = 0.88c.
The more luminous the picture the higher the density. Each picture shown
corresponds to an area of 15× 15 [208]. The dimensionless units are used as
specified in the main text.
Delta-function impurity.- The special case of a single delta-function impurity V =
δ (x− xi + y − yi) can be regarded as a limiting case of the above setup, i.e., a single
point obstacle at which the wave function is zero. In this case at the critical veloc-
ity slightly below the speed of sound rarefaction pulses are generated, see Fig. 4.6,
consistent with considerations in [180]. However, for finite size obstacles in diameter
even smaller than the healing length the vortex pairs can be generated as soon as the
the depletion of the condensate density is steep enough (even if the wave function is
not zero at the obstacle), see Fig. 4.7 where we considered a potential of the form
V = V0 (1− tanh [α((x− xi)2 + (y − yi)2)]), with α > 1.
Our numerical analysis shows that a stronger dip in the condensate makes a vortex
pair favorable, while less deep depletions of small radius favor rarefaction pulses. In
particular for weak BEC-impurity interactions of bigger radius (V0  1, b 1) we have
found that vortex pairs are generated, see Fig. 4.8. Hence, it depends on the energy
put into the system via the potential which excitation can be afforded, i.e., low energy
potentials favour energetically cheaper rarefaction pulses while higher energy potentials
more expensive vortex pairs.
Supersonic flow.- Finally, for completion, we consider the superfluid at supersonic
speed (i.e., for Mach numbers M = v/c > 1) passing a narrow and only weakly inter-
acting obstacle, that does not lead to a complete depletion of the condensate density
at its position. The case of a strongly interacting (delta function) obstacle has been
considered in [181, 212], where oblique dark soliton trains were observed in the wake
of the obstacle. Here in Fig. 4.9 we present the emergence of rarefaction pulses in the
stream of the condensate accompanied by Cherenkov waves outside the Mach cone. The
rarefaction pulse in the wake of the obstacle clearly differs from the solitary waves (or
130
(a) (b) (c) (d)
Figure 4.5: Density and Phase.- Pseudo-color density plots of the phase (a) and density
(b) of the flow around an obstacle specified by V0 = 0.72, b = 1 at t = 150
with flow velocity v = 0.88c forming vortex pairs. It is compared with
pseudo-color density plots of the phase (c) and density (d) of the flow around
an obstacle inducing rarefaction pulses specified by V0 = 0.17, b = 1 at
t = 170 with flow velocity v = 0.95c. The more luminous the picture the
higher the density. Each picture shown corresponds to an area of 15 × 15
[208]. The dimensionless units are used as specified in the main text.
continuous stream of vortex pairs) spotted for heavy (delta function) impurities in [181]
in so far as not a full depletion of the condensate has been observed.
4.4 3. Superfluid regimes in the lattice of impurities
In this section we consider N inserted obstacles at positions (xi, yi) with i ∈ {1, . . . , N}
that generate an external potential of the form
V = V0
N∑
i=1
(
1− tanh [(x− xi)2 + (y − yi)2 − b2]) . (4.4.1)
The potential function of the whole array V depends on the spatial order of the inserted
impurities, the radius b of the atoms and the strength of BEC-atom interaction V0. We
suppose that there are no interactions between the impurities themselves and that they
are stationary, but remark that the additional presence of an atom in the superfluid leads
to a change of the potential generated by the other atoms – their states are squeezed
[213] within the superfluid. In this work, however, we do not address the issue how
the size of the impurities materializes, but elucidate that having a fixed size affects the
superfluid as presented.
131
(a) (b) (c)
Figure 4.6: Rarefaction pulse generation due to a delta-function impurity.- Pseudo-color
density plots of superfluid flow around a delta impurity at t = 21.5 (a),
t = 26.5 (b), t = 41.5 (c), with velocity v = 0.92c. Each picture shown
corresponds to an area of 12× 12 [209]. The more luminous the picture the
higher the density. The dimensionless units are used as specified in the main
text.
4.4.1 Impurities arranged into regular lattices
Let us consider the superfluid’s flow around impurities arranged on a triangular lattice
within a finite sized rectangular area. We distinguish different regimes in superfluid
flow passing the lattice that besides the characteristics of impurity atoms V0 and b
depend on the velocity v, the distance between nearest neighbors a and the size of the
lattice. The various regimes are presented in Fig. 4.10 qualitatively as functions of v
and a: Area I corresponds to the superfluid phase, i.e., no excitations emerge in the
superfluid’s stream. The regime II is characterized by the emergence of vortices from
the boundary of the network. III describes very dense packing of impurity atoms, such
that for those velocities vortices are created on the boundaries of the entire lattice, while
the superfluid is expelled from within the lattice. Region IV is described by excitations
within the lattice, which span from excitations smaller then the healing length and
ones bigger than a few healing lengths, i.e., emergent macroscopic structures. In region
V vortices or rarefaction pulses are generated within the lattice as well as outside the
lattice without forming bigger structures as a consequence of the sparsity of the impurity
atom distribution.
In Fig. 4.11 we present the superfluid flow towards the right hand side around a sparse
triangular lattice in the regions II and V. Above the first critical velocity v1, excitations
are generated at the end of the array and directly move into the wake (b) (region II).
Above the second critical velocity v2 excitations are continuously generated within the
lattice and move - carried by the fluids stream - through the impurity network towards
the wake of the system (c) (region V). In Fig. 4.12 we show the first v1 and second v2
critical velocities as functions of the mean distance between nearest neighbors a at the
triangular lattice.
132
(a) (b)
Figure 4.7: V ortex pair generation due to a potential of diameter less than a healing
length.- Pseudo-color density plots of superfluid flow around a very small
obstacle specified by V0 = 40 and α = 7 at t = 0 (a), t = 22.8 (b) with
velocity at infinity v = 0.92c. Each picture shown corresponds to an area of
10 × 10 [210]. The more luminous the picture the higher the density. The
dimensionless units are used as specified in the main text.
.
Let us take a closer look on the lattice dynamics in region V corresponding to a
lattice of intermediate density of impurities. We could identify different processes of
three body interactions of vortex dipoles with single vortex among that are the flyby
regime and the reconnection regime [78]. The flyby characterizes the scenario where an
incoming vortex pair gets deflected by the third vortex and the reconnection regime
describes the situation where vortex 1 of the pair is coupled to the third vortex and the
other vortex 2 of the pair is left behind. Moreover the transition from vortex pair to
rarefaction pulse and vice versa due to loss or gain of energy through sound waves could
be identified. In Fig. 4.13 an example of a vortex pair acquires a velocity component
transversal to the motion of the fluid by pinning to the potential spikes is shown (see
pictures (a),(b),(c),(d),(e),(f)). Here a vortex pair looses one vortex to an impurity.
As the vortex moves away from the obstacle it pulls the trapped vortex back, while
acquiring an additional transversal velocity component and heading towards the next
impurity. This process can be classified as a flyby, when applying a vortex mirror-vortex
analogy [215]. In addition we could identify a locally stationary single vortex placed at
the centroid of the triangle formed by nearest neighbor impurities. When other vortices
enter the scene they almost form a triangle by connecting with the impurity atoms
at the corners (connecting regime) and due to gain of energy through perturbations
subsequently decay into many vortices.
For weaker BEC-impurity interactions we have found an intermediate regime, where
133
(a) (b) (c)
Figure 4.8: V ortex pair generation due to a potential of diameter much bigger than heal-
ing length and causing only slightly a depletion of the condensate.- Pseudo-
color density plots of superfluid flow around a obstacle specified by V0 = 0.25
and b = 4 at t = 57; (a), t = 75; (b), t = 105; (c) with velocity at infinity
v = 0.8c. Each picture shown corresponds to an area of 40 × 40 [209]. The
more luminous the picture the higher the density. The dimensionless units
are used as specified in the main text.
.
(for example for parameters V0 = 0.1 and b = 1) vortices are generated although single
impurities of the same specification would prefer the generation of rarefaction pulses;
high energy rarefaction pulses absorb energy due to perturbations present in the lattice
and once their energy passes some threshold they transform into an energetically favor-
able vortex pair [202]. For very weak BEC-impurity interactions (V0 is very small) solely
rarefaction pulses are generated in the stream of the condensate within and at the wake
of the impurity lattice, which are slightly deflected as they pass weak potential spikes.
The region IV in Fig. 4.10 is characterized by high density arrays of impurities, which
yield excitations that span over several neighboring impurities. In Fig. 4.14 we show
snakes of excitations moving through the lattice. These excitations are either zeros of
wave function and therewith can be identified as vortices or are more comparable with
rarefaction pulses not fully depleting the condensate. In particular we have observed
that excitations within the lattice might move in opposite direction to the mainstream
direction. With more narrow space between impurities further excitations are present
within the lattice. As there is not enough space for a fully developed vortex pair or
rarefaction pulse between neighboring impurity atoms, excitations in such lattice emerge
as finite amplitude sound waves, i.e, spontaneously occurring density depletions between
two neighboring atoms, which occasionally persist and move within the lattice generally
towards all possible directions.
Finally we have considered very high densities of impurity atoms with V0 large enough
such that the condensate is (almost) expelled from the lattice and the velocity is chosen
such that the superfluid phase is surpassed, i.e., III in Fig. 4.10. Here we have found
that vortices are generated at the boundaries of the lattice. In the wake some vortices
134
(a) (b) (c)
Figure 4.9: Rarefaction pulse generation due to a small and weakly interacting potential
at supersonic speed.- Pseudo-color density plots of superfluid flow around a
obstacle specified by V0 = 0.2 and b = 1 at t = 45; (a), t = 82.5; (b) and
t = 180; (c) with Mach number v = 1.01c. The more luminous the picture
the higher the density. Each picture shown corresponds to an area of 35×35
[211]. The dimensionless units are used as specified in the main text.
enter the slipstream region, such that no significant motion between vortices and lattice is
present or movement towards the lattice might occur - an analog situation as encountered
for moving obstacles generating turbulent flow in normal fluids. In Fig. 4.15 we indicate
the motion of the vortices in the slipstream region for a very slow superfluid with only
few vortices present in the wake, i.e., two counter propagating curls of vortices evolving
from both edges of the lattice.
4.4.2 Uniformly distributed impurities
We now turn to the regimes of a flow passing randomly distributed impurities occupying
a finite area A. These inserted atoms can be regarded as an ideal gas of NA uniformly
distributed noninteracting particles in the plane R2 at an instant of time. We denote
the density of particles (or impurities) given by the total particle number per area by n.
To determine the mean distance between particles we recognize that the probability of
finding another particle within the distance r from its origin is P[r,r+dr] = 2pirndr. The
probability to find a particle outside the disc, i.e., in [r,∞], is P[r,∞] = 1−pir2/A, where
A is the total area. Hence the probability distribution function of the distance to the
nearest neighbor is
PN(r) = 2pirN/A
(
1− pir2/A)N−1 , (4.4.2)
135
Figure 4.10: Schematics of the distinct regions in superfluid flow around regular lattices
on a rectangular area. The boundaries are approximate and are sensitive
to the parameters of the impurity network. Here a denotes the distance
between nearest neighbors, which ranges from 0 to 12 healing lengths and
v the absolute value of the superfluid velocity at infinity ranging up to the
speed of sound c.Hatched areas II, III show regions where excitations are
outside the lattice and V, IV indicate the regimes where excitations are
within the lattice as well.
136
(a)
(b)
Figure 4.11: Pseudo-color density plots |ψ|2(x, y) of qualitatively distinguishable phases
in superfluid flow passing a hexagonal lattice of impurities. Shown is the
condensate density at velocity v = 0.5c (a) and v = 0.555c (b). Other
parameters of the system: Array size A = 240×72, frame shown 480×120,
a = 5, b = 1.5,V0 = 1, all snapshots at t = 125 [214]. The more luminous
the picture the higher the density. The dimensionless units are used as
specified in the main text.
137
òò
ò
ò
ò
ò
ò
ò
ò
ò ò
à
à
à
à
à
à
à
à à
à à
4 6 8 10 12
0.35
0.40
0.45
0.50
0.55
vcr/c
a
Figure 4.12: Critical velocities as functions of the distance between impurity to its near-
est neighbors a. For this comparison the remaining free parameters have
been chosen to be V0 = 1, b = 1.5 while the area at which the triangular
lattice of atoms has been present is 200×120 (measured in healing lengths).
Triangles represent data points of the first and squares data points of the
second critical velocity. The interpolation between data points is linear.
[214].
138
(a) (b) (c)
(d) (e) (f)
Figure 4.13: Pseudo-color density plots of superfluid flow around a obstacles on arranged
on a triangularlattice specified by V0 = 10 and b = 2 at t = 310; (a),
t = 312.5; (b), t = 315; (c); t = 317.5; (d), t = 320; (e), t = 325; (f) with
velocity at infinity v = 0.67c. Each picture shown corresponds to an area of
30 × 30 [216]. The more luminous the picture the higher the density. The
dimensionless units are used as specified in the main text.
which for n fixed becomes in the limit N →∞
P (r) = 2pir exp(−pir2n)n. (4.4.3)
Note that (4.4.3) might be regarded as a good approximation to (4.4.2) for large N .
Thus, the mean distance a is given by considering the expectation value
a ≡ E1[r] =
∫ ∞
0
r2pi exp(−pir
2
n
)ndr =
1
2
√
n
. (4.4.4)
In this sense a random distribution of particles determined by its area and number relates
to the mean distance, i.e., n = NA/A = 14a2 . In Fig. 4.16 we present results showing that
even for systems of less chosen structure than fixed lattices, qualitatively different phases
can be distinguished. That is a phase of dissipationless superfluid flow, the generation
of first excitations in the wake of the lattice and generation of excitations within the
lattice. In contrast to the hexagonal lattice, however, these transitions are generally
not as smooth. Figure 4.16 shows the density of the condensate for the superfluid flow
around inserted impurities uniformly distributed on a finite domain [214], [217]. Fig. 4.16
(a) shows a superfluid’s flow without dissipation of energy and generation of elementary
excitations, (b) corresponds to a flow above criticality carrying excitations in the wake
of the superfluid and (c) an even faster flow at which excitations are generated within
the array.
139
4.5 Conclusions
The generation of vorticity by a moving superfluid has generated a lot of experimental
and theoretical work. In our paper we re-examine this problem by developing an asymp-
totic and analytical methods for finding the flow around an obstacle and for determining
the critical velocity of vortex nucleation. We numerically study the various excitations
generated above the criticality. We determine the regimes when a vortex pair or a finite
amplitude sound wave is generated depending on the energetics of the obstacle. We
described several novel regimes as a superfluid flows an array of impurities motivated by
recent experiments.
140
(a)
(b) (c)
Figure 4.14: Pseudo-color density plots |ψ|2(x, y) of superfluid flow passing dense hexag-
onal distributed atoms with velocity v = 0.5c. The size of atom array is
290 × 180, the frame size shown in picture (a) is 220 × 220 at t = 522.5.
Picture (b) is a detail of frame size 70× 70 at t = 522.5 and picture (c) at
t = 525 (with frame size 70× 70). The atom parameters were chosen to be
V0 = 1, a = 3 and b = 1.5. [216]. The more luminous the picture the higher
the density. The dimensionless units are used as specified in the main text.
141
Figure 4.15: Pseudo-color density plots |ψ|2(x, y) of superfluid flow passing very dense
triangular lattice of impurities with velocity v = 0.2c. The frame size
shown is 150 × 150 at t = 539.25. The impurity parameters were chosen
to be V0 = 1, a = 1.6 and b = 1.5. [216]. The more luminous the picture
the higher the density. Arrows point towards the direction of motion of the
encircled vortices, and the big arrow indicates the direction of the superfluid
mainstream. The dimensionless units are used as specified in the main text.
142
(a) (b) (c)
Figure 4.16: Pseudo-color density plots |ψ|2(x, y) of phases in superfluid flow passing
uniformly distributed atoms with different velocities at t = 165. The atom
number is NA = 80, size of possible atom positions 120 × 130 (measured
in healing length), the frame size shown is 210 × 210 in same units and
the atom parameters were chosen to be V0 = 2 and b = 2. Picture (a)
shows the flow at v = 0.2c, (b) at v = 0.32c, (c) at v = 0.39c. All pictures
show the same (random) distribution of atoms and represent a snapshot at
t = 165 [218]. The more luminous the picture the higher the density. The
dimensionless units are used as specified in the main text.
143
5 On-demand dark soliton train
manipulation in a spinor polariton
condensate
Florian Pinsker and Hugo Flayac, Phys. Rev. Lett. 112, 140405 (2014).
We theoretically demonstrate the generation of dark soliton trains in a one-
dimensional exciton-polariton condensate within an experimentally accessible scheme.
In particular we show that the frequency of the train can be finely tuned fully optically
or electrically to provide a stable and efficient output signal modulation. Taking the po-
larization degree of freedom into account we elucidate the possibility to form on-demand
half-soliton trains.
5.1 Main text
5.1.1 Introduction
The first unambiguous observation of Bose-Einstein condensation in dilute Bose gases
at low temperature [4] set off an avalanche of research on this new state of matter. The
lowest energy fraction of a degenerated Bose gas occupying low energy modes obeys the
property of vanishing viscosity and does not take part in the dissipation of momentum,
a phenomenon referred to as superfluidity [18]. This holds true as long as the condensate
is only slightly disturbed [169]. As soon as strong dynamical density modulations occur,
e.g. when the condensate is abruptly brought out of its equilibrium through an external
perturbation, it responds in a unique way by generating robust elementary excitations
such as solitons or vortices [113].
More recently the concept of macroscopically populated single particle states [12, 13]
was transposed to a variety of mesoscopic systems such as cavity photons [14, 15],
magnons [219], indirect excitons [220], exciton-polaritons (polaritons) [16] and even clas-
sical waves [17]. In the proper regime all those systems can be described by complex-
valued order parameters - the condensate wave functions - with dynamics governed by
nonlinear Schro¨dinger-type equations (NSE) such as the Gross-Pitaevskii (GP) [8, 7]
and the complex Ginzburg-Landau equation (cGLE) [48]. Here the nonlinearity associ-
ated with self-interactions plays an essential role on the possible states with or without
excitations, their dynamics and in particular their stability [221]. Similarly in the slowly
varying envelope approximation light waves can be approximated by complex-valued
144
wave functions governed by NSEs that are formally comparable to those of BECs and
thus show analog dynamical behavior such as stationary and moving optical dark or
bright solitons in quasi 1D settings [222, 223].
For several decades light waves have been utilized in a wide range of applications such
as in nonlinear fibre optic communication [222, 224, 225, 226] while research on new
technologies is thriving in particular on elementary circuit components such as optical
diodes [227], transistors [228] or realizations of analog devices involving exciton-polariton
condensates [229, 230] and conceptually on optical computing schemes [231].
Exciton-polaritons are half-light half-matter quasi-particles formed in semiconductor
microcavities and allow high speed propagation from their photonic part while having
strong self-interaction from their excitonic fraction. They are extremely promising from
both the fundamental and technological point of view given the ease it provides to finely
control the parameters of their condensate now routinely produced in different geome-
tries (see e.g. [232]). Indeed, state-of-the-art technology allows to etch any sample shape
to sculpt the confining potential seen by the condensate at will. It explains the plethora
of recent proposals [233, 35, 36, 37, 38, 39, 40, 41] for polariton devices some of which
have been experimentally implemented [229, 230]. The main advantage with respect to
standard optical systems in nonlinear media is the very large exciton-mediated nonlin-
ear response of the system reducing the required input power by orders of magnitude.
Recently there was a growing interest in demonstrating the formation of (spin polar-
ized) topological defects [234, 235] that are now envisaged as stable information carriers
[249, 236, 237] within a young field of research called spin-optronics [238].
In this letter we shall present experimentally trivial schemes for the intended genera-
tion and manipulation of stable and fully controllable wave patterns within a quasi-1D
microcavity. We will demonstrate the on-demand formation of dark soliton trains within
a quasi-1D channel and the optical and electrical dynamical control of their frequency.
Finally we shall demonstrate the possibility to control the polarization of the soliton
trains.
5.1.2 The model
We shall consider the system modeled in Fig.5.1, namely a wire-shaped microcavity
similar to the one implemented in Ref.[240] that bounds the polaritons to a quasi-1D
channel. A metallic contact is embed over half of the sample to form a potential step seen
by the polaritons and whose amplitude can be tuned on-demand applying an electric
field [241]. The spinor polariton field ψ = (ψ+, ψ−)T evolves along a set of effectively
145
Figure 5.1: (Color online) Model of a potential sample consisting in a quasi-1D micro-
cavity (DBR=distributed Bragg reflectors) embedding a metallic deposition
over half of its length to form a potential step. A gate voltage can be applied
to the metal to tune dynamically the step amplitude.
1D cGLEs coupled to a rate equation for the excitonic reservoir [241],
i~
∂ψ+
∂t
=
[
−~
2∆
2m
+ α1
(|ψ+|2 + nR)+ α2|ψ−|2]ψ+
+
[
U − i~
2
(Γ− γnR)
]
ψ+ − Hx
2
ψ− (5.1.1)
i~
∂ψ−
∂t
=
[
−~
2∆
2m
+ α1
(|ψ−|2 + nR)+ α2|ψ+|2]ψ−
+
[
U − i~
2
(Γ− γnR)
]
ψ− − Hx
2
ψ+ (5.1.2)
∂nR
∂t
= P − ΓRnR − γ
(|ψ+|2 + |ψ−|2)nR. (5.1.3)
This model describes in a simple way the phenomenology of the condensate formation
under non-resonant pumping. We assume a parabolic dispersion of polaritons associated
with an effective mass m = 5×10−5m0 where m0 is that of the free electron and a decay
rate Γ = 1/100 ps−1. U(x, t) = [U(t) + U0]H(x), where H(x) is the Heaviside function
U0 = −0.5 meV is the step height induced by the presence of the metal solely and U(t)
is the potential landscape imposed by the external electric field. α1 = 6xEba
2
B/S =
1.2× 10−3 meV·µm and α2 = −0.1α1 are respectively the parallel and antiparallel spin
interaction strength given x, Eb and aB the excitonic fraction, binding energy and Bohr
radius respectively and S the pump spot area. Hx = 0.01 meV is the amplitude of the
effective magnetic field induced by the energy splitting between TE and TM eigenmodes
that couples the spin components. The excitonic reservoir characterized by the decay
146
rate ΓR = 1/400 ps
−1 is driven by the pump term P = AP exp(−x2/σ2) where σ = 20
µm and AP is taken in the range of hundreds of ΓR. It exchanges particles with the
polariton condensate at a rate γ = 2× 10−2ΓR.
We note that while the stimulated scattering is taken into account by Eqs.(5.1.1-5.1.3),
energy relaxation processes dominant under the pump spot, apart from the lifetime
induced decay of the interactions energy, are neglected in this framework and could be
treated e.g. within the formalisms of Refs.[242]. Energy relaxation would not impact
our results qualitatively especially for the finite pump spot size we consider here.
5.1.3 Soliton train generation
As shown in Ref.[113] a local abrupt change of self-interaction strength of the condensate
leads to the formation of a stable and regular dark soliton train. It happens when the
flow in the direction of decreasing interaction due to particle repulsions is locally crossing
the speed of sound cs(x) =
√
µ(x)/m where µ(x) = α1n(x) (for a scalar condensate)
at the point of abrupt change in self-interactions, solitons are formed from dispersive
shock waves [243] that dissipate the local excess of energy. In polariton condensates
the interaction strength α1 is varied tuning the exciton/photon detuning and therefore
the excitonic fraction, but it can hardly be made inhomogeneous within a given sample
nor tuned dynamically. A valuable alternative we follow here is to introduce the tun-
able potential step U(x, t) in Eqs.(5.1.1,5.1.2). The mechanism for soliton generation
is the following (see supplemental material for a more details). Let us suppose we a
homogeneous density n0 at t = 0 and neglect the finite lifetime and pumping of quasi
particles and the geometry of our pump spot. Then taking the potential U stepwise
for all following t > 0, we get close to the breaking point at x = 0 the density n1 for
x = 0− and n2 as x = 0+ and we say n1 = k · n2 with 1 > k > 0. Using momentum and
mass conservation at x = 0 we find the simple criterion 0.6404 > k to break the speed of
sound in the region x < 0, which is in good agreement with our numerical results. In the
regime of soliton-train generation, the frequency ν increases with the magnitude of the
potential step [?] as the corresponding increase of mass passing the step at x = 0 allows
a more frequent breaking of the local speed of sound. This is analogous to the situation
of a superfluid passing an obstacle above criticality for which greater mass transport is
equivalent to a higher number of generated vortices in 2D [245].
For a given metal type and deposition thickness on top of the microcavity, Tamm
plasmon-polariton modes [246] were predicted to form at the interface inducing a local
redshift of the polariton resonances of amplitude U0 and the required potential step. We
note that in the absence of plasmon the interface would form a Schottky junction known
to blue detune the polariton modes [247]. The application of a voltage to the metal
produces the additional gate redshift U(t) through the excitonic Stark effect up to a few
meVs for voltages lying in the range of tens of kV/cm [248] and standing for the input
modulation of the polariton condensate. The non-resonant excitation of the system is
crucial since in this context the condensate phase is free to evolve under the pump spot
by contrast to a resonant injection scheme that would imprint the phase preventing the
onset of solitons.
147
Figure 5.2: (Color online) Optical control. (a), (b) (c) Shows results obtained by pump-
ing over the potential step with increasing pump amplitude of 200ΓR, 375ΓR
and 600ΓR respectively, we monitor here µ(x, t) (meV) in the colormap. (d)
Sinusoidal modulation of the pump amplitude between 0 and 500ΓR and with
a period T = 100ps resulting in signal frequency modulation.
5.1.4 Optical control
Let us start with the simplest passive configuration where no voltage is applied and
therefore the potential step is fixed. We switch on the pump laser focussed on the step at
t = 0 and wait for the steady state to be reached. The reservoir is filled by the incoherent
pump and the stimulation towards the lowest polariton energy state occurs forming the
condensate with a chemical potential µ = (α1 + α2)n/2 − Hx/2 (corresponding to the
measurable blueshift of the polariton emission) where n = |ψ+|2 + |ψ−|2 = n+ + n− is
the total polariton density. Given the interrelation α1 > α2 the condensate interaction
energy is minimized for a linear polarization meaning that n+ = n− and the condensate
is said to be antiferromagnetic [249]. The linear polarization orientation is homogeneous
at zero temperature and fixed by the Hx contribution namely along the axis of the wire.
In our model we trigger the condensation on the x-polarized ground state with initial
populations n0±. Fig. 5.2 shows numerical solutions to Eqs.(5.1.1,5.1.2,5.1.3). We depict
the chemical potential µ(x, t) for crescent pump amplitudes AP . We clearly see the
decrease in the train frequency ν with increasing pump power [panels (a) and (b)] until
the train vanishes [panel(c)].
The condensate heals from the step forming an asymmetric gray soliton resulting from
the local velocity gradient, as it happens e.g. at the boundaries of condensate trapped in
a square potential. The depth of the soliton is imposed by the local background density
and velocity. For a high enough background density, the flow is superfluid (v < cs) both
around the step and within the soliton that remains pined to the step preventing the
148
train onset [panel(c)]. For lower densities, the speed of sound can be surpassed at the
soliton core which allows the condensate to dissipate the local excess of energy via a
dispersive shock wave [243] (see supplemental movies [?]) that releases the soliton to
the side where the background flow is the highest. Then it takes some time for the
condensate to form a new soliton. The higher the density the stiffer is the condensate
and therefore the more time it takes to form a new density depletion. This response
determines the quasi-linear train frequency ν dependence over the chemical potential
shown in the supplemental material [?].
Our results demonstrate the possibility to modulate passively an optical signal via the
formation of stable dark solitons varying the pump amplitude. The dark soliton signals
shall then be detected experimentally at the output via one of the schemes proposed
in the context of nonlinear optics [250] to encode information. Indeed, as proposed in
[25] soliton trains can be used to store numbers determined uniquely by an adjustable
ν. So far, most of the device proposals involving microcavity polaritons have focussed
on signal transmission but never on its modulation. Nonetheless, as one can see, the
train frequencies lie in the range of THz allowing to perform very high speed processing
due to the polariton photonic part combined with a large exciton-mediated nonlinear
response.
In Fig.5.2(d), we show an example of sinusoidal input power modulation that leads
to a dynamical variation of ν or a modulation of the output on-demand to produce
useful wavepackets. The main advantage of this all-optical input modulation scheme
is that it allows us to reach high speed variation of ν while the drawback is that the
background density of the condensate is obviously affected as well. Finally we note
that this setup involving a fixed potential step doesn’t require specifically a metallic
deposition. A sample split in two parts with slightly different lateral width might be
sufficient to reproduce the effects discussed above.
5.1.5 Electric control
Now let us consider the case where the pump power is fixed and in addition an electric
field is applied to the metallic contact to modulate the potential step height. Under
such assumptions, the chemical potential µ is globally fixed. The higher the step (the
electric field), the larger the density gradient and hence one encounters a greater mass
transport towards lower energy regions. So similarly to [245] we obtain an increase in
dark solitons train frequency as shown in [?]. To demonstrate this behavior, we show in
the Fig.5.3(a) the results obtained by ramping down linearly the potential step from 0
meV to -1.5 meV which corresponds to an increase in the electric field amplitude. We
clearly see the linear increase in ν versus time.
Similarly to the results of Fig.5.2(d) we show in Fig.5.3(b) results obtained from a
sinusoidal modulation of the potential step amplitude producing an efficient dynamical
modulation of the output signal in the form of wavepackets. Such an electric control
of the polariton flow has the advantage of impacting weakly on the background density
but there might be some technological limitation on the switching speed.
149
Figure 5.3: (Color online) Electric control of the dark soliton trains. (a) The potential
step amplitude U(t) is linearly ramped down versus time from 0 meV to -1.5
meV. (b) Sinusoidal modulation of the step between 0 and −1.5 meV with
a period of 50 ps.
5.1.6 Polarization control
So far we have discussed a phenomenology that could be reproduced using a scalar
condensate without any need for its spinor character. Indeed we have considered the ideal
case of a perfectly linearly polarized condensate with no polarization symmetry breaking
namely n+(x) = n−(x) for any x position. The consequence is that the dark solitons
formed in one spin component are perfectly overlapping with the ones in the other
component and are behaving as scalar ones. However in real experimental situations the
fluctuations brought by the structural disorder or the background noise can affect the
linear polarization of the condensate leading to local inhomogeneities slightly breaking
the polarization symmetry or the equivalence between the two spin components. As
discussed in [236] these fluctuations can lead to the separation of dark solitons in each
component to form pairs of half-solitons [234, 235]. As soon as they are split they will
150
repel each other under the condition α2 < 0 and start to feel an effective magnetic force
imposed by Hx and be accelerated or slowed down depending on their linear polarization
texture [236]. This effect produced by local inhomogeneities or random processes would
obviously be harmful to the formation of a deterministic spin signal. However as it was
observed experimentally in Ref.[251], using a polarized excitation laser can lead to the
formation of a circularly/elliptically polarized condensate due to the long characteristic
spin relaxation times of excitons. In Fig.5.4, we show results capitalizing on this effect
to produce a useful spin signal.
We have modeled a slightly elliptically polarized nonresonant pump introducing two
different reservoir/condensate transfer rates γ1 and γ2 in Eqs.(5.1.1,5.1.2). We have
adjusted the ratio γ2/γ1 to 0.90, 1.11 and 0.99 in panels (a), (b) and (c) respectively.
The colormap shows the degree of circular polarization ρc(x, t) = (n+ − n−)/(n+ + n−)
of the polariton emission. We see that the weak ensuing density imbalance between the
two spin components of the condensate leads to a well defined polarization symmetry
breaking inducing either the formation of trains of pairs of half-solitons [panel (c)] or
trains of half-solitons with a well defined polarization for larger imbalances [panels (a)
and (b)]. It means that not only the frequency of the trains can be finely tuned but also
their polarization by variation of the input polarization. It provides another degree of
freedom to code information.
5.1.7 Conclusions
We have shown the strong potential of microcavities for high speed optical signal modu-
lation for information coding. Our proposal involves the all-optical or electric control of
dark soliton trains within a realistic and experimentally trivial scheme. We have demon-
strated the possibility to tune both the train frequency and its polarization. The present
concept could not only play a central role at the heart of future high speed polariton
circuits within the rapidly expanding field of spin-optronics but also allow the very first
observation of dark soliton trains in a quantum fluid.
151
5.2 Supplemental material
5.2.1 Velocities
To clarify the mechanism underlying the dark soliton train formation in our setting, we
have computed the speed of sound cs and the superfluid velocity v = ~/m∇φ dynamics
(φ is the phase of the wave function). Fig. 5.5 shows results obtained for two different
pump powers: In the panel (a) the condensate is superfluid (|v| < cs) all over the step
and no local energy dissipation can occur to regularize the density and phase modulation
(asymmetric grey soliton) imposed by the step. On the contrary in the panel (b) the
core of the soliton at the step crosses the speed of sound (|v| > cs) allowing a shock wave
to develop releasing the soliton to the right of the step. The frequency of the train is
increasing with the stiffness ρs = n~2/m (density) of the condensate since it takes more
time to heal (produce a new modulation at the step) for a higher density leading to a
lower train frequency [see Fig. 5.2]. The full dynamics is shown in the supplemental
movies provided with the manuscript.
5.2.2 Dependencies of the train frequency
We have numerically determined the dependence of the train period T and frequency
ν = 1/T on the chemical potential µ and the potential step amplitude U . The results
are presented in Fig. 5.6 and Fig. 5.7 respectively.
5.2.3 Critical density modulation
Our analytical consideration starts with neglecting the spin degree of freedom (antiferro-
magnetic condensate regime), so we get the governing equation for the condensate wave
function ψ given by
i~
∂ψ
∂t
=
(
−~
2∆
2m
+ U + α′′1
(|ψ|2 + nR)− i~
2
(Γ− γnR)
)
ψ − Hx
2
ψ+. (5.2.1)
First, we note that at the breaking point at x = 0 the complex pumping and decaying
term is positive (for the parameters considered), corresponding to an increase in particles
there. We simplify this equation by neglecting the effect of pumping to the condensate,
which increases the superfluid’s velocity there and by additionally approximating the
reservoir population as nR ' c1+c2|ψ|2. The latter expression is due to an assumed quasi-
stationary reservoir population nR = P/(k1 + k2|ψ|2) and further simplified via a Taylor
expansion to nR = P (k3 + k4|ψ|2) and considered for P = k5. Here c1, c2, k1, k2, k3, k4, k5
are constants. Also note that the nonlinearity is small. Although a constant pump is
far from the geometry of the exponential distribution used in our simulations, it is the
first order approximation to hold at x = 0 and as we will see it allows us to make a
simple statement about the relevant physics in our soliton generator. Now, by absorbing
constant terms into the phase of the wave function via ψ → ψ exp(iθt) we get the
152
Gross-Pitaevskii like equation(
−i~ ∂
∂t
− ~
2∆
2m
+ U ′ + α′1|ψ|2
)
ψ = 0, (5.2.2)
where α′1 is the effective self-interaction strength for our simplified exciton-polariton
equation. Recall that the potential U’ which shall be applied is of stepwise form, i.e.
U ′1 > 0 for x < 0 and U
′
2 = 0 for x > 0 as soon as t > 0. Using the Madelung ansatz
ψ = n1/2eiφ we get the generalized and rescaled quantum hydrodynamical system [244]
∂tρ+ ∂x · (nv) = 0 (5.2.3)
ρ (∂tv + v∂xv) = −∂xP + ∂xΣ (5.2.4)
with P = α′1/(2m2)n2 and Σ = (~/(2m))2 n·∂2x lnn, where the superfluid velocity is given
by v = ∂xφ. The integrated form of mass conservation corresponding to the continuity
equation (5.2.3) is
∂t
∫ x2
x1
ndx+
∫ x2
x1
∂x · (nv)dx = 0. (5.2.5)
Considering the limits x1 → 0 and x2 → 0 one gets mass conservation at x = 0 in ele-
mentary form, n1v1 = n2v2, where we have introduced the abbreviation v1 = v(x1) etc..
From (5.2.4) and by dropping the anisotropic quantum stress tensor we get elementary
momentum conservation n1v
2
1 + α
′
1n
2
1/4 = n2v
2
2 + α
′
1n
2
2/4 [244]. Now, combining both
elementary equations (momentum and mass conservation) we get
v21 = α
′
1
n2
4n1
(n1 + n2) (5.2.6)
and
v22 = α
′
1
n1
4n2
(n1 + n2). (5.2.7)
We write n1 = kn2 and suppose n2 = const. > 0 and k > 0 as both densities are both
positive quantities. In those terms we find criteria for the local velocity to be greater
than the local speed of sound c2s = α
′
1ni (i ∈ {1, 2}) depending on the size of density
modulation. At x < 0 we have
1 > 4k2/(k + 1), (5.2.8)
where the positive part of the solution is 0 < k < 0.6404. For x > 0 we get
k
4
(k + 1) > 1 (5.2.9)
with k > 1.562 being the reciprocal value of the corresponding estimate for x < 0 and
it reflects the symmetry of our argument. Note however, that k < 1, since the applied
potential at x < 0 will reduce the density n1 there compared to n2, hence excluding the
second criterium.
153
Figure 5.4: (Color online) Control over the polarization of the trains. The colormap
shows the degree of circular polarization ρc. (a) γ2/γ1 = 0.90, (b) γ2/γ1 =
1.11 and (c) γ2/γ1 = 0.99.
154
−60 −40 −20 0 20 40 60−1.5
−1
−0.5
0
0.5
1
1.5
x(µm)
Ve
lo
cit
ie
s 
(µm
/ps
)
t=120 ps
−60 −40 −20 0 20 40 60
x(µm)
t=120 ps
Fig.S 5.5: Snapshots of velocity related quantities taken at ts = 120 ps. The blue enve-
lope displays ±cs(x, ts) (speed of sound) and the red line shows the condensate
velocity v(x, ts). The panel (a) shows results from a pump amplitude 600ΓR
where no soliton develop [see Fig.1(c)]. The panel (b) shows the development
of the train for a lower amplitude of 375ΓR [see Fig. 5.1 (b)] due to the local
supersonic regions at the step.
0.65 0.7 0.75 0.8 0.85
15
20
25
30
35
40
45
µ (meV)
T 
(ps
)
0.65 0.7 0.75 0.8 0.85
0.03
0.04
0.05
0.06
0.07
µ (meV)
ν 
(T
Hz
)
Fig.S 5.6: Train (a) period and (b) frequency versus the local chemical potential obtained
from increasing linearly the pump power. Here the step amplitude is fixed to
U0 = 0.5 meV. The data points are linearly interpolated.
155
0.4 0.6 0.8 1 1.2
10
20
30
40
50
U (meV)
T 
(ps
)
0.4 0.6 0.8 1 1.2
0.05
0.1
0.15
0.2
U (meV)
ν 
(T
Hz
)
Fig.S 5.7: Train (a) period and (b) frequency versus the potential step heights at fixed
pump power 300ΓR corresponding to µ=0.7 meV.
156
6 Coupled counterrotating polariton
condensates in optically defined
annular potentials
F. Pinsker and N. G. Berloff in collaboration with the experimenters A. Dreismann, P.
Cristofolini, R. Balili, G. Christmann, Z. Hatzopoulos, P.G. Savvidis and J. J. Baumberg,
doi: 10.1073/pnas.1401988111 (2014).
Polariton condensates are macroscopic quantum states formed by half-matter half-
light quasiparticles, thus connecting the phenomena of atomic Bose-Einstein condensa-
tion, superfluidity and photon lasing. Here we report the spontaneous formation of such
condensates in programmable potential landscapes generated by two concentric circles of
light. The imposed geometry supports the emergence of annular states that extend up
to, yet are fully coherent and exhibit a spatial structure that remains stable for minutes
at a time. These states exhibit a petal-like intensity distribution arising due to the in-
teraction of two superfluids counter-propagating in the circular waveguide defined by the
optical potential. In stark contrast to annular modes in conventional lasing systems, the
resulting standing wave patterns exhibit only minimal overlap with the pump laser itself.
We theoretically describe the system using a complex Ginzburg-Landau equation, which
indicates why the condensate wants to rotate. Experimentally, we demonstrate the abil-
ity to precisely control the structure of the petal-condensates both by carefully modifying
the excitation geometry as well as perturbing the system on ultrafast timescales to reveal
unexpected superfluid dynamics.
6.1 Main Work
6.1.1 Author contributions
The contribution by each author are as follows: A.D., P.C., G.C., Z.H., P.G.S., and J.J.B.
designed research; A.D., P.C., R.B., and G.C. performed experiments; F.P. and N.G.B.
performed theoretical analysis and numerical simulations; F.P. and N.G.B. contributed
new analytic tools; Z.H. and P.G.S. managed design and growth of samples; A.D., P.C.,
R.B., G.C., F.P., N.G.B., and J.J.B. analyzed data; and A.D., P.C., R.B., G.C., F.P.,
N.G.B., and J.J.B. wrote the paper.
157
6.1.2 Significance
Collections of bosons can condense into superfluids, but only at extremely low temper-
atures and in complicated experimental setups. By creating new types of bosons which
are coupled mixtures of optical and electronic states, condensates can be created on a
semiconductor chip and potentially up to 300 K. One of the most useful implementations
of macroscopic condensates involves forming rings, which exhibit new phenomena since
the quantum wavefunctions must join up in phase. These are used for some of the most
sensitive magnetometer and accelerometer devices known. We show experimentally how
patterns of light shone on semiconductor chips can directly produce ring condensates of
unusual stability, which can be precisely controlled by optical means.
6.1.3 Introduction
Circular loops are a key geometry for superfluid and superconducting devices, because
rotation around a closed ring is coupled to the phase of a quantum wavefunction. So
far however they have not been optical accessible, although this would enable a whole
new class of quantum devices, particular if room temperature condensate operation is
achieved. In lasing systems with an imposed circular symmetry an annulus of lasing spots
can sometimes form along the perimeter of the structure [252, 253, 254, 255, 256, 257].
Such transverse modes are often referred to as petal-states [252] or daisy modes [253]
and are interpreted as annular standing waves [254], whispering gallery modes [255] or
coherent superpositions of Laguerre-Gauss (LG) modes with zero radial index [256, 257].
Their circular symmetry makes them interesting for numerous applications such as free
space communication or fibre coupling [258], while their LG-type structure suggests im-
plementations using the orbital angular momentum of light [259], such as optical trapping
[260] or quantum information processing [261]. Petal-states have been reported for var-
ious conventional lasing systems, including electrically and optically pumped VCSELs
[253, 255], as well as microchip [257] and rod lasers[252]. A fundamentally different type
of lasing system is the polariton laser [262, 263]. Polaritons are bosonic quasi-particles,
resulting from the strong coupling between microcavity photons and semiconductor exci-
tons [262, 263, 264, 16, 265, 266, 267, 268, 270]. Their small effective mass (bestowed by
their photonic component) and strong interactions (arising from their excitonic compo-
nent) favour Bose-stimulated condensation into a single quantum state, called a polariton
condensate [16, 265]. These fully coherent light-matter waves spread over tens of mi-
crons [266] and exhibit a number of phenomena associated with superfluid He and atomic
Bose-Einstein condensation, such as the formation of quantized vortices[267, 268] and
superfluid propagation [249]. Their main decay path is the escape of photons out of the
cavity, resulting in the emission of coherent light. Note that unlike their weakly-coupled
counterparts, polariton lasers require no population inversion since their coherence stems
from the stimulated scattering of quasi-particles into the condensate, not the stimulated
emission of photons into the resonator mode [262].
In this work we study the spontaneous formation and characteristics of petal-shaped
polariton condensates (Fig. 6.1) the strong-coupling analogue to the annular modes
158
observed in conventional lasers. We show how the fragile ring-states observed previ-
ously [270] can be stabilized using carefully-prepared optical confinement, resulting in
fully coherent (SI Appendix, SI Text1) and truly macroscopic quantum objects. We
dynamically manipulate the latter both by changes of the pump-geometry and ultrafast
perturbations, revealing rich many-body physics which is numerically modeled using a
complex Ginzburg-Landau equation. We demonstrate an exceptional degree of control
over the system suggesting the relevance of our findings for future applications such
as all-optical polaritonic circuits [271, 272] and interferometers [273]. We furthermore
emphasize differences to the weakly-coupled case, arising as a consequence of the strong
nonlinearities that govern the behaviour of polaritons[274, 275] and the fundamentally
distinct mechanism responsible for the buildup of coherence.
6.1.4 Petal-condensates
Our experimental setup utilizes a high-resolution spatial light modulator (SLM), which
allows phase-shaping of the pump laser into the desired intensity-patterns on the sample
(Fig.6.1 (f)). Results here were obtained from a high-quality, low-disorder microcavity,
incorporating four sets of three quantum wells. We nonresonantly excite our sample
with a single-mode continuous-wave laser, at energies approximately above the bottom
of the lower polariton branch. The resulting polariton emission is detected with a CCD
for imaging, a monochromator plus CCD for spectral analysis and a streak camera for
monitoring its temporal evolution. A Fourier lens and Mach-Zehnder interferometer are
used to probe k-space distribution and first-order coherence. All measurements were
performed at cryogenic temperatures and on sample locations where the exciton mode
is tuned ∼ 7 meV below that of the cavity mode.
The excitation-pattern used to stably generate petal-type condensates consists of two
concentric circles of laser light (Fig. 6.1 (c), white rings), with a local intensity ratio
for inner to outer circle of 1 : 8. The nonresonant excitation results in the formation
of a cloud of hot excitons at the position of the pump, which subsequently relax in
energy, couple to the cavity mode and accumulate at the bottom of the lower-polariton
branch. Due to their repulsive interactions, polaritons experience a local blue-shift at
positions of high exciton density [266, 276]. For the given pump geometry, this induces
a potential landscape resembling an annular waveguide. Strikingly, the resulting polari-
ton condensate forms in the region between the outer and the inner pump ring, thus
unlike conventional lasing systems [257, 277] and contrary to Manni et. al. [270] ex-
hibiting only minimal overlap with the pump laser itself (Fig.6.1 (c)). This effect is a
direct consequence of the specific properties of polaritons, namely their strong nonlinear
interactions with the hot exciton cloud and their ability to propagate over tens of mi-
crometer distances before they decay [266, 276]. The resulting petal-condensates possess
a well-defined energy and exhibit a characteristic annular intensity distribution (Fig. 6.1
(a)-(d)), which remains stable for > 1.000s timescales. The real space image of a ring
with n intensity lobes translates into a reciprocal space image with an identical num-
ber of lobes, where the emission coming from a specific lobe in real space stems from
polaritons with equal and opposite wavevectors around the annulus (Fig.6.1 (h)-(k)).
159
Figure 6.1: Petal-shaped polariton condensates. (a)-(d) Spatial images and spectra (hor-
izontal cut at y = 0) of annular states with increasing azimuthal index l. The
shaded rings in (c) indicate the position of the pump laser. (e) Double-ring
with radial index p = 1. (f) Schematic of the experimental setup, showing
the phase-shaped pump laser and the resulting polariton emission. (g) phase
of the condensate wave function in (b), extracted following the method dis-
cussed in [270]. As constant reference, one of the lobes was magnified and
superimposed over the whole image. (h)(k) real- and k-space images of the
petal state with l = 10 The k-space image in (k) is obtained by spatially
filtering the polariton emission as shown in (i).
160
This observation yields the picture of two counter-propagating superfluids, analogous to
superconducting loops and qubits [278, 279]. Because both waves are subject to periodic
boundary conditions, their superpositions always possess an even number of lobes with
a well-defined phase and a phase-jump of pi in between lobes (Fig. 6.1 (g)). Math-
ematically, these properties are consonant with a description as superpositions of two
LGp,±l modes with radial index p = 0 and azimuthal indices ±l, where the number of
lobes n is given by n = 21 (SI Appendix, SI Text 2). Adjusting the diameter of the
excitation pattern allows the selection of petal-states with an arbitrary even number of
lobes, up to a (power-limited) maximum of n ' 120. However, increasing the separation
between inner and outer pumprings results in the formation of higher-order ring-states
with radial nodes (p > 0) and identicaln (Fig.6.1 (e)). The orientation of the nodal
lines depends on the relative phase of the two counter-propagating waves; for a uniform
pump, it is pinned by local disorder, as can be seen from the rotation of the condensate
as we move across the sample (SI Video 1), matching similar observations for the case
of VCSELs(2). However, azimuthally modulating the pump intensity allows us to freely
select the orientation of the petals, further demonstrating the high degree of control pos-
sible over the system (SI Text 3). These ring-states are highly resilient to irregularities
of the sample surface, maintaining their shape even in the presence of cracks and other
defects (SI Text 4).
6.1.5 Power dependence
To explore the formation process of the reported ring-shaped condensates, we study
the evolution of the system as the excitation power increases (Fig. 6.2). The lat-
ter causes a growth of the density-induced blue-shift potential at the position of the
pump, accelerating polaritons away from the region of their creation and up to a final
velocity vmax '
√
2V0/m∗, where m∗ is the polariton effective mass. For low pump
powers the excited polaritons gain only little momentum and hence cover only short
distances before they decay; consequently pump and polariton luminescence coincide
(Fig.6.2 (a),(e)). However, as pump intensity and blue-shift increase, so does the mo-
mentum of the polaritons. Those travelling towards the centre of the pump-geometry
are eventually slowed down by the potential associated with the inner ring and accu-
mulate in the region between both rings (Fig.6.2 (b),(e)). This accumulation is further
enhanced by stimulated scattering of polaritons directly from the pump, as can be seen
from the non-linear increase of polariton emission from the region between the laser
rings even below threshold (green line in Fig. 6.2 (d)). At sufficiently high powers, the
corresponding polariton population reaches the critical density for condensation and a
petal-state forms (Fig.2c), with the polariton wavevector kc pointing around the annu-
lus. The formation of the condensate is accompanied by a strong nonlinear increase
of emission intensity, while the number of polaritons outside the pump-ring decreases
(Fig.6.2 (d),(e)). The latter effect suggests that the condensate now efficiently harvests
almost all polaritons created at the pump due to stimulated scattering, as reported for
optically confined condensates previously [280]. Increasing the excitation power beyond
the condensation threshold Pthr quickly leads to the breakdown of the single ring-state,
161
which unlike in the work of Manni et. al. [270] is only observed within a narrow power
range up to 1.3Pthr. We attribute this observation to the repulsive nature of polariton-
polariton interactions, which blue-shift the condensate energy with increasing density
and eventually screen the influence of the inner pump ring, allowing a superposition of
higher-order states to fill up the whole excitation geometry (SI Appendix, SI Text5).
6.2 Mode selection
We next use the flexibility of our setup to systematically vary the pump-geometry and
study the mode-selection mechanism linking a specific excitation pattern to the resulting
petal-state. As the diameter of the pump ring is gradually increased, the number of lobes
n is found to grow as n ∼ r2c , where rc denotes the radius of the condensate ring (Fig.6.3
(a), green points). Note that in all cases the condensate forms in the region between the
two pump rings.
A qualitative explanation for this observation can be found by considering that conden-
sation will initially occur at the point of highest polariton density. Just below threshold,
this corresponds to the doughnut-shaped region between the two pump rings, as can be
seen from the distribution of polariton emission in Fig.6.2 (b). The optimum overlap
between this low-energy polariton reservoir with radius rres and the condensate-pattern
is achieved for LG0,±l-states of identical radius, i.e. rc = rres [256, 257]. Consequently,
these states possess the lowest condensation threshold and first start oscillating as the
pumping increases. Because this argument holds equally for the energetically degener-
ate left- and right-handed LG0,±l modes, both are excited simultaneously, resulting in
the observed standing wave patterns. The radius of maximum intensity for a superpo-
sition of LG0,±l modes lies at rc = w
√
l/2, where w denotes the width of the cavitys
fundamental mode (SI Appendix, SI Text2). Taking into account that the number of
lobesn is given by n = 2l, we arrive at the relation n = [2/w]2r2c , which reproduces the
experimental data for w ' 9.3µ m (green line in Fig.6.3 (a)).
6.2.1 Theoretical description
To theoretically approach the observed phenomena we use a complex Ginzburg-Landau
(cGL)-type equation [91] incorporating energy relaxation [281] and a stationary reservoir:
i∂tψ = (1− iη)H~ ψ (6.2.1)
H
~
=
[
−~∇
2
2m
+ V (~x, t) +
i
2
(δRN(~x, t)− γC)
]
(6.2.2)
Here, ψ(~x, t) represents the order parameter of the condensate, η the rate of energy
relaxation and m the effective mass of polaritons on the lower branch of the disper-
sion curve. δRN/2 and γC are the condensate gain and loss rates, respectively; and
V (~x, t) = gRN + gC |ψ|2 + Vdis denotes the potential landscape experienced by the con-
densate, which arises due to interactions with the reservoir (gRN), polariton-polariton
162
Figure 6.2: Sample luminescence with increasing excitation power. (a,b,c) Spatial image
of the polariton emission with increasing pumping power. Dashed blue line
indicates the position of the laser as shown in (c). (d,e) Power dependence
of the intensity distribution along a central horizontal cut.
163
Figure 6.3: (a) Measured number of lobes n (green points) and spatial separation be-
tween lobes (black points) as a function of condensate radius rc. Fits are
obtained from simulations of the cGL equation for different pump radii (SI
Appendix, SI Text 6) or the analytic expression for the density-maximum
of LG-modes (SI Appendix, SI Eq. S4). (b) Simulated spatial density of a
petal-condensate with n = 20 lobes and (c) corresponding phase. (d) Hori-
zontal cut of (b), indicating the relative position of the condensate and the
pumping profile.
164
interactions (gC |ψ|2) and energy fluctuations due to sample disorder (Vdis). The radially
symmetric pumping profile P (~x) is chosen to reproduce the experimental double-ring
excitation-geometry (SI Appendix, SI Text 6), giving rise to a reservoir density distribu-
tion following ∂tN = −(γR+β|ψ|2)N+P , where γR represents the reservoir decay and β
the condensate-reservoir scattering rate. Because the relaxation of the reservoir is much
faster than the decay of the condensate (γR  γC) [281], the reservoir dynamics can be
approximated by the stationary value N ' P (~x, t)/(γR +β|ψ|2) ' P/γR + (Pβ/γ2R)|ψ|2,
where in the second step |ψ| was assumed to be small. Numerical solutions of equation
(6.2.1) for the parameters given in the Materials and Methods section are presented in
Fig.6.3. The simulated condensate density and corresponding phase (Fig. 6.3 (b),(c))
are obtained for a pumping profile matching that of Fig.6.1 (c). The simulation clearly
reproduces the observed petal-structure with n = 20 lobes. Note that energy relax-
ation and disorder prove critical to stabilize the angular orientation of the calculated
petal-modes (SI Appendix, SI Text 6). The maxima of condensate and reservoir are spa-
tially separated (Fig. 6.3 (d)) in accordance with the experiment (Fig.6.2 (e)), although
their separation is less pronounced in the simulation (Fig.6.2 (e)). We attribute this
deviation to the simplicity of our model which does not incorporate the propagation of
uncondensed polaritons. Gradually increasing the radius of the simulated pumping pro-
file results in the formation of states with a higher (even) number of lobes. The relation
between condensate radius and lobe number agrees precisely with the experimental data
(green and black lines in Fig.6.3 (a), SI Appendix, SIText6). An analytic estimation for
the observed relation is provided in the Supporting Information (SI Appendix, SI Text
7). Note that the formation of lobes corresponds to excitations of the systems ground
state. In the linear picture the latter are associated with the larger in-plane wave vectors
of higher-order -modes, whereas from the viewpoint of a (nonlinear) complex Ginzburg-
Landau equation the phase-jumps observed between adjacent lobes and the vanishing
condensate density can be interpreted as a signature of dark solitons or solitary waves.
However, the standing wave pattern appears in simulations even if the real nonlinearity
is set to zero (gC = 0). The weak nonlinearity slightly modifies the density profile, but
does not change the number or position of the lobes. While this implies that the forma-
tion mechanism of the petal-structure can be understood in terms of linear physics, the
condensate itself and each of its lobes still represent a highly non-linear system due to
polariton-polariton interactions. Arrays of the lobes represent excitations of the conden-
sate ground state and can arise due to the interaction of counter-propagating superfluids
in an effective setting [282, 26]. We assume that the reported ring-condensates occupy
excited states instead of their ground states to maintain energy conservation: Polaritons
generated in the blue-shifted pumping regions which scatter into the condensate lack an
efficient mechanism for energy relaxation [276], as evidenced by the fact that both pos-
sess the same energy (SI Text 8). Since the blue-shift at the position of the condensate
is much smaller than that at the pump, the condensate must form in an excited state,
translating into a transverse wave vector kc > 0 (linear picture) or the formation of an
array of dark solitary waves (non-linear picture). Increasing the radius of the pump
rings results in the formation of states with a higher number of lobes, corresponding to
higher condensate energies (SI Appendix, SI Text 9). Qualitatively, one can associate
165
this with an accelerated rotation of the two counter-propagating condensates. A related
phenomenon was studied in [48], where the authors show that inhomogeneous pumping
of large-area trapped condensates can spontaneously induce condensate rotation and the
formation of vortex lattices. Assuming the annular waveguide forms a harmonic trap
with level spacing ~ω, the stable rotation speed is Ω = nV /r˜2 = 1, where r˜ = r/
√
~/mω.
nV represents the number of vortices in the lattice and was shown to grow quadratically
with the radius of the pump. However we note that this intuitive explanation does
not include the presence of 2 counter-propagating condensates, and that given the one
dimensional geometry here, vortex pairs are not seen [24] but instead a standing wave
with the full condensate density depletions.
6.2.2 Condensate dynamics
To explore the time dynamics of the system, we non-resonantly perturb the cw-pumped
petal condensate with a 150−fs pulsed laser, which is focussed on the side of the ring
(Fig.4a, inset). Fig.6.4 (a) shows a cylinder projection of the polariton emission around
the condensate ring at different times t, as reconstructed from a full set of tomographic
streak camera measurements. Before it is perturbed (t < 0ps), the condensate forms
a stable petal-state with n = 22 lobes. The initial effect of the perturbing pulse P at
t = 0ps is a strong reduction of the overall polariton emission after ∼ 2ps (Fig.4a,b). We
attribute this behaviour to photons that are rapidly generated by the laser pulse and
subsequently propagate ballistically across the cavity, where they extract gain from the
exciton reservoir due to stimulated emission into the wave-guided mode. The additional
exciton population locally introduced by the laser pulse creates a localized blue-shift
potential barrier, analogous to the weak links forming Josephson junctions in supercon-
ducting loops and SQUIDs. The potential barrier breaks up the ring state by pushing
apart its lobes while transiently reducing their number, corresponding to a reduction
of the respective vorticity of each counter-propagating wave. Note that the modified
potential landscape at this point no longer imposes periodic boundary conditions on the
system, thus allowing the formation of states with an odd number of lobes as well. The
additional gain provided by the laser pulse creates an imbalance of the polariton density
around the ring (green and pink lines, Fig.4b). The high density of polaritons formed on
either side of the disturbance propagate along the annulus at a velocity v ' 1.3µm/ps
(dashed line in Fig.6.4 (a)), matching that expected for polariton wave packets oscillat-
ing in a harmonic potential [280]. As a consequence, density oscillations with a period
of T ' 14ps are observed when the lobes are perturbed sideways by the impulse (Fig.6.4
(b) and (c)). As the exciton reservoir generated by the laser pulse decays and further
feeds the condensate, the overall polariton emission increases above its initial value due
to the additional gain provided (Fig.6.4 (a) and (b)). The reduction of the correspond-
ing potential barrier at the same time allows the convergence of the separated lobes and
finally the re-formation of the petal-state after t > 400ps. Simulations confirm that this
behaviour is characteristic of the cGL nonlinear quantum dynamics (SI Video 2).
166
Figure 6.4: Time dynamics of a perturbed petal-condensate. (a) Cylinder projection of
the polariton emission around the condensate annulus at different times .
The perturbing pulse P arrives at t = 0ps. The insets show spatial images
of the condensate ring at t = 10ps and t = 54ps. Dotted lines indicate the
position of space cuts depicted in (b) and time cuts depicted in (c). Dashed
line is a guide to the eye. (Insets) Spatial images at different times.
167
6.2.3 Summary and conclusion
We have studied the properties of exciton-polariton condensates in optically-imposed
annular potentials. The pumping-geometry supports the formation of quantum states
that extend up to 100µm, are highly resilient to defects and sample disorder and remain
stable for minutes at a time. The observed phenomena are reproduced by simulations
of a complex Ginzburg-Landau type equation. The spatial separation of pump reservoir
and condensate minimizes dephasing and other perturbations due to interactions with
hot excitons [283], thus making this excitation geometry highly-advantageous for stable
macroscopic quantum devices operating at ultra-low thresholds. Exploitation of such
states on semiconductor chips is analogous to those of superconducting weak-link devices.
Since the pure LG0,±l modes carry a net orbital angular momentum associated with a
helically propagating phase and exhibit a vortex in their centre, lifting the degeneracy
between the counter-propagating modes (e.g. with magnetic fields) can result in a pure
rotating condensate with giant stable vortex core. The ability to sculpt the polariton
potentials into arbitrary shapes (SI Appendix, SI Text 10) opens up new explorations
of condensate superfluid flow in a wide variety of topologies.
6.2.4 Materials and Methods
The sample consists of a 5λ/2 AlGaAs/AlAs microcavity, with a quality factor exceeding
Q > 16, 000, corresponding to cavity-photon lifetimes τ > 7ps. Four sets of three
quantum wells are placed at the antinodes of the cavity optical field, with an exciton-
photon Rabi splitting of 9meV . All reported experiments were performed at sample
positions where the cavity mode is detuned 7meV below the exciton energy. The system
is pumped with a λ = 755− nm single-mode continuous wave Ti:Sapphire laser, which
is projected through a 4x telescope and a 50x high-NA microscope objective acting as
a Fourier lens. The resulting polariton emission of ∼ 800nm is collected with the same
microscope objective and separated from the laser radiation by means of a tuneable
Bragg filter; it is detected by a Si CCD for imaging, a .055m monochromator with a
nitrogen-cooled CCD for spectral analysis and a streak camera (time resolution 2.5ps) for
monitoring the temporal evolution. A Fourier lens and a Mach-Zehnder interferometer
were used to probe its k-space distribution and first-order coherence. The parameters
used in the simulation were chosen in agreement with Refs. [270, 281, 284], where energy
relaxation rate η = 0.02, reservoir decay rate γR = 10ps
−1, condensate decay rate
γC = 0.556ps
−1, reservoir interaction constant gR = 0.072µm2 · ps−1, self-interaction
constant gc = 0.002µm
2 · ps−1, factor of condensate gain rate δC = 0.06µm2 · ps−1,
condensate-reservoir scattering rate β = 0.05µm2 · ps−1. The effective mass of the lower
polariton branch was measured as m = 4.7 · 10−35kg.
168
6.3 Supporting information
6.3.1 Macroscopic coherence
The petal-type condensates discussed in this work extend over tens of microns while be-
ing fully coherent, thus beautifully depicting truly macroscopic quantum-states. Fig.6.5
illustrates this by probing the coherence of a ring with n = 80 lobes and a diameter
of approximately 90µm. The condensate emission is analysed with a Mach-Zehnder in-
terferometer, resulting in two spatially separated images (Fig.6.5 (a)). The coherence
between opposite sides of the ring is probed by overlapping the corresponding parts of
the emission, clearly resulting in the appearance of interference fringes (Fig.6.5 (b)).
6.3.2 Laguerre-Gauss modes
The electric field of a Laguerre-Gauss mode with radial index p and azimuthal index l
is given by (6.5):
LGp,l =
√
2p!
pi(p+ |l|)! ·
1
w(z)
(
r
√
2
w(z)
)|l|
L|l|p
(
2r2
w2(z)
)
· exp
(
r2
(
− 1
w2(z)
− ik 1
2R(z)
)
+ i(2p+ |l|+ 1)ζ(z)− ilφ
)
(6.3.1)
with
ζ(z) = arctan
(
z
zR
)
(6.3.2)
and where r and φ are the radial and azimuthal coordinates respectively. z denotes
the propagation distance, with w(z), R(z) and zR as the width, the radius of curvature
and the Rayleigh range of a Gaussian beam defined in the usual way. L
|l|
p (z) repre-
sents a generalized Laguerre-polynomial. The observed petal-states with a single ring
closely resemble coherent superpositions of two LG modes with zero radial and opposite
azimuthal indices, which are given by:
LG0,±l = LG0,+l + LG0,−l (6.3.3)
The angular dependence of this expression is carried by the last factor, which can be
written as cos(lφ). Collapsing all other factors for the explicitly given function into the
function A(r, z, l), Eq.(4.2.9) takes the form:
LG0,±l = A(r, z, l) cos(lφ), (6.3.4)
giving the characteristic relation between number of lobes and azimuthal index: n = 2l.
Finally, the radius of maximum intensity of the LG0,±l mode is found for ∂r|A|2 = 0,
yielding:
rmax =
√
l/2w(z). (6.3.5)
Three examples for superpositions of LGp,l modes are shown in Fig.6.6.
169
Figure 6.5: Coherence of petal-states. (a)Superposition of two spatial images of one
petal-state with n = 80 lobes, as recorded with a Mach-Zehnder interferom-
eter. The two images are offset to probe the coherence between the opposite
sides of the ring. (b)Magnified view of the area marked by the dashed square
in (a), showing the interference between the two images .
170
Figure 6.6: Superpositions of Laguerre-Gauss modes. (a,b) Modes with zero radial index
l = 5 and azimuthal indices l = 20 and , respectively. (c)Double-ring with
p = 1 and l = 20 .
171
6.3.3 Optical pinning
Spatially modulating the pump laser allows us to overcome the influence of the local
disorder-potential and determine the orientation of a petal-state. Fig. 6.7 (a) shows an
azimuthally modulated laser circle with 8 intensity maxima. This pump configuration
results in the formation of a petal-state with n = 10 lobes (Fig. 6.7 (b)), the orientation
of which can be changed by rotating the pump (Fig. 6.7 (c) and (d)).
6.3.4 Stability with respect to defects and disorder
The observed petal-condensates are highly resilient to defects on the sample surface, as
demonstrated in Fig. 6.8. The location of a crack is revealed by illuminating it with a
single excitation spot (Fig. 6.8 (a)). In this case some of the resulting polariton emission
is scattered along the edge of the crack, as indicated by the dashed white line. In Fig.
6.8 (b) an annular pump is used to generate a petal-state with n = 18 lobes at the same
position, which remains stable despite the fact that the crack intersects it at two points.
Fig. 6.8.(c),(d),(e) illustrates the stability of the ring-condensate with respect to
movement across the sample, i.e. sample disorder. Despite the fact that the pump
is shifted by 80µm between Fig. 6.8 (c) and Fig. 6.8 (e), the observed petal-state
remains unaltered, except for a slight rotation of the orientation of the pattern. Fig. 6.9
demonstrates the influence of a point-defect on a double-ring state with n = 32 lobes
(azimuthal index l = 16) and one radial node (p = 1). The undisturbed state is shown
in Fig. 6.17 (a). As the pump (Fig. 6.9 (i)) is moved across the sample surface, a
point defect perforates the outer ring from the top, suppressing one of its lobes (Fig.
6.9 (b)). Note that at this point the outer ring possesses an odd number of lobes,
suggesting that here the system is no longer object to periodic boundary conditions. As
the pump is moved further and the defect blocks the inner ring as well (Fig. 6.9 (c)), the
petal-state collapses and a complicated polariton distribution fills out the whole ring,
with maximum intensity at the centre of the geometry. Fig. 6.9 (h) shows that a local
disturbance of the condensate can sometimes lead to rearrangement of the full pattern,
in this case resulting in a somewhat rectangular wavefunction.
6.3.5 Power dependence
This section provides additional details for the discussion of the condensate power de-
pendence of the main text. Fig. 6.10 shows spectra corresponding to the spatial images
presented in Fig. 6.2 (a)-(c) (main text), measured along a central horizontal cut. The
spectra demonstrate the increasing blue-shift of the lowest energy polariton states with
increasing pumping power as well as their spatial shift towards the position of the petal-
structure. The weaker intensity luminescence observed at higher energies stems from
the decay of polaritons that have not yet relaxed to the lowest energy state.
The ring-states reported in this work are only stable within a narrow power range of
approximately 1− 1.3Pthr. Fig. 6.11 (a) shows a petal state with 34 lobes, as observed
directly at condensation threshold (Pthr = 47.0mW ). The bottom panel shows the
172
Figure 6.7: Determining the orientation of a petal-state by modulating the pump.
(a),(c)Spatial images of the pump laser, azimuthally modulated with 8 in-
tensity maxima. The geometry in (c) is rotated by 28 with respect to(a).
(b),(c)Corresponding n = 10 petal states, with lobe-orientations determined
by the orientation of the pump. .
173
Figure 6.8: Stability of petal-states with respect to defects and disorder. (a) Single
excitation spot revealing a crack on the sample. Dashed white line illustrates
the location of the crack. (b) Petal-state at the same position. (c-e) Petal-
state as the pump is moved across the sample. Point-defect at the bottom
provides a reference for the movement. .
174
Figure 6.9: Double-ring state with n = 32 lobes and one radial node (a) moving across a
point-defect. Depending on the relative position of the defect, the number of
lobes in the outer or inner ring is reduced (b,d,f,g), the ring-state is destroyed
(c) or changes its shape(h). (i) shows the position of the pump laser.
175
Figure 6.10: Spectra obtained along a central horizontal cut of the near-field images
shown in Fig. 6.2 a-c (main text). Colour scale normalized on each.
176
corresponding spectrum, exhibiting a well-defined single energy. As the excitation power
is increased above threshold P = 60mW , multiple petal-states of different energies and
lobe number form in the optical potential (Fig. 6.11 (b)). This effect is accompanied
by an overall increase of the polariton emission. For even higher excitation powers
(P = 70.7mW ), the ring-regime eventually collapses and superpositions of various states
fill up the whole pump geometry, with a maximum polariton density at the centre of the
ring (Fig. 6.11 (c)). The observed power dependence is well reproduced in simulations
of the complex Ginzburg-Landau (cGL) equation for conditions similar to those of the
experiment (Fig. 6.12).
6.3.6 Simulations of the complex Ginzburg-Landau equation
Fig. 6.13 (a) shows the spatial distribution of the blue-shift potential associated with the
repulsive interaction between polaritons and the exciton reservoir, as used in our simu-
lations of the cGL equation (c.f. paragraph 3 of the main text). Fig. 6.13 (b) depicts
a number of randomly distributed potential spikes accounting for sample disorder. The
latter are necessary to radially stabilize the simulated petal-states, i.e. prevent fluctu-
ations of the lobe orientation which would result in a washed-out intensity distribution
in a time-averaged measurement. To additionally stabilize the simulation, we introduce
the energy relaxation constant η (c.f.Eq.1, main text).
Fig. 6.14 reports the simulated number of lobes n and separation between adjacent
lobes ∆ obtained for a stepwise increase of the radius of the pumping profile. Fitting
of both datasets results in the relations n ' r2C and ∆ ' 1/rc, in agreement with the
experimental data presented in Fig. 6.3 (a) (main text).
6.3.7 Analytic estimate of the relation between number of lobes
and condensate radius
After having demonstrated good agreement between numerical simulations and experi-
mental data (Fig. 6.3, main text), here we provide an analytic estimate for the relation
between number of lobes and condensate radius. In the steady state, the order param-
eter of the condensate satisfies ∂tψ(t) = −iµψ, where µ denotes the chemical potential
and we set ~ = 1 so that:
µ1ψ = (1− iη)
(−∇2 + a0(r) + a1(r)|ψ|2)ψ (6.3.6)
with a0(r) = 2m(gRP (r)/γR + iδcP (r)/(2γR) − iγc/2), a1 = 2m(gc − gRβP (r)/γ2R −
iδcβP (r)/(2γ
2
R)) and µ1 = 2mµ. Including the energy relaxation constant η is essential
for the numerical stability of the lobes, but can be neglected for an analytic approxima-
tion. In Ref. [253] the appearance of lobes as solutions of a linear problem at the low-
est linear threshold has been demonstrated by solving the linear eigenvalue problem for
asymmetric pumping profiles. Identifying the lobes in our set-up with the linear problem
as well we are looking for the standing wave solutions of the form ψ = A exp(iS) cos(nθ).
The term Re[a1(r)]A
2 is small for the experimental parameters and does not affect the
177
Figure 6.11: Effects of increasing the pump above condensation threshold. (a)Petal-
state with n = 34 lobes, as observed at condensation threshold. The white
line depicts the intensity distribution along a horizontal cut through the
centre of the ring. The bottom panel shows the spectrum for the same cut.
(b) As the power increases, multiple petal-states start to overlap, (c) until
the ring-condensate eventually breaks-down and superpositions of different
states fill the whole pump geometry, with the maximum intensity in the
centre.
178
Figure 6.12: Simulated power dependence for a petal-condensate with n = 22 lobes. The
behaviour of the system matches that shown in Fig.6.11 as the excitation
power increases.
179
Figure 6.13: Conditions for simulations of the cGL equation. (a) Blueshift-potential
corresponding to the pumping profile and (b) random disorder potential.
180
Figure 6.14: Simulation of the condensate distribution for varying pump radii. Shown
are the number of lobes (green points) and separation between adjacent
lobes ∆ (black points) as the condensate radius increases. Solid lines are
fits. The special scaling was chosen so that the simulated relation matches
the experimental data, with n = 0.046[µm−2]r2c and ∆ = 140.3[µm
2]/rc.
181
formation of the lobes, so it will be neglected in what follows. The term Im[a1(r)]A
2 does
not affect the formation and number of lobes as well, but leads to density saturation.
The real and imaginary parts of Eq. 6.3.6 then become:
µ1 =
n2
r2
− A
′′ + A′/r
A
+Re[a0(r)] + S
′2 (6.3.7)
0 = 2
A′
A
S ′ + S ′′ +
S ′
r
− Im[a0(r)]. (6.3.8)
At large distances Im[a0(r)] converges to the constant value −mγc, while Re[a0(r)]
decays exponentially, implying S ′ → const. ≡ vmax. From Eqs. (6.3.7), (6.3.8) we have
to the leading order (logA)′ = −mγc/(2vmax)− 1/(2r) or A '
√
r exp(−mγcr/(2vmax)).
Inserting this in Eqs. 6.3.7 gives:
µ1 = v
2
max −
(
mγc
2vmax
)2
. (6.3.9)
Next, we take the linearized Eq.(6.3.6) and its complex conjugate, subtract and integrate
within a circle of radius r1  rc, where rc is the radius of the condensate, to get:
A(r1)
2S ′
∣∣∣∣
r=r1
'
∫
r≤r1
Im(a0)A
2rdr = A(r)2
∫
r≤r1
Im(a0)rdr. (6.3.10)
Here r denotes some point inside the circle r ≤ r1. Taking a sufficiently large radius
r1 we get vmax ' rc, so the velocity of the outflow is proportional to the pumping
radius. (As an alternative way to arrive at this conclusion, one might also observe that
due to mass continuity, the flux of the outflow at a fixed radius is proportional to the
number of particles created. The number of particles is in turn proportional to the area
of the condensate, i.e. the segment between the outer and the inner laser rings, which
is proportional to the radius rc). Together with Eqs. 6.3.7 and Eq. 6.3.9 this finally
gives the number of lobes as n2 ' r2cµ1 ' r4c , in agreement with the experimental and
numerical findings. Eq. 6.3.10 also gives the linear dependence of the pumping power
on the velocity outflow at a large radius. For a fixed pumping radius, Eq. 6.3.9 gives
the criterion for the condensation: the condensate emerges when µ1 becomes positive.
6.3.8 Polariton energies at different positions
Fig. 6.15 demonstrates that the energy of the condensed polaritons remains constant as
they move across the sample. Fig. 6.15 (a),(b) show a petal-condensate with n = 18 lobes
and the corresponding dispersion curve (spectrometer slit horizontally aligned at y = 0).
In images Fig. 6.15 (b) - (d), an iris is employed to spatially select the polariton emission
from different positions on top of and outside of the condensate pattern. Fig. 6.15 (c)
corresponds approximately to the position of the pump ring where polaritons are created
with k = 0. Fig. 6.15 (f) to (h) depict the corresponding dispersion curves, showing
that the energy of the condensed polaritons (intensity maximum) remains approximately
182
constant at different spatial positions. This observation implies that polaritons do not
relax in energy as they move from the pump ring (Fig.6.15 (c) ) to the petal-condensate
(Fig. 6.15 (b)). Similar results have been obtained for freely-flowing condensates in
previous works [253, 254].
6.3.9 Condensate energy and condensation threshold vs radius
Fig. 6.16 (a) depicts the measured condensate blueshift energy for a number of petal-
states with different radii, demonstrating that higher order states are energetically ex-
cited. The solid line is derived from the fitted relation between number of lobes and
condensate radius n = 0.046µm−2r2c , (see main text), and the measured polariton effec-
tive mass m ' 4.5 · 10−35kg. Due to the periodic boundary conditions of the annular
states condensate radius and wavevector are related as 2pirc = nλc/2 = npi/kc, where λc
and kc denote the condensate wavelength and wavevector, respectively. Together with
the polariton dispersion E(kc) = E0 + ~2k2c/(2m) (quadratic approximation), we arrive
at a relation between condensate energy and radius:
E(rc) = E0 +
~2n2
8mr2c
= E0 +
~20.0462r2c
8m
(6.3.11)
6.3.10 Flexibility of excitation method and variations of the
double-ring geometry
Although we focus on annular pump geometries and the resulting petal-states in this
work, our experimental setup in practise allows the generation of polariton condensates
with arbitrary pump distributions and corresponding blue-shift potentials, as demon-
strated in Fig. 6.17.
An alternative approach to generate petal-shaped condensates is explored in Fig.
6.18. Instead of a pump distribution consisting of two concentric laser rings (with the
maximum of the laser intensity at the outer ring), a single spot above condensation
threshold is placed in the centre of a lower power circular pump. If only a single laser
spot is present (Fig. 6.18 (a)) the local blue-shift at the pump leads to the formation
of a radially expanding condensate (Fig. 6.18 (b)) [254, 255]. An additional ring of
laser light around the excitation spot is employed in Fig. 6.18 (c). The ring-shaped
pump is driven below condensation threshold, thus serving as a potential barrier for the
polaritons expanding from the central spot. The latter are slowed down and accumulate
in the doughnut-shaped region between the inner spot and the ring. However, no distinct
intensity distributions like in the case of petal-type states are observed.
Finally, in Fig. 6.19 we utilize the flexibility of our excitation method to study how
the shape of the annular condensates changes depending on the relative intensity of the
inner and the outer pump rings. Fig. 6.19 (b) shows a petal-state with lobes with an
intensity ratio between inner and outer pump of 1 : 8, which is the ratio used for the
examples discussed in the main text. In this configuration, the inner ring creates a weak
183
Figure 6.15: Polariton energy at varying spatial positions. Top row shows spatial im-
ages of a condensate ring with lobes. (a) Full spatial image, (b-d) polariton
emission from different spatial positions as selected with an iris. Dashed
white line indicates the position of the ring-state. The bottom row de-
picts the corresponding dispersion curves, indicating the polariton energy
(logarithmic colour scale).
184
Figure 6.16: (a) Measured condensate blueshift ∆E as a function of the condensate ra-
dius. The solid line depicts their relation as expected from Eq.6.13, namely:
∆E(rc) = E(rc) − E0 ' 4 · 10−7meV µm−2r2c . (b) Condensation thresh-
old as a function of condensate radius. Solid line corresponds to the fit
Pthr ' 1.350mW + 8 · 10−3mWµm−2r2c .
185
Figure 6.17: Different pump geometries and resulting polariton condensates. (a) Spatial
image of a ‘happy condensate’, generated with a pump shaped like a smiley
face. The laser luminescence appears orange, the polariton luminescence
violet. (b) Resonator-style pump, resulting in a localized standing wave.
186
Figure 6.18: Condensate generated by a single excitation spot inside a circular optical
potential. (a)Spatial image of the excitation spot and (b) resulting emission,
corresponding to a radially expanding condensate. (c)Same excitation spot
now surrounded by a circle of laser light below threshold. (d)The formerly
free-flowing condensate created at the spot is confined by the potential and
accumulates in the ring.
187
annular waveguide potential resulting in a clear petal state. If the inner pump ring is
omitted (Fig. 6.19 (a)) polaritons created at the outer ring are no longer prevented from
travelling towards the centre of the excitation-geometry, the whole ring-trap is filled
with polaritons and a superposition of various states is observed. If, on the other hand,
the intensity of the inner ring is increased, the additional gain and corresponding blue-
shift in the central region distort the previously clear petal-state and eventually result
in unwanted filling of the area inside the inner pump with condensate, thus drastically
changing the mode shape (Fig. 6.19 (c)). Thus the inner pump ring blue-shift has to be
high enough to confine the condensate in the multi-lobe configuration but weak enough
to contribute only marginally to the overall gain of the condensate.
6.3.11 Video 1: Sample disorder and lobe orientation
We have developed simulations showing the behaviour of a petal state with n = 8
lobes as the pump is moved across the sample. The video is available in the published
version. The movement of the pump can be tracked from the relative motion of the two
point-defects at the left side of the image. The orientation of the lobe-pattern remains
stable for a stationary pump (first 5s of the video). Moving the pump, however, triggers
fluctuations of the lobe orientation, demonstrating that in the case of a uniform pump
the latter is set by local variations of the potential landscape, i.e. sample disorder.
6.3.12 Video 2: Simulation of the dynamics of a locally disturbed
condensate
The video shows a simulation of the time evolution of a petal-condensate with n = 20
lobes, as it is disturbed by an additional laser pulse on top of the ring. The parameters
for the simulation are the same as the ones listed in the Materials and Methods section
of the main text. The disturbing laser pulse is modeled as additional Gaussian peak in
the pump distribution of the form
Ppulse = (4± τ/τ0) · exp
(−0.8 · (x− x0)2 − 0.8 · (y)2) , (6.3.12)
where the maximum amplitude of the disturbing pulse is 100 times that of the max-
imum of the annular pump. Here τ denotes the time steps of the simulation with
τ = {0, . . . , 2225000} and τ0 = 110000. At τ = τ0 the sign of the second term is changed
from plus to minus, simulating first growth and then decay of the disturbing pulse. The
position of the pulse x0 is at the radius of the maximum of the lobes. The simulated
condensate behaviour is similar to that observed in the experiment. The initial petal-
state observed at τ = 0 is destroyed upon the arrival of the disturbance. The additional
gain provided by the perturbing pulse creates out-of-equilibrium polariton populations
on its sides, which subsequently counter-propagate along the annulus, resulting in the
observed oscillatory behaviour. As the additional blueshift potential of the pulse decays,
the initial petal-state recovers.
188
Figure 6.19: Influence of the inner pump ring. All images were observed for the same
overall pumping power of , but with varying ratios between the intensities
of the inner and the outer pump-rings. The ratios between the inner and
the outer ring for the different measurements were (a) 0:8, (b) 1:8 and (c)
5:8.
189
Bibliography
[1] L. Wittgenstein, ”Logisch-philosophische Abhandlung - Tractatus logico-
philosophicus”, Suhrkamp, Frankfurt am Main, Germany (1998).
[2] J. A. Evans, ”Future Science”, Science 342, 44 (2013); DOI: 10.1126/sci-
ence.1245218
[3] T. Tao, ”Universelle Gesetze”, Spekt. d. Wiss., 60 (2014).
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell,
”Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor”, Science
269, 198 (1995).
[5] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M.
Kurn, and W. Ketterle, ”Bose-Einstein Condensation in a Gas of Sodium Atoms”,
Phys. Rev. Lett. 75, 3969 (1995).
[6] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, ”Evidence of Bose-
Einstein Condensation in an Atomic Gas with Attractive Interactions”, Phys. Rev.
Lett. 75, 1687 (1995).
[7] L. P. Pitaevskii, ”Vortex Lines in an Imperfect Bose Gas”, Zh. Eksp. Teor. Fiz. 40,
646 (1961); Sov. Phys. JETP 13, 451 (1961).
[8] E. P. Gross, ”Structure of a quantized vortex in boson systems”, Nuovo Cimento
20 451 (1961); J. Math. Phys. 4 195 (1963).
[9] N. N. Greenwood and A. Earnshaw, ”Chemie der Elemente”, Wiley-VCH, ISBN
3-527-26169-9, Weinheim, Germany (1988).
[10] A. J. Leggett, ”Bose-Einstein condensation in the alkali gases: Some fundamental
concepts”, Rev. Mod. Phys. 73, 307 (2001).
[11] L.P. Pitaevskii and S. Stringari, ”Bose-Einstein Condensation”, Clarendon, Oxford,
United Kingdom (2003).
[12] A. Einstein, ”Quantentheorie des einatomigen idealen Gases II”, Sitzungsber.
Preuss. Akad. Wiss., 3 (1925).
[13] L. Onsager and O. Penrose, ”Bose-Einstein condensation and liquid helium”, Phys.
Rev. 104, 576 (1956).
190
[14] J. Klaers, J. Schmitt, F. Vewinger and M. Weitz, ”BoseEinstein condensation of
photons in an optical microcavity”, Nature 468, 545 (2010).
[15] I. Carusotto and C. Ciuti, ”Quantum fluids of light”, Rev. Mod. Phys. 85, 299
(2013).
[16] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keel-
ing, F. M. Marchetti, M. H. Szymanska, R. Ande´, J. L. Staehli, V. Savona, P. B.
Littlewood, B. Deveaud and Le Si Dang, ”Bose-Einstein condensation of exciton
polaritons”, Nature 443, 409 (2006).
[17] C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi and J. W. Fleischer, ”Observation of
the kinetic condensation of classical waves”, Nature Physics 8, 437 (2012).
[18] F. London, ”The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degener-
acy”, Nature 141, 643 (1938).
[19] C. Raman, R. Onofrio, J. M. Vogels, J. R. Abo-Shaeer and W. Ket-
terle,”Dissipationless Flow and Superfluidity in Gaseous Bose-Einstein Conden-
sates”, Journal of Low Temperature Physics, 122, 99-116 (2001).
[20] K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. Andre´,
Le Si Dang and B. Deveaud-Ple´dran, ”Quantized vortices in an exciton-polariton
condensate”, Nature Physics 4, 706-710 (2008).
[21] M. Correggi, F. Pinsker, N. Rougerie and J. Yngvason, ”Rotating superfluids in
anharmonic traps: From vortex lattices to giant vortices”, Phys. Rev. A 84, 053614
(2011).
[22] M. Correggi, F. Pinsker, N. Rougerie and J. Yngvason, ”Critical rotational speeds
in the Gross-Pitaevskii theory on a disc with Dirichlet boundary conditions”, J.
Stat. Phys. 143, 261- 305 (2011).
[23] M. Correggi, F. Pinsker, N. Rougerie and J. Yngvason, ”Critical rotational speeds
for superfluids in homogeneous traps”, J. Math. Phys. 53, 095203 (2012).
[24] F. Pinsker, N. G. Berloff and V. M. Pe´rez-Garc´ıa,”Nonlinear quantum piston for the
controlled generation of vortex rings and soliton trains”, Phys. Rev. A 87, 053624
(2013).
[25] F. Pinsker, ”Computing with soliton trains in Bose-Einstein condensates”,
arXiv:1305.4088 (2013).
[26] F. Pinsker and H. Flayac, ”On-Demand Dark Soliton Train Manipulation in a Spinor
Polariton Condensate”, Phys. Rev. Lett. 112, 140405 (2014).
[27] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and
W. Ketterle, ”Observation of Feshbach resonances in a BoseEinstein condensate”,
Nature 392, 151-154 (1998).
191
[28] H. Deng, H. Haug and Y. Yamamoto, ”Exciton-polariton Bose-Einstein condensa-
tion”, Rev. Mod. Phys. 82, 1489 (2010).
[29] J. J. Hopfield, ”Theory of the Contribution of Excitons to the Complex Dielectric
Constant of Crystals”, Phys. Rev., 112, 15551567 (1958).
[30] C. Weisbuch, M. Nishioka, A. Ishikawa and Y. Arakawa, ”Observation of the
coupled exciton-photon mode splitting in a semiconductor quantum microcavity”,
Phys. Rev. Lett. 69, 33143317 (1992).
[31] J. D. Plumhof, T. Sto¨ferle, L. Mai, U. Scherf and R. F. Mahrt, ”Room-temperature
BoseEinstein condensation of cavity exciton-polaritons in a polymer”, Nature Ma-
terials 13, 247-252 (2014).
[32] A. Dreismann, P. Cristofolini, R. Balili, G. Christmann, F. Pinsker, N. G. Berloff,
Z. Hatzopoulosd, P. G. Savvidis and J. J. Baumberg,”Coupled counterrotating po-
lariton condensates in optically defined annular potentials”, Proceedings of the Na-
tional Academy of Sciences, 201401988; doi: 10.1073/pnas.1401988111 (2014).
[33] B. Nelsen, G. Liu, M. Steger, D. W. Snoke, R. Balili, K. West, L. Pfeiffer, ”Coherent
Flow and Trapping of Polariton Condensates with Long Lifetime”, arXiv:1209.4573
[34] T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos, G. Stavrinidis, G. Deligeorgis,
Z. Hatzopoulos, and P. G. Savvidis, ”Polariton condensate transistor switch”, Phys.
Rev. B 85, 235102 (2012).
[35] T. C. H. Liew, M. M. Glazov, K. V. Kavokin, I. A. Shelykh, M. A. Kaliteevski,
and A. V. Kavokin,”Proposal for a Bosonic Cascade Laser”, Phys. Rev. Lett. 110,
047402 (2011).
[36] T. C. H. Liew, A.V. Kavokin, T. Ostatnicky, M. Kaliteevski, I.A. Shelykh and R.A.
Abram,”Optical Circuits Based on Polariton Neurons in Semiconductor Microcav-
ities”, Phys. Rev. Lett. 101, 016402 (2008).
[37] I. A. Shelykh, G. Pavlovic, D. D. Solnyshkov, and G. Malpuech, ”Proposal for
a Mesoscopic Optical Berry-Phase Interferometer”, Phys. Rev. Lett. 102, 046407
(2009).
[38] I. A. Shelykh, R. Johne, D. D. Solnyshkov, and G. Malpuech, ”Optically and elec-
trically controlled polariton spin transistor”, Phys. Rev. B 82, 153303 (2010).
[39] T. Espinosa-Ortega and T. C. H. Liew,”Complete architecture of integrated pho-
tonic circuits based on and and not logic gates of exciton polaritons in semiconductor
microcavities”, Phys. Rev. B 87, 195305 (2013).
[40] H. Flayac, and I. G. Savenko, ”An exciton-polariton mediated all-optical router”,
Appl. Phys. Lett. 103, 201105 (2013).
192
[41] T. C. H. Liew, I.A. Shelykh and G. Malpuech, ”Polaritonic devices”, Physica E 43
(9), 1543-1568 (2011).
[42] D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdre´, E. Giacobino, R. Cingolani,
A. Bramati, G. Gigli and D. Sanvitto, ”All-optical polariton transistor”, Nature
Communications 4 1778 (2013).
[43] C. Anto´n, T.C.H. Liew, J. Cuadra, M.D. Mart´ın, P.S. Eldridge, Z. Hatzopoulos, G.
Stavrinidis, P.G. Savvidis, L. Vina, ”Quantum reflections and the shunting of polari-
ton condensate wave trains: implementation of a logic AND gate”, arXiv:1305.5678
(2013).
[44] C. Sturm, D. Tanese, H.S. Nguyen, H. Flayac, E. Gallopin, A. Lemaitre, I. Sagnes,
D. Solnyshkov, A. Amo, G. Malpuech, J. Bloch, ”Giant phase modulation in a
Mach-Zehnder exciton-polariton interferometer”, arXiv:1303.1649 (2013), to appear
in Nature Communications.
[45] G. Malpuech, A. Di Carlo, A. Kavokin, J. J. Baumberg, M. Zamfirescu and P.
Lugli, ”Room-temperature polariton lasers based on GaN microcavities”, Appl.
Phys. Lett. 81, 412 (2002).
[46] A. Kavokin,”Exciton-polaritons in microcavities: Recent discoveries and perspec-
tives”, physica status solidi 247, 1898-1906, (2010).
[47] N.G. Berloff and J. Keeling, Chapter to appear in Springer and Verlag book on
”Quantum Fluids: hot-topics and new trends” eds. A. Bramati and M. Modugno
[48] J. Keeling and N. G. Berloff, ”Spontaneous rotating vortex lattices in a pumped
decaying condensate”, Phys. Rev. Lett. 100, 250401 (2008).
[49] F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre´, and B. Deveaud-Ple´dran,
”Spontaneous pattern formation in a polariton condensate”, Phys. Rev. Lett. 107,
106401 (2011).
[50] M.O. Borgh, G. Franchetti, J. Keeling and N. G. Berloff ”Robustness and observ-
ability of rotating vortex lattices in an exciton-polariton condensate”, Phys. Rev.
B 86, 035307 (2012).
[51] S. Bose, ”Plancks Gesetz und Lichtquantenhypothese”, Zeitschrift fr Physik 26, 178
(1924).
[52] F. Pinsker, ”A Fast Rotating Bose-Einstein Condensate on a Disc”, Diploma thesis,
University of Vienna (2010).
[53] C.J. Pethick, H. Smith, ”Bose-Einstein Condensation in Dilute Gases”, Cambridge
University Press, Cambridge, UK (2002).
193
[54] W. Ketterle, ”Nobel lecture: When atoms behave as waves: Bose-Einstein conden-
sation and the atom laser”, Rev. Mod. Phys., 74, (2002).
[55] B. Schlein, ”Contribution to the Proceedings of the Conference ’Days on PDEs’”,
Biarritz (2011).
[56] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker
Denschlag and R. Grimm, ”Bose-Einstein condensation of molecules”, Science 302,
2101 (2003); DOI: 10.1126/science.1093280
[57] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, ”Theory of Bose-Einstein
condensation in trapped gases”, Rev. Mod. Phys. 71, 463, (1999).
[58] E.H. Lieb and R. Seiringer, ”Proof of Bose-Einstein condensation for dilute trapped
gases”, Phys. Rev. Lett. 88, 170409 (2002).
[59] C.N. Yang, ”Concept of Off-Diagonal Long-Range Order and the Quantum Phases
of Liquid He and of Superconductors ”, Rev. Mod. Phys. 34, 694 (1962).
[60] V. I. Yukalov, ”Basics of Bose-Einstein condensation”, Physics of Particles and
Nuclei 42, 460-513 (2011).
[61] L.D. Landau and E.M. Lifshitz, ”Quantum mechanics, Non-relativistic Theory”,
Pergamon Press (1965).
[62] E.H. Lieb, J. P. Solovej, R. Seiringer, J. Yngvason, ”The Mathematics of the Bose
Gas and its Condensation.” Oberwolfach Seminars, 34, Birkha¨user, Basel, 184pp.
(2001).
[63] J. Dalibard, ”Bose - Einstein condensation in gases”, Proceedings of the Interna-
tional School of Physics Enrico Fermi, Course CXL, Varenna (1998).
[64] M.L. Harris, ”Realisation of a Cold Mixture of Rubidium and Caesium”, PhD thesis,
Durham University (2008).
[65] M. Theis, ”Optical Feshbach Resonances in a Bose-Einstein Condensate”, PhD
thesis, Innsbruck University (2005).
[66] W. Bao, ”Numerical methods for the nonlinear Schro¨dinger equation with nonzero
far-field conditions” International Press 11, (3), 001022, (2004).
[67] B. Svistunov, E. Babaev, and N. Prokofev, ”Superfluid States of Matter”, Taylor
& Francis, (2014).
[68] J. Bladel, ”On Helmholtz’s Theorem in Finite Regions”, Midwestern Universities
Research Association, (1958).
[69] A. L. Fetter, ”Rotating trapped Bose-Einstein condensates”, Rev. Mod. Phy., 81,
1-45 (2009).
194
[70] E. Madelung, ”Eine anschauliche Deutung der Gleichung von Schro¨dinger”, Die
Naturwissenschaften, 13 (45) 1004 (1926).
[71] E. Madelung, ”Quantentheorie in hydrodynamischer Form”, Zeitschrift fu¨r Physik,
40, Issue 3-4, pp 322-326 (1927).
[72] L. Ambrosio, G. Crippa, A. Figalli and L. V. Spinolo, ”Existence and Uniqueness
Results for the Continuity Equation and Applications to the Chromatography Sys-
tem”, IMA Vol. Math. Appl. 153 (2010).
[73] N.G. Berloff, ”Pade´ approximations of solitary wave solutions of the Gross-
Pitaevskii equation”, Journal of Physics A: Mathematical and General, 37, 1617
(2004).
[74] I. Gasser and P. A. Markowich, ”Quantum Hydrodynamics, Wigner Transforms
and the Classical Limit”, Asymptotic Analysis, 14 pp. 97-116 (1997).
[75] T. C. Wallstrom, ”Inequivalence between the Schrdinger equation and the Madelung
hydrodynamic equations”, Phys. Rev. A 49 Number 3, 1613 (1993).
[76] E. Schro¨dinger, ”Die Mehrdeutigkeit der Wellenfunktion”, Ann. Physik 32, 49
(1938).
[77] D.S. Petrov, D.M. Gangardt and G.V. Shlyapnikov, ”Low-dimensional trapped
gases”, J. Phys. IV France, 1, (2008).
[78] N. Parker, ”Numerical Studies of Vortices and Dark Solitons in Atomic Bose-
Einstein Condensates”, PhD thesis, Durham University (2004).
[79] A. Go¨rlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-
Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle,
”Realization of Bose-Einstein Condensates in Lower Dimensions”, Phys. Rev. Lett.
87, 130402 (2001).
[80] R. Kollar, ”Existence and Stability of Vortex Solutions of Certain Nonlinear
Schro¨dinger Equations”, PhD thesis, University of Maryland (2004).
[81] F. Kh. Abdullaev and J. Garnier, ”Emergent nonlinear Phenomena in Bose-Einstein
condensates: Theory end experiment”, Springer series on atomic, optical and
plasma physics (2008).
[82] J.S. Russell, ”Report on waves”. Fourteenth meeting of the British Association for
the Advancement of Science (1844).
[83] Y.S. Kivshar and G.P. Agrawal, ”Optical Solitons”, Elsevier Science, USA (2003).
[84] N. Parker, ”Numerical Studies of Vortices and Dark Solitons in Atomic Bose-
Einstein Condensates”, Thesis, University of Durham (2004).
195
[85] R. Balakrishnan and I. I. Satija, Pramana, ”Solitons in BoseEinstein condensates”,
J. Phys., 77, pp. 929-947, (2011).
[86] A.A. Abrikosov, ”On the magnetic properties of superconductors of the second
group”, Zh. Exp. Teor. Phz. 32, 1442 (1957).
[87] H. Flayac, ”New Trends in the Physics of spinor exciton-polariton condensates”,
PhD thesis, Universite Blaise Pascal (2013).
[88] F. Pinsker and N. G. Berloff, Phys. Rev. A 89 (5), 11 (2014).
[89] S. Pau, J. Jacobson, and Y. Yamamoto, ”Microcavity exciton-polariton splitting in
the linear regime”, Phys. Rev. B, 51, no. 20, pp. 14 43714 447, (1995).
[90] A. Dreismann, first year report, Cambridge University (2012).
[91] M. Wouters and I. Carusotto, ”Excitations in a nonequilibrium Bose-Einstein con-
densate of exciton polaritons”, Phys. Rev. Lett. 99, 140402 (2007).
[92] Y. Xue, M. Matuszewski, ”Creation and abrupt decay of a quasi-stationary dark
soliton in a polariton condensate”, arXiv:1401.2412 (2014).
[93] S. B. Papp, J. M. Pino and C. E. Wieman, ”Tunable Miscibility in a Dual-Species
Bose-Einstein Condensate”, Phys. Rev. Lett. 101, 040402 (2008).
[94] H. Flayac, I. A. Shelykh, D. D. Solnyshkov, and G. Malpuech, ”Topological stabil-
ity of the half-vortices in spinor exciton-polariton condensates”, Phys. Rev. B 81,
045318 (2008).
[95] A. Aftalion, ”Vortices in Bose-Einstein Condensates”, Progress in Nonlinear Differ-
ential Equations and their Applications 67, Birkha¨user, Basel, 2006.
[96] A.L. Fetter, ”Rotating Trapped Bose-Einstein Condensates”, Rev. Mod. Phys. 81,
647-691, (2009).
[97] E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, ”The Mathematics of the Bose
Gas and its Condensation”, Oberwolfach Seminar Series 34, Birkha¨user, Basel
(2005).
[98] E.H. Lieb, R. Seiringer, J. Yngvason, ”Bosons in a Trap: A Rigorous Derivation of
the Gross-Pitaevskii Energy Functional”, Phys. Rev. A. 61, 043602, (2000).
[99] E.H. Lieb and R. Seiringer, ”Derivation of the Gross-Pitaevskii Equation for Ro-
tating Bose Gases”, Comm. Math. Phys. 264, 505-537 (2006).
[100] J.-B. Bru, M. Correggi, P. Pickl and J. Yngvason, ”The TF Limit for Rapidly
Rotating Bose Gases in Anharmonic Traps”, Comm. Math. Phys. 280, 517-544
(2008).
196
[101] L. Erdo˝s, B. Schlein, H.T. Yau, ”Rigorous Derivation of the Gross-Pitaevskii Equa-
tion”, Phys. Rev. Lett. 98, 040404 (2007).
[102] L. Erdo˝s, B. Schlein, H.T. Yau, ”Derivation of the Gross-Pitaevskii Equation for
the Dynamics of Bose-Einstein Condensate”, Ann. Math. 172, 291-370 (2010).
[103] P. Pickl, ”Derivation of the Time Dependent Gross Pitaevskii Equation with Ex-
ternal Fields”, preprint arXiv:1001.4894v2 [math-ph] (2010).
[104] E.H. Lieb, R. Seiringer and J. Yngvason, ”The Yrast Line of a Rapidly Rotating
Bose Gas: The Gross-Pitaevskii Regime”, Phys. Rev. A 79, 063626 (2009).
[105] M. Lewin, R. Seiringer, ”Strongly Correlated Phases in Rapidly Rotating Bose
Gases”, J. Stat. Phys. 137, 1040-1062 (2009).
[106] M. Correggi, T. Rindler-Daller and J. Yngvason, ”Rapidly Rotating Bose-Einstein
condensates in Homogeneous Traps”, J. Math. Phys. 48, 102103 (2007).
[107] L. C. Evans, ”Partial Differential Equations”, reprinted edition, Amer. Math. Soc.,
RI. (2008).
[108] A. N. and I. Shafrir, ”Minimization of a Ginzburg-Landau Type Functional with
Nonvanishing Dirichlet Boundary Condition” Calc. Var. Partial Differential Equa-
tions 7, 191-217. (1998).
[109] H. Brezis, L. Oswald, ”Remarks on Sublinear Elliptic Equations”, Nonlinear Anal-
ysis, 10, 55-64 (1986).
[110] S. Serfaty, ”On a Model of Rotating Superfluids”, ESAIM: Control Optim. Calc.
Var. 6, 201-238 (2001).
[111] M. Correggi and J. Yngvason, ”Energy and Vorticity in fast Rotating Bose-Einstein
Condensates” J. Phys. A.: Theor. 41, 445002, 19pp. (2008).
[112] E. Sandier, S. Serfaty, ”On the Energy of Type-II Superconductors in the Mixed
Phase”, Rev. Math. Phys. 12, 1219-1257 (2000).
[113] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V.
Shlyapnikov, and M. Lewenstein, ”Dark solitons in Bose-Einstein condensates”,
Phys. Rev. Lett. 83, 5198 (1999); S. Stellmer, C. Becker, P. Soltan-Panahi, E.-
M. Richter, S. Do¨rscher, M. Baumert, J. Kronja¨ger, K. Bongs, and K. Sengstock,
”Collisions of dark solitons in elongated Bose-Einstein condensates”, Phys. Rev.
Lett. 101, 120406 (2008); A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, M. K.
Oberthaler, D. J. Frantzeskakis, G. Theocharis and P. G. Kevrekidis, ”Experimental
observation of oscillating and interacting matter wave dark solitons”, Phys. Rev.
Lett., 101, 130401 (2008).
197
[114] K.E. Strecker, G.B. Partridge, A.G. Truscott, G.B. Hulet, ”Formation and prop-
agation of matter-wave soliton trains”, Nature 417, 150 (2002); L. Khaykovich,
F. Schreck, G. Ferrari, T. Bourdel, J. Cubizoller, L.D. Carr, Y. Castin, C. Sa-
lomon, ”Formation of a matter-wave bright soliton”, Science 296, 1290 (2002); S.
L. Cornish, S. T. Thompson, and C. E. Wieman, ”Formation of bright matter-wave
solitons during the collapse of attractive Bose-Einstein condensates”, Phys. Rev.
Lett. 96, 170401 (2006).
[115] B. Eiermann, Th. Anker, M. Albiez M. Taglieber, P. Treutlein, K.-P. Marzlin,
and M. K. Oberthaler, ”Bright Bose-Einstein gap solitons of atoms with repulsive
interaction”, Phys. Rev. Lett. 92, 230401(2004).
[116] C. Becker, S. Stellmer, P. Soltan-Panahi, S. Do¨rscher, M. Baumert, E.-M. Richter,
J. Kronja¨ger, K. Bongs, and K. Sengstock, ”Oscillations and interactions of dark
and darkbright solitons in BoseEinstein condensates”, Nature Phys. 4, 496 (2008).
[117] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E.
A. Cornell, ”Vortex formation in a stirred Bose-Einstein condensate”, Phys. Rev.
Lett. 83, 2498 (1999).
[118] K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, ”Vortex formation in
a stirred Bose-Einstein condensate”, Phys. Rev. Lett., 84, 806 (1999).
[119] S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Go¨rlitz, T. L. Gustavson,
A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, ”Observation of Vortex Phase
Singularities in Bose-Einstein Condensates ”, Phys. Rev. Lett. 87, 080402 (2001).
[120] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W.
Clark, and E. A. Cornell, ”Watching dark solitons decay into vortex rings in a
Bose-Einstein condensate”, Phys. Rev. Lett. 86, 2926 (2001).
[121] Z. Dutton, M. Budde, C. Slowe and L. V. Hau, ”Observation of quantum shock
waves created with ultra-compressed slow light pulses in a Bose-Einstein conden-
sate”, Science 293, 663 (2001).
[122] N. S. Ginsberg, J. Brand and L. V. Hau, ”Observation of Hybrid Soliton Vortex-
Ring Structures in Bose-Einstein Condensates”, Phys. Rev. Lett. 94, 040403 (2005);
I. Shomroni, I. Shomroni, E. Lahoud, S. Levy and J. Steinhauer, ”Evidence for an
oscillating soliton/vortex ring by density engineering of a BoseEinstein condensate”,
Nature Phys. 5, 193 (2009).
[123] K. M. Mertes, J. W. Merrill, R. Carretero-Gonza´lez, D. J. Frantzeskakis, P. G.
Kevrekidis, and D. S. Hall, ”Nonequilibrium dynamics and superfluid ring excita-
tions in binary Bose-Einstein condensates”, Phys. Rev. Lett. 99, 190402 (2007).
[124] P. Engels, C. Atherton, and M. A. Hoefer ”Observation of Faraday waves in a
Bose-Einstein condensate”, Phys. Rev. Lett. 98, 095301 (2007); J. J. Chang, P.
198
Engels, and M. A. Hoefer,”Formation of dispersive shock waves by merging and
splitting Bose-Einstein condensates”, ibid. 101, 170404 (2008).
[125] C. C. Bradley, C. A. Sackett, and R. G. Hulet, ”Bose-Einstein condensation of
lithium: Observation of limited condensate number”, Phys. Rev. Lett. 78, 985
(1997); J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell,
and C. E. Wieman, ”Controlled collapse of a Bose-Einstein condensate”, Phys. Rev.
Lett. 86, 4211 (2001).
[126] R. Carretero-Gonza´lez, D. J. Frantzeskakis and P. G. Kevrekidis, ”Nonlinear waves
in BoseEinstein condensates: physical relevance and mathematical techniques”,
Nonlinearity 21, R139 (2008).
[127] V. M. Pe´rez-Garc´ıa, N. G. Berloff, P. G. Kevrekidis, V.V. Konotop, and B. A.
Malomed, ”Nonlinear phenomena in degenerate quantum gases”, Physica D 238,
1289 (2009).
[128] C. Yin, N. G. Berloff, V. M. Pe´rez-Garc´ıa, D. Novoa, A. V. Carpentier, and H.
Michinel, ”Coherent atomic soliton molecules for matter-wave switching”, Phys.
Rev. A 83, 051605(R) (2011).
[129] G. K. Batchelor, ”An Introduction to Fluid Dynamics”, Cambridge University
Press, Cambridge, England (1967).
[130] R. J. Donnelly, ”Quantized Vortices in Helium II”, Cambridge University Press,
Cambridge, England (1991).
[131] S. Komineas and N. Papanicolaou, ”Nonlinear waves in a cylindrical Bose-Einstein
condensate”, Phys. Rev. A 67, 023615 (2003); ibid. 68 043617 (2003).
[132] C. A. Jones and P. H. Roberts, ”Motions in a Bose condensate. IV. Axisymmetric
solitary waves”, J. Phys. A: Math. Gen. 15 599 (1982); C. A. Jones, S. J. Putterman
and P. H. Roberts, ”Motions in a Bose condensate. V. Stability of solitary wave
solutions of non-linear Schro¨dinger equations in two and three dimensions”, J. Phys.
A, Math. Gen. 19 2991 (1986).
[133] N. G. Berloff and C. F. Barenghi, ”Vortex nucleation by collapsing bubbles in
Bose-Einstein condensates”, Phys. Rev. Lett., 93 , 090401, (2004).
[134] E. A. Kuznetsov and J. J. Rasmussen, ”Instability of two-dimensional solitons and
vortices in defocusing media”, Phys. Rev. E 51 (5), 4479-4484 (1995).
[135] D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, ”Vortex formation
by merging of multiple trapped Bose-Einstein condensates”, Phys. Rev. Lett. 98,
110402 (2007); R. Carretero-Gonza´lez, B. P. Anderson, P. G. Kevrekidis, D. J.
Frantzeskakis and C. N. Weiler, ”Dynamics of vortex formation in merging Bose-
Einstein condensate fragments”, Phys. Rev. A 77, 033625 (2008).
199
[136] A. Glezer, ”The formation of vortex rings”, Phys. Fluids 31 3532 (1988).
[137] S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos, and R. G. Hulet,
”Extreme Tunability of Interactions in a 7Li Bose-Einstein Condensate”, Phys. Rev.
Lett. 102, 090402 (2009).
[138] F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, B. A. Malomed, ”Controlling
collapse in Bose-Einstein condensates by temporal modulation of the scattering
length”, Physical Review A 67, 013605 (2003); H. Saito, and M. Ueda, ”Dynam-
ically stabilized bright solitons in a two-dimensional Bose-Einstein condensate”,
Phys. Rev. Lett. 90, 040403 (2003).
[139] V. M. Pe´rez-Garc´ıa, V. V. Konotop and V. A. Brazhnyi, ”Feshbach resonance
induced shock waves in Bose-Einstein condensates”, Phys. Rev. Lett. 92, 220403
(2004).
[140] J. Belmonte-Beitia, V. M. Pe´rez-Garc´ıa, V. Vekslerchik, V. V. Konotop, ”Lo-
calized nonlinear waves in systems with time-and space-modulated nonlinearities”
Phys. Rev. Lett. 100, 164102 (2008); V. V. Konotop and P. Pacciani, ”Collapse
of solutions of the nonlinear Schrdinger equation with a time-dependent nonlinear-
ity: Application to Bose-Einstein condensates”, ibid., 94, 240405 (2005); Y. Sivan,
G. Fibich, and M. I. Weinstein, ”Waves in nonlinear lattices: ultrashort optical
pulses and Bose-Einstein condensates”, ibid. 97, 193902 (2006); G. Theocharis, P.
Schmelcher, P. G. Kevrekidis, and D. J. Frantzeskakis, ”Matter-wave solitons of
collisionally inhomogeneous condensates”, Phys. Rev. A 72, 033614 (2005).
[141] A. M. Kamchatnov, A. Gammal and R. A. Kraenkel, ”Dissipationless shock waves
in repulsive Bose-Einstein condensates”, Phys. Rev. A 69, 063605 (2004).
[142] A. V. Gurevich and A. L. Krylov, ”Nondissipative shock waves in media with
positive dispersion”, Zhurnal Eksperimental Noi I Teoreticheskoi Fiziki, 92, 1684-
1699 (1987).
[143] G. A. El and A. L. Krylov, ”General solution of the Cauchy problem for the
defocusing NLS equation in the Whitham limit”, Phys. Lett. A 203, 77-82 (1995).
[144] M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and
V.Schweikhard, ”Dispersive and Classical Shock Waves in Bose-Einstein Conden-
sates and Gas Dynamics”, Phys. Rev. A 74, 023626 (2006).
[145] I. Kulikov and M. Zak, ”Shock waves in a Bose-Einstein condensate”, Phys. Rev.
A 67, 063605 (2003)
[146] C. Wang, P. G. Kevrekidis, T. P. Horikis, D. J. Frantzeskakis, ”Collisional-
inhomogeneity-induced generation of matter-wave dark solitons”, Phys. Lett. A
37, 3863 (2010).
200
[147] T. F. Scott, R. J. Ballagh and K Burnett, ”Formation of fundamental structures
in Bose- Einstein condensates”, J. Phys. B: At. Mol. Opt. Phys. 31 329-335 (1998).
[148] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and
E. A. Cornell,”Vortices in a Bose-Einstein condensate”, Phys. Rev. Lett., 83, 2498
(1999).
[149] M. I. Rodas-Verde, H. Michinel, V. M. Pe´rez-Garc´ıa, ”Controllable soliton emission
from a Bose-Einstein condensate”, Phys. Rev. Lett. 95, 153903 (2005).
[150] A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, V. M. Pe´rez-Garc´ıa, ”Analysis
of an atom laser based on the spatial control of the scattering length”, Phys. Rev.
A 74 013619 (2006).
[151] K. E. Strecker, G. B. Partridge, A. G. Truscott and R. G. Hulet, ”Formation and
propagation of matter-wave soliton trains”, Nature 417, 150-153 (1999).
[152] W. Bao, S. Jin and P.A. Markowich, ”Numerical solution of the GrossPitaevskii
equation for BoseEinstein condensation”, J. Comput. Phys. 187, 318-342 (2003).
[153] In simulations, we used ∆x = 1/6, ∆t = 0.0015, z ∈ (−500, 500), µ = 0.1 and
reflective boundary conditions.
[154] F. Be´thuel, P. Gravejat and J.-C. Saut, ”Existence and properties of travelling
waves for the Gross-Pitaevskii equation”, arXiv:0902.3804v1 (2009).
[155] T. Tsuzuki, ”Nonlinear waves in the Pitaevskii-Gross equation”, J. Low Temp.
Phys. 4, 441(1971).
[156] E. A. Kuznetsov and M. D. Spector, ”Modulation instability of soliton trains in
fiber communication systems”, Theor. and Math. Phys. 120, 997-1008 (1999).
[157] P. F. Byrd and M.D. Friedman, ”Handbook of Elliptic Integrals for Engineers and
Scientists”, Springer-Verlag, Berlin, Germany (1954).
[158] L. D. Carr, W. C. Clark and W.P. Reinhardt, ”Stationary solutions of the one-
dimensional nonlinear Schrdinger equation. I. Case of repulsive nonlinearity”, Phys.
Rev. A 62, 063610 (2000).
[159] L. D. Carr, W. C Clark and W.P. Reinhardt, ”Stationary solutions of the one-
dimensional nonlinear Schrdinger equation. II. Case of attractive nonlinearity”,
Phys. Rev. A 62, 063611 (2000).
[160] W-P. Zhong, M. R. Belic, Y. Lu and T. Huang, ”Traveling and solitary wave solu-
tions to the one-dimensional Gross-Pitaevskii equation”, Phys. Rev. E, 81, 016605
(2010).
[161] V. V. Konotop and L. Pitaevskii, ”Landau dynamics of a grey soliton in a trapped
condensate”, Phys. Rev. Lett. 93, 240403 (2004).
201
[162] J. Brand and W.P. Reinhardt, ”Solitonic vortices and the fundamental modes of
the snake instability: Possibility of observation in the gaseous Bose-Einstein con-
densate”, Phys. Rev. A, 65, 043612 (2002).
[163] N.G. Berloff, ”Evolution of rarefaction pulses into vortex rings”, Phys. Rev. B, 65,
0236031 (2002).
[164] We used a computational window of 240 space units in z and 10 units along the
transverse directions x, y. Typical space steps are about ∆x = 0.25 and the time
steps ∆t ' 1.5× 10−4.
[165] N.G. Berloff, ”Pade approximations of solitary wave solutions of the Gross-
Pitaevskii equation”, J. Phys. A: Math. Gen., 37 (5), 1617-1632 (2004).
[166] Kapitza, P., ”Viscosity of liquid helium below the λ-point”, Nature 141, 74, (1938).
[167] Allen, J. F., Misener, A. D., ”Flow of liquid helium II”, Nature 141, 75, (1938).
[168] L. Tisza, ”Transport phenomena in helium II”, Nature 141, 913 (1938).
[169] C. Raman, M. Kohl, R. Onofrio, D.S. Durfee, C.E. Kuklewicz, Z. Hadzibabic and
W. Ketterle, ”Evidence for a critical velocity in a Bose-Einstein condensed gas”,
Phys. Rev. Lett. 83 (13), pp. 2502-2505 (1999).
[170] R. Onofrio, C. Raman, J. M. Vogels, J. Abo-Shaeer, A. P. Chikkatur and W. Ket-
terle, ”Observation of Superfluid Flow in a Bose-Einstein Condensed Gas”, Phys.
Rev. Lett. 85, 2228-2231 (2000).
[171] P. Engels and C. Atherton, ”Stationary and Nonstationary Fluid Flow of a Bose-
Einstein Condensate Through a Penetrable Barrier”, Phys. Rev. Lett. 99, 160405
(2007).
[172] T. Frisch, Y. Pomeau and S. Rica, ”Transition to dissipation in a model of super-
flow”, Phys. Rev. Lett. 69, 1644-1647 (1992).
[173] C. Josserand, Y. Pomeau and S. Rica, ”Vortex shedding in a model of superflow”,
Physica D 134, 111-125 (1999).
[174] V. Hakim, ”Nonlinear Schro¨dinger flow past an obstacle in one dimension”, Phys.
Rev. E 55, 28352845 (1997).
[175] T. Winiecki, J. F. McCann, and C. S. Adams, ”Pressure drag in linear and non-
linear quantum fluids”, Phys. Rev. Lett. 82, 5186-5189 (1999).
[176] J. S. Stießberger and W. Zwerger, ”Critcal velocity of superfluid flow past large
obstacles in Bose-Einstein condensates”, Phys. Rev. A 62, 061601(R) (2000).
[177] N.G. Berloff and P.H.Roberts,”Motions in a Bose condensate: VII. Boundary-layer
separation”, J. Phys. A: Math. Gen. 33, 4025-4038 (2000).
202
[178] N. Pavloff, ”Breakdown of superfluidity of an atom laser past an obstacle”, Phys.
Rev. A 66, 013610 (2002).
[179] G. Watanabe, F. Dalfovo, F. Piazza, L. P. Pitaevskii and S. Stringari, ”Critical
velocity of superfluid flow through single-barrier and periodic potentials”, Phys.
Rev. A 80, 053602 (2009).
[180] G.E. Astrakharchikab and L.P. Pitaevskii, ”Motion of a heavy impurity through
a Bose-Einstein condensate”, Phys. Rev. A 70, 013608 (2004).
[181] Y.G. Gladush, L.A. Smirnov and A.M. Kamchatnov, ”Generation of Cherenkov
waves in the flow of a BoseEinstein condensate past an obstacle”, J. Phys. B: At.
Mol. Opt. Phys. 41 165301 (6pp) (2008).
[182] A.L. Fetter, ”Rotating trapped Bose-Einstein condensates”, Rev. Mod. Phys. 81,
647–691 (2009).
[183] A.L. Fetter, ”Rotating vortex lattice in a Bose-Einstein condensate trapped in
combined quadratic and quartic radial potentials”, Phy. Rev. A 64, 063608 (2001).
[184] A.L. Fetter, N. Jackson and S. Stringari, ”Rapid rotation of a Bose-Einstein con-
densate in a harmonic plus quartic trap”, Phys. Rev. A 71, 013605 (2005).
[185] D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W. Setiawan, C. Sanner and W.
Ketterle, ”Critical velocity for superfluid flow across the BEC-BCS crossover”, Phys.
Rev. Lett. 99, 070402 (2007).
[186] S. Giorgini, L. P. Pitaevskii and S. Stringari, ”Theory of ultracold atomic fermi
gases”, Rev. Mod. Phys. 80, 1215-1274 (2008).
[187] A. S. Rodrigues, P. G. Kevrekidis, R. Carretero-Gonzlez, D. J. Frantzeskakis, P.
Schmelcher, T. J. Alexander and Yu. S. Kivshar, ”Spinor Bose-Einstein condensate
flow past an obstacle”, Phys. Rev. A 79, 043603 (2009).
[188] D.M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner,
J. Stenger and W. Ketterle, ”Optical Confinement of a Bose-Einstein Condensate”,
Phys. Rev. Lett. 80, 2027 (1998).
[189] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang,
L. You and M. S. Chapman, ”Observation of Spinor Dynamics in Optically Trapped
Rb87 Bose-Einstein Condensates”, Phys. Rev. Lett. 92, 140403 (2004).
[190] A. Amo, J. Lefre´re, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre´, E.
Giacobino and A. Bramati, ”Superfluidity of polaritons in semiconductor microcav-
ities”, Nat. Phys. 5, 805. (2009).
203
[191] A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A.
Lema´ıtre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C. Tejedor and L. Vina,
”Collective fluid dynamics of a polariton condensate in a semiconductor microcav-
ity”, Nature 457, 291-295 (2009).
[192] S. Pigeon, I. Carusotto and C. Ciuti, ”Hydrodynamic nucleation of vortices and
solitons in a resonantly excited polariton superfluid”, Phys. Rev. B 83, 144513
(2011).
[193] J. Keeling and N. G. Berloff, ”Spontaneous Rotating Vortex Lattices in a Pumped
Decaying Condensate”, Phys. Rev. Lett. 100, 250401 (2008).
[194] C. Connaughton, Christophe Josserand, Antonio Picozzi, Yves Pomeau and Sergio
Rica, ”Condensation of classical nonlinear waves”, Phys. Rev. Lett. 95, 263901
(2005).
[195] H. Salman and N. G. Berloff, ”Condensation of classical nonlinear waves in a
two-component system,” Special issue on quantum gasses, Physica D, 238, 1482,
(2009).
[196] J. Keeling and N. G. Berloff, ”Condensed-matter physics: Going with the flow”,
Nature 457, 273-274 (2009).
[197] G. Grimberg, W. Pauls and U. Frisch, ”Genesis of dAlemberts paradox and an-
alytical elaboration of the drag problem”, Phys. D, Nonl. Phen. 237, 1878 - 1886
(2008).
[198] J.O. Hirschfelder, C. J. Goebel and L. W. Bruch, ”Quantized vortices around
wavefunction nodes. II”, J. Chem. Phys. 61, 5456 (1974).
[199] C. Huepe and M.-E. Brachet, ”Scaling laws for vortical nucleation solutions in a
model of superflow”, Physica D 140 126-140 (2000).
[200] N. G. Berloff and B. V. Svistunov, ”Scenario of strongly nonequilibrated Bose-
Einstein condensation”, Phys. Rev. A 66, 013603 (2002).
[201] N. G. Berloff and A. J. Youd, ”Dissipative dynamics of superfluid vortices at
nonzero temperatures”, Phys. Rev. Lett. 99, 145301 (2007).
[202] N.G. Berloff, ”Interactions of vortices with rarefaction solitary waves in a Bose-
Einstein condensate and their role in the decay of superfluid turbulence”, Phys.
Rev. A 69, 053601 (2004).
[203] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson,
”Observation of Vortex Dipoles in an Oblate Bose-Einstein Condensate”, Phys.
Rev. Lett. 104, 160401 (2010).
204
[204] J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, F. Minardi, A. Kantian
and T. Giamarchi, ”Quantum dynamics of impurities in a one-dimensional Bose
gas”, Phys. Rev. A 85, 023623 (2012).
[205] S. Rica, ”A remark on the critical speed for vortex nucleation in the nonlinear
Schro¨dinger equation”, Physica D 148, 221-226 (2001).
[206] C. A. Jones and P. H. Roberts, ”Motions in a Bose condensate. IV. Axisymmetric
solitary waves”, J. Phys. A: Math. Gen. 15 2599 (1982).
[207] We numerically generated solutions using fourth order finite differences scheme
in space and the fourth order Runge-Kutta time integration on a computational
window of 361 space units in x and 121 units along the direction y transverse to the
superfluid flow. Typical space steps are about ∆x = 0.25 and time steps ∆t = 10−3.
At the boundaries we have implemented an absorption layer of 20 grid points to
minimize the reflection of emitted waves.
[208] We have considered a grid of 350 units in the x direction and 200 in the transverse
direction. Step sizes in both directions have been 1/5 healing layer and time steps
have been ∆t = 0.00015.
[209] We have considered a Dirichlet-type obstacle at a computational grid of 700 units
in x direction and 400 in the transverse direction. Step sizes in both directions have
been 1/10 healing layer and time steps have been ∆t = 0.00005.
[210] We have considered a computational grid of 500 units in the x direction and 500
in transverse direction. Step sizes in both directions have been 1/5 healing layer
and time steps have been ∆t = 0.00002.
[211] The grid was 500 units in x direction and 500 in the transverse direction. Step sizes
in both directions have been 1/4 healing layer and time steps have been ∆t = 0.0001.
[212] G. A. El, A. Gammal, and A. M. Kamchatnov, ”Oblique Dark Solitons in Super-
sonic Flow of a Bose-Einstein Condensate”, Phys. Rev. Lett. 97, 180405 (2006).
[213] D.H. Santamore, E. Timmermans, ”Multi-impurity polarons in a dilute Bose-
Einstein condensate”, New J. Phys. 13 103029 (2011).
[214] The computing grid has been chosen to be 440 points in the x direction and 200
points in the y direction. Time steps were about ∆t = 0.004 and space steps ∆x = 1.
At the boundaries of the grid we have implemented an absorption layer of 14 points
in width to simulate open boundary conditions.
[215] P. Mason, N. G. Berloff, and A. L. Fetter, ”Motion of a vortex line near the
boundary of a semi-infinite uniform condensate”, Phys. Rev. A 74, 043611 (2006).
205
[216] The computing grid has been chosen to be 900 points in the x direction and 500
points in the y direction. Time steps were about ∆t = 0.00025 and space steps
∆x = 0.5. At the boundaries of the grid we have implemented an absorption layer
of 14 points in width to simulate open boundary conditions.
[217] Our simulations on randomly distributed atoms relied on the Fortran RANDOM
NUMBER routine and we have used the standard seed.
[218] The computing grid has been chosen to be 880 points in x direction and 480
points in transversal direction. Time steps were about ∆t = 0.0004 and space steps
∆x = 0.5. At the boundaries of the grid we have implemented an absorption layer
of 14 points in width.
[219] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B.
Hillebrands and A. N. Slavin, ”Bose-Einstein condensation of quasi-equilibrium
magnons at room temperature under pumping”, Nature 443, 430 - 433 (2006).
[220] L. V. Butov, ”Cold exciton gases in coupled quantum well structures”, J. Phys.
Condens. Matter 19, 295202 (2007).
[221] I. S. Aranson and L. Kramer, ”The world of the complex GinzburgLandau equa-
tion”, Rev. Mod. Phys. 74, pp. 99-143 (2002).
[222] G. P. Agrawal, ”Nonlinear Science at the Dawn of the 21st Century”, Lecture
Notes in Physics 542, pp. 195-211 (2000).
[223] J. Scheuer, M. Orenstein, and D. Arbel, ”Nonlinear switching and modulational
instability of wave patterns in ring-shaped vertical-cavity surface-emitting lasers”,
JOSA B 19, pp. 2384-2390 (2002).
[224] G. P. Agrawal and H. A. Haus, ”Applications of Nonlinear Fiber Optics”, Physics
Today 55 pp. 58 (2002).
[225] D. Woods, T. J. Naughton, ”Optical computing: Photonic neural networks”, Na-
ture Physics 8, 257-259 (2012).
[226] E. M. Dianov, P. V. Mamyshev and A. M. Prokhorov, ”Nonlinear fiber optics”,
Sov. j. quantum electron 18, 1-15. (1988).
[227] L. Bi, H. Juejun, P. Jiang, H. D. Kim, F. G. Dionne, L.C. Kimerling, C. A.
Ross, ”On-chip optical isolation in monolithically integrated non-reciprocal optical
resonators”, Nature Photonics 5, pp. 758-762 (2011).
[228] W. Chen, K. M. Beck, R. Bu¨cker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki and V.
Vuletic´ ,”All-Optical Switch and Transistor Gated by One Stored Photon”, Science
341 768-770 (2013).
206
[229] D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdre´, E. Giacobino, R. Cingolani,
A. Bramati, G. Gigli and D. Sanvitto ”All-optical polariton transistor”, Nature
Communications 4 1778 (2013).
[230] C. Anto´n, T.C.H. Liew, J. Cuadra, M.D. Mart´ın, P.S. Eldridge, Z. Hatzopoulos,
G. Stavrinidis, P.G. Savvidis and L. Vina, ”Implementation of a logic AND gate
with polariton condensate bullets”, arXiv:1305.5678 (2013).
[231] D. Woods and T. J. Naughton, ”Optical computing”, Applied Mathematics and
Computation 215, pp. 1417-1430 (2009).
[232] G. Tosi, G. Christmann, N. G. Berloff, P. Tsotsis, T. Gao, Z. Hatzopoulos, P. G.
Savvidis and J. J. Baumberg, ”Sculpting oscillators with light within a nonlinear
quantum fluid”, Nature Physics 8, 190-194 (2012).
[233] T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos, G. Stavrinidis, G. Deligeorgis,
Z. Hatzopoulos and P. G. Savvidis, ”Polariton condensate transistor switch”, Phys.
Rev. B 85, 235102 (2012).
[234] R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E.
Giacobino, J. Bloch, A. Bramati, G. Malpuech and A. Amo, ”Half-solitons in a
polariton quantum fluid behave like magnetic monopoles”, Nature Physics 8, 724-
728 (2012).
[235] D. D. Solnyshkov, H. Flayac and G. Malpuech, ”Stable magnetic monopoles in
spinor polariton condensates”, Phys. Rev. B 85, 073105 (2012).
[236] H. Flayac, D. D. Solnyshkov and G. Malpuech, ”Separation and acceleration of
magnetic monopole analogs in semiconductor microcavities”, New J. Phys. 14,
085018 (2012).
[237] H. Terc¸as, H. Flayac, D. Solnyshkov and G. Malpuech, ”Non-Abelian Gauge Fields
in Photonic Cavities and Photonic Superfluids”, arXiv:1310:6588 (2013).
[238] I. A. Shelykh, A. V. Kavokin, Y. G. Rubo, T. C. H. Liew and G. Malpuech, ”Po-
lariton polarization-sensitive phenomena in planar semiconductor microcavities”,
Semicond. Sci. Technol. 25, 013001 (2010).
[239] F. Manni, ”Coherence and Topological Defects in Exciton-Polariton Condensates”,
PhD thesis, E`cole polytechnique fe`de`rale de Lausanne, Lausanne, Switzerland
(2013).
[240] E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lema´ıtre, I.
Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech and J. Bloch,
”Spontaneous formation and optical manipulation of extended polariton conden-
sates”, Nature Physics 6, 860 (2010).
207
[241] H. Flayac, G. Pavlovic, M. A. Kaliteevski and I. A. Shelykh, ”Electric generation
of vortices in an exciton-polariton superfluid”, Phys. Rev. B 85, 075312 (2012).
[242] I. G. Savenko, T. C. H. Liew, and I. A. Shelykh, ”Stochastic Gross-Pitaevskii
Equation for the Dynamical Thermalization of Bose-Einstein Condensates”, Phys.
Rev. Lett. 110, 127402 (2013).
[243] Y. V. Kartashov and A. M. Kamchatnov, ”Two-dimensional dispersive shock waves
in dissipative optical media”, Optics Letters, 38, Issue 5, pp. 790-792 (2013).
[244] N. G. Berloff and Victor M. Perez-Garcia, ”A nonlinear quantum piston for the
controlled generation of vortex rings”, (preprint) arXiv:1006.4426 (2010).
[245] N. G. Berloff and P. H. Roberts, ”Motions in a Bose condensate: VII. Boundary-
layer separation”, J. Phys. A: Math. Gen. 33 4025 (2000).
[246] M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin
and I. A. Shelykh, ”Tamm plasmon-polaritons: Possible electromagnetic states at
the interface of a metal and a dielectric Bragg mirror”, Phys. Rev. B 76, 165415
(2007).
[247] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M. D. Fraser, T.
Byrnes, P. Recher, N. Kumada, T. Fujisawa and Y. Yamamoto, ”Coherent zero-
state and pi-state in an excitonpolariton condensate array”, Nature, 450, 529-532
(2007).
[248] T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh and R.
A. Abram, ”Exciton-polariton integrated circuits”, Phys. Rev. B 82, 033302 (2010).
[249] H. Flayac, H. Terc¸as, D. D. Solnyshkov, and G. Malpuech, ”Superfluidity of spinor
Bose-Einstein condensates”, Phys. Rev. B 88, 184503 (2013).
[250] N. Q. Ngo, ”Proposal for a high-speed optical dark-soliton detector using a mi-
croring resonator”, Phot. Technol. Lett. 19, 7 (2007).
[251] E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, P. Tsotsis, Z. Hatzopoulos,
P. G. Savvidis, A. V. Kavokin and P. G. Lagoudakis, ”Nonlinear Optical Spin Hall
Effect and Long-Range Spin Transport in Polariton Lasers”, Phys. Rev. Lett. 109,
036404 (2012).
[252] Y. Senatsky, J.-F. Bisson, J. Li, A. Shirakawa, M. Thirugnanasambandam and K.
Ueda ”LaguerreGaussian Modes Selection in Diode-Pumped Solid-State Lasers”,
Opt. Rev. 19, (4) 201-221 (2012).
[253] S. F. Pereira, M. B. Willemsen, M. P. van Exter and J. P. Woerdman, ”Pinning
of daisy modes in optically pumped vertical-cavity surface-emitting lasers.” Appl.
Phys. Lett. 73, (16), 2239-2241 (1998).
208
[254] M. Schulz-Ruhtenberg, Y. Tanguy, R. Jger, T. Ackemann, ”Length scales and
polarization properties of annular standing waves in circular broad-area vertical-
cavity surface-emitting lasers”, Appl. Phys. B 97, (2), 397-403 (2009).
[255] Q. Deng, H. Deng and D. G. Deppe, ”Lasing on higher-azimuthal-order modes
in vertical cavity surface emitting lasers at room temperature”, Opt. Lett. 22, (7)
463-465 (1997).
[256] D. Naidoo, K. Ait-Ameur, M. Brunel and A. Forbes, ”Intra-cavity generation
of superpositions of Laguerre-Gaussian beams”, Appl. Phys. B 106, (3), 683-690
(2012).
[257] Y.F. Chen, Y.P. Lan and S.C. Wang, ”Generation of LaguerreGaussian modes in
fiber-coupled laser diode end-pumped lasers”, Appl. Phys. B 72 (2) 167-170 (2001).
[258] M. Grabherr, M. Miller, R. Ja¨ger, D. Wiedenmann, R. King, ”Commercial VC-
SELs reach 0.1 W cw output power”, Proc. SPIE 5364: 174-182 (2004).
[259] L. Allen, M. W. Beijersbergen, R.J.C Spreeuw, J.P. Woerdman, ”Orbital angular
momentum of light and the transformation of Laguerre-Gaussian laser modes”,
Phys. Rev. A 45, (11), 8185-8189 (1992).
[260] J.A. Rodrigo, A.M. Caravaca-Aguirre, T. Alieva, G. Cristo´bal, M.L. Calvo, ”Mi-
croparticle movements in optical funnels and pods”, Opt. Express 19 (6) 5232-5243
(2011).
[261] G. Molina-Terriza, J.P. Torres and L. Torner, ”Twisted photons”, Nat. Phys. 3
(5), 305-310 (2007).
[262] A. Imamoglu, R.J. Ram, S. Pau and Y. Yamamoto, ”Nonequilibrium condensates
and lasers without inversion: Exciton-polariton lasers”, Phys. Rev. A 54, (6), 4250-
4253 (2006).
[263] H. Deng, G. Weihs, D. Snoke, J. Bloch and Y. Yamamoto, ”Polariton vs. photon
lasing in a semiconductor microcavity”, P. Natl. Acad. Sci. 100, (26), 15318-15323
(2003).
[264] C. Weisbuch, M. Nishioka, A. Ishikawa and Y. Arakawa, ”Observation of the
coupled exciton-photon mode splitting in a semiconductor quantum microcavity”,
Phys. Rev. Lett. 69, (23), 3314-3317 (1992).
[265] R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer and K. West, ”Bose-Einstein Conden-
sation of Microcavity Polaritons in a Trap”, Science 316 (5827) 1007-1010 (2007).
[266] E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lema´ıtre, I.
Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech and J. Bloch,
”Spontaneous formation and optical manipulation of extended polariton conden-
sates”, Nat. Phys. 6, (11), 860-864 (2010).
209
[267] K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. Andre´,
Le Si Dang and B. Deveaud-Ple´dran, ”Quantized vortices in an exciton-polariton
condensate”, Nat. Phys. 4, (9), 706-710 (2008).
[268] D. Sanvitto, F. M. Marchetti, M. H. Szyman´ska, G. Tosi, M. Baudisch, F. P.
Laussy, D. N. Krizhanovskii, M. S. Skolnick, L. Marrucci, A. Lema´ıtre, J. Bloch,
C. Tejedor and L. Vina, ”Persistent currents and quantized vortices in a polariton
superfluid”, Nat. Phys. 6, (7), 527-533 (2010).
[269] Amo A, et. al. (2009) Collective fluid dynamics of a polariton condensate in a
semiconductor microcavity. Nature 457(7227):291-296l.
[270] F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre´, B. Deveaud-Ple´dran,
”Spontaneous pattern formation in a polariton condensate”, Phys. Rev. Lett. 107,
(10), 106401 (2011).
[271] T. C. H. Liew, A. V. Kavokin, I. A. Shelykh, ”Optical Circuits Based on Polariton
Neurons in Semiconductor Microcavities”, Phys. Rev. Lett. 101, (1), 016402 (2008).
[272] A. Amo, T. C. H. Liew, C. Adrados, R. Houdre´, E. Giacobino, A. V. Kavokin
and A. Bramati, ”Exciton-polariton spin switches”, Nat. Photonics 4, (6), 361-366
(2010).
[273] G. Franchetti, N. G. Berloff and J. J. Baumberg, ”Exploiting quantum coherence
of polaritons for ultra sensitive detectors”, arXiv:1210.1187 (2012).
[274] G. Nardin, K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, R. Andre´, Le Si
Dang, B. Pietka and B. Deveaud-Ple´dran, ”Dynamics of Long Range Ordering in
an Exciton Polariton Condensate”, Phys. Rev. Lett. 103, (25), 256402 (2009).
[275] R. Spano, J. Cuadra, G. Tosi, C Anto´n, C. A. Lingg, D. Sanvitto, M. D. Mart´ın,
L. Vina, P. R. Eastham, M. van der Poel and J. M. Hvam, ”Coherence properties
of exciton polariton OPO condensates in one and two dimensions”, New J. Phys.
14, 075018, (2012).
[276] G. Christmann, G. Tosi, N. G. Berloff, P. Tsotsis, P. S. Eldridge, Z. Hatzopoulos, P.
G. Savvidis and J. J. Baumberg, ”Polariton ring condensates and sunflower ripples
in an expanding quantum liquid. Phys. Rev. B 85, (23), 235303 (2012).
[277] D. Naidoo, T. Godin, M. Fromager, E. Cagniot, N. Passilly, A. Forbes and Kamel
Ait-Ameurc, ”Transverse mode selection in a monolithic microchip laser”, Opt.
Commun. 284, (23), 5475-5479 (2011).
[278] R. Kleiner, D. Koelle, F. Ludwig, J. Clarke, ”Superconducting quantum interfer-
ence devices: State of the art and applications”, Proc. IEEE, 92 (10), 1534 (2004).
210
[279] T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. S. Levitov, S. Lloyd,
J. J. Mazo, ”Superconducting persistent-current qubit”, Phys. Rev. B 60, (22),
15398-15413 (1999).
[280] P. Cristofolini, A. Dreismann, G. Christmann, G. Franchetti, N. G. Berloff, P.
Tsotsis, Z. Hatzopoulos, P. G. Savvidis and J. J. Baumberg, ”Optical Superfluid
Phase Transitions and Trapping of Polariton Condensates”, Phys. Rev. Lett. 110,
(18), 186403 (2013).
[281] M. Wouters, T. Liew, V. Savona, ”Energy-relaxation in one-dimensional polariton
condensates”, Phys Rev B 82, (24), 245315 (2010).
[282] T. F. Scott, R. J. Ballagh and K. Burnett, ”Formation of fundamental structures
in Bose-Einstein condensates”, J. Phys. B: At. Mol. Opt. Phys. 31, (8), 329 (1998).
[283] Askitopoulos A, H. Ohadi, A. V. Kavokin, Z. Hatzopoulos, P. G. Savvidis, and
P. G. Lagoudakis, ”Polariton condensation in an optically induced two-dimensional
potential”, Phys. Rev. B 88, (4), 041308(R) (2013).
[284] L. Ge, A. Nersisyan, B. Oztop and H.E. Treci, ”Pattern Formation and Strong
Nonlinear Interactions in Exciton-Polariton Condensates”, arXiv:1311.4847 (2013).
[285] D. Naidoo, K. Ait-Ameur, M. Brunel and A. Forbes, ”Intra-cavity generation
of superpositions of LaguerreGaussian beams”, Appl. Phys. B 106, (3), 683-690
(2012).
211