Bose-Einstein Condensation and Superfluidity in
Two Dimensions
Richard Jonathan Fletcher
Trinity College

This dissertation is submitted for the degree ofDoctor of Philosophy

April 2015
Department of Physics

2

To my parents Nick and Mary, my sister Liz and my brother Will.

ii

Declaration
I declare that this thesis is my own work and is not substantially the same as
any that I have submitted or am currently submitting for a degree, diploma or
any other qualification at any other university. No part of this thesis has alreadybeen or is being concurrently submitted for any such degree, diploma or any otherqualification. This thesis does not exceed the word limit of sixty thousand words,
including tables, footnotes, bibliography and appendices, set out by the Facultyof Physics and Chemistry.

iii

iv

Abstract
It was known as early as 1934 that crystallisation is precluded in two dimensions. The requisite long-range order is destroyed by thermal fluctuations at any
non-zero temperature, and therefore ordered phases such as a Bose-Einstein condensate (BEC) cannot exist in a uniform two-dimensional system. However, below
a non-zero critical temperature a two-dimensional Bose fluid becomes superfluid,
despite absence of the long-range coherence normally associated with this phenomenon. Remarkably, no symmetry is broken, but a topological order emerges
which is sufficient to support superfluid behaviour; this Berezinskii-KosterlitzThouless (BKT) transition is driven by the suppression of vortices, topologicaldefects in the system phase.
Ultracold atomic gases offer an ideal system for addressing the low-temperature
properties of a Bose fluid. Coherence properties are conveniently revealed in theatomic momentum distribution, flexible trapping geometries can be engineeredusing light potentials, interactions between atoms are sufficiently weak to allow
a tractable theoretical description and crucially the interaction strength may betuned via molecular resonances. Whilst in a uniform system the BKT transitionis the sole low-temperature phenomenon, most experiments with ultracold atoms
employ a harmonic trapping potential, which modifies the density of states andrestores the possibility of conventional BEC in the ideal gas limit. Whether the
low-temperature transition is BEC- or BKT-like, at the critical point one anticipates a sharp growth in the extent of coherence.
In the first part of this thesis, we offer some theoretical discussion of the lowtemperature phase transitions in a two-dimensional Bose fluid, first in the context
of a uniform trapping potential and subsequently extending this treatment toharmonic confinement. In addition to overviewing pure-two-dimensional physics,
we explore to what extent residual excitations in the restricted dimension should
modify behaviour.
The second part concerns an experimental work, in which we engineer a harmonicallytrapped two-dimensional Bose gas with tuneable interactions, identify the onsetof extended coherence over a decade of interaction strength, and show that theinteraction-driven BKT transition smoothly converges onto the purely quantumstatistical BEC transition in the limit of vanishing interactions. Furthermore, wevalidate the predictions of a classical-field approach for the BKT critical point,
which models the non-perturbative behaviour of interacting bosons using a turv

vibulent matter wave.

Acknowledgements
Very few theses represent the isolated efforts of a lone author, and it is a great
pleasure to thank those who have helped, enabled and supported me over thepreceding four years and acknowledge their part in this work. Foremost amongst
these is Zoran, who has been an ever-present figure in my academic life since 2007.
It has been a long journey from his initial casual offer of a summer placement tothe completion of this thesis, and Zoran’s boundless enthusiasm, sage advice andgenerous encouragement have been a constant and unwavering feature in this. In
my opinion, one of the great features of the group is that members are judged solely
on their input and efforts; merit always wins over seniority, and an undergraduate
on a summer placement is taken as seriously as a wise elder, if they have something
useful to say. This attitude largely (but not solely) originates with Zoran, andmakes the lab a fresh, exciting and dynamic place to work.
Almost as consistent a figure has been Rob, who took Alex and me under his wing
upon our arrival in the group. From the beginning Rob treated us with admirable
patience and as academic equals (at least in Alex’s case entirely justifiably), which
retrospectively seems entirely ludicrous as I stumbled my way through the firstterm of work on superheating BECs. Since then I hope that my abilities havematured somewhat, but Rob’s kind, helpful and ridiculously proficient input havebeen a continual font of support and progress.
Alex and I started our PhDs together, and we enjoyed countless discussions gradually, but not usually irrevocably, convincing ourselves we understood nothing
about physics. His willingness to get to the heart of matters rather than accept
dogmatic statement in place of understanding was a breath of fresh air. He has
also been my (semi-voluntary) full-time consultant on all computing matters, andI hope he will forgive the endless teasing regarding his devotion to Microsoft,in which he remains steadfast despite ridicule, persecution and empirical evidence.
The previous year and a half has academically been dominated by work with Nir
and Martin. Nir and I took the first 2D steps together, installing the new lightand learning how to wield it. Working with him is fantastically good fun; theheady mixture of physics, Rocky clips, debate over food, wine, music and dinners,
stories of mountains and snow and obsessing over the coffee machine makes ‘being
at work’ something of a misnomer. He has been a strong influence on the evolutionof my attitudes to science, research and ‘living the good life’. Martin was my
vii

viii
teammate for the BKT experiment which constitutes this thesis, and which I
sincerely doubt would exist without him. His steadfast enthusiasm, experimental
talent and refusal to admit despair or frustration have been invaluable, and in
addition he is one of the kindest people one could hope to meet. As anotherpermanent feature of my PhD, I am also extremely grateful to our group secretary
Pam, who dealt admirably with all the lost invoices, confused emails and lastminute requests I could muster.
I have been fortunate to spend at least part of my time overlapping with various
group members who have passed on (to greater things), all of whom improved life
in some way. Naaman and Robbie were endlessly willing to help out a haplessundergraduate, and largely built the machine which allowed my own work tohappen. Stuart and Scott welcomed me into the ring team; a cynic would suggestthat a willing new graduate student was exactly what an experiment demanding
unreasonable levels of menial labour demanded, but I choose to adopt a morecharitable view and thank them for adopting me onto their project in my earlydays, essentially teaching me everything I know about the machine, and being
willing competitors in numerous games and challenges.
In addition to our lab, rumours swirled of another experiment across the narrowsea; ruled by Toby and Igor, it was a strange and exotic world of pull-up bars,
balance boards and well-organised, neatly labelled, well-stocked cupboards. I
pillaged more of their supplies than I care to admit, talked of life and science with
Toby over coffee of varying quality but boundless quantity, and marvelled at Igor’stales of cliffs, ravines and the high seas. Aziza was with us for only a year, but
her contribution is a cycle time a full minute faster than the old machine. The
new generation of students, Jay, Konrad and Christoph are already making theirmark. Jay had the pleasure(?) of joining the 2D experiment partway through andsingle-handedly set up the high-field imaging light, with little more than vague
assertions that ‘it’s probably doable’ to guide her. Konrad joined more recently
and immediately threw himself into paper discussions (even at version 20+, which
is an accolade few can claim), whilst installing our new Bragg setup. Christophinhabits the ‘new machine’, and given their rate of progress must be doing asterling job.
Despite what some might claim (new students look away now), life does persist
outside of the lab and mine was populated by people who made my graduateyears rich, absorbing and superbly good fun. There are too many to attempt tolist here, partially for fear of omission but principally for the inability to do people
justice. The life of formals, feasts, brunches, BA events, breakfast club, Portugal
Street parties, cycling adventures, ski trips and gents rowing was a good one, andI thank all those who lived it with me.
The only, and outstanding, exception to this blanket omission is Sun. She offers,
without conditions or demands, a seemingly inexhaustible source of compassion,
care and support, and has listened with astounding humour to more rants about
lasers, air-conditioning units and Matlab than any person has any right to expect.
She has been my partner on innumerable exploits, schemes and adventures, andix
I’m immensely fortunate to have her in my life.
Finally, and surpassing all others, I thank my family: my parents Nick and Mary,
my sister Liz, and my brother Will. It feels trivial and absurd to attempt toelucidate anything specific; their impact on my life cannot be quantified. A certainty is that without their unconditional love, belief and support my life would
be incomparably poorer, and this thesis is as much theirs as mine.

xContents
1 Introduction

1

2 Low-Temperature Phases in Two Dimensions
2.1 Low-Temperature Phases . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . .
2.1.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . .

13
14
14
15

3 The Uniform Two-Dimensional Bose Gas
3.1 BEC and Superfluidity in the Uniform Gas . . . . . . . . . . . . .
3.2 Interactions in Two Dimensions . . . . . . . . . . . . . . . . . . .
3.3 Bogoliubov Analysis of Excitations . . . . . . . . . . . . . . . . .
3.3.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Correlation Function . . . . . . . . . . . . . . . . . . . . .
3.3.3 Consistency of Bogoliubov Approach . . . . . . . . . . . .
3.4 Superfluid Fraction, and the BKT Transition . . . . . . . . . . . .
3.4.1 Superfluid Fraction . . . . . . . . . . . . . . . . . . . . . .
3.4.2 The BKT Transition . . . . . . . . . . . . . . . . . . . . .
3.5 Classical-Field Approach . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Overview of Classical-Field Method . . . . . . . . . . . . .
3.5.2 Critical Point . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . .

19
19
20
23
23
24
26
27
27
30
34
34
35
35

4 Two-Dimensional Physics in Trapped Gases
4.1 Effect of Residual Axial Freedom . . . . . . . . . . . . . . . . . .
4.1.1 Kinematically Two-Dimensional Superfluid . . . . . . . . .
4.1.2 Kinematically Three-Dimensional Superfluid . . . . . . . .
4.1.3 Condition for Two-Dimensional Behaviour . . . . . . . . .
4.2 Effect of Harmonic Trapping . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 The Interacting Gas . . . . . . . . . . . . . . . . . . . . .
4.2.3 Applicability of Uniform System Results . . . . . . . . . .
4.2.4 Comparison with Quantum Monte-Carlo . . . . . . . . . .
4.2.5 Unification of BEC and BKT . . . . . . . . . . . . . . . .

41
41
42
44
46
47
48
48
53
55
56

5 Experimental Overview
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57
57

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CONTENTS
5.2
5.3

Previous Experimental Work . . . . . . . . . . . . . . . . . . . . .
Identifying the Critical Point . . . . . . . . . . . . . . . . . . . . .

58
60

6 Experimental Setup and Characterisation
6.1 Optical Trap for Two-Dimensional Confinement . . . . . . . . . .
6.1.1 Axial Trapping . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Planar Trapping . . . . . . . . . . . . . . . . . . . . . . .
6.2 Tuning Interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Momentum Focusing . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Conceptual Outline . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Focusing our Two-Dimensional Bose Gas . . . . . . . . . .
6.3.3 Effect of a Finite Momentum-Space Resolution . . . . . . .
6.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 High-field State Structure . . . . . . . . . . . . . . . . . .
6.4.2 Rescaling Function . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Imaging Characterisation . . . . . . . . . . . . . . . . . . .

63
64
65
69
72
74
74
77
80
83
84
87
91

7 Mapping the BKT Phase Diagram
7.1 Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Loading of the Two-Dimensional Trap . . . . . . . . . . .
7.1.3 Scanning of Atom Number . . . . . . . . . . . . . . . . . .
7.1.4 Image Aquisition and Processing . . . . . . . . . . . . . .
7.2 Image Analysis and Parameter Extraction . . . . . . . . . . . . .
7.2.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Number . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3 Thermodynamic Variables . . . . . . . . . . . . . . . . . .
7.2.4 Critical Parameter Extraction . . . . . . . . . . . . . . . .
7.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Critical Number . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Critical Chemical Potential . . . . . . . . . . . . . . . . .
7.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93
93
93
94
96
97
98
99
99
99
104
107
107
109
110

8 Conclusions and Outlook

113

A Publications
Connecting Berezinskii-Kosterlitz-Thouless and BEC Phase Transitions
by Tuning Interactions in a Trapped Gas . . . . . . . . . . . . . .
Stability of a Unitary Bose Gas . . . . . . . . . . . . . . . . . . . . . .
A Superheated Bose-Condensed Gas . . . . . . . . . . . . . . . . . . .
Persistent Currents in Spinor Condensates . . . . . . . . . . . . . . . .

115
117
127
135
145

B Light Potentials
153
B.1 Light Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2 Sculpting Potentials via Fourier Optics . . . . . . . . . . . . . . . 154
C Trap Characterisation

157

CONTENTS

xiii

C.1 Axial Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.2 Planar Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.3 Focusing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
D Absorption Imaging and Calibration
D.1 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . .
D.2 Magnification of Imaging Systems . . . . . . . . . . . . . . . . . .
D.3 Rescaling Function . . . . . . . . . . . . . . . . . . . . . . . . . .
D.4 Imaging Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .

161
161
162
164
165

E Anti-Gravity Field

167

F Error Analysis
169
F.1 Statistical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
F.2 Systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

What is a scientist after all?
It is a curious man looking through a keyhole,
the keyhole of nature,trying to know what’s going on.
JACQUES YVES COUSTEAUChapter 1
Introduction
Quantum mechanics can have a somewhat undeserved reputation; it is the physics
of the microscopic, the esoteric, the unfamiliar. But regardless of their ability toshock and surprise, many-body interacting quantum systems constitute the universe we inhabit and consequently understanding their physics is of both fundamental and practical importance. Remarkably, whilst the richness of observed
quantum phenomena seems almost limitless - from superconductivity in MRI
magnets to quantum magnetism, stability of neutron stars to behaviour of thequark-gluon plasma - such apparent diversity arises from the interplay of very
few ingredients; quantum statistics, the form and strength of the inter-particle
interactions, temperature, gauge fields and any external potentials.
Many of these interesting behaviours arise out of the interactions between particles. In an ideal gas, every constituent member follows a single-particle trajectory,
and the behaviour of the whole cannot be more than the sum of its parts. Onceinteractions are introduced, the concept of collective behaviour becomes meaningful, whereby a macroscopic phenomenon such as superfluidity emerges fromthe complexity of the underlying microscopic physics. One of the great goals ofmodern physics is to elucidate how this emergence occurs.
There is however an inherent difficulty in both theoretical and computational approaches to this question. Theoretically, the high densities of real-world quantum
systems create great difficulties; interactions between particles are strong, of various and complicated forms, and are generally many-body. Computationally, it isextremely demanding to simulate many-particle quantum systems of even modest
size. Whereas the state-vector of an N -particle classical system has length 6N , aquantum system of N particles each with M possible states requires M N complex
numbers to specify its state completely, which rapidly becomes computationallyinfeasible1 .
1

As a concrete example, the state of 3 spin-1/2 particles is naturally written in the basis|ψi = c1 | ↑↑↑i + c2 | ↑↑↓i + c3 | ↑↓↑i... which could be denoted by a 8-dimensional vector cn . For
N particles, the length of cn would be 2N . Assuming that storage of a complex number involves∼ 10 bytes, the quantum state of N = 100 spins would consume 1019 TB.

1

2

CHAPTER 1. INTRODUCTION

Ultracold Gases as Quantum Simulators
A measure of the importance of quantum behaviour is provided by the phase-space√density D = nλ3th , where n is the particle number density and λth = h/ 2πmkB T
is the thermal wavelength associated with a particle of mass m at temperature
T . This approximately corresponds to the occupation of individual states, andhence quantum statistics become relevant for D & 1. In macroscopic, ‘real-world’
quantum systems such as liquid He or the electron sea in a metal, n ∼ 1029 m−3 ,and the onset of quantum behaviour occurs at relatively high characteristic temperatures of ∼ 10 K and ∼ 104 K respectively1 .
In ultracold atomic gases, one can access this same quantum limit of D  1,but at much lower densities, and commensurately lower temperatures of order
100 nK. This enables many-body quantum phenomena to be addressed at densities ∼ 1010 times lower than real-world systems; as will be outlined in the next
Section, this greatly simplifies the treatment of interactions, permitting tractable
theoretical progress. Furthermore, the control and tuneability afforded by thesesystems make them ideal ‘simulators’ of quantum physics [1]. Intricate sculptingof optical and magnetic fields create arbitrary confining potentials, interactions
may be tuned by means of molecular resonances [2], and the recent advent of artificial gauge potentials enables neutral atoms to effectively behave as particles ofarbitrary charge subjected to electromagnetic fields [3]. In the spirit of the quantum simulator envisaged by Feynman [4], the philosophy is to engineer custom
Hamiltonians and observe the subsequent behaviour in a clean, controllable andaddressable environment.
The exquisite control over geometry and interactions afforded by the techniquesof ultracold atomic gases has enabled researchers to address a wide variety of interesting quantum states. Bose-Einstein condensation (BEC) was first observedin 1995 [5], and requires the fewest ingredients, being driven purely by quantum statistics. Once the complications of geometry and interactions are included,
numerous phase transitions are possible, including the superfluid-to-Mott insulator [6], normal-to-superfluid [7–9], Anderson localisation [10], and the Dicke
transition [11].

Tuning Interactions
One of the most important tools in the ultracold atom arsenal is the ability to arbitrarily tune the strength and sign of interactions between atoms. The quantummechanical problem of scattering is in general rather involved. If two atoms
located at r1 and r2 interact via a potential V (r2 − r1 ), the natural coordinates
are centre-of-mass position R and relative position r. The overall wavefunction
straightforwardly takes the separable form Ψ(r, R) = exp(iK · R)ψ(r), where K
For a more extreme example, the number density inside a neutron star is n ∼ 1044 m−3 ,meaning that quantum behaviour is dominant for temperatures below 1012 K!
1

3

Figure 1.1: Two atoms colliding. In order to interact, their impact parameter b mustbe smaller than the range of interaction rVdW . This limits collisional angular momentum
to L . prVdW , where p is the typical atomic momentum.

is the centre-of-mass wavevector, and the relative wavefunction ψ is the solution
to the time-independent Schrödinger equation
"

#

−~2 2
∇ + V (r) ψ(r) = Eψ(r),
2µ r

(1.1)

where µ = m/2 is the reduced mass. In general, the potential V (r) may be rather
complex; however, the interatomic potentials via which ultracold gases interact
have two properties that afford a simplified approach.
1. The Van der Waals scattering potentials are short-ranged, falling faster thanr−1 [12].
2. Collisional energies are sufficiently low that the scattering wavefunction contains an extremely small contribution from l > 0 angular momentum states1 .
For two particles colliding with momentum p and impact parameter b, theirrelative angular momentum is L = pb; this is illustrated in Fig 1.1. For interaction range r0 , the relative momentum necessary to appreciably admix
states with l > 0 into the scattering wavefunction is
p&

~
.
r0

(1.2)

For typical interactions ranges2 of ∼ 100 a0 , where a0 is the Bohr radius,
this momentum scale corresponds to temperatures of T & 100 µK. This islarger than typical temperatures in ultracold gases by a factor ∼ 103 , andscattering is dominated by the s-wave channel l = 0.
1
2

Eigenstates of the angular momentum operator L̂2 labelled by quantum number l.
For 39 K, the Van der Waals radius rVdW = 64 a0 [13].

4

CHAPTER 1. INTRODUCTION

The scattering length If both simplifying assumptions are valid, far from thescattering centre at r = 0 the wavefunction ψ takes the form
ψ(r) ≈

eikz|{z}Incident
plane wave

+

f ikr
e
|r {z }.

(1.3)

Scatteredspherical wave

An unperturbed plane wave exp(ikz) contains an admixture of l-components,
with l = {0, 1, 2...}. In general, each component is modified by the presence ofa potential, resulting in an outgoing scattered wave with amplitude and phase
encoded in the scattering amplitude f (θ, φ). If scattering is s-wave then only
the l = 0 component is coupled into the scattered wave, which is consequently
spherically-symmetric. Furthermore, in the case of elastic scattering the only effectof a potential can be to impart a phase shift δ on the l = 0 component.
A phase shift is dimensionless; the incident plane wave provides an associatedwavevector k, and so the scattering potential must be characterised by a scattering
length a, defined as1(1.4)
lim k cot δ = − .
k→0a
The scattering amplitude is related to the phase-shift δ via1f=

1 iδe sin δ,k=−a.
1 + ika(1.5)

The scattering potential, which in general may be extremely complex, is thus
parametrised by a single scalar parameter a. Furthermore, any two potentials
characterised by the same value of a will result in completely identical behaviour,
regardless of how different their specific functional forms are2 . The ability to replace the complicated actual potential V (r) by a simplified model potential greatly
facilitates tractable theoretical descriptions of many-body, interacting quantum
systems, and is a tremendous advantage of ultracold atomic systems.

Feshbach resonances The scattering length a is thus simply set by the phaseshift δ imparted by a scattering potential. In ultracold gases one may tune δ via
a Feshbach resonance [2], whereby two free atoms in one spin state are brought
into resonance with a bound molecule in another spin state. Since different spinstates possess different magnetic moments, the relative energy of the two can betuned simply by application of a uniform magnetic field, and the scattering phase1

This is calculated within the theoretical framework of partial-wave decomposition, see forexample [14].
2
Typically one considers contact interactions V (r) = gδ(r), where the coupling constantg = 4π~2 a/m [12].

5Figure 1.2: A Feshbach resonance. On the left is the interatomic potential V (r) for
two-particle spin states |ai and |bi, which have energy difference ∆ε at zero magnetic
field. We assume the atoms are in state |ai; a magnetic field can be used to tune thedifference between the free-atom energy (dashed black line) and bound molecular state(dashed blue line). In the vicinity of the resonance a can take any value, illustrated on
the left for a resonance at 402.5 G in the |4 2 S1/2 , F = 1, mF = 1i state of 39 K.

shift set to an arbitrary value. In the vicinity of a resonance, the scattering length
varies with field strength B according to!

a(B) = a∞∆,1−
B − B0(1.6)

where a∞ is the scattering length far from the resonance, which is located at B0
and has width ∆. In our experiment we employ a resonance exhibited by 39 K
in the hyperfine state |4 2 S1/2 , F = 1, mF = 1i, for which these parameters are
a∞ = −29 a0 , B0 = 402.5 G and ∆ = 52 G [15, 16]. This B-dependence isillustrated in Fig. 1.2.
This flexibility allows all interaction regimes - from non-interacting, to attractive,
to repulsive - to be addressed in a highly controllable manner, within a single
experimental arena. Over the course of my PhD, working with 39 K in a varietyof trapping geometries I addressed interaction-driven physics in various regions ofFig. 1.2, indicated by circled numerals; an overview of these works [17–19] is givenin the next Section.

This Thesis
The principal topic of this thesis concerns the nature of the low-temperaturephases exhibited by a two-dimensional Bose system. Low-temperature phase transitions are commonly associated with the appearance of some order in a system.
When water freezes into ice, the previously random and chaotic arrangement ofmolecules is replaced by a regular configurational order. Likewise, more exotic
transitions display comparable behaviour; BEC is characterised by a global phasecoherence and magnetism results from a regular ordering of spins.

6

CHAPTER 1. INTRODUCTION

In a harmonically-trapped two-dimensional gas, tuning interactions can qualitatively change the nature of this transition. For an interacting gas, the emergenceof a topological order below some critical temperature gives rise to superfluidity,
mediated by the Berezinskii-Kosterlitz-Thouless (BKT) transition, whereas BEC
is precluded by the requisite critical density being infinite. On the other hand,
an ideal gas cannot exhibit superfluid dynamics, but can undergo BEC resulting
in a homogeneous phase. Using a harmonically-trapped two-dimensional Bosegas, we identify the low-temperature phase transition over a decade of interaction strengths, and show that the interaction-driven BKT transition smoothly
converges onto the purely quantum-statistical BEC transition in the limit of vanishing interactions; this corresponds to region Á of Fig. 1.2 [19].
In addition to this work, during my PhD we also addressed two extreme interaction
regimes in a three-dimensional gas, denoted by regions À and Â. The ability
to entirely switch off interactions grants access to the ideal gas limit in which
momentum states are decoupled from each other, prohibiting the redistributionof particles in response to changing global variables such as energy and number.
We were able to engineer a ‘superheated’ BEC which was decoupled from thesurrounding thermal reservoir, persisting at temperatures much higher than thecritical temperature [17].
In the opposite limit, turning attention to the strongest interactions allowed byquantum mechanics we accessed the unitary regime; rather than interactions
becoming ever more important, one in fact observes an emergence of universal
physics where the exact strength of interactions becomes irrelevant [18]. These
two works on tuneable interactions in a three-dimensional gas are briefly summarised below, before we turn our attention to two-dimensional physics for theremainder of this thesis.
All papers resulting from my PhD work, including a study of the effect of spinor
degrees of freedom on the stability of superfluid flow [20, 21], are presented forcompleteness in Appendix A.
Stability of a Unitary Bose Gas Richard J. Fletcher, Alexander L. Gaunt,
Nir Navon, Robert P. Smith, and Zoran Hadzibabic
The scattering length, which provides a lengthscale for all interaction-driven behaviour of a system, can be tuned to arbitrarily large values close to a Feshbachresonance. Rather than interactions becoming ever more important, as a diverges the scattering amplitude f → i/k. Quite remarkably, in the limit of thestrongest interactions allowed by quantum mechanics, one uncovers a situation
where no details whatsoever of the interaction potential appear in any systemdescription.
This is illustrated in Fig. 1.3, which shows the probability density |ψ|2 of Eq. (1.3)
around a scattering potential, along with the s-wave component |ψl=0 | for ka =0.2,
1 and 1000. As the scattering length increases, both the strength of scatteringand the phase shift δ rise commensurately. However, for ka & 1 the dependence

7

Figure 1.3: The relative-coordinate wavefunction ψ(r) in the vicinity of a scattering
potential. The density plots depict |ψ|2 in the plane y = 0, with the colour range
corresponding to modulations of ±20% compared to an unperturbed plane wave. Theline plots show the l = 0 component |ψl=0 | of both an unpertubed plane wave (dottedgreen line) and the wavefunction of Eq. (1.3) (solid blue line). The vertical red lines
mark the scattering length a. With increasing ka, initially the strength of wavefunction
modulation and phase shift both rise. As ka exceeds unity, the system remains frozen;
behaviour is ‘unitarity limited’.

on a vanishes, and ψ remains static, being set purely by the wavevector k. In
this unitary limit, for a thermal gas the diverging a is replaced by the naturallengthscale λth [22–25].
In a degenerate gas, the interparticle spacing n−1/3 provides the natural lengthscale at which the scattering length saturates [26–29]. This can also be viewed as
a competition of energy scales [22]; the characteristic energy of a thermal gas is
kB T ∼ ~2 /(mλ2th ), whereas for a degenerate gas it is interaction energy ~2 an/m.
These exceed the energy scale associated with the scattering length ~2 /(ma2 )when a & λth and a & n−1/3 respectively.
The irrelevance of interaction strength in this regime leaves λth or n−1/3 as the
only lengthscale that can appear in the system description, and behaviour is set
by simple parameters such as temperature or density; in this sense the physics
becomes universal. For example, in a degenerate unitary Bose gas one shouldobserve an effective ‘fermionization’, in which density sets all energy, time andlengthscales [26–29], and the equation of state becomes that of an ideal Fermi
gas [26–28, 30]
Over the past decade, there has been extensive theoretical and experimental work
concerning the unitary Fermi gas [31]. In the vicinity of a Feshbach resonance one
observes a crossover between a BCS superfluid of momentum-paired fermions, anda BEC of fermionic molecules. Despite extensive theoretical interest [22, 24–28,
32–41], experimental studies of the unitary Bose gas have been more limited [25,29, 30, 42–44]. This is principally due to an inherent instability of such systems;
whilst two-body collisions are necessarily elastic due to energy and momentum
conservation, three bosons can react to form a tightly-bound molecule and a free
atom. The energy released in the process leads to heating of the gas, and generally

CHAPTER 1. INTRODUCTION

1 0 -2 3( L 3 / L M3 ) - 1 / 4

L 3 (c m

66

s -1 )

8

1 0 -2 40

41

1 0 0 0 a 0 / a

2
1
0
-2

0

2

4

6

8

λ/ a
Figure 1.4: The three-body loss coefficient L3 , scaled to the theoretical unitarity-limited
4
upper bound LM
3 [25] (green line). Away from unitarity L3 displays the anticipated ascaling (red line), before saturating at ∼ 30% of the upper limit. Inset is a plot of L3 as
a function of a for two different temperatures; as unitarity is approached, losses exhibit
a temperature-dependent saturation.

ejects all participants from a trapped sample. The rate of recombination rises as
a4 [45], dramatically shortening attainable lifetimes as the enticing unitary regime
is approached.
In this study we measured the general scaling laws relating the particle-loss andheating rates to the temperature, scattering length, and atom number. The threebody loss coefficient is defined as L3 = −Ṅ /(hn2 i N ), where N is the total atom
number and h...i denotes an average over a trapped gas. In addition to confirming
the L3 ∝ a4 scaling for a  λth , in the regime a & λth we confirm for the
first time the expected saturation at a unitarity-limited upper bound L3 ∝ T −2 ,illustrated in Fig. 1.4. This is a dramatic manifestation of universal physics,
in which temperature rather than interaction strength controls a fundamentallyinteraction-driven process. Furthermore, species-specific Efimov physics [25] arefound to give a unitarity-limited L3 three times lower than the theoretical upperbound, making 39 K particularly promising for studies of unitary physics in the
Bose gas.
Additionally, we characterise the recombination heating of the gas in various interaction regimes, discerning the effect of molecular state binding energy andunitarity-enhanced losses for low-momentum atoms. Our work is one of the firstexperimental efforts to address the stability of the unitary Bose gas [25, 29].

9

Figure 1.5: A partially-condensed Bose gas, which can be viewed as two coupled subsystems; a BEC and thermal reservoir, with chemical potentials µ0 and µ0 respectively.
Kinetic equilibrium is mediated by Landau damping, whereby the population of collective
BEC excitations comes to thermal equilibrium with the surrounding thermal bath. However, phase equilibrium is achieved via elastic scattering, which redistributes particles
amongst states. Each subsystem also experiences some inelastic loss rate Γ.

A Superheated Bose-Condensed Gas Alexander L. Gaunt*, Richard J.
Fletcher*, Robert P. Smith, and Zoran Hadzibabic
Formulating a quantitative description of systems containing very many particles might at first seems rather daunting. However, statistical physics turns thiscomplexity to its advantage, concerning itself with the average behaviour of an
ensemble rather than particular constituents. The ergodic hypothesis posits thata system explores all available1 microscopic configurations equally; this effective
averaging over individual arrangements means that whilst the behaviour of a single particle within an ensemble might be beyond theoretical reach, the physics ofthe system as a whole follows simple, powerful relations.
The study of non-equilibrium processes is a major challenge, since they do not
obviously exhibit simple, universal governing laws. A familiar example of a nonequilibrium system is superheated water; in the absence of nucleation centres,
water may be heated to T > Tc = 100◦ C whilst remaining liquid. The water,
and the surrounding environment, can be viewed as two subsystems in mutual
kinetic equilibrium, both being characterised by the same temperature T > Tc .
However, the system is not in global phase equilibrium, since the phase transition
from water to steam is energetically inhibited.
Cold atomic gases are ideal for studies of non-equilibrium physics. The distribution of particles amongst energy states is easily accessible via the momentum
distribution, and interactions can be tuned by molecular resonances. In this1

Under the appropriate constraints, for example conservation of particle number or energy.

10

CHAPTER 1. INTRODUCTION

Figure 1.6: Temporal phase diagram. For each value of a we plot the equilibriumtime at which the BEC should vanish (green points) and the time at which the BEC
actually vanishes (red points). Solid curves are spline fits to the data. For a ≈ 0 the
BEC survives in the superheated regime for a whole minute. The inset shows the resultsof numerical calculation which simulates the gas as two partially-decoupled subsystems,
and reproduces our data (red points) extremely well.

experiment, a gas of 39 K in the vicinity of a Feshbach resonance was used todemonstrate dynamic stalling of a continuous phase transition. We initially prepare a partially-condensed gas, before rapidly reducing the interaction strength
to a low value. In the subsequent evolution, the energy-per-particle rises throughlight scattering, and atoms are lost via background gas collisions and three-bodyrecombination.
One can view the Bose gas as two subsystems; a BEC, and the surrounding thermal
reservoir, illustrated in Fig. 1.5. These are coupled in two ways, both dependent
on the scattering length a. First, the kinetic thermal equilibrium between thecollective excitations in the BEC (phonons)
and the thermal bath is ensured by√Landau damping, whose rate is ∝ a [12,46]. Second, elastic scattering mediates
the redistribution of particles amongst momentum states in response to changing
global parameters and drives global phase equilibrium, at a rate ∝ a2 [47]. The
different rate scalings enable one to engineer a superheated Bose-condensed gas,
which displays kinetic, but not phase, equilibrium. The BEC and thermal bathare both at the same temperature T > Tc , but the condensed fraction remains
non-zero.
The lack of phase equilibrium is manifested as different chemical potentials foreach component. Remarkably, these are observed to diverge with time, meaning

11
that we are not simply observing a relaxation towards equilibrium, but rather apersistent out-of-equilibrium state in which a coherent object is protected against
annihilation by the closure of its decay channel. In Fig. 1.6 we show a temporalphase diagram, indicating the time at which the BEC would vanish in an equilibrium system (green points), and the time at which the observed condensed fraction
reaches zero (red points). At our lowest interaction strengths we observe a BEC
persisting into the superheated regime for up to minute. In terms of temperature,
the weakest-interacting systems display a BEC up to temperatures T ≈ 1.5Tc ,equivalent to water remaining in a liquid state at 287◦ C.

12CHAPTER 1. INTRODUCTION

Chapter 2
Low-Temperature Phases in Two
Dimensions
It was known as early as 1934 [48] that crystallisation is precluded in two dimensions. The requisite long-range order (LRO) is destroyed by thermal fluctuations at any non-zero temperature; this is formalised as the Mermin-Wagner
theorem [49], and implies that ordered phases such as BEC cannot exist in a uniform two-dimensional system. However, below a non-zero critical temperature a
two-dimensional Bose fluid becomes superfluid, despite absence of the long-range
coherence normally associated with this phenomenon. Remarkably, no symmetryis broken, but a topological order emerges which is sufficient to support superfluid
behaviour; this BKT transition is driven by the suppression of vortices, topologicaldefects in the system phase [50–52].
Whilst in a uniform system this is the sole low-temperature transition, the modified density of states induced by a harmonic trapping potential restores the possibility of conventional BEC in the ideal gas limit. It is this interplay between
BKT and BEC physics which is considered in this thesis.
In the current Chapter we present a theoretical overview of two phenomena which
may be exhibited by a system of degenerate, interacting bosons: Bose-Einsteincondensation and superfluidity. This discussion is made without any speculationon their possibility, which is considered in the case of a uniform two-dimensional
system in Chapter 3. In contrast to experiments on liquid 4 He [53] and othercondensed matter systems [52, 54] which are typically uniform in-plane, mostexperiments with ultracold bosonic gases confine atoms in-plane using a harmonic potential [9, 55–68]. The relationship between BEC and superfluidity inharmonically-trapped two-dimensional gases is the subject of Chapter 4, where
we also consider the effect of residual freedom in the ‘frozen’ dimension and ask
what it means to be ‘two-dimensional’.
Having completed our discussion from a theoretical perspective, the second partof the thesis concerns an experimental work on a harmonically-trapped twodimensional ultracold Bose gas. By tuning the strength of interactions, we were13

14

CHAPTER 2. LOW-TEMPERATURE PHASES IN TWO DIMENSIONS

able to observe the unification of BKT and BEC transitions in the limit of vanishing interactions, and quantitatively compare the measured critical parameters
to theoretical predictions made over a decade ago. An overview of the experimentis given in Chapter 5, the experimental setup is presented in Chapter 6, and themethod and results are discussed in Chapter 7. Finally, in Chapter 8 we give someexperimental conclusions and offer an outlook on future research directions.

2.1

Low-Temperature Phases

Our general problem under consideration is the following: we imagine a uniform
system of bosons of mass m confined to a two-dimensional world with number
density n and temperature T , interacting via some repulsive contact potential
characterised by a dimensionless interaction strength g̃. Our question is, whatstate (if any) should emerge as the phase-space density D = nλ2th is increased
to values O(1)? This thesis is concerned with the interplay between BEC andsuperfluidity in two-dimensional systems, and in this Section we overview these
phenomena.

2.1.1

Bose-Einstein Condensation

The quantum statistics of bosons are described by the Bose-Einstein distribution,
which controls how bosons should occupy the energy levels of a system:f (ε) =1eβ(ε−µ) − 1

,

(2.1)

where β = (kB T )−1 and kB is the Boltzmann constant, µ is the chemical potential, ε is the energy of the state under consideration, and f (ε) the occupation
number.
Unlike their fermionic counterparts, multiple bosons may occupy a single energy
state which offers the possibility of BEC, whereby a macroscopic fraction O(1)of particles accumulate in a single eigenstate. Unlike many familiar phase transitions, BEC is not driven by the interactions between particles but is a naturalconsequence of their quantum statistics. Put quantitatively, the single-particle
density matrix exhibits an eigenvalue of order the total particle number N , with amacroscopically-occupied single-particle wavefunction given by the corresponding
eigenstate [69].
In a uniform system, the density matrix ρ̂ is diagonal in momentum-space; letting|ki denote a state of momentum ~k,ρ̂ =X

Nk |kihk|,

(2.2)

k

where Nk is the occupation number of state |ki. The existence of a BEC implies
that N0 ∼ N , which is necessarily accompanied by the appearance of long-range

2.1. LOW-TEMPERATURE PHASES

15

order. This is quantified by the first-order correlation function,DE

g (1) (r) = Ψ̂† (r) Ψ̂ (0) ,(2.3)

where the bosonic field operator Ψ̂† (r) creates a particle in the position state |ri.
Evaluating g (1) for the density matrix of Eq. (2.2) one finds1
g (1) (r) =

X Nk
k

=

L2

eik·r ,

N0 Z d2 k+n(k)eik·r ,
2
2
L
(2π)

(2.4)

where n(k) is the system momentum distribution2 , and the conversion of a sum to
an integral is valid assuming the system extent L → ∞. The condensate density
N0 /L2 is therefore revealed as the infinite-distance limit of g (1) , and the existence of a BEC implies a well-defined phase relation between infinitely-separated
points3 .

2.1.2

Superfluidity

The presence of phase
D Ecoherence generally engenders a non-zero ‘macroscopic
wavefunction’ ψ = Ψ̂ , where the average is over a region much larger thaninterparticle spacing but smaller than the lengthscale of any macroscopic flows
or potentials [70]. The case of ψ 6= 0 is intimately linked to the phenomenonof superfluidity, first observed in liquid 4 He [71, 72] in which it was posited to
be a consequence of BEC by London [73]. Whereas bulk flow of a classical fluid
gradually decays into elementary excitations, a superfluid flows without dissipation.
To understand the origins of this behaviour, one can consider the available dissipation channels.
1. Decay into excitations with linear momentum, which transfers energy andmomentum of the bulk flow into creation of quasi-particles4 . The Landaucritical velocity [74] relates the form of the quasi-particle dispersion relation
ε(k) to a critical velocity vc , below which excitations cannot be created
whilst conserving both momentum and energy,
!

ε (k)
.
vc = min
~k
1

(2.5)

D E
We make use of the relation Â = Tr(ρ̂Â) for some observable Â, and the position-space

representation of a momentum state hr|ki = L−1 exp(ik · r).
2
Defined as the number of particles in the momentum-space element d2 k/(2π)2 .
3The definition of the BEC density as n0 = lim g (1) (r) is known as the Penrose-Onsagerr→∞

criterion [69].
4
The label ‘quasi-particle’, refers to a collective excitation of the system with a definite energy
and momentum. In that sense an excitation can be treated as particle-like.

ϵ(k)CHAPTER 2. LOW-TEMPERATURE PHASES IN TWO DIMENSIONS

ϵ(k)

16

k

vc

k

Figure 2.1: Examples of dispersion relations corresponding to a normal fluid (red) anda superfluid (blue). Whereas the Landau critical velocity of Eq. (2.5) vanishes in thecase of a normal fluid, in a superfluid one obtains some vc > 0, given by the gradientof the black dashed line. Below this velocity the bulk motion cannot transfer energy into
an excitation whilst conserving energy and momentum.

Examples of the dispersion relations for a normal fluid, and superfluid areshown in Fig. 2.1.
2. Angular momentum of the bulk fluid can dissipate into elementary rotational
excitations (vortices). In the case of ψ 6= 0, vortices become quantised [70,75]. Fixing the density n to be constant, one may include dynamics in themacroscopic wavefunction by permitting modulation of its phase θ(r), giving√
ψ(r) = neiθ(r) .
(2.6)
The fluid velocity can be defined via the quantum-mechanical flux1 j =(~/(2mi)) [ψ ∗ ∇ψ − ψ∇ψ ∗ ], and making the identification j = nv yields thevelocity field~v (r) = ∇θ (r) ,(2.7)
ma simple looking equation with a profound consequence; the requirement forphase θ(r) to be single-valued forces quantisation of flow around any closed
loop,
I
~
,
ν ∈ Z.
(2.8)
v · dr = 2πνm
For the case of rotation around some point at radius r = 0, the azimuthal
velocity is then v(r) = ν~/(mr). The flow associated with a single vortex
has energy Evortex and angular momentum Jvortex given by
Evortex =Jvortex =
1

Z R
ξ

Z R

!

1
π~2
R
nmv 2 (r) d2 r = ν 2
n ln
,
2
m
ξ

(2.9)

nmv (r) rd2 r = ν~N,

(2.10)

ξ
2This is derived by demanding that the probability density ρ = |ψ| obeys the continuity
equation ∂ρ∂t = −∇ · j.

2.1. LOW-TEMPERATURE PHASES

17

where R is the system radius, N the total particle number and ξ the radius of
the central zero-density region demanded by the undefined phase θ(r = 0).
The quantisation displayed by Eqs. (2.9, 2.10) means that below some finite
critical velocity [70], vortex excitation is not possible whilst conserving both
energy and angular momentum.
In a classical fluid, very many single-particle states are occupied and there isno macroscopic wavefunction. Whilst individual single-particle states musthave quantised circulation, the bulk fluid can exhibit arbitrary vorticity
leading to dissipation at any flow velocity.
If neither decay channel outlined above is available, bulk motion cannot dissipate
and the system is superfluid.
A familiar example is provided in three dimensions, where the appearance of
BEC in an interacting system heralds the onset of superfluid behaviour [7, 8];
the existence of a macroscopic wavefunction results in quantised vortices, andinteractions suppress density fluctuations on long lengthscales such that the lowk excitations are phase-waves (phonons) with a linear dispersion relation. As
a consequence, below some non-zero velocity neither vortices nor phonons may
be excited, and the energy associated with macroscopic flow cannot decay intoelementary excitations.
It is worth mentioning that although an interacting BEC exhibits superfluid flow,in general one neither equates to nor even entails the other. The canonical example
is that a non-interacting BEC has a 100% condensed fraction, but does not display
superfluid flow which may be seen by application of Eq. (2.5) to a free-particle
dispersion relation. Conversely, liquid 4 He, whilst possessing a superfluid fractionof order unity as T → 0, has a condensed fraction (defined through the largest
density matrix eigenvalue) of ∼ 10% [69, 76]. As we shall see, the possibility
for superfluidity without condensation at all is dramatically demonstrated by theBose gas in two dimensions.

18

CHAPTER 2. LOW-TEMPERATURE PHASES IN TWO DIMENSIONSChapter 3
The Uniform Two-Dimensional
Bose Gas
Having outlined possible low-temperature phases for a collection of bosons in
Chapter 2, we now turn our attention to their feasibility in a uniform twodimensional Bose gas. The possibility of BEC and superfluidity is addressedin Section 3.1, which necessitates a description of the collective excitations; aBogoliubov-type approach is the topic of Section 3.3, which predicts the interacting two-dimensional Bose gas to support superfluid behaviour. The superfluid
critical point is identified in Section 3.4, where vortex excitations are included‘by-hand’, and the BKT transition is introduced as the mechanism behind thesuperfluid-normal transition. A more quantitative classical-field description ispresented in Section 3.5, which makes explicit predictions for the superfluid critical point and equation of state.

3.1

BEC and Superfluidity in the Uniform Gas

The original conception of BEC by Einstein was as a phase transition driven by
the saturation of thermal states [77], and is a natural consequence of bosonicquantum statistics. For a particular system, one can imagine gradually adding
particles (increasing µ) and ask whether a single state becomes macroscopicallyoccupied.
In a uniform system the eigenstates are momentum states |ki, with corresponding
energies ε0 (k) = ~2 k 2 /(2m) in the case of an ideal gas. The particle density nth
corresponding to the thermally-populated states |k > 0i is obtained by integrating
the Bose distribution over momentum states,
nth =

Z

i−1


d2 k h β(ε0 (k)−µ)
−2
βµ
e
−
1
=
−λ
ln
1
−
e
,
th
(2π)2

(3.1)

where |k = 0i is excluded by the integration measure d2 k. As the chemical
potential µ approaches its upper bound of ε0 (0) = 0, this density diverges. The
19

20

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

thermal states may therefore hold an arbitrarily large number of particles, andthe state |k = 0i is never macroscopically occupied for any finite phase-spacedensity. This result is reassuring: the presence of a BEC would imply long-ranged
coherence lim g (1) (r) = const., violating the Mermin-Wagner theorem [49].
r→∞

It is worth commenting on the effect of interactions on this argument. As weshall see in Section 3.2 they can be introduced at a mean-field (MF) level via an‘interaction potential’ Vint = 2gn, accomplished by the replacement µ → µ−2gnth
in Eq. (3.1). Our conclusion is unchanged; thermal density remains divergent
with increasing µ, which may now grow indefinitely rather than being restricted
to µ < 0.
Given the absence of BEC in the uniform two-dimensional system, one might
wonder whether this precludes superfluidity. Without phase coherence, how is
one to define superfluid flow v ∼ ∇θ? However, the absence of BEC merely
implies that lim g (1) (r) = 0. Whilst coherence over specifically infinite distancer→∞
is precluded, if phase coherence falls slowly enough this shall turn out to be
sufficient for superfluidity.
Even qualitatively, we can argue that BEC and superfluidity must be distinct
from each other. Condensation is driven by the quantum statistics of particles;interactions certainly modify a system’s behaviour [78–80], but are not intrinsic
to the transition. Superfluidity however is a collective, dynamic phenomenon;
for a macroscopic wavefunction to support interesting dynamics such as vortices
or superfluid flow, one requires a spatially-varying phase θ(r). This necessitates
dynamic admixing of multiple eigenstates, which is driven by interactions1 . Furthermore, the low-k dispersion relation of an ideal gas is quadratic in any numberof dimensions; a non-zero critical velocity arises out of interaction-induced deformation of this spectrum.
Any discussion of superfluidity therefore requires a formalism for interactions,which is the topic of Section 3.2. Once equipped with this, the coherence properties and quasi-particle excitations for the uniform Bose gas are considered withina Bogoliubov approach, which provides a framework from which to address superfluid behaviour.

3.2

Interactions in Two Dimensions

Whilst the form of interatomic interactions is generally complicated and systemdependant, the low temperatures and densities of ultracold gases afford an assumption of two-body contact interactions2 . The Hamiltonian is then given by
H=
1

Z "

#

g
−~2 2
∇ Ψ̂(r) + Ψ̂† (r)Ψ̂† (r)Ψ̂(r)Ψ̂(r) − µΨ̂† (r)Ψ̂(r) d2 r,
Ψ̂ (r)
2m
2
†

(3.2)

More specifically, the interaction energy must be greater than the typical spacing between
single-particle eigenenergies for an appreciable state admixture.
2
This was justified in the Introduction.

3.2. INTERACTIONS IN TWO DIMENSIONS

21

where g is a coupling constant setting the strength of interactions. The field
operator Ψ̂(r) is two-dimensional, and must have dimensions length −1 . ThereforeΨ̂† ∇2 Ψ̂ and Ψ̂† Ψ̂† Ψ̂Ψ̂ are dimensionally identical, and the coupling constant g
must on dimensional grounds be ∝ (~2 /m). We can thus parameterise interactions
by a dimensionless strength g̃, such that
g≡

~2
g̃.
m

(3.3)

For the purposes of future discussions, several salient points are worth making
here.
Scale invariance In contrast to its counterpart in three dimensions, the Hamiltonian of Eq. (3.2) is scale-invariant, being unchanged under the rescaling of spacer → Λr. This means that interactions cannot provide any absolute length (orenergy) scale; we already showed that interactions are parameterised by a dimensionless strength g̃, whereas in three dimensions the scattering length a sets anabsolute lengthscale1 [12]. The remaining energy scales are the chemical potentialµ and temperature T , and only their ratio rather than absolute values can bephysically meaningful.
Effect of bosonic exchange symmetry The average interaction energy density of a system with Hamiltonian (3.2) is given by (g/2) hn2 i. An ideal nondegenerate Bose gas displays Gaussian density fluctuations, such that hn2 i = γn2with γ = 2, whereas in the case of suppressed fluctuations γ = 1. The energy costVint of locally adding a particle is therefore given by∂ g 2
γn = γgn.
Vint =
∂n 2




(3.4)

Density fluctuations can be simply understood as arising from the demand ofbosonic exchange symmetry: defining a pairwise contact potential of strength A
as U (r1 , r2 ) = Aδ(r2 −r1 ), it is straightforward to evaluate the interaction energy of
two bosons in properly symmetrised wavefunctions. For a d-dimensional uniform
system of extent L, let two single-particle states be denoted ψ = L−d/2 exp(ik1 · r)and φ = L−d/2 exp(ik2 · r): in the degenerate case,Ψ(r1 , r2 ) = φ(r1 )φ(r2 )→hΨ|U |Ψi = A,

(3.5)

→

hΨ|U |Ψi = 2A.

(3.6)

whereas in the non-degenerate case,Ψ(r1 , r2 ) =φ(r1 )ψ(r2 ) + φ(r2 )ψ(r1 )
√
2

22

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

Figure 3.1: Particle density |Ψ(x1 , x2 )|2 for two particles in symmetrised degenerate(left) and non-degenerate (right) wavefunctions (see text). Axes are in units of system
extent L. The white dashed lines denote x1 = x2 .

This is illustrated in Fig. 3.1, where we plot the particle density |Ψ(x1 , x2 )|2 for
two particles in one dimension. The chosen eigenstates are φ = L−1/2 exp(i2πx/L)and ψ = L−1/2 exp(i4πx/L), which obey periodic boundary conditions. In the
non-degenerate case, particle density is enhanced along the line x1 = x2 .

Suppression of density fluctuations As the degeneracy of a system increases,
neglecting interactions one might expect γ to smoothly reduce from its fully fluctuating value of γ = 2 to the non-fluctuating value γ = 1. In fact, without BEC
this reduction is small: for simplicity, we imagine K states each containing anequal, and very large, number of bosons. If we randomly select two particles, theprobability of them being in the same state is 1/K. The expected value of γ isthen1hγi = 1 ×
K




K −1
1
+ 2×
=2− .
K
K




(3.7)

Thus without the presence of a BEC (K = 1), suppression of fluctuations is rather
small.
There is however an energetic advantage to reducing fluctuations: assuming fixed
average density n, minimising the interaction energy density (g/2)γn2 equates
to minimising γ. A careful population of the phase-density excitations [51] in a
two-dimensional gas finds this is significant for D  1.
1Specifically, in three dimensions the relevant coupling constant is given by g3D = (~2 /m)4πa.

3.3. BOGOLIUBOV ANALYSIS OF EXCITATIONS

3.3

23

Bogoliubov Analysis of Excitations

In order to establish whether a given system is superfluid, one must have someknowledge of the dispersion relation of excitations. In the case of a BEC, one
typically uses a Bogoliubov analysis [81] to obtain the collective quasi-particle excitations; one assumes a single mode to contain the majority of particles, and expands the Hamiltonian in a small parameter corresponding to the weakly-occupied
modes. In the case of a two-dimensional Bose gas there is generally no BEC at any
non-zero temperature, and no single mode dominates. However, if one assumes
that density fluctuations are small, these provide a convenient small parameterin which to expand the Hamiltonian, enabling a Bogoliubov-type analysis despite
the absence of BEC [51, 82].
Such a treatment was performed within a classical-field approach in [51]. At
T = 0 all particles√ occupy the lowest eigenstate, giving a macroscopic wavefunction ψ(r) = n. One includes spatial variation in phase and small density
perturbations,ψ(r) =qn (1 + 2η (r))eiθ(r) ,(3.8)

and substitutes into the Hamiltonian of Eq. (3.2). Replacement of the quantum
field operator with the classical-field Ψ̂ → ψ is justified if quantities of interest
involve only modes with energies ε(k)  kB T , the occupation numbers of which
are much larger than unity [83–85].
Application of the Gross-Pitaevskii equation (GPE) [86, 87] yields equations of
motion for η(r) and θ(r) which characterise the collective excitations supportedby the system.

3.3.1

Dispersion Relation

From the equations of motion for η(r) and θ(r) one obtains the quasi-particle spectrum [51], which sets the energy cost ε(k) to create an excitation with momentum
k:
v
!
u 2 2
2k2
u~ k
~
+ 2gn .
(3.9)
ε(k) = t2m
2m
This spectrum is illustrated in Fig. 3.2; one can understand it in two limits:q• A linear low-k regime ε(k) ≈ k ~2 gn/m of phonon excitations.
• A quadratic high-k regime ε(k) ≈ ~2 k 2 /(2m)+gn of free-particle excitations.
One might wonder why the free particles see a repulsive potential 1gn rather
than the 2gn expected due to Bose exchange symmetry; this is because theexcitation spectrum describes the energy cost associated with creating√aquasi-particle, and so is referenced to the interaction energy of ψ(r) = n,equal to 1gn.

24

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

ξ -1

ϵ(k)

Free-particle regime

vc

Phonon regimekFigure 3.2: The quasi-particle spectrum of a two-dimensional Bose gas with small
density fluctuations. At k  ξ −1 the spectrum is that of phonons, and is linear in k.
This gives a non-zero superfluid critical velocity vc according to the Landau criterionof Eq. (2.5). At higher k  ξ −1 the quasi-particles become free-particle-like, with aquadratic dispersion relation.

The effect of repulsive interactions is to modify the free-particle ε0 (k) ∝ k 2 , whichdevelops a linear regime at low-k. The crossover between the linear and quadratic
regimes is given by the healing length ξ,
s

ξ=

~2
=
nmg

s

1
.
g̃n

(3.10)

According to Eq. (2.5) we expect the low-T uniform
Bose gas to be superfluid,qwith critical velocity set by the phonon speed gn/m.
Composition of excitations The quasi-particles of Eq. (3.9) are collective
phase-density excitations, and it is interesting to consider their composition. Intuitively, the energy cost of modulating the density is finite even in the limit
k → 0, whereas the cost of phase fluctuations vanishes and phonons should consequently be dominated by phase modulation. Repulsive interactions thus suppress
density fluctuations on lengthscales > ξ [51].

3.3.2

Correlation Function√
The zero-temperature state ψ(r) = n displays true LRO, characterised byg (1) (r) = n. As the temperature T is increased from zero, quasi-particles will be3.3. BOGOLIUBOV ANALYSIS OF EXCITATIONS

25

thermally generated, and consequently decrease the coherence between separated
points. Since the long-distance correlations will be dominated by long-wavelength
phonons which are primarily phase-waves, a reasonable approximation is to neglect density fluctuations1 . The Hamiltonian of Eq. (3.2) reduces to the simpler
form,
~2 Z
ns |∇θ (r)|2 d2 r.
(3.11)
HXY =2m
The ‘by-hand’ substitution of the total density n for the superfluid density ns
accounts for the fact that density fluctuations are never totally suppressed, so weheuristically include them via the (unknown) quantity ns [51, 88]. This is consistent within the two-fluid model2 ; the effect of T > 0 is to generate Bogoliubov
quasi-particles which together constitute the normal component. Therefore theexistence of a normal component and the presence of density fluctuations botharise from the same physical source. Additionally, one can define the superfluid
density via a system’s resistance to twisting of its phase [89], and so it is plausible that the energy cost associated with ∇θ 6= 0 is associated with a superfluid
component.
The Hamiltonian HXY is the two-dimensional XY model for a lattice of spins si
with interactions ∝ si · sj . The correlation function for this model at low temperatures is known [90, 91], and gives the important result for the low-T uniform
Bose gas:g (1) (r) = ns

ξ
r

!

1
ns λ2
th

,

(3.12)

where the healing length ξ provides an upper momentum cutoff; modes withk > ξ −1 have been incorporated into n → ns . The lack of true coherence at any
T > 0 is a manifestation of the Mermin-Wagner theorem. As opposed to threedimensions, the density of states for phonon modes goes as k rather than k 2 . The
weaker suppression of k → 0 modes in the two-dimensional space is insufficient topreserve phase order at distances → ∞.
However, the extent of coherence is sufficient to support superfluidity. As notedin [92, 93], Josephson provides a direct relation between the correlation function
and superfluid density [94],n0 m,
2k→0 k G(k, 0)

ns = lim(3.13)

where G(k, z) is the single-particle Green’s function at imaginary frequency z. For
spatial correlation function r−η one has G(k, 0) ∼ k η−2 [92]. In a system of extent
L the condensate density n0 ∼ L−η and the lowest k ∼ 1/L. Taking the limit
L → ∞ Eq. (3.13) approaches a non-zero value, indicating a non-zero superfluid
density.
1
2

See [51] for a more complete discussion of the justification for this assumption.
See Section 3.4.1.

26

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

3.3.3

Consistency of Bogoliubov Approach

At the start of the present Section, we noted that the Bogoliubov analysis of thetwo-dimensional Bose gas relies upon density fluctuations being small. Additionally, the classical-field of Eq. (3.8) has a phase which is assumed to vary in awell-defined way. However, an ideal Bose gas would display large density fluctuations hn2 i = 2n2 , and we have shown above that at any T > 0 the two-dimensional
Bose gas is not coherent. Each of these points is addressed here.
• The presence of repulsive interactions suppresses density fluctuations onlengthscales > ξ. If total density fluctuations are significantly reduced,
expansion around a state with fixed density remains reasonable. A simple
estimate of this regime comes from demanding that only phonon modes(principally phase fluctuations) are significantly populated1 . This condition
kB T  gn yields D  (2π/g̃). More carefully, populating the spectrum of
Eq. (3.9) and calculating the resulting density fluctuations finds that D  1
is in fact sufficient [51].
At finite temperature, we can associate the residual density fluctuations withthe presence of a normal component [51], heuristically replacing n → ns and√expanding around a state ψ(r) = ns eiθ(r) . This assumes that only the superfluid component supports Bogoliubov quasi-particles; unlike the normal
fluid, the energy cost associated with a phase gradient is non-zero [89].
• The correlation function of Eq. (3.12) decays algebraically with distance,
vanishing as r → ∞. This is consistent with the absence of BEC; we saw
in Eq. (2.4) that a macroscopic occupation of |k = 0i would give a finite
limit [69].
Whilst over sufficiently large separations the phase order is destroyed, one
can always define a local phase, and coherence decays slowly enough togive a non-zero macroscopic wavefunction. The algebraic form of Eq. (3.12)
means there is no lengthscale associated with the decay of correlations; one
therefore cannot define a low-k validity cutoff in the derived collective excitations.
These results provide a framework for introducing excitations to the two-dimensional
Bose gas. The question we are concerned with is how the presence of these excitations should modify superfluid properties, and is the topic of the next Section.
1

This approach simply indicates the typical D at which density fluctuations are suppressed.
One might worry about the validity of the classical-field approximation if occupations of phononmodes become small.

3.4. SUPERFLUID FRACTION, AND THE BKT TRANSITION

3.4

27

Superfluid Fraction, and the BKT Transition

Previously, we showed that the two-dimensional Bose gas should be superfluid at
T = 0. In this Section, we imagine slowly increasing the temperature and ask how
the superfluid fraction should change under the thermal generation of Bogoliubov
quasi-particles. However, regardless of its internal consistency the Bogoliubov
approach is inherently perturbative, only accounting for smooth deformations of
ψ. The inclusion of topological excitations is addressed in Section 3.4.2, where we
introduce the topological BKT transition.

3.4.1

Superfluid Fraction

At finite temperature, quasi-particles will be thermally generated according to
the Bose distribution, and will not themselves exhibit superfluid behaviour. This
problem is conveniently treated within a ‘two-fluid model’, with two constituent
parts [12]:1. A superfluid component of density ns , which exhibits dissipationless flow√
and is described by the macroscopic wavefunction ψ(r) = ns eiθ(r) [70].
2. A normal component of density nn , consisting of a thermal gas of quasiparticles. Since these are in general not free particles, this density is definedaccording to the momentum density j of a quasi-particle gas moving withvelocity v.
j = nn mv(3.14)

We introduce the effect of finite temperature by thermally populating the dispersion relation of Eq. (3.9), and integrating to find the total momentum of theresultant excitation gas. Consider a quasi-particle of momentum p in the superfluid frame, which has an energy cost of ε(p). If the normal component has avelocity v, then the energy cost in the normal frame is ε(p) − p · v; this is depicted
in Fig. 3.3.
We can assume that quasi-particles interact sufficiently that the normal component is in thermal equilibrium; quasi-particles are then populated according to
the Bose distribution,
1,(3.15)
f (p) = β(ε(p)−p·v)e−1
and the net momentum density of the quasi-particle gas is
j=

Z

d2 p
pf (p).
(2π~)2(3.16)

28

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

Figure 3.3: The two-fluid model, consisting of a superfluid component of density Ds ,and collective excitations which constitute a normal fluid of density Dn . If the superfluid is in motion, there is some relative velocity v between the superfluid and normalcomponents; for convenience of calculation we work in the frame of the superfluid. Theenergy cost to create a quasi-particle of momentum p is ε(p) in the superfluid frame,
whereas in the normal frame it is ε(p) − p · v.

From Eq. (3.14) we obtain the normal density
nn = j · v/(mv 2 ),=Z

1
d2 p p · v.
22β(ε(p)−p·v)(2π~) v e
−1

(3.17)

If the normal fluid velocity is much smaller than ε(p)/p for any populated excitation1 , the Bose distribution in Eq. (3.17) can be written as an expansion about
the stationary case:
nn =

=

Z

d2 p (p · v)2 ∂f0,(2π~)2 v 2∂εZ

d2 p p2 ∂f0,(2π~)2 2 ∂ε

(3.18)

where f0 is the Bose distribution of Eq. (3.15) evaluated at v = 0, and we havemade use of the average √absolute projection of any two-dimensional vector p alongsome direction being p/ 2.
We can now insert the Bogoliubov excitation spectrum of Eq. (3.9) with the substitution n → ns . Transforming variables into the dimensionless normal density
Dn = nn λ2th and superfluid density Ds = ns λ2th , we obtain the integral equation√Z ∞ue u(u+g̃Ds /π)Dn (Ds ) =du  √(3.19)
2 ,0
u(u+g̃Ds /π)e−11Equivalently, the velocity difference between superfluid and normal components should bemuch less than the critical velocity of Eq. (2.5).

3.4. SUPERFLUID FRACTION, AND THE BKT TRANSITION29
˜
log10 g

1.0

0
0.8
-1
-2

0.4

-3

ℱ

0.6

-4
0.2
-5
0.0
0.00

0.05

0.10

0.15

0.20

0.25

-6

-1

Figure 3.4: The decay in superfluid fraction F with decreasing phase-space density
D. Curves are generated by thermal population of Bogoliubov quasi-particles, which
form a normal component (see text). Progressive shades of blue indicate the strength ofinteractions g̃.

which assumes that the quasi-particle spectrum depends solely upon Ds ; thisshould be valid for high superfluid fractions.
In experiments, the quantity most easily controlled and measured is the total
density D. As this is varied, one can use Eq. (3.19) to obtain an implicit equation
for the superfluid density Ds ,D = Dn (Ds ) + Ds .

(3.20)

This can be solved numerically, and Fig. 3.4 plots the variation of superfluid
fraction F = Ds /D with total density D for various interaction strengths g̃. As one
would intuitively expect, the superfluid fraction falls from unity as D decreases.
Larger values of g̃ make the superfluid more resilient to thermal depletion, due tothe stronger deformation of the low-k limit of ε(k) from quadratic to linear. By
contrast, the ideal gas superfluid fraction would be zero for any T > 0.
At this stage, one might naively feel that the picture is complete. Starting from thesuperfluid T = 0 limit, with increasing T the normal density smoothly increases,
until the superfluid fraction has fallen to zero. However, a flaw is revealed byconsidering the Hamiltonian HXY of Eq. (3.11). At low temperatures, the correlation function g (1) (r) takes the algebraic form of Eq. (3.12). At high-T however,correlations decay exponentially g (1) (r) ∼ exp(−r/ζ) where ζ is some temperature dependent lengthscale [95]. This can be simply understood in a lattice modelby noting that at very high temperatures, adjacent lattice sites are very weakly
correlated. Therefore correlations should fall by some fixed fraction per lattice
site, resulting in exponential decay.

30

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GASWhat is troubling is that there is no smooth deformation which maps an algebraic
function onto an exponential, and it is unclear how g (1) (r) should evolve with increasing T ; as noted by Berezinskii this is suggestive of a phase transition [90].
Furthermore, an algebraic function possesses no lengthscale; this is again suggestive of some phase transition, at which the correlation length diverges andg (1) switches from exponential to algebraic. This reasoning is correct; a phasetransition does occur, but is not predicted within the perturbative Bogoliubov
approach.

3.4.2

The BKT Transition

The Bogoliubov calculation of the superfluid fraction does not predict any phasetransition. However, such a perturbative approach can only capture smooth deformations of the order parameter; the topology of ψ(r) is unchanged by theaddition of Bogoliubov quasiparticles, which deform it in a continuous manner.
By construction, this approach cannot account for topological excitations suchas vortices, singularities in ψ(r) around which θ(r) winds by ±2π, which involvea discrete change in the structure of ψ(r). The global phase winding associatedwith a vortex is very destructive to coherence, and the presence of vortices leads
to a breakdown of the slow, algebraic decay of Eq. (3.12).
The favourability of thermally nucleating a vortex depends on the resultant change
in the system free energy ∆F = Evortex − T Svortex where Evortex is the kinetic
energy of a vortex velocity field and Svortex the entropy associated with the vortex position. From Eq. (2.9) we obtain the energy of a single-charged vortex as
Evortex = (1/2)Ds kB T ln(R/ξ) where R is the system radius, and ξ the healing
length of Eq. (3.10). This provides the length scale over which |ψ(r)| can varyfrom its bulk value to zero at the vortex centre, and gives the size of a vortex core.
The vortex entropy Svortex depends on the number of ways one can place a single
vortex into the gas, giving Svortex = kB ln(R2 /ξ 2 ). Consequently,!

R
1.
∆F = (Ds − 4) kB T ln
2ξ

(3.21)

One therefore recovers a critical superfluid density, below which ∆F becomes negative and the proliferation of free vortices becomes energetically favourable,
DsBKT = 4.

(3.22)For a superfluid with Ds > 4, whilst the formation of single vortices is energeticallyunfavourable, vortices may exist as bound pairs of opposite circulation. There isno net phase winding around a pair, hence whilst its associated entropy diverges
with system size, its energy does not [51,52]. Vortex pairs have ∆F < 0 at all nonzero temperatures; they have no long-range effect on the phase, but will reduce
the superfluid density compared with a phonon-only model.
Once vortices are included, a modified superfluid evolution emerges as T is increased from zero. The superfluid density Ds initially falls smoothly due to risingSuperfluid phaseg(1)(r)Normal phaseg(1)(r)3.4. SUPERFLUID FRACTION, AND THE BKT TRANSITION

31

∼r-η

∼exp(-r/ζ )Figure 3.5: The BKT transition. For superfluid densities Ds > 4 vortices can only
exist in bound pairs (black ovals), which have no long-range effect on coherence. The
correlation function g (1) (r) decays algebraically, driven by the phase distortions associated with collective excitations (red lines). As temperature is increased, the superfluid
fraction progressively falls until the BKT criterion Eq. (3.22) is reached. The proliferation of free vortices becomes energetically favourable, leading to destruction of thesuperfluid and exponentially-decaying correlations.

temperature and generation of quasi-particle excitations. Free vortices are energetically suppressed, and vortices can only exist in bound pairs of opposite
circulation which do not affect coherence over large distances. In accordance with
the Mermin-Wagner theorem true LRO is absent, but the correlation functiong (1) (r) falls to zero only algebraically, giving a ‘quasi-LRO’ sufficient to support
superfluidity [92]. At the critical point Ds = DsBKT , a proliferation of free vortices destroys the superfluid and leads to a universal jump in superfluid density∆Ds = 4. The long-range phase distortions of unbound vortices give rise to an
exponentially-decaying correlation function [95]. This defect-driven loss of superfluidity is known as the Berezinskii-Kosterliz-Thouless (BKT) transition [50, 90],
and is illustrated in Fig. 3.5.
It is interesting to consider the behaviour of vortex pairs just before the BKTtransition is reached. The energy of a pair of oppositely charged vortices separated
by a distance l is [52],l
Ep (l) = kB T Ds ln .
ξ

(3.23)

Noting that a pair of separation l is ∼ l-fold degenerate, we find the average pair32

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

separation hli as
R ∞ 2−Dξ

hli = R ∞ξ=l s dl,l1−Ds dlDs − 2ξ.
Ds − 3(3.24)For large superfluid densities Ds  1 vortex pairs are tightly bound, with size oforder the healing length. As the density falls, the pair size starts to diverge. Even
this simple argument would predict the pair size to diverge at Ds = 3; in reality
the presence of other pairs reduces the energy cost [51] of Eq. (3.23), making
Eq. (3.24) a lower bound and the transition occurs at Ds = 4.
Critical density Although Eq. (3.22) gives a universal value for the critical superfluid density, it does not set the critical total density DBKT . This will depend
upon system-specific microscopic physics, which controls the superfluid fraction
F(D). Experimentally, in ultracold gases D is more straightforwardly measured
and controlled than Ds , and knowledge of the critical total density DBKT is desirable.
Quantitatively predicting DBKT requires an accurate description of the non-perturbative
behaviour of interacting bosons in the fluctuation region near the critical point.
Classical-field simulations [96] model this behaviour by a turbulent matter-wave
field [97]; this approach is introduced in Section 3.5. As a toy model, we canuse our Bogoliubov model to make an estimate by fixing Ds in Eq. (3.20), andevaluating DBKT = Dn (4) + 4. This can be evaluated for various value of g̃, andthe results are plotted in Fig. 3.6.
We empirically find a logarithmic dependence of DBKT upon g̃, illustrated by theexcellent linear fit (black line) of DBKT (g̃) = 4.9 − 0.99 ln g̃. For comparison, theclassical-field result [96] is DBKT (g̃) = 5.9 − ln g̃ (dashed red line); the two are
in excellent agreement, with a small constant offset. It is unsurprising that our‘phonon-only’ model of Eq. (3.19) over-estimates the superfluid density, since itneglects the reduction in superfluid fraction associated with vortex-pairs.
Graphically, in Fig. 3.7 we plot the smooth, quasi-particle driven decay of superfluid density identically to Fig 3.4, in addition to the BKT criterion (black dashedline) of Eq. (3.22). The BKT critical point is identified as where this is crossed,
causing a g̃-dependent jump in superfluid fraction F.

3.4. SUPERFLUID FRACTION, AND THE BKT TRANSITION

33

20

BKT

15

10

5

0

-10

-8

-6
˜
ln(g)

-4

0

-2

Figure 3.6: BKT critical density from the ‘phonon-only’ model of Section 3.3 (seetext). Black points denote numerical values for the critical density at specific interaction
strengths, and the black line is an empirical linear fit. For comparison, the red dashedline denotes the results of a classical-field prediction [96] introduced in Section 3.5.

˜
log10 g

1.0

0
0.8
-1
-2

0.4

-3

ℱ

0.6

-4
0.2
-5
0.0
0.00

0.05

0.10

0.15

0.20

0.25

-6

-1

Figure 3.7: The decay in superfluid fraction F with decreasing phase-space density D,
including the BKT transition point of Eq. (3.22). When the black dashed line is reached,
the proliferation of free vortices becomes energetically favourable leading to destructionof the superfluid.

34

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS

3.5

Classical-Field Approach

Although the arguments of Section 3.3 provide an intuitive picture of the superfluid critical point, there are several approximations that one may have trouble
with. Perhaps foremost, the dispersion relation of Eq. (3.9)√was derived for T = 0
and based on small perturbations about the state ψ(r) = n; it is therefore unclear how far it should support a T > 0 description. Second, we have entirely
neglected the influence of bound vortex pairs in the superfluid state. Third, ourapproach does not automatically predict the BKT transition, and says nothing
regarding the normal side of the transition or the equation of state.
A more careful approach was made in [96, 98], where a classical-field method wasused to address the low-T properties of the two-dimensional Bose gas. In this Section we give an overview of their method, and present some salient results.

3.5.1

Overview of Classical-Field Method

In a homogeneous ideal gas, eigenstates are momentum states |ki, whose dispersion relation is free-particle-like. Interactions couple different momenta [12], andmodify this dispersion relation1 . If this coupling is weak, then the problem can betreated perturbatively within MF theory; the energy of each |ki is simply shiftedby a MF potential 2gn.
This approach is not sufficient for a two-dimensional gas close to the BKT criticalpoint. It does not predict any transition, does not contain any suppression offluctuations and cannot include superfluid dynamics or vortices. These phenomena arise out of strong coupling between momentum states; the system can nolonger be described via small corrections to the ideal gas picture and the problem
becomes non-perturbative [96, 98, 99]. This regime is reached when interactions
become of order the energy of a particular |ki; therefore non-perturbative effects
are confined to low-k states2 [99].
The authors of [96,98] employ a classical-field approach, describing the system bya classical-field ψ:
H=

Z "

g
~2
|∇ψ|2 + |ψ|4 − µ|ψ|2 d2 r.
2m2#(3.25)

By evolving this field under the GPE until steady state is observed, equilibrium
properties of the gas are obtained. The classical-field method accounts for the nonperturbative dynamics of all strongly occupied low-k modes of ψ [83, 85, 100]; thedescription should fail for low-occupation higher-k modes, but these are weakly
coupled and can be described within an ideal gas picture [96, 100].
1

If the system remains translationally-invarient, k should remain a good quantum number.
We already saw a similar phenomenon in the Bogoliubov approach of Section 3.3; stateswith k > 1/ξ simply had their energies shifted, whereas those with k < 1/ξ were coupled into
phonon modes with a linear rather than quadratic dispersion relation.
2

3.5. CLASSICAL-FIELD APPROACH

3.5.2

35

Critical Point

The BKT transition was identified by the requirement of Ds = 4, giving a critical
total density DBKT and chemical potential µBKT :
D

BKT

380
= ln
g̃

!

!

,

BKT

µ

g̃
13.2
= kB T ln.
πg̃(3.26)

Several points deserve to be made about these expressions:• The critical phase-space density is reasonably close to the value obtainedby our simple Bogoliubov model of Section 3.3; the comparison between thetwo is illustrated in Fig. 3.6.
• In the limit of vanishing interactions g̃ → 0, DBKT diverges. This makes
it manifest that the BKT transition is fundamentally interaction driven;in an ideal gas, superfluid dynamics are prohibited and the healing length
diverges, making vortices conceptually impossible.
• The critical chemical potential is an interaction-dependent rescaling of thetemperature. This is a consequence of scale-invariance1 . The only relevantenergy scales are µ and kB T , therefore only their ratio µ̃ can be physicallymeaningful,µ.
(3.27)µ̃ =kB T
The divergence of DBKT as g̃ → 0 illustrates the breakdown of the BKT picturein the ideal gas limit. According to Eq. (3.1), in an ideal gas D diverges as µ → 0.
Therefore the g̃ → 0 limits of DBKT and µBKT are mutually consistent, and it isinteresting that the BKT critical point, derived from considering the superfluid
transition of an interacting gas, approaches the critical point for ideal gas BEC.
It is this connection which is the topic of the experimental work presented in
Chapters 5 - 7.

3.5.3

Equation of StateIn addition to locating the BKT critical point, classical-field simulations also
provide the equation of state for the two-dimensional Bose gas [98]. Naively thiswould take the form D(µ, T, g̃), but two factors simplify matters.
• The scale-invarience of the two-dimensional Bose gas Hamiltonian means
D(µ, T, g̃) reduces to D(µ̃, g̃).
• By referencing phase-space density and chemical potential to their criticalvalues, which absorb all interaction-dependence, in the vicinity of the BKT
1See Section 3.2.

36

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS
point the equation of state takes the universal form ∆D = (D − DBKT ) =2πλ(X), where λ(X) is a universal function of the dimensionless parameter,X=µ̃ − µ̃BKT.
g̃

(3.28)

The function λ(X) is tabulated in [98] and provides the corresponding D for
any set (µ, T, g̃). It is illuminating to consider an intuitive limit for X  0,where one expects a low density gas exhibiting full bosonic density fluctuations.
A MF approach is therefore reasonable, introducing interactions perturbatively
via a potential 2gn. Inclusion in our semiclassical equation of state Eq. (3.1)
yieldsD = − ln 1 − eµ̃−Dg̃/π .
(3.29)
Changing to the variables X and ∆D, we obtain


13.2

1



∆D = − ln 1 − eg̃[X− π (∆D+ln 380 )] − ln

380
.
g̃

(3.30)One immediately notices that that Eq. (3.30) does not exhibit a universal form.
It does however approach a universal form close to the transition, which can beused to quantify where a universal description becomes valid within a MF picture.
This requires the exponent in Eq. (3.30) to become small, specifically
13.2
1∆D + lng̃ X −
π
380






 1.

(3.31)In this limit, our MF equation of state reduces to∆D = − ln −380 X −13.2
1∆D + ln
π
380




,

(3.32)

where all dependence on g̃ is absorbed into the parameter X.
In Fig. 3.8, we plot the universal, classical-field equation of state ∆D = 2πλ(X)along with the MF approximation of Eq. (3.32). This result is important, andseveral points are worth making:• The intuitive MF approach (green dashed line) gives an excellent descriptionof the classical-field results (red line) for X  0. As X = 0 is approached
the classical-field curve rises above the MF; this is a consequence of thegradual suppression of fluctuations, which reduces the interaction potentialfrom its fully fluctuating value of 2gn. This enhances the density at a givenµ̃ compared with a MF approach. Empirically, this suppression is significant
within a fluctuation region −1 . X < 0 denoted by the shading. A fully
fluctuating, MF description fails whenµ̃BKT − µ̃ . g̃,(3.33)

which could have been predicted on purely dimensional grounds [96]. The
fluctuation region marks the onset of non-perturbative physics in the low-k
states, and the failure of MF theory [99].

3.5. CLASSICAL-FIELD APPROACH

37

10Classical-field
Fully fluctuating MF

Δ

5

0

-5

-4

-2

0
X

2

4

Figure 3.8: The universal equation of state for a two-dimensional Bose gas, showing
the numerical classical-field results of [98] and a MF approximation of Eq. (3.32). The
blue shading demarcates the ‘fluctuation region’ in the vicinity of the critical point, where
suppression of density fluctuations becomes significant and a MF approach fails.

• There is no discontinuity in either D or its derivative at the BKT point
X = 0. This is a consequence of the ‘infinite-order’ of the BKT transition; no symmetry is broken, unlike second-order transitions such as BEC
in three dimensions where the system arbitrarily chooses a global phase. All
derivatives of the free energy are continuous across the transition.
• For X  0 one recovers a linear growth in D, characteristic of a ThomasFermi regime where the chemical potential is set by interaction energy alone.
It is also enlightening to consider the equation of state at fixed g̃, for comparison
with approximations lacking a universal form. In Fig. 3.9 we show the equationof state D(X), for g̃ = 0.05. The plot shows the classical-field equation of state(red line) and the MF model for a fully fluctuating Bose gas of Eq. (3.30) (green
dashed line). In addition, the equivalent MF curve for the case of fully suppressed
density fluctuations (green dotted line) is shown, obtained by replacing 2gn → 1gn
in Eq. (3.29). There are several points of interest to be made:• As X increases into the fluctuation region, the classical-field result rises awayfrom the fully fluctuating MF curve, towards the non-fluctuating MF curve.
The non-fluctuating model provides an upper bound for the equation ofstate; to rise above would require density anti-bunching (γ < 1 in Eq. (3.4)).
• For large X  0 the classical-field result approaches the non-fluctuating
MF result.

38

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GAS
35Classical-field
30
25

Fully fluctuating MF
Non-fluctuating MF



20
15
10
5
0
-4

0
X

-2

2

4

Figure 3.9: The equation of state for a two-dimensional Bose gas with g̃ = 0.05 (redline). For comparison, also plotted are MF curves for a gas with full fluctuations (green
dashed line) and fully suppressed fluctuations (green dotted line).

As Figs. 3.8 and 3.9 illustrate, it is the gradual suppression of density fluctuations
as the BKT transition is approached that makes an analytic form for the equationof state difficult. This is in sharp contrast to three dimensions, where fluctuation
suppression is usually negligible in a normal gas, and complete in a BEC. In thiscase, suppression is driven by degeneracy. In the two-dimensional case however,
it is primarily the energetic advantage to reducing density fluctuations that drives
fluctuation suppression in an interacting gas.
In terms of the interaction potential γgn of Eq. (3.4), the case γ = 2 corresponds toa fully fluctuating Bose gas, and γ = 1 to fully suppressed fluctuations. One might
be tempted to consider fractional 1 < γ < 2 to heuristically represent the partial
suppression of fluctuations. However, we recall that the interaction potential isdefined as the cost in interaction energy associated with adding a particle (seeEq. (3.4)). For partial suppression of fluctuations, one can write hn2 i = γ(n)n2 ,where γ is now some function of n. This gives an interaction potential∂ gγ(n)n2 ,∂n 2= gγ(n)n +g ∂γ 2n.
2 ∂nVint =(3.34)

Therefore one cannot in this case introduce an interaction potential linear inn; there is an additional quadratic term, with a generally complicated prefactor∂γ/∂n which lacks a simple analytic form. Only in the two extremes of fullfluctuations and fully suppressed fluctuations does the second term disappear,
permitting a simple Vint ∝ n.

3.5. CLASSICAL-FIELD APPROACH

39

35
Superfluid density, s
30

Quasi-condensate density, qc

25



20
15
10
5
0
-4

Universal jump Δs = 4
-2

0
X

2

4

Figure 3.10: The variation in superfluid density (blue line) and quasi-condensate
density (purple line) across the BKT transition.

Finally, the classical-field results of [98] additionally provide the superfluid densityqDs , and the ‘quasi-condensate’ density Dqc . The latter is defined as Dqc ≡λ2th 2 hni2 − hn2 i. In the limit of vanishing fluctuations, one simply obtains
Dqc = D, whereas for a fully fluctuating Bose gas Dqc = 0. The quasi-condensate
density therefore provides a measure of the extent to which fluctuations have been
suppressed. These quantities are plotted in Fig. 3.10. First, the jump in Ds isrecovered within the classical-field approach. This jump is universal, independentof interactions strength or particle specifics. Second, the quasi-condensate density
takes a universal form across the transition. In particular, at the BKT transition
point X = 0, one obtains Dqc = 7.16 [98]. For experimentally realistic values ofg̃ ∈ [0.01, 1], the critical density DBKT ≈ 6 → 10, and so already at the transition
Dqc is a significant fraction1 of D.
This now completes our discussion of the infinite, uniform Bose gas in two dimensions. Having identified and characterised the BKT superfluid-normal transition,and introduced the uniform-system equation of state, we now turn our attention
to more experimentally realistic systems. In Chapter 4 we consider how to apply
understanding of a pure-two-dimensional uniform gas to a quasi-two-dimensional,
harmonically-trapped system.

1

The quasi-condensate density exceeding the total density at the critical point suggests an
inconsistency with the classical-field results [51, 101]. This occurs for DBKT < 7.16, or g̃ > 0.3.

40

CHAPTER 3. THE UNIFORM TWO-DIMENSIONAL BOSE GASChapter 4
Two-Dimensional Physics inTrapped Gases
The discussion of Chapter 3 focused upon a pure-two-dimensional system of infinite, uniform extent. Whilst this model is theoretically accessible, experimentswith cold gases differ in two crucial ways. First, the cold-atom researcher unavoidably inhabits a three-dimensional world, and no system can ever be perfectly twodimensional. Second, in most experimental studies using ultracold atomic gases,
the planar trapping potential is harmonic rather than uniform. These considerations are the topic of this Chapter; in Section 4.1 we ask how the presence of higher
axial states should modify pure-two-dimensional behaviour, and in Section 4.2 weconsider both how to apply uniform system results to the case of harmonic trapping, and how the interplay between BEC and BKT might be modified.

4.1

Effect of Residual Axial Freedom

In real-world experiments on two-dimensional systems, axial confinement is provided by some trapping potential, with a discrete spectrum of eigenenergies. For
specificity we consider a harmonic trapping potential of frequency ωz ; for an ideal
gas this potential has energy levels Ej = j~ωz 1 and eigenstates |ji. However,interactions modify both the energies and eigenstates, which are denoted E j and|ji.
In the limit of ωz becoming very large2 the trap provides truly two-dimensional
confinement, with no excitation or dynamics possible in the axial direction. However, in most ultracold atomic systems this spacing is comparable in magnitude
to temperature and interaction energy.
One can identify two distinct limits of superfluid behaviour, depending upon thetrap spacing ωz :1
2

We neglect zero point energy (1/2)~ωz to simplify the notation.
This will be quantified later in the Section.

41

42

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES• A kinematically two-dimensional gas. In-plane, excitations are collective Bogoliubov modes, labelled by a two-dimensional wavevector k = (kx , ky ). Axially, excitations take the form of thermal promotion to higher axial states,
which are well-spaced compared to other energy scales.
• A kinematically three-dimensional gas. In this case, all possible excitations
are collective Bogoliubov modes, labelled by a three-dimensional wavevectork = (kx , ky , kz ).

The crossover between these regimes is set by comparing interaction energy withaxial level spacing. Dynamics or collective excitations in the axial direction require
admixture of multiple axial states; conversely, occupation of a single state ensuresthat the superfluid phase cannot vary axially, and the superfluid order parameteris two-dimensional. It is the superfluid interaction energy gns which admixes
axial trap levels; one therefore expects a crossover between kinematically twoand three-dimensional behaviour whengns ∼ ~ωz .

(4.1)

This can also be simply understood as when the superfluid
healing length ξ =q√
1/ g̃ns becomes comparable with the axial lengthscale ~/(mωz ).
In this Section, we consider each of these limits, and ask how the ‘pure-twodimensional’ BKT picture should be modified.

4.1.1

Kinematically Two-Dimensional Superfluid

Taking our inspiration from Section 3.3 we start at T = 0, where atoms form asuperfluid in the axial ground state |j = 0i. We now imagine switching on T > 0,
and ask how the system behaviour should differ from a pure-two-dimensionalgas.
In Section 3.3 we calculated the spectrum for quasi-particles, collective excitationsof the superfluid which are thermally populated by increasing temperature. This
resultant gas of excitations defined the normal component (Eq. (3.14), and thesuperfluid fraction steadily reduces as T rises. With the presence of higher axial
levels, we now have two channels by which the superfluid fraction falls.
1. Thermal generation of quasi-particles, identically to the pure-two-dimensionalcase. These are collective excitations of the superfluid, labelled by a twodimensional wavevector k. We assume that the presence of atoms in higher
axial states |j > 0i does not modify the dispersion relation of Eq. (3.9).
Therefore the normal density Dn (Ds ) defined in Eq. (3.19) should remain
valid.
2. Thermal promotion of atoms to higher axial levels |j > 0i. Making theassumption that the density in these higher levels is small, their dispersion

4.1. EFFECT OF RESIDUAL AXIAL FREEDOM

43

Figure 4.1: A cartoon depicting a slab of superfluid (blue cuboid), which occupies theaxial ground state |j = 0i. The effect of finite temperature is to thermally generate bothcollective quasi-particles in the superfluid (red lines), and promote atoms to higher axial
states |j > 0i (red points). Each of these channels reduces the superfluid density.

relation relative to the superfluid should be free-particle-like,

εj>0 (k) = Ej − E0 +~2 k 2,2m(4.2)

where k is the planar momentum.
We employ Eq. (3.19) to evaluate the normal density associated with theseatoms. Substituting the dispersion relation Eq. (4.2), and making use of
integration by parts we find that

Dn =

=

XZ ∞
j>0 0

XZ ∞
j>0 0

=−

X

du 

du 



ueu+β (Ej −E0 )2 ,eu+β (Ej −E0 ) − 11,eu+β (Ej −E0 ) − 1
ln 1 − e−β (Ej −E0 ) .

(4.3)

j>0

This rather obvious result states that the normal density associated withatoms in excited axial states is equivalent to their density, assuming a freeparticle dispersion relation and that µ ≈ E0 .
These two channels via which the superfluid fraction falls are illustrated in Fig. 4.1.
To calculate the superfluid fraction at fixed total density, we calculate and subtract
the normal densities associated with each channel. Generalising Eq. (3.20) to the

44

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES

case of a quasi-two dimensional system gives an implicit equation for Ds :
D0

+

|{z}

Density in axial
ground state

Dj>0| {z }

Density in axial
excited states

=

Ds
|{z}

Superfluid
density+

Dn (Ds )|{z}

Normal density due totwo-dimensional collective
excitations of superfluid

+

Dj>0

.

| {z }

Normal density
due to thermal
population of axial
excited states(4.4)
It trivially follows that the axial excited state population appears identically on
both sides. Subtracting, one uncovers a modified implicit equation,
D0 = Ds + Dn (Ds ) .

(4.5)

This looks identical to the superfluid density equation for a pure-two-dimensional
system, Eq. (3.20), with the simple substitution D → D0 . Importantly, the function Dn (Ds ) accounts for two-dimensional collective excitations of the superfluid,
and so is unchanged by the presence of atoms in excited levels. The total twodimensional superfluid density Ds is determined solely by the axial ground state
density D0 , which is therefore the relevant density that determines when the BKTtransition is crossed. One thus expects the axial ground state to support puretwo-dimensional physics, with the only caveat being that the superfluid should bekinematically two-dimensional.
In terms of comparing experiment with the classical-field theoretical expectations of [96, 98], we note that the classical-field ψ represents the heavily-occupied,
low-k modes of a pure-two-dimensional system. Since ψ is two-dimensional, its
dynamics cannot account for axial excitations. In the case of a kinematically twodimensional system, there are no axial collective excitations available; ψ is thena valid representation of the axial ground state. Excitation to higher axial statesis free-particle-like; these atoms are well-described by MF theory [99] and can be
included/removed in theories ‘by-hand’. The critical parameters and equation ofstate provided by [96, 98] therefore correspond to the axial ground state.
This argument that pure-two-dimensional BKT physics should be observed in
the axial ground state is supported by comparison with the results of Quantum
Monte-Carlo (QMC) simulation [99, 102] of quasi-two-dimensional gases. Thesesimulations are for the more experimentally common case of harmonic in-plane
trapping; BKT physics in a harmonic trap is the topic of Section 4.2, and the QMC
results are presented in Section 4.2.4. The assumption that the axial ground state
supports pure-two-dimensional physics, in accordance with the arguments of thisSection, is found to give superb agreement with the BKT transition point obtained
from QMC.

4.1.2

Kinematically Three-Dimensional Superfluid

One can imagine making the superfluid progressively more three-dimensional, bydecreasing the axial level spacing ~ωz to values comparable to the superfluid4.1. EFFECT OF RESIDUAL AXIAL FREEDOM

45

Figure 4.2: A ‘slab’ of superfluid (blue cuboid) of axial extent L. Vortices (red andgreen circles) can exist as topological defects, whose energy is minimised by aligning
their cores normal to the superfluid plane. The arrowheads indicate the vortex charge.
We imagine increasing the axial thickness L, which increases the admixture of higher
axial states into the superfluid wavefunction and results in the onset of axial dynamics(see text).

interaction energy gns . The resultant admixture of higher axial levels into the superfluid wavefunction will progressively permit superfluid dynamics and collective
excitations in the axial direction.
We can immediately note that the classical-field predictions for a pure-two-dimensional
system should not be valid, since the field ψ does not incorporate axial dynamics.
We can no longer circumvent this problem by separating the system into an axial
ground state supporting two-dimensional excitations, and a set of free-particle
excitations in higher axial states. Therefore we can no longer apply pure-twodimensional physics, unlike in the kinematically two-dimensional case.
However, it is still interesting to ask how the BKT picture is modified in thisregime. For a simple toy model, we consider a uniform ‘slab’ of superfluid, with
infinite extent in-plane and axial thickness L, illustrated in Fig. 4.2. Reducing
ωz is equivalent to increasing L. In reality, the superfluid axial profile is not a
top-hat; we do not expect this to qualitatively alter our conclusions.
The quantities of interest are the two-dimensional density1 D2D = n2D λ2th , threedimensional density D3D = n3D λ3th , and superfluid fraction F. One may triviallyrelate these densities by
L 3D
D .
(4.6)
D2D =
λth
Assuming that vortex cores remain axially-aligned, the energetic arguments for the
proliferation of vortices in Section 3.4.2 should still apply to the two-dimensional
superfluid density, and one expects the universal jump of Eq. (3.22) to remain
valid. However, we are interested in how the jump in superfluid fraction varies
with increasing thickness L.
1

In all other Sections, densities are assumed to refer to the two-dimensional density. Here we
explicitly include the superscript ‘2D’ since discussion also concerns three-dimensional densities.

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES
1.0

1.0

L/λth

0.8

0.8

10

0.6

0.6

8

0.4

6

0.2

0.2

4

0.0
0.00 0.05 0.10 0.15 0.20 0.25

0.0
0.0

ℱ

ℱ

46

0.4

(2 D)-1

0.1

0.2

0.3

0.4

0.5

2

(3 D)-1

Figure 4.3: The decay in superfluid fraction F as a function of two-dimensional density D2D (left) and three-dimensional density D3D (right), for a kinematically threedimensional gas with (a/λth ) = 0.1. The black dashed line illustrates where the BKT
criterion of Eq. (3.22) is satisfied. Whereas the righthand curve has no L-dependence,
the lefthand curve does, illustrated by progressive shadings. The coloured dashed lines
on the righthand plot indicate the BKT criterion of Eq. (3.22) expressed in terms of
three-dimensional density.

For a kinematically three-dimensional gas, the superfluid fraction F is a functionof D3D ; it is derived entirely analogously to Eq. (3.19), with the result√
Z ∞u(u+4Ds3D a/λth )
3/2


u
e
4
3D
3D
du  √
Dn Ds = √
(4.7)
2 .
3 π 0
u(u+4Ds3D a/λth )−1eThis function F(D3D ) has no L-dependence; we can make use of the mappingof Eq. (4.6), to infer the variation with two-dimensional density, F(D2D ) which
is now dependent on L. The resulting sketches of superfluid fraction are illustrated in Fig. 4.3. In this regime, as (L/λth ) is increased, the magnitude of
the jump in superfluid fraction ∆F falls. As a system enters the kinematicallythree-dimensional regime, one should observe the smooth disappearance of any
discontinuity in superfluid fraction as the transition is crossed.

4.1.3

Condition for Two-Dimensional Behaviour

Based on the arguments above, the possibility to observe pure-two-dimensional
BKT physics in a quasi-two-dimensional system depends upon two factors:
1. The superfluid must be kinematically two-dimensional. According to Eq. (4.1),
this demands
gns  ~ωz
(4.8)
to be satisfied at the critical point; substituting the critical value of ns =4/λ2th , we obtain
2g̃kB T ~ωz .
(4.9)
π

4.2. EFFECT OF HARMONIC TRAPPING47q

In terms of axial extent L = ~/(mωz ), this becomes (L/λth )  2g̃ −1/2
which is of order unity for typical values of g̃ ∈ [0.01, 1].
2. The occupation of excited states should be small, such that atoms in higher
axial states do not influence the behaviour of axial ground state atoms. ThisrequireskB T . ~ωz .
(4.10)For example, assuming Ej = j~ωz , at a temperature kB T = ~ωz the axial
ground state contains 72% of atoms1 . In a uniform system this condition
equates to (L/λth ) . (2π)−1/2 .
It is unsurprising that L/λth  1 sets the condition for two-dimensional behaviour; the typical coherence length should be larger than the gas thickness,such that the phase on one axial face is strongly correlated with the opposite face,resulting in a two-dimensional order parameter ψ(r).
Summary of uniform system results All previous discussion has focusedon the case of an infinite uniform in-plane potential. In this geometry, we havedemonstrated that conventional, saturation-driven BEC is impossible. Instead,an interacting gas will undergo a normal-superfluid transition via the BKT mechanism. The critical phase-space density for this transition2 diverges in the ideal
gas limit, g̃ → 0, and therefore the ideal gas should not undergo a phase transition
for any finite D.
In the following Section 4.2, we shall apply the uniform system results of Chapter 3
to the case of a harmonic planar trapping potential. Not only is this situation
more relevant to experiments with ultracold atoms, but also qualitatively changes
the nature of the ideal gas limit.

4.2

Effect of Harmonic Trapping

Harmonic potentials are the most natural trapping geometry if atoms are confinedwith optical or magnetic fields, being the leading-order power close to a potential
minimum. Depending upon the question addressed this can either be a help or hindrance; inhomogeneity means that different regions do not cross phase transitions
coincidentally, complicating the study of critical phenomena or transition dynamics. On the other hand, equations of state can be obtained ‘single-shot’, since thespatially varying potential effectively scans the local chemical potential [1, 103].
In this Section, we ask how the presence of a harmonic potential changes the lowtemperature phases available to a two-dimensional Bose gas. The case of an ideal
gas is addressed in Section 4.2.1, and an interacting gas in Section 4.2.2. Finally,
1

This is evaluated for an ideal gas with µ = 0, such that the atomic fraction in the axial∞
Pground state = g2 (1)/g2 (e−j ), where g2 (z) is the polylogarithm of order two.
j=0
2

See Eq. (3.26).

48

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES

in Section 4.2.5 we identify a crossover from interacting BKT physics to ideal gas
BEC physics as interactions are taken to zero.

4.2.1

The Ideal Gas

In Section 3.1 we imagined gradually increasing the chemical potential of a uniform
ideal two-dimensional gas, and showed that the density D of thermally-populated
states diverges as µ → 0. This means physically that if one were to continually
increase D, no single state would ever become macroscopically occupied.
In a harmonic trap, the relevant global quantity is the atom number N ; to ascertain whether BEC is possible, we ask how the number of thermal particles should
scale with the chemical potential µ. Within the LDA one can straightforwardly
generalise the thermal density of Eq. (3.1) to a harmonic trap with in-plane potential V (r), via the substitution µ → µ − V (r). This results in

D (r) = − ln 1 − eβ(µ−V (r)) .

(4.11)

Assuming a harmonic potential V (r) = (1/2)mωr r2 integration is straightforward,
yielding,
!2


kB T
g2 eβµ ,(4.12)
N=~ωrwhere g2 (z) =∞P

z n /n2 is the polylogarithm function of order two. Setting µ = 0

n=0

gives an upper bound to the number of thermal atoms,
π
Nc0 =

2

6

kB T
~ωr

!2

.

(4.13)The presence of the harmonic trapping potential therefore restores the possibility
for conventional BEC, driven by the saturation of thermal states. The transition
occurs when the central phase-space density D(r = 0) reaches infinity, just as ina uniform system. However, the presence of a trap permits the infinite uniform
gas critical density to be locally attained at the centre, whilst requiring a finite
total number of atoms.

4.2.2

The Interacting Gas

BEC In a three-dimensional gas, ideal-gas BEC occurs at a critical atom numberof Nc0 ≈ 1.202(kB T /(~ω))3 , and a central phase-space density D3D ≈ 2.612. If
weak repulsive interactions are switched on, both of these numbers are slightly
shifted [79, 80]. To lowest order in a/λth , the critical atom number rises sincemore atoms are required to attain the critical central density [79]. However,interactions do not alter the BEC-like nature of the transition, which remains
statistically driven by the saturation of excited states [78].

4.2. EFFECT OF HARMONIC TRAPPING49

In a two-dimensional gas, even qualitatively a problem rears its head. The divergence of the critical (µ = 0) density profile of Eq. (4.11) at the trap centre is
acceptable for an ideal gas, but non-zero interactions would entail locally infinite
energy density. More quantitatively, one may introduce interactions at a MF levelby the substitution µ → µ − 2gn in Eq. (4.11), yielding an implicit equation forD,
D (r) = − ln 1 − eβ(µ−V (r))−g̃D/π .
(4.14)The total atom number N is found by integration of D over the trap; this caneither be done numerically, or by noting that integration over a two-dimensional
area is equivalent to linear integration over the trap potential energy [104], sinceboth vary as r2 . We introduce an effective potentialg̃1[D (r) − D (0)] ,u (r) = mωr2 r2 +2
βπ(4.15)

which for simplicity of subsequent integration limits we have defined such that itvanishes at r = 0. In terms of u(r), the equation of state Eq. (4.14) becomes

D (u) = − ln 1 − eβ(∆−u) ,(4.16)

where the modified chemical potential ∆ = µ − g̃D(0)/(βπ) is referenced to the
MF interaction potential at the trap centre.
The total atom number may now be obtained by converting the spatial integral
into an integral over the effective potential u(r),
!

π Z∞
∂r2
π Z ∞  2d r D (r) = 2
duD (u) ,
N= 2
λth r=0λth u=0∂u!

g̃ ∂D
2π Z ∞
du 1 −
D (u) ,=22mωr λth u=0
βπ ∂u
=

kB T
~ωr

!2 





g2 eβ∆ +g̃
D (0)2 .
2π
(4.17)

Conventional, saturation-driven BEC should occur in the limit ∆ → 0, whichcorresponds to the chemical potential reaching the lowest trap energy level. In theideal gas case g̃ = 0, Eq. (4.17) reassuringly recovers Nc0 of Eq. (4.13). However,for any finite interactions g̃ > 0 we note that the second term of Eq. (4.17)
diverges with the central density. Therefore N grows indefinitely with increasingµ, the thermal states can accommodate any number of particles and BEC neveroccurs.
Another viewpoint is that the density of a critical ideal gas reaches infinity, but
over an infinitesimal area resulting in a finite total atom number. However, in aninteracting gas, the central curvature of the effective potential smoothly vanisheswith increasing µ [92]. The gas becomes progressively ‘more uniform’ at the trap
centre, leading to an infinite critical atom number and suppressing BEC. This

50

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES
˜
g
1.0

8

0.8

6


0.6
4
0.4
2

0

0.2

-4

-2

0
Radius (Rth)

2

4

0

Figure 4.4: The radial density profiles for a harmonically trapped two-dimensional
Bose gas of various interactions strengths. The black dashed profile is an ideal gas withµ̃ = −10−4 . The blue lines are profiles for interacting gases, with µ̃ chosen so that thecentralpdensity coincides with the ideal gas. The radius is in units of thermal radius
Rth = kB T /(mωr2 ), where ωr is the planar harmonic trap frequency.

is illustrated in Fig. 4.4, where we plot density profiles for various values of g̃
obtained by numerical solution of Eq. (4.14). All central phase-space densities
are fixed to that of an ideal gas close to BEC, with µ̃ = βµ = −10−4 . As theinteraction strength is increased, the density profile broadens and more atoms are
required to reach a given central density.
BKT Having seen that interactions suppress the occurrence of BEC, we now
turn our attention to the possibility of a superfluid transition. Intuitively, in aharmonically trapped gas we anticipate a local BKT transition when the central
density reaches DBKT ; within the LDA, the trap centre should locally behave asa uniform gas.
The critical number for BKT to occur in a harmonic trap, NcBKT , is therefore
the trap population such that the central density reaches DBKT . An intuitive
approach would be to use our MF relation Eq. (4.17) which relates the central
phase-space density D(0)
 and total atom number N . For large central densitiesβ∆β|∆|  1, and g2 e
≈ g2 (1) = π 2 /6. This yields the MF result for the BKT
critical number [92, 104],
!

3g̃
380
NcBKT
= 1 + 3 ln2
.
0
Nc
π
g̃

(4.18)In the ideal gas limit g̃ = 0 one recovers the ideal gas critical number for BEC,
a consequence of the central density for both BEC and BKT being infinite. This

4.2. EFFECT OF HARMONIC TRAPPING

51

approach assumes that the gas displays full bosonic density fluctuations, implicit
in our inclusion of a MF interaction potential 2gn. One might worry about thisfor two reasons:
1. As the classical-field results of Section 3.5 indicate [96, 98], the suppressionof density fluctuations is already significant at the BKT critical point. Thisenhances the density at a given chemical potential µ̃, which permits DBKT
to be reached at the trap centre with fewer total atoms than a MF approach
would suggest.
2. An inconsistency is already manifest between the MF equation of state
Eq. (4.14) and classical-field critical point Eq. (3.26). If one demands a
critical phase-space density DBKT according to classical-field predictions,
MF theory predicts a critical chemical potential,µ̃MF =

g̃DBKT
BKT+ ln 1 − e−D
,
π

=



g̃ 380
BKT
ln+ ln 1 − e−D,πg̃≈ µ̃BKT + g̃



380
1
1
ln
−
,
π 13.2 380= µ̃BKT + 1.07g̃.

(4.19)

Thus a MF approach overestimates the chemical potential predicted byclassical-field theory for any non-zero interactions, again suggesting thatthe critical number of Eq. (4.18) should be an overestimate.
An attempt to improve upon the MF prediction of Eq. (4.18) was made in [99]. A
hybrid model is adopted, which assumes that most of the trapped cloud is welldescribed by MF theory; from Eq. (3.33) we see that MF theory fails when thelocal chemical potential satisfiesµBKT − [µ − V (r)] < g̃kB T.

(4.20)

When µ = µBKT the trapqcentre is critical. This condition then defines a fluctua√tion radius Rfluct = 2g̃ kB T /(mωr2 ), inside of which MF theory fails.
Within the fluctuation radius, the authors introduce a local correlation density,
defined as the difference between the classical-field density of [98], and MF theory.
The total density profile can then be constructed by summing a MF profile withchemical potential µ = µBKT , and the correlation density within the fluctuation
radius.
This hybrid profile is illustrated in Fig. 4.5, where we plot the density profiles atg̃ = 0.2 corresponding to a MF critical profile neglecting suppression of fluctuations (green dashed line), a MF profile with chemical potential µBKT (blue dashedline) and the correlation density (red line).

52

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES
8
BKT
6

MF profile at μBKT+Correlation densityPure-MF critical profile
42
MF profile at μBKT
0
-2.0

-1.5

-1.0

-0.5

0.0
0.5
Radius (Rth)

1.0

1.5

2.0

Figure 4.5: The critical radial density profile at g̃ = 0.2 for a fully fluctuating twodimensional gas (green dashed line) and a model which takes into account suppression offluctuations near the critical point; this hybrid profile is constructed from a MF profileat µ = µBKT (blue dashed line) and a correlation density (red line) which accounts forthe discrepancy between the MF and classical-field equations of state. Both profiles havea peak density DBKT (black dashed line) at r = 0 where the BKT transition occurs. The
small discrepancy between the hybrid profile and critical density is due to non-exactnessof the heuristic analytic function for the correlation density (see text, and [99]).

The MF critical number of Eq. (4.18) corresponds to the area underneath thegreen dashed curve, which overestimates the atom number required. To obtainthe critical atom number accounting for suppression of fluctuations, we sum thecontributions from a MF profile at µ = µBKT , and the correlation density. The former can be easily obtained from Eq. (4.17), by setting µ = µBKT . The correlation
density is well-described by a heuristic analytic function of chemical potential [99],which is then integrated over the central region r < Rfluct . The result is,
3g̃ 2 g̃
6g̃
g̃
NcBKT
=
1
+
ln
+
ln
+
3
2
Nc0
π
16
16π
16
|
{z
}








90g̃
2
|16π
{z }

Contribution from MF

Contribution fromprofile at µ = µBKT

correlation density.

(4.21)

This result is plotted in Fig. 4.6, along with the MF result of Eq. (4.18). The
discrepancy between the two is dramatic, and the curves only converge for exponentially small g̃.
The significant difference for small values of g̃, is a consequence of the slow divergence of DBKT as g̃ → 0 and the universal value for the quasi-condensate density.
The quasi-condensate fraction at the BKT critical point is 7.16/ ln(380/g̃) [96],4.2. EFFECT OF HARMONIC TRAPPING

53

4.5
Mean-field critical number
4.0Classical-field critical number

0
NBKT
c /Nc

3.5
3.0
2.5
2.0
1.5
1.0
0.0

0.2

0.4

˜
g

0.6

0.8

1.0

Figure 4.6: The BKT critical number NcBKT for a harmonically trapped twodimensional gas, scaled to the ideal gas BEC number Nc0 . The green dashed line ispredicted using a MF model for a fully fluctuating Bose gas. The red line is based ona hybrid profile which accounts for the suppression of density fluctuations close to the
BKT transition (see text).

which falls to zero only logarithmically with reducing g̃; for illustration, it only
falls below 0.5 at g̃ = 2 × 10−4 .
The suppression of fluctuations is therefore significant at the critical point even for
extremely small interaction strength, allowing the central density to peak strongly
above the MF theory. The majority of atoms lie outside the fluctuation radius
Rfluct , are well-described by MF theory, and experience the full 2gn interaction
potential. The net result is that the cloud density peaks strongly in excess of MF
theory within a small central region, attaining the critical density with (globally)significantly fewer atoms than MF theory would suggest.

4.2.3

Applicability of Uniform System Results

The understanding of BKT in a harmonic trap of this Chapter arose from applying
infinite uniform system results within the LDA. However, ‘real-world’harmonicq
trapping potentials are of finite extent, with a lengthscale set by ~/(mωr ) andthe planar eigenstates have a finite energy spacing ~ωr .
The BKT mechanism is mediated by the proliferation of free vortices in a superfluid, which requires the possibility of in-plane dynamics in order to have a spatially varying superfluid phase. This necessitates that the superfluid wavefunction
has contributions from multiple planar trap eigenstates; thus at the critical point

54

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASES

the superfluid interaction energy gns = 2g̃kB T /π should be much larger thanthe planar eigenenergy spacing. One can therefore apply infinite uniform system
results to the centre of a harmonic trap if~ωr 

2g̃kB T.
π

(4.22)The legitimacy of LDA is also discussed in [57]. The authors demand that when
the trap centre is locally critical, the healing length should be much smaller than
the radius Rfluct of the fluctuation region in which beyond-MF effects contribute.
This gives an essentially identical condition to Eq. (4.22).
BEC-BKT crossover in a finite system Whereas in the thermodynamic
limit of ω → 0, N ω 2 = const. the transition would be BKT-like for all g̃ > 0,and BEC-like strictly for g̃ = 0, Eq. (4.22) implies that in a finite system thetwo transitions actually meet in a crossover region of finite width in g̃-space. The
BEC transition also extends into g̃ > 0; for finite planar level spacing ~ωr BEC isreached1 for a finite critical D.
For our planar trapping frequency2 of ωr ≈ 2π × 37.7 Hz the critical phasespace density for BEC is ∼ 13 [51], which is smaller than DBKT for g̃ . 10−3 .
Furthermore, Eq. (4.22) indicates that the BKT transition should become badly
defined for g̃ . 10−2 . In our experiment of Chapters 5 - 8 we investigate 0.05 .
g̃ . 0.5, with the lower limit set by our finite momentum resolution3 and thiscrossover is not relevant in our case.
Fig. 4.7 illustrates the various criteria which must be satisfied if one is to observe the pure-two-dimensional BKT transition in the axial ground state of aharmonically-trapped gas. The horizontal lines denote the criterion that the superfluid be kinematically two-dimensional, quantified by Eq. (4.9). The curvedlines denote the criterion that planar superfluid dynamics are possible at the BKT
critical point, quantified by Eq. (4.22). Each criterion is interaction-dependent,
and are sketched for a range of g̃. The horizontal dashed black line marks thecondition kB T = ~ωz , which gives an approximate temperature scale such thatthe presence of atoms in axial excited states do not significantly affect the groundstate physics. The red point denotes our specific experimental conditions which
satisfy all criteria; we therefore expect pure-two-dimensional BKT physics to beaddressable in our system for g̃ & 10−2 .

1The definition of BEC outside the thermodynamic limit is slightly difficult; here it is taken
as a point beyond which the ground state occupation grows sharply whilst the occupation of
excited states remains approximately constant.
2
See Chapter 6.
3This is discussed in Section 6.3.3.

4.2. EFFECT OF HARMONIC TRAPPING

55

4

˜
g
0.5

kBT/ℏωz

3

2

Kinematically-2D superfluid

0.4
0.3

LDA valid

0.2

1
Our experiment
0
0

20

40

60
ωz/ωr

80

100

0.1

120

Figure 4.7: Necessary criteria for a ‘pure-two-dimensional’ BKT transition to occur ina harmonically-trapped gas (see text). Three conditions must be satisfied: the superfluid
must be kinematically two-dimensional (horizontal coloured lines), planar superfluid dynamics must be possible (curved coloured lines), and the majority of atoms should residein the axial ground state (dashed black line). The green shading denotes the region inwhich all conditions are satisfied. Our experiment (red point) satisfies all conditions forg̃ & 10−2 . The minimum and maximum values of g̃ plotted, 0.05 and 0.5 respectively,
correspond to the bounds of interaction strength addressed in our experiment.

4.2.4

Comparison with Quantum Monte-CarloOur understanding of the BKT transition in a harmonically trapped gas is based
upon two assumptions, which have been addressed and justified in preceding Sections.
• That in the case of a kinematically two-dimensional system, the axial groundstate should display pure-two-dimensional physics. This was discussed in
Section 4.1.
• That uniform gas results for the BKT transition can be applied to a harmonic gas, within the LDA. We expect that this should be valid in the limitof Eq. (4.22).
As a check of this approach, we can compare our predictions for the BKT criticalpoint to the results of QMC simulations for a harmonically trapped, quasi-twodimensional system [99, 102]. Two results are provided for different trap parameters, based upon specifics of experiments at ENS [9] and NIST [57]. These specifics
are noted in Table 4.1, along with the QMC predictions for the BKT critical temperature T BKT .
These results can be compared with the predictions based on applying pure-twodimensional theory to the axial ground state. Setting T = T BKT from Table 4.1,

56

CHAPTER 4. TWO-DIMENSIONAL PHYSICS IN TRAPPED GASESg̃ωz /2πωr /2πN
T BKT (nK)

ENS
0.13
3 kHz
50 Hz
20, 000
186

NIST
0.02
1 kHz
20 Hz
17, 900
74

Table 4.1: BKT critical temperatures T BKT , calculated by QMC [99, 102], for two
different trap configurations.

Eq. (4.21) gives the critical axial ground state numbers as 13,200 and 10,600 for the
ENS and NIST experiments respectively. The axial excited state population canbe calculated within a MF model for the axial levels (presented in Section 7.2.3),
and one finds excited state numbers of 6,800 and 7,500 respectively.
This results in total critical numbers of 20,000 and 18,100 for the two experiments, in superb agreement with the results of QMC. This lends support to theapplicability of pure-two-dimensional results to the axial ground state, and thevalidity of Eq. (4.21) which applies uniform system classical-field results to thecase of harmonic trapping.

4.2.5

Unification of BEC and BKTThe results presented in this Section suggest that the two very different transitions, the statistically-driven BEC and interaction-driven BKT, should continuously connect as interactions are tuned to zero. Whereas in a uniform system one
is precluded from following the BKT transition to the ideal gas limit g̃ → 0 by
the diverging critical atom number, the presence of a harmonic trap results in a
finite critical number at all interaction strengths. The harmonic potential of mostultracold atomic systems therefore offers an opportunity to observe this crossover,
and ascertain exactly how the BKT critical point converges onto that for BEC.
An experimental work confirming this unification is the topic of the remainder ofthis thesis.

Chapter 5
Experimental Overview
5.1

Introduction

In Chapters 3 and 4, we discussed the low-temperature phases available in a twodimensional Bose gas. In a uniform planar trapping potential, BEC is precluded
since the associated critical phase-space density is infinite. However, an interacting
gas undergoes the BKT superfluid-normal transition at a critical phase-space density DBKT = ln(380/g̃) where g̃ is a dimensionless measure of interaction strength.
This is a fundamentally-interaction driven transition mediated by the proliferation of free vortices, topological defects which can only exist in an interacting
system.
As the interaction strength approaches zero, the critical density DBKT diverges,
and approaches the infinite critical density for BEC. Furthermore, the criticalchemical potential µBKT → 0, again recovering the BEC result. This suggeststhat the critical points of the two conceptually very different phase transition
should be continuously connected as a function of g̃, where the BEC critical pointis recovered as the ideal-gas limit of the BKT transition. In a uniform system it is
not possible to follow this connection, due to the divergent atom number required.
However, as Section 4.2 addressed, in the case of harmonic in-plane trapping thecritical atom number for both ideal-gas BEC and interacting-gas BKT is finite.
The transition occurs when the trap centre locally reaches the uniform-system
critical density and one is effectively probing uniform system physics; however,the presence of the harmonic trapping potential results in a finite critical atom
number at all interaction strengths.
Whereas most condensed matter experiments in two-dimensional systems suchas superfluid He films [53], exciton-polariton condensates [54] and thin-film superconductors [52] utilise a uniform trapping potential which precludes explicit
observation of this unification, experiments with ultracold atomic gases typically
employ harmonic trapping potentials [9, 55–68]. In this experimental work, we
directly observe the unification of the BKT transition with the ideal-gas BEC
limit in a harmonically-trapped two-dimensional gas with tuneable interaction57

58

CHAPTER 5. EXPERIMENTAL OVERVIEW

strength.
In Section 5.2 we outline previous work on trapped two-dimensional atomic gases,in particular studies of the BKT transition. Section 5.3 overviews the relationshipbetween the correlation function of a gas and its momentum distribution, whichforms the basis of our method for identifying the onset of extended coherence;
this provides an experimentally convenient signature of either the BKT or BEC
transition.
Our experimental setup for trapping and probing two-dimensional gases is described in Chapter 6. The experiment itself, extraction of relevant gas quantities
and resulting critical parameters for the BKT transition are presented in Chapter 7, and finally Chapter 8 summarises the conclusions which may be drawn fromour work, and offers an outlook on future research possibilities.

5.2

Previous Experimental Work

The BKT transition was first conclusively observed experimentally in thin-films ofliquid 4 He [53], where the characteristic discontinuous jump in superfluid density
was detected as a shift in the frequency of a torsional pendulum. In the following
years, aspects of BKT physics have been addressed in systems as diverse as thinfilm superconductors [52], Josephson-junction arrays [52] and exciton-polariton
systems [54].
Whilst well-suited to superfluid fraction measurements, condensed matter experiments are somewhat limited in the parameters which may be controlled. In liquid
He experiments for example, temperature can be set by the film substrate, andtwo-dimensional density varied by changing the film thickness, but neither density,interaction strength nor trapping geometry can easily be tuned. Furthermore, theinteraction parameter g̃ in liquid 4 He films is estimated to be of order unity [98],making theoretical approaches difficult.
By contrast, both the density and temperature of two-dimensional ultracold gases
can easily be controlled. Trapping potentials are provided by sculpted optical andmagnetic fields; the resultant potentials are extremely flexible, and have enabledexperiments to address the cases of harmonic [9, 55–68], uniform [105] and twodimensional lattice [66] planar trapping geometries.
Importantly, the precise control over interaction strength afforded by Feshbachresonances [2] opens the possibility of addressing any interaction regime, fromstrongly interacting to the ideal gas limit [61, 66], within a single experimental
arena. The realisation of two-dimensional cold gases within the weakly-interacting
limit g̃  1 opens a regime in which quantitative, theoretical descriptions become
possible [51, 96, 98], and which is precluded by condensed matter systems.
In the context of investigative ease, cold two-dimensional gases can be addressedin both real-space and momentum-space, and interference techniques give access5.2. PREVIOUS EXPERIMENTAL WORK
(a)

59
(b)

(c)

Figure 5.1: Various signatures of the BKT transition have been observed in ultracold gases. (a) Interference between two quasi-two-dimensional planes of atoms revealed a proliferation of vortices above a critical temperature [9]; these are revealed as
discontinuities in the interference fringes. (b) Complimentary momentum-space (left)
and real-space (right) images show a sharp increase in the extent of coherence [58],whilst spatially nothing dramatic is observed. (c) Stirring a harmonically trapped twodimensional gas at various radii reveals a discontinuous jump in the critical velocity forheating, signifying a jump in superfluid density at some critical chemical potential [63].

to the phase of the order parameter, yielding information regarding correlation
functions [9, 57], proliferation of vortex phase defects [9, 65] and phase fluctuations [64, 106]. Cold atom systems therefore offer a tantalising combination ofexperimental feasibility and theoretical tractability, permiting quantitative predictions [51, 96, 98] to be rigorously tested in a controllable, tuneable system.
The BKT transition was first observed in ultracold gases in 2006 at ENS [9]. The
proliferation of vortices above a critical temperature was revealed by interfering
two two-dimensional atomic planes; vortices manifested themselves as dislocationsin the interference pattern. In the following years, experiments identified theanticipated growth in the extent of coherence close to the transition [9, 56–58, 60,
61, 68], and the discontinuous appearance of a non-zero superfluid fraction [63];some of these experiments are depicted in Fig. 5.1.
In addition to confirming important features of the BKT transition, cold gas
experiments have made manifest the importance of beyond-MF critical physics.
As the transition is approached a marked departure from MF physics is apparent even on the normal side; various studies identified a gradual suppression ofdensity fluctuations [61], reduction in compressibility [58], reduction in interaction energy [62] and enhanced occupation of low-momentum states [57, 60] withrespect to MF theory.

60

CHAPTER 5. EXPERIMENTAL OVERVIEWIn Section 3.5 we introduced the classical-field analysis of weakly-interacting twodimensional gases [96, 98], which attempts to account for such non-perturbative,
beyond-MF critical behaviour. The resultant equation of state [98], was observed
to provide a superb description of quasi-two-dimensional trapped Bose gases, inexperiments at ENS [59, 62] and Chicago [61, 66]. Not only were the measured
density profiles at fixed interaction strength reproduced very well by theory, but
the ability to tune interaction strength allowed confirmation of the predicted universal form for the equation of state [61, 66]. Additionally, density was observed
to depend purely upon µ̃ = µ/(kB T ), confirming the predicted scale invariance of
the weakly-interacting two-dimensional Bose gas.
Whilst many experiments provided information about the behaviour of a twodimensional gas in the vicinity of the BKT point, relatively few quantitatively
identified the critical parameters. This is rendered difficult by two factors: First,
the infinite order of the BKT transition precludes any dramatic or sudden behaviour in real-space [57, 58, 61, 62, 68]. The equation of state displays no discontinuity or sharp feature at the critical point, aside from the superfluid fraction
which is not straightforward to quantify in an ultracold gas1 . Second, the wide
fluctuation region surrounding the BKT critical point [96, 98, 99] means that MF
theory gradually becomes invalid as the transition is approached, rather than any
sudden failure at the transition [60].
Therefore, whilst various aspects of the BKT transition have been addressed at
specific interaction strengths, a systematic study of the BKT critical point, how
it varies with interaction strength, and in particular how it approaches the ideal
gas limit, has been lacking. The only previous experiment to employ tuneableinteractions, in Chicago, has focused on the interaction dependence of the equationof state [61, 66]. Recent experiments offered confirmation of both scale invarianceand universality, but being limited to in situ measurements could not directly
reveal the critical point.
In the next Section, we overview our method for identifying the BKT transition,
before moving on to our particular experimental work, which forms the remainderof this thesis.

5.3

Identifying the Critical Point

There are several experimental signatures of the BKT transition that might beemployed to identify the critical point. As outlined in Section 5.2, previous experiments have observed the onset of superfluid behaviour [67], the proliferationof vortices [9], the width of the in situ density profile [57] and the appearance ofa significant low-momentum fraction [56–58, 60, 68].
In this work, we systematically identify the critical point for the onset of extended
coherence for a wide range of interaction strengths, 0.05 < g̃ < 0.5. Whether the1

This discontinuity was directly observed in a trapped Bose gas in [63].

5.3. IDENTIFYING THE CRITICAL POINT

61

low-temperature transition is BEC- or BKT-like, at the critical point one anticipates a sharp growth in the range of coherence [51, 100, 102, 107]. In the caseof BEC this is a consequence of condensation into a single state1 , whereas in the
BKT case it is due to the transition to a correlation function that decays algebraically rather than exponentially with separation2 . The appearance of extended
coherence provides an experimentally convenient probe of the critical point, sincein a uniform system the correlation function g (1) (r) is revealed as the Fouriertransform of the momentum distribution n(k).
This relation is slightly modified for the case of an inhomogenous system. The momentum distribution n(k) is the number of atoms in a momentum-space element
DE
d2 k/(2π)2 , and is related to momentum occupation numbers by n(k) = A â†k âkwhere A is some system area which will not appear explicitly in our final result.
The momentum distribution can be rewritten in spatial field operators,
DEn (k) = A â†k âk ,
=
=

Z

d2 r

Z

d2 r0 eik·(r−r ) Ψ̂† (r) Ψ̂ (r0 ) ,

Z

d2 r

Z

d2 a eik·a Ψ̂† (r) Ψ̂ (r − a)

0

D

E

E

D

D

a = r − r0
.

(5.1)

E

We can identify g (1) (r, a) = Ψ̂† (r) Ψ̂ (r − a) as a local correlation function,
whose Fourier transform with respect to a integrated over the entire sample isrevealed in the momentum distribution.
A change in the functional form of g (1) (r) leads to a dramatic change in its values at
distances much larger than the thermal wavelength λth [51]. The Fourier relationof Eq. (5.1) implies that the appearance of increased coherence over a large lengthscale L will manifest itself as enhanced population of momentum states k . 2π/L.
Any change in the functional form of g (1) should be increasingly pronounced atlarger L, and the consequent change in the momentum distribution should be
progressively more dramatic with lower k. Thus, unlike the in-trap density distribution, which varies very smoothly through the BKT critical point [57, 58, 61, 62],n(k) can provide a dramatic signature of the phase transition [58].

1
2

See Section 2.1.1.
See Section 3.3.

62

CHAPTER 5. EXPERIMENTAL OVERVIEWChapter 6
Experimental Setup andCharacterisation
Our experiment was carried out with a gas of 39 K atoms at a temperature T ≈170 nK, confined in an optical harmonic trap with radial and axial trapping frequencies (ωr , ωz ) ≈ 2π × (37.7, 4040) Hz, giving a trap-frequency ratio > 102 .
As illustrated in Fig. 4.7, the axial level spacing is sufficiently large such thatthe superfluid is kinematically two-dimensional1 , planar superfluid dynamics are
possible2 , and the majority of atoms reside in the axial ground state3 .
A great advantage of a large aspect ratio ωz /ωr is that the critical atom numbers
scale quadratically with this quantity. Since NcBKT ∼ (kB T /(~ωr ))2 , the aspectratio (ωz /ωr )2 provides an upper bound to the allowed critical numbers if thetransition is to be crossed in the quasi-two-dimensional limit. In our system thisresults in NcBKT . 104 , sufficiently high to be easily observable, whilst ~ωz /kB ≈193 nK, an easily attainable temperature regime.
Interactions were tuned by means of a Feshbach resonance centred on 402.5 G [15,16]. The momentum distribution n(k) was obtained via the ‘momentum focusing’
method [58,68,108,109] which obtained the momentum distribution exactly, withno admixture from the in situ spatial distribution. Imaging of the cloud was
performed along the axial direction, giving access to the full planar momentum
distribution.
Each of these components shall be addressed in turn. For each, we first give a conceptual overview, followed by more specific experimental characterisation.

1

See Section 4.1.
See Section 4.2.3.
3
See Section 4.1.
2

63

64

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

Figure 6.1: The constituent beams of our optical trapping setup. Tight axial confinement is provided by two ‘blades’ of blue-detuned light, whereas a crossed red-detuned
beam (ODT) provides in-plane confinement; surfaces represent equipotentials of their respective beams. A typical vertical absorption image of a trapped two-dimensional atomic
cloud (dark disk) is projected below the trap.

6.1

Optical Trap for Two-Dimensional Confinement

Our atomic trapping potential is generated by light potentials, which are a central tool in ultracold atomic physics. By tuning a laser to be either red- or
blue-detuned from an atomic resonance, atoms experience either an attractive or
repulsive potential proportional to the light intensity. Available trapping potentials are limited only by the ability to sculpt light into the desired geometry; suchdiverse trapping configurations as harmonic traps, uniform box traps [110] andoptical lattices in one, two and three dimensions [111] have been realised. An
overview of light potentials and the use of Fourier optics to generate intricate
optical patterns is given in Appendix B.
In our system, two-dimensional confinement is generated by two repulsive ‘blades’of light at λ = 532 nm which tightly confine atoms in the axial direction. To
provide harmonic in-plane trapping, we superimpose our axial potential withan attractive optical dipole trap (ODT) consisting of a red-detuned beam at
λ = 1064 nm crossed in a bow-tie configuration. The complete optical trapping setup is illustrated in Fig. 6.1, along with an absorption image of a trapped
two-dimensional cloud, taken along the axial (vertical) direction.

6.1. OPTICAL TRAP FOR TWO-DIMENSIONAL CONFINEMENT

65

Figure 6.2: The phaseplate used to generate our axial trapping potential. The plate imprints a 0-π phase pattern onto a Gaussian beam, and a lens projects the Fourier transform of the resulting amplitude pattern E(x, z) onto its focal plane, giving Ef (xf , zf ).
The atoms occupy the resulting potential landscape; the intensity pattern is illustrated,along with a 1D cut through xf = 0.

6.1.1

Axial Trapping

Our axial potential is derived from the Fourier spectrum of a 0-π phase pattern
imprinted on a blue-detuned Gaussian beam. The beam is first passed through aphaseplate, which imposes a relative retardation of λ/2 between vertical halves,
and the resulting spectrum is projected onto the atoms by a ‘Fourier lens’. The
phaseplate is depicted in Fig. 6.2. Qualitatively, one expects destructive interference between the two phaseplate halves to produce a nodal intensity plane between
two sheets of light. This technique for two-dimensional trapping is also employedin [59, 112], and our optical setup was designed by Naaman Tammuz [113].
Quantitatively, for a Gaussian incident beam with waist wz in the z-direction, theassociated 1D amplitude profile at the phaseplate is,
Ep.p. (z) ∼ e−z 2 /wz2

×


1

:z<0
−1 : z > 0(6.1)

Decomposing this amplitude pattern into wavenumbers q in the z-direction, we
obtain the spectrum:Ẽp.p. (q) ∼ e−q 2 wz2 /4∼e−q 2 wz2 /4"Z−iqwz /2

dt e

−∞

−

Z ∞

dt e

−t2

#

,

−iqwz /2

qwz
.
erfi
2


−t2



(6.2)

where erfi(z) denotes the imaginary error function1 . By placing a lens after thephaseplate, a focal distance f before the atoms, this spectrum is mapped onto1

erfi(z) = − √2iπ

R iz
0

2

e−t dt.

66

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

amplitude at the atoms Ef (zf ) according Eq. (B.7):−zf2 /Wz2

Ef (zf ) ∼ e!

zf πwz
erfi
,
λf

(6.3)

which is a modulated Gaussian of waist Wz = λf /(πwz ). The atoms occupy the
resulting nodal plane, which provides a tight axial confinement.
The prefactor of Eq. (6.3) is fixed by the total power P of the beam. Including
x-dependence gives an intensity pattern:
2
2
2
2
zf
2P
e−2xf /Wx e−2zf /Wz erfi
πWx WzWz

If (xf , zf ) =

2

.

(6.4)

The trapping frequency can be found by relating light intensity to the induced
atomic potential√via Eq. (B.3) and expanding about zf = xf = 0. We make useof erfi(z) ≈ 2z/ π for z  1, and equate the resulting quadratic-zf potential to(1/2)mωz2 zf2 . This yields a trapping frequency:ωz (xf ) = 2π × 0.721

x2
1 − f2
Wx

! s

P
MHz,
Wx Wz3(6.5)

where P is the total beam power in watts, and waists Wx , Wz are in microns.
The full optical setup is illustrated in Fig. 6.3. A compact diode laser1 provides
1.8 W of light at 532 nm, with a beam waist of 750 µm. The laser output is firstpassed through an AOM, and the first-order diffracted beam used for the axialtrapping which permits fast (< 1 µs) switching of the axial trap. A telescope
expands the beam to a 1.5 mm waist, and a λ/2 waveplate followed by PBS
ensures that a clean and stable polarisation propagates to the atoms.
It is beneficial for the axial light to propagate along the horizontal imaging axis,
to facilitate alignment. Two dichroic mirrors2 , one before the Fourier lens and one
after the atoms, are used to direct the green light whilst remaining transparent
to λ = 767 nm which is used for the horizontal imaging.
In the x-direction, a cylindrical lens (f = 500 mm) focuses the beam and together
with the Fourier lens (f = 45 mm) gives an x-collimated beam at the atoms, with
Wx = 135 µm.
At the Fourier lens, the beam is collimated in the z-direction with waist wz =1.5 mm, giving a waist at the atoms of Wz = λf /(πwz ) = 5.1 µm. We therefore
anticipate an axial frequency upper bound3 from Eq. (6.5) of ωz ≈ 2π × 7.2 kHz.
This frequency would correspond to a temperature ~ωz /kB = 344 nK, setting aneasily attainable temperature scale for quasi-two-dimensional behaviour.
1

Laser Quantum, Excel 532 nm.
Semrock Inc. LM01-552-25.
3The laser outputs 1.8 W, some fraction of which is lost en route to the atoms.
2

6.1. OPTICAL TRAP FOR TWO-DIMENSIONAL CONFINEMENT

67

Horizontal
CameraMonitorPhotodiode

PID
Photodiode

Slide

Slide

λ=767 nm

Dichroic

Dump

Atoms

f=45 mm
Dichroic
0-πphaseplateλ/2
λ=532 nm
1.8 W

PBS
f=50 mm

f=150 mm

80 MHzCylindrical
f=500 mm

Dump

Figure 6.3: The optical setup for the axial trapping potential (see text). For clarity
all reflective mirrors are omitted.

Anti-trapping The maximum ωz is found exactly at xf = 0, in the Fourier lens
focal plane. Away from this point, the zero-point energy E0 (xf , y) = ~ωz (xf , y)/2
will fall, giving an effective ‘anti-confinement’. From Eq. (6.5), the zero-point
energy displays a quadratic x-dependence:x2
E0 (x) = E0 (0) 1 − f2 ,Wx!

1
= E0 (0) − mω 2x x2f .
2

(6.6)

We
q can therefore identify an anti-trapping transverse potential, of frequency ω x =~ωz (0)/(mWx2 ). For the numbers given above, this evaluates to ω x ≈ 2π ×10.1 Hz.
Along the propagation direction of the beam, ŷ, a similar anti-trapping potential
arises due to divergence of the z-focused beam. Wz displays a variation with
distance y away from the focus,sWz (y) = Wz (0) 1 +

y2
,
yR2

(6.7)

68

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATIONσy (pixels)

40
35
30
25
0.0

0.5 1.0 1.5 2.0
Hold time (ms)Figure 6.4: Measurement of axial trapping frequency. The black points depict thevariation in axial width σy of a BEC, after 17 ms ToF, for various trap hold times
after exciting a monopole breathing mode. The red dashed lines are superimposed on theimages to guide the eye.

where yR = πWz2 /λ = 154 µm is the Rayleigh length for a Gaussian beam focussed
to Wz . Inserting into Eq. (6.5) and keeping quadratic terms in y 2 /yR2 , yields avariation in zero-point energy with y,!

3y 2
E0 (y) = E0 (0) 1 − 2 ,4yR
1= E0 (0) − mω 2y y 2 .
2

(6.8)

The anti-trapping
frequency in the propagation direction therefore evaluates toqω y = 3~ωz (0)/(4myR2 ), which for the beam parameters above gives ω y ≈ 2π ×
7.7 Hz.

Power stabilisation After passing through the atom chamber, the axial trapping beam is terminated at a beam dump. A glass slide picks off a small beam
fraction, which is directed onto a photodiode (see Fig. 6.3). The resulting photodiode output voltage is passed to a PID1 whose error signal is fed back to theamplitude control of the initial AOM. This feedback permits control and stabilisation of the axial trapping power. We empirically find a bandwidth of & 5 kHz,
which is sufficient to avoid significant heating of atoms via fluctuations in thetrapping beam power.

Axial trap frequency The frequency of axial trapping was determined fromthe breathing mode of an ideal gas BEC, and the experimental procedure is given
in Appendix C. Fig. 6.4 shows a typical set of breathing mode ToF images, together with the variation in observed vertical Gaussian width σy with hold time.
1

Made in-house by A. L. Gaunt.

6.1. OPTICAL TRAP FOR TWO-DIMENSIONAL CONFINEMENT

69

We obtain1 an axial frequency ofωz = 2π × (4040 ± 6) Hz,(6.9)

at a laser output power of 1.8 W. From the analytic frequency of Eq. (6.5) this
implies a power of 560 mW incident on the atoms. Between the laser output andatoms, the axial trapping beam passes through 14 air-glass interfaces; some of thesurfaces are AR coated for 532 nm, but we can estimate a potential compounded
reduction of 0.9614 = 0.56. Furthermore, the AOM diffraction efficiency of ≈ 0.8
would lead to a total power of 1.8 × 0.56 × 0.8 = 0.8 W reaching the atoms. We
also note that the axial trapping power after exiting the science cell is ∼ 100 mW;
however, this light has passed through the horizontal imaging optics, which are
AR coated for 780 nm, and so this number can only provide a lower bound on the
power at the atoms. Whether the lower axial frequency is purely due to power
reduction, or whether imperfect beam profile and optic performance reduce thefrequency, we make no definite conclusion on.
Nevertheless, this trapping frequency corresponds to an axial temperature scaleof 193 nK, still easily achievable. Furthermore, the damping of the breathingmode is negligible over 18 oscillation cycles; fits were not improved by the inclusion of a damping envelope, implying excellent harmonic quality of the axial
potential.

6.1.2

Planar Trapping

The axial trapping potential induces a large trapping frequency (∼ 4 kHz) in theaxial direction, and weak anti-trapping (∼ 10 Hz) in-plane. Harmonic confinement
is provided by two crossed attractive Gaussian beams; the optical setup of thisODT is outlined in [113, 114]. On each pass, the beam is focused to a waist of≈ 140 µm. By design, the ODT waist, axial trapping transverse waist, and axialtrapping Rayleigh length are all comparable. This ensures good matching between
the two traps, such that atoms can inhabit the harmonic region of both beams
without unnecessarily wasting laser power.

Planar trap frequency The planar trapping characteristics are determined byexciting an in-plane centre-of-mass oscillation of a BEC, and observing the resulting motion on the axial imaging system. From this evolution we can extract botheigenfrequencies of the planar trapping potential, and the eigenaxis orientations
which are not generally along the camera (and hence image) axes. We assume
that the planar trap is harmonic, with axes x̂0 , ŷ 0 that lie at some angle θ to thecamera axes. The trapping potential is then11
V (x0 , y 0 ) = mωx0 x02 + mωy0 y 02 .
2
2
1

Taken 02.10.2014.

(6.10)

70

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

x (pix)36yy'Camera axes

34
32
30

θ

y (pix)x'Trap axes

x

44
42
40
38
0

0.1
thold (s)0.2

Figure 6.5: In general, we assume trap axes to be related to camera axes by a rotationby θ. On the right we plot the displacement of an oscillating cloud along the horizontal(x) and vertical (y) camera axes. The red line is the coupled function of Eq. (6.12),
fitted to an oscillating cloud in our focusing potential.

A point in trap coordinates is related to camera coordinates via a rotation transformation,!
!
!
xcos θ − sin θ
x0.
(6.11)=y0ysin θ cos θ
We assume that the cloud oscillates independently along the trap eigenaxes; theobserved motion along the camera axes should then be,!

x(t)cos θ − sin θ=y(t)sin θ cos θ!

!

A cos (ωx0 t + φx0 ).
B cos (ωy0 t + φy0 )(6.12)

This yields a coupled fit function for (x(t), y(t)) with {A, B, ωx0 , ωy0 , φx0 , φy0 , θ} as
fit variables1 ; we find trap parameters2 ofωx0 = 2π × (33.9 ± 0.5) Hz,
ωy0 = 2π × (42.0 ± 0.3) Hz,
θ = 64 ± 6◦ .

(6.13)We also repeated the frequency extraction in the absence of the axial trapping
light3 . This oscillation is illustrated in Fig. 6.5 and the extracted trap parameters
1

A second method was to rotate the observed cloud coordinates (x(t), y(t)) by a variable
angle, until the fitted amplitude of a single frequency component was maximised. The eigenfrequencies are obtained by single frequency fits to the rotated data sets. This approach yields
identical frequencies to the first method, but slightly different eigenaxis angle θ. The two methods yield θ = 58◦ and 70◦ with axial light present, θ = 41◦ and 34◦ without. In practice we used
an average of the two θ values, which corresponds to the value given in Eqs. (6.13) and (6.14).
2
Taken 02.10.2014.
3
Taken 03.10.2014.

6.1. OPTICAL TRAP FOR TWO-DIMENSIONAL CONFINEMENT
Axial trap present
33.9 Hz

71

Axial trap absent
38.8 Hz33.8 Hz42.0 Hz
64°

37°

Figure 6.6: The in-plane eigenaxes (green dashed lines) and frequencies both in thepresent of the axial trapping beam, and in their absence. The black lines are the axes ofthe vertical imaging camera. In the case of axial trapping being present, we also show
an in situ image of a trapped cloud; the extracted eigenaxes agree excellently with theobserved cloud asymmetry.

areωfoc,x0 = 2π × (33.8 ± 0.2) Hz,ωfoc,y0 = 2π × (38.8 ± 0.1) Hz,θ = 37 ± 3◦ .

(6.14)

These parameters will later be necessary to characterise the momentum focusing,used to extract the momentum distribution of a trapped gas (see Section 6.3).
We illustrate in Fig. 6.6 the planar trap geometry with and without axial trapping
light present. In the case of axial trapping, we superimpose an in situ image of a
two-dimensional cloud; the extracted eigenaxis orientation agrees excellently withthe observed cloud asymmetry.
It is somewhat unanticipated that the presence of axial trapping light significantly modifies the planar trapping potential. In Section 6.1.1 we concluded
that spatial variation of the axial trapping beam should lead to weak, approximately symmetric anti-trapping of ≈ 10 Hz; instead we observe an additional
asymmetric trapping of ≈ 16 Hz. This may be caused by inhomogeneities in
the axial trapping light, leading to an imperfect nodal plane. In the absence ofthe phaseplate, a Gaussian beam of 560 mW1 , wavelength 532 nm, and waists(Wx , Wz ) = (135, 5) µm would induce a potential of height 31 µK. For cloud temperatures of ∼ 100 nK, one needs imperfect destructive interference in the axial
1

Estimated power at the atoms inferred from our measured axial frequency.

72

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

trapping nodal plane of only ∼ 0.3% in order to appreciably distort the cloudshape.
Having characterised our two-dimensional trap, we conclude trapping frequenciesof (ωr , ωz ) ≈ 2π × (37.7, 4040) Hz. Here, ωr is the geometric mean of the in-plane
trapping frequencies. Although the in situ cloud is anisotropic, we note that for
any quantity Φ which depends upon the local potential V (x, y) ∝ ωx2 x2 + ωy2 y 2 ,an average or sum over the whole cloud can be recast as a circular integral:==ZΦ ωx2 x2 + ωy2 y 2ZΦ ωr2 x̄2 + ωr2 ȳ 2ZΦ ωr2 r̄2



dx dy,ωr ωrdx̄ dȳ,
ωx ωy





x̄ =

ωx
x ,
ωr

ȳ =

d2 r̄.

ωyyωr(6.15)

Therefore the momentum distribution and critical atom number in our anisotropic
trap should be identical to a circular cloud of radial frequency ωr , assuming validity of the LDA.

6.2

Tuning Interactions

In addition to sculpting the geometry of the confining potential, exploring the
BEC-BKT crossover requires the ability to tune interatomic interactions. As discussed in Section 3.2, for a three-dimensional dilute Bose gas at low temperatures,the complicated actual interatomic potential can be represented by a contact potential, of strength g3D [12]:V (r1 , r2 ) = g3D δ (r2 − r1 ) =4π~2 aδ (r2 − r1 ) ,m(6.16)

where a is the s-wave scattering length, a scalar scattering amplitude that is
independent of the two-body relative momentum k. In a two-dimensional system,
in which atoms are confinedqto the ground state of an axial harmonic potential
with oscillator length lz = ~/(mωz ), the scattering amplitude is in general kdependent [115, 116]. However, in the limit lz  a, usually well-satisfied byexperiments, one can treat the gas as collisionally three-dimensional [51], whilst
remaining kinematically two-dimensional.
The two-dimensional coupling constant g is then proportional to the three-dimensional
scattering length: g is defined such that the interaction energy density is (g/2)n2 ,where as usual n denotes the two-dimensional density. Theng 2 g3D Z
2π~2 a Z2n =dz n3D (z) =dz n3D (z)2 .
22m(6.17)

6.2. TUNING INTERACTIONS

73

Figure 6.7: The Feshbach coil arrangement, which produces a uniform magnetic fieldalong the axial direction at the atoms (red sphere). On the graph we plot the variationin g̃ around a Feshbach resonance in the |F, mF i = |1, 1i state of 39 K, labelled in the
low-field basis. The resonance is centred at 402.5 G, illustrated by the dashed black line.
We assume an axial trapping frequency of 4 kHz.

√
2 2
Inserting the axial ground state density n3D (z) = ne−x /lz /(lz π), yieldsg=~2 √ a
8π
m
lz

→

g̃ =

√

a
8π .
lz

(6.18)The tuneability of a via a Feshbach resonance1 translates into the ability to tuneg̃, allowing us to explore the crossover from ideal-gas BEC to interacting gas BKT
within a single experimental setup2 .
Feshbach resonances were reviewed in the Introduction; we repeat some salient
details here for convenience. For 39 K, the hyperfine state |4 2 S1/2 , F = 1, mF = 1i
(expressed in the low-field basis) displays a resonance centred at 402.5 G, with awidth of 52 G [15, 16]. This is an easily attainable field strength, and the wide
resonance permits accurate tuning of the scattering length.
The resultant variation in g̃ in the vicinity of the resonance is sketched in Fig. 6.7;
for a sense of scale, we assume an axial trapping frequency of 4 kHz, and calculateg̃ via Eqs. (1.6) and (6.18).
Feshbach field setup Our Feshbach field is produced by a Helmholtz coil pair,which provides a uniform field at the atoms orientated in the axial direction. The
coil setup is described in detail in [113, 114] and is illustrated in Fig. 6.7.
The coils are driven by a current-controlled power supply [113, 114], and thetimescale for changing the field strength is ≈ 5 ms. Our analogue control voltage
is accurate to 10−3 V, with the resonance centred on a control voltage of 3.805 V.
This suggests a field error of ∼ 0.1 G, leading to errors of 1% and 4% for oursmallest (g̃ = 0.05) and largest (g̃ = 0.5) interaction strengths addressed.
1

See Introduction, and [2].
From Eq. (6.18) √we note that the assumption of a collisionally-three-dimensional gas should
break down for g̃ ≈ 8π = 5, an order of magnitude larger than values addressed in our work.
2

74

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

6.3

Momentum Focusing

Having addressed our setup for confining atoms to a quasi-two-dimensional environment and tuning their interactions, we now turn to our method for pinpointing
the BKT critical point. As discussed in Section 5.3, we identify the onset of extended coherence by monitoring the in-plane momentum distribution, n(k). The
canonical technique for accessing the momentum distribution in trapped ultracold gases is time-of-flight (ToF) imaging [117], whereby the trapping potential
is abruptly switched off, and the gas allowed to expand for a time tToF before
recording the density distribution. Assuming ballistic expansion, the final position r(tToF ) of an atom is given by r(tToF ) = r(0) + p(0)tToF /m, where p(0) is
the initial momentum. Therefore, as tToF is increased, the imaged distribution
maps smoothly from the initial spatial distribution onto the initial momentum
distribution. One expects behaviour at the BKT transition to smoothly become
more dramatic.
Since observing the exact momentum distribution would require waiting an infinite ToF, we instead employ a ‘momentum focusing’ method whereby the momentum distribution can be obtained exactly. This was previously demonstrated
in 1D [108] where details of phase fluctuations within a condensate were extracted
from the consequent broadening of the momentum distribution. The method was
extended to harmonically-trapped two-dimensional gases in [58,68] where a sharp
growth in low-k states was observed below a critical temperature, and attributed
to the BKT transition. A theoretical overview of the focusing method is givenin [109].
In this Section, we first outline the conceptual details of momentum focusing andoffer some comparison to traditional ToF imaging. Some practical details of applying momentum focusing to our two-dimensional Bose gas are described in Section 6.3.2, and Section 6.3.3 concerns itself with the effect of a finite momentumspace resolution on identifying the BKT transition.

6.3.1

Conceptual Outline

Momentum focusing takes advantage of the symmetry of position and momentum
in the harmonic oscillator Hamiltonian. For a particle undergoing harmonic motion, momentum is mapped over a quarter-period onto position, and vice versa.
Quantitatively, for an oscillator of frequency ωfoc with Hamiltonian1
p2
2+ mωfoc
x2 ,
HHO =
2m 2

(6.19)

the time evolution is given by Hamilton’s equations:
ẋ =

p
m

,

2
ṗ = −mωfoc
x.

(6.20)

6.3. MOMENTUM FOCUSING

75
p

p(0)

Final state vector

Initial state vector
R(ωfoct1/4=π /2)p(0)mωfocxFigure 6.8: The evolution of state vector (mωfoc x, p) for a particle moving in a harmonic oscillator. The time-evolution is described by a rotation matrix R(ωfoc t), andafter a quarter period, the spatial distribution reflects the initial momentum distribution.

For some initial position and momentum (x(0), p(0)), the solution to Eq. (6.20)
is,
!
!
!
mωfoc x(0)mωfoc x(t)cos ωfoc t sin ωfoc t.
(6.21)=p(0)p(t)− sin ωfoc t cos ωfoc t|{z

R(ωfoc t)}

The time-evolution matrix R(ωfoc t) is a rotation matrix in the oscillator phasespace. Its action is to rotate the vector (mωfoc x, p) by an angle ωfoc t, illustrated
graphically in Fig. 6.8.
After a quarter-period t1/4 = π/(2ωfoc ), the position isx(t1/4 ) =

p(0)
.
mωfoc

(6.22)

This is the essence of momentum focusing; the initial momentum is mapped onto
final position.

Comparison with ToF The great advantage of momentum focusing is thatit grants experimental access to the infinite ToF limit, providing the exact momentum distribution. The disadvantage compared with ToF imaging is that the
momentum resolution is fixed, whereas with ToF imaging resolution increases
with expansion time. Since we expect the BKT transition to be revealed in theoccupation of low-momentum states, the criterion of interest for the two methods
is the lowest resolvable momenta in an ensemble.
For an imaging system of resolution δx, and a focusing trap of frequency ωfoc , theresolution in momentum-space follows straightforwardly from Eq. (6.22), ∆p =

76

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

mωfoc δx. The focusing method therefore gives the occupation of all momentum
states p < mωfoc δx, from the number of atoms within δx of an image centre.
If ToF imaging is used, the initial cloud size R sets a lower bound for the sizeof the imaged cloud. Even in the case of all atoms having zero momentum,
after any ToF the spatial distribution would be of extent R. After a flight time
tToF , the central image region of radius R approximately contains atoms with
momenta p < mR/tToF . Therefore, ToF imaging provides superior resolution of
low-momentum atoms compared with focus imaging if
tToF >

1 R.
ωfoc δx(6.23)

The
q definition of the cloud size R is somewhat arbitrary; the thermal radius Rth =kB T /(mωr2 ) is a reasonable candidate. Generally speaking, for any imaging
system which can reasonably resolve in situ clouds, the ratio (R/δx)  1 leading
to tToF being significantly larger than the focusing trap timescale.
For our system, T ≈ 170 nK1 , ωr ≈ 2π × 38 Hz, and resolution is limited by thecamera pixelsize2 ∆x = 2.95 µm. The same potential provides planar trapping
and a focusing potential, hence ωfoc ≈ ωr . This qleads to a required tToF > 36 ms,
at which point the expanded cloud size ∼ tToF kB T /m ∼ 10R, and the opticaldensity of an image has fallen by a factor of ∼ 100 compared to an in situ image. The resulting poor signal-to-noise ratio3 , especially in the distribution wings,
makes it difficult to extract cloud parameters such as number, temperature andchemical potential.
On the other hand, momentum focusing does not result in significant planar expansion of the cloud. It preserves good signal-to-noise in the wings of the momentum distribution whilst providing better low-momentum resolution than any
ToF under 36 ms.
A subtlety is worth mentioning here: there is an additional advantage of momentum focusing over ToF imaging if we wish to observe a sharp feature in
momentum-space. For either method, we require the cloud axial extent to not
exceed the imaging system depth-of-field. For imaging resolution of δx, the depthof field b is approximately twice the Rayleigh length4 , b = 2πδx2 /λ. For a trapped
quasi-two-dimensional cloud, the axial evolution after release is approximatelythat of a non-interacting
q Gaussian wavepacket, and after release time t the axialq22extent is ≈ 1 + ωz t ~/(mωz ). Demanding that this extent is less than the1

See Section 7.1.
See Section 6.4.3.
3
In principle, no information should be lost by imaging the same number of atoms spreadover more pixels. In practice, the dominant source of imaging noise is photon count shot noiseon each camera pixel. Since optical density (OD) is calculated from the number of photons
absorbed by atoms, the ratio of shot noise to absorbed photons increases with falling OD. This
causes signal-to-noise to fall with falling OD.
4
This sets the extent of the focal point at the atoms, in the direction of imaging.
2

6.3. MOMENTUM FOCUSING

77

depth of field for our system parameters1 restricts our release times to t . 11 ms.
Our focusing time of ≈ 7 ms is within this limit. On the other hand, using alengthy ToF would result in the axial extent of the cloud greatly exceeding thedepth of field, resulting in blurring and distortion of the low-k peak.

6.3.2

Focusing our Two-Dimensional Bose Gas

In the case of an ensemble of particles, interactions must be negligible during
focusing, which is a single-particle process. In a two-dimensional system, this canbe accomplished by switching off the tight axial confinement of frequency ωz [58,68], whilst leaving the weak planar potential of frequency ωr . Axial expansion
causes interactions to vanish on a timescale ωz−1  ωr−1 , and the in-plane evolution
is essentially free-particle-like. Atoms must remain within the harmonic region of
the ODT during focusing, and not be excessively displaced vertically. Focusing
is therefore performed in the presence of a vertical magnetic field gradient whichcancels the force downwards due to gravity; this is described in Appendix E. Aftera quarter-period, an absorption image taken along the axial direction reveals theplanar momentum distribution.
Residual scattering Despite axial expansion being much faster than in-plane
evolution, some residual inter-atomic collisions are inevitable during focusing.
This can be estimated by noting that the axial
q expansion is driven by kinetic
2
energy and so the density should fall as 1/ 1 + ωz2 t2 after release. The scatq

tering rate at time t after release is then γ/ 1 + ωz2 t2 , where γ is the in-trap
scattering rate, and for a focusing time of tfoc the number of collisions per atomisNcoll =

Z tfoc0dt γ q1
1 + ωz2 t2,q1= γ ln ωz tfoc + 1 + ωz 2 t2foc .
ωz(6.24)

Whilst this integral diverges with time, it does so logarithmically.
In our experiment a typical trapped atom number is 2 × 104 giving an in situ
scattering rate3 of ≈ 4000g̃ 2 . Our focusing trap frequency4 is ωfoc ≈ 2π × 36 Hz,
and the range of interaction strengths considered is 0.05 < g̃ < 0.5. This results in
Ncoll ≈ 0.2 for our largest interaction strength, and Ncoll ≈ 0.002 for our smallest,
both much smaller than unity.
1

The trap frequency ωz = 2π × 4040 Hz (Section 6.1.1) and we take our resolution to be
dominated by the pixelsize of our CCD, giving δx ≈ 3 µm (Section 6.4.3).
2
The definition of two-dimensional trapping is that the axial trap level spacing ~ωz exceeds
other energy scales in the gas.
3
See Section 7.1.3.
4
See Eq. (6.14).

78

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

4 ms

5 ms

6 ms
5
4.5

4.5

σy (pix)

σx (pix)

5

4
3.5
4

7 ms

4
3.5
3

5

6
7
8
9
Focus time (ms)

10

2.5
4

5

6
7
8
9
Focus time (ms)10

Figure 6.9: Evolution of central peak with focusing time. As the focusing time isscanned, the sharp low-momentum peak displays a minimum width after a quarterperiod of the focusing trap. We first rotate images by θ = 37◦ such that trap and camera
axes align, then perform a two-dimensional fit using some heuristic two-curve function
to extract a measure of the peak widths σx,y . A quadratic fit to the resulting evolution
provides the focusing time at which the peak is minimised.

Determination of focusing time We empirically determine the focusing timeat which the momentum distribution is revealed, by preparing a weakly-interacting
degenerate cloud at a = 10 a0 and monitoring the width of the sharp lowmomentum peak as a function of focusing time; the experimental procedure isoutlined in Appendix C and the evolution plotted in Fig. 6.9.
We fit a heuristic quadratic dip function, obtaining the focusing times t0 as thosefor which the peak width is minimal. These are;tx0 = 7.5 ± 0.1 ms,

ty0 = 6.6 ± 0.1 ms.

(6.25)To check that these are not overly sensitive to the choice of interaction strength,
we repeat with a scattering length of 242 a0 , obtaining
tx0 = 7.6 ± 0.1 ms,

ty0 = 6.5 ± 0.1 ms.

(6.26)As representative focusing times we choose 7.5 ms and 6.5 ms respectively. From
our measured trap frequencies of Eq. (6.14) we would predict focusing times of
7.4 ms and 6.4 ms; the slightly longer empirical times suggest the presence ofa delay mechanism. One axial trap period is 0.25 ms, so a measured delay of
0.1 ms is entirely consistent with the time for interactions to vanish via axial
expansion.
Focusing trap anisotropy The anisotropy of the focusing trap means that
perfect focusing cannot be attained simultaneously in both directions. To first6.3. MOMENTUM FOCUSING

79pi

Spread

Shift
mωxf
ϕ
mωxi

Figure 6.10: The initial phase-space of a trapped gas; each point is populated according
to the appropriate thermal distribution. The solid line is defined by Eq. (6.29); all atoms
along this line will be mapped by momentum focusing onto the same final position.
Imperfect focusing corresponds to finite φ = ωfoc ∆t; this has the effect of both shifting
the line, and introducing a non-zero gradient. Therefore there is a small overall shift inthe momenta mapped onto some final position, and also a non-zero spread. The dottedline would be the result for φ = 0, corresponding to the ideal mapping of Eq. (6.22).

order, the inherent blurring of the momentum distribution will be isotropic if the
focusing time error is an equal fraction of each axis period; we therefore choose afinal focusing time tfoc according to
7.5 − tfoc
tfoc − 6.5
=
7.5
6.5

→

tfoc = 6.96 ms.

(6.27)

We can estimate the consequent blurring of the momentum distribution. For anerror ∆t in the focusing time, an initial state vector (mωfoc xi , pi ) evolves onto a
final state vector (mωfoc xf , pf ) according to,mωfoc xf
pf

!







π
+φ
2

=
π
− sin 2 + φ

cos



sin



!

mωfoc xi
,
pi

π
+φ
2

π
cos 2 + φ

− sin φ cos φ=− cos φ − sin φ!

mωfoc xi,pi!

(6.28)

where φ = ωfoc ∆t. Some final position xf therefore corresponds to any initial statesatisfying,mωfoc xf = cos φ pi − sin φ mωfoc xi .
(6.29)
This can conveniently be graphically represented, as in Fig. 6.10 where we plot
the initial phase-space. Atoms populate this phase-space according to the Bose
distribution, and any atoms along the line defined by Eq. (6.29) will be mapped
onto the same final position xf .
The effect of imperfect focusing φ 6= 0 is thus twofold:

80

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION
1. The central momentum mapped onto some final position is shifted by afactor cos φ. In our system, T ≈ 170 nK and ωfoc ∆t ≈ 0.11 along eachfocusing trap axis. The systematic shift of momentumis then of the order√−1cos(ωfoc ∆t)pth ≈ 0.03 ~ µm , where pth = mkB T is the thermal momentum. This is an order of magnitude smaller than our momentum-spaceresolution set by the pixelsize, mωfoc ∆x = 0.4 ~ µm−1 .
2. The momenta of atoms at a particular final position will have a finite spread,given by sin φ mωfoc δxi , for some spreadof initial position δxi . This will beqof order the cloud thermal radius, kB T /(mωr2 ) for trap planar frequency
ωr . In our system ωr ≈ 2π × 37.7 Hz1 , giving a momentum spread of
0.4 ~ µm−1 . By coincidence, this is equal to the accuracy enabled by our
momenta resolution.

6.3.3

Effect of a Finite Momentum-Space Resolution

Our momentum resolution is ∆k ≈ mωfoc ∆x/~ = 0.4 µm−1 , imposed by the finite
pixelsize ∆x of our imaging system, and the anisotropy of the focusing trap. The
rationale behind measuring the momentum distribution is that a change in thefunctional form of g (1) should be revealed in the occupation of low-momentum
modes2 . The lowest momentum atoms addressable are those with k . ∆k, which
are focused to within 1 pixel of the centre of an imaged cloud. Our best probeof extended coherence is then the measured peak height of the momentum distribution, denoted P0 . In terms of lengthscale, we are probing correlations overL ≈ 2π/∆k = 15.2 µm. At typical temperatures T ≈ 170 nK, this correspondsto (L/λth ) = 22.4, much larger than unity.
In the case of uniform trapping, L/λth should be as large as possible for detectionof the transition. In the case of harmonic trapping, the critical point is reachedwhen only the infinitesimal central region becomes superfluid; hence we should
not expect to see any discontinuous jump in P0 , but rather a rapid growth as thecloud crosses the critical point. Importantly, in this case the cloud size sets anupper limit on desirable values of L. If we probe coherence over a long lengthscale
L, then the BKT transition will only become apparent once a region of diameter∼ L has become superfluid. We thereforeqrequire L to be much smaller than thethermal diameter of the cloud, 2Rth = 2 kB T /(mωr2 ). This results in an overall
condition on L (or equivalantly ∆k),λth  L  2Rth .

(6.30)

In our system λth ≈ 0.7 µm, L ≈ 15 µm and 2Rth ≈ 51 µm. The cloud inhomogeneity will unavoidably impose a shift in the apparent critical point, which is
discussed in Appendix F.
12

See Eq. (6.13).
See Section 5.3.

6.3. MOMENTUM FOCUSING81

How large should L/λth be? For the BKT transition to be detectable itmust be accompanied by a sharp feature in P0 . Although we are working with aharmonic trap, we can take the uniform case as a toy model and ask how P0 shouldchange as the gas crosses from normal to superfluid. Measurement of P0 equates
to measuring the shaded area underneath the correlation function spectrum, n(k),illustrated in Fig. 6.11.
For a degenerate two-dimensional ideal gas, the low-k momentum distribution
is [51]:√
1
4π −D/2
4π
,
kc ≈
e
.
(6.31)
nk ≈ 2 2
2
λth k + kc
λth
We note that this function becomes progressively peakier with increasing D, leading to a diverging correlation length ζ = kc−1 [51, 105]. For total system area A,
the number of atoms with k < ∆k is
N∆k =



2

1
dk
∆k
√
nk = A 2 log 1 +2
λthkc
k<∆k (2π/ A)

Z



λth
1
= A 2 log 1 + π
λth
L

!2

!2 
,



eD  .

(6.32)

The case of a BKT superfluid is more difficult; from Eq. (3.12) the correlation
function g (1) (r) ∼ r−η , where η = 1/Ds . On dimensional grounds the momentum
distribution n(k) ∼ k η−2 , but being more precise is difficult since the Fouriertransform of the algebraic g (1) is not convergent. However, we can assume somemaximum momentum kmax and write the fraction of atoms within the central
region as!ηR ∆k η−1
∆k
N∆k
dk
0 k
=
(6.33)≈ R kmax η−1
NTotalkmax
k dk
0
A convenient measure of the short-range lengthscale in the gas is the thermal wavelength, and so a reasonable estimate of the peak number is N∆k ≈ An(λth /L)η .
At the BKT point η = 1/4, and so this expression is only very weakly dependent
on the exact cutoff kmax . We thus expect a relative change in P0 at the BKT
critical point,
λth 1/4L.
Relative change in P0 =
 2
BKT
λth
D
log 1 + π L e

DBKT



(6.34)

This ratio is plotted in Fig. 6.11 for various values of g̃. As expected, the transition
is revealed at distances L  λth , and becomes progressively more dramatic as
longer lengthscales are probed.
In reality, this ratio should approach unity as L → 0; this corresponds to counting
atoms with k < ∞, and an ‘L = 0’ measurement equates to summing the total
atom number which does not change across the transition! This limit is not perfectly recovered by Eq. (6.34); the assumed upper momentum cutoff kmax = 2π/λth

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATIONSuperfluid

Normalg(1)(r)

ℱ

n(k)

˜
g

3.0
Peak Ratio

82

1.0
0.8
0.6
0.4
0.2

2.5
2.0
1.5
1.0
0

5

10 15 20 25 30
L/λth

Figure 6.11: The BKT transition causes g (1) (r) to change from exponential to algebraic decay, with a corresponding change in n(k). Any change in the functional formof g (1) should be most dramatic at longer lengthscales, and lower momenta. The vertical dashed lines denote our momentum resolution ∆k and the corresponding lengthscale
L = 2π/∆k on the appropriate plots. Measuring P0 is equivalent to the number of atoms
with k < ∆k, denoted by the shaded red regions. P0 should offer a clear transition signature if this area changes by an appreciable fraction at the critical point. Eq. (6.34)
estimates the ratio of N∆k after crossing the BKT point to its value before; this ratio isplotted on the right for various values of g̃.

is somewhat arbitrary, and the algebraic BKT superfluid correlation function is
only valid for L  λth . However, it is reassuring that the observed limit is ≈ 0.9,
reasonably close to unity.
We note that our finite coherence lengthscale L implies that we cannot reliably
detect the BKT transition below some value of g̃; physically, this is a consequenceof the diverging normal gas correlation length with increasing D. As a roughguide, the value of DBKT such that P0 changes by 10% across the transition is
DBKT = 8.67, corresponding to g̃ = 0.07. A slightly different argument√asks thevalue of g̃ for which the normal gas correlation length λth exp(DBKT /2)/ 4π = L.
At our typical temperatures of 170 nK, this results in DBKT = 8.75 and g̃ = 0.06,
which gives an almost identical estimate.
The results of Eq. (6.34) and Fig. 6.11 offer good support for using the onsetof coherence as a signature of the BKT transition; at least in a uniform system,
crossing the transition leads to a significant jump in P0 for sufficiently large L/λth .
In a harmonic system this should translate into a sharp (but continuous) growthin P0 .
An important point is worth mentioning here; we are specifically concerned withthe peak height P0 of the momentum distribution, not with its functional form.
The former provides a model-free method to identify the onset of extended coherence, which in turn offers an experimental signature of the BKT phase transition [51, 102, 107, 118]. However, we make no theoretical assumption about theform of n(k) at low-k. As a normal cloud approaches the critical point, n(k)becomes narrower than a Gaussian fitted to the distribution wings, indicating
coherence over lengthscales > λth [51, 57, 60] and even peaks above a pure-MF6.4. IMAGING

83

model [60, 99]. Thus unlike in three dimensions, where the presence of a bimodal
distribution in n(k) is a signature of BEC, such ‘peakiness’ does not provide areliable signature of BKT [60].

6.4

Imaging

All information about our trapped gases comes from images of the atomic density.
Imaging is principally along the vertical (axial) direction, which reveals the fullplanar atomic distribution. Our system also incorporates an imaging setup along
the horizontal direction [113, 114], and is not discussed further here, being usedsolely for trap alignment. For ease of reference, the essential characteristics ofeach imaging direction are tabulated here in Table 6.1.
In this Section, we first briefly review the absorption imaging used to map thecolumn density of an atomic sample. Since imaging of our cloud primarily takes
place in the presence of the (∼ 400 G) Feshbach field, Section 6.4.1 addresses
the state structure of 39 K in this ‘high-field’ regime. At these magnetic fields, a
residual admixture of |mJ i states leads to a ‘depumping’ effect, where the apparent OD falls with exposure to imaging light; a rescaling function to account forthis effect is derived in Section 6.4.2. The characterisation of magnification andabsolute imaging efficiency is the topic of Section 6.4.3.

Absorption imaging We probe our cloud using low-intensity absorption imaging, where the imaging light intensity I is much lower than the atomic transition
saturation intensity Is . Absorption imaging and our imaging procedure are described in more detail in Appendix D. Using this method, the optical density is
obtained as!
Catom (r),(6.35)OD ≈ − ln
Cref (r)where Catom (r) is a map of imaging light photon counts with atoms present, andCref (r) the corresponding map without. Optical density is related to the number
column density n via OD = nσ0 , where the zero-intensity cross section σ0 =3λ2 /(2π) for a closed cycling transition [119].

Optimised reference image In practice, the light pattern Cref (r) obtained
from the atom-free image will not be exactly identical to that incident on theatom cloud. The intensity of imaging light varies with time, and small frequency
drifts lead to a changing interference pattern on propagation through the imaging
optics. This leads to ‘fringing’ and distortion of calculated OD patterns. We
eliminate this effect, via an optimal reference algorithm developed in [120]. For
a particular data set, all of the atom-free images provide a basis {Cref,i }. For aparticular Catom , we select a region free from atoms, but with imaging light. Thisregion is then decomposed into the basis {Cref,i }, giving a set of coefficients for

84

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

Horizontal
Vertical

Model
QE
∆x
PCO imaging, pixelfly QE≈ 0.25 ≈ 2.43 µm
Princeton Instruments, PIXIS 1024-BR 0.98
2.95 µm

M
2.65
4.41

Table 6.1: Essential characteristics of the horizontal and vertical cameras and imaging
systems. QE denotes quantum efficiency, ∆x the in-trap length mapped onto a single
CCD pixel, and M the magnification.

Figure 6.12: A critical measured momentum distribution at g̃ = 0.28, before (left) andafter (right) applying the optimal reference algorithm.

the admixture of each individual reference image. This set of coefficients is used
opt
to build an optimal reference image Cref.
Qualitatively, we use a region of light free from atoms to make a ‘best guess’ of
the imaging light pattern originally incident on the atoms. By evaluating the OD
opt
of Eq. (6.35) using Cref
, both offsets and fringing are significantly reduced; thisis illustrated in Fig 6.12.

6.4.1

High-field State Structure

It is desirable to image our cloud in the presence of the Feshbach field, for two reasons. Foremost, our momentum focusing technique necessitates the cancellationof gravity during focusing which is provided by the superposition of the Feshbach field and a weak quadrupole field1 . Furthermore, the timescale of ∼ 5 ms
to switch off the Feshbach field would make it impossible to image our cloud in
situ; this is useful for alignment of the two-dimensional trap. We therefore use
our Feshbach field as an imaging guide, and in this Section calculate the relevant
atomic eigenstates.
At our field strengths of ∼ 400 G, the magnetic energy of an atom will be ∼ µB B =1

See Appendix E.

6.4. IMAGING

85

0.6 × h GHz. The fine-structure splitting for 39 K is ∼ 1700 × h GHz [121], so we
expect J to remain a good quantum number and there should be no admixtureof the D1 and D2 line.
The zero-field hyperfine splitting on the other hand is ≈ 460 × h MHz for the
ground state 4 2 S1/2 [121], so F should no longer be a good quantum number at
our field strengths. The hyperfine and Zeeman terms of the atomic Hamiltonianare,
Hmag = AI · J + gJ µB Jz B + gI µB Iz B.
(6.36)
Here we assume the magnetic field to define the z-axis, A is the hyperfine coefficient, gJ is the electronic g-factor, and gI is the nuclear g-factor. This Hamiltonian
is conveniently diagonalised in states |mI , mJ i for a particular J state. We image
on the D2 line;
4 2 S1/2 → 4 2 P3/2 .
(6.37)For 39 K the nuclear spin I = 3/2; this gives a basis of 8 spin-states |φi i = |mI , mJ ii
for the ground manifold, and 16 for the excited manifold. For reference, the groundand excited state parameters are tabulated in Table 6.2.
4 2 S1/2
4 2 P3/2
A (MHz ~ −1 ) 1450.5352809
38.283gJ
2.0023193043622 4/3
gI-0.00014193489
Table 6.2: Hyperfine and Zeeman term parameters for the ground and excited statesof the D2 line in 39 K [121].

The Hamiltonian can be recast as a matrix Hij = hφi |Hmag |φj i for each manifold
and diagonalised, yielding both the level energies and eigenstates at a particularfield strength.
In Fig. 6.13 we plot the results for the ground and excited manifolds of the D2
line. At high-fields, the second term of Hamiltonian (6.36) dominates, and atomic
eigenstates are approximately given by the basis states |mI , mJ i. Each mJ manifold is labelled, with the varying mI splitting each into a number of parallellines.
Choice of imaging states Our ground state at low-field is |F = 1, mF = 1i;
at high field this connects to |mI = 3/2, mJ = −1/2i which is plotted in blue in
Fig. 6.13. As an upper state for absorption imaging we choose |mI = 3/2, mJ =−3/2i. This is (almost) a cycling transition when driven by σ − -polarised imaging
light, maximising photon scattering per atom. This upper state is plotted ingreen.
Regardless of the Feshbach field at which the experiment is performed, at the
beginning of the momentum focusing stage we jump the Feshbach field to its zerocrossing value of 350.5 G. Not only does this minimise the atomic interactions

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

E (h×1MHz)

86

500

4 2S1/2

2

mJ =1/20-5000mJ =-1/2
100

200 300
B (G)

400

500

mJ =3/2

1000 4 P3/2
500
0
-500
-1000
0

100

mJ =1/2
mJ =-1/2
mJ =-3/2
200

300

400

500

Figure 6.13: The energy level structure corresponding to hyperfine and Zeeman energies, for the D2 ground state (4 2 S1/2 ) and excited state (4 2 P3/2 ). The states at highfield are approximately those of pure mJ , which are labelled. The blue (green) state
denotes our ground (excited) state for absorption imaging. Although the blue and green
states form an almost closed cycling transition, there is a small probability on each cycleof decaying to the purple state (see text).

during the focusing, but also ensures that we always image at the same fieldstrength, with the same eigenstate composition. At the zero-crossing, the groundeigenstate is,|groundi = 0.987 |mI = 3/2, mJ = −1/2i − 0.158 |mI = 1/2, mJ = 1/2i. (6.38)
This means there is a small, ≈ 2.5% admixture of mJ = 1/2 into the ground state.
Conversely, the mJ = 1/2 ground manifold contains a state approximately given
by |mI = 1/2, mJ = 1/2i, but with a 2.5% admixture of |mI = 3/2, mJ = −1/2i;
this state is plotted in purple. On each photon absorption/emission cycle, thisleads to a 2.5% probability of an atom decaying into the dark mJ = 1/2 lower manifold. This ‘depumping’ leads to an effective cross section that falls with exposure
to imaging light; calibration of this effect is discussed in Section 6.4.2.
Experimental imaging setup From Fig. 6.13, the energy splitting of ourground and excited states is -299 MHz relative to the D2 line at the zero-crossing
field of 350.5 G. Frequency variation around this point is −µB /~ = −1.4 MHz G−1 .
We wish to work over a field range of ≈ 50 G, leading to a frequency tuneability
requirement of ≈ 70 MHz.
Our high-field imaging light is derived from the MOT cooling light, detuned by≈ −7.6Γ from the zero-field |F = 2i → |F 0 = 3i D2 transition [113,114]. This lightis therefore already detuned by ≈ −205 MHz from the D2 line, and generating
high-field imaging light requires a further negative shift of approximately 94 →164 MHz.
The setup for preparation and delivery of high-field imaging light was designed
and constructed by Jay Man [122]. After passing vertically through the atom
cloud, it is focused onto the CCD camera using a objective and imaging lens setup
designed and built by Alexandre Dareau [123], which has an imaging resolution

6.4. IMAGING
4 2P3/2

4 2S1/2

87|mI =3/2,mJ =-3/2>

2.5%

σ - 97.5%∼|mI =1/2,mJ =1/2>∼|mI =3/2,mJ =-1/2>

Figure 6.14: The atomic states of 39 K as a function of magnetic field strength. Axes
are identical to Fig. 6.13, labels are omitted here for clarity. The absorption imaging
transition is from the ground state (blue) to an excited state (green), driven by a σ −photon (solid red arrow). Atoms are most likely to decay back to the imaging groundstate (solid blue arrow). However, the ground manifold state ∼ |mI = 1/2, mJ = 1/2i
(purple) contains a small admixture of |mI = 3/2, mJ = −1/2i; there is therefore asmall probability of an atom decaying into a dark state (dashed blue arrow) and exiting
the cycling transition. All transitions and states are assumed to be at a field strength of
350.5 G; the transition arrows are displaced horizontally for clarity only.

of . 2µm [21, 123]; these references contain a comprehensive description of thevertical imaging optics.

6.4.2

Rescaling Function

The residual impurity of our high-field imaging states results in a depumping
effect, whereby some fraction of atoms are transferred to a dark state. This will
reduce the apparent OD, by a factor which depends on the absolute number ofabsorption cycles.
In ‘ideal’ absorption imaging, the imaging light drives a perfectly closed cyclingtransition. Although there is a weak dependence of cross-section on imaging intensity1 , the measured OD is independent of the exact number of absorption/emission
cycles each atom undergoes. However, the transition for our high-field imaging is
not perfectly closed, leading to a ≈ 2.5% probability for an atom to relax into adark state on each absorption cycle2 . This is illustrated in Fig. 6.14 and leads toan effective cross-section that depends upon the number of imaging photons. Thiscan vary not merely shot-to-shot, but even pixel-to-pixel within a single image.

Empirical rescaling function To correct for this effect, we experimentallycalibrate the relative variation of the observed OD with the camera counts in
1
2

See Eq. (D.2).
See Eq. (6.38).

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATIONOD0/OD

88

7
6
5
4
3
2
1
0
0

2

4

6

8

10

C/C0

Figure 6.15: Measurement of the OD rescaling function. The top row shows unprocessed absorption images of a thermal cloud at (C/C0 ) =1, 1.5 and 2.5 (left to right).
For three particular radii, denoted by coloured rings, we construct a correlation plot of(OD0 /OD) vs (C/C0 ), which are plotted on the right. The bottom row shows the top
images after OD rescaling (see text).

the corresponding reference image, C. For this, we image a three-dimensional
thermal cloud after expansion from the ODT. Assuming cylindrical symmetryof the cloud, any variations in OD observed around a ring concentric with thecloud centre we attribute to the variations in the probe intensity. Averaging over
several images, we build, pixel by pixel, a correlation plot of (OD0 /OD) versus(C/C0 ), where C0 is a fixed reference value and OD0 the corresponding OD. We
construct such plots for various ring radii (i.e. various atomic column densities)
and various (mean) probe intensities, and find that to a very good approximation
the scaling of OD due to depumping depends only on the light intensity andnot on the atomic column density. Finally, we construct a numerical interpolating
function (OD0 /OD) = F(C/C0 ) from the compiled correlation plots from all radii,
illustrated in Fig. 6.16.
In practice, for an experimental shot we first obtain the optimal reference image
opt
Cref(r), and then rescale each pixel of the OD image OD(r) according to
opt
Cref
(r)
.
C0

!

OD(r) → OD(r) × F(6.39)

This transformation accounts for any spatial or temporal variation in the imaging
photon count. It not only takes the depumping effect into account, but also
any other dependencies on imaging intensity, such as Doppler detuning of atoms
during the imaging pulse and the effect of finite I/Is .
In Fig. 6.15 we illustrate three identical thermal clouds, imaged with mean probe
intensities (C/C0 ) = 1, 1.5 and 2.5. We also plot the rescaling function extracted
from three different ring radii (coloured rings) over ∼ 40 images with a range ofprobe intensities. The radii of the rings are 50, 65 and 80 pixels, and a Gaussian
fit gives the cloud radius as 36 pixels. The true OD therefore falls by a factor of∼ 2 between consecutive rings; however, the rescaling functions extracted at eachradius overlap excellently. At the bottom of the figure are shown processed images,
after applying the mapping of Eq. (6.39) pixel-by-pixel to the top images.

6.4. IMAGING

89

Analytic toy model As a sanity check, we can construct a simple model for the
rescaling function F. For a two-level transition illuminated by light of intensity
I and frequency detuning δ, the resulting scattering rate and absorption crosssection are [119],
Rscatt =1sΓ
2 1 + s + (2δ/Γ)2

σ=

,

σ0
,
1 + s + (2δ/Γ)2(6.40)

where s = I/Is . Assuming that δ = 0 for a stationary atom, after scattering
N photons the detuning experienced by an atom will be ≈ (2π/λ)N vr , where
vr = h/(mλ) is the recoil velocity induced by a single photon emission1 . From
Eq. (6.40) we have a differential equation for N (t):∂N (t)sΓ1= Rscatt =.
∂t
2 1 + s + 4πvr 2 N (t)2

(6.41)λΓThis can be solved numerically. From Eq. (6.38) we anticipate a γ = 97.5%
survival probability of an atom per absorption event. The visible column density
should then decay during the imaging pulse as n(t) = n(0)γ N (t) . This gives theeffective OD as a function of time:n(t)σ(t) = [n(0)σ0 ] γ N (t)= ODtrue γ N (t)2 ∂N (t),sΓ ∂t2 ∂N (t),sΓ ∂t(6.42)

where ODtrue = n(0)σ0 would be the ideal optical density in the limit of zeroimaging intensity and no depumping. Over an imaging pulse of duration τ , theaverage transmitted intensity fraction is
hIit
1Zτ
=
dt e−n(t)σ(t) ,I0
τ 0(6.43)

giving an overall apparent ODmeas ofhIit
ODmeas = − ln
I0

!

1Zτ
= ln
dt e−n(t)σ(t) .
τ 0




(6.44)

Assumpting that the optical density is small, expanding to first order gives,"Z#−1
τ
ODtrueN (t) 2 ∂N (t)
≈τ
dt γ
.
ODmeassΓ ∂t
0(6.45)

This results in an OD scaling function that is independent of atomic density, as
we assumed for our empirical function. By normalising Eq. (6.45) to its value at
1

We are neglecting the effect of photon re-emission. This is isotropic, so leads to a random
walk in velocity space. Therefore
the contribution of photon emission to net velocity along thepimaging beam will scale as N (t), and will be much smaller than the contribution of absorption,scaling as N (t).

90

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION
10

OD0/OD

8

6

4

2

0

2

4

6

8

10

C/C0

Figure 6.16: The empirical rescaling function (OD0 /OD) = F(C/C0 ) obtained frommapping the correlations between imaging photon count and observed optical density(black points). The rescaling function is referenced to the optical density at a reference
imaging photon count of 1300, equivalent to (I/Is ) ≈ 0.028. The red band is the toymodel result of Eq. (6.45) referenced to its value at C = C0 . The band thickness reflects
the uncertain ratio between imaging intensities at the atoms and camera (see text).
This model assumes a depumping rate of 2.5% per absorption cycle, and has no otheradjustable parameters.

.
our reference count C0 , we obtain a model for our rescaling function F(C/C0 ).
In Fig. 6.16 we show both the empirically measured rescaling function, and this
toy result. The band illustrates the uncertain ratio between the observed photon
count and the photon count incident on the atoms; there are 12 air-glass interfaces
between the atoms and camera, giving a potential compounded intensity reductionof 0.9612 = 0.61. We observe a generally good agreement between the model andobservations.
It is difficult to be more quantitative; our arguments assume that all atoms are
depumped at the same rate, which is equivalent to assuming that all atoms experience the same light intensity1 . In addition, this model neglects other factors
which can reduce the observed OD, such as imperfections of the imaging polarisation, and the finite linewidth of the imaging laser. However, it is reassuring thatit nevertheless reproduces our observations to within a few percent.
1

We can make a rough estimate of when this should break down: the ratio of surviving
fractions at opposite faces of a cloud is ∼ γ N /γ N exp(−OD) ∼ γ N ×OD . Considering the typicalnumber of scattered photons is N ∼ 10, this leads to an extremely small OD limit for our simple
model to be quantitatively accurate.

6.4. IMAGING

6.4.3

91

Imaging Characterisation

It is essential to properly characterise the imaging setup if quantitative information regarding trapped gases is to be obtained. First, to provide a lengthscale forimages the imaging system magnification is required; this is essential for determining both atom number and temperature. Second, the effective cross-section
is usually some fraction of σ0 ; this can result from a multitude of factors, suchas acceleration of atoms during the imaging pulse, imperfect imaging polarisation
and a finite linewidth of the imaging laser. Knowledge of this imaging efficiency
is essential for absolute measurement of atom numbers and densities.
Magnification The magnification of our horizontal imaging system is obtainedby calibration against the vertical displacement of a free-falling cloud. The magnification of the vertical imaging system is then obtained by imaging a quasi-pure
BEC on both imaging systems, and deducing the vertical magnification by relation to the horizontal. The experimental procedure is given in Appendix D, andresults are in Table 6.1.
Imaging efficiency Although the rescaling function of the previous Section
accounts for any spatio-temporal variations in imaging light intensity, it does notgive an absolute cross-section for absorption of imaging light. Good knowledge of
our imaging efficiency is essential if we are to quantitatively confirm the crossoverfrom interacting gas BKT critical conditions, to the ideal gas BEC limit.
We obtain the absolute imaging efficiency by measurement of the critical point
for conventional three-dimensional BEC, comparing the observed critical number
with its theoretical value. The experimental procedure is described in Appendix D,
and we find a ratio of theoretical to observed critical number of 4.3 ± 0.2 after the
OD rescaling of Eq. (6.39).

92

CHAPTER 6. EXPERIMENTAL SETUP AND CHARACTERISATION

Chapter 7
Mapping the BKT Phase
Diagram
In Chapter 5, we outlined an experiment for identifying the BKT critical point ina harmonically-trapped Bose gas, over a range of interaction strengths. Chapter 6
overviewed our experimental setup for creating quasi-two-dimensional Bose gases,
tuning their interactions and probing the resulting momentum distribution. In
this Chapter, we describe the experimental method in more detail, and present
our measurements of the critical point as a function of interaction strength.
Section 7.1 gives a typical experimental sequence for preparing a two-dimensional
Bose gas and scanning the BKT critical point. The transition is identified bymonitoring the extent of coherence as atom number slowly decays; measurement of
this extent, along with thermodynamic variables such as temperature and chemical
potential, is described in Section 7.2. Our results for the critical atom number
and chemical potential are the topic of Section 7.3, with an analysis of systematic
errors in Section 7.4.

7.1

Experimental Sequence

7.1.1

Sample Preparation

We initially prepare a cold gas of 39 K identically to the sequence described in [16],which results in ≈ 6 × 106 atoms in the ODT at a temperature of ≈ 4.4 µK. The
geometric mean of trap frequencies is ω̄ ≈ 2π × 164 Hz; the gas is thermal, witha central phase-space density of ≈ 0.03.
Our two-dimensional axial trapping beam1 has a vertical waist of ≈ 5 µm; loadingof an initially spherical cloud would be inefficient, resulting in only a thin ‘slice’of atoms being captured. We therefore first cool and load our spherical ODT
1See Section 6.1.1.

93

94

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

Figure 7.1: The red sheet used for loading our cloud into the green axial trapping
light. For clarity the ODT is not shown; in practice a combination of the red sheet andODT provides a highly-flattened harmonic trap, which compresses the atomic cloud tolie between the green blades.

cloud into a flattened trap, formed by superimposing the ODT with an attractive
Gaussian beam propagating horizontally, with vertical waist 10 µm and transverse
waist 500 µm. The optics for this ‘sheet’ beam are described in [21], and it isillustrated in Fig. 7.1.
We evaporatively cool the cloud by an exponential ramp-down of the ODT power.
Over the first 3 s of this ramp, the sheet is increased from zero to its full powerof 600 mW. The cloud is therefore simultaneously cooled and loaded into the
combined ODT/sheet trap; a small condensate is visible at this point, indicating
central phase-space densities O(1).

7.1.2

Loading of the Two-Dimensional Trap

In the ODT/sheet trap, the cloud temperature is ∼ 300 nK, and the axial frequency provided by the red sheet is ∼ 1 kHz. This gives an axial thermal diameter∼ 2.5 µm which is much smaller than the separation between the green blades of
the axial trapping light; by differentiating Eq. (6.4) the blade separation is calculated as 9.4 µm. This permits loading of the green nodal plane without capturing
any atoms above or below the blades. Loading of the two-dimensional trap is
accomplished by ramping on the green (axial) light to approximately a third of
its eventual final power and then ramping the sheet beam to zero. Our loading
efficiency from the sheet into the two-dimensional trap is approximately 50%. We
do not observe any trapped atoms above or below the green blades, and so attribute these losses to heating and resultant in-plane losses during the transfer
sequence.
After this transfer, the anti-gravity gradient is introduced, and the green lightramped to full power. The final in-trap cooling of the atoms is accomplishedby a ramp-down of the ODT power resulting in planar evaporation. This yields≈ 40, 000 atoms at ≈ 170 nK in our experimental trap configuration. Finally, theGreenControl(V)
Quadrupole Scattering
LengthControl
(a0)
(V)

95

3
2
1
0
3
2
1
0
3
2
1
0
300
200
100
0

TAKE AXIAL IMAGE

Sheet
Control(V)

ODT
Control(V)

7.1. EXPERIMENTAL SEQUENCE

0.2

thold

0.1
0
0

2

4

6

8

10 12 14 16
Experiment Time (s)

tfoc

18

Figure 7.2: The time evolution of ODT power, sheet beam power, axial green trap
power, scattering length and anti-gravity coil control voltage.

Feshbach field is adjusted to bring the interaction strength to its value of interest.
The entire two-dimensional trap loading sequence is illustrated in Fig. 7.2.
Fig. 7.3 depicts the atom cloud after loading into the two-dimensional trap, andafter final evaporative cooling. Images along both horizontal and vertical directions are shown; whilst in situ images are possible vertically thanks to the use of
our Feshbach field as an imaging guide, the horizontal images use a small (∼ 4 G)
imaging guide along the imaging direction. We therefore ramp the Feshbach fieldto zero over 100 ms before taking horizontal in situ pictures; if a BEC is present
this results in heating and depletion of the atoms. However, the method remains
useful as an alignment tool.

96

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM
30 μm

30 μm

30 μm

30 μm

Figure 7.3: Absorption images of the atom cloud immediately upon loading into thetwo-dimensional trap (left) and after the final ODT evaporation ramp (right). The main
panels show images taken along the axial direction, with the corresponding horizontal
images on top. Integration across the trapping plane leads to very high OD, and thehorizontal images are saturated for a large fraction of the cloud (see Eq. (D.5)).

7.1.3

Scanning of Atom Number

The cloud is prepared on the superfluid side of the BKT transition; this is apparent
from a dramatic low-k peak in momentum-space [58, 68]. To scan the criticalpoint, we simply wait a variable time in the final trap configuration. Atoms arelost from the trap via background gas collisions, scattering from trapping beams
and three-body recombination. These result in a smooth and reproducible decay
in atom number, and eventually the cloud centre crosses the superfluid-normal
BKT transition.
It is important that the gas equilibrates faster than the rate of atom loss. We canestimate the elastic scattering rate by assuming an ideal gas density profile1 ; theaverage two-dimensional density hni is thenrdr [ln (1 − exp (µ̃ − V (r)))]2,
hni = Rrdr ln (1 − exp (µ̃ − V (r)))
R

(7.1)

where V (r) = (1/2)mωr2 r2 is the in-plane trapping potential, and µ̃ is fixed togive a characteristic critical central phase-space density D ≈ 8.
A similar approach provides the average wavevector: integrating the Bose distri1

See Eq. (4.11).

7.1. EXPERIMENTAL SEQUENCE

97

bution over the trap yields the average in-plane momentum,RDE

kr2 = RRh  2 2~ k
2m

+ V (r) − µRh  2 2~ k
2m

+ V (r) − µ

rdr k 3 dk exp β
rdr kdk exp β

i−1



−1



−1

i−1 .

(7.2)

The axial momentum distribution follows trivially from noting that the harmonic
oscillator ground state
wavefunction is ψ(z) ∼ exp(−z 2 /(2lz )), where the oscilq
lator length lz = ~/(mωz ). The momentum distribution is simply the modulus square of the Fourier transform of ψ(z), hence n(kz ) ∼ exp(−kz2 lz2 ) andhkz2 i = 1/(2lz2 ).
√ q
We take the mean relative velocity between two atoms as vr = 2 hkr2 i + hkz2 i(~/m).
The mean elastic scattering rate is approximately 8πa2 vr hni [47], which is conveniently written as
Mean scattering rate ≈ 0.2N g̃ 2 .
(7.3)
To be confident that our obtained value for a particular quantity Φ represents theequilibrium value, we require multiple thermalising scattering events in the time
taken for Φ to change by its measurement error bar [79]. This limitation would
make it impossible to reliably extract the critical point at g̃ = 0; in our experimentN ∼ 2 × 104 and for weak interaction strengths we find gas parameters to varymeasurably over1 ∼ 1 s. Demanding 5 scattering events in this time limits us tog̃ & 0.04.

7.1.4

Image Aquisition and Processing

After a variable wait time, during which the atom number smoothly falls, theaxial potential is abruptly (< 1 µs) switched off whilst leaving the ODT present
and unchanged. The Feshbach field is simultaneously jumped to its zero-crossing
value2 of 350.5 G. After waiting for the momentum focusing time3 tfoc = 6.96 ms
we take an absorption image along the vertical direction. This reveals a ‘freezeframe’ image of the momentum distribution at the moment the axial trapping wasremoved. By repeated this sequence for various hold times, the full evolution of
the momentum distribution across the BKT transition is mapped out.
Typically, a full evolution sequence consists of images taken at ≈ 70 different
hold times across the BKT transition. For each hold time we take two images;
one at a relatively low intensity I/Is ≈ 0.023 and one at a higher intensity of
1

See Fig. 7.6.
As the discussion of Section 6.3.2 indicates, interactions vanish on a timescale ωz−1 , which ismuch faster than the timescale for the Feshbach field to change. The Feshbach field is changed
during focusing because the OD rescaling function of Eq. (6.39) was measured for this imaging
field only, and would be inapplicable to images taken at different fields. Furthermore, minimisinginteractions should reduce any effect of interactions upon the development of a low-momentum
peak.
3
See Eq. (6.27).
2

98

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

I/Is ≈ 0.2. The imaging depumping effect1 leads to a trade-off between signal in
the distribution wings and resolution of high OD at the distribution peak. We
therefore use the ‘low’ intensity images for extraction of N , µ and T , and our‘high’ intensity image for extraction of P0 .
There are three post-processing procedures applied to the raw experimental images:
1. Optimal reference images are calculated for each shot2 . This provides a‘best-guess’ for the imaging light pattern incident on each atomic sample.
2. The OD of each image is rescaled according to the transformation Eq. (6.39).
This accounts for the dependency of imaging efficiency upon imaging photon
count.
3. The entire image is scaled to take our imperfect imaging efficiency into
account3 .

7.2

Image Analysis and Parameter Extraction

The output from a critical point scan is a sequence of images, showing the evolution of the momentum distribution across the BKT critical point. In Fig. 7.4 weshow momentum distributions for an above-critical gas, critical gas, and normal
gas. At high atom numbers, a sharp peak is evident in momentum-space, signifying extended coherence. As the number smoothly decays, this peak initially
falls sharply, before decaying slowly with the profile wings. For comparison, we
also show the corresponding in situ profiles; our imaging is not quantitatively
reliable in this mode, but gives a qualitative impression of the spatial evolution.
As expected, there is no dramatic signature of a transition in real-space4 .
For quantitative comparison with theoretical expectations, each image sequencemust be converted into the evolution of various quantities of interest. In particular,
we are interested in the degree of extended coherence, the total atom number N ,the temperature T and chemical potential µ. The first two parameters can beextracted in a fit-free manner; the quantities (T, µ) however require a model for thegas distribution. In this Section, the extraction of each quantity is outlined.

1

See Section 6.4.2.
See Section 6.4.
3
See Section 6.4.3.
4
See Sections 5.2 and 5.3.
2

7.2. IMAGE ANALYSIS AND PARAMETER EXTRACTION

99

Figure 7.4: The momentum-space (top) and real-space (bottom) distributions for a
super-critical, critical, and normal gas. The interactions strength is g̃ = 0.28, and these
profiles correspond to the data of Fig. 7.6, with the hold times being 6 s, 11 s and 16
s respectively. Whereas momentum-space shows a sharp peak which rapidly falls andabruptly vanishes, the spatial distribution evolved smoothly across the transition.

7.2.1

Coherence

To follow the evolution of extended coherence, we simply monitor the height of
the low-momentum peak, P0 , by averaging the OD within 1 pixel of the peakcentre1 . This measurement is entirely free of any fitting assumptions.
The effect of our finite momentum resolution ∆k = 0.4 µm−1 was discussed in
Section 6.3.3. The momentum distribution should show a sharp change at the
BKT point for (L/λth )  1; our system probes2 (L/λth ) ≈ 22.4, and we anticipate
that the BKT transition should be reliably identified for g̃ & 0.06.

7.2.2

Number

The total atom number is obtained by summing the OD over all pixels of an image,relying upon our absolute calibration of imaging√efficiency. In practice, we restrict
this sum to within 4 thermal momenta, p < 4 mkB T , to avoid contributions tonoise from atom-free regions of an image. Again, extraction of this parameter
makes minimal assumptions.

7.2.3

Thermodynamic Variables

In order to extract the temperature T and chemical potential µ, a fitting modelfor the momentum distribution is necessary. In three dimensions, for typical interaction strengths ideal-gas models are usually applicable; the ratio of interaction
1
2

See Chapter 5 and Section 6.3.3.
See Section 6.3.3.

100

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

to thermal energy is,
4π~2 an3D
g3D n3D
=
,
kB T
mkB T
=2

a 3D
D .
λth

(7.4)

For typical values a ≈ 100a0 and λth ≈ 0.5 µm, even for degenerate thermal
clouds D3D ≈ 1 this ratio evaluates to ∼ 1% and ideal-gas models remain appropriate.
However, in two dimensions the situation is more complicated. For typical interaction strengths, interactions noticeably modify the gas distribution even in thethermal phase. In this case, the ratio of interaction to thermal energy is,~2 g̃n
gn
=
,
kB T
mkB T
=

g̃
D.
2π

(7.5)

In this case, close to the BKT point D ≈ DBKT ≈ 10, and for typical interaction
strengths 0 < g̃ < 1 the ratio is close to unity meaning that interactions must be
accounted for.
Moreover, the suppression of bosonic density fluctuations close to the BKT critical point1 means that constructing a comprehensive model for the momentum
distribution is very difficult. It is therefore necessary to restrict our fitting to momenta for which the occupations at a particular (T, µ) can be straightforwardly
calculated.
First, we consider several possible candidates for the momentum distribution,and the momenta for which they are valid. Second, we outline the chosen modelfor the momentum distribution, based on a mean-field-Hartree-Fock (MFHF) approach [101].
Which model is appropriate? There are several approximations to the momentum distribution that we could consider.
1. Boltzmann gas, neglecting interactions and quantum statistics. This assumes both that the occupation per state is very small such that quantum
statistics are irrelevant, and neglects interaction energy. In a harmonic system of trapping frequency ωr , this distribution is,n(p) =

2πkB T
mωr2



! ∞
p2
X β µ− 2m
−j~ωz

e

.

(7.6)

j=0
2

p
 (1 + µ̃), assumingThis approach should be valid for momenta 2mkBTthermal energy to exceed interaction energy.
1

See Section 3.5.3.

7.2. IMAGE ANALYSIS AND PARAMETER EXTRACTION101

2. Ideal quantum gas, neglecting interactions. Integrating the Bose distribution
over position, one obtains n(p) as,
2πkB Tn(p) = −
mωr2

! ∞
X



2

p
β µ− 2m−j~ωzln 1 − e

!

,

(7.7)

j=0

which reduces to the Boltzmann case for very large momenta. Such an
approach is valid if the kinetic energy is much larger than any interactionp2
 (µ̃ + g̃D/π).
energy scales, and hold for momenta 2mk
BT
3. Quantum gas with mean-field interactions. The simplest way to account forinteractions is to introduce an interaction potential, defined by the increase
in interaction energy associated with adding a particle; according to Eq. (3.4)
this is 2gn for a fully fluctuating Bose gas. Physically, this is a perturbative
approach to interactions, where we assume planar eigenstates |ki retain aquadratic dispersion relation, and calculate their first-order shift in energy
due to the interaction term of Eq. (3.2). One can numerically calculate themomentum distribution in this case; such a calculation is the topic of thefollowing Section.
In terms of validity, this approach should fail for those momenta whose occupation numbers are significantly modified by correlations. Comparison withthe classical-field results [98] of Section 3.5 showed that a fully fluctuating
MF equation of state fails when (µ̃BKT − µ̃) . g̃. Assuming that this failure
is dominated by the lowest momenta [99], we infer a momentum-space MF
p2& (g̃ + µ̃ − µ̃BKT ). This argument is not particularly
validity limit 2mkBTrigorous; we shall later empirically show in Section 7.2.4 that this cutoff isappropriate.
Going beyond mean-field, correlations make a straightforward model of the momentum distribution difficult; strong coupling between momentum states changes
both the eigenenergies and eigenstate themselves. We therefore adopt a MF perturbative model, the construction of which proceeds as follows.

Mean-Field-Hartree-Fock (MFHF) model Our momentum distribution model
follows [101]. We work within the LDA, and assume that at in-plane position r
within the trap the wavefunctions |ψi are locally separable into planar momentum
states |ki, and axial oscillator states |φj i,|ψik,j = |ki|φj i.

(7.8)

This separable form means that eigenenergies are simply summations of planar
kinetic energy ~2 k 2 /(2m), and axial eigenenergies εj , which are taken relative tothe zero-point energy ~ωz /2.
The momentum distribution is found by integrating the Bose-Einstein distribution
over the in-plane coordinate r. For a planar trapping potential V (r) we define

102

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

the local chemical potential µL (r) = µ − V (r), and take into account thermal
occupation of higher axial states. This gives,n(k) =

XZ

"
2

dr e

β

#−1



 2 2

~ k
+εj (r)−µL (r)
2m

−1

.

(7.9)

j

For a non-interacting system the axial energies would simply be εj = j~ωz , butthese are modified by the presence of interactions.
To evaluate the set {εj }(µL ) at a particular radius r0 we solve the axial Schrödinger
equation,"mωz2 z 2 ~ωz−~2 ∂ 2−
+ 2g3D n3D (z) φj (z) = εj φj (z),+2m ∂z 22
2#

(7.10)

where the three-dimensional coupling constant g3D = 4π~2 a/m. The threedimensional density is constructed from a particular set of axial eigenstates {|φj (z)i}and energies {εj } by integrating the Bose distribution of Eq. (7.9) over momentum
rather than position, giving at the radius r0 ,
1 X 
ln 1 − eβ(µL (r0 )−εj ) |φj (z)|2 .
n3D (z) = − 2
λth j(7.11)

Eqs. (7.10, 7.11) must be solved iteratively; an appropriate starting guess is a
non-interacting system of states and energies. This method takes into accountthermal populations of excited states, interaction-induced deformation of axial
eigenstates1 and interaction-induced shifts in eigenenergies including the differing
overlaps between different axial states.
Practically, the Schrödinger equation of Eq. (7.10) is conveniently solved by matrix
diagonalisation, in the basis of non-interacting axial states. The first three terms
then simply give a contribution to on-diagonal elements
ideal
Hi,j
= hφi (z)|mωz2 z 2 ~ωz−~2 ∂ 2+−|φj (z)i = i~ωz δij ,2m ∂z 22
2

(7.12)and the only laborious step is the evaluation of the density term,
intHi,j= hφi (z)|2g3D n3D (z)|φj (z)i.

(7.13)

Once the set of eigenenergies {εj }(µL ) has been determined, the momentum distribution for a (µ, T ) pair can be constructed via Eq. (7.9). By repeating this forappropriate ranges of µ and T , we construct a numerical interpolation functionn(p, T, µ) for each interaction strength which can be used as a fit function forexperimental images.
1

At the BKT critical point, for g̃ = 0.5 the critical chemical potential is µBKT ≈ 0.5kB T .
Our typical temperatures are 170 nK and the trap spacing ~ωz /kB = 193 nK, hence interactionenergy is roughly half the level spacing, resulting in some admixture between axial states.

7.2. IMAGE ANALYSIS AND PARAMETER EXTRACTION

103

Extracting µ and T Having equipped ourselves with numerical fit functionsn(p, T, µ) for each interaction strength addressed, we can proceed to extract µ andT from images. To avoid the computational laboriousness of a two-dimensional
fit, we first average each image over iso-momenta. A pixel at (x0 , y 0 ) in focusing
trap coordinates maps onto momentum-space according to
px0
py 0

!

!

mωfoc,x0 x0=,mωfoc,y0 y 0(7.14)and therefore atoms with equal momentum magnitude p lie along a contour
p2 = p2x0 + p2y0 = (mωfoc,x0 x0 )2 + (mωfoc,y0 y 0 )2 after momentum focusing. We canrelate trap coordinates to image coordinates (x, y) by Eq. (6.11), giving image
iso-momenta defined by


p2
2
2
2
2
2
=x
ω
cos
θ
+
ω
sin
θ
0
0
foc,x
foc,y
m2


2
2
2
2
+ y 2 ωfoc,x0 sin θ + ωfoc,y 0 cos θ





2
2
+ xy ωfoc,x0 − ωfoc,y 0 sin 2θ.

(7.15)

This is simply an ellipse, orientated at an angle θ relative to image coordinates. The values of (ωfoc,x0 , ωfoc,y0 , θ) are given by the focusing trap parametersof Eq. (6.14).
The resulting one-dimensional profiles are straightforwardly fitted with our numerical functions n(p, T, µ). To avoid a MF validity cutoff which depends upon
˜ and fitthe (initially unknown) µ̃, we assume a ‘worst-case’ scenario of µ̃ = µBKT
only momenta p2 > 2mkB T g̃. We set T = 220 nK, greater than the temperatureof any single image.
In addition, since our one-dimensional data are derived from elliptically averaging√
a two-dimensional image, we weight points by p. Our fitting procedure seeks
to minimise the sum of squared errors between the data and fit model. Thisweighting artificially increases the contribution of each data point linearly with
momentum, effectively simulating a two-dimensional fit.
An example of an elliptically-averaged momentum distribution n(k) for a critical
cloud with g̃ = 0.28 is shown in Fig. 7.5, along with the fitted MFHF profile(green line). For illustration, we also indicate both the Boltzmann distribution of
Eq. (7.6) and ideal quantum distribution of Eq. (7.7), shown as blue and purple
lines respectively, for whom µ = 0 and T is fixed to the result of the MFHF fit.
It is clear that the Boltzmann model fails furthest into the wings, with the ideal
quantum model performing better and the MFHF visibly superior to both. The
vertical red line indicates the momentum bound below which we expect MF to
fail.

104

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

Figure 7.5: The momentum distribution n(k) of a critical cloud (black points) withg̃ = 0.28, extracted by elliptically-averaging an image over increasing iso-momenta. The
green solid line is the result of fitting a MFHF model for n(k) (see Section 7.2.3), where
the red vertical line denotes the lower bound to momenta for which this model should beapplicable. Also plotted are distributions corresponding to a Boltzmann gas (blue) andideal quantum gas (purple), with µ = 0 and T fixed by the MFHF fit results.

7.2.4

Critical Parameter Extraction

For a particular image, we obtain N , P0 , T and µ as described above. From a fullsequence of images at various in situ hold times t, we obtain their evolutions N (t),P0 (t), µ(t) and T (t). An example set of evolution curves is depicted in Fig. 7.6,
for a sample at g̃ = 0.28. Whilst N , µ and T display a smooth evolution withhold time, P0 shows two clear regimes.
We attribute the sharp feature in the evolution of P0 as indicating the disappearance of extended coherence. A change in the functional form of g (1) in a uniform
system would induce a discontinuity in P0 ; in a trapped system however, any transition first occurs at the centre over a vanishingly small region, and P0 evolves
continuously.
From the P0 (t) curve, we extract the critical hold time tc at which the transition
occurs. We do not attempt to apply a specific model to the evolution of P0 ; theexact evolution is influenced by changes in N , T and the specifics of our imaging
system. We therefore choose a heuristic fit function, consisting of two piecewise
segments joined at tc .
(P0 (t) = Pc +a 1 − e−b|t−tc |c |t − tc |: t < tc
: t > tc

(7.16)

0.6
0.4
0.2
0.0
0

4
2
0
200
100
0

μ/ℏωz

P0 (a.u.)

0.8

N (104)tc

1.0

T (nK)

7.2. IMAGE ANALYSIS AND PARAMETER EXTRACTION

5

10
15
20
Hold time (s)

25

30

105

0.5
0.0
-0.5
-1.0
0 5 10 15 20 25 30
Hold time (s)Figure 7.6: The evolution of P0 , N , T and µ with hold time, for a sample withg̃ = 0.28. The critical time tc is identified via a piecewise fit to P0 (t) (see text), indicatedby the red solid line. Once determined, other parameters are fitted with third orderpolynomials (solid red lines) and their values at tc interpolated. The red dashed lines
denote tc , and the interpolated critical values of each parameter.

Close to tc , the exponential segment simply reduces to a linear evolution ab|t − tc |.
This piecewise function is effectively two linear segments, but allowing for OD
saturation at early times.
The P0 fit is superimposed on the example data set of Fig. 7.6 (solid red line).
Having determined tc , we fit N (t), T (t) and µ(t) with third-order polynomials
and read off their critical values. A cubic is chosen merely to provide a smoothly
evolving curve, and the method is effectively an interpolation with the demand ofslow variation.
Finally, we arrive at a critical set of parameters Nc = N (tc ), Tc = T (tc ) and µc =µ(tc ). By repeating for various values of the interaction strength, we construct
the variation of these critical parameters with g̃; this variation is presented anddiscussed in the following Section.

Validity of MF momentum model We note that the only theoretical assumption in our entire methodology is the validity of MF theory in the momentum
distribution wings. We recall from Section 7.2.3 that we assumed a MF approach
to be valid for momenta p2 > 2mkB T g̃. If this is invalid, or worse if the MF modelis always inappropriate, we should see variation in the fitted variables (µ, T ) as
this cutoff is varied.
We therefore repeated the extraction of critical parameters for various valuesof momentum cutoff. Fig. 7.7 shows the dependence of (µc , Tc ) on the fitting
momentum cutoff for g̃ = 0.44. Reassuringly, a plateau is observed, with thecritical parameters being insensitive to cutoff over a reasonable range of values.

106

CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

Figure 7.7: The left plots depict the extracted critical temperature Tc and chemical
potential µc for interaction strength g̃ = 0.44, as a function of the lower bound in fitted
momenta. The vertical red dashed line depicts our suggested cutoff p2 = 2mkB T g̃ forthe validity of a MF fitting model, calculated for T = 220 nK, which is greater thanany observed temperatures. Reassuringly this lies within a plateau for both quantities,
illustrated by oranges bands which represent the mean and standard deviation of T andµ plateau values. The plateau boundaries are estimated as the blue/red solid lines. On
the right we show estimated upper (red) and lower (blue) plateau bounds as a functionof g̃. The red dashed line is the same MF cutoff, and lies within the plateaux for allg̃ . 0.5.

The lower plateau boundary is set by beyond-MF effects, which enhances themomentum occupation numbers. This results in a momentum distribution which
is more peaky than a pure-MF approach would suggest, increasing the fitted
temperature. Such beyond-MF peakiness was experimentally observed in [60].
At extremely small cutoffs this trend reverses due to saturation of the high-OD
central region; this reduces the fitted µ and increases the fitted T .
The upper plateau boundary is more confusing. As the fit is restricted further
to the momentum distribution wings, it is reasonable to expect that the fit noise
would increase but we see a clear systematic trend of increasing fitted temperature. We attribute this effect to the anharmonicity of our momentum focusing
trap, due to the Gaussian profile of the ODT beams which provide the focusing
potential. Qualitatively, this is a reasonable assumption; the Gaussian potential
would lead high-momentum atoms to travel further during focusing than an ideal
harmonic trap would permit. Our momentum mapping p = mωfoc x will thenoverestimate their momentum, leading to a higher apparent ‘local’ temperaturein the high-momentum wings. More quantitatively, we performed numerical simulations of a cloud evolving in a Gaussian focusing potential rather than harmonic,
and observed the same behaviour. This upper bound imposed by our anharmonic
focusing potential sets an upper limit of the values of g̃ for which we can reliably
extract (µ, T ). We therefore restrict our study to g̃ . 0.5.
In Fig. 7.7 we also show the lower and upper plateau bounds for various values ofg̃. Reassuringly, our cutoff of p2 > 2mkB T g̃ (red dashed line) always lies within
the plateau. This gives us good confidence both that our MF model is valid, and7.3. EXPERIMENTAL RESULTS

107

our momentum cutoff is appropriate.

7.3

Experimental Results

Having obtained the critical number Nc , chemical potential µc and temperature
Tc we can proceed to compare them with theoretical predictions for BKT and
BEC over a range of interaction strength. As we showed in Section 4.1, in aquasi-two-dimensional system we should expect pure-two-dimensional physics tooccur in the axial ground state. We therefore first remove the contribution toNc from excited axial states. This is straightforwardly accomplished within our
MFHF model1 ; using (µc , Tc ) we can calculate the excited state number Nj>0 by
integrating Eq. (7.9) over k, for axial levels j > 0. This yields the ground state
number N̄c = Nc − Nj>0 .
In the following Sections 7.3.1 and 7.3.2 we present our results for N̄c and µc as afunction of g̃.

7.3.1

Critical Number

In Fig. 7.8, we show N̄c for various g̃, scaled to the ideal gas BEC critical number
Nc0 of Eq. (4.13). Our finite momentum resolution means that the BKT transition
cannot be reliably identified for g̃ . 0.06; this is denoted by a orange shaded
region2 . This region is wider than the finite crossover region between BKT and
BEC (g̃ . 0.01) resulting from the finite size of the system3 , and comparable withthe region lacking a sufficiently high thermalisation rate4 , g̃ . 0.04. At higherg̃ & 0.5, the focusing trap anharmonicity should distort the ratio; when fitting the
far wings of the momentum distribution, the apparent temperature rises5 . This
both increases the calculated occupation Nex of axial excited states, and increases
the calculated ideal gas critical number Nc0 ; both of these effects reduce the ratio
plotted in Fig. 7.8.
Without any adjustable parameters, our data display excellent agreement withthe critical point prediction of Eq. (4.21), denoted by the red line. This offers
strong support for the validity of modelling the non-perturbative fluctuation region
surrounding the BKT critical point using the classical-field approach of [96, 98].
Whilst the classical-field results were anticipated by the authors to be valid for
1

See Section 7.2.3. One might worry about using a MF model, which we have already asserted
should fail for low momenta. However, whilst beyond-MF physics affects the distribution of
atoms between in-plane momentum states, it should not change the population of each axial
level which is dominated by MF physics [99].
2This limit can be estimated as when the normal gas correlation length exceeds the distance
L over which we probe coherence; this is discussed in Section 6.3.3.
3
See Section 4.2.3.
4
See Section 7.1.3.
5
See Section 7.2.4.

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CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM
2.0
1.8

Nc/N0c

1.6
1.4
1.2
1.0
0.8

★
0.0

0.1

0.2

˜
g

0.3

0.4

0.5

Figure 7.8: The critical atom number scaled to the ideal gas BEC result Nc0 , as afunction of interaction strength g̃. The solid red line denotes a classical-field prediction
for the BKT critical point, without any free parameters, whereas the dashed green line
is a MF approximation which neglects suppression of bosonic density fluctuations in thenormal state. The star denotes the ideal gas BEC critical point, and the shaded region
the values of g̃ for which our identification of the onset of coherence is not reliable.
Error bars are statistical (see Section 7.4).

g̃ . 0.1 [98], the critical point prediction describes our data extremely well for allg̃ addressed.

Furthermore, the agreement with Eq. (4.21) over an order of magnitude in g̃, andthe proximity of our lowest reliable g̃-values to zero, indicates that the criticalatom number smoothly converges onto the ideal gas BEC result as g̃ → 0. This
convergence is observed without any free parameters, relying upon an accurate
absolute number calibration, and confirms the connection between the BKT and
BEC critical point in the ideal gas limit.

It is also interesting to contrast our data with the MF prediction of Eq. (4.18),
denoted by the green dashed line. The severe overestimation of Nc provided
by this approach emphasises the extent of fluctuation suppression at the criticalpoint.

7.3. EXPERIMENTAL RESULTS

109

0.6
0.5

˜
μc

0.4
0.3
0.2
0.1
0.0

★
0.0

0.1

0.2

˜
g

0.3

0.4

0.5

Figure 7.9: The critical chemical potential as a function of interaction strength g̃.
Values are shown scaled to T , in accordance with classical-field predictions and the scale
invariance of the two-dimensional Bose gas. The dashed line is a MF result assuming
no suppression of density fluctuations, whereas the dotted line assumes full suppression.
Error bars are statistical.

7.3.2

Critical Chemical Potential

For a more direct comparison with the uniform system BKT theory1 we also
consider the critical chemical potential µc . As a consequence of the scale invarianceof the two-dimensional Bose gas2 we expect a critical µ̃c = (µ/(kB T ))c , which isplotted in Fig. 7.9 against g̃. We observe very good agreement with the classicalfield prediction of Eq. (3.26), denoted by a red line. Just like the theoreticalprediction, our data approach µ̃c = 0 as g̃ → 0, which recovers the ideal gas BEC
result. The small overall shift between the data and classical-field prediction iscomparable with our estimated systematic error3 .
It is illuminating to compare our observed µ̃c with two intuitive MF approximations. We introduce an interaction potential γgn, where γ = 2 corresponds toa fully fluctuating Bose gas and γ = 1 to full suppression of fluctuations4 . Although MF theory alone does not predict the BKT transition, we can define a
critical chemical potential µ̃γc such that the density reaches DBKT . Using the MF
1

See Chapter 3.
See Section 3.2.
3
See Section 7.4.
4
See Section 3.2.
2

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CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM
StatisticalN̄c /Nc0µ̃c± 0.08± 0.03

Calibration± 0.14± 0.05

Systematics
Finite size Anharmonicity
+0.04
−0.01
+0.005−0.04

Table 7.1: Absolute standard uncertainties (68% confidence interval) and estimatedsystematic shifts affecting the measured N̄c /Nc0 and µ̃c . Statistical errors are shown aserror bars in the main paper. All values are typical for the experimentally explored rangeof g̃. The positive (negative) systematic shifts correspond to the measured values being
higher (lower) than the true ones.

equation of state (Eq. (3.1)), this yieldsµ̃γc =


γg̃DBKT
BKT+ ln 1 − e−D,
2π(7.17)

which are plotted as dashed (γ = 2) and dotted (γ = 1) lines in Fig. 7.9. Generally
the γ = 1 curve provides a much better approximation to µ̃c , highlighting how
strong the suppression of fluctuations is at the critical point.
Although it is tempting to consider a fractional 1 < γ < 2 to account for the partial
suppression of fluctuations, this approach does not physically make sense; introducing interactions via a potential Vint ∝ n is valid only for a density-independent
γ. This is only the case for the extremal values γ = 2 or 1 (see Section 3.5).
Finally, we note that in previous experiments on the in situ equation of state [61,66], µ̃c was deduced by defining it so as to satisfy the theoretical expectation [98]that the phase-space density should be a universal function of (µ̃ − µ̃c )/g̃. Our
measurements of µ̃c defined through the emergence of extended coherence showthat the two definitions indeed lead to very similar values.

7.4

Error Analysis

Since we are seeking to compare our measured critical points with BKT and BEC
predictions without any free parameters, it is important to properly characterise
their associated errors. All error bars shown in Figs. (7.8, 7.9) are purely statistical, illustrating how precisely critical parameters can be determined given
experimental noise in measured quantities and the finite number of images per
critical point sequence. In addition to these statistical errors, there are various
sources of systematic error. These can be divided into arising from system calibration uncertainties, finite-size physics, the inhomogeneity of our trapped gas
and anharmonicity of our focusing trap. The procedure for estimating statistical and systematic errors is given in Appendix F, with a summary presented inTable 7.1.
Cumulatively, our critical parameter uncertainties are sufficiently small that wemay strongly exclude fully fluctuating MF predictions, denoted by green dashed7.4. ERROR ANALYSIS

111

lines in Figs. 7.8 and 7.9. Whilst our critical atom numbers display excellent
agreement with classical-field predictions for the BKT critical point, the measured
critical chemical potentials are shifted with respect to the classical-field prediction
by 0.05 . ∆µ̃ . 0.1. This shift is comparable with our combined statisticaland systematic errors; within experimental uncertainty we cannot conclude any
significant discrepancy.

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CHAPTER 7. MAPPING THE BKT PHASE DIAGRAM

Chapter 8
Conclusions and Outlook
In this experimental work, we have systematically identified the onset of extended
coherence in a harmonically-trapped quasi-two-dimensional gas over a wide range
of interaction strengths. Without any free parameters we extracted the criticalatom number and chemical potential, comparing them with various predictions
for both the BKT and BEC phase transitions. Our central conclusions are twofold.
First, by identifying the critical point over a wide range of g̃, we showed that thecritical point for the normal-superfluid BKT transition does indeed converge onto
the ideal gas BEC result. This connection between the interaction-driven BKT
and statistically-driven BEC transitions would not be observable in an infinite
uniform potential.
Second, we observe excellent agreement of the measured critical atom number
and chemical potential with the predictions of a classical-field analysis of the
BKT transitions [96, 98, 99]. This agreement is sustained over an order of magnitude in interaction strength 0.05 . g̃ . 0.5; the agreement may persist forhigher values of g̃, but we were precluded from addressing these by experimental
limitations1 . This agreement emphasises the crucial importance of beyond-MF behaviour in the vicinity of the BKT transition, and offers support for the validity ofusing classical-field simulations to model the non-perturbative critical behaviourof interacting bosons. Mean-field predictions for the critical parameters, whichassume full bosonic density fluctuations, give a poor description of the criticalparameters.
Furthermore, a subtlety worthy of mention is that the classical-field predictions
were made for a pure-two-dimensional system, with no excitations or dynamics included in the axial direction. Our measurements were (unavoidably) forthe case of quasi-two-dimensional trapping, with . 30% of atoms occupying excited axial states. The quality of agreement between the calculated ground state
atom number and the pure-two-dimensional classical-field predictions supports
1

Experiments on the equation of state [61, 66] suggested the classical-field results to remain
accurate at least up to g̃ ≈ 3.

113

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CHAPTER 8. CONCLUSIONS AND OUTLOOK

the conclusions of Section 4.1 that the axial ground state should display puretwo-dimensional physics, with the caveat that the superfluid be kinematically
two-dimensional. In our case the ratio of trap spacing to superfluid interactionenergy ~ωz /(gns ) ≈ 3.6 for our highest interaction strength g̃ ≈ 0.5, and thiscondition is well-satisfied.
In future work, it would be very interesting to address the dimensional crossoverfrom two-dimensional to three-dimensional behaviour, and investigate at which
point pure-two-dimensional predictions become invalid. In particular, in Section 4.1 we suggested that as collective superfluid excitations become graduallypossible in the axial direction, the jump in superfluid fraction at the BKT transition should smoothly disappear.
The recent creation of two-dimensional traps within a uniform planar trapping
potential at ENS [105] offer exciting opportunities for studies of non-equilibrium
critical phenomena such as the nucleation of phase defects [105]; one could extend recent experiments on the Kibble-Zurek mechanism in the three-dimensional
homogenous Bose gas [124] to two dimensions. Another experiment facilitatedby a uniform potential would be a conclusive measurement of the predicted algebraic correlation function for a BKT superfluid. Whilst several experiments
have addressed the form of the correlation function in harmonic systems [9, 57]
measurements were not conclusive; the use of a Ramsey-interferometric method
for obtaining correlation functions [57, 124] would offer a quantitative measure ofg (1) for a uniform system.

Appendix A
Publications

115

116

APPENDIX A. PUBLICATIONS

117

Connecting Berezinskii-Kosterlitz-Thouless and BEC
Phase Transitions by Tuning Interactions in a Trapped
Gas
Richard J. Fletcher*, Martin Robert-de-Saint-Vincent*, Jay Man, Nir Navon,
Robert P. Smith, Konrad G. H. Viebahn, and Zoran Hadzibabic
Phys. Rev. Lett. 114, 255302 (2015)

118

APPENDIX A. PUBLICATIONS

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PHYSICAL REVIEW LETTERS

PRL 114, 255302 (2015)

Connecting Berezinskii-Kosterlitz-Thouless and BEC Phase Transitions
by Tuning Interactions in a Trapped Gas
Richard J. Fletcher,* Martin Robert-de-Saint-Vincent,† Jay Man, Nir Navon, Robert P. Smith,
Konrad G. H. Viebahn, and Zoran Hadzibabic
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom(Received 12 January 2015; revised manuscript received 22 March 2015; published 26 June 2015)
We study the critical point for the emergence of coherence in a harmonically trapped two-dimensional
Bose gas with tunable interactions. Over a wide range of interaction strengths we find excellent agreementwith the classical-field predictions for the critical point of the Berezinskii-Kosterlitz-Thouless (BKT)
superfluid transition. This allows us to quantitatively show, without any free parameters, that theinteraction-driven BKT transition smoothly converges onto the purely quantum-statistical Bose-Einsteincondensation transition in the limit of vanishing interactions.
DOI: 10.1103/PhysRevLett.114.255302

PACS numbers: 67.85.-d, 03.75.Hh, 05.30.-d

Reducing the dimensionality of a physical systemincreases the importance of thermal fluctuations and can
profoundly affect the type of order that the system can display
at low temperatures [1–4]. In a uniform two-dimensional(2D) Bose gas, the true long-range order associated with
Bose-Einstein condensation (BEC) is precluded at any nonzero temperature by the Mermin-Wagner theorem. However,an interacting 2D Bose gas still undergoes the BerezinskiiKosterlitz-Thouless (BKT) transition to a superfluid state at a
critical phase-space density DBKT [5,6]. Quantitatively predicting DBKT requires an accurate description of the nonperturbative behavior of interacting bosons in the fluctuation
region near the critical point. Classical-field simulations [7]model this behavior by a turbulent matter-wave field [8],and predict DBKT ¼ ln ð380=~gÞ, where g~ is a dimensionlessmeasure of the interaction strength [9]. This result makes
it manifest that the transition is interaction driven; the criticaltemperature T c ∝ n=DBKT , where n is the gas density,
vanishes in the noninteracting limit g~ → 0 for any noninfinite
n. While for phase-space density D > DBKT true long-rangeorder is still absent, the first order correlation function g1 ðrÞ
decays only algebraically at large distance, in contrast to
the exponential decay in the normal degenerate gas. Suchextended coherence is sufficient for superfluidity [10], and inpractice offers a signature of the phase transition [11–14].
In contrast to the infinite uniform system, in a 2D
harmonic trap, pertinent to most ultracold-atom experiments
on BKT physics [15–29], the modified density of statesallows for a BEC transition to occur in the ideal gas (~g ¼ 0)
[30,31] (see also Ref. [32]). In an isotropic trap of frequency
ωr , it should occur at a critical atom number [30,31]
N 0c ¼
π 2 kB T 2
:
6 ℏωr

ð1Þ

This ideal-gas BEC transition is similar to the familiar
condensation in three dimensions; it is purely quantum
0031-9007=15=114(25)=255302(5)statistical and follows from Einstein’s standard argument of
the saturation of the excited states. However, the importanceof dimensionality emerges if interactions are introduced. In
three dimensions, ideal-gas BEC occurs when in the trap
center D ≈ 2.612; weak interactions slightly shift the criticalatom number [33–35], but do not alter the BEC-like natureof the transition. In two dimensions, while N 0c is finite, thephase-space density required in the trap center for theexcited states to saturate is infinite, just as in a uniform
system [14]. For any g~ > 0 this is unattainable and theexcited states can accommodate any number of particles[36]. The BEC transition is thus suppressed and one expects
it to be replaced by the BKT transition with noninfinite
DBKT [10].
These arguments suggest that the two conceptually very
different phase transitions, the interaction-driven BKT andthe saturation-driven BEC, are in fact continuously connected as g~ → 0 [10,37]. The harmonic trapping potential
offers the opportunity to experimentally observe thisunification of BKT and BEC physics. While in an infinite
uniform gas no transition occurs for g~ ¼ 0, in a harmonic
trap a transition always occurs at a finite N c and always
results in significantly extended coherence of the gas.
The nature of this transition is quantitatively encoded in
the value of N c . This picture is supported by the calculations of the critical atom number N BKT[10,38], based oncthe classical-field simulations [7,39], which suggest that
N BKT
smoothly connects to N 0c in the limit g~ ¼ 0. In acfinite-size system the change from the BKT to the BEC
transition is a crossover that spans a nonzero range of g~values, but for realistic experimental parameters the widthof this crossover region is very small, g~ ≲ 10−2 [18,40].
Various signatures of a phase transition, including
emergence of extended coherence [15,17–19,21] andsuperfluidity [24], have been observed in trapped 2D gaseswith different specific g~ values. On the other hand,systematic studies with a tunable g~ have focused on in situ

255302-1© 2015 American Physical Society

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PHYSICAL REVIEW LETTERS
(a)

6s

(b)

11 s

16 s

tc

1

P0 (a.u.)measurements of the equation of state [22,27], which do notdirectly reveal any striking signatures of the infinite-order
BKT transition.
In this Letter, we systematically study the critical point
for the emergence of extended coherence in a harmonicallytrapped 2D Bose gas over a wide range of interaction
strengths, 0.05 < g~ < 0.5. We show, without any freeparameters, that N c generally agrees very well with thebeyond-mean-field calculation of N BKT[38], and convergesconto N 0c of Eq. (1) as g~ → 0. The critical chemical potential
μc , which directly reveals uniform-system conditions for aphase transition to occur in the trap center, also agrees with
the BKT theory and converges onto the BEC value, μc ¼ 0,
for g~ → 0. Our measurements also reiterate the importanceof the suppression of density fluctuations in the normal
state near the BKT critical point, previously observed inRefs. [18,19,21–23].
The experiment was carried out using a 39 K gas, in theapparatus described in Ref. [41]. For 2D trapping, thetight axial (vertical) confinement is provided by two
repulsive “blades” of blue-detuned light, formed by passing
a 532-nm Gaussian beam through a 0-π phase plate [20,42],
while a red-detuned 1064-nm dipole trap provides thein-plane (horizontal) confinement. The radial and axialtrapping frequencies are ðωr ; ωz Þ ≈ 2π × ð38; 4100Þ Hz.
For all of our measurements T ∈ ½140 nK; 190 nK and
μ=kB < 100 nK, resulting in a small (< 30%) occupationof thepﬃﬃﬃﬃﬃﬃexcited axial states. The interaction strengthg~ ¼ 8π a=lz [14], where a is the s-wave scattering length
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
and lz ¼ ℏ=ðmωz Þ, is controlled via a Feshbach resonance centered at 402.5 G [41,43].
To characterize long-range coherence of a gas we study
its (in-plane) momentum distribution nðkÞ [19]. A change inthe functional form of g1 ðrÞ leads to a dramatic change in its
values at
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ much larger than the thermal wavelength
pdistances
λ ¼ h= 2πmkB T [14], and an increase of coherence oversome large distance L manifests itself in enhanced population of the low-momentum states k ≲ 2π=L. Thus, unlike
the in-trap density distribution, which varies very smoothly
through the BKT critical point [18,19,22,23], nðkÞ canprovide a dramatic signature of the phase transition [19].
As illustrated in Fig. 1, to identify the critical point for agiven g~ , we start with a highly coherent 2D gas and measurenðkÞ after holding the cloud in the trap for a variable time t.
During the hold time, the atom number N slowly decays
through various inelastic processes [44], while the elasticcollision rate (≈ 0.2N g~ 2 s−1 ) remains sufficiently high toensure that the gas is in quasistatic equilibrium. To measurenðkÞ, we employ the “momentum focusing” technique[19,29,45,46]. We turn off just the tight z confinement,so the rapid vertical expansion (predominantly driven by
the zero-point motion along z) removes all the interactionenergy on a time scale 1=ωz ≪ 1=ωr . The subsequenthorizontal ideal-gas evolution in the remaining in-plane
harmonic potential reveals nðkÞ as the spatial distribution04
N (104)

PRL 114, 255302 (2015)

APPENDIX A. PUBLICATIONS

Nc
0
0

10

20
Hold time (s)30FIG. 1 (color online). Determination of the critical point for the
onset of coherence, for g~ ¼ 0.28 and T ≈ 140 nK. (a) Evolutionof the momentum distribution nðkÞ with the hold time t (see text).
Extended coherence is revealed as a sharp peak in nðkÞ. Each
image is an average of three experimental realizations. (b) Evolution of the momentum-distribution peak P0 and the smoothly
decaying total atom number N. We associate the thresholdlike
behavior of P0 with the critical time tc and deduce thecorresponding N c . The solid line is a heuristic piecewise fit
function used to determine tc [47].

after a quarter of the trap period. We probe this distribution
by absorption imaging along z [see Fig. 1(a)].
Our k-space imaging resolution, Δk ≈ 0.4 μm−1 , sets
the largest distance over which we can probe coherence
to L ¼ 2π=Δk ≈ 15 μm, which is much larger thanλ ≈ 0.7 μm. To probe coherence on this length scale, we
simply monitor the peak value of the momentum distribution, P0 , without making any theoretical assumptions about
the exact shape of nðkÞ at low k. To get the corresponding
atom number N we do a simple summation over the image.
Importantly, we eliminate the systematic error due to theuncertainty in the absorption-imaging cross section by
independently calibrating our imaging system throughmeasurements of the BEC critical point in a 3D gas [47].
In Fig. 1(b) we show a typical evolution of P0 and N(here g~ ¼ 0.28). While N decays smoothly, P0 shows two
distinct regimes, which allows us to identify the criticalhold time tc and the corresponding N c . We note that evenfor N significantly below N c the peak of nðkÞ rises above a
Gaussian fitted to the wings of the distribution, indicatingsome coherence on a length scale > λ [18,21]. The smoothevolution of such non-Gaussian “peakiness” of nðkÞ does
not reveal a phase transition [21], and only P0 corresponding to L ≫ λ shows a clear change in behavior at awell-defined N c [51]. Our large L is stillp
small
compared
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
to the thermal diameter of the cloud, 2 kB T=ðmω2r Þ ≈50 μm, so the observed N c is closely linked to the255302-2

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PRL 114, 255302 (2015)

occurrence of a phase transition in the center of thetrap [47].
For comparisons with theory, we also fit μ and T to eachnðkÞ image. Unlike in three dimensions, in two dimensions
interactions affect nðkÞ appreciably even in the normal
state, and near the critical point it is in general insufficient
to treat them at a mean-field (MF) level. However, beyondMF correlations primarily affect the highly populated low-k
states [38]. We restrict our fits to the high-k wings ofthe distribution (ℏ2 k2 > 2~gmkB T), where we expect the
beyond-MF effects to be small, and still carefully include
the effects of interactions at a MF level [47]. Following
Ref. [37], we also account for the residual thermal
occupation of the axial excited states and the interactioninduced deformation of the axial eigenstates.
In Fig. 2 we summarize our measurements of the criticalatom number for a wide range of interaction strengths. To
compare our data with the strictly 2D theoretical calculations, we correct the observed “raw” N c by subtracting the
calculated population of the excited axial states [47]. We
scale this corrected critical number N̄ c to the BEC criticalatom number N 0c of Eq. (1) and plot it versus g~ . Our Δklimited value of L imposes a lower bound on g~ for which we
can reliably identify the critical point. In the absence ofany phase transition, in a weakly interacting degeneratepﬃﬃﬃﬃﬃﬃ gas
g1 ðrÞ ∼ expð−r=l0 Þ, with l0 ¼ λ expðD=2Þ= 4π [14].
We thus do not expect our measurements to reliably identify
N c if l0 > L for some D < DBKT . This occurs forg~ < 380λ2 =ð4πL2 Þ ≈ 0.06, indicated by the shaded areain Fig. 2. Our measurements also stop being reliable forg~ ≳ 0.5; in that regime our MF temperature fits are
restricted to very high k values, which are affected by the

Nc/N0c

2

1
BKT
BEC

0
0.0

0.1

0.2

0.3

0.4

0.5

g

FIG. 2 (color online). Critical atom number as a function of theinteraction strength g~ . All numbers are scaled to the ideal-gas
BEC critical number N 0c, defined in Eq. (1). The solid line is theclassical-field BKT prediction of Eq. (2), without any freeparameters. The dashed line is an approximation that neglects
suppression of density fluctuations in the normal state. The starð⋆Þ denotes the critical point for BEC, which only occurs in theideal-gas limit. The shaded region, g~ < 0.06, indicates the regime
in which our measurements stop being reliable (see text). The
error bars are statistical.

week ending
26 JUNE 2015

anharmonicity of the optical trap. The error bars in Fig. 2 arestatistical, while the systematic uncertainty in N̄ c =N 0c is≲0.2 [47].
Without any free parameters, we find generally excellent
agreement with the prediction of Ref. [38]:
 

 
N BKT
3~g 2 g~
6~g
g~
c
þ
;
15
þ
ln
≈
1
þ
ln
0
3
2
16
16
π
16π
Nc

ð2Þ

which is based on fixing the phase-space density in the trap
center to DBKT and integrating a uniform-system equationof state over the trap, using the classical-field resultsof Ref. [39].
The agreement with Eq. (2) over a very broad range ofinteraction strengths and the proximity of our lowest
reliable g~ values to zero allow us to conclude that thecritical atom number, without any free parameters, indeedsmoothly converges onto the BEC result of Eq. (1).
It is instructive to also compare our data with theapproximation N BKT=N 0c ¼ 1 þ 3~gD2BKT =π 3 [10,12], showncby the dashed line in Fig. 2. Here, the critical phase-spacedensity is again set to DBKT , but the suppression of bosonicfluctuations in the normal state is neglected; i.e., the density
profile is calculated using MF theory with an interaction
potential 2gnðrÞ, where g ¼ ðℏ2 =mÞ~g. Our data strongly
exclude this result, confirming the importance of the
suppression of density fluctuations near the critical point
even for our lowest g~ values.
For a more direct comparison with the uniform-systemtheory, we also consider the critical chemical potential forthe onset of coherence. Like N c in Fig. 1, μc is experimentally defined via the critical hold time tc . The classicalfield simulations [7] predict DBKT to be reached for
μBKT¼ kB Tð~g=πÞ ln ð13.2=~gÞ, which reduces to the BEC
cprediction, μc ¼ 0, for g~ ¼ 0.
In Fig. 3 we plot μ~ c ¼ μc =ðkB TÞ versus g~ , and again
observe generally good agreement with the classical-field
prediction (solid line), all the way down to g~ ≈ 0.06, i.e.,very close to the expected BEC limit. The small systematicdifference between the data and the theory is comparable
to our systematic uncertainty in μ~ c of ∼0.05 [47].
We also compare our data with two intuitive approximations to μ~ c . We consider interaction potentials γgn withγ ¼ 2, corresponding to a fully fluctuating Bose gas, and
γ ¼ 1, corresponding to a complete suppression of density
fluctuations. In both of these extremes one can analytically write Dγ ðμÞ~ ¼ − ln ½1 − exp ð~μ − γgn=ðkB TÞÞ [14].
Defining μ~ γc so that Dγ ðμ~ γc Þ ¼ DBKT we obtain the dashed(γ ¼ 2) and dotted (γ ¼ 1) lines in Fig. 3. Generally,
γ ¼ 1 provides a better approximation, highlighting how
strong the suppression of density fluctuations in the normal
state is.
Finally, we note that in previous experiments [22,27], on
the in-trap equation of state, μ~ c was deduced by defining it
so as to satisfy the theoretical expectation [7,39] that the255302-3

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APPENDIX A. PUBLICATIONSPHYSICAL REVIEW LETTERS

PRL 114, 255302 (2015)
0.6

µc

0.4

0.2

BKT
BEC

0.0
0.0

0.1

0.2

0.3

0.4

0.5

g

FIG. 3 (color online). Critical chemical potential as a functionof the interaction strength g~ . The solid line is the classical-field
BKT prediction. The dashed line is derived assuming a fully
fluctuating Bose gas, while the dotted line corresponds to acomplete suppression of density fluctuations. The error bars are
statistical.

phase-space density should be a universal function of
ð~μ − μ~ c Þ=~g. Our measurements of μ~ c , defined throughthe emergence of extended coherence, show that the two
definitions indeed lead to very similar values.
In conclusion, by studying the critical point for the
emergence of extended coherence in a harmonicallytrapped 2D gas with tunable interactions, we have quantitatively confirmed the predictions of classical-field theory
and observed the expected unification of BKT and
BEC transitions in the limit of vanishing interactions.
The in-plane harmonic potential enables this observation.
However, to quantitatively study the exact functional formof the slowly decaying correlations in a BKT superfluid, inthe future it would be very interesting to study coherence ofa tunable 2D gas in a uniform potential [52–54]. Just below
T c this should reveal an interaction–strength–independent
algebraic decay of the first-order correlation function,corresponding to a universal jump in the superfluid
density [55].
We thank Naaman Tammuz and Alex Gaunt forexperimental contributions in the early stages of the
project, Shannon Whitlock for help with image processing,and Andre Wenz, Markus Holzmann, and Jean Dalibard
for useful discussions. This work was supported by
AFOSR, ARO, DARPA OLE, and EPSRC [Grant
No. EP/K003615/1]. N. N. acknowledges support from
Trinity College, Cambridge, R. P. S. from the Royal
Society, and K. G. H. V. from DAAD.
R. J. F. and M. R. dS. V. contributed equally to this work.

*

rjf45@cam.ac.uk
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APPENDIX A. PUBLICATIONS

125

Connecting Berezinskii-Kosterlitz-Thouless and BEC phase transitions
by tuning interactions in a trapped gas
Supplemental Information
Richard J. Fletcher? , Martin Robert-de-Saint-Vincent? ,Jay Man, Nir Navon, Robert P. Smith, Konrad G. H. Viebahn, and Zoran Hadzibabic

Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

Absorption image analysis

Fringe removal: Optical density (OD) images are usuallyaffected by fringes due to the temporal variations in the probebeam. We eliminate this effect using the fringe-removal algorithm of Ockeloen et al. [1].
Imaging at high magnetic field: Near the Feshbach resonance our atoms are in the electronic ground state approximately described by |4S1/2 , mJ = −1/2, I = 3/2, mI =3/2i, and are imaged on the transition to the excited state|4P3/2 , mJ = −3/2, I = 3/2, mI = 3/2i. However, due
to the residual hyperfine mixing in the ground state manifold,the imaging transition is not perfectly cycling, and the probability of optical pumping into other ground states is ≈ 3%
per cycle. Consequently, the relation between the observed
OD and the atomic column density depends on the numberof photons scattered per atom, which in turn varies with theinhomogeneities of the probe beam.
To correct for this effect, we experimentally calibrate therelative variation of the observed OD(~r) with the camera
counts, C(~r), in the reference image taken without atoms.
For this, we image a 3d thermal cloud after expansion froma dipole trap. Assuming cylindrical symmetry of the cloud,any variations in OD observed along a ring concentric withthe cloud centre we attribute to the variations in the probe intensity. Averaging over several images, we build, pixel by
pixel, a correlation plot of OD/OD0 versus C/C0 , where C0
is a fixed reference value and OD0 the corresponding optical density. We construct such plots for various ring radii (i.e.
various atomic column densities) and various (mean) probe intensities, and find that to a very good approximation the scaling of the optical density due to the optical pumping depends
only on the light intensity and not on the atomic column density. We thus finally create an interpolated rescaling function
F (C/C0 ) = OD0 /OD, and correct our images according to
ODscaled (~r) = OD(~r) · F (C(~r)/C0 ). This relative calibration corrects for both the optical pumping and for the Doppler
shifting of atoms with increasing number of scattering events.
Imaging-intensity optimisation: In our main experimental sequences we take two sets of interlaced images, underidentical conditions, but with two different (mean) imagingbeam intensities: a “low” intensity I0 ≈ 0.08 mW/cm2 anda “high” intensity I1 ≈ 0.3 mW/cm2 (which is still well below the saturation intensity Isat ≈ 1.8 mW/cm2 ), for which
the scaling function F is larger by a factor of 2.7. The low-

intensity images are used for determining N , T and µ, while
the high-intensity images are used just to determine, with alarger dynamic range, the high-OD momentum peak P0 .
Absolute atom-number calibration

To calibrate our absolute imaging detectivity, we measure the BEC critical point in a 3d gas [2], prepared in
an optical dipole trap of frequencies (ωx , ωy , ωz ) = 2π ×(64.3(1.0), 54.0(6), 93.9(2)) Hz. We calibrate the observed
critical atom number on the theoretical value:
Nc = A

Ncidealfs
3 + δNc .

(1)[1 − 3.426(a/λ) + 42(a/λ)2 ]Here, Ncideal = ζ(3)(kB T )3 /(~3 ωx ωy ωz ), where ζ is the Riemann function, the denominator in Eq. (1) corresponds tomean-field (MF) and beyond-MF corrections [2], A ≈ 1.1 isthe trap-anharmonicity correction for our Gaussian potential,and δNcfs is the finite size-correction [3]. For probe intensity
I0 and image-pulse duration of 80 µs we obtain an effective
absorption cross section σeff = 0.25(1) × 3λ20 /(2π), where
λ0 ≈ 767 nm is the imaging wavelength.
Extraction of thermodynamic parameters

Model functions: To extract µ and T from our images,
we fit a model to the azimuthally-averaged momentum distributions. By constraining the fit to the wings where ~2 k 2 >2mkB T g̃, we ensure that interaction effects are described wellwithin a mean-field (MF) framework [4].
The model we use includes the thermal occupation of theexcited axial states and the interactions between atoms in all
axial states, as in [5, 6], and also the axial deconfinement of
the cloud due to the repulsive interactions, as in [7].
We construct the model 2d momentum distribution for anin-plane potential V (x, y) within the 2d local density approximation. Defining the local chemical potential µL (x, y) =µ − V (x, y), and writing ~r = (x, y), we haven(~k) =

XZ
j

2

"

d r exp~2 k 2r) − µL (~r)
2m + εj (~

kB T

!

#−1

−1

,

(2)

126

APPENDIX A. PUBLICATIONS
2
Statistical

Systematics
Calibration Finite size Anharmonicity

N̄c /Nc0µ̃c± 0.08± 0.03± 0.14± 0.05

+0.04

−0.01

+0.005−0.04

TABLE I. Absolute standard uncertainties (68% confidence interval)and estimated systematic shifts affecting the measured N̄c /Nc0 andµ̃c . Statistical errors are shown as error bars in the main paper. All
values are typical for the experimentally explored range of g̃. The
positive (negative) systematic shifts correspond to the measured values being higher (lower) than the true ones.

where {εj }(~r) are the eigenenergies of the axial states of
motion, {|φj (z)i}(~r), measured from the zero-point energy~ωz /2. Due to interactions, they are not simply j~ωz , but
depend on µL (~r).
To evaluate the set {εj }(µL ) we solve the 1d axial problem
defined by two coupled equations:µL − εj1 X
ln 1 − exp|φj (z)|2 , (3)n3 (z) = − 2
λ jkB Twhere n3 is 3d density, and−~2 d2mωz2 z 2
~ωz
+
−
+
2g
n
(z)
φj (z) = εj φj (z),332m dz 22
2
(4)
where g3 = 4π~2 a/m. We iteratively find a self-consistentsolution to Eqs. (3-4), and then compute n(~k) in Eq. (2).
Critical parameters: To identify the critical time tc , we fit
the following heuristic function to P0 (t):P0 (t) = Pc + Θ(t − tc ) · c · |t − tc |(5)

+ Θ(tc − t) · a [1 − exp(−b|t − tc |)] ,where Θ(t) is the Heaviside function. In the limit b  1/|t −tc |, this is simply a piecewise linear function, while the nonzero b allows for the saturation of the peak OD at short t.
To determine N (tc ), T (tc ) and µ(tc ) we fit the smooth time
evolutions of N , T and µ by third-order polynomials.
Finally, we use the above MF model to compute the occupations of the j > 0 axial states for (T (tc ), µ(tc )), and correct
Nc to get N̄c . This operation assumes that beyond-MF effects
affect only the axial ground state. This approximation is justified by the fact that ~ωz /kB T > g̃ [4].
Momentum-space resolution

Our k-space resolution is limited by the camera pixel sizeand by the slight anisotropy of the in-plane harmonic potential in which we perform the momentum focusing. After removing the green “blade” beams, we have ωx,y = ω̄ ± ∆ω,with ω̄/(2π) = 36.3(1) Hz and ∆ω/(2π) = 2.5(1) Hz. Thismeans that the focusing is not obtained simultaneously in both
directions, but at the focusing time tf = π/(2ω̄) ≈ 7 ms
the blurring
p of the momentum distribution is isotropic, with∆k ≈ mkB T /~2 sin(∆ω tf ) ≈ 0.4 µm−1 . This happens
to closely coincide with the resolution set by the pixel size. To
determine the height of the momentum peak, P0 , we average
the image within one pixel of the fitted cloud centre.
Confidence intervals

In Table I, we summarise the main uncertainties in the critical parameters N̄c /Nc0 and µ̃c in Figs. 2-3 in the main paper.
The statistical errors, defined as 68% confidence interval
and arising from experimental noise and fitting errors, areshown by the error bars in the main paper. Here they are averaged over the experimental points.
Systematic uncertainties in Table I fall into three categories:(i) calibration errors and drifts in the trapping frequencies,
imaging magnification and absolute detectivity, (ii) finite-size(non-zero ωr ) effects, and (iii) anharmonicity of the opticaltrapping potentials. These affect the calculated Nc0 and also
the model functions used for extracting µ and T . The calibration errors we calculate by propagating errors, the finite-size
effects are calculated following [3], and we assess the typicalmagnitude of the anharmonicity shifts by numerical simulations in the ideal-gas limit.
Additionally, two more fundamental systematic effects
could shift the apparent critical point. If we assume that the
emergence of coherence over lengthscale L requires the whole
region near the cloud centre, of radius L/2, to be superfluid,then the trap inhomogeneity leads to apparent positive shiftsof N̄c /Nc0 and µ̃c , of . 0.2 and ≈ 0.05, respectively. On theother hand, the critical divergence of the correlation length
above Tc has an opposite effect, which we estimate to be ofcomparable absolute size [8]. We thus expect these two effects to partially cancel, and lead to overall shifts that do not
exceed our statistical and systematic errors (see also [9]).

[1] C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, Phys. Rev. A 82, 061606 (2010).
[2] R. P. Smith, R. L. D. Campbell, N. Tammuz, and Z. Hadzibabic,
Phys. Rev. Lett. 106, 250403 (2011).
[3] T. Haugset, H. Haugerud, and J. O. Andersen, Phys. Rev. A 55,
2922 (1997).
[4] M. Holzmann, M. Chevallier, and W. Krauth, Phys. Rev. A 81,
043622 (2010).
[5] R. N. Bisset, D. Baillie, and P. B. Blakie, Phys. Rev. A 79,
013602 (2009).
[6] S. Tung et al., Phys. Rev. Lett. 105, 230408 (2010).
[7] Z. Hadzibabic et al., New J. Phys. 10, 045006 (2008).
[8] Z. Hadzibabic and J. Dalibard, Rivista del Nuovo Cimento 34,
389 (2011).
[9] R. N. Bisset, M. J. Davis, T. P. Simula, and P. B. Blakie, Phys.
Rev. A 79, 033626 (2009).

127

Stability of a Unitary Bose Gas
Richard J. Fletcher, Alexander L. Gaunt, Nir Navon, Robert P. Smith, and Zoran
Hadzibabic
Phys. Rev. Lett. 111, 125303 (2013)

128

APPENDIX A. PUBLICATIONS129
PRL 111, 125303 (2013)

PHYSICAL REVIEW LETTERS

week ending
20 SEPTEMBER 2013

Stability of a Unitary Bose Gas
Richard J. Fletcher, Alexander L. Gaunt, Nir Navon, Robert P. Smith,* and Zoran Hadzibabic
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom(Received 10 July 2013; published 20 September 2013)
We study the stability of a thermal 39 K Bose gas across a broad Feshbach resonance, focusing on theunitary regime, where the scattering length a exceeds the thermal wavelength !. We measure the generalscaling laws relating the particle-loss and heating rates to the temperature, scattering length, and atom
number. Both at unitarity and for positive a ! ! we find agreement with three-body theory. However, fora < 0 and away from unitarity, we observe significant four-body decay. At unitarity, the three-body loss
coefficient, L3 / !4 , is 3 times lower than the universal theoretical upper bound. This reduction is aconsequence of species-specific Efimov physics and makes 39 K particularly promising for studies ofmany-body physics in a unitary Bose gas.
DOI: 10.1103/PhysRevLett.111.125303

PACS numbers: 67.85."dThe control of interactions provided by Feshbach resonances makes ultracold atomic gases appealing for studies
of both few- and many-body physics. On resonance, thes-wave scattering length a, which characterizes two-bodyinteractions, diverges. At and near the resonance a gas is in
the unitary regime, where the interactions do not explicitly
depend on the diverging a. Instead, a is replaced byanother natural length scale. In a degenerate gas this lengthscale is set by the interparticle spacing;pinﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
a thermal
ﬃ gas it isset by the thermal wavelength ! ¼ h= 2"mkB T , where mis the particle mass and T is the temperature.
Over the past decade, there have been many studies of
the unitary Fermi gas [1]. More recently, there has been an
increasing interest in both universal and species-specificproperties of a unitary Bose gas [2–15]. It is however anopen question to what extent this state can be studied in(quasi-)equilibrium, since at unitarity three-body recombination leads to significant particle loss and heating [16].
The severity of this instability is not universal [10], as it
depends on the species-specific few-body Efimov physics[8,18–28]. Characterizing and understanding the stabilityof a unitary Bose gas is thus important both from theperspective of Efimov physics and for identifying suitable
atomic species for many-body experiments.
The per-particle loss rate due to three-body recombination is given by_#3 $ "N=N¼ L3 hn2 i;(1)where N is the atom number, L3 is the three-body loss
coefficient, n is the density, and h% % %i denotes an averageover the density distribution in a trapped gas. Away fromunitarity, L3 & @a4 =m [29,30], with a dimensionlessprefactor exhibiting additional variation with a due toEfimov physics [19,27]. At unitarity L3 should saturate at&@!4 =m / 1=T 2 . Experimental evidence for such saturation was observed in [8,10,18]. More quantitatively, at
unitarity we expect
0031-9007=13=111(12)=125303(5)

pﬃﬃﬃ
pﬃﬃﬃ
9 3@ 4
36 3"2 @5
L3 ' $! ¼$ 3;m
m ðkB TÞ2(2)

where $ * 1 is a species-dependent, nonuniversal dimensionless constant [10] (see also Refs. [31–33]).
Similar scaling arguments apply to the two-body elastic
scattering rate, #2 , which drives continuous re-equilibrationof the gas during loss and heating. Away from unitarity#2 / hni@a2 =ðm!Þ; hence, at unitarity #2 / hni@!=m. The
possibility to experimentally explore many-body physics ofa quasiequilibrium unitary Bose gas depends on the ratio#3 =#2 . Remarkably, at a given phase-space density, n!3 ,this ratio depends only on the species-specific $.
Recently, $ ' 0:9 was measured for 7 Li [10]. The gas
was held in a relatively shallow trap, so that continuous
evaporation converted heating into an additional particle
loss, and the extraction of $ relied on theoretically modelingthis conversion and assuming the 1=T 2 scaling of Eq. (2).
In this Letter, we study the stability of the 39 K Bose gas
in the jF; mF i ¼ j1; 1i hyperfine ground state, across a
broad Feshbach resonance centered at 402.5 G [25]. We
perform experiments in a deep trap and verify the predicted
recombination-heating rate both at unitarity and for
positive a ! ! [10,30]. At unitarity we measure L3 /T "1:7+0:3 and $ ' 0:3, a value that makes 39 K particularly
promising for studies of an equilibrium unitary gas.
Additional measurements at a < 0, away from unitarity,
reveal the importance of four-body processes [20,23],
consistent with previous studies in 133 Cs [22], 39 K [25],and 7 Li [26].
Our experimental setup is described in Ref. [34]. We
start by preparing a weakly interacting (!=a ' 35) thermal
gas in a harmonic optical trap. The trap has a depth of
U ' kB , 30 %K and is nearly isotropic, with the geometric mean of the trapping frequencies ! ¼ 2" , 185 Hz.
We then tune a close to a Feshbach resonance, by ramping
an external magnetic field over 10 ms. At this point
we have N ' 105 atoms at T ' 1 %K, corresponding to125303-1! 2013 American Physical Society

130

APPENDIX A. PUBLICATIONS! ' 5 , 103 a0 , where a0 is the Bohr radius. At the trap
center n ' 3 , 1012 cm"3 and n!3 < 0:1, so even at unitarity and assuming $ ¼ 1, we still always have #2 - #3 .
We let the cloud evolve for a variable hold time, t, of up to
4 s, and then simultaneously switch off the trap and the
Feshbach field (within &100 %s [35]). Finally, we image
the cloud after 5 ms of time-of-flight expansion.
Figure 1 shows the particle loss and heating in a resonantly interacting gas (!=a ¼ 0). Restricting our measurements to T < 2 %K ensures that evaporative losses andcooling are negligible. We have taken 19 similar data series,
each at a fixed a, spanning the range "12 < !=a < 12.
We first study the relationship between T and N during
the evolution of the cloud. One expects three sources ofheating related to three-body recombination [10,30].
(i) For any a, losses preferentially occur near the centerof the cloud, where the atoms have lower potential energy.
(ii) For a > 0, recombination results in a shallow dimer
with binding energy " ¼ @2 =ðma2 Þ, and the third atom
carries away ð2=3Þ" as kinetic energy. In all our experiments " < U, so this atom remains trapped and increases
the energy of the cloud. (iii) At unitarity, three-body recombination preferentially involves atoms that also havelower kinetic energy.
To a good approximation, in our experiments we cancapture all these effects by a simple scaling law:
NT & ¼ const;(3)

with the exponent & varying across the resonance. Ignoring
unitarity effects, & ¼ 3 for a * 0, and & ¼ 3=½1 þ!2 =ð9"a2 Þ0 for a > 0 (see also [30]). In the latter case& changes as the cloud heats, but in our measurements
this variation is small enough that a constant & ¼"d½lnðNÞ0=d½lnðTÞ0 describes the data well (see inset of
Fig. 2). At unitarity, a universal value of & ¼ 1:8 waspredicted in Ref. [10].

In Fig. 2 we show our measured values of &. For!=a - 1 we find agreement with the nonunitary prediction shown by the red dashed line. However, approachingunitarity we see gradual deviation from this theory. On
resonance, we measure & ¼ 1:94 + 0:09, close to the unitary prediction of & ¼ 1:8 (indicated by the red star), andfar from the nonunitary & ¼ 3.
Moving away from unitarity into the a < 0 region (opensymbols in Fig. 2, corresponding to "2000 < a=a0 <"400), & rises further, but does not reach the expected
nonunitary limit. By analyzing the dynamics of the particle
loss, NðtÞ, we find that in this region four-body decay isalso significant (see Fig. 3); in this case our prediction
for & is not applicable. Previously, indirect evidence forfour-body decay in this region was seen in Ref. [25], but
not in Ref. [28], where the initial cloud density was significantly lower.
We fit the NðtÞ data by numerically evolving a loss
equation featuring both three- and four-body decay [22],N_ ¼ "L3 hn2 iN " L4 hn3 iN;

3

2

β

1

50
1

0

N/N0

N (103)(4)

where L3 and L4 are fitting parameters and we use themeasured TðtÞ to evaluate the thermal density averages. To
obtain purely three- (four-) body fits we fix L4 (L3 ) to zero.
In Fig. 3 we show NðtÞ for a ¼ "850a0 . The model
including both L3 and L4 provides an excellent fit to thedata, with '2 ' 1. In comparison, pure four- and threebody fits have '2 ' 5 and 7, respectively. We observe fourbody effects for all our data with "2000 < a=a0 < "400.
However, we find that they are relevant only at densities* 1012 cm"3 , which reconciles the observations of

100

0.3
1

2.0

T (µK)

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PRL 111, 125303 (2013)

0

-10

T/T0

1.7

0

10

λ/a

1.5
1.0
0.0

0.5

1.0

t (s)

FIG. 1 (color online). Particle loss and heating in a resonantlyinteracting Bose gas (!=a ¼ 0). Each point is an average of 5
measurements and error bars show standard statistical errors.
Solid red lines are fits based on Eqs. (5) and (3).

FIG. 2 (color online). Heating exponent &, as defined in
Eq. (3). The red dashed line is a result of nonunitary three-bodytheory, while the red star indicates the predicted value of 1.8 atunitarity. Open symbols indicate the region where four-bodydecay is significant (see text and Fig 3). Note that ! '5 , 103 a0 and horizontal error bars reflect its variation during ameasurement sequence at a fixed a. Vertical error bars show fitting
uncertainties. Inset: Log-log plots of N vs T (scaled to their values
at t ¼ 0) for the data series at !=a ' "5:3 (open) and 8.5 (solid).

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160

0.25

80

10-2

60

0.20

a = 700a0

3 body4 body3+4 body

t (s)

1/ν

N (103)N (103)100

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PHYSICAL REVIEW LETTERS

PRL 111, 125303 (2013)

1

0.15

0.10

1

0

5

t (s)

FIG. 3 (color online). Three- vs four-body decay for a < 0(away from unitarity). N decay at a ¼ "850a0 is fitted to amodel including both three- and four-body losses (green solidline), as well as to pure three- and four-body models (red dashedand black dot-dashed line, respectively). Inset: For comparison,
at a ¼ 700a0 , the solid green and the dashed red lines areindistinguishable, showing that four-body decay does not playa detectable role.

Refs. [25,28]. A more detailed study of this region, including any four-body resonances [22], is outside the scope of
this Letter.
For a > 0 the same analysis does not reveal any four-bodydecay (see inset of Fig. 3). In this case the pure three-body fit
and the fit including both L3 and L4 are indistinguishable,with '2 ' 1, and give the same L3 (within the 10% fitting
errors), while the pure four-body fit has '2 ' 2. Thisstrongly excludes L4 as a relevant fit parameter. Using asimilar procedure, we have also checked that for both positive and negative a we do not detect any five-body decay.
We henceforth focus on the three-body decay dynamics
at unitarity, using the a > 0 nonunitary regime for comparison. Invoking Eq. (3), in both regimes the particle loss
should be described by:
N_ ¼ "AN ( ;

10

λ/a

(5)

where A and ( are constants. Here, ( absorbs all the N and
T dependence of L3 and hn2 i. Integration gives a fitting
function NðtÞ ¼ ½Að( " 1Þt þ Nð0Þ1"( 01=ð1"(Þ . For a ! !
we expect ( ¼ 3 þ 3=&, whereas at unitarity L3 / 1=T 2
implies ( ¼ 3 þ 5=&. To test this hypothesis in an
unbiased way, we analyze our data using ( as a freeparameter.
Note that here we invoke Eq. (3) merely to anticipate thevalidity of Eq. (5) and the ( values; experimentally, ouranalysis of NðtÞ and ( is decoupled from the measurementsof TðtÞ and &. The validity of our approach is seen in Fig. 1,
where the fit of NðtÞ is based on Eq. (5). The fit of TðtÞ isthen obtained by inserting the fitted NðtÞ and & into Eq. (3).
Our fitted values of ( are summarized in Fig. 4. We see acrossover from nonunitary to unitary behavior as the

FIG. 4 (color online). Particle-loss exponent (, as defined in
Eq. (5). The red dashed line shows the nonunitary theory, ( ¼3 þ 3=&, assuming nonunitary & values. The red star shows theunitary prediction, ( ¼ 3 þ 5=&, corresponding to L3 / 1=T 2
and the measured &. Error bars are analogous to those in Fig. 2.

resonance is approached, confirming the appearance of atemperature-dependent L3 . Now combining our measurements of & and (, at unitarity we get L3 / T "1:7+0:3 , inagreement with the expected 1=T 2 scaling.
Next, using the fitted A and (, for each data series at aparticular a, and for any evolution time t, we extractpﬃﬃﬃ"2"kB TðtÞ#3
L3 ðtÞ ¼ 3 3
NðtÞ("3 A:(6)m!2
Combining all our data series, we reconstruct L3 ða; TÞ.
In Fig. 5 (main panel) we show L3 at a fixed T ¼
1:1 %K, scaled to the theoretical upper bound LM
3 ðTÞ,
6

L3 (cm6s-1)

10-3(L3 / LM
)-1/4
3

20

a = -850a0

10-23
10-24
0
1000 a0 / a

4

1

2
1
0
-2

0

2

4

6

8

λ/a

FIG. 5 (color online). Three-body loss coefficient. Main panel:"1=4(see text) at T ¼ 1:1 %K. Horizontal green line
ðL3 =LM3 Þmarks the theoretical upper bound on L3 , while the red dashedline is a guide to the eye showing the L3 / a4 nonunitary scaling.
At unitarity, L3 =LM
3 ' 0:27. Inset: L3 at 1:1 %K (open symbols)and 1:7 %K (solid symbols). The expected ratio between the two
unitary plateaux is indicated by the green vertical bar.

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APPENDIX A. PUBLICATIONSPHYSICAL REVIEW LETTERS"1=4
obtained by setting $ ¼ 1 in Eq. (2). Plotting ðL3 =LM
3 Þversus !=a clearly reveals two key effects. First, for!=a * 3, we see the nonunitary scaling L3 / a4 [37].
Second, close to the resonance, L3 saturates at ' 0:27LM
3 .
In the inset of Fig. 5 we focus on the region close to theresonance and compare L3 for two different temperatures,
T ¼ 1:1 %K and 1:7 %K. Away from the resonance,
L3 does not show any T dependence. At unitarity, theratio of the two saturated L3 values is close to the expected
1=T 2 scaling.
Finally, to refine our estimate of $, we fix ( ¼ 3 þ 5=&(i.e., L3 / 1=T 2 ) and reanalyze the three data series taken
closest to the resonance, for which j!=aj < 0:6 at all times.
This gives us a combined estimate of $ ¼ 0:29 + 0:03,
while the systematic uncertainty in $ due to ourabsolute atom-number calibration [38,39] is about 30%.
Writing L3 ¼!3 =T 2 , this corresponds to !3 ' 4:5 ,10"23 ð%KÞ2 cm6 s"1 . In the context of Efimov physics,$ ¼ 1 " e"4) [10], where ) is the Efimov width parameter[40]. We deduce ) ¼ 0:09 + 0:04 (see also [25]).
In conclusion, we have fully characterized the stabilityof a 39 K gas at and near unitarity. We have experimentally
verified the theoretically predicted general scaling laws
characterizing particle loss and heating in the unitary
regime, confirmed the relevance of four-body decay on
the negative side of the Feshbach resonance, and measured
the species-specific unitarity-limited three-body loss coefficient, L3 / 1=T 2 . The unitary value of L3 , 3 times lower
than the universal theoretical upper bound, makes 39 K a
promising candidate for experimental studies of manybody physics in a unitary Bose gas.
We thank F. Chevy, J. Dalibard, A. Grier, P. Massignan,
B. Rem, and F. Werner for useful discussions and comments on the Letter. This work was supported by EPSRC(Grant No. EP/K003615/1), the Royal Society, AFOSR,
ARO and DARPA OLE.
Note added.—Recently, a study of a degenerate unitary
85 Rb gas was reported [41].

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APPENDIX A. PUBLICATIONS135

A Superheated Bose-Condensed Gas
Alexander L. Gaunt*, Richard J. Fletcher*, Robert P. Smith, and Zoran Hadzibabic
Nature Phys. 9, 271-274 (2013)

136

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137

LETTERSPUBLISHED ONLINE: 24 MARCH 2013 | DOI: 10.1038/NPHYS2587

A superheated Bose-condensed gas
Alexander L. Gaunt† , Richard J. Fletcher† , Robert P. Smith* and Zoran Hadzibabic
Our understanding of various states of matter usually relies on
the assumption of thermodynamic equilibrium. However, thetransitions between different phases of matter can be strongly
affected by non-equilibrium phenomena. Here we demonstrateand explain an example of non-equilibrium stalling of a continuous, second-order phase transition. We create a superheatedatomic Bose gas, in which a Bose–Einstein condensate (BEC)
persists above the equilibrium critical temperature1,2 , Tc , if
its coupling to the surrounding thermal bath is reduced bytuning interatomic interactions. For vanishing interactions the
BEC persists in the superheated regime for a minute. However, if strong interactions are suddenly turned on, it rapidly
boils away. Our observations can be understood within a twofluid picture, treating the condensed and thermal componentsof the gas as separate equilibrium systems with a tunableinter-component coupling. We experimentally reconstruct a
non-equilibrium phase diagram of our gas, and theoretically
reproduce its main features.
Non-equilibrium many-body states can persist for a very longtime if, for example, a system is integrable, the transition to thelower free-energy state is inhibited by an energy barrier, or the target
equilibrium state is continuously evolving owing to dissipation.
Ultracold atomic gases offer excellent possibilities for fundamental
studies of non-equilibrium phenomena3–14 and have been used tocreate counter-intuitive states such as repulsively bound atom pairs5and Mott insulators with attractive inter-particle interactions12 .
Our superheated Bose gas is reminiscent of superheated distilledwater, which remains liquid above 100 C. Specifically, as thetemperature characterizing the average energy per particle andthe populations of the excited states rises above Tc , the cloud
remains in the partially condensed phase, which in true equilibrium
should exist only below Tc . However, there are also importantdifferences. Boiling of water is a first-order phase transition andis stalled in clean samples by the absence of nucleation centres.
In that case the transition is inhibited by an energy barrier. For
a second-order phase transition such a barrier does not exist andthe superheating we observe is a purely dynamical non-equilibrium
effect, which arises because different properties of the system evolveat different rates. In this respect our gas also bears resemblance
to the long-lived non-equilibrium spin structures observed inspinor condensates6,9 , pre-thermalized states in quenched onedimensional Bose gases14 and supercritical superfluids predicted tooccur in quenched two-dimensional gases15 . In all of those cases,
however, non-equilibrium states are observed owing to the system’sslow approach to true equilibrium. Here, the system actually evolves
away from equilibrium.
In Fig. 1 we summarize the basic idea of our experiments andthe key concepts needed to understand them. In an equilibriumgas, a BEC is present only if T < Tc , where Tc depends on the
total particle number N , or equivalently if the chemical potentialµ > µc . In a standard experiment, after a BEC is produced, it

gradually decays because T rises, owing to technical heating, and/or
Tc decreases, because N decays through various inelastic processes.
As T /Tc increases, elastic collisions redistribute the atoms between
the thermal and condensed components, aiming to ensure theequilibrium particle distribution. The BEC atom number, N0 , cantherefore decay in two ways: by direct inelastic loss, and throughelastic transfer of atoms into the thermal component. Here we
reduce the rate of the elastic particle transfer by tuning the strengthof inter-particle interactions, characterized by the s-wave scattering
length a. This protects the BEC deep into the superheated regime,
where N0 > 0 even though T > Tc .
We can understand our observations within the two-fluid
picture outlined in Fig. 1b. Here we treat the thermal and condensed
components as two coupled subsystems with atom numbers N 0 and
N0 , chemical potentials µ0 and µ0 , and instantaneous per-particle
inelastic decay rates 0 and 0 , respectively. In equilibrium µ0 = µ0 ;note that µ0 is defined only if N0 > 0, so µ0 > µc .
The two components are coupled in two ways, both dependent
on the scattering length a. First, the local kinetic thermal equilibrium between the collective excitations in the BEC (phonons) andthe thermalbath is ensured by Landau damping, the rate of whichpis / a (refs 16,17). Second, the global phase equilibrium (thatis, the equilibrium condensed fraction N0 /N ) is ensured by theelastic scattering with a rate / a2 . Crucially, owing to the different
scalings with a, we find a large parameter space where the twocomponents can be considered to be in kinetic equilibrium while
the system is not in global phase equilibrium. In other words, thetwo components are at the same temperature, but have different
chemical potentials.
In our optically trapped 39 K gas18 , we control a by an external
magnetic field tuned close to a Feshbach resonance at 402 G(ref. 19), the dominant source of 0 and 0 is spontaneous scatteringof photons from the trapping laser beams, and 0 has an additional
contribution from three-body recombination.
The key steps in our experimental sequence are summarized
in Fig. 1c. We start by preparing a partially condensed gas in
the |F ,mF i = |1,1i hyperfine ground state by evaporative coolingat a = 135a0 , where a0 is the Bohr radius18 . We then reduce a(over 50 ms) and follow the subsequent evolution of the cloud,probing the atomic momentum distribution by absorption imaging
in time-of-flight expansion. Reducing a (at constant N0 ) initiallyreduces µ0 below µ0 (ref. 13), but subsequently µ0 decays slower.
In Fig. 2 we quantitatively contrast the equilibrium evolution of
a cloud at a = 83a0 and the non-equilibrium evolution at 5a0 . In
both cases we start at time t = 0 (Fig. 1c) with N0 ⇡ 2 ⇥ 104 andN ⇡ 2⇥105 at T ⇡ 160 nK. In both cases Tc decreases at a similar rate
owing to similar N decay. At 5a0 , the temperature rises faster owing
to less effective evaporative cooling at a fixed optical trap depth.
Whether the gas is in equilibrium or not, it can always be
characterized by two extensive variables, the total particle number
N and energy E. We measure these quantities by direct summation

Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK. † These authors contributed equally to this work.
*e-mail: rps24@cam.ac.uk.
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APPENDIX A. PUBLICATIONSLETTERSa

NATURE PHYSICS DOI: 10.1038/NPHYS2587
b

βµ

c

a

Cooling

Non-equilibrium
evolution

Elastic

κ

µ′

βµc

1

µ0

βµ

LandauT/T c(N)

Γ′

µ0

Γ0
βµc

Superheated

µ′

0

Time

Figure 1 | Creating and understanding a superheated Bose-condensed gas. a, In equilibrium, a BEC is present if T < Tc or equivalently µ > µc (here,= 1/(kB T)). The arrow indicates the cooling trajectory along which a BEC is produced. The insets show measured momentum distributions, with thecondensed component indicated in red. b, Two-component picture. The thermal and condensed components have chemical potentials µ0 and µ0 , andinelastic decay rates 0 and 0 , respectively. The net flow of particles between the two components, , depends on µ0 , µ0 and the scattering rate / a2 . Inpequilibrium µ0 = µ0 = µeq . The Landau damping of the collective modes in the BEC has a rate / a. c, Time sequence of the experiment. Reducing a after
preparing a BEC reduces the coupling between the two components and extends the condensate lifetime. In the superheated regime µ0 < µeq < µc butµ0 > µc .

N 0 (103)

b

83 a0

20

N0

N0eq

¬
t = tc

10

5 a0

20
N 0 (103)

a

¬
t

tc

10

0

0

T eq

240

240
T (nK)

180

120

0.0

0.0

µ0
¬0.4

µ′
µ eq

¬0.8
0

Superheated

180

120

β (µ − µ c)

β (µ − µ c)

T (nK)

Tc

–0.4

–0.8
20

40
Time (s)

60

0

20

40
Time (s)60

Figure 2 | Equilibrium versus non-equilibrium BEC decay. a, At a = 83a0 the cloud is always in quasi-static equilibrium. The measured N0 is in excellent
eq
agreement with the predicted N0 and vanishes when T eq = Tc ; the three separately calculated chemical potentials, µ0 , µ0 and µeq , all agree with eachother. The dotted green line marks the equilibrium critical time, tc , and the dashed red lines show the experimental bounds on the time t̄ when the BEC
actually vanishes. b, At 5a0 , the BEC persists in the superheated regime (T eq > Tc ) for t̄ tc ⇡ 40 s.

of the momentum distribution and its second moment. To measure
N0 we count the atoms within the central peak rising above thesmooth thermal distribution.
From the measured N (t ) alone we calculate the equilibrium
Tc (t ) (ref. 2). From N (t ) and E(t ) we calculate the equilibrium
intensive thermodynamic variables µeq (t ) and T eq (t ), and theeqequilibrium number of condensed atoms, N0 (t ) (refs 2,20,21);eq
in these calculations N0 > 0 if and only if T eq < Tc . For
comparison, we also directly fit a temperature T f to the wings ofthe momentum distribution. In addition, supposing only that thetwo components are separately in equilibrium, from the measured
N0 and N 0 we calculate µ0 and µ0 . (For theoretical details see
Supplementary Information.)
272

At 83a0 (Fig. 2a) we find excellent agreement between theeqmeasured N0 and the N0 predicted without any free parameters.
The BEC vanishes exactly at the equilibrium critical time tc (dottedgreen line), at which T eq = Tc . Note that the dashed red lines show
the experimental bounds on the time t̄ when the BEC vanishes. The
separately calculated µ0 , µ0 and µeq are all consistent and we havealso checked that the fitted T f coincides with the calculated T eq . All
this gives us full confidence in our equilibrium calculations.
At 5a0 (Fig. 2b) we observe strikingly different behaviour. The
BEC now survives much longer than it would in true equilibrium;t̄ tc ⇡ 40 s. We also see that µ0 and µ0 diverge from each otherfor t > tc , so the system is moving away from the global phaseequilibrium rather than towards it. The observed superheating can
NATURE PHYSICS | VOL 9 | MAY 2013 | www.nature.com/naturephysics© 2013 Macmillan Publishers Limited. All rights reserved

139

LETTERS

NATURE PHYSICS DOI: 10.1038/NPHYS2587a20

N0 (103)

1.6

10
0
0

10

2
Time (s)1.4¬
T / TcN0 (103)

20

4

1.2

1.0

0
20
Time (s)030Figure 3 | Quenching the superheated Bose-condensed gas. Filledeq
symbols show the evolution of N0 at a = 3a0 , the green solid line shows N0
and orange shading indicates the superheated regime. Open symbols show
the rapid decay of the BEC after it is strongly coupled to the thermal bathby an interaction quench to a = 62a0 at time tq . We show two experimental
series in which tq = 20 s (black) and 30 s (red). Inset: an interaction quencheq
at a time when T < Tc and N0 > 0 does not kill the BEC. (Note that here we
make an even stronger quench, to 100a0 .)

b

40
60
a (a0)

80

100

50

30

T < TcNo BEC

30
10
¬10

0

40
80
Time (s)10Superheated0thus not be understood as just a transient effect. (Note that µ0 –µc
is always very small owing to weak interactions.)
At 5a0 the gas is not in global phase equilibrium, but it isstill a good approximation to view its two components as two
equilibrium subsystems at the same (kinetic) temperature, as inFig. 1b. We have checked that the momentum distribution in
the non-condensed component is still fitted well by a thermal
distribution, with T f always within 10% of the calculated
T eq (N , E) (see Supplementary Information). For the BEC, ina weakly interacting gas the equilibrium relation µ0 (N0 ) relieson the macroscopic occupation of a single quantum state1 ,rather than on global equilibrium. Moreover, even for thelowest-energy collective modes we estimate the Landau damping
time to be <1 s (refs 16,17; see Supplementary Information),that is, much shorter than the characteristic timescale of ourexperiments. Thus, although this distribution is not directly
measurable, we expect the distribution of collective excitationsin the BEC to be characterized by a temperature T0 that is
also close to T eq ⇡ T f .
These conclusions hold for any a >⇠ 1a0 (see Supplementary
Information). Exactly at a = 0 our theoretical picture does break
down, because the Landau damping rate vanishes and the BEC has
no equilibrium features; the two components are simply completely
decoupled. Bearing this small caveat in mind, from here on we refer
to T0 ⇡ T f ⇡ T eq simply as the temperature of the system T .
A way to directly see that the gas is superheated is to suddenly
increase the coupling of the BEC to the thermal bath. In Fig. 3 weshow the results of two experimental series in which a is quenched(within 10 ms) from 3a0 to 62a0 at different times in the superheatedregime. The filled (open) symbols show N0 measured before (after)
the quench. The small 0 is almost unaffected by the change in a,and the sudden N0 decay is due to the increase in  (Fig. 1b). For
eqreference, the green line shows the calculated N0 at 3a0 and orange
shading indicates the superheated regime. As shown in the inset of
Fig. 3, we have checked that an interaction quench at t < tc (that is,eqwhen T < Tc and N0 > 0) does not kill the BEC.
As shown in Fig. 4, we have explored the limits of superheating
for a range of interaction strengths, including small negative valuesof a. For a < 0, a BEC is stable against collapse only for N0 < C/a,with C ⇡ 2 ⇥ 104 a0 for our trap parameters22–24 . However, after N0drops below this critical value, at small |a| it decays slowly.
In Fig. 4a we plot the highest temperature at which we stillobserve a BEC, T̄ ⌘ T (t = t̄ ), scaled to the equilibrium Tc at20

a (a0)10a (a0)

0

¬10

0

20

40
Time (s)6080Figure 4 | Limits of superheating. a, The highest temperature at which weobserve a BEC, T̄, scaled to the equilibrium Tc . Close to a = 0 the BEC
survives up to ⇡ 1.5Tc . The red line shows the results of our numericalcalculations, with the shaded area indicating the theoretical uncertainty.
Experimental error bars are statistical. The point at 62a0 is fixed to unity by
the absolute atom number calibration2,27 . b, Temporal phase diagram. For
each value of a we plot the equilibrium tc (green points) and the time t̄ atwhich the BEC actually vanishes (red points). The t̄ errors correspond todashed lines in Fig. 2 and the uncertainty in tc is indicated by the scatter ofpoints. Solid curves are spline fits to the data. For a ⇡ 0 the BEC survives inthe superheated regime for a whole minute. Inset: numerically calculated
phase diagram, with t̄ data overlaid.

the same N . For a ! 0, the BEC survives up to T ⇡ 1.5Tc . (Forcomparison, this is analogous to superheated water at 280 C.)
In Fig. 4b we reconstruct the temporal phase diagram of our
non-equilibrium gas. Here, a horizontal cut through the graphcorresponds to a time series such as shown in Fig. 2. For eacha, we plot the measured t̄ (red points) and the equilibrium tc(green points). The solid curves are spline fits to the data. The
width of the orange-shaded region corresponds to the time thatthe BEC survives in the superheated regime. For a ⇡ 0 this regionspans a whole minute.
The phase diagram in Fig. 4b is measured by always startingwith N0 ⇡ 2 ⇥ 104 . In general, non-equilibrium behaviour canstrongly depend on the initial conditions. However, we find that t̄ isessentially constant (within experimental errors) for initial N0 in therange (1–5)⇥104 . The primary reason for this is that the three-bodycontribution to 0 grows with N0 ; this leads to self-stabilization of
the condensed atom number on timescales much shorter than t̄ .
We theoretically reproduce our non-equilibrium observations
using a two-component model directly corresponding to Fig. 1b.
Starting with the measured initial N0 , we numerically simulate theevolution of a BEC coupled to a thermal bath characterized by µ0 (t ).
To do this we calculate 0 from our experimental parameters, andfor  we use the form25⇥
 = A el N0 e (µ0 µc )NATURE PHYSICS | VOL 9 | MAY 2013 | www.nature.com/naturephysics© 2013 Macmillan Publishers Limited. All rights reserved0

e (µ µc )

⇤

(1)
273

140

APPENDIX A. PUBLICATIONSLETTERS

NATURE PHYSICS DOI: 10.1038/NPHYS2587

Here el / a2 is the elastic collision rate and A is a dimensionlesscoefficient. The largest uncertainty in our calculations comes fromthe theoretical uncertainty in A ⇡ 1–10 (ref. 26). (For details see
Supplementary Information.)
In Fig. 4a we show the calculated T̄ /Tc . The red line corresponds
to A = 3 and the shaded area to the range A = 1–10. The calculation
generally captures our experimental observations well. With A = 3
we obtain quantitative agreement with the data, except exactly at
a = 0, where the model is not valid. In the inset of Fig. 4b we show
the calculated temporal phase diagram, with A = 3, together withthe experimental t̄ data. Again the general features of the diagram
are captured well for a 6= 0.
The success of our calculations supports a conceptually simple
way to think about dynamical non-equilibrium effects near acontinuous phase transition. Extending the BEC lifetime bytuning interactions could also have practical benefits for precision
measurements and quantum information processing.
Received 3 September 2012; accepted 14 February 2013;
published online 24 March 2013; corrected online
28 March 2013

References
1. Dalfovo, F., Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of Bose–Einsteincondensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999).
2. Smith, R. P., Campbell, R. L. D., Tammuz, N. & Hadzibabic, Z. Effects ofinteractions on the critical temperature of a trapped Bose gas. Phys. Rev. Lett.
106, 250403 (2011).
3. Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Colloquium:
Nonequilibrium dynamics of closed interacting quantum systems.
Rev. Mod. Phys. 83, 863–883 (2011).
4. Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum Newton’s cradle. Nature
440, 900–903 (2006).
5. Winkler, K. et al. Repulsively bound atom pairs in an optical lattice. Nature
441, 853–856 (2006).
6. Sadler, L. E., Higbie, J. M., Leslie, S. R., Vengalattore, M. &Stamper-Kurn, D. M. Spontaneous symmetry breaking in a quenchedferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).
7. Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T. & Schmiedmayer, J.
Non-equilibrium coherence dynamics in one-dimensional Bose gases. Nature
449, 324–327 (2007).
8. Haller, E. et al. Realization of an excited, strongly correlated quantum gas phase.
Science 325, 1224–1227 (2009).
9. Guzman, J. et al. Long-time-scale dynamics of spin textures in a degenerate
F = 1 87 Rb spinor Bose gas. Phys. Rev. A 84, 063625 (2011).
10. Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum
many-body system. Nature 481, 484–487 (2012).
11. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated
strongly correlated one-dimensional Bose gas. Nature Phys. 8, 325–330 (2012).
12. Mark, M. J. et al. Preparation and spectroscopy of a metastable Mott-insulator
state with attractive interactions. Phys. Rev. Lett. 108, 215302 (2012).

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13. Smith, R. P., Beattie, S., Moulder, S., Campbell, R. L. D. & Hadzibabic, Z.
Condensation dynamics in a quantum-quenched Bose gas. Phys. Rev. Lett. 109,
105301 (2012).
14. Gring, M. et al. Relaxation and pre-thermalization in an isolated quantum
system. Science 337, 1318–1322 (2012).
15. Mathey, L. & Polkovnikov, A. Light cone dynamics and reverse Kibble–Zurek
mechanism in two-dimensional superfluids following a quantum quench.
Phys. Rev. A 81, 033605 (2010).
16. Fedichev, P. O., Shlyapnikov, G. V. & Walraven, J. T. M. Damping of
low-energy excitations of a trapped Bose–Einstein condensate at finite
temperatures. Phys. Rev. Lett. 80, 2269–2272 (1998).
17. Pethick, C. & Smith, H. Bose–Einstein Condensation in Dilute Gases(Cambridge Univ. Press, 2002).
18. Campbell, R. L. D. et al. Efficient production of large 39 K Bose–Einsteincondensates. Phys. Rev. A 82, 063611 (2010).
19. Roati, G. et al. 39 K Bose–Einstein condensate with tunable interactions.
Phys. Rev. Lett. 99, 010403 (2007).
20. Tammuz, N. et al. Can a Bose gas be saturated? Phys. Rev. Lett. 106,
230401 (2011).
21. Smith, R. P. & Hadzibabic, Z. Effects of interactions on Bose–Einsteincondensation of an atomic gas. Preprint at http://arxiv.org/abs/1203.
2063 (2012).
22. Ruprecht, P. A., Holland, M. J., Burnett, K. & Edwards, M. Time-dependent
solution of the nonlinear Schrödinger equation for Bose-condensed trapped
neutral atoms. Phys. Rev. A 51, 4704–4711 (1995).
23. Gerton, J. M., Strekalov, D., Prodan, I. & Hulet, R. G. Direct observation of
growth and collapse of a Bose–Einstein condensate with attractive interactions.
Nature 408, 692–695 (2000).
24. Donley, E. A. et al. Dynamics of collapsing and exploding Bose–Einsteincondensates. Nature 412, 295–299 (2001).
25. Gardiner, C. W., Zoller, P., Ballagh, R. J. & Davis, M. J. Kinetics ofBose–Einstein condensation in a trap. Phys. Rev. Lett. 79, 1793–1796 (1997).
26. Gardiner, C. W., Lee, M. D., Ballagh, R. J., Davis, M. J. & Zoller, P. Quantum
kinetic theory of condensate growth: Comparison of experiment and theory.
Phys. Rev. Lett. 81, 5266–5269 (1998).
27. Smith, R. P., Tammuz, N., Campbell, R. L. D., Holzmann, M. & Hadzibabic, Z.
Condensed fraction of an atomic Bose gas induced by critical correlations.
Phys. Rev. Lett. 107, 190403 (2011).

Acknowledgements
We thank S. Beattie and S. Moulder for experimental assistance. This work was
supported by EPSRC (Grant No. EP/K003615/1), the Royal Society, AFOSR, ARO andDARPA OLE.

Author contributions
All authors contributed extensively to this work.

Additional information
Supplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to R.P.S.

Competing financial interests
The authors declare no competing financial interests.

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141

SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS2587

A superheated Bose-condensed gas
Supplementary Information

A superheated Bose-condensed gas
Alexander L. Gaunt, Richard J. Fletcher, Robert P. Smith, and Zoran Hadzibabic

Cavendish Laboratory, University of Cambridge, J. J. Thomson Ave., Cambridge CB3 0HE, United Kingdom(Dated: February 8, 2013)

Determination of N , E and N0

We take an absorption image of the atom cloud after τ = 18 ms of time-of-flight (TOF) expansion from a nearly isotropic trapwith a geometric mean of trapping frequencies ω̄/2π ≈ 70 Hz.
For absolute calibration of our atom numbers we use a Tc measurement at a = 62 a0 , assuming that at this a the cloud is inequilibrium [1, 2].
For a < 100 a0 we assess the interaction-energy contribution to the total energy E to be <∼ 1% and thus E ≈ 2Ek , where
Ek is the kinetic energy. We obtain Ek from the second moment of the atom distribution after TOF, and correct it for the small
effect of the initial in-trap cloud size. This amounts to rescaling the energy by a factor ω̄ 2 τ 2 /(1 + ω̄ 2 τ 2 ).
In Fig. S1 we show N and Ek for the same two experimental series shown in Fig. 2 in the main text.
To improve the detection of small N0 values we always switch a to zero at the start of TOF [1].
83 a0

5 a0

3

N (10 )

200

100

Ek/NkB (nK)

0
300
200
100
0

0

20

40

60

0

20

time (s)40

60time (s)Fig. S1: Parametrization of N and Ek . The total atom number N and kinetic energy per particle Ek /N for two experimental series at 83 a0
and 5 a0 . The experimental data are fitted with polynomial forms to obtain smooth functions N (t) and Ek (t).

Equilibrium calculations

For an equilibrium ideal Bose gas in a spherically symmetric harmonic trap of frequency ω, the thermal component satisfies:
##
" ′
! ∞ "
u2
N0
µ − µ0c−u du,(1)
g2 expN ′ (µ′ , T ) = c
ζ(3) 0
kB T
2
& 2
$∞ %
2
µ′ −µ0g2 exp( kB T c − u2 ) u2 u du03
Ek (µ′ , T )&= kB T $ ∞ %,2µ′ −µ0c
N ′ (µ′ , T )2
g exp(− u ) u du
0

2

kB T

(2)

2

&−3
' T (3 %ζ(2) h̄ωB
1 − 2ζ(3)includes the finite size correction, µ0c = 32 h̄ω, and g2 (z) is the dilogarithm
where Nc0 = ζ(3) kh̄ω
kB T
function. Note that Eqs. (1) and (2) are obtained by integrating the three-dimensional momentum distribution in a harmonically
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APPENDIX A. PUBLICATIONSSUPPLEMENTARY INFORMATION

DOI: 10.1038/NPHYS2587

2

trapped Bose gas, described by the polylogarithm function g3/2 [3], along one direction. For N > Nc0 , the chemical potential iscapped at µ′ = µ0c and also µ0 = µ0c . The thermal atom number is saturated, N ′ = Nc0 , and any additional particles must go
into the condensate, N0 = N − Nc0 .
Interactions modify this picture in two ways:
!
2

2πh̄
(i) The critical point is shifted. At mean-field (MF) level µc = µ0c + 4ζ(3/2)a/λ, where λ =mkB T is the thermal
wavelength. A small beyond-MF correction is quadratic in a/λ and has an additional logarithmic correction. Experimentally,
the corresponding Nc is [1, 2]"# a $2 %−3a.
(3)
Nc = Nc0 1 − 3.426 + 42λλ(ii) Due to interactions, in the presence of a BEC, N ′ is no longer saturated at Nc . Empirically,
N ′ = Nc + S0 (N0 )2/5 + S2 (N0 )4/5 ,(4)where the non-saturation parameters S0 and S2 depend on a and T [4, 5]. The excess number of thermal atoms, N ′ − Nc , can bedirectly attributed to the shift of the chemical potential above µc ; for an interacting BEC µ0 > µc and in equilibrium µ′ = µ0 .
For µ0 we use a modified Thomas-Fermi law [8]:
&"
'
%2/5
h̄ω
N0 a
5/2
15
µ0 − µ c =
+3
−3 ,
(5)
2
aosc
(
where aosc = h̄/mω is the harmonic oscillator length.
Eq. (5) and Eqs. (1) and (2) modified to include interaction effects (Nc0 → Nc and µ0c → µc ) form a complete set needed forour calculations [6]. We proceed in two ways:(1) Assuming that the system is in global equilibrium, µ′ = µ0 = µeq , we use only N (t) and E(t) to calculate µeq (t), N0eq (t)and T eq (t) (green lines in main-text Fig. 2).
(2) Additionally, from N , E and the measured N0 we calculate µ0 and µ′ without assuming µ′ = µ0 (red and blue points inthe bottom panel of main-text Fig. 2).
Justification of the two-fluid picture for a gas out of global phase equilibrium

In our theoretical picture (Fig. 1(b) in the paper), we assume that in the superheated regime the thermal and condensed
components can still to a good approximation be assumed to be separately in equilibrium. Moreover, we assume that they are at
the same temperature, but just have different chemical potentials.
Here we provide a more detailed justification of these assumptions.
First, for the thermal component we show in Fig. S2 that the radial velocity distribution still looks like a thermal distribution
at a temperature very close to the calculated T eq (N, E). Here we show data for the same 5 a0 series as shown in Fig. 2(b) in
the main paper. In Fig. S2(a) we show the distribution measured at t = 45 s, i.e. deep in the superheated regime. The data (red)is fitted almost perfectly by an equilibrium thermal distribution (green) constrained to be characterised by N, E and T eq . An
unconstrained fit (blue) gives only a very slightly different shape with T f within few % of T eq . Note that full thermalisationof the thermal component, in the sense that occupations of all the individual momentum states acquire their equilibrium values,
should have a rate ∝ a2 . Still, experimentally, the overall “coarse-grained” shape of the distribution is essentially thermal. In
Fig. S2(b) we compare the calculated T eq (green) and the fitted T f (blue) for the whole 5 a0 series. For comparison we also
show the equilibrium Tc (black line). Note that this is the same plot as in Fig. 2(b) in the main paper, with just the T f points
added.
Second, for the collective excitations in the BEC to be in equilibrium with the thermal bath, the Landau-damping time τL [3, 7]
must be short compared to the characteristic time scale of the experiment. For a uniform system at a temperature higher than theinteraction energy per particle [3]:
1/2

n λ2
τL ω
≈ 0 1/2 ,
2π
4πa

(6)

where ω is the excitation frequency, n0 is the condensate density, and λ is the thermal wavelength. For our assessment we use
this uniform-system result with our peak n0 ; this only overestimates τL for our harmonically trapped gas [7].
2

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SUPPLEMENTARY INFORMATION

DOI: 10.1038/NPHYS2587

3

a

b
240

T (nK)

80

3n(vr) (10 s/cm)

120

180

40
120
0
0

1

2

3

4

0

vr (cm/s)

20

40
time(s)60

Fig. S2: Thermal distribution in a gas out of global phase equilibrium. (a) For a gas at 5 a0 in the superheated regime we show the radial
velocity distribution (red), the distribution corresponding to N , E and T eq (green) and the unconstrained fit (blue) giving T f . Even though
the gas is not in true equilibrium the distribution still looks thermal and T f and T eq agree to within few %. (b) Comparison of T f (blue) andT eq (green) for the whole 5 a0 data series shown in Fig. 2(b) of the main paper. The solid black line shows the equilibrium Tc .

Exactly at a = 0 the damping time diverges and our theoretical picture breaks down. However, already for a = 1 a0 , for all
our experimental parameters the RHS of Eq. (6) is < 100. Then, even for our lowest-energy modes, with ω/2π ∼ 100 Hz, we
get τL < 1 s.
Non-equilibrium evolution of N0

The non-equilibrium evolution of N0 is described by the differential equationṄ0 = −κ − Γ0 N0 ,(7)

where κ is the coupling to the thermal bath due to elastic collisions and Γ0 is the instantaneous inelastic loss rate per particle.
Following [9] we use
κ=A
2

"
!
"
′
8m(akB T )2 2β(µ′ −µc ) ! β(µ0 −µ′ )e− 1 N0 = Aγel N0 eβ(µ0 −µc ) − eβ(µ −µc ) ,
e
3
πh̄

(8)

′

BT )
eβ(µ −µc ) is essentially the elastic scattering rate for a thermal cloud at µ′ and A ≈ 1 − 10 is awhere γel = 8m(akπh̄3theoretically uncertain prefactor [8].
(1)
(3)
In the inelastic loss term we include contributions from one-body scattering and three-body recombination, Γ0 = Γ0 + Γ0 .
In our system, one-body loss is dominated by spontaneous scattering of photons from the trapping laser beams. We calculate itfrom the known wavelength and intensity of the beams. For all the reported experiments

(1)

Γ0 ≈

1 −1
s .
35

(9)

The loss rate of condensate atoms due to three-body recombination in the presence of a thermal cloud is given by [10]
(3)

Γ0 =

$
K3 (a) # 2⟨n0 ⟩ + 6⟨n0 n′ ⟩ + 6⟨n′2 ⟩ ,6(10)

where n0 is the condensate density, n′ the thermal density, K3 (a) the known a-dependant three-body coefficient [11, 12], and⟨...⟩ stands for an average over the density distribution. We set n′ to its value in the centre of the trap and for the condensate weagain use a modified Thomas-Fermi approach:⟨n0 ⟩GS⟨n0 ⟩ = !
"3/5 ,
5/3
1 + (⟨n0 ⟩GS /⟨n0 ⟩TF )NATURE PHYSICS | www.nature.com/naturephysics

(11)

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APPENDIX A. PUBLICATIONSSUPPLEMENTARY INFORMATION

DOI: 10.1038/NPHYS2587

4⟨n20 ⟩GS⟨n20 ⟩ = !
"6/5 .
5/6
1 + (⟨n20 ⟩GS /⟨n20 ⟩TF )

(12)

Here
GS refers to the non-interacting Gaussian ground state
approximation, ⟨n0 ⟩TF =√ and 2TF to the Thomas-Fermi√
2π(15/7)(15N0 a/aosc )−3/5 ⟨n0 ⟩GS and ⟨n20 ⟩TF =3π(15 /56)(15N0 a/aosc )−6/5 ⟨n20 ⟩GS , where ⟨n0 ⟩GS =N0 /(2πa2osc )3/2 and ⟨n20 ⟩GS = N02 /(3π 2 a4osc )3/2 . Eqs. (11) and (12) smoothly interpolate between the ground state result (forN0 a/aosc ≪ 1) and the Thomas-Fermi approximation (for N0 a/aosc ≫ 1). Note that other forms which smoothly interpolate
between these two limits give essentially the same results.
Using Eqs. (7) - (12) we simulate the evolution of the condensate atom number, N0 (t), from its initial value N0 (t = 0). We
use the measured N (t) and E(t) and the numerically evolved N0 (t) to obtain µ′ (t) for use in Eq. (8). To determine t̄ from ourcalculations we define the condensate to be present if N0 is larger than N0min = 3kB T /(h̄ω), the thermal occupation of the firstexcited state.
In the main text we show the results of our calculations for T̄ and t̄ (Fig. 4), with A = 3. In Fig. S3 we show that our
calculations (with the same value of A) also describe well the full dynamics N0 (t).

measured N020

calculated N0eq

3

N0 (10 )

N0

10

0
0

20

40

60

time (s)Fig. S3: Non-equilibrium N0 dynamics. We plot the calculated N0 (t) (dashed red line) together with the measured N0 (red points) for thesame 5 a0 data series as in Fig. 2(b) in the main text. For comparison we also show the calculated N0eq (t) (solid green line).

[1] Smith, R. P., Campbell, R. L. D., Tammuz, N. & Hadzibabic, Z. Effects of interactions on the critical temperature of a trapped Bose gas.
Phys. Rev. Lett. 106, 250403 (2011).
[2] Smith, R. P., Tammuz, N., Campbell, R. L. D., Holzmann, M. & Hadzibabic, Z. Condensed fraction of an atomic Bose gas induced by
critical correlations. Phys. Rev. Lett. 107, 190403 (2011).
[3] Pethick, C. & Smith, H. Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002).
[4] Tammuz, N. et al. Can a Bose gas be saturated? Phys. Rev. Lett. 106, 230401 (2011).
[5] Smith, R. P. & Hadzibabic, Z. Effects of interactions on Bose-Einstein condensation of an atomic gas. arXiv:1203.2063 (2012).
[6] In Eqs. (1) and (2) we also replace the function g2 with a slightly modified dilogarithm function g̃(z) = (3/4)|g2 (z)|+(1/4)Re {g2 (z)},which remains real for z > 1. This is done just for mathematical convenience and does not affect our results since the integrals are strongly
dominated by the region where x2 /2 > (µ′ − µc )/kB T and g̃ = g2 .
[7] Fedichev, P. O., Shlyapnikov, G. V. & Walraven, J. T. M. Damping of low-energy excitations of a trapped Bose–Einstein condensate at
finite temperatures. Phys. Rev. Lett. 80, 2269 (1998).
[8] Gardiner, C. W., Lee, M. D., Ballagh, R. J., Davis, M. J. & Zoller, P. Quantum kinetic theory of condensate growth: comparison ofexperiment and theory. Phys. Rev. Lett. 81, 5266–5269 (1998).
[9] Gardiner, C. W., Zoller, P., Ballagh, R. J. & Davis, M. J. Kinetics of Bose–Einstein condensation in a trap. Phys. Rev. Lett. 79, 1793–1796(1997).
[10] Söding, J. et al. Three-body decay of a rubidium Bose–Einstein condensate. Appl. Phys. B 69, 257 (1999).
[11] Zaccanti, M. et al. Observation of an Efimov spectrum in an atomic system. Nature Physics 5, 586–591 (2009).
[12] Zaccanti, M. Tuning of the interactions in ultracold K-Rb quantum gases. Ph.D. thesis, University of Florence (2007).
4

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Persistent Currents in Spinor Condensates
Scott Beattie, Stuart Moulder, Richard J. Fletcher, and Zoran Hadzibabic
Phys. Rev. Lett. 110, 025301 (2013)

146

APPENDIX A. PUBLICATIONS

147
week ending
11 JANUARY 2013

PHYSICAL REVIEW LETTERS

PRL 110, 025301 (2013)

Persistent Currents in Spinor Condensates
Scott Beattie, Stuart Moulder, Richard J. Fletcher, and Zoran Hadzibabic
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom(Received 18 October 2012; published 9 January 2013)
We create and study persistent currents in a toroidal two-component Bose gas, consisting of 87 Rb atoms
in two different spin states. For a large spin-population imbalance we observe supercurrents persisting forover two minutes. However, we find that the supercurrent is unstable for spin polarization below a welldefined critical value. We also investigate the role of phase coherence between the two spin components
and show that only the magnitude of the spin-polarization vector, rather than its orientation in spin space,is relevant for supercurrent stability.
DOI: 10.1103/PhysRevLett.110.025301

PACS numbers: 67.85.!d, 03.75.Kk

0031-9007=13=110(2)=025301(5)created by intersecting a 1070 nm ‘‘sheet’’ laser beam andan 805 nm ‘‘tube’’ beam [7]. The sheet beam confines theatoms to the horizontal plane with a trapping frequency of
350 Hz. In plane, the tube beam confines the atoms to thering with a trapping frequency of 50 Hz. The trap depth isabout twice the BEC chemical potential, !0 =h " 0:6 kHz,and varies azimuthally by <10%.
Our tube trapping beam is a Laguerre-Gauss LG3 lasermode in which each photon carries orbital angular
(a)

(b)
B

G
LG
RF
q=0

(c)

(d)

q=3

100

(72%)

50

0
6
(28%)

q2 Er

q=3

2R
countsPersistent currents are a hallmark of superfluidity andsuperconductivity, and have been studied in liquid heliumand solid state systems for decades. More recently, it
has become possible to trap an atomic Bose-Einsteincondensate (BEC) in a ring geometry [1–8] and induce
rotational superflow in this system [3,5,7,8]. This offers
new possibilities for fundamental studies of superfluidity ina flexible experimental setting. Both long-lived superflow[5,7] and quantized phase slips corresponding to singly
charged vortices crossing the superfluid annulus havebeen observed [7,8].
So far experiments on persistent currents in atomic
BECs were limited to spinless, single-component condensates. Extending such studies to multicomponent systems,
in particular those involving two or more spin states[9–11], is essential for understanding superfluids with avectorial order parameter and for applications in atominterferometry [12,13]. Persistent flow in a two-component
Bose gas has been studied theoretically [14–17] but many
issues remain open. Even the central question of whether,and under what conditions, this system supports persistentcurrents has not been settled.
In this Letter, we study the stability of supercurrents in a
toroidal two-component gas consisting of 87 Rb atoms in
two different spin states. For a large spin-population imbalance we observe superflow persisting for over two minutes
and limited only by the atom-number decay. However at asmall population imbalance the onset of supercurrent
decay occurs within a few seconds. We demonstrate theexistence of a well-defined critical spin polarization separating the stable- and unstable-current regimes. We also
study the connection between spin coherence and superflow stability, and show that in our system only the modulus of the spin-polarization vector is relevant for thestability of the supercurrent. The existence of a critical
population imbalance was anticipated in Refs. [15–17], but
quantitative comparison with our measurements will
require further theoretical work.
Our setup is outlined in Fig. 1(a). We load a BEC of
N " 105 atoms into an optical ring trap of radius 12 !m,

q=2
q=1
10
R (µm)

14

FIG. 1 (color online). Preparation and detection of supercurrent in a two-component gas. (a) The ring trap is formed by ahorizontal ‘‘sheet’’ beam and a vertical Laguerre-Gauss (LG)‘‘tube’’ beam. B is the external magnetic field. (b) Supercurrent
is induced by a Raman transfer of atoms between two spin states,
j"i and j#i, using the LG beam and an auxiliary Gaussian (G)beam. During the transfer each atom absorbs 3@ of angularmomentum from the LG beam. Two-component gas is createdby coupling j"i and j#i states with an rf field. The characteristic
rotational energy is Er =h " 0:4 Hz. (c) Time-of-flight image ofthe atoms, with spin states separated using a Stern-Gerlach
gradient. The rotational state q is deduced from the radius R
characterizing the central hole in the density distribution. The
image shown was taken after t ¼ 4 s of rotation; the longitudinal
spin polarization is Pz ¼ 0:44 and q ¼ 3 for both spin states.
(d) Histogram of "900 measurements of R at various Pz and t.

025301-1! 2013 American Physical Society

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PHYSICAL REVIEW LETTERS
(a)

t=3s

5s

40s

20s

(b)

80s

120s

(c)

q=3

q=2

q=0

3s

5s

20s

Atom number (103)momentum 3@. We use the same beam to induce a supercurrent via a two-photon Raman process [3,7,18]. We
briefly (200 !s) pulse on an auxiliary TEM00 Gaussianbeam, copropagating with the LG beam, to transfer all
atoms between two spin states, j"i and j#i [Fig. 1(b)].
Each atom absorbs angular momentum 3@ from the LG
beam and we thus create a (single-component) current
corresponding to a vortex of charge q ¼ 3 trapped at thering centre. Such current can persist for over a minute anddecays in quantized q ! q ! 1 steps, corresponding to 2"
phase slips in the BEC wave function [7].
The j"i and j#i states also define the spin space for our
two-component experiments. To create a two-component
current we prepare a pure jq ¼ 3; #i state and then couple
j"i and j#i by a radio frequency (rf) field, which carries noorbital angular momentum and does not affect the motional
state of the atoms. The j"i and j#i are two F ¼ 1 hyperfine
ground states, mF ¼ 1 and 0, respectively. The mF ¼ !1state is detuned from Raman and rf resonances by the quadratic Zeeman shift in an external magnetic field B of 10 G.
After preparing a rotating (q ¼ 3) cloud in a specificspin state, we let it evolve in the ring trap for a time t andthen probe it by absorption imaging after 29 ms of timeof-flight expansion. We separate the two spin components
with a Stern-Gerlach gradient and directly measure thelongitudinal spin-polarization Pz ¼ ðN" ! N# Þ=ðN" þ N# Þ,
where N" (N# ) is the number of atoms in the j"i (j#i) state[Fig. 1(c)]. The rotational state, 0 ' q ' 3, is seen in thesize R of the central hole in the atomic distribution [7],arising due to a centrifugal barrier [3]. As shown in
Fig. 1(d), the R values are clearly quantized [7,8], allowing
us to determine q with >99% fidelity [19].
In Fig. 2 we illustrate the dramatic difference between
superflow stability in a Pz ¼ 1 single-component gas and aPz ¼ 0 two-component system. The two different Pz states
are created, respectively, by a (140 !s) " and a (70 !s)"=2 rf pulse at t ¼ 0. In the pure j"i state [Fig. 2(a)] thecurrent persists for over two minutes, with the BEC always
remaining in the q ¼ 3 state for "90 s. In contrast, at
Pz ¼ 0 [Fig. 2(b)] the first phase slip occurs within 5 s
and the current completely decays within 20 s. During the
decay we always observe the two spin components to be in
the same q state.
Supercurrent stability generally depends on the numberof condensed atoms [5,7] and at Pz ¼ 0 the atom number
per spin state is halved. However, from the N-decay curves
in Fig. 2(c) we see that this alone cannot explain the
difference in superflow stability. At Pz ¼ 1 rotation still
persists for N " 104 while at Pz ¼ 0 it stops alreadyat N > 4 ( 104 . Moreover, if we apply a "=2 rf pulse at
t ¼ 0 but then immediately remove all the j"i atoms fromthe trap with a resonant light pulse, the current again
persists for over a minute. This unambiguously confirmsthat in Fig. 2(b) the superflow is inhibited by the presenceof both spin components.

80

40

0
0

50
Time (s)100

FIG. 2 (color online). Single- versus two-component supercurrent. (a) In a pure j"i state (Pz ¼ 1) supercurrent persistsfor over 2 min, with no phase slips occurring for "90 s. (b) At
Pz ¼ 0 the first phase slip occurs within 5 s and we observe norotation beyond 20 s. (c) Total atom number decay for Pz ¼ 1(open symbols) and Pz ¼ 0 (solid symbols). Dashed lines aredouble-exponential fits.

We now turn to a quantitative study of the supercurrent
stability as a function of the spin-population imbalance(Fig. 3). We tune Pz by varying the length !t of the rf pulse
applied at t ¼ 0, and measure the q state of the majority(j"i) spin component as a function of t. Whenever theradius R is fittable for the minority component we get thesame q for both spin components in >99% of cases.
However, for N# < 104 we cannot determine q for theminority component.
Based on "1600 measurements of qðPz ; tÞ, in Fig. 3
we reconstruct the complete current stability diagram for
0 ' Pz ' 1 [20]. The contour plot of hqðPz ; tÞi is obtainedby spline interpolation through a 3D mesh of data points
1

3
stable

0.8

2

0.6

Pz

PRL 110, 025301 (2013)

APPENDIX A. PUBLICATIONS

0.4

1
unstable

0.2
00Mean q148

50

Time (s)

100

150

0

FIG. 3 (color online). Supercurrent stability in a partially spinpolarized gas. The statistically averaged supercurrent state, hqi,of the majority spin component is shown as a function of Pz andthe evolution time t. The contour plot is based on "1600
measurements of qðPz ; tÞ. The transition between stable- andunstable-current regimes occurs at 0:6 < Pz < 0:7. In the stableregime the current eventually decays due to the atom-number
decay.

025301-2

149

(a)

z
t

P
x

y

(c)

x

z

y

0

0

t
x

y

diffusion [21] and small magnetic field inhomogeneities[22]. Pz is a constant of motion but the transverse polarization decays and P ! Pz [Fig. 4(a)].
We study the transverse-polarization decay in a Ramseytype experiment. Starting in the j#i state we apply two "=2
rf pulses separated by time t and then measure Pz . The first
pulse creates a purely transverse P~ ¼ ð1; 0; 0Þ and thesecond one maps the decaying P into Pz [23]. As seen in
Fig. 4(b), we observe a very long spin-coherence time,
tcoh , 10 s. This means that in the unstable regime in
Fig. 3 phase slips occur already at t & tcoh , when we cannot
equate P and Pz .
We also perform a complementary experiment in which
we adiabatically dress the rotating BEC with the rf field.
In presence of the rf field of frequency !, the effective
magnetic field is B~ eff ¼ ð2@=!B Þð0; "R ; !$Þ, where !B is
the Bohr magneton and $ ¼ ! ! !B B=ð2@Þ is the detun^ At t ¼ 0
ing from resonance. On resonance, B~ eff / y.
we adiabatically (in 100 ms) sweep $ from a large value( -"R ) to zero, thus preparinga Pz ¼ 0 superpositionpﬃﬃﬃ
state jyi ¼ ðj"i þ ij#iÞ= 2 [Fig. 4(c)]. At this point, jyi
is equivalent to the j"=2i state prepared by an rf pulse,which does not show long-term current stability [Fig. 2(b)].
However, if we leave the rf field on during the in-trap
evolution, jyi is an eigenstate of the Hamiltonian and thecoherence between j"i and j#i components does not decay[24]. In this case Pz ¼ 0 supercurrent is stable and persists
for more than a minute [Fig. 4(d)].
These experiments clearly show that for analyzing
current stability in a partially polarized gas we must
~ With this understanding, we now
distinguish Pz and jPj.
quantitatively characterize the onset of the supercurrent
decay in Fig. 3 by the time % at which the probability
that the first phase slip (q ¼ 3 ! 2) has occurred is 50%;
this closely corresponds to the border of blue- and whiteshaded regions.
In Fig. 5(a) we see that % rapidly increases for Pz *0:64,
saturating at 100 s due to the N decay. We now combine
our measurements of % and the transverse-polarization~ at the onset of superdecay fðtÞ [Fig. 4(b)] to calculate jPj
current decay:

Time (s)

20

(d)

z

B
Beff

(b) 1

z

P2

B

q=3
x

y
t=60s

FIG. 4 (color online). The role of spin-coherence in superflow
stability. (a) An rf pulse at t ¼ 0 creates a spin-superposition
state in which Pz < 1 but P ¼ 1. As the superposition decoheres,transverse spin-polarization decays and P ! Pz . (b) Transversepolarization decay (see text). Double-exponential fit (solid redline) gives the decay function fðtÞ. (c) Adiabatic dressing of thespin state. In presence of a resonant rf field B~ eff / y^ andthe dressed Pz ¼ 0 state jyi is stable against decoherence.
(d) In the dressed jyi state the supercurrent is also stable.

with integer q values. The blue-shaded region corresponds
to rotation times for which no phase slips occur. We clearly
distinguish two qualitatively different regimes. For large
Pz the superflow is fundamentally stable and limited onlyby the atom-number decay; for low Pz the current starts todecay within a few seconds. A sharp transition between thetwo regimes occurs at 0:6 < Pz < 0:7.
To fully understand these observations, we need to distinguish a coherent superposition of j"i and j#i states froman incoherent mixture. The rf pulse at t ¼ 0 corresponds
to rotation around the y axis on a Bloch sphere and puts
the BEC in a superposition state j#i ¼ sinð#=2Þj"iþcosð#=2Þj#i. Here, # ¼ "R !t, where "R is the rf Rabi
frequency. In this state Pz ¼ !cosð#Þ but the gas is still
fully spin polarized; the polarization vector is P~ ¼
~ ¼ 1. Subsequently the½sinð#Þ; 0; !cosð#Þ* and P + jPj
spin superposition decoheres, due to both intrinsic spin
(a)

(b)

120

(c)

1.0

Pc

(s)

P( )

80
0.5

Pc

40
0
0.0

week ending
11 JANUARY 2013

PHYSICAL REVIEW LETTERS

PRL 110, 025301 (2013)

0.5
Pz

1.0

0.0
0.0

P

Pc
0.5
Pz

1.0

FIG. 5 (color online). Critical spin polarization Pc . (a) Characteristic time of the first phase slip, %, versus Pz . Vertical blue
~ at the onset of the supercurrent decay. Fit to the points with Pð%Þ > Pz (horizontal
line marks Pc , accurately determined in (b). (b) jPj
blue line) gives Pc ¼ 0:64ð1Þ. (c) Stability diagram on the Bloch sphere. The blue-shaded region in Fig. 3 maps into the outer shell~ > Pc .
jPj025301-3

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APPENDIX A. PUBLICATIONSPRL 110, 025301 (2013)

PHYSICAL REVIEW LETTERS

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Pð%Þ ¼ P2z þ ð1 ! P2z Þfð%Þ:(1)

In Fig. 5(b) we clearly distinguish two regimes: one where
Pð%Þ is constant (within errors) and one where Pð%Þ ¼ Pz .
We thus complete our physical picture and accurately
determine the critical spin polarization Pc . (1) If Pz >Pc ,~ never drops below Pc ; the supercurrent is fundathen jPj
mentally stable, % - tcoh and Pð%Þ ¼ Pz . (2) If Pz < Pc ,supercurrent decay starts at % & tcoh , when the decaying P
becomes equal to Pc . From all the data in this regime we
get Pc ¼ 0:64ð1Þ.
For 0 ' Pz ' Pc the value of % varies from 4 to 15 s andqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ~
the orientation of Pð%Þ
¼ ð P2c ! P2z ; 0; Pz Þ in spin spacevaries from purely transverse to purely longitudinal, but
the onset of the supercurrent decay always occurs at the~ We thus conclude that the region of supercurrent
same jPj.
stability is in fact the outer shell of the Bloch sphere where~ > Pc [Fig. 5(c)]. This spin-rotational symmetry is
jPj
intuitive but we note that it need not be universal. In our
87
Rb gas the strengths of intra- and intercomponent interactions are almost identical, so the Hamiltonian is almost
invariant under rotations in spin space. In the future it
would be very interesting to study supercurrent stability
as a function of both intra- and intercomponent couplingstrengths.
The existence of a critical population imbalance forsuperflow stability was predicted in Refs. [15–17], assuming equal intra- and intercomponent interactions and nointercomponent coherence. The current instability was
associated with out-of-phase density fluctuations in thetwo components. However, the agreement on the value of
Pc has not been reached. In Refs. [15,16] it was predictedthat any q > 1 flow is unstable for essentially any P < 1,
but according to Ref. [17] such current is stable abovesome nontrivial interaction-dependent Pc . The latter conclusion qualitatively agrees with our observations.
However, none of the existing theories is quantitatively
applicable to our experiments, since they are limited to thesimplified cases of reduced dimensionality and very weakinteractions. Moreover, the interplay of the spin and rotational degrees of freedom may involve new physical effects. Specifically, the dynamics of the local spin vector on
the Bloch sphere can result in a Berry phase and unwindthe ‘‘scalar’’ phase describing the rotational flow; the timescale for this process would be the same as the spin
decoherence seen in a Ramsey experiment [25].
In summary, we have observed persistent currents in
multiply connected spinor condensates, demonstrated theexistence of a critical spin polarization for stable superflow, and elucidated the role of spin coherence in supercurrent stability. Our results should stimulate further
theoretical work on this fascinating many-body problem
and are also relevant for applications in trapped-atominterferometry. An important next step would be to study

week ending
11 JANUARY 2013supercurrents in a two-species system with significantly
different intra- and intercomponent interactions.
We thank A. Gaunt and R. Smith for experimental assistance and N. Cooper, S. Baur, E. Demler, T. Kitagawa, G.
Kavoulakis, J. Smyrnakis, J. Dalibard, and J. Thywissen for
discussions. This work was supported by EPSRC (Grants
No. EP/G026823/1 and No. EP/I010580/1).

[1] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy,
and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201(2005).
[2] A. S. Arnold, C. S. Garvie, and E. Riis, Phys. Rev. A 73,
041606 (2006).
[3] C. Ryu, M. F. Andersen, P. Cladé, V. Natarajan, K.
Helmerson, and W. D. Phillips, Phys. Rev. Lett. 99,
260401 (2007).
[4] K. Henderson, C. Ryu, C. MacCormick, and M. G.
Boshier, New J. Phys. 11, 043030 (2009).
[5] A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan,
W. T. Hill III, C. J. Lobb, K. Helmerson, W. D. Phillips,
and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[6] B. E. Sherlock, M. Gildemeister, E. Owen, E. Nugent, andC. J. Foot, Phys. Rev. A 83, 043408 (2011).
[7] S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and
Z. Hadzibabic, Phys. Rev. A 86, 013629 (2012).
[8] K. C. Wright et al., arXiv:1208.3608.
[9] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).
[10] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822
(1998).
[11] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner,
A. P. Chikkatur, and W. Ketterle, Nature (London) 396,
345 (1998).
[12] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys.
Rev. Lett. 78, 2046 (1997).
[13] P. L. Halkyard, M. P. A. Jones, and S. A. Gardiner, Phys.
Rev. A 81, 061602 (2010).
[14] T.-L. Ho, Phys. Rev. Lett. 49, 1837 (1982).
[15] J. Smyrnakis, S. Bargi, G. M. Kavoulakis, M.
Magiropoulos, K. Kärkkäinen, and S. M. Reimann, Phys.
Rev. Lett. 103, 100404 (2009).
[16] S. Bargi, F. Malet, G. M. Kavoulakis, and S. M. Reimann,
Phys. Rev. A 82, 043631 (2010).
[17] K. Anoshkin, Z. Wu, and E. Zaremba, arXiv:1207.3449.
[18] M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri,
K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97,
170406 (2006).
[19] The histogram in Fig. 1(d) was obtained for t < 20 s. At
longer t the characteristic value RðqÞ slowly decreases [7].
We take this into account when determining q.
[20] Based on "500 measurements we find essentially symmetric behavior for negative Pz . One small difference isthat no phase slips occur for about 70 s for Pz ! !1 and90 s for Pz ! 1.
[21] A. Widera, S. Trotzky, P. Cheinet, S. Fölling, F. Gerbier,
I. Bloch, V. Gritsev, M. D. Lukin, and E. Demler, Phys.
Rev. Lett. 100, 140401 (2008).
[22] In the presence of in-plane field gradients the twocomponent gas is extremely susceptible to phase025301-4

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PRL 110, 025301 (2013)

PHYSICAL REVIEW LETTERS

separation. By balancing j"i and j#i in-trap
density distributions we minimized such gradients to<5 ( 10!5 G=cm.
[23] Because of long separation the two rf pulses have a random
relative phase ’. After the second pulse Pz ¼ P cosð’Þ

week ending
11 JANUARY 2013and hP2z i ¼ P2 =2. For each t we get hP2z i from about
40 measurements.
[24] We checked for t up to 100 s that we can always convert
jyi into pure j #i by sweeping ! away from resonance.
[25] S. Baur (private communication).

025301-5

152

APPENDIX A. PUBLICATIONSAppendix B
Light Potentials
In this Appendix, we first overview the relationship between light intensity andthe potential experienced by a 39 K atom in the relevant hyperfine state for theexperiments in this thesis. Next, we introduce the concept of using Fourier optics
to craft atomic potentials.

B.1

Light Potentials

If far-detuned light of frequency ω and intensity I illuminates a two-level quantum
system {|1i, |2i} with resonant frequency ω0 , a potential is induced via the AC
Stark effect [12]:V (I) = |h1|er|2i|2

1
1
1
+
I,
20 ~ ω0 − ω ω0 + ω




(B.1)

where er is the dipole operator. The coupling matrix element can be related tothe decay linewidth Γ = lifetime−1 via |h1|er|2i|2 = 3π0 ~c3 Γ/ω03 [125].
In the case of a real atom, transitions are generally between two manifolds ofmany spin states. For two levels with angular momentum numbers |J, F, mF i and|J 0 , F 0 , m0F i the appropriate matrix element is [126],2|hJ, F, mF |er|J 0 , F 0 , m0F i| =3π0 ~c3 Γ
ω03
|

{z

}

Matrix element
for two-levelsystem2J 0 + 1
2J + 1Λ2 (J, F, mF , J 0 , F 0 , m0F ) .

|

|

{z

}

Accounts fordegeneracy ofeach J state{z

Clebsch-Gordon
coefficient, gives
spin-state overlap

}

(B.2)
In our experiment, axial confinement is produced by π-polarised light at 532 nm.
Labelling atomic states in the low-field basis, experiments are performed in |4 2 S1/2 , F =1, mF = 1i; this is coupled to excited states {|F = 1, mF = 1i, |F = 2, mF = 1i}in both D1 (J = 1/2) and D2 (J = 3/2) upper manifolds. Inserting the generalised matrix element of Eq. (B.2), the two-level potential of Eq. (B.1) is modified
153

154

APPENDIX B. LIGHT POTENTIALS

Figure B.1: The effect of a lens on an incident plane wave, with wavevector component
q normal to the optic axis. The plane wave is brought to a point at a position zf = qf /kon the focal plane.

to:V (I) =

1 3πc2
3
3 ωD1

"

#

ΓD1
ΓD1
2 3πc2++3
ωD1 − ω ωD1 + ω
3 ωD2

"

ΓD2
ΓD2
+
ωD2 − ω ωD2 + ω

#!

I.

(B.3)
The detuning of our axial light is much larger than the fine-structure splitting, andso the above expression effectively reduces to the simple two-level result. Specifictransition parameters for 39 K are tabulated in [121].

B.2

Sculpting Potentials via Fourier Optics

If one can shape light into a desired geometry, then the light potential of Eq. (B.3)
induces a corresponding atomic potential. One particularly useful phenomenon
for crafting intricate optical patterns is the ability of a lens to project the Fouriertransform of an incident optical field onto its focal plane. For monochromatic lightof wavenumber k, the wavevector component parallel to the plane of the lens, q,
is mapped onto spatial position zf at the focal plane.
This can straightforwardly be seen by considering a plane wave incident on a lens,
illustrated in Fig. B.1. For simplicity we work in 1D. The incident amplitude
pattern on the lens is ∼ eiqz ; the effect of the lens is to impart a phase change−kz 2 /(2f ). To obtain the amplitude Ef (zf ) at some point zf on the focal plane,
one integrates over all possible paths taken from the lens plane to zf .
Ef (zf ) ∼

Z

iqz

dz

e

e

| {z }|{z}|

Integrate over Incident
lens plane amplitude−ikz 2 /(2f ) ik

e

{z

Lens

}|

√ 2

1

f +(zf −z)2q

f 2 + (zf − z)2{z}|

Propagate phaseto focal plane{z

Decay ofamplitude

.

(B.4)

}

Making the assumption that |zf − z|  f , the path length can be expandedB.2. SCULPTING POTENTIALS VIA FOURIER OPTICS

155

as,
q

f 2 + (zf − z)2 ≈ f +

zf2
z2
zf
+
− z.
2
2
2f
2f
f

(B.5)

The first two terms are independent of z and can be taken outside of the integral.
The third term is exactly cancelled by the lens, leaving only the linear fourthterm. Our amplitude integral therefore simplifies considerably, toE(zf ) ∼=Z

dz eiqz e−ikzf z/f ,Z

dz eiz(q−zf z/f ) ,!

zf∼δ q− y .
f

(B.6)

For some incident collection of plane waves, each is mapped onto position in the
focal plane according to,
zf k
q→
.
(B.7)
f
This reciprocal relationship allows intricate, micron-scale patterns of light to beconstructed from their macroscopic counterparts in Fourier space1 .
One trivial example is focusing a Gaussian beam; an incoming amplitude profile22
2 2
E(r) ∼ e−r /w has a Fourier spectrum Ẽ(q) ∼ e−q w /4 . Making use of the lens
22
mapping of Eq. (B.7), in the focal plane of the lens Ef (rf ) ∼ e−rf /W , where thefocused waist W = λf /(πw).

1

In terms of amplitude, the pattern on one focal plane of a lens is the Fourier transform of
the pattern on the opposing focal plane. In the case of an object being some arbitrary distancefrom a lens, the intensity pattern on the focal plane still corresponds to the square modulusof the object’s Fourier transform. However, the amplitude itself will in general differ by somephase from the exact Fourier transform. This means that the distance from object to lens canbe freely varied, whilst leaving the focal intensity pattern unchanged.

156

APPENDIX B. LIGHT POTENTIALSAppendix C
Trap Characterisation
This Appendix provides the experimental procedure for measurement of trapping
frequencies in both the axial and planar direction, and the appropriate focusing
time for the momentum focusing of our two-dimensional gas.

C.1

Axial Frequency

The axial trap frequency was obtained from observing the axial breathing mode ofan ideal gas BEC. For a gas of non-interacting particles oscillating symmetricallyin a harmonic potential, the cloud distribution repeats every half trap-period. The
observed breathing mode frequency is therefore 2ωz , and provides a clean measureof trapping frequency. This mode was excited as follows:
1. Prepare an almost-pure BEC in the two-dimensional trap (see Section 7.1).
2. Switch interactions to zero, by ramping the Feshbach field slowly to its zerocrossing.
3. Flash the axial trapping light off for 17 µs, corresponding to 7% of a trap
period.
4. Wait for a variable hold time.
5. Release the BEC from the trapping beams, and observe horizontally after
17 ms ToF to access the momentum distribution of the breathing BEC.
The evolution of the BEC vertical extent with hold time then reveals the frequencyof the breathing mode, and is illustrated in Fig. 6.4.

C.2

Planar Frequency

To determine the in-plane frequencies, we again prepare an almost-pure BEC
identically to the axial frequency sequence of Section C.1. The BEC is then
157

158

APPENDIX C. TRAP CHARACTERISATION

displaced in-plane using a red-detuned ‘tweezer’ beam propagating in the axial
direction, of wavelength 820 nm and waist Wtw = 240 µm. The optical setup forthis beam is described in [21]. The following sequence is used to determine theplanar frequencies:
1. Prepare an almost-pure BEC, identically to the axial frequency sequence.
2. Slowly ramp up the tweezer beam, centred Wtw /2 = 120 µm away from the
ODT centre. At this displacement, the maximum in-plane force is exerted
at a given tweezer power.
3. Rapidly (< 1 µs) switch off the tweezer beam.
4. Hold the BEC in-trap for a variable wait time, thold .
5. Take an in situ image on the vertical imaging system.
From the resulting evolution of the in-plane position of the cloud, we obtain boththe trap eigenfrequencies and the orientation of the trap eigenaxes; these are given
in Eqs. (6.13) and (6.14).

C.3

Focusing TimeThe momentum distribution of our two-dimensional gas is obtained using momentum focusing, outlined in Section 6.3, in which free particle evolution in a harmonic trap maps a particle’s momentum onto position over a quarter-oscillation.
We empirically determine the focusing time at which the momentum distribution
is revealed, by preparing a degenerate cloud and monitoring the width of the sharp
low-momentum peak as a function of focusing time. The experimental sequenceis as follows:
1. Prepare a degenerate two-dimensional cloud; see Section 7.1.
2. Switch off the axial trapping.
3. Wait some variable hold time, before taking a vertical absorption image.
When taking critical point data (Section 7.1.4), the Feshbach field is jumped toits zero-crossing value before taking the image, ensuring an identical spin statecomposition for imaging regardless of the in-trap interaction strength. This step isomitted for determining focusing time; the field takes several ms to stablise to somenew strength, and we wish to avoid the risk of any inadvertent variable detuning
as the focusing time is scanned. We therefore hold the scattering length fixed
throughout both trapping and focusing at a low value a = 10 a0 , (equivalentlyg̃ = 0.01), and image with a frequency resonant at this field. The absolute imaging
efficiency is unknown, but we are interested solely in the variation of peak widthwith hold time.

C.3. FOCUSING TIME

159

Our focusing trap is anisotropic, with the eigenaxes at an angle of θ = 37◦ to thecamera axes1 . We therefore rotate our cloud images by θ, such that the fitted
x and y widths on an image correspond to those along trap eigenaxes. Finally,
we fit the rotated image with some heuristic two-curve model, for example two
Gaussians, to account for the low-momentum peak and broad thermal wings.
Evolution of the peak width with focusing time is plotted in Fig. 6.9, and we takethe appropriate focusing times as the location of a width minimum.

1

See Eq. (6.14).

160

APPENDIX C. TRAP CHARACTERISATION

Appendix D
Absorption Imaging andCalibration
D.1

Absorption Imaging

We probe our cloud using absorption imaging, in which a beam of resonant light ofwavelength λ is directed through the atoms, which are at the focal plane of a CCD
camera. As the resonant imaging light passes through a cloud, it drives transitions
from the initial ground state to some excited state. The atom subsequently decays
over a timescale Γ−1 , emitting a photon isotropically, and reducing the net fluxof imaging light through the cloud. From the amount of transmitted light as afunction of position over the cloud, the column density can be mapped.
For an atomic transition of saturation intensity1 Is , the intensity I of the imaging
light decays with propagation distance z through the sample [119], according
to,
σ0
dI
= −n3DI(D.1)
dz
1 + IIs
where the zero-intensity cross section σ0 = 3λ2 /(2π) for a closed cycling transition [119]. This equation can be integrated to yield
OD = − lnI
I0 − I
+
,
I0
Is

(D.2)

where I0 is the incident imaging intensity, and the optical density OD is definedin terms of the column density n as OD = σ0 n.
Typically cold atoms experiments make use of two limits. For low intensities(I/Is )  1, the log term dominates and the linear term can be neglected. At
high intensities the opposite situation arises, and the transmitted intensity varies
Defined as πhcΓ3λ3 , this is the intensity for which the Rabi frequency of transfer from theground to excited state equals the spontaneous decay rate Γ.
1

161

162

APPENDIX D. ABSORPTION IMAGING AND CALIBRATION

linearly with OD. Practically, photon shot noise provides an experimental limitation to which method is used, giving an upper bound to the OD which can bereliably resolved by low-intensity imaging, and a lower bound to high-intensity
imaging. For extraction of (µ, T ) from a cloud image good accuracy is requiredin the low-OD wings of the distribution, and so we make use of low-intensity
imaging.
Practically, for a single atomic sample we take three images:
1. An initial image with atoms present, whose shadow is projected onto the
CCD. This takes the form of a map of detected photon counts Catom (r) +Cback (r) over the cloud.
2. A reference image Cref (r) + Cback (r), with an identical pulse of imaging lightbut no atoms present.
3. A background image Cback (r), with no imaging light but all other experimental settings identical. This accounts for any contribution to camera
counts from sources other than imaging light.
The optical density is then obtained as,!

Catom (r).
OD ≈ − ln
Cref (r)(D.3)

It is worth estimating our upper bound to measurable OD; for a total number ofdetected imaging photons γ on a camera pixel (in the absence of atoms), the maximum OD that can be reliably measured is found by equating the corresponding√
detected photon count to the shot noise γ. Thus,√γ
ODmax = − ln
.
γ

(D.4)

For an imaging intensity I, pulse duration τ , imaging wavelength λ, pixelsize ∆x,
and CCD quantum efficiency q, one has γ = Iqτ ∆x2 λ/(hc). This results in a
maximum observable OD of,
ODmax =

D.2

1 Iqτ ∆x2 λ
ln
.
2
hc

(D.5)

Magnification of Imaging Systems

The magnification of our horizontal imaging setup is obtained by observing a freefalling cloud at various times; it accelerates at a known g = 9.81 ms−2 downwards.
The pixelsize ∆x is the in-trap length which maps to a single CCD pixel on ourcamera. We find a small spatial variation in ∆x, summarised in Table D.1. We do
not attribute this variation to a particular source; possibilities are some residual
curvature of the CCD, or aberrations of our imaging system. We heuristicallyD.2. MAGNIFICATION OF IMAGING SYSTEMSToF (ms)
2
4.5
7.5
10.5
12.5

M
2.65
2.71
2.71
2.72
2.73

163

Pixelsize ∆x (µm)
2.43
2.38
2.38
2.37
2.36

Table D.1: Magnification M and pixelsize for the horizontal imaging system at various
free-fall times, determined from free-fall under gravity.

account for it via a linear fit, giving a pixelsize ∆x = 2.43 − 0.0056 × ToF, where∆x is in µm and ToF in ms.
Having calibrated the horizontal pixelsize against gravity, the magnification of thevertical imaging system can now be obtained by relation. We prepare an almostpure BEC in the ODT [16], and compare images taken on both the horizontal andvertical imaging systems. The Thomas-Fermi density profile [12] of an interacting
BEC has a sharp-edge, giving a robust cloud lengthscale. The absorption image istaken along the horizontal direction only, and the resulting fluorescence is observedon the vertical camera; this ensures that precisely the same object is observed oneach camera. We take several cautionary steps to improve accuracy:• For imaging intensity I and imaging pulse duration τ , the acceleration of
an atom during a pulse scales as I, displacement as Iτ 2 and fluorescencesignal as Iτ . The cloud distortion in the vertical image at a fixed signal istherefore minimised by decreasing the pulse duration whilst preserving the
product Iτ .
We therefore shorten our imaging pulse from our typical value of 80 µs to
40 µs, whilst increasing our imaging beam intensity to ≈ 0.6Is from ourtypical value of ≈ 0.3Is .
• We desire as large a BEC as possible, to reduce the fractional error in radius
measurements. However, the resultant peak OD is high enough to saturate
absorption images, leading to a distortion in the fitted TF profile. For
our imaging intensity, pulse duration and horizontal camera parameters1
Eq. (D.5) gives a maximum observable ODmax ≈ 4, much less than typical
BEC densities.
To circumvent this problem, we selectively prepare only a small atomic fraction in a visible state. Experiments are performed with 39 K in the state|F = 1, mF = 1i and horizontal imaging is on the cooling transition, withground state |F = 2, mF = 2i. Exposure to repump light could be used
to controllably transfer atoms to the visible state before imaging; however,for optically thick samples the non-uniform repumping would lead to a distorted transferred cloud. Instead we employ a microwave (MW) pulse of
duration 200 µs immediately prior to imaging, approximately resonant with1

Quantum efficiency ≈ 0.25, ∆x = 2.43 µm from Table D.1.

164

APPENDIX D. ABSORPTION IMAGING AND CALIBRATION
the |F = 1, mF = 1i → |F = 2, mF = 2i transition. We empirically tune
the MW frequency to 468.4 MHz, such that the horizontal image OD is justbelow saturation.

The horizontal and vertical images are each integrated along one direction (vertically on the horizontal image, and along the imaging beam on the vertical image).
The two resulting 1D profiles n(x) should correspond to exactly the same doublyintegrated cloud density profile, and are each fitted with a doubly integrated
Thomas-Fermi profile,
!2
x2
(D.6)
n(x) = A 1 − 2 .
Rx
Averaging over 23 images, we find a size ratio of 1.21 ± 0.01, giving a verticalmagnification of 4.41 and pixelsize ∆x = 2.95 µm.

D.3

Rescaling Function

As discussed in Section 6.4.1, on each absorption imaging cycle there is a small(≈ 2.5%) probability than an atom relaxes into a dark state. We empiricallyaccount for this via a rescaling function F, scaling the OD on each image pixel
by a photon-count-dependent factor. This mapping is obtained as follows:
1. Prepare a thermal cloud of 39 K in our ODT [16] at T ≈ 600 nK, in thepresence of an anti-gravity gradient.
2. Release the cloud for 10 ms ToF in total.
3. Jump the Feshbach field to its zero crossing value tfoc = 6.96 ms (seeEq. (6.27)) before the image is taken. This ensures magnetic conditions
are identical to those after momentum focusing of a two-dimensional cloud.
4. Repeat for a wide range of imaging intensities.
5. For each image, obtain an optimal reference pattern1 of imaging photon
opt
counts Cref(r).
6. Assume that on each ring of pixels, concentric with the cloud centre, thevariation in OD is due to the variation in the incident imaging photon count.
For a single ring-radius, from all images construct a plot of (OD0 /OD) vs(C/C0 ), where C0 is a fixed reference count and OD0 the corresponding
optical density. In our case we choose C0 = 1300, equivalent to (I/Is ) ≈0.028.
7. Repeat the previous step for a range of ring radii; the lower bound should
be set so as to avoid any regions of saturated OD, and the upper bound set
to lie within the atomic cloud.
1

See Section 6.4.

D.4. IMAGING EFFICIENCY

165

8. Finally, construct a numerical interpolating function (OD0 /OD) = F(C/C0 )from the compiled correlation plots from all radii.
In principle the function F(C/C0 ) could be obtained from a single ring radius;
reassuringly, we find that the functions from each radius overlap extremely well,which is illustrated in Fig. 6.15. This implies that the true rescaling function is aseparable function of actual atomic density and imaging count.
The rescaling function, along with an analytic toy model which explains its behaviour well, is given in Fig. 6.16.

D.4

Imaging Efficiency

The absolute imaging efficiency is determined by comparison of the observed
BEC critical number with its theoretical value. In a harmonic trap of frequencies(ωx , ωy , ωz ), this critical number is
Nc3D = A ζ(3)kB T~(ωx ωy ωz )1/3

!3

1
+ δNcfs . (D.7)(1 − 3.426(a/λth ) + 42(a/λth )2 )3Here A is a correction due to anharmonicity of the Gaussian trapping beams, andδNcfs the finite-size shift of the critical point [127]. We also take into account theinteraction-induced shift of the critical point [79].
The critical point is extracted in the following manner. An anti-gravity gradientis present for all stages, to ensure identical magnetic conditions to experimentswith two-dimensional gases.
1. Evaporatively cool a thermal cloud in the ODT [16], until a small BEC isproduced.
2. Wait a variable hold time at a scattering length of 59 a0 . During this holdatoms are lost via background gas collisions and scattering from trapping
beams.
3. Release the cloud for 10 ms ToF in total.
4. Jump the Feshbach field to its zero crossing value tfoc = 6.96 ms (seeEq. (6.27)) before the image is taken. This ensures magnetic conditions
are identical to those after momentum focusing of a two-dimensional cloud.
5. Take an absorption image of the cloud. Optimal reference images are calculated1 , and the OD rescaling of Eq. (6.39) applied.
6. Extract the thermal number N , temperature T , and condensed number N0 .
2/5

7. Plot N vs N0 , and T vs N0 . The critical point (Nc3D , Tc ) is found as thecorresponding intercepts.
1

See Section 6.4.

166

APPENDIX D. ABSORPTION IMAGING AND CALIBRATION
2/5

The N0 x-axis of the number plot ensures that total number N varies linearly with the x-coordinate [78], making extrapolation to the intercept more reliable.
The anharmonicity factor A is found by numerical integration of the Bose distribution at µ = 0 over our true trapping potential, which is the sum of crossed attractive Gaussian beams. This method is outlined in [114], and yields A ≈ 1.09.
Our trap frequencies at the ODT power used are (ωx , ωy , ωz ) = 2π×(53.2, 63.1, 89.5) Hz.
The planar frequencies are obtained identically to Section 6.1.2, and the verticalfrequency by displacing the cloud using an Ioffe coil [113, 114] and observing thesubsequent vertical oscillation on the horizontal imaging system.
We find a ratio of theoretical to observed critical number of 4.3 ± 0.2. The error isestimated from taking five similar critical points over the course of one month. We
also repeated the above sequence for various in-trap interaction strengths ranging
59 − 242 a0 , which led to a similar spread.

Appendix E
Anti-Gravity Field
During the momentum focusing, it is essential that atoms experience minimalvertical displacement so that they remain within the harmonic region of the ODT.
During focusing, the ODT is insufficient to support the atoms against gravity. We
therefore perform our experiments within a vertical magnetic field gradient, which
compensates the gravitational potential gradient.
This is provided by an anti-Helmholtz coil pair, which provides a quadrupole
magnetic field at the atoms. This coil pair is also used for our MOT field, magnetictransport, and provides the quadrupole component of our magnetic QUIC trap.
Extensive details of the coil setup can be found in [113, 114].
The atoms experience a superposition of the uniform Feshbach field (0, 0, BF ),and a quadrupole field (Ax, Ay, −2Az), where BF and A are functions of thecurrent in the Feshbach and quadrupole coils respectively. The total field, and its
magnitude, are then given by




Ax


B =  Ay
BF − 2Az

,

v
u
u
4Az
A2 2
+
(x + y 2 + 4z 2 ).
|B| = B t1 −
F

BF

BF2

(E.1)

Knowing from dimensional analysis that we require µB A ∼ mg, where g is thegravitational acceleration and µB the Bohr magneton, for typical cloud radii R ∼100 µm the quadrupole field varies by ∼ AR ∼ mgR/µB = 0.07 G. This isextremely small compared with the Feshbach field BF ≈ 400 G, and keeping firstorder terms in A
|B| = BF − 2Az.
(E.2)
The atoms are located at the quadrupole centre (z = 0) where the field strengthis set solely by the Feshbach coils. However, |B| now varies linearly with verticaldisplacement z; empirically, we find the necessary quadrupole current by creating
a BEC in our ODT [16], releasing it in the presence of our anti-gravity field,
and imaging along the horizontal direction. Tuning the current such that verticaldisplacement over very long (30 ms) ToF is less than 1 pixel (= 2.43 µm) ensuresthat the residual vertical acceleration is . 10−3 g. This would induce a vertical
167

168

APPENDIX E. ANTI-GRAVITY FIELD

displacement . 0.2 µm during our focusing time of ≈ 7 ms, which is negligible
compared with the ODT waist of 140 µm.
As expected from Eq. (E.2) the necessary quadrupole current is independent of
BF . To confirm that the quadrupole field does not shift the field strength at
the atoms, we measured the resonant imaging frequency of a thermal cloud in
the ODT, both with and without the anti-gravity field. Imaging was along the
Feshbach field (axial) direction, between the ground state |4S1/2 , I = 3/2, mI =3/2, mJ = −1/2i and excited state |4P3/2 , I = 3/2, mI = 3/2, mJ = −3/2i; thedetails of this ‘high-field’ imaging is discussed in Section 6.4. The field strength
used was 305.5 G.
The resonant imaging frequency in the presence of the anti-gravity field was shiftedby −0.1 ± 0.1 MHz compared to its absence. The energy splitting of our imaging
two-level system varies as −µB B (see Section 6.4.1), which implies a shift inmagnetic field of 0.07 ± 0.07 G, less than our Feshbach field resolution of ≈ 0.1 G.
At our highest g̃ ≈ 0.5, using Eq. (1.6) this field shift would correspond to anerror of ∆g̃ ≈ 0.01; this source of error was subsequently neglected.

Appendix F
Error Analysis
The errors on our measured critical atom number and chemical potential for the
BKT transition can be divided into two categories: statistical, and systematic.
Statistical errors are a measure of how well we can determine critical parameters
given the scatter in experimental data. In addition to these statistical errors,
there are various sources of systematic error. These can be divided into arisingfrom system calibration uncertainties, finite-size physics, the inhomogeneity of ourtrapped gas and anharmonicity of our focusing trap. In this Appendix we outline
our estimates of both categories of error, and a summary of their values is given
in Table 7.1.

F.1

Statistical

Each critical point measurement consists of M ≈ 72 images of a gas crossing
the BKT point; we estimate the statistical error in critical parameters via thefollowing ‘bootstrapping’ approach.
1. Randomly select M images from the entire critical point measurement; some
images may be selected multiple times, others not at all.
2. Apply the analysis of Section 7.2.4 to obtain Nc , Tc and µc for this selected
image set.
3. Repeat a large number of times, in our case 1000.
4. Take the standard deviation of the critical values to be representative of thestatistical uncertainty in their determination.
This method avoids the use of fit uncertainties; our fit functions for P0 (t), µ(t),T (t) and N (t) are somewhat arbitrary.
169

170

F.2

APPENDIX F. ERROR ANALYSIS

Systematic

Calibration uncertainty The accurate extraction of atom numbers and temperatures require good knowledge of our trap and imaging parameters. Uncertainties in trap frequencies, imaging magnification and absolute imaging efficiency allcontribute to uncertainty in critical parameters N̄c /Nc0 and µ̃c . These uncertainties
are given at appropriate points within Chapter 6 but are tabulated in Table F.1
for convenience. They are propagated to our final critical parameters N̄c /Nc0 andParameter
Planar trap frequency ωrAxial trap frequency ωz
Focusing trap frequency ωfoc
Imaging pixelsize ∆x
Imaging efficiency

Nominal Value
37.7 Hz4040 Hz36.2 Hz2.95 µm
0.23

Uncertainty±0.3 Hz±6 Hz±0.1 Hz±0.02 µm
±0.01

Table F.1: Calibrated system parameters and their uncertainties.

µ̃c , assuming an ideal gas result for the number of atoms in excited states.
The end result is a symmetric systematic error of ∆µ̃c ≈ ±0.05 and ∆(N̄c /Nc0 ) ≈±0.14
Finite size The ideal gas critical number Nc0 of Eq. (4.13) is exact in the thermodynamic limit [128], in which N → ∞ and ωr → 0 with N ωr2 held constant.
For a finite size system, this critical number is slightly shifted. This effect canbe understood as µ needing to reach the lowest trap energy ~ωr /2 for BEC tooccur, rather than zero, and increases the critical atom number with respect tothe infinite system result.
This can be estimated straightforwardly, since the shift in µ̃c for the ideal gas
is simply ∆µ̃c ≈ ~ωr /(2kB T ). Evaluating this for a characteristic T = 170 nK
gives ∆µ̃c ≈ +0.005. The corresponding shift in critical number is calculated
following [127], giving ∆(N̄c /Nc0 ) ≈ +0.04.
Inhomogeneity Exactly at the critical point of a harmonically-trapped twodimensional gas, the gas becomes critical at the trap centre only, over a vanishingly
small area. Since we are probing coherence over a finite distance L ≈ 15 µm, at
the critical point coherence cannot exist over this length. One expects that the
momentum distribution peak P0 should grow rapidly once a region of diameter L
has become superfluid. This results in a shift ∆µ̃ in the apparent critical chemical
potential:
 2
1
L
∆µ̃ =
mωr2
.
(F.1)
2kB T
2
For our trap parameters, this evaluates to ∆µ̃ ≈ +0.05.

F.2. SYSTEMATIC

171

In terms of the critical number ratio N̄c /Nc0 , a rough estimate comes from assuming that the shift in atom number associated with some chemical potential shift∆µ̃ is dominated by atoms in the distribution wings, which are well-described by
MF theory. We can then use the MF critical number expression of Eq. (4.17),
evaluated at µ̃ = µ̃c and µ̃ = µ̃c + ∆µ̃. This yields a positive shift in the criticalnumber ratio of . 0.2.
Critical divergence of ξ As a normal gas approaches the BKT transition, thecorrelation length ξ diverges exponentially [51, 95],s

ξ = λth exp

!

aTBKT
,
T − TBKT(F.2)

where TBKT is the critical BKT transition temperature, and a is a model-dependent
dimensionless parameter. In our experiment this divergence corresponds to a
sharp growth in P0 as the transition is approached; in principle P0 should still
exhibit a discontinuous gradient at the BKT transition, but experimentally thediverging ξ makes detection of this change difficult and will shift the apparent
critical temperature by ∆T .
It is worth pointing out that this critical divergence in ξ is entirely distinct fromthe rapid growth in a normal gas as phase-space density is increased1 . The former
divergence is ‘pinned’ to the BKT transition temperature, and would shift theapparent onset of coherence regardless of the phase-space density or g̃ in question.
The latter represents a growth in ξ independent of any transition, and precludes
reliable measurement of the transition if ξ  L before the system is even close tocriticality2 . The two separate phenomena are illustrated in Fig. F.1.
We can estimate the criticality-induced shift by setting a = 1 and asking when
ξ = L. This yields a fractional temperature shift ∆T /TBKT ≈ 0.1. The weak
logarithmic dependence of ∆T upon L means that decreasing this criticalityinduced shift to 1% would requires us to increase (L/λth ) by a factor of 103 !
A positive fractional temperature shift of 0.1 corresponds to a negative shift in
the critical phase-space density ∆D ≈ −0.1DBKT . Noting that the critical chemical potential is dominated by interaction energy and assuming fully suppressed
fluctuations at the critical point, we can deduce a corresponding shift in µ̃c :µ̃c ≈g̃DBKT,2π=⇒ ∆µ̃c ≈ −g̃
0.1DBKT ,
2π

(F.3)

giving a shift of ∆µ̃c ≈ −0.05 at our maximal g̃ = 0.5. Fortuitously, this is comparable in magnitude and of opposite sign to our shift due to trap inhomogeneity,and so we expect these systematic shifts to approximately cancel each other.
1
2

See Eq. (6.31).
See Section 6.3.3.

172

APPENDIX F. ERROR ANALYSIS
40

Normal gas ξ, in absence of BKT

BKT

Critical divergence of ξ

ξ/λth

30
L/λth
20

10

0

2

4

6

8

10



Figure F.1: The behaviour of the correlation length ξ in the vicinity of the BKTtransition, which occurs at phase-space density DBKT (red dashed line). In the absenceof any transition, the normal gas ξ grows exponentially with phase-space density D.
If ξ exceeds our experimentally-probed coherence lengthscale L (green dashed line) for
D < DBKT , reliable identification of the transition is precluded. However, regardless ofthe normal gas behaviour, in the vicinity of the BKT point ξ diverges and will always
lead to an apparent shift in the critical point; this shift is illustrated by an arrow.

Anharmonicity of planar potential The planar potential for both in-plane
trapping and momentum focusing is provided by our ODT. The beam waist is
140 µm; in comparison the thermal diameter of a 170 nK cloud in a 38 Hz planarpotential is 50 µm. The atoms therefore experience a small deviation from perfectharmonic trapping; there are two principle consequences to consider:
1. The atom number required for the central density to become critical isshifted relative to harmonic trapping.
2. The momentum focusing no longer provides a perfect mapping between initial momentum and final position.
We are interested in the order of magnitude of these shifts, and so consider an ideal
gas case for simplicity. The critical atom number shift is calculated by numericallyintegrating the Bose distribution over our true ODT potential of crossed Gaussian
beams at µ = 0, and comparing the obtained atom number with the harmonic
trap result; this method is outlined in [114], and gives a positive shift to the criticalatom number of ∼ 10%.
Our fit functions for extraction of T and µ assume a harmonic focusing trap; theanharmonicity therefore leads to a shift in the measured values with respect totheir true values. An ideal gas is invariant under momentum focusing; there areF.2. SYSTEMATIC

173

no interaction effects, and so switching off the axial trapping has no effect upon
the planar distribution. We therefore simply populate our true ODT potential atµ = 0 and a typical T ≈ 170 nK, according to the Bose distribution for an ideal
gas; this gives the atom distribution that would be obtained in an image. We
then fit the distribution with an ideal gas model for a perfect harmonic trap, andcompare the fitted µ and T with their true values.
Fortuitously, the two effects oppose each other; the actual critical number is increased, but the fitted temperature is higher than the true and so we normalise
to an increased Nc0 . Overall, we obtain a shift in the critical chemical potential∆µ̃c ≈ −0.04, and atom number ∆(N̄c /Nc0 ) ≈ −0.01.

174

APPENDIX F. ERROR ANALYSIS

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