Degenerate Bose
Gases: Tuning Interactions &Geometry

Alexander Lloyd Gaunt
Trinity College, October 2014

A dissertation submitted for the
degree of Doctor of Philosophy

Cavendish Laboratory

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 other qualification. This thesis does
not exceed the word limit of sixty thousand words, includingtables, footnotes, bibliography and appendices, set out by
the Faculty of Physics and Chemistry.

2014 Alex Gaunt

Cavendish Laboratory
University of Cambridge
J J Thomson Avenue
Cambridge CB3 0HEUKii

Abstract

This thesis describes experiments carried out on a dilute cloud of Bosonic atoms
cooled to quantum degeneracy. It is divided into two parts: Part I describes experiments in which we tune the inter-particle interactions to explore the system’s
dynamics in both weakly and strongly interacting limits. Part II describes novel
optical trapping geometries for cold atomic gases, with an emphasis on the first everrealisation of an atomic Bose-Einstein condensate in a uniform potential.
Part I employs a method in which inter-particle interactions in a nanokelvin gas of
39 K atoms are tuned via a Feshbach resonance. Using this technique we explore

the non-equilibrium dynamics of a weakly interacting Bose-condensed gas in adissipative system. The highlight of this study is the observation of a “superheated”
Bose-Einstein condensate persisting at temperatures up to 1.5 times higher thanthe equilibrium critical temperature. In a second study, we investigate the opposite
regime of maximal interaction strength (the “unitary” regime). Here we experimentallydemonstrate scaling laws governing the dynamics of clouds undergoing atom loss
and heating by multi-particle collisions.
In Part II, we demonstrate holographic techniques for shaping laser beams to produce
new atom trap geometries. We concentrate on the theoretically important andexperimentally novel uniform geometry and review our first experiments performedin this system. In equilibrium we study both the thermodynamics of a homogeneous
87 Rb gas held in our novel trap and perform spectroscopic measurements of its
ground-state. In a dynamical study we investigate the coherence properties of a
Bose-Einstein condensate formed after a rapid thermal quench of a homogeneousgas. We observe Kibble-Zurek dynamics following this quench, and are able makethe first measurement of the dynamical critical exponent for the 3D Bose-Einsteincondensation transition using our homogeneous system.

iii

Acknowledgements“Writing is difficult: only some people are good at it.” This diplomatic phrase positedby a wise man rings in my ears as I struggle to eloquently pen my heartfelt gratitude
to the people who have supported me over the last three years. My Ph.D. has taughtme that where the pen can stutter, the keyboard can triumph. Therefore I will attempt
to forego the unfamiliar world of vocabulary, sentence structure and punctuation by
distilling my gratitude into a more manageable Excel bar chart.
Fig. 0.1 shows the average flux of emails that I received from contemporaneous lab
members normalised to a reference flux from my wife, Beth. This bar chart provides
a good starting point for objectively dividing my acknowledgements among the key
characters who helped me with the work presented in this thesis1 .

Fig. 0.1.: Normalised email flux from lab members. We count the total number of emails received
from each lab member and divide by the overlap time with my Ph.D. For reference, theresulting fluxes are normalised to that from my wife, Beth.

Zoran’s unerring dedication and enthusiasm irrefutably shines out of Fig. 0.1. As mysupervisor, his active influence and impact on every aspect of the lab’s progress is
exemplified by the only score greater than unity. An analysis of the length and depthof the emails is beyond the scope of this thesis, but I am certain that Zoran would
1

In my analysis of this data I will mitigate any potential offense by mentioning systematic errors
arising from the use of email volume as a metric for gratitude.

v

again claim victory in these statistics. There is nothing more reassuring for a Ph.D.
student than knowing that your supervisor is always available with deeply insightfulcomments on the experiments. Furthermore, Zoran has gone out of his way tosupport the progress of my career and I am truly grateful to Zoran for everythingthat he has done to help me over the years.
The next highest bars in Fig. 0.1 are the postdocs, Rob Smith and Nir Navon, whowere directly involved in the experiments presented in this thesis. Rob’s contribution
is vastly under-represented in Fig. 0.1. As a man of quiet brilliance, he exemplifies
the systematic problem with the email metric. His uncanny ability to understandeverything and make anything work, has been pivotal to every success in this thesis.
Nir has injected great enthusiasm and “rock and roll” into the lab. As one of the mostbadass physicists in the group, he played an essential role in the uniform-system
experiments, and was a key antagonist in the on-going dialogue concerning thequality of my software, PIDs, and general persona.
I have been proud to be part of a motley band of Ph.D. students alongside Richard
Fletcher, Tobias Schmidutz and Igor Gotlibovych. Rich formed part of the original
Dream Team and many happy nights were spent watching cartoons as we superheated some BECs. A lot of thanks have to go to Tobias and Igor (as a strong but
silent type, Igor’s contribution is again underestimated by the email tally). Theirmonths of hard work (including an unfortunate incident involving my laser and adrill) meant that I could hit the ground running in Part II of this thesis. Thanks also
go to the former members of the lab, Naaman Tammuz, Robbie Campbell, Stuart
Moulder and Scott Beattie who built the machine used in Part I of this thesis.
Aziza Suleymanzade and Martin Robert-de-Saint-Vincent have been busy working
on experiments which are not covered in this thesis, but their presence has completed
the daily life that I have enjoyed in the group. Regardless of experimental allocation,
all group members (including the Ghost) are bonded by TheQuiz, and it has been apleasure to MC TheQuiz for such an enthusiastic bunch of budding historians.
Finally, even though “working from home doesn’t count”, I have been supported
outside the lab by my parents, Mary and Stephen, my siblings, Jo and Steph andmy wife Beth. Beth’s contribution to this work is difficult to overstate. Without
her concern for my well-being, I would have rapidly devolved into a cave-dwelling
creature surviving solely on Cherryade and prawn cocktail crisps. Her unerring
optimism that I will one day share her delight in a picturesque sunset or a brightrainbow and her contagious enthusiasm for life infinitely brighten up my life outside
the lab. I am truly grateful to her for being my companion through everything.

Contents

Outline1I

5Tuneable Interactions

1 Introduction

7

1.1 Tuneable s-wave scattering . . . . . . . . . . . . . . . . . . . . . . .
1.2

Producing 39 K condensates. . . . . . . . . . . . . . . . . . . . . . .

11

1.3 Extreme interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2 A Superheated Bose-condensed Gas

15

2.1 Qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3 Experimentally measuring chemical potentials . . . . . . . . . . . . .

21

2.4 Observation of a superheated state . . . . . . . . . . . . . . . . . . .

28

2.5 Dynamical model of superheating . . . . . . . . . . . . . . . . . . . .

31

2.6 Limits of superheating . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3 Stability of a Unitary Bose Gas

39

3.1 Can a unitary bose gas be stable at degeneracy? . . . . . . . . . . .
3.2

Scaling Law 1: N T β = const.

41. . . . . . . . . . . . . . . . . . . . . .

43∝ 1/T 2 at unitarity . . . . . . . . . . . . . . . . . .

50

3.4 Extraction of the rate parameter ζ . . . . . . . . . . . . . . . . . . . .

51

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Trap Geometry55

3.3 Scaling Law 2: L3

II

7

4 Introduction

57

4.1 Atom-light interactions . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5 Robust Digital Holography for Ultracold Atom Trapping

65

5.1 Digital holography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.2 Vector-holography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.3 Raster-holography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75vii

6 Bose-Einstein Condensation of Atoms in a Uniform Potential
6.1 Building an optical box . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 First realisation with atoms . . . . . . . . . . . . . . . . . . . . . . .
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77
78
86
96

7 Observing Equilibrium Properties of a Homogeneous Bose Gas
97
7.1 Thermodynamic experiments . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Ground state measurements . . . . . . . . . . . . . . . . . . . . . . 105
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8 Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose gas
117
8.1 Domain size in Kibble-Zurek theory . . . . . . . . . . . . . . . . . . . 118
8.2 Quantitative measurement of domain sizes . . . . . . . . . . . . . . 122
8.3 Direct observation of Kibble-Zurek freeze-out . . . . . . . . . . . . . 131
8.4 Kibble-Zurek scaling and universal critical exponents . . . . . . . . . 136
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9 Conclusions and Outlook
139
9.1 Thermodynamics with tuneable interactions . . . . . . . . . . . . . . 139
9.2 Kibble-Zurek with tuneable interactions . . . . . . . . . . . . . . . . . 140
9.3 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Appendices

145

A Experimental Sequence Overview
147
A.1 Rubidium-87 D2 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B Raster Holography151
B.1 MRAF: Mixed-region amplitude freedom . . . . . . . . . . . . . . . . 151
B.2 Optical vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.3 Experimental realisation with laser light . . . . . . . . . . . . . . . . 154
B.4 Fourier vs Helmholtz . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C Graphics Card Acceleration of Computations for Ultracold Atom Experiments
159
C.1 Image processing: AnalysisGpUI . . . . . . . . . . . . . . . . . . . . 160
C.2 Physical simulation: EvolveGPuE . . . . . . . . . . . . . . . . . . . . 162
D Numerical Simulation of Kibble-Zurek Dynamics
167
D.1 Dynamical evolution of ξ . . . . . . . . . . . . . . . . . . . . . . . . . 167
D.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Bibliography

173

Outline

Single atoms are well understood quantum objects. However, when a macroscopic
number of interacting atoms come together, an exact treatment of the ensemble canbecome impossible. The fundamental difficulty with many-body quantum mechanics stems from the exponential scaling of the system’s Hilbert space with particle
number2 . Some analytical and computational methods such as the mean-field approximation or Monte-Carlo techniques allow us to probe weakly interacting systems.
However, strongly interacting many-body systems are at the heart of many of themost fascinating and least understood phenomena in modern physics, ranging fromhigh temperature superconductivity and superfluidity to exotic forms of magnetism.
Moreover, we are often interested in dynamical effects in out-of-equilibrium systems,
which can again be difficult to describe exactly with theory.
One fruitful approach to understand many-body quantum systems is to use experimental rather than theoretical simulations, as envisioned by Feynmann in 1982 [1,2].
This is the route persued here, where we use an experimental simulator basedon an ultracold atomic gas cloud. A "quantum simulator" is simply a very cleanquantum system which is easily accessed and tuned in the laboratory. The dynamic
and static properties of our simulator are defined by a many-body Hamiltonian overwhich we have excellent control. By observing the behaviour of our simulation as
we tune the parameters of the Hamiltonian, we can probe the many-body physics of
our system, and any other quantum system described by the same Hamiltonian.
At the core of our experimental simulator is a dilute gas of ultracold Bosonic atoms.
In Einstein’s textbook picture of a 3D Bose gas (discussed in detail in chapter 7),
this system undergoes a phase transition to form a Bose-Einstein condensate (BEC)at the critical point nλ3T ≈ 2.612, where n is the atom density and λT is the thermal
2
The intractability of macroscopic quantum systems is best exemplified by trying to write down anexact description of a state, even before we start to tackle any evolution. Whereas in the classicalcase, recording the 3D position and velocity of N particles requires only 6N real numbers, to write thestate vector for a quantum system containing N distinguishable particles occupying M levels in anarbitrary superposition requires M N complex numbers. Computationally, if we have 1 GB of computer
RAM, this means we can store a classical state containing N ∼ 107 particles, but only the state ofN ≈ 26 particles in the simplest M = 2 level system.

1

wavelength3 . Below the critical temperature, the characteristic wavepacket size, λT ,is larger than the interparticle spacing, n−1/3 , meaning quantum effects dominate
and our simulator probes many body quantum mechanics.
In general, the anatomy of a many-body Hamiltonian, Ĥ, to be simulated in our
experimens is as follows:Ĥ = ĤKin +| {z }
Kinetic
energyĤInt|{z}Interactionenergy+ĤPot| {z }(0.1).

Potential
energyAn essential property of a useful quantum simulator is the ability to implement custom
Hamiltonians, Ĥ in the lab. Dilute atomic gases fulfil this requirement because theresponse of the individual atoms to electromagnetic fields is well understood. This
allows us to use a toolbox of magnetic and optical techniques to synthesise tailored
Hamiltonians, and for a dilute Bose gas, any of the three terms in Eq. 0.1 can beexperimentally tuned4 . This thesis describes techniques for tuning the last two terms
in Eq. 0.1: the interaction term, ĤInt , and the potential term, ĤPot . We then explore
the phenomenon of Bose-Einstein condensation in the unconventional Hamiltonians
that we produce. The thesis is divided into two parts, each of which concentrates
on tuning one term, as summarised below.
Interactions

Potential Geometry

Species

Part I

Tuneable

Traditional harmonic

39 K

Part II

Fixed

Uniform

87 Rb

OutlookTuneable

Uniform

39 K

Tab. 0.1.: Summary of the emphasis throughout the thesis. A different machine is used for each partwith specialisations to handle new interaction regimes or geometries. In the outlook section,
we briefly comment on the prospect of a new machine which combines these researchstrands.

Part I: HInt - Tuneable interactions
Since the presence of inter-particle interactions poses many theoretical difficulties
in many body quantum physics, it is extremely useful to experimentally tune theinteractions and empirically explore the resulting physics. The ability to tune interparticle interactions in ultracold gases is unparalleled in other systems, and the firsthalf of this thesis will be dedicated to exploring Bose gases in the two extremes of
3

The ideal value 2.612 is modified slightly in finite-size and interacting systems [3, 4]
The kinetic energy term is not strictly tuneable, however using synthetic gauge fields [5–7], or bytrapping the atoms in a lattice, we can modify the dispersion relation of our Bose gas.
42

interaction strength: In chapter 2, we will explore a very weakly interacting system,and demonstrate a novel non-equilibrium state of a “superheated Bose condensed
gas". In chapter 3, we study the opposite limit where the interaction strength is the
maximum allowed by quantum mechanics. This strong coupling limit is particularlyinteresting because the behaviour of the system ceases to depend on the nature of
the interactions and universal behaviour emerges.
Part II: HPot - Flexible geometries
Perhaps the most intuitive term to change in the Hamiltonian is ĤPot . The geometryof the potential energy landscape can play a key role in the physics displayed bya system. By sculpting appropriate potentials for our atoms, we can create race
tracks for superfluids [8–10], lattices to model solid-state materials [11–13], andmany other interesting geometries. In chapter 5, we present two very versatile
optical methods for creating custom atom traps, and demonstrate experimental
realisation of high-quality, intricate trapping potentials using these techniques.
The development of this optical toolbox inspired us to create one of the mostlong-sought, and theoretically simplest, potentials: a homogeneous box potential.
This potential is abundant in theoretical work because it often offers the simplest
understanding of many-body phenomena, and is appropriate for describing several
naturally occuring many-body systems. However, this potential is difficult toachieve in experiments with atomic gases. In chapter 6, we overcome these
difficulties and present the first realisation of an atomic BEC in a three-dimensional
homogeneous potential, and in chapters 7 and 8 we outline the first equilibrium anddynamical results from the fruitful research paths that our uniform trap has opened up.

3

The following publications are derived from work presented in this thesis:
Part I
Chapter 2 A Superheated Bose-condensed Gas
A. L. Gaunt∗ , R. J. Fletcher∗ , R. P. Smith and Z. Hadzibabic
Nature Phys. 9, 271-274 (2013).

Chapter 3 Stability of a Unitary Bose Gas
R. J. Fletcher, A. L. Gaunt, N. Navon, R. P. Smith and Z. Hadzibabic
Phys. Rev. Lett. 111, 125303 (2013).

Part II
Chapter 5 Robust Digital Holography For Ultracold Atom Trapping
A. L. Gaunt and Z. Hadzibabic
Sci. Rep. 2, 721 (2012).
Chapter 6 Bose-Einstein Condensation of Atoms in a Uniform Potential
A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith, and Z. Hadzibabic
Phys. Rev. Lett. 110, 200406 (2013).

Chapter 7 Quantum Joule-Thomson Effect in a Saturated Homogeneous Bose
Gas
T. F. Schmidutz, I. Gotlibovych, A. L. Gaunt, R. P. Smith, N. Navon and Z.
Hadzibabic
Phys. Rev. Lett. 112, 040403 (2014).

Chapter 7 Observing properties of an interacting homogeneous Bose-Einsteincondensate: Heisenberg-limited momentum spread, interaction energy,
and free-expansion dynamics
I. Gotlibovych, T. F. Schmidutz, A. L. Gaunt, N. Navon, R. P. Smith, and Z.
Hadzibabic
Phys. Rev. A 89, 061604(R) (2014).

Chapter 8 Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose gas
N. Navon∗ , A. L. Gaunt∗ , R. P. Smith, and Z. Hadzibabic
Science (in press) (2015).

4

Part I
Tuneable Interactions

Introduction

1

Ultracold atoms, and in particular BECs of atoms, give us a unique opportunity toexperimentally access many-body physics in different limits of inter-partical interaction strength. The ability to continuously tune the interaction strength makes cold
atom systems both insightful simulators of less accessible quantum systems, andalso interesting quantum systems in their own right.
Chapters 2 and 3 will separately detail our experiments exploring two opposite
regimes of very weak and very strong interactions. Throughout these chapters
we attempt to highlight novel physical phenomena without reviewing an undue
volume of background theory. In particular, an understanding of standard methodsof laser and evaporative cooling [14] is not required to understand the results inthese chapters (the novel stages of the experiments start after a degenerate cloudhas been prepared). Therefore we have attempted to limit the discussion of thecooling stage of the experiment into section 1.2 and Fig. 1.2. An short pictorialsynopsis of the basic steps for laser cooling can also be found in Appendix A.
Aside from the cooling process, another textbook concept, which underpins the novel
aspects of these experiments, is the phenomenon of Feshbach resonances. Thisphenomenon provides the mechanism by which we tune the inter-atomic interactions.
Since this is of central importance in our work, we devote a short section below tooutlining the basic theory of inter-atomic interactions and Feshbach resonances.

1.1 Tuneable s-wave scattering
In order to understand Feshbach resonances, we must first clarify exactly what wemean by "inter-atomic interactions". Below, we briefly describe how the potentiallycomplex process of one atom scattering off another can be entirely characterised bya single parameter known as the s-wave scattering length, a. We then move on todescribe how a Feshbach resonance can be used to tune a.

1.1.1 s-wave scattering
One could set about the daunting task of writing down a very detailed microscopic
model for the the Van der Waals interaction between neutral bosonic atoms. However,7

we often work in the dilute regime where the typical length scale, rint , of the interaction
potential is much smaller than the other length scales in the gas (namely the mean
distance between atoms, n−1/3 , and the thermal wavelength, λT ). This meansthat the microscopic properties of the interaction potentials are largely irrelevant
for describing physical phenomena in an ensemble. In this section, we outline thestandard formalism which coarse-grains these microscopic details into a single
parameter [3, 15, 16].
The first step of this formalism is to solve the Schrödinger equation, (∇2 + k 2 )ψ =2mr V (r)ψ, in relative coordinates, r, for the wavefunction ψ describing the relative~2
motion of two colliding particles with energy E and reduced mass mr in a generalpinteraction potential, V (note that we have defined k = 2mr E/~2 ). Using theGreen’s function for the Helmholtz operator (∇2 + k 2 ), and applying the first order
Born approximation, the solution is as follows1 :

f (k)

eikr

ψ(r) ≈ φk (r) +r| {z } |incident

z


mr
−
2π~2

Z

}|

{
e
V (r )φk (r ) d r ,{z}−ikr̂·r0

0

0

3 0

(1.1)

scattered

where φk = eik·r is the particles’ wavefunction before the scattering event andr̂ = r/|r|. We can now make the following simplifications:• The angular momentum of a pair of atoms colliding with impact parameter R
is hR/λdB where λdB is the deBroglie wavelength associated with the relative
motion. In a thermal ensemble at temperature T , we write λdB ∼ λT , where
λT is the thermal wavelength of the atoms. At degeneracy, λT ∼ n−1/3 , andfor the colliding particles to interact, we require R < rint . Since we assume
rint  n−1/3 , we conclude that hR/λdB  h, and thus that the colliding
pair’s angular momentum is small [14]. This means that the amplitude of thescattered spherical wave must be a function, f (k), with no angular dependence.
This limit is known as the s-wave scattering limit, with higher order angular2 ∼ k × 100 µK.
terms only significant for clouds at kB T > ~2 /2mrint
B
For k → 0, we can expand 1/f (k) = −1/a + bk + O(k 2 ), with suitable coefficients a and b. In the simplest case (|ka|  1), we take f (k) = −a, thereby
reducing the entire scattering problem to the single parameter, a, known as the1

To see this qualitatively, note that the Schrödinger equation is identical to the optical Helmholtz
equation with a source term ∝ V ψ. By analogy with Huygens’ wavelets and Fraunhofer diffraction,
we quote that the far-field solution to this Schrödinger equation is a spherical wave with amplitude
proportional to the Fourier transform of this source term. Since the source term explicitly contains ψitself, we use the first order Born approximation, ψ ≈ φk to obtain Eq. 1.1.

8

Chapter 1

Introduction

“s-wave scattering length". We will return to the second term in this expansion
in chapter 3.
• Eq. 1.1 states that the amplitude of the scattered wave is proportional to the
Fourier transform of V . Since rint is the smallest length scale in the scattering
problem, V has a very broad Fourier spectrum, which can be treated asa constant for experimentally relevant k. Therefore, regardless of the true
microscopic details of V , we obtain good results using a delta-function pseudopotential V (r) = V0 δ(r).

Using these simplifications in Eq. 1.1, we find:V0 =4π~2 a,m(1.2)

where we have used mr = m/2 for atoms of mass m. Importantly, in this expression,both the sign and magnitude of the scattering potential have been parameterised interms of a single number, a.

1.1.2 Feshbach resonances
Experimental tuning of a is achieved by the phenomenon of Feshbach resonances[17]. Feshbach resonances arise from spin-dependent terms in the interaction
potential which couple different spin states during elastic scattering. The details of
this mechanism are provided below.
When two alkali atoms are brought close together, their atomic orbitals overlap andstart to form a molecular orbital. If the (fermionic) electrons in the atom are in asymmetric spin-triplet state, this molecular orbital must be spatially antisymmetric,
and the converse is true for electrons in antisymmetric spin-singlet states. Since
the spatially symmetric and antisymmetric orbitals experience different degrees
of attraction to the nuclei, their energies are different. This means that there is aspin-dependent exchange term χ(r)S1 · S2 in the scattering Hamiltonian, which splits
the energy depending on the alignment of the electron spins S1 and S2 (χ captures
the dependence of this term on inter-atomic separation). The total Hamiltonian, H,
for the two atoms can then be written:

1
2
H = Hhf,Z+ Hhf,Z+ χ(r)S1 · S2 + HVdW (r),1.1 Tuneable s-wave scattering

(1.3)

9

Fig. 1.1.: Using a Feshbach resonance to tune the s-wave scattering length. (a) Interaction potential
for two states of definite F at large separation. The red curve shows the incident "open"
channel, and the blue curve is the "closed" channel containing a bound state (horizontalline). Using a magnetic field, B, we can tune the bound state above or below the freeparticle energy (since the different F states have a differential magnetic moment, δµ). (b)The bound state energy (blue) with respect to the free particle energy (red) as a function of
B at fixed particle separation. The dotted lines show the energy ignoring coupling between
these states, and the solid lines show the coupled states. (c) Second-order perturbation
theory for the full scattering event gives a resonance in the scattering length when the boundstate energy equals the free particle energy. We illustrate the parameters of this resonancefor 39 K atoms in the |F, mF i = |1, 1i state.
iwhere Hhf,Z
are the hyperfine and Zeeman terms for the ith atom, and HVdW contains
any spin-independent parts of the Van der Waals interaction. For large separations,
χ is small and the spin coupling is a perturbation on the separate eigenstates of thehyperfine terms. For small separations, χ dominates, and the hyperfine terms are
perturbations to the singlet and triplet eigenstates.

Now suppose we have a collision between two atoms which are both in the hyperfine
state |F, mF i at large separation. The S1 · S2 term is not diagonal in the hyperfine
basis, and therefore, can couple different hyperfine states and cause transitions
between them. The symmetry of the Hamiltonian means that the total projectionof the angular momentum on a quantisation axis is conserved, but F can change
during a scattering event. At low temperatures (kB T  the hyperfine splitting),
spontaneous increases in F are energetically forbidden. This means that there may
be a "closed" channel of higher F which the atoms cannot make a real transition
to.
The closed channel may contain one or more bound states (Fig. 1.1(a)), and even if
a real transition to these states is energetically forbidden, a virtual transition can bemade during a scattering event. This is a second order process (transition in to andthen back out of the bound state), and we diagrammatically illustrate the resulting

10

Chapter 1

Introduction

energy shift in Fig. 1.1(b). We can include this energy shift as an effective change,
δa, in the scattering pseudo-potential parameterised by a. Analytically, we write themagnitude of the shift using the familiar second order perturbation theory "energy
denominator" to obtain [3, 18, 19]:

δa ∝

1
,
E − Ebs(1.4)

where E is the incident energy and Ebs is the energy of the virtual bound state.
Since the incident and virtual states have different magnetic moments (they havedifferent F ), the (linear) Zeeman effect allows us to write the energy denominator interms of an external magnetic field B:

δa(B) = −∆,B − B∞(1.5)

where B∞ is the resonant magnetic field and ∆ is called the width of the resonance2 .
We plot this form in Fig. 1.1(c) where it is clear that the scattering length can betuned to any value by varying an external B.

1.2 Producing 39K condensates
To harness the mechanism in the previous section, we need to work with a species
which has a suitably broad Feshbach resonance at an experimentally accessible
magnetic field strength. 39 K fits these criteria and is the species which we will use
throughout the first part of this thesis. Unfortunately, despite all of its attractive
properties, 39 K is comparatively difficult to cool to degeneracy. The methods toovercome this problem are now standard in our lab [24–26], and we describe them
only briefly here.
The difficulties with cooling 39 K are:
1. Standard sub-Doppler laser cooling techniques are ineffective for 39 K (due tothe poorly resolved hyperfine structure in the excited electronic state). This
makes the starting conditions for evaporative cooling unfavourable.
2. Evaporative cooling relies on the inter-atomic interactions to continuously reequilibrate the gas to lower temperatures as we remove the highest energy
2

In our experiment, we use 39 K atoms in the |F, mF i = |1, 1i state. In this case, ∆ = 52 G, andB∞ = 402.5 G [20–23]

1.2

Producing 39 K condensates

11atoms. Alongside the Feshbach shift, δa(B), each atomic species has abackground scattering length, abg such that a = abg + δa(B). Unfortunately,abg = −29 a0 for 39 K [23] (where a0 is the Bohr radius). This means thatpotassium is a very weakly interacting gas at B = 0, which makes conventional
evaporative cooling in low-field magnetic traps inefficient. Moreover, even if
a BEC could be achieved by standard low field evaporation, the negative
scattering length means it would be unstable against collapse [3, 27–29].
A combination of these two facts makes 39 K comparatively hard to cool to degeneracy.
While these problems can in principle be overcome, and both sub-Doppler cooling of
K and direct evaporation of this species to degeneracy have recently been reported[30–32] (see also chapter 9), we take a different approach, originally inspired by [20].
We use an easy-to-cool gas of 87 Rb to sympathetically cool 39 K down to a point(∼ 5 µK) where we can effectively trap it in a standard optical dipole trap. In the
dipole trap, we can exploit the Feshbach resonance to tune a to a large positive
value (135 a0 ). We finally achieve condensation by standard evaporative cooling at
this high scattering length. The stages of this cooling are schematically outlinedin Fig. 1.2. Further details of the apparatus can be found in the PhD theses ofRobert Campbell [25], Naaman Tammuz [26] and Stuart Moulder [33], who, alongwith post-doctoral researchers Scott Beattie and Robert Smith, are credited with theconstruction of most of the setup. Since all the novel experiments described in thefollowing two chapters start after the cooling process has been completed, we will
not include technical details of the inherited apparatus here and instead direct thereader towards the builders’ theses for description beyond the schematic presented
in 1.2.

1.3 Extreme interactions
The two experiments discussed in the first part of this thesis concentrate the extremesof a very weakly interacting and very strongly interacting gas:
In chapter 2 we will tune the interactions to sufficiently small values to observe a gas
far from equilibrium. By stalling the transfer of atoms between the condensate andits thermal surroundings, we will prevent the decay of the BEC even as the systemrises above the critical temperature. We refer to this BEC stranded above the criticaltemperature as a "superheated" BEC.
In chapter 3, we tune the interactions to the opposite regime, where the gas is
as stongly interacting as quantum mechanics allows. In this so-called "unitary"
limit, we can probe universal physics which is hard to describe theoretically. In thislimit, collisions involving more than two particles become more frequent and can

12Chapter 1

Introduction

Time/s

Magnetic trap

Laser coolingchamber

Optical trap

Magneto-optical trap
Optical radiation
RF/Microwave radiation
K atoms
Rb atoms

0

15

15

1. Magneto-optical trap loaded
from background vapour of K andRb.

2. Optical molasses followed bypumping to magneticallytrappable |F,mF=|2,2 state forboth species.

107

1mK
100μK

106

10μK

105
104

1μK
100nK

3. Physical transportof atoms to lower
pressure chamber ina magnetic
quadrupole trap.

17

6.8 GHz

82

84

108

N (log)T (log)1m
Evaporative cooling/
Science chamber

469 MHz

4. Microwave evaporation of Rb
in a Quadrupole-Ioffe
configuration (QUIC) trap using
microwaves to induce |2,2→|1,1 transitions. K is
sympathetically cooled by elastic
collisions with Rb.

5. Transfer to an optical dipole
trap. Remaining Rb atoms are
removed with a resonant lightpulse. Transfer K to |1,1 byLandau-Zener sweep ofmagnetic field while applying RF
radiation.
6. Tune K scattering length to
135a0 and evaporatively cool in
optical trap by reducing opticalpower.

105
200nK

90

Fig. 1.2.: Summary of our experimental sequence for cooling 39 K [24–26]. At the top we show aschematic of the experimental apparatus, containing two vacuum chambers (we omit allelectromagnetic coils and optics for clarity). Cartoons of the trapping potential for eachcooling stage are drawn below the part of the apparatus where the stage takes place. The
bars on the right show the approximate atom number and temperature (log scales) at theend of each step.
1.3

Extreme interactions

13

destabilise the cloud. We observe the dynamics of a cloud where this few-body
physics dominates, and by characterising this dynamics (particularly concentratingon three-body physics), we address the question of whether a degenerate unitary
Bose cloud can ever be stable.

14

Chapter 1

Introduction

A Superheated Bose-condensed
Gas

2

Without any inter-particle interactions, individual particles in a closed system travelindefinitely on a fixed trajectory. Only by introducing interactions can particles redistribute their energy and change their dynamics, ultimately tending towards anequilibrium steady state. Interactions therefore play a key role in attaining and maintaining thermodynamic equilibrium, and systems can be stranded out of equilibrium
if interactions are turned off or modified.
In this chapter we will study the non-equilibrium dynamics of a BEC of very weaklyinteracting 39 K atoms. The most striking phenomenon we will observe in this regimeis the persistence of a condensate stranded in a system far out of equilibrium at
a temperature approximately 1.5 times higher than the equilibrium BEC transition
temperature. We can draw parallels between this novel superheated state andtraditional superheated systems such as very pure water heated to over 100°C. In a
sample of superheated water, the particle motion is microscopically characteristicof a temperature over 100°C, whereas macroscopically, the system is not in thethermodynamically stable phase. This is directly analogous to what we observe inour superheated Bose-condensed gas.
An important difference is that whereas superheated water is an example of a stalledfirst-order transition, our Bose gas is superheated relative to the second-order BEC
transition. First order transitions can be easily stalled by rendering the system
incapable of overcoming the activation energy for the transition (in pure water weremove all nucleation sites for the formation of steam bubbles). Meanwhile, second
order transitions have no activation energy, and instead, we exploit a dynamical
method for making our novel superheated state. Specifically, our experimental
system is dissipative (technical heating and atom loss gradually deplete the cloud),and by tuning the inter-particle interaction strength, we render the system incapableof keeping up with the rate of dissipation, and we allow the system to diverge awayfrom equilibrium.
This chapter is organised as follows: In the first section, we will clarify the processes
which move atoms and energy around our system, and describe a simple qualitative
picture of our dissipative system. We then move on to present the experimental
procedure for realisation of the superheated state in section 2.2, and the quantitative15

techniques for analysing the experimental data in section 2.3. We show experimental
results comparing the dynamics of a superheated state to an equilibrium system in
2.4. Finally, we build a quantitative model for the non-equilibrium dynamics of our
superheated state in section 2.5, and test it by predicting the limits of superheating
seen in the experiment.

2.1 Qualitative picture
To construct a simple picture in which superheating may occur, we consider a “two-fluid" modelof our partially condensed Bose gas held in a
trap (Fig. 2.1). In this model, we have two equilibrium subsystems - the thermal bath and thecondensate. We consider the dissipation of eachof these subsystems and the mechanisms foratom and energy transfer between them in thefollowing sections.

Fig. 2.1.: The two fluid model for ourBose gas. The components of this model
will be explained individually in sections
2.1.1 - 2.1.4 and figures 2.2 - 2.5.

2.1.1 Equilibrium subsystems
In the two-fluid model, we conceptually divide ourdegenerate cloud into (1) the condensate and (2)its surrounding thermal cloud. In the Bogoliubov
picture, we crudely imagine this division to be at
the transition from the linear phonon spectrum tothe free particle spectrum. In real space the condensate appears as a dense object embedded in
the thermal cloud. We schematically represent
the two subsystems in Fig. 2.2.
Empirically (see section 2.4.3 for more justification), we find that these subsystems can be
individually described by equilibrium properties(excluding |a| < 1 a0 ), and it is the balance between these two systems which can be driven outof equilibrium (even at much larger interaction
strengths  1 a0 ). Specifically, there are two
key aspects to the equilibrium between thesesubsystems:

16

Chapter 2

A Superheated Bose-condensed Gas

Fig. 2.2.: Division of the system into degenerate and non-degenerate subsystems. We treat each subsystem as individually in equilibrium and study nonequilibrium effects arising from imbalance between the subsystems.

1. Thermal equilibrium: In thermal equilibrium, the temperature, T 0 , describing
the thermal cloud is the same as the temperature, T0 , describing the excitationsof the condensate. We explain in section 2.1.4 that in almost all cases that westudy (more precisely, for |a| > 1 a0 ), thermal equilibrium holds. this means wealways consider a state with a single well defined temperature, T 0 = T0 ≡ T .
2. Phase equilibrium: In phase equilibrium, the chemical potential, µ0 , describing
the thermal cloud is the same as the chemical potential, µ0 , describing thecondensate. In particular, phase equilibrium forbids there to be a condensate
present if the number of atoms in the system, N , is less than a critical number,
Nc , or equivalently if T is greater than a critical temperature, Tc (N ). This formof equilibrium will be violated in our superheated system.

2.1.2 Dissipation
At the start of the experiment, the two subsystems are in global equilibrium. However, dissipation plays a key role in their subsequent evolution.
Dissipation in our experiment is primarily causedby scattering of atoms with photons from thelaser beams used to trap the cloud. This drives
Fig. 2.3.: Dissipation in our systemboth particle loss and heating on a characteristic (namely technical heating and atom loss)
timescale of 35 s in our system (see Eq. 2.25).
causes the chemical potentials of the two
By continuously changing the atom number andtemperature of the two subsystems at different
rates, dissipation causes their chemical potentials to diverge away from the phase equilibrium
condition µ0 = µ0 . In the absence of any further
processes, µ0 always falls faster than µ0 (seeEq. 2.19). This imbalance breaks phase equilibrium unless an additional process acts to continuously re-equilibrate the chemical potentials(see below).

subsystems to become unbalanced at a
rate dictated by the per particle loss rates
Γ0 and Γ0 . An additional imbalancing ofµ0 < µ0 due to heating of the cloud by thelaser beams is also taken into account inour calculations.

2.1.3 Atom transfer - Elastic scattering
If the two subsystems interact, they will exchange
atoms with the aim of equilibrating their chemical
potentials. This exchange proceeds by elastic

Fig. 2.4.: Elastic scattering between thesubsystems proceeds at a rate κ to try tocontinuously re-equilibrate the chemical
potentials in response to dissipation.

2.1

Qualitative picture17

s-wave scattering between atoms, with the dissipation-induced imbalance µ0 < µ0
tending to drive a net flow of atoms out of the BEC into the thermal cloud.
The rate, κ, at which atoms are transferred depends on the strength of inter-atomic
interactions, parameterised by a. Indeed, we will see in Eq. 2.24 that κ ∝ a2 .
For sufficiently large a, re-equilibration by elastic scattering will be faster than thedissipation rate. In this large-a case, we will simply observe quasi-static equilibrium
decay of the cloud and the condensate will vanish when the atom number falls below
the equilibrium critical number Nc . However, in a very weakly interacting system,
elastic scattering cannot keep up with the dissipation and a non-equilibrium state
will form (for reference, the characteristic time scale for elastic scattering at a = 5 a0
is 16 s in our experiment). In this non equilibrium state, atoms are stranded in thecondensed state, and the phase transition to a purely thermal state is dynamicallyinhibited even as the atom number falls below the equilibrium critcal number Nc (orequivalently the temperature rises above Tc ). We then refer to the state in whichatoms are still stranded in the condensate as a “superheated Bose-condensed
gas".

2.1.4 Energy transfer - Landau damping
The final process we consider in the two-fluid
picture is concerned with thermal equilibrium: If
we take a thermos flask of cold water and put
it in a room full of steam at over 100°C, thenwe have successfully stranded the water out ofphase equilibrium with the steam by reducing theinteractions between the two phases. However,this is not an interesting superheated systembecause microscopic inspection of the particles
reveals that the water in the flask is still below its
boiling point. Crucially, this analogy tells us that
it is not only the absence of phase equilibrium,
but also the presence thermal equilibrium which
defines a superheated state.

Fig. 2.5.: Landau damping maintains
thermal equilibrium between the two subsystems for all interaction strengths (excluding |a|  1 a0 ).

In our Bose-condensed gas, we define the temperature of the thermal cloud fromthe occupation of free particle energy levels, and we define the temperature of the
BEC from the occupations of collective modes (i.e. phonons). Consider artificially
amplifying one phonon mode in a condensate originally at thermal equilibrium with
its surrounding thermal cloud. Through mean-field interactions, this condensate
phonon modifies the thermal cloud density distribution, and in turn, the mean-field

18

Chapter 2

A Superheated Bose-condensed Gas

Fig. 2.6.: Comparision of the characteristic rates of relevant processes in our system for typical initialconditions. In this plot we adjust the elastic scattering rate based on the assumption that∼ 3 collisions are required to achieve equilibrium [36–38]. This very simple picture taken
at the start of the cloud’s evolution predicts phase equilibrium for a > 6a0 . As the systemevolves, the atom number and temperature change, which changes the elastic scatting rateand extends the range of a in which we can observe non-equilibrium behaviour. In order tocapture these effects, we need the fully dynamical model in section 2.5.

effect of this modified thermal cloud alters the spectrum of the condensate. The
ultimate result of this “back-action" on the condensate is that the original phonon isdamped back to equilibrium. The rate at which this damping occurs depends on
the interaction strength, a. A more complete description of this process (known as
Landau damping) can be found in [3, 34, 35]. By analytically modelling this backaction in a homogeneous system1 , one arrives at the following phonon damping
rate2 :

1/2

8 n λ2 1τL (ωp ) = 3/2 0 T 1/2 ,ωp |a|
3π(2.1)

where τL (ωp ) is the Landau damping time for a phonon at frequency ωp , n0 is thecondensate density and λT is the thermal wavelength at the equilibrium temperature.
We insert our empirical peak density, n0 ≈ 5 × 1014 cm−3 , into the homogeneousgas result in Eq. 2.1 to estimate the thermal equilibration time scale in our trapped√system3 . The lowest frequency phonon mode in our trapped system is ωp = 2ωho[3] where ωho ≈ 2π × 71 Hz is the trap frequency of our isotropic harmonic trap. At
1

Note that here we are resorting to homogeneous system results to avoid the complications of theinhomogeneous trapping potential. Exactly these sorts of simplifications motivate the work to produce
an experimental uniform system in chapter 6.
2Here we assume that the kinetic energy of the thermal cloud is greater than the interaction energyof the condensate, which is appropriate for our experiments.
3
Handled in this way, the homogeneous expression should give an upper bound on τL for ourharmonically trapped gas [35].

2.1

Qualitative picture19

typical experimental temperatures of ∼ 180 nK, this gives a Landau damping timeof τL < 1 s even for a as low as 1 a0 .
Since Landau damping occurs on a timescale (< 1 s) much faster than the dissipation(≈ 35 s), we can be confident that our gas is almost always in thermal equilibrium.
Only for a → 0, where the Landau damping time diverges, is the assumption ofthermal equilibrium unjustified (exactly at a = 0 we have a situation similar to thethermos flask analogy). Therefore, throughout this investigation, we assume thermal
equilibrium T0 = T 0 ≡ T , and anticipate that this assumption will only fail to capture
the correct behaviour for |a| < 1 a0 .
√We stress that the key point in our experiments is that 1/τL ∝ a, while κ ∝ a2 . It is
this difference in scaling with a that allows us to find a large parameter space where
the two subsystems are in thermal equilibrium, but not in phase equilibrium (see Fig.
2.6).

2.1.5 Summary
We have described how the inter-particle interaction strength sets the rate at which
our system can respond to being driven out of phase equilibrium by dissipation.
For sufficiently strong interactions, particle transfer between the BEC and thermal
cloud proceeds quickly enough to keep the system in global equilibrium. Under
these conditions, the BEC disappears when the system crosses the equilibrium
critical point. However, in a weakly interacting gas, particles are stranded in the
BEC, meaning we can observe a BEC at a temperature higher than the equilibrium
critical temperature - i.e. we see a superheated state.

2.2 Experimental procedure
To put our discussion on a concrete footing, it is helpful to immediately introduce theexperimental procedure which is illustrated in Fig. 2.7(a).
The experiment starts after cooling the atoms to degeneracy at T ≈ 160 nK (seesection 1.2). We stop the evaporative cooling by raising the dipole trap depth to 2 µK,
and wait at a = 135 a0 for 2 s to ensure the cloud has reached global equilibrium.
Then we use a Feshbach resonance (see section 1.1.2) to tune the interactions by
ramping an external uniform magnetic field over 50 ms to create a weakly interacting
system. Finally we wait for a hold time, t, to allow the system to evolve under the
influence of dissipation.

20

Chapter 2

A Superheated Bose-condensed Gas

Fig. 2.7.: Experimental sequence. (a) We prepare a BEC and then reduce a over a period of 50 ms
to study the cloud as it evolves at low a. (b) Schematic illustration of µ0 (red) and µ0(blue) in an equilibrium system (large a). µ0 and µ0 are locked together until µ0 = µc ≈ 0
and the BEC disappears. (c) Schematic illustration of the chemical potentials when a isinsufficient to maintain phase equilibrium. We define the superheated regime as the region
with µ0 < µeq < µc but µ0 > µc .

To probe the properties of the gas created in this procedure, we reduce the interaction
strength all the way to 2.2 a0 (i.e. essentially non-interacting) and release the cloudfrom the trap, allowing it to expand ballistically for a time-of-flight (ToF) of tToF =18 ms. After this time, an atom with momentum p is at r ≈ ptToF /m (ignoring thesmall initial size of the cloud). Therefore, when we image the cloud using absorption
imaging [25, 26, 39, 40], we obtain it’s momentum distribution.
By repeating this whole sequence for different hold times, t, we can measure themomentum distribution as a function of time in the presence of dissipation. In thenext section, we will see how the momentum distributions can be converted into
the chemical potentials µ0 and µ0 which we use to track the superheating of oursystem.

2.3 Experimentally measuring chemical potentials
We already indicated that the balance between chemical potentials µ0 and µ0 isof central importance in defining whether a system is in phase equilibrium. In thelanguage of chemical potentials, the arguments of our qualitative picture run as
follows: For large interaction strengths4 , we expect µ0 = µ0 always, and a BEC
exists whenever µ0 > µc where µc is the critical chemical potential (see section
2.3.2). For "small" interaction strengths, we will show in Eq. 2.19 that dissipation
causes µ0 to diverge away from µ0 . Eventually µ0 < µeq < µc while µ0 > µc (whereµeq is the chemical potential that a cloud of the observed total atom number and4

We empirically find that (approximately) a > 50 a0 is sufficiently large in our system to observe
predominantly equilibrium effects (see Fig. 2.13).

2.3

Experimentally measuring chemical potentials

21

energy would have in phase equilibrium). These inequalities define the superheatedregime, and this behaviour is schematically illustrated in Fig. 2.7.
Since we cannot assume that our partially condensed gas is in global equilibrium,
extraction of µ and µ0 from the absorption images that we record is not immediately
simple. We proceed by considering that regardless of whether a system is inequilibrium or not, two extensive variables are always well defined: the total atom
number, N = N 0 + N0 , and the total cloud energy, E. We will first explain how we
extract these two extensive variables from the absorption images in the following
section, and then in section 2.3.2 we take the next step of calculating µ and µ0 from
N and E.

2.3.1 Extraction of N and E
We use absorption imaging to probe our atom clouds. The light absorbed from an
imaging beam of intensity I propagating in the z direction through an atom cloud
can be linked to the cloud of density n(r), using the Beer-Lambert law:dI
= −n(r)σ(I)I,
dz

(2.2)

where the cross section σ is given on resonance by the Optical-Bloch equations[14]:

σ = σ01.
1 + I/Isat(2.3)

Here Isat is the saturation intensity for the imaging transition and5 σ0 = 3λ2 /2π
where λ is the wavelength of the resonant imaging light. Integrating 2.2 gives:


− log

I
I0


+

I − I0
= σ0
IS

Z
n dz,

(2.4)

where I0 is the initial intensity of the beam before entering the cloud, which wemeasure by taking an additional image with no atoms present. For our low intensity
beam (I < IS ), to good approximation, we can keep only the logarithmic term on
the left-hand-side of Eq. 2.4 (in section 6.2 we will keep both terms).
5

σ0 can deviate from this theoretical value due to (among other factors) components of incorrect
polarisation in the imaging light. To eliminate these uncertainties we calibrate the value of σ0 by
measuring the critical atom number for a cloud at a = 62a0 and comparing to the theoretical value
assuming that the cloud is in equilibrium at this value of a [41, 42].

22

Chapter 2

A Superheated Bose-condensed Gas

8 3 a 0

5 a 0

3(1 0 )

2 0 0

N

1 0 0

0

E k/N k B (n K )

3 0 0
2 0 0
1 0 0
0
0

2 0

4 0

6 0

0

2 0

4 0

tim e ( s )6 0tim e ( s )Fig. 2.8.: Raw measurement of N and Ek after tuning a0 as in Fig 2.7 (single experimental run perpoint; scatter indicates random error scale). These series at 83 a0 and 5 a0 are put throughthe analysis in section 2.3.2 to extract T eq and µ0 in Fig. 2.9. To calculate µeq and thetheoretical curves in Fig. 2.13, we parameterise N and Ek /N using the polynomial fits
shown by solid lines.

With this link between n and I, the extensive variables N and E can be calculated
from the image as follows:Z

1nd r =
σ0
3

N=



Z
log

r2 n d3 r
3πm
E ≈ 2Ek = m=
2
tToF
σ0 t2ToF
R

I0
I

Z



(2.5)

dxdy

3

ρ log



I0
I


dρ,

(2.6)

where m is the mass of a 39 K atom and ρ is the radial coordinate in the 2D camera
image centred on the cloud. Note that we have assumed spherical symmetry andignored the interaction energy in the expression for E (we assess its contribution to
be < 1% for a < 100 a0 ). Furthermore, we have assumed that the total energy, E, issimply twice the kinetic energy, Ek (the mean kinetic and potential energy are equal
in a non-interacting harmonic oscillator). Any initial size effects from the finite in-trap
size can be accounted for by rescaling the energy by a factor ωho t2ToF /(1 + ωho t2ToF ),which effectively un-does the effect of convolving the momentum distribution with
the in-trap cloud size (assuming gaussian-like cloud profiles).
Example plots of N and Ek /N curves during experimental sequences for clouds
evolving with strong (83a0 ) and weak (5a0 ) interactions are shown in Fig. 2.8. These
curves can be converted into chemical potentials and temperatures (see Fig. 2.9)by the operations summarised in the next section.

2.3

Experimentally measuring chemical potentials

232.3.2 Calculation of chemical potentials
The process to obtain µ0 and µ0 , from the extensive variables, N and E, is not simple
and requires several numerical steps. Below we outline the basic procedure (which
could be omitted on first reading) and in section 2.3.3 we use some of the resultspresented here to provide a satisfactory answer to the question of why dissipation
should necessarily tend to drive particles in the direction from the BEC to the thermal
cloud. We leave a full dynamical calculation of the rate of this particle flow until
section 2.5.
We start the task of relating µ0 to N 0 and E by integrating the Bose distribution for a
non interacting gas to obatin theoretical form for the density, n0 , of the excited states
under the semi-classical approximation [3]:

0

d3 p
1
i
 2
h
p
1
(2π~)3 exp 1
2 r 2 − µ0 )
−1
+
mω
ho
kB T 2m
2



2 r2
mωho
1
µ0
= 3 g3/2 exp −
+
,
2kB T
kB T
λT
Z

0

n (r, µ , T ) =(2.7)

where gα (z) is the polylogarithm function. This allows us to calculate the theoreticalforms for N 0 and Ek as functions of µ0 and T :

0

0

Z

N (µ , T ) =


 0
Nc0
g3 eµ /kB T
ζ(3)

(2.8)

 0

µ /kB Tge413mωho r2 n(r, µ0 , T )/, d3 r = kB T22
g3 eµ0 /kB T(2.9)

n(r, µ0 , T ) d3 r =Z
Ek (µ0 , T )1=
N 0 (µ0 , T )
N0


3
kB T
where Nc0 = ζ(3) ~ω
is the critical atom number, and ζ(α) = gα (1) is thehoRiemann zeta function. These non-interacting expressions are modified in a finitesized interacting system by two corrections:• Finite-size corrections: The maximum value of N 0 = Nc0 in Eq. 2.8 occurs atµ = 0. However, the semi-classical description does not capture the quantum
mechanical zero-point energy 0 = 3~ωho /2 of a trapped system. When we
include this in the Bose factor, we find that the maximum (i.e. critical) valueof N 0 is actually reached when µ rises to µ = µ0c ≡ 0 . The correction to the24

Chapter 2

A Superheated Bose-condensed Gas

critical number, Ñc0 in a trapped system can then be calculated by expandingN 0 (µ0c , T ) = N 0 (0, T ) + µ0c ∂µ0 N 0 :Ñc0 = Nc0ζ(2) µ0c
1+ζ(3) kB T(2.10)• Interaction shift: Mean-field interactions shift the critical chemical potential ina harmonic trap to µc = µ0c + 4ζ(3/2)a/λT , and the resulting shift in the criticalnumber has been measured (including beyond mean-field terms) as [41, 42]:a+ 42
1 − 3.426
λT

Nc = Ñc0aλT2 !−3(2.11)

These corrections transform Eq. 2.8 and 2.9 into:

N 0 (µ0 , T ) =Nc
g3 expζ(3)µ0 − µ

h

c


,

kB T



i

0
c
g4 exp µkB−µT
Ek (µ0 , T )3 0i= kB T hc
N 0 (µ0 , T )2
g3 exp µkB−µT(2.12)

Finally, the key step in this procedure is to numerically invert this pair of equations: N 0 (µ0 , T )invert−−−→ E(µ0 , T ) µ0 (N 0 , E).

(2.13)

 T (N 0 , E)

Since these expressions for µ0 and T are in terms of N 0 rather than the measured
N , there are two routes to proceed:
1. In the column densities retrieved from the absorption images, the BEC stands
out as a sharp peak at low momentum on top of the broader thermal distribution(see Fig. 2.11(a)). We can remove the thermal cloud by fitting and subtracting
the theoretical thermal profile:

Z

r
0

0

n (r, µ , T ) dz =


 0

2πkB Tµmω 2 ρ2
g2 exp
−
mω 2
kB T
2kB T

2.3

(2.14)

Experimentally measuring chemical potentials

25leaving only the residual BEC atoms, N0 , to be counted. We can then calculate
N 0 = N − N0 to find µ0 (N, N0 , E) and T (N, N0 , E).
To establish whether the system is in phase equilibrium, we also need toknow µ0 . To find µ0 , we use a modified Thomas-Fermi law for the condensate
subsystem, which interpolates between µ0 = µ0c = 3~ωho /2 for N0 = 0 and theinfinite particle Thomas-Fermi limit [43]:


2/5
~ωho
N0 a
3~ωho
5/2
µ0 − µc =
15
+3
−
,
2
aho
2

(2.15)

where aho is the harmonic oscillator length.
So, by this route, we measure {N, N0 , E} and calculate {µ0 , µ0 , T }.
2. An alternative route would be to ask how the measured N atoms would be
distributed if the system was in global equilibrium (µ0 = µ0 ≡ µeq ) at energy E.
In this case, we know from [44, 45] that the harmonically trapped gas has anequilibrium equation of state6 :N0ζ(2) µ0≈1+Ncζ(3) kB T(2.16)

More precisely, in our calculations we include the next order term ∝ (µeq )2 ∼4/5
N0 , found experimentally in [45] as:

N 0 = Nc + S0 (N0 )2/5 + S2 (N0 )4/5 ,(2.17)

where S0 and S2 depend on a and T , and we used Eq. 2.15 to link µ0 and
N0 . Using N0 = N − N 0 , Eq. 2.17 becomes an implicit equation for N 0 (N, T ).
Inserting this into Eq. 2.13 allows us to calculate {µeq , T eq } from {N, E} via{N 0eq , N0eq }.
Empirically, we find T eq ≈ T (i.e. T (N 0eq , E) ≈ T (N 0 , E)) for all the experimental sequences we measure (because the condensate atom number isalways small). This allows us to drop the notation T eq and simply refer to thetemperature of the cloud as T .
626

The derivation of Eq. 2.16 is outlined in section 7.1.1

Chapter 2

A Superheated Bose-condensed Gas

By comparing the ‘true’ chemical potentials (calculated via route 1) with their equilibrium counterparts (calculated via route 2), we are able to see which range of a is welldescribed by the equilibrium theory and which range of a shows large deviations
from it.

2.3.3 Direction of the particle flow
Having introduced the basic expressions for calculating chemical potentials, it isworth returning to the question raised in section 2.1.2 and 2.1.3 of why the dominantmotion of atoms in response to dissipation is out of rather than into the BEC.
Imagine a system which is infinitesimally displaced from equilibrium (µ0 ≈ µ0 ). In
the absence of elastic processes, we can write: µ̇0 /µ0 = −α0 Γ0

where


 µ̇ /µ = −α Γ
0
0
0 0



 Γ0 = −Ṅ 0 /N.

(2.18) Γ = −Ṅ /N000

We can calculate α0 ≈ 2/5 (from Eq. 2.15), and α0 ≈ (1 − Nc /N 0 )−1 (from Eq. 2.16).
Therefore, without any elastic effects, the instantaneous ratio of the rates of decayof µ0 and µ0 when the system starts to depart from equilibrium is:2µ̇0 /µ0
=
0
0
µ̇ /µ
5
|



Nc Γ 0
1− 0
.
N
Γ0
{z
} |{z}<1(2.19)≈1

Assuming three-body effects do not dominate (see Eq. 2.26 and chapter 3), energy
inselective inelastic processes should give a ratio of Γ0 /Γ0 ∼ 1, and since we startwith an equilibrium condensate (N 0 > Nc ), the right-hand side of Eq. 2.19 is always
smaller than 1, meaing µ’ always falls faster than µ0 ’. Atoms will tend to flow down
this dynamically growing chemical potential gradient by elastic processes in the
direction from the BEC to the thermals.
Now that we have shown that elastic processes necessarily deplete our BEC in thepresence of energy-inselective dissipation, we are justified in our notion that limiting
the elastic scattering rate will preserve the condensate in a superheated state.

2.3

Experimentally measuring chemical potentials

27

Fig. 2.9.: Equilibrium vs. non-equilibrium BEC decay. (a) At a = 83 a0 the cloud is always in quasistatic equilibrium. The measured N0 is in excellent agreement with the predicted N0eq , andvanishes when T eq = Tc ; the three separately calculated chemical potentials, µ0 , µ0 andµeq , all agree with each other. 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 5 a0 , the BEC persists in the superheated regime (T eq > Tc ) fort̄ − tc ≈ 40 s. Note that we are careful to start both the 83 a0 and 5 a0 series with the same
initial conditions. The raw data for N0 are shown with one experimental run per point, andthe extracted values of T and µ0 are shown with three experimental runs per point; error
bars indicate the uncertainty in the mean.

2.4 Observation of a superheated state
The previous sections take us from absorption images to chemical potentials for thetwo subsystems. In this section, we present experimental trajectories of µ0 and µ0
for the examples of an equilibrium decay at 83 a0 and a non-equilibrium decay at
5 a0 . After analysing this data within the two-fluid model, we then verify the validityof this model a posteriori in section 2.4.3.

2.4.1 Equilibrium decay vs. superheating
Using the methods of sections 2.2 and 2.3.2, Fig. 2.9 shows the trajectories ofN0 , T , µ0 and µ0 as we hold the gas at fixed a (this figure can be compared withthe predictions of Fig. 2.7). By contrasting data taken at a "high" scattering length(a = 83 a0 ) and at a "low" scattering length (a = 5 a0 ) we can compare the cases of
equilibrium and non-equilibrium decay, which are discussed separately below.

28

Chapter 2

A Superheated Bose-condensed Gas

3

N0 (10 )

20

10

0
0

10

20
time (s)30

Fig. 2.10.: Quenching the superheated Bose-condensed gas. Solid symbols show the evolution of N0
at a = 3a0 (single experimental run per point), the green solid line shows N0eq , and orange
shading indicates the superheated regime. Open symbols show the rapid decay of the
BEC after it is strongly coupled to the thermal bath by an interaction quench to a = 62a0 attime tq . We show two experimental series in which tq = 20 s (black) and 30 s (red).

Equilibrium: a = 83 a0
At a = 83 a0 (Fig. 2.9(a)) we find excellent agreement between the measured N0
and the predicted N0eq . The BEC vanishes exactly at the equilibrium “critical time",tc (dotted green line), at which T = Tc . Furthermore, the separately calculated µ0 ,µ0 and µeq all coincide exactly as predicted for a system in global equilibrium.
Superheated: a = 5 a0
At a = 5 a0 (Fig. 2.9(b)) 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 other for t > tc , so the system is moving
away from the global phase equilibrium rather than towards it. The non-equilibrium
behaviour is thus not just a transient effect. Explicitly, since T > Tc for t > tc while
N0 > 0, we have direct evidence that our non-equilibrium state is a superheated
Bose-condensed gas.

2.4.2 Quenching the superheated state
We can further confirm our observation of a superheated state by suddenly reintroducing strong interactions into the system. In Fig. 2.10, we show the effect ofquenching the interaction strength from 3 a0 to 62 a0 . The quench is performed within∼ 10 ms, and for the experimental sequences shown, we either perform the quench
at hold times t = 20 s or 30 s.
When we suddenly increase in the elastic scattering rate, we are giving the atoms
which were stranded in the BEC a mechanism (elastic scattering) to rapidly flow

2.4

Observation of a superheated state29

Fig. 2.11.: Thermal distribution in a gas out of global phase equilibrium. (a) Absorption image of theof a BEC in a system at 5 a0 in the superheated regime after 18 ms ToF. Inset: the sameimage azimuthally averaged. The red peak shows the condensed part of the cloud, withthe thermal part in blue. The dotted blue line shows the expected profile for the criticalnumber of atoms at the temperature of our superheated cloud. (b) From the profile in(a), we can extract (1) the radial velocity distribution (red), (2) the theoretical distribution
corresponding to the measured N , E and calculated T eq (green), and (3) the temperaturefrom a fit, T f , (blue). Even though the gas is not in true equilibrium, the distribution still
looks thermal and T f and T eq agree to within a few percent. Inset: Comparison of T f(blue) and T eq (green) for the whole 5 a0 series (error bars analogous to Fig 2.9). The
solid black line shows the equilibrium Tc (N ). Note that T f (blue) and T eq agree within a
few percent and both differ singificantly from equilibrium.

down the chemical potential gradient into the thermal cloud, leading to rapid decayof the BEC in < 1 s. This is the second-order transition analogy to sprinkling gritinto a sample of superheated water: the water suddenly boils when the grit surface
provides a nucleation site with lower activation energy.

2.4.3 Verification of the two-fluid modelThroughout the data analysis above, we employed a key feature of our two-fluid
model: the assumption that the thermal and BEC subsystems are individually inequilibrium. This is essential because out of equilibrium temperature and chemical
potential are potentially ill-defined concepts. In this section, we give evidence which
supports this assumption.
In our experiment, we only measure extensive variables, and then calculate intensive
variables from these. If the thermal subsystem is individually in equilibrium, thenwe should recover the same intensive variables (e.g. temperature) by fitting thecloud with a suitable polylogarithm function (Eq. 2.14). In Fig. 2.11(b) we show theradial velocity distribution (n(r)r/tToF ) of a cloud deep in the superheated regime(a = 5 a0 , t = 45 s). Overlaid on this figure are two curves:

30

Chapter 2

A Superheated Bose-condensed Gas

• A fitted profile (Eq. 2.7) with temperature T f as one of the free parameters(blue).
• The distribution expected based on the temperature, T eq , derived from thetotal kinetic energy of the cloud (Eq. 2.13) (green).
We find that the data is fitted almost perfectly by the equilibrium distribution characterised by T eq , and the fit for T f gives only very slightly different shape. In Fig.
2.11(c), we compare the calculated T eq with the fitted T f for the whole 5 a0 series;
T f always agrees with T eq to within 10%. This indicates that the themal cloud isitself thermal equilibrium. The very fast rate for Landau damping calculated in 2.1.4
ensures the condensate excitations are also separately in thermal equilibrium (fora > 1 a0 ), as required for our two-fluid picture.

2.5 Dynamical model of superheating
So far we have only observed the superheating effect at a = 5a0 . To add quantitative
understanding, we now explore how the superheating phenomenon varies across a
range of interaction strengths, a. In this section, we first re-cast the two-fluid modelin a quantitative framework, and then demonstrate that our quantitative picture
captures the main features of the experimental dynamics for almost all values ofa.
The non-equilibrium evolution of N0 in the two-fluid model is governed by the following
equation which combines the dissipative and elastic terms from Fig. 2.3 and 2.4:Ṅ0 = −κ − Γ0 N0 .

(2.20)

By evaluating the rates of both the elastic (κ) and dissipative (Γ0 ) processes in thecloud, we can integrate this expression to extract a predicted trajectory for N0 (t).
The forms for κ and Γ0 are described in the following sections:2.5.1 2-body elastic term, κFirst consider the process in which two thermal atoms collide to form one condensate
and one thermal atom. The rate at which this process occurs is calculated in [46].
Here we will give a simple outline which explains the physical origin of each of theterms.

2.5 Dynamical model of superheating

31

The semi-classical s-wave collision rate, nσv sets the time scale for elastic scattering:nσv =8m(kB T )2 2a .
π~3

(2.21)

2 is the s-wave
Here n ∼ λ−3
T is the atom density of our degenerate gas, σ = 8πapscattering cross section for indistinguishable bosons7 and v ∼ 8kB T /πm is themean velocity of the atoms in the gas.

Alongside this characteristic rate, we need a factor to account for the availability ofcollision partners and final states. The rate at which atoms in thermal states withenergies 1 and 2 can collide is proportional to the number of atoms, f (1 ) andf (2 ), in those states. Indeed, the full collision integral describing the availability ofstates for this process is:Zκ∝d1 d2 d3 f (1 )f (2 )(1 + f (3 ))(1 + N0 )∆(1 , 2 , 3 ),(2.22)

where the explicit factors of 1 represent spontaneous collision and their accompanying occupation factors represent stimulated collisions. ∆(1 , 2 , 3 ) represents a setof delta functions which ensure energy and momentum conservation. For simplicity,rather than evaluate this collision integral, we will extract the main functional form of
the most dominant term: The condensate is macroscopically occupied, so we retain
only the stimulated part of (1 + N0 ) → N0 , and the thermal cloud levels have small
occupation factors so we retain only the spontaneous part8 of (1 + f (3 )) → 1.The(µ0 −µc )+(µ0 −µc )RkB Tcollision integral then becomes N0 d1 d2 f (1 )f (2 ) ∼ N0 e
, wherefor simplicity we use a Boltzmann form for f , and we deliberately leave the exponent
ungathered for later comparison.
Combining Eq. 2.21 and 2.22, we find the total scattering rate into the condensateis:κin ∝ γel N0 e

µ0 −µckB T(2.23),

7
For distinguishable bosons, the s-wave scattering cross section is 4πa2 . The extra factor of 2 in
the indistinguishable case arises from symmetrisation.
µ0

8

By ignoring f (3 ), we are taking the lowest order term in e kB T . Higher order terms have little
role once the system becomes superheated (µ0  0)32Chapter 2

A Superheated Bose-condensed Gas

(µ0 −µc )where we define γel = nσve kB T .
By a similar argument, we can write the collision integral of the reverse process(scattering out of the condensate) as ∼ N0 e
body processes then becomes:(µ0 −µc )+(µ0 −µc )kB T. The total rate for two µ −µµ0 −µc
c
0
kB T
kB T
,
κ = Aγel N0 e−e(2.24)

where A is a theoretically uncertain prefactor in the range 1 − 10 [43].
Eq. 2.24 sets the rate at which system can re-equilibrate after being driven out of
equilibrium. This rate is to be compared with the dissipation term in the next section
to establish whether superheating will take place.

2.5.2 Dissipation term
The dominant source of dissipation in our system is spontaneous scattering of atoms
with photons in the trapping beam (a “one-body" process). Additional dissipation
comes from the finite probability of three atoms simultaneously colliding, with theresult that all three atoms ultimately leave the trap. This three-body process becomes
more dominant as the interaction strength increases, and will be discussed morewhen we consider strongly interacting systems in chapter 3. Below we give sufficient(1)(3)
details to calculate the rates, Γ0 and Γ0 , of the one- and three-body processes(1)(3)
respectively to recover the full dissipation rate Γ0 = Γ0 + Γ0
One-body
Our optical trap is made from two crossed 1.2 W laser beams with waists of 140 µm.
The laser light is at λ = 1070 nm, which is far red-detuned from the potassium
λ0 = 767 nm D2 transition in an effort to minimise the photon-atom scattering rate
while maintaining a deep dipole trap. In this far-detuned limit, the scattering rate isgiven by [15, 47]:

3πc2
(1)
Γ0 =
2~ω03



ω
ω0

3 

Γ
Γ
+
ω0 − ω ω0 + ω

2
ICDT ≈

1 −1
s ,
35

(2.25)

where Γ = 2π × 6 MHz is the linewidth of the transition, and ICDT is the intensityof the crossed dipole trap experienced by the atoms. Since the thermal radius at
200 nK (≈ 14 µm) is much smaller than the beam waist (140µm) we use the peakintensity of the beams to obtain the 35 s time scale in Eq. 2.25.

2.5 Dynamical model of superheating

33

Three-body
The probability of finding three atoms in the same unit volume scales as the cube of
the atomic density. Accounting for all of the thermal-thermal-condensate, thermalcondensate-condensate and condensate-condensate-condensate collisions, theper-particle three-body loss rate from the condensate is [48]:

(3)

Γ0 =


L3 (a)
hn20 i + 6hn0 n0 i + 6hn02 i .
6(2.26)The behaviour of the rate coefficient, L3 , for a strongly interacting system is the
subject of considerable study in chapter 3. In this section we use the empiricalresults of [23] for a weakly interacting gas. For the average densities, hni, we use
the followingRR
hn0 i: In the term hn02 i = n3 (r) d3 r/ n(r) d3 r, we use the expression for n0 (r) in
Eq. 2.7. In the term hn00 n0 i we use the local value of the thermal density at thetrap centre, n0 (0), (where the three-body rate is highest):
hn0 i: We use a functional form which smoothly interpolates9 between the noninteracting Gaussian ground state, hn0 iGS (applicable for N0 a/aho  1) andthe interacting Thomas-Fermi result, hn0 iTF (applicable for N0 a/aho  1):hn0 i = 1+

hn0 iGShi5/3 3/5
hn0 iGShn0 iTF,

hn20 i = 1+hn20 iGSh 2 i5/6 6/5 ,(2.27)

hn0 iGShn20 iTF

where:

hn0 iGS = N0 2πa2ho−3/2hn20 iGS = N02 3π 2 a4ho

−3/2

√

hn0 iTF = 15 7 2π



15N0 aaho

−3/5

hn0 iGS
−6/5
√ 
0a
hn20 iTF = 67556 π 15N
hn20 iGSaho(2.28)

2.5.3 Integration of non-equilibrium dynamics
Now that we have all the required rates in Eqs. 2.24, 2.25 and 2.26, we can integrate
the trajectory of N0 in Eq. 2.20. The method for this integration is as follows:
934

The results do not strongly depend on the exact form of the interpolation between these limits.

Chapter 2

A Superheated Bose-condensed Gas

m e a s u re d N
2 00

c a lc u la te d N
0

e q
0

1 0

N

0

3

(1 0 )

N

0
0

2 0

4 0

6 0

tim e ( s )Fig. 2.12.: Non-equilibrium N0 dynamics. We plot the calculated N0 (t) (dashed red line) together withthe measured N0 (red points; single experimental run per point) for the same 5 a0 data
series as in Fig. 2.9. For comparison, we also show the calculated N0eq (t) (solid greenline).

1. Take the starting measurements of N (t = 0), E(t = 0), N0 (t = 0) and evaluateµ0 (t = 0) as described in section 2.3.2.
2. Using this initial value for µ0 , evaluate κ in Eq. 2.24 (since the theoretical valueof A is unknown, we repeat this whole integration scheme for several values
in the range 1-10, which appear in Fig. 2.13).
3. With this initial κ, take an Euler step to N0 (t = δt), where δt is the time interval
between the experimental measurements on the decay curves.
4. Using the measured N (t = δt), E(t = δt) and this calculated N0 (t = δt),calculate µ0 (t = δt) and iterate back to step 2 to produce entire N0 and µ0
decay curves.
The success of our model in reproducing the observed condensate decay at 5 a0
(using A=3) is shown in Fig. 2.12.
Since we only included 2-body effects which are stimulated by the macroscopic
condensate, the theoretical solutions of Eq. 2.20 can only asymptote to N0 = 0, andwe can never predict the N0 = 0 point. A more physical cut-off is to define that thecondensate has disappeared when N0 falls below the thermal occupation of theBTfirst excited state, i.e. N0cut ≈ 3k
~ωho . This cut-off allows us to calculate the time t̄ atwhich the condensate vanishes.

2.5 Dynamical model of superheating35

In the following section, we test the quantitative model derived here by investigating
the extremes of the superheating as a function of a.

2.6 Limits of superheating
It is natural to ask how far the system can be heated above the critical temperatureand how long this non-equilibrium state can live for. These limits of our superheatedstate are treated in the sections below where we will see that our dymaical model
captures the basic behaviour very well.

2.6.1 Maximum temperature
In Fig. 2.13(a) we plot the highest temperature at which we still observe a BEC,
T̄ ≡ T (t = t̄) scaled to the equilibrium critical temperature Tc corresponding tothe observed N (t = t̄). For a → 0, the BEC survives up to T ≈ 1.47 Tc . This isanalogous to observing superheated water at 275 °C.
We also show the calculated T̄ /Tc derived from our quantitative model. The red line
shows the results for A = 3 and the shaded area corresponds to A = 1 − 10. Ourmodel captures the data well, with A = 3 agreeing everywhere except exactly at
a = 0. We already anticipated that the model would not be accurate at a = 0 because
Landau damping vanishes in this limit, so the assumption of thermal equilibrium is
no longer valid. Furthermore, our simple model does not include interactions beyonds-wave scattering (e.g. weak dipolar interactions [49]) which become relevant for
a → 0. The effect of these residual interactions is to drive T̄ /Tc below the predictedvalue, as observed.

2.6.2 Maximum lifetime
In Fig. 2.13(b), we reconstruct the temporal phase diagram of our non-equilibrium
gas. Here, a horizontal cut though the graph corresponds to a time series such asthat shown in Fig. 2.9. For each a, we plot the measured t̄ (red points) and theequilibrium tc (green points) together with spline fits to the data. The width of theorange region corresponds to the time that the BEC survives in the superheatedregime. For a ≈ 0, this region spans a whole minute. In addition, in the inset, weplot the calculated phase diagram with A = 3 for comparison with the data. Again,
the main features are captured well for a 6= 0.

36

Chapter 2

A Superheated Bose-condensed Gas

Fig. 2.13.: Limits of superheating. (a) The highest temperature at which we observe a BEC, T̄ , scaled
to the equilibrium Tc . Close to a = 0 the BEC survives up to ≈ 1.47 Tc . The red line shows
the results of our numerical calculations, with the shaded area indicating the theoreticaluncertainty (A = 1 − 10). Experimental error bars are statistical. The point at 62a0 is fixedto unity by the absolute atom number calibration [41, 42]. (b) Temporal phase diagram.
For each value of a we plot the equilibrium tc (green points) and the time t̄ at which the
BEC actually vanishes (red points). The t̄ errors correspond to dashed lines in Fig. 2.9
and the uncertainty in tc is indicated by the scatter of points. Solid curves are spline fits tothe data. For a ≈ 0 the BEC survives in the superheated regime for a whole minute. Inset:
numerically calculated phase diagram, with t̄ data overlaid.

2.6.3 Independence on the initial conditions
We have now presented the main results of this work: observation of a long lived
non-equilibrium BEC above the equilibrium critical temperature. However, before
concluding this chapter, we highlight a fortuitous universality in our results.
Non-equilibrium phenomena are usually strongly dependent on the initial conditions.
For the data presented here, we were careful to always start with N0 (t = 0) ≈ 2 × 104
atoms by holding the cloud at a = 135 a0 to allow the atom number to decay (inquasi-static equilibrium) to reach this constant starting point. However, empirically,
we find that t̄ is essentially constant, within experimental error bars, for initial N0 in
the range (1 − 5) × 104 . Our quantitative model captures this unexpected universality(3)(3)
well, and for a > 0, we attribute this effect to the three body term Γ0 . Γ0 grows with
N0 , and therefore it has a “self-stabilizing" effect, rapidly depleting large condensates
to bring them in line with the behaviour of smaller condensates on timescales much
shorter than t̄.
For a < 0, we attribute the universality to the “Bosenova" phenomenon [3, 27–29],
where attractive interactions violently collapse the cloud to a dense point accompanied by rapid atom loss (see Fig. 2.14(a)). By studying the balance of kinetic, potential and interaction energy it can be shown that the collapse continues until the condensate atom number falls below a critical number Nnova ≈ 0.57aho /|a| [3, 27] (seeFig. 2.14(b)). This rapid Bosenova collapses all initial conditions with N0 > Nnova to
N0 = Nnova .

2.6 Limits of superheating37

Fig. 2.14.: Observation of Bosenovae for a < 0. (a) We record violent collapse and revival (red points)
of N0 during the early times in of a series at a = −5a0 (single experimental run per point).
N0 eventually stabilises at N0 ≈ Nnova (blue dotted line) and then decays slowly. (b) The
observed Nnova agrees well with the theoretical prediction Nnova ≈ 0.57aho /|a|. Horizontal
error bars are associated with the uncertainty in the Feshbach resonance position andvertical error bars indicate the error in Nnova from the extrapolation of data for t > 5 s backto t = 0. (see black line in (a)).

2.7 Conclusion
In this chapter, we explored and understood the dynamics of a weakly interacting
Bose gas. The highlight of our investigation was the observation of a novel state:a superheated Bose-condensed gas. We demonstrated that this phenomenon
arises because the phase transition from a BEC to a normal gas can be dynamicallyinhibited by the slow atom transfer out of the BEC at low interaction strengths. The
state is long lived, lasting up to one minute, and reaches temperatures up to 1.47
times the critical temperature. We mapped out a non-equilibrium phase diagram
for this system, and successfully reproduced our measurements using a numericalmodel based on a two-fluid picture of a partially condensed gas. The successof our calculations supports a conceptually simple way to think about dynamical
non-equilibrium effects near a second order transition.

38

Chapter 2

A Superheated Bose-condensed Gas

Stability of a Unitary Bose Gas

3

The previous chapter concentrated on the dynamics of an extremely weakly interacting Bose gas. Now, we use our ability to tune a to arbitrarily large values toinvestigate the opposite limit where interactions are as strong as quantum mechanics
allows: the so-called “unitary" regime. This regime is experimentally particularly
exciting because the strongly interacting gas is difficult to understand analytically.
Many condensed matter phenomena (most famously high temperature superconductivity) exist only in strongly correlated systems. While this physics has beenstudied extensively in unitary Fermi gases over the past decade (see e.g. [50] andreferences within), work is only now starting on the unitary regime in a Bosonic gas.
The reason for the lack of experimental work on unitary Bose gases is that dissipative three- or more-particle collisions (which are forbidden by the Pauli principle in
Fermionic gases) can rapidly destabilise a strongly interacting cloud of Bosons. It isthus an open question to what extent and in what regimes a unitary Bose gas has
well defined equilibrium properties. Here we experimentally study a thermal unitarygas of 39 K bosons, and address the viability of bringing this gas to degeneracy at
unitary.
In the unitary limit, a diverges to infinity. Recall that we defined a in section 1.1.1
from the expansion of the s-wave scattering amplitude, 1/f (k) = −1/a + bk + O(k 2 ).
A full treatment of the scattering problem with a regularized δ function pseudopotential [16, 51] shows that |b| = 1. This means that in the unitary regime (ka  1),the scattering amplitude takes the value |f (k)| ≈ 1/k; i.e. the scattering length iseffectively replaced by 1/k.
Extending this argument to a thermal gas of many atoms, we say that if a is greaterthan the thermal wavelength, λT , then we should replace a with λT . For a condensate,
the natural length scale cutoff which replaces a is the interparticle spacing. Thishighlights the universality of the physics in the unitary regime: Once a has dropped
out of the problem, the details of the very strong inter-atomic interactions cease toplay any direct role in the phenomena we observe, and the two-body interactions are
instead characterised by parameters of the gas which are not species-specific.
For Bosons, a particularly important phenomenon arising in this very strongly interacting regime is the dominance of multi-particle collisions. Outside unitarity, we
can model low energy collisions as colliding billiard balls of radius a. The probability

39

Fig. 3.1.: Final states available to three interacting particles. For a < 0, three particles can form an
Efimov trimer which rapidly decays to a deep dimer and a high energy third atom. For a > 0,
two particles can form a shallow dimer with the third carrying away a large share of theexcess binding energy. Note that the shallow dimer forms from s-wave interactions which
are characterised by the length scale a and the energy scale ~2 /ma2 , which is linear onthese axes.

of a third atom being present in the volume ∼ a3 surrounding the colliding pairgrows strongly with a (∝ na3 ) until a is replaced with λT at unitarity. The increased
likelihood of the presence of a third particle dramatically influences the physics
of the collision: At a simple level, conservation of energy and momentum allows
many different final states if the conserved quantities can be divided between threeparticles (e.g. the atoms can leave as three free atoms or as an atom and a dimer).
At a more complicated level, the particles can form a 3-particle bound state, known
as an Efimov trimer under suitable conditions (see Fig. 3.1) [23, 52–64].
We will see that the net effect of a three-body collision on the atom cloud involves
both particle-loss and heating. In this chapter we will study the dynamics of noncondensed Bose gases in and around the unitary regime where we expect thesethree- (or more generally, few-) body effects to dominate. We will focus on the generalscaling laws relating the three-body particle-loss and heating rates to temperature,scattering length and atom number. The dynamics that we uncover can be used to
speculate on the stability of a unitary gas at different phase space densities.
The chapter is organised as follows: in section 3.1, we summarise the theory which
addresses whether a degenerate unitary bose gas can be studied in equilibrium. In
this theory, a single species-specific parameter, ζ, determines whether equilibrium,
can be reached at a given phase space density. We measure this parameter for our
39 K system by experimentally recovering two general scaling laws in sections 3.2
and 3.3. The conclusions in section 3.5 show that our measured value of ζ makes

40

Chapter 3

Stability of a Unitary Bose Gas

39 K a promising species for further studies of many-body physics in a unitary Bosegas.

3.1 Can a unitary bose gas be stable at degeneracy?
In the previous chapter, we assessed the dynamics of a weakly interacting gas bycomparing the rate of elastic (two-body) processes with the (one-body) dissipation
rate. The same logic can be applied in the opposite limit of strong interactions, but inthis limit dissipation is dominated by three-body processes. Below we examine thebalance of the two- and three- body scattering rates to quantitatively address whether
rapid three-body loss can destabilise a cloud approaching the critical point.
Three-body loss rate
For a non-condensed cloud (N 0 = N ) we already defined the three-body loss
coefficent, L3 , in Eq. 2.26 such that:Γ(3) = −Ṅ
= L3 hn2 i.
N

(3.1)

In this chapter, we will be primarily interested in measring values of L3 and its scaling
with a and temperature, T . Theoretically, close to a Feshbach resonance, but stillaway from unitarity, L3 can be shown to scale as ∼ ~a4 /m [65, 66]. At unitarity wereplace a → λT to reach the asymptotic form L3 ∼ ~λ4T /m ∝ 1/T 2 . We will verify
this scaling law in section 3.3.
Additionally, species-specific Efimov physics gives a dimensionless prefactor as
follows [67]:√
9 3~ 4L3 = ζ
λ ,
m T

(3.2)

(where we have ignored additional weak log-periodic modulation arising from theEfimov spectrum [23, 54, 62]). The non-universal constant ζ ≤ 1 [67–70] will play animportant role in the stability calculations below. Physically, ζ describes the fractionof three particle interactions which result in atom loss. A fraction 1 − ζ of three bodycollisions form a transient trimer which decays back into free atoms which remain in
the trapped gas1 .
1
ζ can be written ζ = 1 − e−4η where η is known as the Efimov width parameter. η is definedsuch that the lifetime of a deeply bouind effimov trimer is ∼ ~/ηET , where ET is the trimer stateenergy [71, 72].

3.1

Can a unitary bose gas be stable at degeneracy?

41

Two body scattering rate
To asses the stability of a unitary gas, we compare Γ(3) to the the two-body elastic
scattering rate, γel = nσv. We are careful to define v as the relative velocity of twopthermal particles, 4 kB T /πm, and σ = 8πλ2T at unitarity. The ratio of the two- andthree-body rates is then locally:√
16 2 1
γel
= √
.
Γ(3)
9ζ 3 nλ3T(3.3)

Equilibrium criterion
For a system near the ideal-gas critical point2 nλ3T ≈ 2.612, and the ratio of twoto three- body rates is γel /Γ(3) ≈ 0.56/ζ. This means that for the case ζ = 1, a
critical gas cannot exist in equilibrium, and if we wish to study the universal strongly
correlated physics in a unitary degenerate gas, we need to search for a species
which has a non-universal parameter ζ  1. Until we have verified ζ  1, we workwith thermal clouds (nλ3T  1) for which the equlibrium criterion is satisfied.
Experimental scheme
The experimental difficulty in finding ζ lies in the fact that three-body effects cause
both atom number decay and heating. Since temperature varies with time, Eq. 3.1
cannot be analytically integrated3 (temperature enters via hn2 i, and L3 at unitarity).
One solution offered in [67] is to hold the atoms in a shallow trap such that any
heating leads to evaporation which keeps the temperature constant. By modelingthis evaporation and assuming L3 ∝ 1/T 2 , the value ζ ≈ 0.9 was measured for
7 Li. Here, we aim to extract ζ for 39 K, and we will use a method which allows us toexperimentally verify (rather than assume) L3 ∝ 1/T 2 along the way. The scheme
for our approach is summarised as follows:
1. Scaling Law 1: We first eliminate T from the problem by empirically observing
a simple scaling law, N T β ≈ const. in a three-body decay process, and we
experimentally determine the parameter β.
2. Scaling Law 2: Using scaling law 1, Eq. 3.2 reduces to an integrable formṄ = −AN ν , with a combination of the experimental exponents β and ν
revealing L3 ∝ 1/T 2 at unitarity.
2

Note that it is not known whether a phase space density of 2.612 is the condition for criticaliyat unitarity. We just use this weak-interaction result as a scale where interesting effects may start tooccur.
3
We need to integrate Eq. 3.1 because numerical differentiation of an experimental decay curveof atom number with respect to time leads to large uncertainties in ζ.

42

Chapter 3

Stability of a Unitary Bose Gas

3. ζ Extraction: Using the values of A and ν obtained by fitting data to theintegrated form of Ṅ = −AN ν , we can calculate ζ.
We now consider each of these steps in the following sections.

3.2 Scaling Law 1: N T β = const.
The first scaling law can be considered to be purely empirical for our purposes,
but we will attempt to provide some theoretical justification for the values of β we
observe in different regimes. We begin by outlining the experimental sequence,and then move on to display results supporting this scaling law. To emphasise theempirical nature of this law, we only suggest theoretical justification for the observed
values of β after presenting the experimental results.

3.2.1 Experimental sequence
The experimental procedure is similar to the sequence in chapter 2, but here we
ramp interactions up rather than down, and watch the system’s evolution at the highinteraction strength. More precisely, the sequence is:
1. Using the apparatus described in [24], we cool a cloud of 39 K in the |F, mF i =|1, 1i state with the interaction strength tuned to 135 a0 . We stop the evaporative
cooling when the cloud is still not condensed and then ramp up the trap depth
to U ≈ kB × 30 µK to prevent any further evaporation. We choose the point at
which we stop the evaporation such that we achieve a temperature T ≈ 1 µK
(λT ≈ 5000a0 ) after the ramp up. At the final trap depth, the trap has an
isotropic trapping frequency of ωho = 2π × 185 Hz.
2. We then tune a to the desired value using the Feshbach resonance centred
at 402.5 G. The external magnetic field which achieves this turns on in 10 ms.
At this point, we have N ≈ 105 atoms. Note that these parameters given ≈ 3 × 1012 cm−3 , so nλ3T < 0.1, and we can guarantee equilibrium even forζ = 1.
3. We let the cloud evolve at the chosen interaction strength for a variable holdtime, t, of up to 4 s.
4. We simultaneously switch off both the trap and the Feshbach field (within∼ 100 µs)4 and image the cloud after 5 ms ToF.
4

Note that we must switch off the Feshbach field quickly, especially when starting at a < 0, toprevent atoms adiabatically following the molecular branch as a changes (see Fig. 1.1(b)). Such

3.2 Scaling Law 1: N T β = const.

43

3.2.2 Experimental verification of the scaling law
We have taken 19 data series, each at a fixed a, which span the range −12 <λT /a < 12. Fig. 3.2(a) shows an example the particle loss and heating during thesesequences. Specifically, we show results for a resonantly interacting (λT /a = 0)series in this figure. Note that we restrict our measurements to T < 2 µK  U/kB
to ensure that evaporative losses and cooling are always negligible.
In the inset of Fig. 3.2(b), we give an example series at λT /a = −5.3 and 8.5 which
illustrate that the variation of log(N ) with log(T ) is well captured by a straight line.
This linear relationship is true for all our measurement series, which indicates thevalidity of scaling law 1. We extract the constant slope, β = −d[log(N )]/d[log(T )],for all our series and plot this as a function of λT /a in Fig. 3.2(b). Note that in theunitary region (|λT /a| < 1) all the measured values of β are the same to within theexperimental errors. This is characteristic of the universal (a-independent) behaviourof a unitary gas.
Having observed evidence for the empirically driven scaling law 1, we now introduce some theoretical concepts to help to explain the numerical values of β in Fig.
3.2(b).

3.2.3 Theoretical estimates of βThroughout our discussion, it is important to note that in our final analysis the exact
values of β are not essential for extracting ζ: only the existance of scaling law 1 isrequired. A very basic understanding of why three body loss should cause heating
comes from the fact that three body processes preferentially occur in the densest
central part of the trap. Since the potential energy is lowest here, this atom loss leads
to heating through what could be considered as the opposite process to evaporative
cooling.
Quantitatively, the rate at which the potential energy, U, of the cloud varies as atoms
are lost is:

Z 
U̇ =



Z 
1
1
2
2
mωh.o.
r2 ṅ(r) d3 r = −L3mωh.o.
r2 n3 d3 r.
2
2

(3.4)

2 r 2 /2k T ),Using a Gaussian model for the non-degenerate density, n ∝ exp(mωhoBgives U̇/Ṅ = 12 kB T . The mean kinetic energy per atom is 32 kB T , so the rate ofadiabatic following is undesirable becase any molecules which form would not be resonant with theimaging light, and therefore the atom number would appear reduced.

44

Chapter 3

Stability of a Unitary Bose Gas

Fig. 3.2.: Particle loss and heating in a strongly interacting Bose gas. (a) The time evolution of theatom number and temperature of a resonantly interacting gas (λT /a = 0) held in a deep
optical trap. Each point is an average of 5 measurements and error bars show statisticalerrors. Solid red lines are fits based on Eq. 3.10 and scaling law 1. (b) Heating exponent
β, as defined in scaling law 1. The red dashed line is a result of non-unitary three-bodytheory, while the red star indicates the predicted value of 1.8 at unitarity [67]. Open symbols
indicate the region where four-body decay is significant (see text and Fig. 3.4). Note that
λT ≈ 5×103 a0 and horizontal error bars reflect its variation during a measurement sequenceat a fixed a. Vertical error bars show fitting uncertainties. Inset: Log-log plots of N vs. T(scaled to their values at t = 0) for the data series at λT /a ≈ −5.3 (open) and 8.5 (solid).

3.2 Scaling Law 1: N T β = const.

45

change of the total energy, E = 3N kB T , of the cloud is Ė/Ṅ = 2kB T . This gives
the following scaling law:

3Ṅ kB T + 3N kB Ṫ
= 2kB TṄ=⇒

N T 3 = const.

(λT /a  0).

(3.5)

This very simple model predicts the form of the scaling law we observe, and predicts
β = 3. However, referring to Fig. 3.2(b), we see that it does not correctly capture
the experimental data which shows wide variations of β with λT /a. We consider themodifications to this simple theory for λT /a  0, λT /a  0 and λT /a → 0 cases
separately below.
β for λT /a  0
From Fig. 3.1, the closest three-body state to free atoms for a > 0 is the shallow
molecular dimer state, with the third atom gaining most of the released binding
energy. The fate of the molecule and third atom can be summarised as follows (seeFig. 3.3):• Third Atom: When the dimer forms, it releases binding energy  which canbe calculated from the length scale a of the shallow dimer:  = ~2 /ma2 . For
a/λT > 1, we ignore the initial kinetic energy of the atoms, meaning thatenergy and momentum conservation dictate the energy of the third atom after
the three-body collision to be 23 . For a > 300 a0 (i.e. λT /a < 17), this energy
is less than our trap depth (≈ kB × 30 µK) and the atom will remain in the trap.
Therefore the cloud is heated by an energy contribution 23  per collision event.

Fig. 3.3.: The fate of thee atoms colliding with a > 0. (a) Approach of three atoms within a distance a.
(b) Two atoms form a shallow dimer and the third atom (3) flies away with 2/3 of the dimer
binding energy (insufficient to knock the atom out of the trap). (c) A fourth atom (4) collides
with the dimer allowing it to relax to a deeply bound state. The fourth atom picks up ∼ 2/3of the deep dimer binding energy and is ejected from the trap, along with the deep dimer.

46

Chapter 3

Stability of a Unitary Bose Gas

• Shallow dimer: The shallow dimer will collide with a fourth atom to decay intoa deeply bound state and leave the trap. The fourth atom gains most of the
deep binding energy which is sufficient to eject it from the trap.
The net result of this process is to modify our calculation in Eq. 3.5. Whereas in
Eq. 3.5, we stated that that three atoms each originally carry average energy 2kB T ,and all this energy is lost when the atoms leave, we have now argued that atom 3 inFig. 3.3 remains in the trap with energy 23 . Therefore, the total energy of the cloudis reduced by [3(2kB T ) − 23 ] in each three body event. Using  = ~2 /ma2 , we canwrite:"  #Ė
1 λT 2= kB T 2 −
9π a
Ṅ

=⇒

"
  #
dT
1 λT 2
3N kB= −kB T 1 +
dN
9π a
(3.6)

Strictly speaking, we should integrate Eq. 3.6 as before to extract the law governing
the decay. However, the dependence of λT on T complicates this integration. We
note that our experimental decay curves cover at most a factor of 2 in temperature
range (1 − 2 µK). For a modest λT /a = 5, as T increases by a factor of 2, the factor[1 + 1/9π(λT /a)2 ] only falls to 80% of its original value. We therefore treat λT /a asa constant in this integration to obtain β = 3/[1 + 1/9π(λT /a)2 ].
We find that this approximate form for β is a good match for the data in the range
5 < λT /a < 10. For λT /a < 5, the data bends away from this crude prediction
because the unitarity effects discussed in the next section start to play a role.
β for λT /a → 0 (unitarity)
Considerable theoretical work was carried out in [67], leading to the prediction
β = 1.8 at unitarity. This value arises from two contributions• As before, preferential removal of atoms from the lowest potential energy
regions of the trap gives U̇/Ṅ = 12 kB T .
• At unitariy, three body effects also preferentially remove low kinetic energy
particles. The scattering length, a, is capped by the deBroglie wavelength, λdBof the atoms, meaning that at unitarity high kinetic energy atoms experience alower effective interaction strength than low kinetic energy atoms. Thus there
is a higher loss rate for low kinetic energy atoms, and it can be shown that the
total effective rate of loss of kineitc energy is Ėkin /Ṅ = 56 kB T [67].

3.2 Scaling Law 1: N T β = const.

47

Combining these contributions gives Ė/Ṅ = 56 kB T which can be integrated as in
Eq. 3.5 to give β = 95 . Experimentally, we find β = 1.94 ± 0.09 on resonance, which
is close to the unitary prediction.
β for λT /a  0
Referring to Fig. 3.1, the closest three-body state to free atoms for a < 0 is an
Efimov trimer state. This state is very short lived, rapidly decaying into an un-trapped
deeply bound dimer with the third atom gaining most of the released binding energy
and also escaping the trap [73]. On its own, this loss mechanism should lead toβ = 3 as predicted above, but in Fig. 3.2(b), we see that the experimental value ofβ does not rise to meet this expected value.
One possible reason for this difference is the presence of four-body processes at
a < 0. Studies using 133 Cs have previously observed four-body loss features in thea < 0 region [58], and indirect evidence of four-body decay in a 39 K gas at a < 0
was seen in [23] (but not in [64], where the initial cloud density was significantly
lower). We find evidence of four-body processes by analysing the dynamics of theparticle loss in the following way:
In the presence of dominant three- and four-body processes, the atom number
should evolve according to:Ṅ = −L3 hn2 iN − L4 hn3 iN,(3.7)

Where L3 and L4 are the three- and four-body rate coefficients respectively, which, farfrom unitarity, depend only on a. We numerically evolve Eq. 3.7 for each data series
at constant a, using the measured values of T (t) to evaluate hni i for a calculated
N (t) in a Gaussian model. We then fit the constants L3 and L4 to our experimental
decays. This allows us to evaluate the relevance of each term. The results are
summarised in table 3.1, and illustrated in Fig. 3.4. We compare the cases of a < 0and, a > 0 below:
For a = −850a0 , only the decay model which includes both three- and four-bodydecay captures the data well (χ2 ≈ 1), and a pure four- (three-)body decay in which
L3 (L4 ) is fixed to 0 does not. This means that both the three- and four- body terms
are relevant. We find a similar conclusion for all our data with −2000 < a/a0 < −400.
Furthermore by comparing the fitted values of L3 and L4 in the ratio L4 n/L3 , wefind that L4 is relevant for densities > 1012 cm−3 (note that this criterion on density
reconciles the observations of [23, 64] in which the experiments were carried out atdifferent densities).

48

Chapter 3

Stability of a Unitary Bose Gas

+700

-850

a/a0

L3
free
free
0
free
free
0

χ2
1
7
4
1
1
2

L4
free
0
free
free
0
free

Tab. 3.1.: Evidence of four-body decay for a < 0. We try to fit our data at a = −850a0 and +700a0
using the numerically evolved form of Eq. 3.7. We allow either or both of L3 and L4 to be
free parameters or fixed to zero, and measure the goodness-of-fit metric, χ2 .

1 6 0

N (1 0 3 )

N (1 0 3 )

1 0 0

8 0

1 0 -26 0

2 0

a = 7 0 0 a 0

3 b o d y
4 b o d y
3 + 4 b o d yt (s )

1

a = -8 5 0 a 0
1 0 -31
t (s )

Fig. 3.4.: Three- vs four-body decay for a < 0 (away from unitarity). The decay of N at a = −850a0 isfitted to a model including both three- and four-body losses (green solid line), as well as
to pure three- and pure four-body models (red dashed and black dot-dashed line, respectively). Inset: For comparison, at a = 700a0 , the solid green and the dashed red lines areindistinguishable, showing that four-body decay does not play a detectable role.

3.2 Scaling Law 1: N T β = const.

49

For a = +700a0 , we find that the model which includes three- and four-body decay
gives essentially the same fit as a pure three-body decay (within our 10% fitting
errors on L3 ), with χ2 ≈ 1 in both cases. A pure four-body decay does not capture
the data well (χ2 ≈ 2). This excludes L4 as a relevant fitting parameter for a > 0,
meaning the decay is three-body dominated. Using a similar procedure, we also
checked that we do not detect any five-body decay for either a < 0 or a > 0.
Since we are ultimately interested in the ζ parameter for three-body processes,
we exclude the four-body region (a/λT  0) from the remainder of the analysis.
Having now experimentally established scaling law 1 between N and T , and having
understood the limits of β, we may now proceed to the second scaling law in thenext section.

3.3 Scaling Law 2: L3 ∝ 1/T 2 at unitarity
Armed with the experimental result N T β = const, we eliminate the varying temperature from the differential equation in Eq. 3.1. Using a Gaussian model for thethermal cloud gives2

hn i = N

2



2
mωho
√
2π 3kB T

3
,

(3.8)

and we parameterise the scaling of L3 with temperature as L3 ∝ T −α . Combining
these parametrisations in Eq. 3.1, we can write Ṅ ∝ −N 3 T −3−α . Now invoking
scaling law 1, this becomes:Ṅ = −AN ν ,(3.9)

where we introduced ν = 3 + 3+α
β , and A is constant. Away from unitarity, we
expect L3 to be temperature independent (depending only on a), while at unitarity,
we expect L3 ∝ 1/T 2 . Explicitly, the onset of this unitary scaling can be detected bya change of ν from 3 + 3/β to 3 + 5/β.
Eq. 3.9 can be integrated to give:1/(1−ν)N (t) = A(ν − 1)t + N (0)1−ν
.

50

Chapter 3

Stability of a Unitary Bose Gas(3.10)

0 .2 51 / ν

0 .2 00 .1 50 .1 0

0

5

1 0

λ/ a
Fig. 3.5.: Particle-loss exponent ν, as defined in Eq. 3.9. The red dashed line shows the nonunitary
theory, ν = 3 + 3/β, assuming nonunitary theoretical β values. The red star shows theunitary prediction, ν = 3 + 5/β, corresponding to L3 ∝ 1/T 2 and the measured β. Error
bars are analogous to those in Fig. 3.2(b).

We fit the atom decay to this form with A and ν as free parameters. An example fit
is shown for the resonant (λT /a = 0) series in Fig. 3.2(a) together with a fit to T (t)found by combining the fitted N (t) and the fitted β in scaling law 1.
Fig. 3.5 shows the values of ν for all our three-body dominated decays. The
dashed line in this figure corresponds to the non-unitary theory, ν = 3 + 3/β using
the theoretical values of β from section 3.2.3. Again, we see surprisingly good
agreement in the range 5 < λT /a < 10. For λT /a < 5, we see the data start
to diverge away from the non-unitary theory, and instead approach the prediction
ν = 3 + 5/β (using the measured value of β on resonance). From the measured
value of ν on resonance, we obtain α = 1.7 ± 0.3 (i.e. L3 ∼ 1/T 1.7±0.3 ), which is inagreement with the expected 1/T 2 scaling.
The remaining fit parameter, A, will be discussed in the next section, where we use
its value to extract ζ.

3.4 Extraction of the rate parameter ζComparing the definition of A to the definition of L3 in Eq. 3.1, we obtain:√
L3 (t) = 3 3



2πkB T (t)mωh.o.



3.4

N (t)ν−3 A.

(3.11)

Extraction of the rate parameter ζ51

s -1 )

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

L 3 (c m

6

6

1 0 -2 40

41

1 0 0 0 a 0 / a

2
1
0
-2

0

2

4

6

8

λ/ a
−1/4
Fig. 3.6.: Three-body loss coefficient. Main panel: (L3 /LM(see text) at T = 1.1 µK. The
3 )horizontal green line marks the theoretical upper bound on L3 , while the red dashed line
is a guide to the eye showing the L3 ∝ a4 nonunitary scaling. At unitarity, L3 /LM
3 ∝ 0.27.
Inset: L3 at 1.1 µK (open symbols) and 1.7 µK (solid symbols). The expected ratio between
the two unitary plateaux is indicated by the green vertical bar.

Inserting all of our raw data, N (t) and T (t), with the fitted parameters, ν and A, into
this expression, allows us to reconstruct L3 (a, T ). In Fig. 3.6 we show L3 at a fixed
T = 1.1 µK, scaled to the theoretical upper bound, LM
3 (T ), obtained by setting ζ = 1
M−1/4
in Eq. 3.2. Plotting (L3 /L3 )vs. λT /a clearly reveals two key effects. Firstly,for λT /a > 3, we see the non-unitary scaling L3 ∝ a4 . Secondly, we observe thesaturation of L3 for |λT /a| < 1 to an a-independent value ≈ 0.27LM
3 (i.e. ζ ≈ 0.27).
This saturation is characteristic of the universal behaviour at unitarity, and we havechecked that the ratio of the saturated values at different temperatures is in line withexpected 1/T 2 scaling (see Fig. 3.6 inset).
We refine our estimate of ζ by fixing ν = 3 + 5/β and re-analysing the three
data series taken closest to the resonance, (|λT /a| < 0.6). By this method, we
obtain ζ = 0.29 ± 0.035 , with a systematic uncertainty (due to absolute atomnumber calibration) of about 30%. This value of ζ (approximately three times smaller
than the theoretical maximum) makes 39 K a promising candidate for many-body
physics in a unitary Bose gas because the ratio of two- to three-body rates becomes
γel /Γ(3) ≈ 5/nλ3T . There are still open questions regarding magnitude of phasespace densities at which exotic states first form at unitarity, and the exact numberof collisions required to recover equilibrium at unitarity. These questions make it
difficult to give precise statements about whether this balance of two- and threebody rates is sufficient to achieve a stable degenerate 39 K gas in unitarity. However,5

Recalling the definition of the Efimov width parameter, η, as ζ = 1 − e−4η , we deduce η =0.09 ± 0.04.

52

Chapter 3

Stability of a Unitary Bose Gas

we can say that a gas of 39 K atoms can be used to probe phase space densities
three times higher than 7 Li (ζ = 0.9) while remaining in equilibrium at unitarity.

3.5 Conclusion
In this chapter, we have characterised the stability of a 39 K gas at and near unitarity.
We experimentally verified two general scaling laws characterising particle loss
and heating in the unitary regime, namely N T β = const. and L3 ∝ 1/T 2 . Using
these laws we were able to measure the species-specific three-body loss parameter,
ζ. We find a value ζ = 0.29 ± 0.03 for 39 K, meaning the three body rate in thisspecies is roughly 3 times smaller than the theoretical maximum. This could allow adegenerate unitary 39 K to be observed in (quasi)equilibrium.
Note that since this work was carried out, a study of a degenerate unitary 85 Rb gas
was reported in [74]. 85 Rb has a 10 G wide Feshbach resonance at 155.05 G [75],and a value of ζ ≈ 0.21 [63] (similar to our observed ζ). This gives us further
encouragement that our 39 K system will be useful for future studies into the universal
physics at unitarity. However, the work in [74] highlights a further problem withstudying unitary systems: the absolute time-scales in a unitary gas are extremely
short. For example, three body loss limits the lifetime of the gas in [74] to ≈ 600 µs.
Such rapid decays pose many technical challenges, the most notable being thedifficulty of ramping the Feshbach magnetic field to the resonance in a time much
shorter than the cloud lifetime.

3.5

Conclusion

53

Part II
Trap Geometry

Introduction

4

All of the experiments in the first part of this thesis were carried out in parabolic
traps. In these experiments considerable effort is put into controling the frequency,
polarisation, intensity, and alignment of the many laser beams which manipulate andprobe an atom cloud. However, little attempt is made to sculpt a laser beam’s profile
beyond the simple Gaussian mode which comes straight out of a commercial source.
When used for optical trapping, these Gaussian modes produce a parabolic potential
to first order. While a parabolic trap may be the easiest optical geometry to produce,
it is not always the most revealing trap shape in which to study many-body quantum
mechanics. New trap geometries can produce different forms for the density ofstates, different dispersion relations [13], different transport properties [76, 77], and(of relevance to chapter 3) different three-body loss rates. This can lead to interesting
physics being either highlighted or obscured by the experimental trap geometry.
In this part of the thesis, we will use an optical technique described in chapter 5
for sculpting complex atom traps beyond simple parabolae. In chapters 6, 7 and
8 we will concentrate on one particular novel geometry - the uniform trap - andwe will present the first work which experimentally probes a truly homogeneous
Bose gas. Many quantum theories, including Einstein’s original BEC proposal, arephrased in terms of homogeneous systems, but until now it has been experimentallychallenging to produce the sharp walls needed to box off a uniform region of spacein which to trap an atom cloud. Our uniform box trap opens many future researchpaths, and here, we will present two contrasting routes:
Chapter 7 Equilibrium studies of condensation and the ground state in a uniform potential

Chapter 8 Dynamical studies of the BEC phase transition in a uniform potential

Additional ideas for future research directions in our uniform system are presented
in chapter 9.
Since optical methods are crucial for realising and probing our flexible trap geometries, we will use this chapter to briefly review the theory of atom-light interaction
before embarking on any experimental details.

574.1 Atom-light interactions
A full treatment of atom-light interactions can become very complicated, so we will
work under a careful set of assumptions which allow us to develop a useful intuition
without undue theoretical labour. These assumptions are enumerated explicitly
below in the context of a laser beam of frequency ω driving an atomic transition
between a ground state |gi and excited state |ei separated by energy ~ω0 (see Fig.
4.1(a)).
1. Semi-classical approximation: In our treatment we will quantise the atom’s
energy, but we will treat the laser radiation as a classical electric field E =E0 cos (ωt). Qualitatively, this is justified by the large occupation of the laser
mode.
2. Dipole approximation: Since the atom is smaller than the wavelength of incident
radiation (a0  λ), we ignore the spatial variation of the electric field and treat
the microscopic theory as a local picture with spatial variation inserted by handat the end.
3. Rotating wave approximation: We assume that the laser is tuned sufficiently
close to resonance such that the detuning δ = ω − ω0 is much smaller thanthe laser frequency. Specifically, the rotating wave approximation requires|δ|  ω + ω0
4. No scattering limit: From Eq. 2.25, we see that after applying the rotating wave
condition, the rate at which an atom incoherently scatters incident photons is
proportional to (Γ/δ)2 , where Γ is the linewidth of the atomic transition. Here,we will work in the limit Γ  δ such that we can ignore these spontaneous
scattering events.
Using these assumptions, we can build a framework for describing the behaviour of
atoms in off resonant laser light. In particular, in section 4.1.1, we will describe thecoherent single photon process which will be used for constructing the potentials andin section 4.1.2, we will discuss two photon processes used to probe the propertiesof the trapped atom clouds.

4.1.1 1-Photon: The AC Stark effect
The AC Stark effect is a phenomenon in which an optical field creates either atrapping or anti-trapping potential for the atoms. The techniques described in thefollowing chapters all rely on this effect to link the optical intensity to the depth of58

Chapter 4Introduction

the potential experienced by the atoms. Once this link is established, the task of
producing potentials of complex geometry is reduced to the task of sculpting opticalfields. The derivation below explicitly reveals the link between intensity and potential,
and is included in detail for comparison with the two-photon theory in section 4.1.2.
The arrangement we use to describe the AC stark effect is illustrated in Fig. 4.1(a),consisting of a single laser illuminating a two-level atom. Under the semi-classical approximation, the laser introduces a term −d̂ · E into the two-level atomic Hamiltonian,where d̂ = qr̂ is the electric dipole operator for the atom. In a simple hydrogen-likeatom q is the electron charge and r̂ is the position operator for the electron. For
basis states with definite parity the diagonal matrix elements of d̂ are zero, and we
can write:hiĤ = ~ω0 |ei he| − d̂ · E0 cos (ωt) |ei hg| + h.c.

(4.1)

The Schrödinger equation for the time evolution of the two-level atom wavefunction|ψi = cg |gi + ce |ei in Ĥ is then:

 

 
0
Ω cos (ωt) cg ċg 

i~   = ~ 
 ,
ċe
Ω cos (ωt)
ω0
ce

(4.2)

where Ω = − hg| d̂ · E0 |ei /~ is the Rabi frequency for the interaction. Now wemake the unitary transformation (c̃g , c̃e )T = (cg , eiωt ce )T and apply the rotating wave
approximation to remove rapidly oscillating terms1 :

 

 
c̃
0
Ω
d  g 1  c̃g i  = 
 
dt
2
c̃e
Ω −2δ
c̃e



=⇒


0
Ω
~

Ĥeff = 
,
2
Ω −2δ

(4.3)

1
Strictly speaking, we should first perform the transformation (c̃˜g , c̃˜e )T = (cg , eiω0 t ce )T which
produces terms oscillating at frequency δ and ω + ω0 . At this point we can use the rotating wave
approximation to drop the ω + ω0 terms before applying a second unitary transformation (c̃g , c̃e )T =(c̃˜g , eiδt c̃˜e )T . For clarity, we combine the unitary transformations into a single operation in the main text
and appeal to the intuitive interpretation of the rotating wave approximation as a removal of "rapidly
oscillating terms". This produces the same results.

4.1

Atom-light interactions

59

Diagonalising the effective Hamiltonian, Ĥeff in Eq. 4.3 reveals a new ground stateenergy,

U=

~Ω2
I
∝ ,
4δ
δ

(4.4)

Where we have taken the limit of large detuning: δ  Ω and I ∝ Ω2 is the intensityof the laser. This shift of the ground state provides an effective potential for theatoms due to the interaction with the light. Equation 4.4 provides all our qualitative
understanding of the optical trapping potential, U :• The depth of the potential is directly proportional to the local intensity of thelight field.
• The sign of the potential depends on the detuning, δ. Red-detuned light(i.e. ω < ω0 ) produces attactive potentials, and blue detuned light (ω > ω0 )produces repulsive potentials.
Although Eq. 4.4 is conceptually useful, we often do not work in the |δ|  ω0 regime
required for the rotating wave approximation. Instead, we use large detunings
in order to minimise technical heating due photon scattering (see Eq. 2.25). A
more careful derivation of the optical potential, which is valid for large detunings,
gives [15, 47]:3πc2
U =− 3
2ω0



Γ
Γ
+
ω0 − ω ω0 + ω


I,

(4.5)

where c is the speed of light and Γ is the linewidth of the transition. This expression
retains all the qualitative properties of Eq. 4.4 discussed above, the key result beingthat the potential experienced by the atoms is proportional to the local intensity of
the light. Since the intuition that U ∝ I is sufficient to understand that we can directly
sculpt a potential landscape for atoms by shaping the optical field, we can now moveon from our 1-photon discussion to start describing 2-photon processes.

4.1.2 2-Photon: Stimulated Raman scattering
In chapters 7 and 8, we will use a technique known as Bragg spectroscopy to probe
atom clouds produced in our novel trap geometries. At the heart of this technique isa 2-photon process in which the atom absorbs a photon from one laser beam andthen emits into a second beam (Raman scattering). Since the theory of this process
follows along similar lines to the 1-photon AC Stark effect already described, we

60

Chapter 4Introduction

Fig. 4.1.: Illustration of atom-light interactions discussed in this chapter. (a) Level diagrams illustrating
the AC stark effect. States are labelled with |atomic state, photon numberi, and ~ is omitted
for clarity. (a.i.) The bare states illuminated by laser light of frequency ω. (a.ii.) Performinga unitary transformation gives the dressed state picture described by Ĥeff in Eq. 4.3. The
dashed lines illustrate the effect of the off-diagonal coupling terms in Ĥeff on the eigenstatesof the diagonal part. (a.iii.) Transforming the coupled states back to the bare state picture
reveals a shift U to the ground state energy, which we treat as an external potential. (b)Illustration of stimulated Raman two-photon scattering. (b.i.) The level diagram for a threestate atom absorbing frequency ω1 and emitting at ω2 . (b.ii.) Solution for the occipationof the ground (blue) and excited state (red) as a function of time according to Eq. 4.12 fordifferent detunings, δ.

will derive the basic expressions for the behaviour of an atom in the presence of
two laser beams here and then later expand upon theory for application in Bragg
spectroscopy in section 7.2.1.
Consider the arrangement shown in Fig 4.1(bi.) where two lasers with frequencies
ω1 (laser 1) and ω2 (laser 2) illuminate a three-level system, {|gi , |ii , |ei} withenergy levels {0, ~ωi , ~ωe } respectively, (0 < ωe  ωi ). We assume that the energy
levels are sufficiently resolved to allow us to treat the transition |gi → |ii (driven bylaser 1) separately from |ii → |ei (driven by laser 2). Under this arrangement, theHamiltonian for the atom in the laser fields is:hiĤ = ~ωi |ii hi| + ~ωe |ei he| − d̂ · E10 cos (w1 t) |ii hg| + d̂ · E20 cos (w2 t) |ii he| + h.c. ,
(4.6)

4.1

Atom-light interactions

61

where En0 is the electric field envelope produced by laser n (n ∈ {1, 2}). As inthe discussion of the AC Stark effect, we immediately write the time evolution of
the atomic wavefunction |ψi = cg |gi + ci |ii + ce |ei accoring to the Schrödinger
equation:

 

 
0
Ω1 cos (w1 t)
0ċg 
 cg 
 

 
 

 
i~  ċi  = ~ Ω1 cos (w1 t)
  ci  ,
ω
Ω
cos
(w
t)
i
2
2
 

 
 

 
ċe
0
Ω2 cos (w2 t)
ωe
ce

(4.7)

where Ωn = − hg| d̂ · En0 |ei /~ is the Rabi frequency for the transition driven by laser
Tn. Now using the unitary transformation (c̃g , c̃i , c̃e )T = cg , eiω1 t ci , ei(ω1 −ω2 )t ce ,and applying the rotating wave approximation, Eq. 4.7 becomes:

 

 
Ω1
0  c̃g c̃g 
0
 
 1
d 
 

 
i  c̃i  = Ω1 −2∆ Ω2   c̃i  , dt   2 
 

 
0
Ω2 −2δ
c̃e
c̃e

(4.8)

where ∆ = ω1 − ωi and δ = (ω1 − ω2 ) − ωe . In the case ∆  Ωn , δ Eq. 4.8 predicts
that c̃i will oscillate much faster than the amplitudes of the other states, implying thatdc̃i /dt will average to zero over timescales in which the other amplitudes change.
This allows us to perform a so-called "adiabatic elimination" [78] of the intermediate
state, |ii:

c̃i =

Ω1
Ω2
c̃g +
c̃e
2∆
2∆

=⇒

  
Ω21
d c̃g   4∆i  =
dt
Ω1 Ω2
c̃e
4∆

 
 c̃g 
 .
Ω22
−
δ
c̃e
4∆
Ω1 Ω2
4∆

(4.9)

The terms Ω2n /4∆ in the diagonal elements of the matrix above arise from the AC
Stark shift of the atom’s energy levels. Therefore, we define a new detuning, δ 0 ,taken with respect to these shifted levels:

δ0 = δ −

62

Ω22
4∆

Chapter 4

+

Ω21
4∆

Introduction

=⇒

  
Ω21
c̃
g
   4∆i  = 
Ω1 Ω2
c̃e
4∆

 
 c̃g 
 
Ω21
0
c̃e
4∆ − δ
Ω1 Ω2
4∆(4.10)

Solving the coupled differential equations in Eq. 4.10 gives:



 
c̃
(t)
 g  cos=c̃e (t)1
2 Ωeff t0+ i Ωδeff sin

Rsin−i ΩΩeff

1
2 Ωeff t1
2 Ωeff t






R
sin 12 Ωeff tc̃(0)−i ΩΩeff g 

,


1
1
δ0
cos 2 Ωeff t − i Ωeff sin 2 Ωeff tc̃e (0)(4.11)q
where ΩR = Ω1 Ω2 /2∆, Ωeff = Ω2R + δ 02 and we have ignored any overall phasefactors. This solution is illustrated in Fig. 4.1(bii.) for different values of δ 0 /ΩR where
we see that the two laser beams drive Rabi oscillations between the states |ei and|gi at frequency Ωeff . A particularly important case is the evolution of the excitedstate population after starting with all the atoms in the ground state:Ω2|c̃e (t)| = 2R sin2
Ωeff
2




1
Ωeff t2(4.12)

This expression, along with Eq. 4.11 will be used as the starting point for building
our theory of Bragg spectroscopy in chapters 7 and 8.

4.2 Outline
The remaining chapters of this thesis are based on a holographic trap shaping
technique to produce light fields with arbitrary custom profiles (chapter 5). Using
the AC Stark effect described in this chapter, we can translate these light fields into
tuneable potential geometries of our atom clouds. With recent developments inprojector technology (e.g. digital mirror devices and liquid-crystal-on-silicon spatial
light modulators), there has been an increased interest in atom traps of complex
geometry [12, 76, 77, 79], and our methods provide a very versatile and robust
approach.
We will concentrate on a uniform trap geometry and produce the first homogeneous
trapped gases (chapter 6). Within this novel geometry, we will study the equilibrium
properties of thermal clouds, partially condensed clouds and pure BECs (chapter 7).
Using the 2-photon theory presented here, we will also study the dynamics of thehomogeneous 3D BEC phase transition (chapter 8). Since six different experiments
will be discussed in the remaining chapters, we provide a graphical roadmap in Fig.
4.2 of the topics covered to show the links between the experiments. Throughout
these studies, we will highlight the (often dramatic) difference between the behaviourof our homogeneous gas and more conventional harmonically trapped clouds.

4.2

Outline63Fig. 4.2.: A graphical roadmap for the chapters in this part of the thesis. Colours indicate the content
of each chapter and the bounding boxes indicate thematically linked subjects

64

Chapter 4Introduction

Robust Digital Holography for
Ultracold Atom Trapping

5

Whereas theoreticians are free to write down any form for the potential term, HPot , intheir Hamiltonian and explore the resulting physics, experimentalists usually requiresignificant investment in new optical and magnetic equipment to construct a new
potential geometry. The physics of a quantum gas depends on the connectivity,
dimensionality, symmetries and exact form of the geometry in which it is trapped,and considerable work has been put into trapping gases in novel chambers [79–81],
channels [9, 76, 77] and lattices [11–13]. One versatile experimental approachis to use configurable arrays of wires or magnetic videotape [82–85] to produce
novel magnetic trapping geometries on so-called “atom chips". However, it is often
undesirable to use a purely magnetic trap, since these deny the use some of thetechniques which make cold atoms an appealing system to study in the first place (astrong external magentic trapping potential makes tuning interactions via a Feshbachresonance or manipulating atoms’ hyperfine states more complicated). Instead, we
ideally need a purely optical method for sculpting the potential geometry.
In this chapter, we describe and experimentally demonstrate two algorithms for
producing optical traps which can have arbitrary 2D profiles. These two algorithms
serve two complementary roles in cold atom experiments, and we can understandtheir differences by comparison with vector- and raster-graphics in computer science(see Fig 5.1):• Vector-Holography: Here we first build a toolbox of analytic holograms which
sculpt laser beams into simple geometrical profiles in the far field e.g. a ringshaped beam, or a line-shaped beam. We then find a way to combine these
simple shapes to construct our final desired trap shape. This method is very
efficient, with almost all the input laser power appearing in the desired trap,
but it is only possible to produce relatively simple trap shapes. This is similar
to the memory-efficient manner in which simple graphics are stored in vector
format on a computer.
• Raster-Holography: Here we use a numerical method to directly find thehologram which produces our desired trap shape in the far field. This method
is (by construction) not as efficient as vector-holography, but we can produce
very intricate trap shapes with details down to the diffraction limit. This is65

Fig. 5.1.: Holographic analogues of vector- and raster-graphics. In a vector-graphics program, we are
given a set of basic shapes, and we can arrange them in a complex image (top left). The
bottom left row shows a simulation of a complex light fields prepared by a vector-hologram.
In contrast, raster-graphics are typically used for storing more complex images, such as
the one in the top right, at the expense of memory efficiency. The bottom right image
shows a real experimental light field prepared by our optical raster method (the image isapproximately 1mm across).

similar to the memory-inefficient manner in which complex graphics can be
stored in a rastered format with pixel-level detail on a computer.
Once a light field has been sculpted by either of these holographic techniques, we
can directly associate the optical geometry with a potential landscape using Eq.
4.4.
After defining what we mean by "holography" in section 5.1, we will devote one
section to each of the two holographic techniques outlined above. We will place
greater emphasis on vector-holography since this technique will be used in later
chapters. Technical details of the raster holography method will be relagated toAppendix B with simple proof-of-principle experiments with atoms in raster holograms
being outlined in section 5.3.

5.1 Digital holography
Holography was originally demonstrated as a means to record both the phase andamplitude of light scattered from an object in order to later reconstruct this light field
when the object is absent [86]. Digital holography has now evolved to encompass
more than this object reconstruction technique, and refers to any process whichuses a computer to reconstruct a desired light field by altering the light’s phase
and/or amplitude in a single control plane.

66

Chapter 5

Robust Digital Holography for Ultracold Atom TrappingFig. 5.2.: Apparatus for producing holograms using an SLM. The SLM imprints a phase on the incident
Gaussian laser beam, and the far field diffraction pattern of this imprinted beam is brought
to z = 2f by the lens.

In this chapter, we will use a phase-only spatial light modulator1 (SLM) to imprint aphase pattern, φ(r⊥ ), on an incident Gaussian laser beam with the aim of producing
a desired custom trapping pattern in the far-field diffraction plane (r⊥ spans theplane perpendicular to the propagation direction). We choose an SLM which only
affects the laser’s phase and does not modify the amplitude of the incident lightbecause we want to produce deep atom traps with lasers of reasonable power (i.e.
we do not want to block out any of the incident light by amplitude moduation). Since
our phase-only method is vulnerable to any phase aberrations in our system, we
keep the apparatus accompanying the SLM in this chapter very simple: A single
lens of focal length f acts to bring the far-field diffraction pattern from infinity to thetrapping plane at z = 2f (see Fig. 5.2).
The techniques presented in this chapter rely on a good understanding of the transfer
function which maps the imprinted light field in the SLM plane to the resulting
diffraction pattern in the trapping plane. Fraunhofer diffraction theory states that
this mapping is approximately a Fourier transform [87]. This approximation will
suffice for most of our work, however, in Appendix B.4 we will find that full numericalsimulation of the Helmholtz equation describing the light propagation can lead tobetter experimental results.
Note that we can only imprint the phase φ onto a single 2D plane in the light field.
Since we can only control one plane of light, we can only make traps which havean arbitrary 2D profile, and we rely on the beam propagation to extrude this patternin the third direction. Simulations and experiment suggest that the 2D profile wecreate is preserved up to at least one Rayleigh length on either side of the focus(see Fig. 5.3).
1

Part number: Hamamatsu X10468-04

5.1

Digital holography67

Fig. 5.3.: A simulation (left) and experimental images (right) of light fields within δz < 2zR of the focal
plane (z-axis not to scale, zR ≈ 1 mm is the Rayleigh range of our apparatus). Using thesetechniques we qualitatively determine that the trap shape persists for ∼ zR either side of
the focal plane before degrading significantly.

The first form of holography that we will consider here is vector-holography, which,despite its simplicity, has great practical use and will form the basis for the work inchapter 6.

5.2 Vector-holography
A typical vector-graphics program on a computer provides a toolbox of simple shapes
and a method to arrange these shapes on a canvas. In this section we build opticalanalogues to these features which will allow us to produce and combine several
simple atom traps.

5.2.1 Toolbox of shapes
We do not intend to give an exhaustive list of primitive shapes constructed in lightfields. Instead, we will only illustrate the phase patterns for the shapes which wewill use to produce the uniform potential in chapter 6. Alongside the trivial case of a“dot", the important shapes are the “line" and the “circle". In table 5.1 we give thedetails of the analytical holograms for these shapes together with the theoreticaland experimental diffraction patterns that they produce. These holograms form theprimitive building blocks for our more complex atom traps (note that the face in Fig.
5.1 is built from these primitive shapes).

5.2.2 Arranging shapes
Once we can make the basic shapes, we need a method to move the shapes aroundon our optical canvas and a method to place more than one shape on the canvas
using a single hologram. This is the “arrange" stage depicted in Fig. 5.1. We handle
the tasks of moving and compositing shapes separately in the sections below:

68

Chapter 5

Robust Digital Holography for Ultracold Atom TrappingTab. 5.1.: Example primitive beam shapes for use in vector-holography. Here we only mention thoseshapes which will be useful in chapter 6. In particular, we note that there are many ways
to produce a circular beam, and we will discuss the relative merits of two methods which
we refer to as the "Axicon beam" and the "Pseudo-Laguerre-Gauss (Pseudo-LG) beam" inchapter 6.

5.2

Vector-holography

69

Moving shapes
Appealing to a familiar property of Fourer transforms, adding a phase gradient in
the SLM plane will shift the position of the diffraction pattern in the trapping plane.
Explicity, we can follow through the Fraunhofer theory to obtain the following result(we illustrate the effect of the gradient on an example φ = `θ):φ → φ − 2πv · r⊥

Shifts the pattern corresponding
to φ by λf v in the trapping planeWe can also move the shapes in and out of the focal plane by adding a quadratic
radial phase pattern (analogous to a lens):

φ→φ+

Shifts the focal point of the pattern

2
kr⊥
2F

by δz = f 2 /F

Compositing shapes
Using the shifts above, we can now place a single shape anywhere on the trapping
plane, but often we will want to produce a trap which contains several shapes (e.g.
the face hologram in Fig. 5.1 consists of five individual component shapes). Suppose
we want to produce a trap containing two shapes corresponding to holograms φ1
and φ2 . For any individual hologram, φi , the light field at the SLM is given by Geiφiwhere G is the incident Gaussian beam amplitude. The addition rule for Fouriertransforms then says that a light field G eiφ1 + eiφ2 will produce the desired twoshape diffraction pattern. However, since in general eiφ1 + eiφ2 6= 1, our phase-only
SLM cannot prepare the required light field since it cannot modulate the amplitude.

Instead, to composit several shapes in the diffraction pattern from a single phase-only
hologram, we must appeal to another property of Fourier transforms: the concept of“phase-dominance" [88–90]. This concept is illustrated using an example in Fig 5.4,
where we see that the phase part of a Fourier transformed function seems to have aqualitatively greater impact on the modulus of the transform than the amplitude part.
There are counter examples to this phenomenon [91, 92], but phase-dominance has
proved to be a very useful experimental tool in all the discussions below.

70

Chapter 5

Robust Digital Holography for Ultracold Atom TrappingFig. 5.4.: The phenomenon of phase-dominance in Fourier transforms. The green panels represent
function amplitudes, and the gray panels give their corresponding phase. If we Fouriertransform the two pictures in (a), we obtain the spectra in (b). Now we take the phaseof one of these spectra and the amplitude of the other and combine these into a hybrid
function, (c). Inverse Fourier transforming this hybrid function gives (d), the amplitude ofwhich closely resembles the picture from which we took the transform’s phase information.
This phenomenon has counter examples, but it is very useful in almost all experimental
cases.

Fig. 5.5.: Ghost and alias traps. We make a hologram with three composited shapes: a single circle
displaced upwards and two short lines displaced to the left (this trapping pattern will become
very relevant in chapter 6). Panel (a) shows a computer simulation of the trapping intensity
produced by this hologram on an intensity scale 0 → 1. Inspecting the trapping plane ona scale 0 → 0.02 in (b) reveals faint ghost traps which were generated by discarding theamplitude of the hologram sum in Eq. 5.1. (c) Inspecting the simulated trapping plane on alarger scale reveals an array of aliases of the trapping pattern which are artefacts of thefinite-sized pixels on the SLM screen.

5.2

Vector-holography71
If we simply discard the troublesome amplitude part of G eiφ1 + eiφ2 , and just
apply the phase: φ1+2 = arg eiφ1 + eiφ2 , then phase-dominance says that the(allowed) light field Geiφ1+2 almost invariably propagates to produce a diffraction
pattern containing the both of the desired shapes (see e.g. Fig. 5.5(a)) [93–95].
In general, to composite n holograms, we use the expression:φΣ = argnX!
iφm +iχm

αm e

,

(5.1)

m=1

where {αm } is a set of parameters used to empirically tune the intensity of theindividual shapes, and {χm } is a set of random numbers between [0, 2π] which
reduce cross-talk between patterns [93] (see next section).

5.2.3 Ghost traps
A side effect of discarding the amplitude in our compositing step is the generationof faint “ghost traps". These are spurious shapes which resemble the desired trap
but are severely distorted and misplaced on the canvas (see Fig. 5.5(b)). The peakintensity of the brightest ghost traps is typically < 2% of the peak intensity of thedesired trap.
In addition to these ghost traps which arise from our compositing technique, theFourier components of the discrete square pixels on the SLM cause both the desired
and ghost traps to be aliased across the trapping plane (see Fig. 5.5(c)). This diverts
light away from the desired trap, but even with the combined losses of ghost traps
and aliasing, we still retain ≈ 80% of the incident power in the desired trap. Thishigh efficiency, combined with the method’s simplicity, make this the most suitable
technique to produce the homogeneous optical box discussed in detail in the next
chapter.

5.3 Raster-holography
The major limitation of the vector-holography method described above is that wecannot make a trap in which shapes overlap. Since all the shapes are derived fromthe same laser, they are all coherent and generally produce interference patterns if
overlapped (see Fig. 5.6). To make complex traps with overlapping geometries, weneed to devise an alternative algorithm.

72

Chapter 5

Robust Digital Holography for Ultracold Atom TrappingFig. 5.6.: Limitations of the vector-holography method. Suppose we want a trap which consists ofan overlapping circle and line (a). Since the two patterns are coherent, they interfere when
they overlap and we simulate the resulting pattern in (b). We need to use the raster method
in section 5.3 for any complex pattern with overlapping geometries.

Figure 5.7(a) summarises the problem that we are trying to solve: we want to findthe phase pattern which when applied to our Gaussian input beam produces thedesired trapping intensity in the trapping plane. Since the potential depends only
on the intensity of the light, we place no constraint on the phase in the trapping
plane. This gives us sufficient degrees of freedom to use a numerical method toself consistently solve for the phase in both the SLM and trapping planes subject tothe amplitude constraints in each plane. Many numerical methods have been used
to approach this problem, ranging from steepest-decent optimisation [96–98] togenetic algorithms [99]. We choose the iterative Gerechberg-Saxton algorithm [100](illustrated in Fig. 5.7(b)) because it is very computationally efficient, and is easily
executed on a fast graphics processing unit.
It is well documented that the convergence of the naïve Gerchberg-Saxton algorithm
is highly erratic because the amplitude constraints in each plane are non-convex

Fig. 5.7.: The Gerchberg-Saxton algorithm. (a) A cartoon of the constraints and the variables to besolved for. The modulus of the light field is fixed in the SLM plane, and we wish to solve forthe phase hologram which produces the correct modulus with no constraints on the phasein the trapping plane. (b) Diagrammatic representation of the iterative Gerchberg-Saxton
algorithm.

5.3

Raster-holography73

Fig. 5.8.: Motivation for an improved algorithm. We try to make a light field in the shape shown on
the far left (this shape is also used in [103]). Real experimental light fields produced using
different algorithms are recorded on the right. A naïve implementation of the GerchbergSaxton algorithm fails to converge to the desired pattern. MRAF improves on this by only
asking for convergence inside regions C and D. In the original MRAF paper, the canvas
region, C was shrunk to a tight fit around D, which is experimentally impractical. We
developed an improved algorithm, OMRAF, which allows C to be expanded to approach the
Nyquist limit.

[101, 102]. For this reason, Pasienski and DeMarco modified the algorithm to theso-called MRAF algorithm [103–106], and we have made further improvements inAppendix B to create the OMRAF algorithm. Fig. 5.8 shows the evolution of the trap
quality between these algorithm upgrades.
Using OMRAF we have generated complex optical patterns in laser light with only
∼ 7% root mean square deviation from the target potential. Furthermore, since ourphase-only method essentially "steers" light into the desired pattern, and does notblock out any of the incident light, we manage to direct approximately 20 − 30% of theincident laser power into the final trapping potential (the remaining light is scatteredoutside region C shown in Fig. 5.8). This is considerably higher than the efficiency
of more traditional amplitude-modulation techniques (e.g. digital projectors) which
would be less than 1% efficient for the target pattern shown in Fig. 5.8.
While our OMRAF algorithm, has the potential to produce more complex trap geometries than can be achieved with only vector-holography, it has not yet been put
to use in an extended project. However by adapting and repurposing the apparatus
described in Stuart Moulder’s PhD thesis [33], we have produced some preliminary
87 Rb atom clouds trapped in simple OMRAF traps (see Fig. 5.9). These very basic
trials serve to show that this moderately sophisticated holographic technique hasgenuine promise for trapping real atom clouds in flexible geometries. Note that thequality of the traps shown in Fig. 5.9 is at present primarily limited by our current
optical setup, which was originally designed to create small ring-traps for study ofsuperfluid flow [9, 33]. Re-purposing this apparatus without modification meant thata typical BEC only covers ∼ 10 × 10 pixels of the trapping plane, so we can currently
only trap atoms in simple shapes.

74

Chapter 5

Robust Digital Holography for Ultracold Atom TrappingFig. 5.9.: Preliminary trials trapping 87 Rb BECs using OMRAF traps in the apparatus describedin [33].The top row shows the target potentials, and the bottom row shows a BEC of atoms
trapped in 805 nm laser light shaped into these potentials (blue to red indicates opticaldensity of 0 to 3). The apparatus in [33] was not explicitly designed for testing OMRAF, andthe geometry of this apparatus is such that the atom cloud covers only ∼ 10 × 10 pixels in
the trapping plane.

5.4 Conclusion
The two holographic methods in this chapter have opened an entirely new research
path with an unprecedented control over the trap geometry. In the following chapters,
we will concentrate on one particular geometry - the uniform trap - where ourvector holographic method proves to be a very versatile technique for trapping andcondensing real clouds of atoms.

5.4

Conclusion

75

Bose-Einstein Condensation ofAtoms in a Uniform Potential

6

In an undergraduate quantum mechanics course, the first system that students tackle
is the uniform, square well. Later, as an introduction to the BEC transition, students
study quantum statistics in an infinite uniform gas of Bosons. Even for difficult
concepts in this thesis, such as Landau damping rates in chapter 2, we have quoted
the simplest uniform system results. It seems that whenever we want to understandcomplex phenomena, it is often best to approach them in a uniform environment
where there are no additional length scales to complicate the problem.
With its simple and fundamental description, the uniform system has a very special
place in theoretical physics. However, experimentally, it is very difficult to achieve athree-dimensional homogeneous potential for cold atoms. Most existing experiments
instead use simple optical or magnetic apparatus to produce parabolic traps. While
the apparatus for parabolic traps may be simple, the many-body physics we wish
to study can be complicated by the inhomogeneous trapping potential, and thefundamental phenomena at play can become obscured by trapping artifacts [41, 45,
107–109].
Various methods have been developed to overcome the problems with inhomogeneous traps and to extract uniform-system properties from a harmonically trappedsample [42, 110–117]. These methods generally rely on the local density approximation (LDA) [42, 110–115] or selective probing of a small central portion of thecloud [42, 116, 117]. In some cases, this can work extremely well, particularly if asingle experimental run in a harmonic system can be mapped onto many uniformsystem runs with different (local) particle densities. However, some experimental
signatures are still difficult to separate from relics of the trap geometry, and somesignals simply cannot be recovered using the LDA. In particular, the LDA breaks
down close to phase transitions where the correlation length diverges (see chapter8). In these situations it is of central importance to construct a genuinely uniform
atom trap.
Motivated by this, we will describe the preparation and observation of the first atomic
BEC in a truly uniform potential. We start in section 6.1 by designing a holographic
method based on the techniques in the previous chapter to create a uniformly dark
3D optical box with steep repulsive walls. Atoms are loaded into this trap using a77

single-chamber apparatus described in detail elsewhere [118], and schematicallyillustrated in Fig. 6.1 and Appendix A. This machine was originally constructedby Igor Gotlibovych and Tobias Schmidutz to create harmonically trapped BECs
of 87 Rb atoms in a hybrid magnetic and optical trap (introduced in [6]). A more
complete description of the apparatus can be found in the PhD theses of Igor andTobias [39, 40]. We adapt this apparatus to transfer atoms out of this harmonic
trap into to the novel uniform trap as described in section 6.2. Since this is thefirst realisation of a homogeneous atomic Bose gas, we will spend most of thischapter explaning and benchmarking the technical details of the uniform trap. In
particular, we will verify the uniformity of our trap (section 6.2.2) and verify that we
can condense in our trap (section 6.2.3). Chapters 7 and 8 will give more details of
the physics that we have explored using this apparatus. These experiments reveal
phenomena which could not have been observed in a more conventional harmonictrap.

6.1 Building an optical box
We will use the simpler vector-holography method from chapter 5 to produce theuniform trap which is used throughout the remaining sections of this thesis. Ourchosen trapping arrangement must fulfill the following criteria:• 3D: In chapter 5, we concentrated on 2D optical traps, and assumed thatthese 2D profiles are typically extruded in the third direction by ∼ 1 Rayleighlength before significant distortion. In section 6.1.2, we will describe themodifications to the holography apparatus which allow us to properly handle
the third direction and produce a 3D region of uniform potential.
• Dark traps. We can either trap the atoms in a uniformly dark region surroundedby steep, blue-detuned (repulsive1 ) walls, or in a uniformly bright, red-detuned(attractive) region. On the bench we can produce dark regions uniform to < 1%of the total trap depth (see Fig. 6.4(e)), or bright regions uniform to within 6%of the total trap depth (see Fig. B.3). We therefore strongly favour a method
which creates a uniformly dark region surrounded by sharp, high, repulsive(blue detuned) walls.

Below we briefly describe the historic progression of the technique which satisfies
both of these criteria.

78

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Time/s

Magnetic trap
Optical trap
Magneto-optical trap

Single chamber

0109

1. Magneto-optical trap loaded
from background vapour of Rb.

N (lin)T (log)Optical radiation
RF radiation
Rb atoms

0.25m

200μK
10μK
1μK

15

15

15

17

30

0

2. Optical molasses followed bypumping to magneticallytrappable |F,mF=|2,2 state.

100nK

3. Transfer atoms into
quadrupole magnetic trap

4. RF evaporation in themagnetic trap using a sweep ofRF frequency to induce
transitions to anti-trapped mF
states.
~10 MHz
5. Decompress magnetic trap tojust below gravity compensation
and allow atoms to fall into
optical trap. Then perform opticalevaporation to BEC.

g105

Fig. 6.1.: Summary of the experimental sequence for cooling 87 Rb in a simple single-chamber apparatus [118] inspired by the hybrid optical/magnetic trapping technique in [6]. We show
a schematic of the experimental apparatus at the top (we omit all electromagnetic coils
and optics for clarity), and underneath we show cartoons of the trapping potentials which
the atoms experience. The bars on the right show the approximate atom number andtemperature (log scale) at the end of each step.

6.1

Building an optical box

79Fig. 6.2.: Possible methods for producing dark optical traps in 532 nm (green) light. (a) and (b)show co-propagating beam methods: Using an appropriate hologram, we can (a) create
co-propagating coherent Gaussian beams of different waists, or (b) radially offset the focus
of a beam in the axially symmetric manner shown. These both produce the dark traps
indicated by white shading. However, the trap in (a) is parabolic in the propagation direction,and the walls of the trap in (b) are very diffuse which wastes most of the input power. Instead,we will use a crossed-beam method illustrated in (c). The original proposal was to use twocrossed tubes of differing diameter to create a dark arena at their intersection as shown
here. This was later modified to a geometry in which a tube of light is capped by two sheet
beams (see Fig. 6.3).

6.1.1 The evolution of our box
To minimise the number of optics which can introduce aberrations into our laser
beam, and for ease of alignment, we want to make the trap using only one SLM.
Furthermore, ideally, we would create our dark 3D optical box using only one opticalaxis. This can be achieved using methods which we refer to as "co-propagating beam
methods" (see Fig. 6.2(a) and (b)). Unfortunately, these methods either produce
parabolic dark traps [119] or are extremely inefficient in their use of light [120, 121](see Fig. 6.2), meaning we are forced to use a second optical axis to make a 3D
trap.
Originally we planned to generate two "tube" beams using a single SLM and thenintersect these tubes at right-angles to trap atoms in the central cavity (see Fig.
6.2(c)). If the tube walls are sufficiently sharp, then the enclosed region will be
uniformly dark. Much of the preliminary optical work was carried out based on this
two-tube geometry, and the first realisation of a uniform Bose-condensed gas wasachieved using this arrangement. However, as work progressed with atoms, theoriginal choice of optical geometry evolved to optimise the atomic cloud shape (seesection 6.1.3). The final geometry used is constructed with the followng beams:
180

δ > 0 in 4.4 gives a repulsive potential

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Tube beam Along one optical axis we shine a "pseudo-LG" beam (see table 5.1) with acircular profile to form a steep-walled tube.
Capping beam The tube is capped at both ends by sheets of light formed from the extrusionof the "line" beam in table 5.1 and propagating perpendicular to the tube.
This capped tube forms a uniformly dark "optical tin-can" in which to trap atoms (seeFig. 6.3(b)). Since all the experimental images from the preliminary optical trials are
based on the two-tube approach, we include simulated images of the final geometry
alongside the experimental images in Fig. 6.3.
In either the two-tube or final geometry, the beams along both optical axes areshaped using a single SLM as described below.

6.1.2 Beam-shaping apparatus
The central principle behind our apparatus is to use a single vector-hologram on
the SLM to generate a plane (referred to as "focal plane A") which contains two
spatially separated sets of optical geometry: one set for the tube beam and one forthe capping beam (see Fig. 6.3(a)). We then use a sharp-edged mirror covering
half of focal plane A to split the capping beam away from the tube beam. The simple
optical components shown in Fig. 6.3 steer and refocus the two beams until theyintersect at right angles to form the optical box in focal region B which coincides
with the atoms in the vacuum chamber.
Approximate positions of the relay lenses are given in Fig. 6.3. However as we haveseen in section 5.2.2, a particularly elegant feature of the SLM is that each beam
can be independently digitally focused by adding a lens-like parabolic phase term of
focal length Fm to each component hologram:







n
X

2 
kr⊥



φΣ = arg 
αm exp φm + 2πvm · r⊥ −
 ,
2F 
m=1

| {z } | {zm}position

(6.1)

focus

where we have used the notation of Eq. 5.1. This precise digital control means that
we can empirically correct for small errors in the relay lens positions (the SLM pixel
size sets a lower bound Fm > 400 mm, which gives us several mm tolerance on thelens positions).

6.1

Building an optical box

81Fig. 6.3.: The optical apparatus used to sculpt and cross the trapping beams. We show approximate
distances and focal lengths in mm. A single hologram on the SLM creates the spatiallyseparated optical geometry for both the tube and capping beams in the focal plane A. A
knife-edge mirror is used to steer the capping beam perpendicular to the tube beam. (a)We show both an experimental image of plane A for the original two-tube geometry, and asimulation of the final geometry, indicating the position of the splitting mirror in white. Simple
optics image the patterns in focal plane A into focal region B. (b) By moving a CCD throughregion B (without the vacuum chamber in place), we can map out the 3D equipotential
surface of the two-tube trap and identify the trapping volume. We also present a cartoonof the final trapping geometry with the area occupied by the atoms highlighted in red. The
diffraction limited spot size in region B is 3 µm radius, and the Rayleigh length is 54 µm.

82

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential6.1.3 The final geometry
Both the tube and the capping beams were revised after observing their performance
when trapping atoms. Below we highlight the major design choices and reasons forthe final geometry:
Tube beam: Axicon vs. pseudo-LG
In table 5.1 we saw two different holograms for producing circular (tubular) beams:
a spiral phase winding, φ = `θ produces a "pseudo-Laguerre-Gauss" (pseudoLG) beam2 parameterised by integer `, and a radial phase gradient, φ = kR
f r⊥ ,
3
produces an "axicon" beam . The analyical form for the diffraction pattern of these
holograms in the focal plane can be found by appropriate Fourier transforms. For
these calculations we imagine 2  illuminating the hologram with a Gaussian beam,r
λf
w0
E0 (r⊥ , z = 0) ∝ w exp − w⊥2 , of waist w (note that we have defined w0 = πw
asthe diffraction limited spot size in the Fourier plane). The results of the transforms
are as follows:

Pseudo-LG

ELG (r⊥ , z = 2f) ∝  
 2 
 2 
r2
r⊥
r⊥
r⊥
I `−1
−
I
exp (i`θ)exp − ⊥2
`+1
2
2
2
w0
2w0
2w0
2w02
Z ∞

AxiconEax (r⊥ , z = 2f ) ∝−e(ρ−iR)2
2
w0


J0

0

2ρr⊥
w02


ρdρ

where Jn (z) is a Bessel function, and In (z) is a modified Bessel functions of thefirst kind. We also consider a hybrid beam, produced by the phase pattern φ =kR
f r⊥ + `ax θ, which we refer to as the "spiral-axicon". In the limit R/w0  0.5 `ax , aspiral-axicon beam still resembles an axicon beam, but, unlike a pure axicon beam,a spiral-axicon is truly dark at r⊥ = 0, which satisfies our desire for a strictly dark
trap.
Spiral-axiconspEax(r⊥ , z = 2f ) ∝ exp (i`ax θ)

Z ∞
e

−

(ρ−iR)2
2
w0

0


J`ax

2ρr⊥
w02


ρdρ

2

The origin of this name is the Laguerre-Gauss solution to the cylindrically symmetric paraxial
Helmholtz equation describing laser beam propagation. Pure Laguerre-Gauss laser modes areindexed by integers ` and p where ` determines the azimuthal phase winding number of the beam, andp determines the number of radial nodes [122]. Since we imprint a phase winding onto a Gaussianbeam, we obtain a beam which is a superposition of several p modes all with the same `. The dominantmode is p = 0, but since our beam is not a pure Laguerre-Gauss mode, we refer to it as a pseudo-LG
beam.
3
The origin of this name is the refractive optical element, known as an axicon. An axicon is a
glass cone which can be modelled as a thin optical element which imprints a radial phase gradient
onto an incident beam.

6.1

Building an optical box

83Fig. 6.4.: A theoretical comparison between the ` = 12 pseudo-LG beam and the R = 6w0 , `ax = 3
spiral-axicon beam. (a) The profiles of the two beams in the direction of propagation andin the focal plane. Here zR is the Rayleigh length. (b) Analysis of the wall thickness andsteepness for a fixed peak radius. The pseudo-LG beam (red shading) has much broader
walls than the spiral-axicon (blue shading), and therefore requires more laser power toachieve the same trap depth. For reference, we show a diffraction limited spot (gray shading)and an r10 power law fit (blue line) to the pseudo-LG for η = 10. (c) Comparison of the fitted
steepness, rn for different η. (d) Experimental focal plane of an ` = 12 pseudo-LG beamwith a zoom showing the level of noise in the bottom of the trap. The circular fringes aroundthe main peak come from a circular aperture that we impose on the hologram to eliminate
the Fourier components of the square edge of the SLM screen (see Fig. 6.6). (e) A plot of
the potential along the dashed line in (d) showing < 1% noise on the trap floor.

Fig. 6.4 shows a comparison between pseudo-LG and spiral-axicon beams, with` and R set appropriately to achieve the trap dimensions we will use later. There
are three main beam properties to consider when comparing these two methods ofgenerating tubes. These are considered separately below:

1. Wall steepness: A perfectly uniform trap would have infinitely steep walls. Thiscan never be achieved for any real optical system with a finite diffraction limit.
To assess how close our beams come to the ideal case, we fit the analytic
n . As n → ∞, we approach
intensity patterns with a radial power law |E|2 ∝ r⊥
the ideal uniform case, and for n → 2 we have the traditional case of a parabolic
trap. Fig. 6.4(c) compares the fitted power n(η) for spiral-axicon and pseudoLG beams with radii 6w0 . Note that η is a parameter which describes how far
up the sides of the wall we explore with the fit. If the peak depth of the trap
(i.e. the height of the walls) is Upeak = ηkB T where T is the temperature of
the trapped cloud then we are only concerned with fitting the region exploredby the bulk of the atoms, i.e. up to ∼ Upeak /η. We find that for η  1, both84

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

beams are very much steeper than parabolic, but the spiral-axicon beam is
steeper for all η.
2. Efficiency: Both beams produce rings of light of finite thickness. If the ringis too thick then we will not have enough laser power to reach high η traps.
Numerically examining the beam profiles reveals that the radius of a pseudoLG ≈ 0.5 `w , and from a parabolic expansion around this radius
LG beam is r⊥0of peak intensity, we obtain the ring thickness as δr⊥ ≈ 0.35 `w0 . For a spiralax ≈ R and δr ≈ 1.66 w . Therefore, the ratio of theaxicon, these values are r⊥
0ax
LG ≈ 0.4R/w . Since wetrap depths created by the two beams is: Upeak /Upeak0
will be working at R ≈ 6 w0 , the spiral-axicon will produce deeper traps for agiven total laser power.
3. 3D profile: Using the Helmholtz propagator in Eq. B.2 we can simulate thepropagation of the shaped beams near z = 2f . We find that the pseudo-LG
beam produces the expected tube of roughly constant radius over a Rayleighlength, zR , on either side of the focus. However, the radius of the axicon tube
monotonically falls with z near the focus (see Fig. 6.4(a)). This varying radiusdoes not threaten the uniformity of our trap, but the loss of symmetry does
make analysing our images more difficult and is an unnecessarily complication
to the otherwise simple message of our atom cloud images (see Fig. 6.5(a)).
In summary, the spiral-axicon wins points 1 and 2: the beam is steeper and more
efficient than the pseudo-LG beam However, for clarity of presentation, we choose
to make our trap using a pseudo-LG tube beam which has a more favorable 3D
structure (point 3). In the final experiment, we use an ` = 12 beam, which gives atube of diameter `ω0 ≈ 35 µm (w0 ≈ 3 µm).
Capping beam: Tube vs sheets
To cap the tube, we can either use a second, larger diameter tube or two line beams.
Both of these arrangements are shown in Fig. 6.3.
Experimentally, we find that if we try to perform evaporative cooling in the twotube arrangement, then atoms which evaporate radially out of the inner tube getstuck in the larger outer tube (see Fig. 6.5(a),(b)). Since evaporating atoms needto completely escape the trap, this confinement in the capping tube may affectevaporation and distorts the images we take. Therefore we choose to use two line
beams to cap the tube. This geometry is achieved by compositing two F = 7.5 m
2cylindrical lens patterns, φ = − ky
2F , with different horizontally displacing phasegradients. This produces two sheet beams 54 µm tall and 3 µm wide separated
horizontally by an easily adjustable 70 µm (see Fig. 6.3(a)).

6.1

Building an optical box

85Fig. 6.5.: Evolution of the final trap geometry. The main trap is the dense, red region in the centre ofeach image, and we have ramped down the trap depth to spill atoms which map out thepath of the laser beams. The white lines show the inferred points of peak light intensity. (a)and (b) show that the cloud shape formed with a spiral-axicon tube beam is distorted by theconverging beam walls, and a pseudo-LG tube beam does not show any distortion. In (a)and (b), some radially evaporating atoms escape from the tube beam but are still trappedin the larger diameter capping beam. Since our picture integrates atom density along the
capping beam propagation direction, these externally trapped atoms will distort our images.
To avoid this, we choose to cap the beam with two sheets of light. The final geometry is
shown in (c).

The final hologram to produce the tube and two linear capping beams is shown in Fig.
6.6. Note that this hologram includes the empirically selected phase gradients andlens terms which correct for the imperfect placement of the relay optics. Simulationsof the pattern produced by this hologram in Focal plane A are shown in Fig. 5.5 and6.3(a).

6.2 First realisation with atoms
Having established the optical technique for producing a uniform trap, we now
describe the results of the experiment using this apparatus to create the first(quasi)homogeneous atomic BEC. We briefly describe the loading procedure, andthen verify that we have succeded in our aim of (a) producing a uniform cloud and(b) condensing that cloud.

86

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Fig. 6.6.: The phase hologram used to produce both the tube and capping beam geometry. To prevent
distortion by the Fourier components of the sharp corners of the SLM screen, we only apply
the phase ramps, 2πvm · r⊥ , inside the circular region which is the same radius as theincident beam waist and centered on the incident beam.

6.2.1 Loading procedure
Using the apparatus described in [118], we cool a gas of 87 Rb atoms in a harmonic
hybrid magneto-optical trap until the cloud is small enough to fit inside our cylindricaluniform box trap. At this point, the cloud of ∼ 7 × 105 atoms is partially condensed
at a temperature T = 120 nK. To transfer these atoms into our uniform trap, we
ramp on the 532 nm laser as illustrated in Fig. 6.7 to create the uniform opticalbox surrounding the harmonically trapped cloud. Out of the 1 W of power from thegreen laser source, 700 mW reach the atoms, achieving a maximum trap depth ofUpeak = kB × 2 µK. Once the green laser is on we ramp down the harmonic trappingbeam.
At this point we need to counter the effect of gravity, which if left uncompensated willturn our uniform trap into a linear trap with potential U = mgz. We use a magnetic
field gradient of 15.276 G/cm produced by the combination of a quadrupole field anda uniform bias field to cancel gravity at the 10−4 level. The transfer into the uniform
box is completed once the harmonic trap has fully turned off and the anti-gravity
field is fully on.
Approximately 80% of the atoms originally in the harmonic trap are successfullytransferred into the uniform trap, but the transfer is not adiabatic. After the transfer,the temperature of the gas is 130 nK. Under these conditions, the cloud is no longer
condensed. Nevertheless, it is sufficiently degenerate to show the effects of Bosestatistics. In the next section, we use this non-condensed gas to verify the uniformityof the trap. We will then progress to the task of condensing in the uniform trap insection 6.2.3.

6.2

First realisation with atoms

87Fig. 6.7.: Experimental apparatus for preparing a uniform BEC. (a) A schematic of the optical andmagnetic components of the trap showing the 532 nm green (uniform) and 1070 nm red(harmonic) lasers and the quadrupole (Q) and bias (B) coils. The location of the atoms isshown by the central red cylinder, and the laser detunings are shown (not to scale) on a
cartoon of the 87 Rb D2 lines. (b) The experimental sequence used to create and probe
the uniform BEC. The uniform box trap is loaded from a harmonic trap, and a magnetic
field gradient cancels gravity. Images from Figs. 6.8 and 6.10 are shown (not to scale) toindicate when in the sequence these pictures are taken.

88

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Fig. 6.8.: Preparing a quasiuniform Bose gas. (a) The atoms are loaded into the box trap after
precooling in a harmonic trap. (b) In situ images of the cloud just before (left) and after(right) loading into the box and corresponding line-density profiles along x (bottom plots)
and z (side plots) directions. The line densities along x (z) are obtained by integrating theimages along z (x). The blue dashed lines in the left panel are fits to the thermal componentof the harmonically trapped gas. The green dashed lines in the right panel are fits based on
the expected profiles for a uniform-density gas. The rounding off of these profiles is due tothe imaging resolution, which is taken into account in the fits.

6.2.2 Non-condensed atoms in a uniform potential
The non-condensed cloud produced immediately after transfer into the uniform
trap can be investigated either in situ or after release from the trap to gather both
qualitative and quantitative evidence for the uniformity of our optical box. These twocases are considered below:

In situ images – Qualitative uniformity
We take images of the optically dense clouds immediately before and after thetransfer using a 500 ns exposure of intense laser light with saturation parameter
I/IS = 150. These high intensity images are analysed using the full form of Eq.
R
2.4 [113, 114, 123] to give the optical density, OD = σ0 n dz profiles shown in Fig.
6.8. The contrast between the shape of the atom cloud in a harmonic trap and our
trap is our first qualitative evidence that the novel trap is indeed uniform.
For each image, we show the line-density profiles, ñ, obtained by integrating theimage along either x (across) or z (down). For a perfectly cylindrical volume oflength L and radius R uniformly filled with atoms, the profiles in these two directions
would be:Zñ(x) =OD dy ∝ TH

xL

~ P SF

6.2

First realisation with atoms

(6.2)

89

r

Z
ñ(z) =OD dx ∝

1−

 z 2R~ P SF,(6.3)

where TH(x) is a top hat function which is unity for −1/2 < x < 1/2 and zero
elsewhere, and ~P SF indicates convolution with the imaging point spread function(which we take to be a Gaussian of waist 5 µm). The green dashed lines in Fig.
6.8(b) are fits to the data using the functions above. These fits describe the datawell, and give L = 63 ± 2 µm and R = 15 ± 1 µm, which are consistent with thecalculated separation of the green walls based on the hologram in Fig 6.6, reducedby the diffraction-limited wall thickness.

Time-of-flight images – Quantitative uniformity
We now turn to a more quantitative study of the uniformity of our gas, which weaccess via the momentum distribution of trapped thermal clouds. To measure themomentum distribution, we release the non-condensed cloud from the uniform trap
and image it after 50 ms ballistic expansion in ToF (see in section 2.2). For a thermal
cloud, this gives an image of the momentum distribution of the atoms convolved
with the (relatively large) initial density distribution in the box4 .
For a quantum gas, the momentum distribution of a cloud contains the fingerprint of
the uniformity of the trap from which that cloud was released. To quantitatively extract
this fingerprint, we parameterise our trap by considering it to be an effective isotropic 5
potential of the form U0 rn . Under this assumption6 , the momentum distribution is:Z
f (p) =

d3 r

1


h 2

i

p
exp kB T 2m
+ U0 rn − µ − 1

!
 
p2
1
µ−
4π U0
3
k T
2m
=
Γ
g
e B
.
n kB T
n 3/n
1

(6.4)

4
Note that we ignore the interaction effects for these non-condensed clouds. The interactionenergy per particle is ∼ kB × 10 nK which is much less than the cloud temperature (refer forward to
Fig. 7.5)
5
The anisotropy
P of the potential does not affect the scalings discussed here: for
Pan anisotropic
trap of the form i Ui rni , the parameter α (defined after Eq. 6.5) becomes α = 1 + i 1/ni , and theremainder of the subsequent analysis is unchanged.
6Note that we have also used the semi-classical approximation (replacing a sum over eigenstateswith an integral over momenta). We used this approximation for harmonic traps in Part I of this thesis.
In the context of a uniform box-trap is exact in the case of an infinite box size. Performing a numericalcomparison between an exact eigenstate summation and the semi-classical momentum integral for a
finite box [39] shows that semi-classical approximation reproduces the momentum state occupations
to within 1% relative error for experimentally relevant parameters.

90

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Note that the trap parameter n has a strong influence on the form of the momentum
2
distribution (as n increases, g3/n (e−x ) becomes more sharply peaked). Using thefact that an atom with momentum p will travel a distance x0 = pt/m in time t, we
can calculate the line-density profile for the cloud (ignoring the initial size of the box)
as:

Z


xm 
f p=
dydx
t
 α+1/2 


r
2
1
T
m
µ− mz2
kB T
2t
,
=
gα e
2t2 kB T θαñToF (z) =(6.5)

where α = 1 + 3/n and the constant θα absorbs various factors such as the imaging
magnification and cross section. To account for the initial size of the uniform box,
we convolve ñToF with the in situ profile from Eq. 6.3.
The limiting cases of a parabolic trap (n = 2) and a truly uniform trap (n → ∞)correspond to α = 5/2 and α = 1 respectively. In Fig. 6.9(a), we attempt to fit our
ToF images with these two limiting cases, using θα , µ and T as free parameters. The
g1 function fits ñToF very well, with the reduced χ2 ≈ 1. The g5/2 fit is comparatively
poor: χ2 ≈ 7 and the systematic patterns in the fitting residuals ∆ñ = ñToF − ñfit
clearly show that this is fundamentally a wrong functional form. Qualitatively, themeasured momentum distribution is more "peaky" than that of a harmonically trapped
degenerate gas (this arises because the density of states in a uniform box allows agreater occupation of low energy states than in a harmonic trap).
As shown in the inset of Fig. 6.9(a), the measured ñ can be fitted well using α in therange 1 − 1.7, whereas higher values can be clearly excluded. Note, however, that
θα was one of our free parameters in these fits. Allowing this theoretical constant tovary gives the fitting function an unphysical freedom and overestimates the rangeof suitable α values. Crucially, only for the correct α is the best-fit value of θα atemperature-independent constant. Below we use this fact to accurately determine
the leading-order correction to the flatness of our trapping potential.
Cooling in the uniform trap – Precise evaluation of flatness
Above, we have looked at qualitative evidence from in-situ density profiles andsimple quantitative evidence from ToF images which suggest that our box is uniform
to within α < 1.7 (i.e. a power law certainly steeper than r4 ). However, we havenoted that we used too many degrees of freedom in these simple fits, and we canextract a better estimate of the true uniformity by finding the value of α for which thefit parameter θα (theoretically constant for fixed α) does not vary with temperature.

6.2

First realisation with atoms

91

Fig. 6.9.: Momentum distribution in a (quasi)uniform degenerate Bose gas. (a) ToF profile ñ(z) of anoncondensed cloud (red) and gα fits to the data, with α = 1 (green) and 5/2 (blue). We
also show the fitting residuals ∆ñ. Inset: fitting χ2 versus α. (b) Flatness of the box potential:
the condition S = d[log(θ)]/d[log(T )] = 0 (see text) gives α = 1.23 ± 0.03, corresponding
to an effective r13±2 potential.

This process is outlined below to reveal that the trap is in fact uniform up to a leading
order correction of the form ∼ r13 .
To vary the temperature of the cloud, we cool the atoms by forced evaporation (i.e.
we reduce the green laser intensity to lower the trap walls and let high energy atoms
escape). Since θα varies with the the total depth of the trap (i.e. with U0 ), we always
recompress the trap back to a depth of 2 µK to keep θα theoretically constant, before
releasing the atoms to take ToF images7 .
The drift of the fitted θα with the corresponding temperature, T , is shown for different values of α in Fig. 6.9(b) (each point represents a single experimental
run). We empirically observe that this drift is described well by a constant slope
Sα = d [log (θα )] /d [log (T )], and we see that Sα varies monotonically8 with α. From
the no-drift condition, Sα = 0, we get α = 1.23 ± 0.03, and therefore conclude thatthe leading-order correction to the flatness of our box is captured by an effective
r13±2 power law.
7

At sufficiently low temperatures, a condensate may form in the trap which will appear as a sharp
peak at low momentum in the centre of our ToF image. Since we do not want to address the additional
complication of the condensate until the next section, we exclude the central 180 µm region of thecloud (i.e. a region larger than all the BECs observed below) from the fits in this section.
8
Physically, if the α used for fitting is too high, as the gas cools the actual momentum distribution
narrows faster than gα . The fit then increasingly underestimates T and compensates by decreasing
θα . Conversely, if α is too low, the fitted θα increases as the gas is cooled. The fitted temperatures
also show the expected systematic drifts.

92

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform PotentialIn reality, our trap is anisotropic: from Fig. 6.4(c), we theoretically expect our potential
cap 0nx
to resemble U0LG r0 10
for η ≈ 10, where nx is the steepness parameter⊥ + U0 xof the capping beams which we numerically find to be nx  10. These values
predict α ≈ 1.2, which corresponds to an effective r15 power law, consistent with theexperimental observation.

Although our careful analysis has enabled us to distinguish an r13 power law fromthe ideal case, we will see in section 7.1 that this distinction is minor for mostthermodynamic and many-body studies.

6.2.3 A BEC of atoms in a uniform potential
Having verified the uniformity of our trap, we now turn our attention to observing
evidence that a phase transition to a condensed state occurs when we evaporatively
cool the cloud using the full experimental sequence in Fig. 6.7. Below, we discuss
two pieces of evidence which indicate that a phase transition does indeed occur
when we cool.

Evidence for BEC – Aspect ratio inversion in time-of-flight

The most immediate evidence for a BEC in our uniform box comes from the sharp
increase in occupation of low momentum states after evaporative cooling. We
observe these states as a dense central peak when we release the cloud for 50 ms
ToF. This macroscopic occupation is obvious in momentum space (ToF) pictures,but importantly, it cannot be seen in situ. Such behaviour is expected for a uniform
system, where the condensate is delocalised across the entire trap volume, andcannot be easily discerned from the homogeneous thermal atoms in situ. Thisphenomenon of condensation in momentum space but not in real space is shown in
Fig. 6.10(b), where it is contrasted with the harmonic case which shows obvious
signs of condensation in both spaces.

In ToF images from our homogeneous trap, the thermal cloud expands with anisotropic velocity distribution, while the BEC peak undegoes an aspect ratio inversion
from the wide, short trap to a narrow, tall ToF peak. This inversion is characteristicof a condensed state9 .

6.2

First realisation with atoms

93

Fig. 6.10.: Condensation in real space and/or momentum space. (a) Harmonic trap. We show
absorption images taken after 50 ms of ToF (main panels) and in situ (inset panels, with thesame colour scale as in Fig. 6.8). The top panels show cuts through the in situ distributions(green points) together with fits to the thermal wings (red line). The condensate is revealedin the colder cloud as a sharp peak rising above the thermal cloud in both real space andmomentum space, with the aspect ratio of the BEC inverting during ToF. (b) Uniform trap.
Cooling is achieved by lowering the trapping laser power P . Again, we show images after
50 ms of ToF and in situ (insets). The bottom panels show cuts through the momentum
distributions recorded in ToF. In contrast to the case of a harmonic trap, no dramatic effectsof cooling are observed in situ. However, BEC is clearly seen in the anisotropic expansion.

Fig. 6.11.: Matter-wave interference with a (quasi)uniform BEC. (a) In situ image of a two-trap configuration, together with a sketch of the green-light arrangement used to create it. (b)Interference pattern observed after 100 ms of ToF expansion. (c) Numerical simulation of
the expected interference pattern.

94

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Evidence for coherence – Interference
A BEC is a phase-coherent object, and we prove the presence of coherence byobserving interference fringes when we overlap our BEC with another coherent
matter-wave. This is very easily achieved in our apparatus as follows:
The versatility of our vector-hologram method allows us to digitally insert a third
capping wall further down the tube beam. This traps a small satellite cloud next
to the large uniform cloud (see Fig. 6.11(a)).We can fill both of these dark traps
with atoms and then simultaneously evaporate in the green light to produce two
spatially separated condensates. Since the satellite trap is very small, the roundingof the potential by the diffraction limit means that it is not uniform. Nevertheless, thesatellite BEC is a coherent object which we can use to interferometrically probe thecoherence of the larger, uniform BEC.
We produce a quasi-pure BEC in both traps (T < 30 nK), and then release theatoms, allowing the two clouds to overlap in ToF. After 100 ms ToF, we observe theinterference fringes seen in Fig. 6.11(b). The contrast of these fringes is up to≈ 50%, limited by our imaging resolution and the fact that the satellite BEC contains
considerably fewer atoms than the uniform BEC. This contrast, and the shape of thefringes are reproduced in a numerical simulation of the Gross-Pitaevskii equation(see Appendix C) shown in Fig. 6.11(c).
Note that the fringes become narrower from right to left in Fig. 6.11. This can bequalitatively understood by considering that the uniform BEC expands very slowly
in the x direction, meaning that we can just consider the expansion of the satellite
BEC, which expands rapidly in all directions. When we take an image after time
tToF of time-of-flight, the momentum component, ~kx , originating in the satellite
will have moved a distance δx = ~kx tToF /m. This gives a parabolic variation ofphase kδx ≈ (m/~tToF )δx2 for the components of the satellite BEC in the x direction.
When this interferes with the uniform BEC (which is essentially not expanding alongx), we obtain fringes with parabolic spacing.
Observation of interference is firm evidence that the cloud in our trap is a coherent
condensate. We will return to the opportunities that this versatile interferometrictechnique offers for future experiments in section 9.3.

9The inversion can be caused either by the anisotropic momentum distribution of an anisotropic
spatial wavefunction or by interactions which tend to expand the cloud fastest in the most constrained
trapping direction where the interaction force −V0 ∇n is largest. In our case, interaction effects
dominate the BEC expansion because the interaction energy of the our BEC with 105 atoms is∼ kB × 1 nK, which is much greater than the kinetic energy of the box ground state (approximately
kB × 10−2 nK).

6.2

First realisation with atoms

956.3 Conclusion
In this chapter, we have discussed the first realisation of an atomic BEC in a uniform
potential. So far, we have simply outlined the novel apparatus, illustrated the profilesof thermal clouds in our trap, and verified that we can produce BECs by evaporative
cooling. Furthermore, we have benchmarked the uniformity of our trap by measuring
the leading order correction from a flat potential to be a steep power law ∼ r13 .
This new system has great potential for new experiments where parabolic trapping
geometries complicate or entirely mask many fundamental many-body phemonena
which we wish to investigate. In the next chapter, we will summarise a series ofexperiments which probe the equilibrium properties of Bose gases in our uniform
trap both near and far below the critical BEC transition temperature. Throughout
these experiments, we will contrast our homogeneous gas results with the more
traditional harmonic trap experiments.

96

Chapter 6

Bose-Einstein Condensation of Atoms in a Uniform Potential

Observing Equilibrium Propertiesof a Homogeneous Bose Gas

7

While benchmarking our novel trap we immediately saw considerable contrast
between the behaviour of our homogeneous gas and that of a harmonically trappedsample. Most notably, in the previous chapter we observed that the momentum
distribution of a degenerate homogeneous gas is "more sharply peaked" at any
given temperature and chemical potential, and we saw that condensation occurs inmomentum space only. In this chapter we will continue to highlight the equilibrium
properties of a gas trapped in our novel system and contrast these with the behaviourof gases in more conventional harmonic traps.
Our discussion will be split into two parts (see Fig. 4.2): Section 7.1 will concentrateon the equilibrium thermodynamics of a thermal or partially condensed homogeneous Bose gas, while 7.2 will specifically study the equilibrium properties of theground state BEC. Since we designed our homogeneous gas as an idealised system,
the experiments discussed in this chapter can all be described accurately by textbook
theories, and they will not not highlight many unexpected surprises (except perhaps
section 7.1.3). This allows us to progress rapidly through many topics with the aimof with creating a solid foundation on which to build more complex experiments later(see chapter 8).

7.1 Thermodynamic experiments
The thermodynamic studies that we present here will concentrate on a central concept in BEC theory: Einstein’s textbook picture of condensation as a purely statisticalphase transition. Using our new trap, we are able to give the best experimental observation of the statistically-driven phenomenon of excited-state saturation (section
7.1.1), and we uncover experimental evidence for a novel cooling effect driven byquantum statistics - the quantum Joule-Thomson effect - which was theoretically
predicted more than 70 years ago [124] (section 7.1.3). Both of these effects werepreviously obscured by the inhomogeneity of parabolic traps.

977.1.1 Saturation of a Bose gas
We will begin our discussion with a theoretical summary of Einstein’s picture of a
non-interacting Bose gas in a potential U (r), and present our experimental support
for this picture in later subsections. Einstein said that the density of atoms in the
excited states is given by:

1
d3 p

h 2
i
p
2π~ exp 1
+
U
(r)
−
µ
−1
kB T 2m
n0 (r)λ3T = g3/2 e[µ−U (r)]/kB T ,n0 (r) =

=⇒

Z

(7.1)

where gα (z) is the polylogarithm function, and the remaining symbols are the standard notation of the semi-classical approximation.
Inspecting Eq. 7.1, we find that the excited state density (in a infinite1 , non interactinggas) cannot rise above nc λ3T = ζ(3/2) = 2.612. This sets a strict limit on the excitedstate populations at a given temperature, which occurs for the case µ → min[U (r)].
Once the excited states are saturated in this way, any additional atoms must pile
up in the ground state, leaving the excited state population constant at constanttemperature (see Fig. 7.1(a)). This is capping of the thermal population is Einstein’ssaturation picture.

The trapping geometry can modify this picture in the following way: The criticaldensity, nc is first achieved at the point(s) satisfying U (r) = min[U (r)], and adding
further atoms will cause a condensate to form at these points. The mean fieldinteraction between the thermal cloud and the newly-formed dense condensate
modifies the effective potential experienced by the thermal atoms. Hartree-Focktheory gives the modification as [3] (see also footnote 8).

Veff (r) = U (r) + 2V0 [n0 (r) + n0 (r)],

1

(7.2)

Here we ignore the zero-point energy of the trap which shifts the critical value of µ in traps with afinite atom number [3]98Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

where n0 is the condensate density and V0 (> 0) is the interaction pseudo-potential
strength defined in Eq. 1.2. Ignoring the dilute thermal-thermal interaction term, andinserting the Thomas-Fermi solution n0 (r) = [µ − U (r)]/V0 gives:(7.3)

Veff = |U (r) − µ| + µ.

Now we make the contrast between the harmonic and uniform cases:
Harmonic case2 r 2 , then Eq. 7.3 describes a “Mexican hat" potential (see Fig.
If U (r) = 12 mωho
7.1(b)). Crucially as the condenate appears (i.e. as µ changes sign) the locus ofpoints U (r) = min[Veff (r)] changes from a single point (µ < 0) to a 3D shell of points(µ > 0). Therefore, the growing condensate effectively pushes the critical region
into a larger volume, and hence more thermal atoms must be added to maintainthe critical density nc . Due to the geometry of the trap, as the condensate grows,
the critical region is pushed further out of the trap and by the volumetric argument
presented above, more and more thermal atoms are required to support the growing
condensate. The growth of the thermal occupation as the condensate grows means
we cannot say that the excited states are saturated on a global scale2 . The trap
geometry has played a key role in this non-saturation argument [45].

For completeness, we briefly quantify this this non-saturation to first order by modelling the thermal atoms as entirely excluded from the region r < RTF occupied by
the condensate [25, 26]. Under this assumption, integrating 7.1 gives:

N 0 λ3T = 4π

Z ∞



1
2
g3/2 e[µ− 2 mωho r ]/kB T r2 dr

(7.4)

RTF

2 = 2µ/mω 2 is the Thomas-Fermi radius of the condensate. Making thewhere RTF
ho
2 and expanding r to first order3 in R
substitution R2 = r2 − RTF
TF /R allows us toevaluate this integral as:

N0~ωho≈ 1 + 2.02
0
Nc
kB T



N0 a
aho

2/5
,

(7.5)

2

Note that the excited states are locally saturatedp
The main contribution to the integral comes from r ≈ 2kT /mωho , so this expansion is valid in
the regime kT  µ.
37.1 Thermodynamic experiments

99

Fig. 7.1.: Saturation of the thermal component. (a) Einstein’s textbook picture of saturation in a uniform
Bose-gas at fixed temperature. Once the total atom number, N passes a temperaturedependent critical number, Nc , the excited state population, N 0 saturates and all additional
atoms appear in the condensate. (b) Mean-field repulsion of thermal atoms from the dense
condensate. We present cartoons of the effective potential (solid line) experienced bythermal atoms in the presence of a condensate. The black dotted lines are the potential
shapes in the absence of a condensate. The orange dotted lines mark the radial extent
of the condensate in the harmonic case and the condensate spans the box in the uniform
case. (c) In the box trap, a partially condensed gas follows the ideal prediction, N0 = Nc ,whereas in the harmonic trap the thermal component is strongly non-saturated. Each point
represents the average of approximately six experimental runs with error bars indicating
the uncertainty on the mean. The dashed red line shows the theoretical prediction for the
harmonic trap. (For the harmonic-trap data T ≈ 110 nK and Nc ≈ 65 × 103 .) 2/5
152/5 N awhere we have used the Thomas-Fermi result µ = 2~ωho [3], aho isahocthe harmonic oscillator length, and N0 is the non-interacting critical atom number(compare this result with Eq. 2.16). This completes our demonstration that the
combination of geometrical and interaction effects cause N 0 to increases with N0
for a < 0 in a harmonic trap.
Uniform case
In contrast to the harmonic case, there is no geometrical change to a uniform
potential as µ varies in Eq. 7.3. The presence of a condensate only causes a global
shift in Veff (see Fig. 7.1(b)). This means that there is no geometrical argumentfor non-saturation in a homogeneous bose gas. A full treatment of an interacting
homogeneous gas, should strictly replace p2 /2m in the Bose factor in Eq. 7.1 withthe full Bogoluibov dispersion relation (p) = [(p2 /2m + V0 n0 )2 − (V0 n0 )2 ]1/2 . This
leads to the thermal phase-space density actually dropping below nc λ3 once the
BEC forms in a uniform system [125–127]. For our parameters this effect is small,and we do not hope to measure it in our experiment [39, 125–127]. Therefore, we
hope to observe a saturated gas even at finite interaction strengths in our system.
We experimentally test this saturation of a homogeneous gas as follows: By varying the number of atoms initially captured in the magneto-optical trap from thebackground gas (see Fig. 6.1), we can change the number of atoms loaded into
the uniform trap. We then evaporate to empirically chosen points such that we

100

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

achieve clouds at a constant temperature of 45 ± 1 nK with different condensed
fractions N0 /N . The thermal atom number is counted by fitting the gα form to a ToF
image, excluding the condensate peak, and we then count N0 by and integrating
the residual of this fit at low momenta. We find, in stark contrast to the harmonic
case, that N 0 does not increase as N0 increases (i.e. the dashed blue line in Fig.
7.1(c) is horizontal to within our experimental errorbars), indicating that our gas isglobally saturated. This is the best demonstration of Einstein’s picture of excitedstate saturation to date.
Two further thermodynamic experiments arise from this saturation property. These
are briefly outlined below:

7.1.2 The critical condition
By varying the final depth of the evaporation in the uniform trap during the sequencein Fig. 6.7, we simultaneously vary the temperature and condensed fraction of the
final cloud. We can extract a critical point by plotting N 0 and N0 vs. T in Fig. 7.2, andlooking at the conditions at which the condensate first appears. For the experimental
sequence shown, we find the critical point to be when N 0 ≈ 5 × 105 atoms, and
Tc = 92 ± 3 nK. From the measured box volume, we get a consistent theoreticalvalue for a critical temperature, Tc0 = 98 ± 10 nK. This calculation includes a small
numerically caluclated finite-size shift and an interaction shift of Tc0 [128–130], of
about 6% and 1%, respectively on top of the ideal gas result nλ3T = ζ(3/2). The error

Fig. 7.2.: The number of condensed (N0 ) and thermal (N 0 ) atoms versus T . The critical temperature
Tc = 92 ± 3 nK is in agreement with the prediction for a uniform Bose gas. Below Tc , a
power-law fit (solid red line) gives N0 ∝ T 1.73±0.06 .(single experimental run per point; scatter
indicates random error scale)

7.1 Thermodynamic experiments101

on the theoretical value includes the 10% uncertainty in our absolute atom-number
calibration (calibrated by measuring Tc in a harmonic trap [41]).
Now we harness the saturation of our gas to verify Einstein’s prediction Nc ∝ λ−3
T in
0
Eq. 7.1. Cooling beyond the critical point gives a gas with N0 > 0 and N = Nc (i.e.
saturation). We measure N 0 and T as we cool and fit our experimental data below
Tc to obtain Nc ∝ T 1.73±0.06 . The measured exponent of 1.73 ± 0.06 is close to theperfectly uniform behaviour of 3/2, and agrees very well with the predicted powerlaw in an r13 potential (for a trap of fixed α, we expect4 Nc ∝ T α+1/2 , so our powerlaw is consistent with our value of α ≈ 1.23).

7.1.3 The quantum Joule-Thomson effect
So far, we have presented studies which give the best experimental evidence for thewell known phenomenon of saturation at nc λ3T = ζ(3/2). While working on theseexperiments, we uncovered a more surprising effect: when a condensed cloud isheld in the imperfect vacuum of our apparatus, it spontaneously cools down.
This phenomenon again boils down to the saturation of the gas: If we remove anatom from the thermal cloud in a partially condensed sample, then, instantaneously,
N 0 < Nc . This inequality is not allowed for a saturated gas with N0 > 0, and so thebalance N 0 = Nc must be restored by transferring an atom from the condensate to
the thermal cloud. This transfer alone does not conserve energy (the condensate
atom is gaining energy), and must be accompanied by a redistribution of the existing
thermal atoms into a lower energy state (see Fig. 7.3). In summary, this means that
if we hold a saturated cloud and remove particles in an energy-independent way,then the cloud temperature will decrease. In our imperfect vacuum, it is collisions
with the background gas which removes atoms from our sample.
We can phrase this process as the quantum analogue of the classical Joule-Thomson
rarefaction of a gas. The Joule-Thomson effect (which is used in industrial refrigerators) describes the change in temperature of an interacting classical gas when it
undergoes isoenthalpic rarefaction: (∂T /∂P )H . In an ideal classical gas, the enthalpy, H, depends only on T , and thus (∂T /∂P )H = 0. Therefore, classically, onlyinteracting gases can show any change in temperature in an isoenthalpic process.
On the other hand, a quantum Bose gas will show Joule-Thomson cooling as a purely
quantum-statistical effect even in the absence of interactions (see below) [124]. We
therefore refer to the process in which (∂T /∂P )H 6= 0 in a system without interactions as the quantum Joule-Thomson effect. This quantum-mechanical cooling effect
was predicted more than 70 years ago [124], and it is only with the experimental
4102This scaling can be obtained by integrating Eq. 6.5 over z

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

Fig. 7.3.: A cartoon of the quantum Joule-Thomson effect (a) Isoenthalpic rarefaction. In a classicalJoule-Thomson process the system volume is increased, whereas in our experiment, theatom number is reduced. For an ideal gas, both processes conserve specific enthalpy
and are thermodynamically equivalent. (b) Microscopic origin of JT cooling in a saturated
quantum gas. Removal of a thermal atom requires a zero-energy particle to come out of
the BEC in order to maintain N0 = Nc , while energy conservation requires redistribution of
atoms between energy levels. Cooling is seen in the change of the relative populations of
the excited states (in this cartoon we neglect the fact that Nc also slowly decreases).

advantages of our uniform system that it has been finally experimentally observed(see also [131] for the fermionic case).
Predicted cooling trajectory
For a non-interacting equilibrium Bose gas, the process of atom loss by backgroundgas collisions is isoenthalpic by the following argument: Within the ideal-gas approximation, the pressure is P = 23 E/V [3], where E is the total energy of the system.
The enthalpy is then H = 53 E. Since the collisions with the background gas are notenergy selective, the average energy per particle, E/N , remains constant, so thespecific enthalpy, h = H/N , is conserved.
We can calculate the temperature change as atoms are lost in this isoenthalpic
process by starting with the expression for the energy of a partially condensed ideal
α+1/2 , toBose gas: E = (α + 1/2) ζ(α+3/2)
ζ(α+1/2) Nc kB T [3], and substituting in Nc = T
obtain:E
T α+3/2
∝
N
N


=⇒
=⇒

∂T
∂N



1
T
α + 3/2 Nh 1/(α+3/2)
T
N
=
.
Ti
Ni
=

7.1 Thermodynamic experiments

(7.6)

103

Fig. 7.4.: The quantum Joule Thomson effect in a partially condensed and a thermal, weakly degenerate Bose gas. Each point represents the average of approximately five experimental runs
with error bars indicating the uncertainty on the mean. The dashed and solid blue line are
predictions based on α = 1 (Eq. 7.6) and a more realistic α ∼ 1.2 for our trap, respectively.
The dotted red line is a numerical calculation (see text).

This provides a theoretical prediction for the cooling phenomenon that we observe.

Experimental results
Fig. 7.4 shows the cooling trajectory for an experiment in which we hold a partiallycondensed cloud at initial temperature Ti = 45 nK in a uniform trap with a vacuum
lifetime5 of τ ≈ 10 s. The cooling as N varies in this plot from Ni = 1.7Nc (Ti ) to
0.3Nc (Ti ) is well captured by Eq. 7.6.
Classically, the Joule-Thomson effect is often phrased in terms of a coefficientµJT = (∂T /∂P )h . For our quantum system, we write:


µJT =
=⇒

∂T
∂N

 
h

∂N
∂P



3 N
= V
2 E
h



∂T
∂N


h

λ3T
2
µJT =
.
5 ζ (5/2) kB

5

(7.7)

Note that due to the low particle density (< 5 × 1012 cm−3 ) the three body recombination rate is
negligible [48], and throughout the decay, the elastic scattering rate remains sufficiently high for thegas to re-equilibrate on a timescale ≤ 2 s  τ [36]. This ensures that we do not form a superheatedstate.

104

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

where we have taken the perfectly uniform case (α = 1). At our lowest temperature,µJT ≈ 4 × 109 K/bar, which is 10-11 orders of magnitude larger than typical values
observed in classical gases [132].
The effect we observe is purely quantum mechanical (µJT → 0 as ~ → 0) and driven
by Bose statistics (not by interactions6 ). Note that this physics is masked in thecase of a harmonic trap because (a) the gas is not saturated for any experimentally
accessible interaction strengths, and (b) the increased densities at the centre of
the trap lead to greater three-body effects which heat the cloud faster than theJoule-Thomson effect cools it (see the equlibrium decay curves in chapter 2).
We do not expect this dramatic cooling to occur in a non saturated cloud since
the qualitative understanding we gave relies on the clamping N 0 = Nc . A minor
complication to this picture is that for a degenerate but not condensed cloud, we do
expect some cooling effect from energy-unselective atom loss. This arises from thefact that the ratio ξ ≡ E/[(α + 1/2)N kB T ] decreases from 1 to ζ(α+3/2)
ζ(α+1/2) (≈ 0.514 for
α = 1) between the classical limit and the critical point. As background collisions
reduce N , the gas becomes less degenerate, and ξ grows. Hence, if E/N is kept
constant T must slightly decrease. The dotted red line in Fig. 7.4 shows a numericalcalculation of this effect for a thermal cloud with Ni = 0.6 Nc (Ti ) [124], and again,
this fits the data very well.

7.2 Ground state measurements
We now shift our attention away from the the thermodynamics of thermal and partially condensed homogeneous gases to concentrate on the BEC itself. In order
to investigate this low energy state, we will introduce the technique of Bragg spectroscopy. Bragg spectroscopy is a valuable tool for accessing precise informationon the momentum distribution of our trapped gas. This will allow us to perform more
complex studies such as the dynamical study in chapter 8. Before embarking onthese dynamical studies, we will invest time in this section to discuss and benchmark
our Bragg spectroscopy apparatus in a simple equilibrium context. We start withthe theory of Bragg spectroscopy (section 7.2.1), and then briefly illustrate this withexperimental applications in static equilibrium (section 7.2.2 and 7.2.3).

6

The total drop in temperature of 22 nK cannot be explained by the < 4 nK change in interaction
2 N −Na ienergy (calculated from the upper bound 8π~). In fact the interaction energy change actuallymVpredicts a small increase in temperature.

7.2 Ground state measurements

1057.2.1 Bragg spectroscopy
Throughout our thermodynamic studies, we were able to extract quantitative information from ToF profiles of partially condensed gases. This is because the kinetic
energy of the excited states in a cloud near the critical temperature is much higher
than the interaction energy (see fig: 7.5), meaning the time-of-flight dynamics isdominated by the kinetic energy. Therefore, after an experimentally achieveable
time-of-flight we can obtain a reasonable measure of the excited-state momentum
distribution of the partially condensed gas.

Fig. 7.5.: Typical energy scales for the experiments in this chapter. For thermal and partially condensed
clouds, ToF expansion is governed by the atoms’ kinetic energy (when we cross Tc , wehave ∼ 5 × 105 atoms). For a pure BEC of ∼ 105 atoms, the interaction energy dominates.

However, if we have a pure BEC, the interaction energy dominates the time-of-flight
dynamics, so we cannot directly image the momentum distribution of a BEC in aninteracting system. Instead, interactions drive the BEC to adopt a characteristic
diamond shape in ToF for which we have no analytical form. So-far we only understand this shape in numerical simulations (see Fig. 7.6), and through the qualitativeargument that that the expansion occurs fastest where the contours of |∇n0 | haveleast curvature.
The difficulty in extracting the momentum distribution can be overcome using Bragg
spectroscopy [133–139]. The essence of the technique is to pick out and spatiallyseparate the atoms in a particular momentum state from the bulk cloud, allowing us
to count the occupation of that individual momentum state. By addressing different
momenta in turn, we can reconstruct the complete in-situ momentum distributionof the cloud. For a pure condensate, this will give the ground state momentum
wave-function, |ψ(p)|2 .
Below we will describe how this momentum selection can theoretically be achieved
using two light beams, and later we will outline the apparatus required for Bragg
spectroscopy.
The theory of Bragg spectroscopy
Since a photon’s energy/momentum ratio is much larger than that of a non-relativistic
atom, single photon interactions produce "near vertical" transitions on an energy-

106

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

Fig. 7.6.: ToF expansion of a gas from a uniform box. (a) Experimental images of a freely expanding
gas taken 0, 25 and 50 ms after release from the box (delineated in orange). Since thecloud becomes more dilute as it expands, we adjust the colour scale as shown by themultiplicatative factor in the top left of each panel. (b) A simulation based on the GrossPitaevskii equation including realistic interactions for 1.5 × 105 atoms (see Appendix C)
matches the experiment well. (c) Simulation of a non-interacting gas under the same
conditions as (b). This illustrates that the ground state expansion in our experiment isdominated by interaction effects.

momentum diagram (see Fig. 7.7(a)). However, combining two laser beams withwavevectors k1 and k2 = k1 + ∆k, we can arrange "near horizontal" transitions
which impart momentum ~∆k to the atom (the atom absorbs a photon from the firstbeam and emits into the second). Importantly, energy and momentum conservation
ensure that for fixed k1 and k2 there is only one class of momentum states (a single
plane in momentum space7 ) which is resonant with this two photon transition. In thisway, a two photon transition can be used to precisely select atoms in a particular
momentum class and kick them out of the bulk cloud.
Two photon transitions have been discussed in section 4.1.2 in the context of Raman
scattering in a system with three discrete levels. For the present application, thewavefunction of the condensate can be written in a continuous momentum basis (i.e.
plane waves) which can all be coupled by the two lasers. To develop an intuition for
the effect of the two laser beams on the occupations of this continuum, we artificiallyselect one particular momentum state |pi and its corresponding state |p + ~∆ki
and imagine applying our Raman theory to only these states. In our experiments,|pi and |p + ~∆ki are in the F = 2 internal state, and the intermediate state, |ii, in
the Raman process will be played by states in the F 0 = 3 hyperfine level (see Fig.
7.7(a)).
7

The resonant plane is given by p · ∆k = 2πm(f − fr − fint ) using symbols defined in Eq. 7.8

7.2

Ground state measurements

107

Fig. 7.7.: Theoretical picture for Bragg spectroscopy. (a) The laser arrangement for transferring
momentum ~∆k. (b) A non-interacting 1D numerical (Gross-Pitaevskii) simulation illustrating
the real and momentum space effect of a resonant Bragg pulse with Ω/2π = 8 Hz, t = 35 ms
on a top-hat wavefunction of width L. After tToF = 140 ms ToF (bottom panels) diffracted
cloud (red) is at position vr tToF . In momentum space, the Bragg pulse has transfered a very
narrow band of momenta out of the the states {P} (blue shading) to the states {P + ~∆k(red shading). Thus, by counting the diffracted atoms as we address different momenta
with the Bragg pulse, we obtain the momentum distribution to a momentum resolution of∼ ∆fRes . (c) A (3D) experimental image corresponding to the (1D) simulation in (b).

As shown in Fig. 4.1(bii.), the detuning δ is a key parameter defining the Rabi
oscillations between the coupled states (note that we will use two lasers with equal
intensities such that δ 0 = δ in the notation of Eq. 4.10). From Fig. 7.7(a), we canwrite δ as:1
δ = ∆ω +~p20+ Eint
2m



1
−
~



(p + ~∆k)2∆+ Eint2m= 2π(f − fr − fint − fp ),(7.8)

where:• f is the frequency difference between the two laser beams: 2πf = (ω1 − ω2 )• fr is the recoil energy Er = ~2 |∆k|2 /2m expressed in frequency units:
fr = Er /h108

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas∆ − E 0 )/h is the difference in per-particle interaction energy for the• fint = (Eintint
two momentum states expressed in frequency units• fp is the atomic momentum expressed in Bragg frequency units: 2πfp =∆k · p/m
Inserting Eq. 7.8 into Eq. 4.12 gives the population of state |p + ~∆ki after illuminating state |pi with light for time t as:2|c̃p+~∆k (t)| = |c̃p (0)|

2



ΩR t
2

2

 q

t
2
2
2
sinc
ΩR + 4π (f − fr − fint − fp ) . (7.9)
2{z}|"Resolution kernel", Res(f − fp )2

This describes Rabi oscillations in time in our artificial 3-level system. When we
return to the continuous momentum picture, we can interpret this result by dividing
the continuum into two relevant sets of states:• States {P}, which are significantly occupied by atoms in the trap (for a pure
BEC, we expect this set to extend ∼ ~/L either side of zero momentum, where
L is the trap length).
• States {P + ~∆k}, which are the displacement of {P} by ~∆k in momentum
space.
For ∆k  1/L, these two sets can be treated as entirely separate (i.e. orthogonal)(see Fig. 7.7(b)), and we further posit that we can treat two photon interactions withthe continuum of momentum states as a set of independent Raman interactions
between each state in {P} exclusively with its corresponding state in {P + ~∆k}.
Under these assumptions, the total occupation of the states {P + ~∆k} after a
Bragg pulse of time t is:
N{P+~∆k} (f ) =

ΩR t
2

2 Z

|ψ(p)|2 Res (f − fp ) d3 p,(7.10){P}

where, in the continuum limit, we have identified |c̃p (0)|2 with the initial momentum√
wavefunction, |ψ(p)|2 . By symmetry, we can separate ψ(p) = N ϕ(pr , pθ )φ(px )where pr , pθ and px are momentum space cylindical polar coordinates aligned withthe symmetries of our cylindrical trap, ϕ and φ are both normalised to unity, and

7.2 Ground state measurements109

N is the total atom number. If we align ∆k parallel to the px axis, we can write thebragg-diffracted fraction as the convolution:

N{P+~∆k} (f )
=
N



ΩR t
2

2 Z

|φ(px )|2 Res (f − ∆kpx /m) dpx .

(7.11)

Therefore, plotting the diffracted fraction as a function of the relative laser detuning,f , we obtain the momentum wave-function, |φ(px )|2 , at a resolution set by the
convolution kernel, Res(f ). Res(f ) is a function centered at f = fc = fr + fint witha zero-zero width, ∆fRes , of:1∆fRes =
π

s

2π
t

2

− Ω2R .

(7.12)For maximum momentum space resolution, we want Res(f ) → δ(f ) (i.e. ∆fRes → 0).
Therefore we minimize ΩR and maximise t while empirically ensuring that (i) we
maintain a measurable diffracted fraction and (ii) the bragg pulse remains shorter
than any characteristic time for the free evolution of φ(px ) (this is particularly relevant
for the ToF measurements in section 7.2.3).
The diffracted atoms move at velocity vr ≈ ~∆k/m, which is much faster than
the ground state BEC expansion velocity by virtue of the arranged ∆k  1/L.
Therefore, after sufficient ToF, the diffracted atoms appear as a cloud separated
from the non-diffracted atoms. This allows us to simply count N{P+~∆k} (f ) and N(see Fig. 7.7(c)), and by changing the relative frequency f of the two lasers, weachieve our initial goal of mapping out |φ(px )|2 via Eq. 7.11.
Apparatus for Bragg spectroscopy
We now put the theory outlined above on a more concrete footing by outlining theapparatus used to realise Bragg spectroscopy in our experiment.
We use two 3 mm diameter, intensity stabilized [40] laser beams of approximately
780 nm which are ∼ 6.6 GHz blue-detuned from the |F = 2i → |F 0 = 3i transition.
The frequencies of these beams are further shifted one by −80 MHz and the otherby (−80 + f ) MHz using AOMs. They are then steered with mirrors to intersect at
the atoms at a total angle of 30° (see Fig. 7.8). This arrangement gives vr ≈ 3 mm/s
and fr ≈ 1 kHz.
Unless otherwise stated, we will apply the Bragg pulse in situ and allow the diffracted
atoms to escape by lowering the walls of the uniform trap below the recoil energy,

110

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

Fig. 7.8.: Apparatus for Bragg spectroscopy. (a) The arrangement of the bragg beams (red) illuminating the uniform trap. The detunings of the lasers are shown (not to scale) on a cartoon of the87
Rb D2 line. (b) A trapped atom with initial momentum p absorbs recoil momentum ~∆kaligned with x-axis. The trap wall height (green) is lower than the energy of the recoilingatoms. The Bragg beams and interference lattice are not drawn to scale; in reality thebeams are much larger than the box and the lattice spacing is much smaller.

Er , while still trapping the BEC (see Fig. 7.8(b)). Atoms diffracted out of the trap
are then easily counted. Since the trapped BEC momentum distribution does not
evolve with time (ignoring the Bragg perturbation), we can apply long Bragg pulses(t = 35 ms) with a low Rabi-frequency (ΩR = 8 Hz) to minimize ∆fRes . In dynamical
experiments (e.g. probing |φ(px )| in ToF) we will reduce the length of the Bragg
pulse and increase ΩR to improve the time resolution of our measurements while
sacrificing spectral resolution.
The experimental results using this apparatus to probe an equilibrium BEC are
outlined in the next sections.

7.2.2 Ground-state wavefunction & The Heisenberg limit
We will start our experimental discussion with a simple example of a momentum
spectrum recorded for an in situ homogeneous BEC in a trap of length L = 30 ± 1 µm
and radius R ≈ 16 µm (see Fig. 7.9(a)). Depending on the interaction strength,
there are two limits for the shape of this ground state wavefunction, summarisedin table 7.1. Referring to Fig. 7.5, we expect our BEC to be deeply in the ThomasFermi regime. This is supported experimentally by both in situ images which reveal
the top-hat profile of the trapped condensate, and by the Bragg spectrum whichclosely fits the sinc2 profile, |φTF (px )|, defined in table 7.1 convolved with Res(f ).
As expected for a fully coherent BEC spanning the uniform box-trap, we find that

7.2 Ground state measurements

111

Non-interacting
Real Space
Momentum SpaceFWHM

φ0 (x) = cos
4L|φ (px )| = 3π02 πx 



L

TH

Thomas-FermixLcos(πfp L/vr )1 − (2fp L/vr )2

W 0 ≈ 1.19

vr
L

φTF (x) = TH
2
|φ

TF

x

L
(px )| =
sinc2
2π
2

W TF ≈ 0.89

L


πfp L
vr

vr
L

Tab. 7.1.: Key results for the real and momentum space wave functions in the non-interacting andThomas-Fermi limits. TH(z) is a top hat function equal to unity for −1/2 < z < 1/2 andzero elsewhere. FWHM indicates the full-width-at-half-maximum for the absolute-squared
momentum space wave functions.

the full-width-at-half-maximum (FWHM), W = 100 ± 3 Hz, is independent of N andclosely matches the expected width (after convolution with Res(f )) of 93 Hz. Notethat since Res(f ) is centered at fc ≈ 1 kHz, we plot the spectrum against Bragg
beam detuning relative to fc .
In Fig. 7.9(b), we plot the FWHM of the Bragg spectra measured for different box
sizes, L (we use the SLM techniques in chapter 5 to move the walls of the opticalbox). This reveals the expected Heisenberg 1/L scaling of the momentum width forthe range of boxes studied.

Fig. 7.9.: Heisenberg-limited momentum spread in a uniform interacting BEC. (a) Main panel: Bragg
spectrum of a trapped BEC of length L = 30 µm. The solid red line shows the theoretical
Heisenberg-limited spectrum |φTF (px )|2 and the dashed blue line shows the theoretical
spectrum after convolution with the Bragg resolution kernel, Res(f ). Left inset: L is determined by fitting the in-trap BEC density profile, accounting for the imaging resolution.
Right inset: FWHM of the spectrum, W , versus the atom number, N , for the same L. The
filled square corresponds to the data in the main panel, with W = (100 ± 3) Hz. The solid
red line shows W TF = 0.89vr /L = 87 Hz and the dashed blue line shows the FWHM
after convolution: W̃ TF = 93 Hz. The dotted blue line shows W̃ 0 = 123 Hz, expected
for a noninteracting BEC. (b) W versus inverse box length, 1/L, showing the expected
Heisenberg scaling. Solid red, dashed blue, and dotted blue lines show W TF , W̃ TF , andW̃ 0 respectively.(All error bars represent 1σ fitting uncertainty on W ).

112

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas7.2.3 Ground state interaction energy
So far, we have only looked at the widths of the Bragg spectrum. However, information can also be extracted from the position fc = fr + fint of the spectral peak.
Since the peak position depends on fint , we can use shifts in the peak position tomonitor variations in the ground state interaction energy. From exchange symmetry,
the interaction energy between atoms in distinguishable states is twice that of atoms
in the same state8 . Therefore we can evaluate the shift fint as:∆0
hfint =Eint− Eint=(2V0 n0 + 2V0 n0 ) − (V0 n0 + 2V0 n0 )(7.13)=V0 n0 ,where V0 is the interaction parameter defined in Eq. 1.2, n0 is the density of atoms in
the BEC, and n0 is the thermal atom density. We rewite this expression as fint = αN
where N is the total atom number, andα=N0 2~a.
N mπR2 L(7.14)

We can estimate the condensed fraction N0 /N by numerically simulating the expected diffracted fraction for a pure condensate at f = fc (see Appendix C) andcomparing this to the experimental value (∼ 15%) measured in Fig. 7.9(a). Thismethod gives N0 /N = 0.8 ± 0.1, which is also supported by "BEC filtering" experiments [40,140] (the uncertainty indicates variation between experimental runs). This
allows us to calculate the theoretical value for α as α = (24 ± 2) × 10−5 Hz/atom.
Experimental Results
To experimentally measure the slope α = dfint /dN we can vary the final value of N
by changing the time during which we catch atoms from the background gas at thestart of an experimental run (i.e. the duration of the magneto-optical trap step in Fig.
6.1). Plotting measured values of fint as a function of N gives an empirical value
α = (20 ± 1) × 10−5 Hz/atom (see Fig. 7.10(a)). This is slightly below the theoretical
8
Briefly, this ’Bosonic factor of 2’ can be derived in the two-particle case by comparing the
expectation of the interaction potential V0 δ(r1 − r2 ) for atoms in identical states |φ1 (r1 )i |φ1 (r2 )i anddifferent states 2−1/2 [|φ1 (r1 )i |φ2 (r2 )i + |φ1 (r2 )i |φ2 (r1 )i]. In the latter case, exchange terms of theform hφ1 (r1 )| hφ2 (r2 )| V0 δ(r1 − r2 ) |φ2 (r1 )i |φ1 (r2 )i contribute a total additional energy V0 /V where V
is the box volume. It is simple to extend this to the n0 -particle case to explain extra interaction energy
V0 n0 .

7.2 Ground state measurements

113

Fig. 7.10.: Bragg spectrum shifts and widths as functions of N and ToF. (a) We plot fc versus N
for Bragg pulses applied in situ (solid squares) and after 50 ms of ToF (open diamonds)
for L = 30 µm. Dashed line: fc ≈ fr ≈ 975 Hz, solid line: fc = fr + αN , with α ≈20 × 10−5 Hz/atom. (b) Evolution of fc during ToF. The solid square corresponds to the intrap data from Fig. 3. The horizontal error bars indicate our temporal resolution, limited by
the Bragg-pulse duration, and vertical error bars indicate fitting uncertainties. (c) Spectral
width for the ToF data presented in panel (a). The solid line shows a numerical simulation
based on the Gross-Pitaevskii equation. (d) Spectral width during ToF for the data in panel(b). In (b) and (d), the solid lines show numerical simulations based on the GP equation
for N = 1.25 × 105 atoms.

value due to the ≈ 15% depletion of the condensate atom number during the Bragg
pulse9 .
To obtain the baseline (i.e. limN →0 fc ) in Fig. 7.10(a), we perform a different
experiment where we release the cloud in ToF before applying the Bragg pulse.
Fig. 7.10(b) shows that the interaction energy (as measured by fc ) drops as thecloud becomes more dilute in ToF10 . This allows us to assume that after 50 ms ToF,
the interaction energy is negligible. Experimentally, we observe this effect in the
independence of fc on N after 50 ms ToF in Fig. 7.10(a). These measurments give
us an accurate measurement of the recoil energy fr = fc (N = 0) ≈ 975 Hz, and abaseline for the in situ measurements.
9

We can repeat the measurement with a reduced ΩR and extrapolate to the vanishing diffractedfraction limit to obtain an improved estimate α = (23 ± 1) × 10−5 Hz/atom [40].
10Note that since the BEC momentum distribution evolves with time in ToF, we must apply shorter
pulses (t = 10 ms) with a higher Rabi-frequency (ΩR = 28 Hz) to "instantaneously" measure themomentum. We thus sacrifice specrtal resolution to gain temporal resolution in the data in Figs 7.10(b)and (d).

114

Chapter 7

Observing Equilibrium Properties of a Homogeneous Bose Gas

For completeness, we finally show the FWHM, W, of the Bragg spectrum as afunction of N after 50 ms ToF and as a function of ToF for N = 1.25 × 105 atoms in
Figs 7.10(c) and (d) respectively. These again show the conversion of interactionenergy to kinetic energy (i.e. broadening of the momentum distribuition) in ToF, withthe size of the effect increasing with N . Furthermore, they demonstrate the excellent
agreement of our experimental system with simple GP simulations11 (see Appendix
C).
Before leaving this topic, it is worth briefly contrasting our results with those obtained
for a harmonically trapped system [141, 142]. The most striking difference is thatin an inhomogeneous cloud, fint is a function of position, and therefore the Bragg
spectrum is broadened by the spread of fint in the cloud. This makes interpreting
Bragg spectra of inhomogeneous systems more difficult than the homogeneous
case: the true momentum distribution is only recovered by processing out theinteraction broadening. Hence we claim that the measurements presented in these
sections have been the clearest demonstration of Heisenberg scaling in cold atomic
gases. Furthermore, we have seen that without any post-processing we have beenable to resolve interaction energies with an astonishing precision of ≈ h × 2Hz(corresponding to a temperature scale of 100pK).

7.3 Conclusion
This section has provided a brief review of the first experiments carried out with ahomogeneous atomic Bose gas. We have discussed both the equilibrium thermodynamics of thermal and partially condensed gases, and also the quantum mechanicalproperties of the ground state in equilibrium. These experiments have alreadyshown that the uniform system has great potential for new experiments whereparabolic trapping geometries complicate or entirely mask fundamental many-body
phemonena which we wish to investigate. In the next chapter, we will perform amore sophisticated experiment which again requires our uniform system, and also
employs the technique of Bragg diffraction introduced here.

11

note that we include a residual 1.5Hz harmonic trapping potential in the simulations to model thecurvature of the imperfect gravity compensating magnetic field

7.3

Conclusion

115

Critical Dynamics of Spontaneous
Symmetry Breaking in a
Homogeneous Bose gas

8Prehistoric cultures long perfected the art of cooling metals across the liquid-solid
phase transition. Crucially, they found that it is not only the transition itself but also
the approach to the transition (via different quenching, tempering and annealing
protocols) which is important for defining the mechanical properties of the final solid.
Nowadays, we interpret these metallic properties in terms of the grain structure of
the material (i.e. the size of the domains of common crystallographic axes), andwe know that the dynamics of the nucleation and growth of grains during a specificcooling trajectory dictates the properties of the resulting solid phase.
This metallurgic example concerns a first order phase transition, but we have alreadyseen that dynamics are also important for determining the properties of systems
near second order transitions. For example, in chapter 2, we saw that a superheatedstate can be dynamically generated when the phase equilibrium point moves fromthe ordered to disordered state at a rate faster than the system can follow. In thischapter, we consider crossing the BEC transition in the opposite direction: we will
drive a homogeneous Bose gas below the critical temperature at a finite rate andexplore the coherence properties of the resulting BEC. Importantly, we will observethat the growth of correlations is stalled by the systems diverging response timenear the critical point. This leads to the formation of finite sized domains of phasecoherence in the condensate, with each domain choosing a different value for thesymmetry breaking phase of the BEC order parameter. Note that unlike the metallic
grains formed in the thermal quench of a metal, we expect the domain boundaries
in a quenched BEC to be a smooth region in which the phase interpolates fromone area (domain) of approximately constant phase to another area (domain) of
approximately constant phase. We will measure the size of these domains using an
adaptation of the spectroscopic techniques introduced in the previous chapter.
The chapter is organised as follows: In section 8.1 we describe the theory of domain
formation in quench-cooled homogeneous BECs, then in section 8.2 we present anexperimental method for accurately extracting the size of these domains. We willcompare the experimentally measured domain sizes to the theoretical predictions in117

8.3 and 8.4, ultimately allowing us to extract the first measurement of the dynamical
critical exponent for the universality class of the homogeneous 3D BEC transition.

8.1 Domain size in Kibble-Zurek theory
The key result in this chapter will be experimental verification of a link between therate at which we cool a homogeneous Bose gas through the BEC transition and thesize of the domains of phase coherence in the resulting condensate. Theoretically,a simple framework for predicting this link is provided by the Kibble-Zurek (KZ)
mechanism [143, 144]. Signatures of this theory have already been studied in arange of experimental systems, including liquid crystals [145], liquid helium [146,147],
superconductors [148–150], atomic BECs [151–156], multiferroics [157] and trappedions [158–160]. We will briefly discuss the KZ theory below before presenting our
experimental results on homogeneous BECs.
When performing a thermal quench through the transition we define the "distance"
to the critical point as the reduced temperature  = (T − Tc )/Tc . A fundamental
characteristic of second order transitions is that as  → 0 several macroscopic
observables diverge as scale-invariant power laws in . Since short-range physics
becomes largely irrelevant near the critical point (due to long-range fluctuationsof the order parameter) all second-order transitions can be classified into a small
number of universality classes depending on their generic features such as symmetries, dimensionallity and range of interactions [161, 162]. All transitions within auniversality class display the same critical exponents for the power-law divergences.
In particular, we will be interested in the critical behaviour of two quantities:• The asymptotic divergence of the correlation length, ξ, as  approaches 0:(ξ∼X+ −ν
X− −ν

>0
<0(disordered),(ordered)(8.1)

where the exponent ν and the ratio of the critical amplitudes X+ /X− areuniversal quantities within a universality class.
• The characteristic rate at which correlations can build up in our system. Thisrate is given by the inverse of the relaxation time, τ , which by the dynamical

118

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

scaling hypothesis [163] follows another universal power law near the criticalpoint:(8.2)

τ ∼ ξz .

Combining Eq. 8.1 and 8.2, we see that the system exhibits a critical slowing
down as it approaches the transition. This means that if we cool quickly acrossthe transition, correlations cannot grow fast enough to span the system before the
BEC forms. The system is then frozen in a condensed state with a finite phasecoherence length given by the equilibrium correlation length at the freeze time (seeFig. 8.1(a)).
To predict the frozen coherence length within the KZ theory, we imagine driving thesystem through the critical point with a linear quench of :=tc − t,τQ(8.3)

where the time t = tc is the time at which we cross the critical point, and the timescaleτQ parametrises the rate of the quench. The KZ hypothesis states that the system
dynamics become frozen when the rate of change of ξ is faster than the system’s
relaxation rate:

ξ˙
1
> .
ξ
τ

(8.4)

Using Eq. 8.1, 8.2 and 8.3, we can solve for the time tfreeze = |tc − t̂| at which thesystem freezes:

1/νt̂± ∼ (X± |τQ |)νz/(1+νz) .

(8.5)

Within t̂ of tc , the correlation length can no longer adiabatically grow in accordancewith Eq. 8.1. In the KZ framework, we assume that ξ is frozen at the length,d = ξ|t=tc −t̂+ , and coherence length (i.e. the size of the domains) for  < 0 will begiven by d. Inserting Eq. 8.5 and 8.3 into 8.1 gives:d = λ0

τQ
τ0

b
b=

8.1

ν
,
1 + νz

Domain size in Kibble-Zurek theory

(8.6)

119

Fig. 8.1.: Domain formation during spontaneous symmetry breaking in a homogeneous Bose gas. (a)Red circles depict thermal atoms and blue areas coherent domains, in which the U (1) gauge
symmetry is spontaneously broken. The black arrows indicate the independently chosencondensate phase at different points in space, and dashed lines delineate domains overwhich the phase is approximately constant. The average size d of the domains formed at
the critical point depends on the cooling rate. Further cooling can increase the population ofeach domain before the domain boundaries evolve. (b) Cartoons illustrating the calculationof the Kibble-Zurek scaling law. The system is frozen in the blue shaded region where˙|ξ/ξ|> 1/τ . The lower panel shows the equilibrium value of ξ (orange), and the Kibble
Zurek hypothesis states that ξ stalls at the value ∼ d (blue dashed line) when the system isfrozenwhere we have explicitly inserted dimensionful constants λ0 and τ0 , which dependon the microsopic details of a particular system. In our system λ0 is expected to be
set by the thermal wavelength at the critical point, and τ0 by the elastic scattering
time τel [152, 164]; for our experimental parameters, λc ≈ 0.7 µm, and a classicalestimate gives τel ≈ 30 ms.
Fig. 8.1(b) graphically summarises the steps in the derivation of Eq. 8.6. In essence,
we have shown that due to the phenomenon of critical slowing down, crossing
the BEC transition at a finite rate causes domains of predictable size to form withindependent choices of the symmetry-breaking order parameter1 (i.e. the BEC
phase).
Our ultimate aim is to quantitatively measure the power-law d ∼ τQb for the homogeneous 3D BEC transition. This transition resides in the same universality class as the
3D XY model and the lambda transition of liquid 4 He, for which the critical exponents
can be calculated in a mean-field (i.e. Ginzburg-Landau) theory [163, 166], or ina beyond mean-field dynamical critical model, known as the F Model [163]. The
relevant exponents in these theories are summarized in table 8.1.

1Note that at the domain boundaries, rare long lived topological defects can also form [165], theirnature and density depending on the physical system. We will not explicity consider these defects inour discussion.

120

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Mean Field

Beyond Mean Field

z

2

3/2

ν

1/2

2/3

X+ /X−

0

∼0.3

b

1/4

1/3

Tab. 8.1.: Crtical exponents and amplitude ratios for time-dependent-Ginzburg-Landau mean field (MF)[163, 166] and beyond MF [163] models. The beyond MF critical exponents are calculated
according to the so-called F Model, and the amplitude ratio is calculated numerically [167].

We hope to measure these critical exponents in later sections, however it is useful
to first discuss the qualitative evidence for domain formation and introduce the basic
experimental parameters. This will be the task for the next section.

8.1.1 Qualitative evidence for domain formation
Experimentally, we will use our uniform box trap to study the Kibble-Zurek theory.
Using individual ToF images it is remarkably simple to observe the qualitative
message of Fig. 8.1(a) - i.e. the faster the temperature quench, the smaller thedomains. The procedure for this qualitative study is outlined below:
We prepare a homogeneous Bose gas by loading 3 × 105 87 Rb atoms into our
cylindrical optical-box trap with length L ≈ 26 µm along the horizontal x-axis, andradius R ≈ 17 µm. Initially T ≈ 170 nK, corresponding to T /Tc ≈ 2. We thenevaporatively cool the gas by lowering the trap depth linearly from ∼ 2 µK to ∼ 30 nK
in time tQ . We cross Tc ≈ 70 nK with 2 × 105 atoms, and at the end of the cooling
we have 105 atoms at T . 10 nK (T /Tc . 0.2).

Fig. 8.2.: Phase inhomogeneities in a deeply degenerate gas are revealed in time-of-flight expansion
as density inhomogeneities. (a) The gas is cooled in 1 s from T ≈ 170 nK, throughTc ≈ 70 nK, to . 10 nK. Each realization of the experiment results in a different pattern, andaveraging over many images results in a smooth featureless distribution. (b) Preparing a
T . 10 nK gas more slowly (over 5 s) results in an essentially pure BEC with a spatially
uniform phase.

8.1

Domain size in Kibble-Zurek theory

121

When we release this condensate from the trap, any phase inhomogeneities are
revealed in ToF expansion as density inhomogeneities [153, 168]. In Fig. 8.2, wecompare the ToF profiles for a gas cooled to T  Tc "quickly" (tQ ≈ 1 s) and "slowly"(tQ & 5 s). Gases cooled quickly reveal density modulations in ToF indicative of
the presence of small domains in situ. In contrast, in our finite-sized box, slow
cooling can produce an essentially pure and fully coherent (single-domain) BEC
which develops the characteristic diamond shape in ToF (c.f. Fig. 7.6). This contrast
supports the qualitative picture illustrated in Fig. 8.1(a).

8.2 Quantitative measurement of domain sizes
We now study the KZ mechanism more quantitatively. Specifically, we will develop
an experimental technique for measuring the size, d, of the domains in our quenchcooled clouds.
To measure the size of the domains, imagine a pure condensate described in
the classical field approximation2 by the complex wavefunction ψ(r), and consider
inspecting this wavefunction at two points separated by r0 . If |r0 | < d then the twopoints may lie in the same domain, so there will be a correlation between the phasesof ψ(r) and ψ(r + r0 ). Conversely if |r0 | > d, we expect little correlation between
the phases. These phase correlations which decay over the length-scale d can be
captured by the spatially averaged two point first order correlation function, definedashψ(r)∗ ψ(r − r0 )i d3 r
1
C1 (r ) = qR=RN
h|ψ(r)|2 i d3 r h|ψ(r − r0 )|2 i d3 r0RZhψ(r)∗ ψ(r − r0 )i d3 r,(8.7)

where h...i denotes averaging over different domain configurations of ψ, and ψ isR
normalised to d3 r |ψ|2 = N .
In our experiment, ψ is only non-zero inside our uniform cylindrical box trap. This
artificially introduces the length scales of the finite-sized box, L and R, into the
problem which complicates the extraction of d from the decay of C1 . Below, weoutline a technical derivation of how the finite sized box affects the measurement ofd. While we attempt to be as formally correct as possible at each step, it is only the
final result in Eq. 8.13 which is important for the later sections. This final result is
2Note that we could treat the Bose gas using the field
R operators Ψ̂(r) and inspect the spatiallyaveraged first order correlation function proportional to d3 r hΨ̂(r)† Ψ̂(r − r0 )i. However for large
occupation of the BEC, it is simpler to approximate Ψ̂(r) as a complex function ψ(r) (i.e. ignore thefield operator commutator).

122

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

relatively intuitive when one considers the limits of d → L and d  L, and simple
experimets reproduce these intiuitve limits (see Fig. 8.6).

8.2.1 Infinite system correlations from a finite-sized experiment
We begin by constructing a function ψ∞ (r) defined in a homogeneous system ofpvolume V∞ → ∞ with a constant modulus N/πR2 L and a phase Θ(r):

r
ψ∞ =

N
exp [iΘ(r)] .
πR2 L

(8.8)

The function Θ is chosen such that ψ∞ everywhere has the same characteristicphase correlations as ψ. This means that we can obtain ψ from ψ∞ by "masking
out" all points not inside the cylindrical trap:(8.9)

ψ(r) = Cyl(r)ψ∞ (r)Fig. 8.3.: The relationship between ψ∞ and ψ. We imagine cutting a finite sized cylinder out of an
infinite system ψ∞ to obtain ψ. The infinite system has characteristic domain size d (phasedomains are illustrated in different colours).

where Cyl(r) is a unity inside the cylindrical trap and zero elsewhere. Inserting Eq.
8.9 into 8.7 gives:

1
C1 (r ) =N
0Z∗
Cyl(r)Cyl(r − r0 )hψ∞(r)ψ∞ (r − r0 )i d3 r,(8.10)V∞

By symmetry, since ψ∞ describes an infinite homogeneous system, we can replace∗ (r)ψ (r + r0 )i with it’s spatial average:
hψ∞
∞

∗
hψ∞
(r)ψ∞ (r − r0 )i =

1
V∞

8.2

Z

∗
hψ∞
(s)ψ∞ (s − r0 )i d3 s.

(8.11)V∞

Quantitative measurement of domain sizes

123Now, noting from Eq. 8.8 that
8.11 into Eq. 8.10 to obtain:

R

3
2
2
V∞ d s |ψ∞ (s)| = N V∞ /πR L, we can insert Eq.

R
Z
∗
0
3
1
V∞ hψ∞ (s)ψ∞ (s − r )i d sqCyl(r)Cyl(r−r0 ) d3 r,
C1 (r ) =πR2 L R h|ψ (s)|2 i d3 s R h|ψ (s − r0 )|2 i d3 s V∞
∞
∞
V∞
V∞
{z
}
|
0

C1∞ (r0 )(8.12)

Where we have identified the central term as the infinite system correlation function
C1∞ (r). For the case r0 = xx̂ (where x̂ is a unit vector along the axis of the cylindrical trap) we can calculate the geometric overlap integral in Eq. 8.12 under theassumption of a perfectly uniform trap with no healing length. This gives the final
result:x
.
C1 (xx̂) = C1∞ (xx̂) × 1 −L(8.13)

I.e. if we measure C1 (as described in the next section), then we will be measuring
the infinite system correlation function, C1∞ (which contains the length-scale d)modulated by a linear geometric envelope arising from the finite trap size.

8.2.2 Homodyne interference for measuring domain size
So far we have said that the length scale d which we want to measure is encoded in
the decay of C1∞ , and we have shown that C1∞ can be deduced from C1 . Now wemust find a way to experimentally measure C1 itself.
Inserting the Fourier transform of ψ(r) into Eq. 8.7 immediately gives C1 as theFourier transform of the momentum distribution of the BEC. Therefore, it is tempting
to refer to our momentum-space measurements chapter 7 and say that we cansimply Fourier transform a Bragg spectrum to obtain C1 . However, for very fast
quenches (τQ ∼ τ0 ), we expect d ∼ λc , corresponding to a Bragg spectrum spreadover W ∼ 3 kHz. Such a wide spectrum requires the Bragg pulse to have a large ΩR t
in order to diffract a measureable fraction of atoms. From Eq. 7.12, this generally
leads to broadening of the Bragg spetrum for experimentally achievable valuesof t. This broadening begins to significantly affect the spectrum as d grows (i.e.
W shrinks), and it was found to be difficult to find a suitable protocol which could
measure both large (∼ 30 µm) and small (∼ 0.7 µm) domains. For this reason, we
devise an alternative experimental approach based on homodyne interference.

124

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

The approach is inspired by [142], and is based on three steps:"Copy"Create a "copy" of ψ(r){ψ(r), ψ(r)}Arrange

Displace one copy by xx̂{ψ(r), ψ(r − xx̂)}Interfere

Create a superposition of the displaced copies

ψ(r) + ψ(r − xx̂)If we count the number of atoms in the homodyne interference pattern arising in the
final step, we recover C1 (xx̂):Z|ψ(r) + ψ(r + xx̂)|2 d3 r = 2N {1 + Re [C1 (xx̂)]} .

(8.14)In the next subsections, we will describe how the steps above can be achieved using
two separated Bragg pulses.

Step 1: "Copy"
After releasing the cloud from the trap, suppose we apply a very short, powerfulpulse of Bragg light with ΩR /2π ≈ 700 Hz and t = 0.1 ms. With these parameters,
the Bragg resolution kernel in Eq. 7.9 becomes very broad with ∆fres ≈ 20 kHz.
Even for correlations of size λc , the Bragg spectrum of the atomic cloud is 3 kHz
 20 kHz wide. Therefore, the resolution kernel can be treated as flat, and all
momenta in the cloud experience the same population transfer in time t.
Since the Bragg process is now effectively momentum-independent, we can treat
the system as a two level system with the lower level, |{P}i, representing the bandof states in the stationary cloud and a higher level |{P + ~∆k}i representing thediffracted atoms. In this basis, we write the initial state as ψ(r) |{P}i ≡ (ψ(r) , 0)T ,which we represent pictorially as follows:

8.2

Quantitative measurement of domain sizes

125Now we can set the frequency difference between the Bragg lasers as f = fc ≈1 kHz such that δ = 0 in Eq. 4.11. The Bragg pulse then couples the |{P}i and|{P + ~∆k}i states according to the matrix

B(ΩR ) =!
cos 2θ
−i sin 2θ,−i sin 2θ
cos 2θ

(8.15)

where for our parameters, θ = ΩR t ≈ π/7. Therefore, after the first Bragg pulse, wehave the state:!
ψ(r)B(ΩR )=
0!

cos 2θ ψ(r)−i sin 2θ ψ(r)(8.16)

Thus we have "copied" the wavefunction ψ(r) into the state(s) |{P + ~∆k}i.
Step 2: Arrange
If we turn off the Bragg beams and wait for time tA , then the diffracted cloud in themoving states |{P + ~∆k}i moves by vr tA . We represent the action of this wait
using the matrix:

A(tA ) =1 0
0 T̂tA!
(8.17)

where T̂ is the translation operator defined by T̂t f (x) = f (x − vr t). After the wait,tA , we have the state!
ψ(r)A(tA )B(ΩR )
=
0

126

Chapter 8!

cos 2θ ψ(r),−i sin 2θ ψ(r − vr tA )Critical Dynamics of Spontaneous Symmetry Breaking(8.18)

And thus we have achieved the aim of displacing one of the copies.
Step 3: Interfere
Finally, we interfere the displaced copies by applying a second Bragg pulse identicalto and in phase with the first pulse. This gives the final state:!
ψ(r)B(ΩR )A(tA )B(ΩR )
=
0

!


cos2 2θ ψ(r) − sin2 2θ ψ(r − vr tA )
, (8.19)−i sin 2θ cos 2θ [ψ(r) + ψ(r − vr tA )]In the region where the copies overlap, the final density of atoms in the |{P + ~∆k}istate depends on the relative phase of the overlapping domains in the two copies. If
we wait a further time-of-flight of tToF = 140 ms (i.e. apply A(tToF )) we obtain theinterference term ∝ [ψ(s) + ψ(s − vr tA )] in state |{P + ~∆k}i spatially separated
by vr tToF from the stationary cloud in state |{P}i. Therefore, we can count thefraction of atoms in this diffracted cloud, and using Eq. 8.14 and 8.13, we obtain:oN|{P+~∆k}i
1n
x
=
1+ 1−
Re [C1∞ (xx̂)] sin2 (θ),
N
2
L

(8.20)

where xx̂ = vr tA . Eq. 8.20 is the key result of this section, stating that we canmeasure the C1∞ correlation function (encoding the length scale d) by simply counting
the fraction of atoms diffracted by our homodyne interferometric scheme. We will
use this technique in our quantitative studies in sections 8.3 and 8.4, however, firstwe will outline one final technical point in the section below.

8.2

Quantitative measurement of domain sizes

127

Fig. 8.4.: Level diagram for the -1 Bragg diffraction order (red). The lasers are tuned such that astationary atom is resonant with the +1 order transition (shown in gray). This results in alarge detuning, δ, for the -1 order.

Aside: Bragg diffraction orders
Note that so far we have only considered transitions from {P} to {P + ~∆k} (the socalled +1 order). Using the same laser beams but swapping the roles of absorbtion
and stimulated emission, it is possible to couple {P} to {P − ~∆k} (the −1 order).
Since the lasers are resonant with the +1 transition, the −1 transition is detuned by
δ/2π = 2fc (see Fig. 8.4). To find the population of the −1 order, we can repeat theanalysis above with δ 6= 0 using the full matrix B(ΩR , δ) from Eq. 4.11 and including
the free evolution of the states3 in A(ΩR , δ). This gives the diffracted fraction in the−1 order as:

N|{P−~∆k}i
2Ω2
= 2R
N
δ



 
 

 
x
x ∞
2 δt
1 + cos δ t +1−
C1 (xx̂) sin(8.21)
vrL2q
where we have assumed Ω2R + δ 2 ≈ |δ| and that C1∞ is a real function (which
is true for any system with an even momentum distribution). We can draw two
conclusions from Eq. 8.21:
3

It is important to note that when the lasers are off (ΩR = 0) during the wait time tA , B(ΩR = 0, δ)
does not reduce to the idenitity for δ =6 0. Therefore, in the rotating frame, we must include theevolution of the states at relative frequency δ during the "arrange" step:






exp iδt2A t
0
0

 1
A(tA , δ) = .
A
0 T̂tA0
exp −iδt
2|{z}
B(ΩR =0,δ)|t→tA

128

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Fig. 8.5.: Relevant Bragg diffraction orders. The arrows show the cumulative population transfer
between the orders with the lines of decreasing thickness indicating relative populations
O(1), O[(ΩR t)2 ], and O[(ΩR t)4 ] respectively. Note that the homodyne interference circuit(highlighted in blue) is only affected to O[(ΩR t)4 ] by the presence of other Bragg orders.
The ToF image shows the result of an experimental run from section 8.3 in which we cool
the gas in tQ = 0.8 s, apply Bragg pulses separated by tA = 0.4 ms, and then wait 140 ms
ToF.

• The population in the −1 order cloud oscillates with tA = x/vr at frequency δ.
• The population in the −1 order is as significant as the +1 order (ignoringosillations). This can be seen by taking the envelope of the osillations of
N|{P−~∆k}i /N , and expanding sin(θ) for small angles, giving N|{P±~∆k}i /N ∼(ΩR t)2 . Qualitatively, the symmetry in the amplitude of the ±1 orders is a resultof the broad Bragg resolution kernel 2π × ∆fres  2δ.
The significance of the −1 order brings into question the validity of our treatment of{P} and {P + ~∆k} as a closed two-level system coupled by B. However, following
the schematic diagram in Fig. 8.5, we see that the −1 order only modifies our
expression in Eq. 8.20 by terms O(Ω4R t4 ), provided that we count all the −1, 0 and+1 orders in N . Since (ΩR t)2 ≈ 0.2  1, we ignore all these higher order terms(4-photon processes and depletion of the zero-order cloud) and work to just firstorder with Eq. 8.20.
This concludes our technical discussion of the derivation and validity of Eq. 8.20
which is key to the experimental work discussed in the remainder of this chapter.

8.2.3 Experimental measurement of C1
In this section, we introduce the various forms for C1 which we observe in theexperiment using our homodyne interference scheme. Fig. 8.6 shows C1 (xx̂)

8.2

Quantitative measurement of domain sizes

129Fig. 8.6.: Correlation function C1 (x) = (1 − x/L)C1∞ (x) measured in equilibrium (blue) and after aquench (red) for, respectively, two different T /Tc values and two different quench times(four experimental runs per point; error bars indicate standard deviation in mean). Inset: 1D
calculation of G1 for a fragmented BEC containing D = 10 (red) and 20 (light red) domainsof random sizes and phases. The solid lines correspond to C1∞ = exp(−xD/L)

functions measured in equilibrium (blue) and after a quench (red), and we compare
three experimental cases:
Pure BEC In an essentially pure equilibrium BEC (prepared slowly, as for Fig. 8.2(b)),
C1∞ (xx̂) = 1 and C1 (xx̂) is simply given by the linear overlap function 1 − x/L(dark blue line).
Partial BEC Cooling slowly in equilibrium, but stopping at T /Tc ≈ 0.7, gives a C1 with afast initial decay, corresponding to the significant thermal fraction. Importantly,
the coherence still spans the whole system, with the slope of the long-ranged
part of C1 giving the condensed fraction (light blue line is a guide to the eye).
Quenched clouds By comparison, the C1 functions for quenched clouds have no equilibrium
interpretation. For the curves shown, T /Tc ≈ 0.2, corresponding to a phasespace density > 25, and yet coherence extends over only a small fraction of L.
Furthermore, in Fig. 8.6, we immediately see agreement with the qualitative
picture that faster quenches lead to C1 functions with a shorter decay length.
The data for the quenched clouds are fitted well by C1∞ ∝ exp(−x/`) (red lines),
which provides a simple and robust way to extract the coherence length. This
exponential form is further supported by a 1D calculation shown in the inset of
Fig. 8.6. Here we generate a wavefunction with a fixed number of domains D,
randomly positioning the domain walls and assigning each domain a random phase.
Averaging over many realizations, we obtain C1∞ (x) that is fitted very well by an
exponential with ` = L/D = d. (In our 3D experiments the total number of domains
is ∼ D3 and C1 (x) is effectively averaged over ∼ D2 1D distributions.).

130

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Since the exponential fit gives us direct access to ` = d, we can now move on to a
fully quantitative study of the KZ theory.

8.3 Direct observation of Kibble-Zurek freeze-out
We are hoping to observe the scaling of d with τQ , but as we will see, in our
experiment we also directly observe the freeze-out phenomenon central to the KZ
hypothesis. For the KZ scaling law of Eq. 8.6, a crucial assumption is that ξ isessentially frozen within t̂± of tc . While intuitively appealing, this assumption is inprinciple only approximate, and the dynamics of the system coarsening (i.e., merging
of the domains) at times t > tc is still a subject of theoretical work [169].
In our experiment we use two different quench protocols outlined in Fig. 8.7. We
start by describing the first quench protocol, QP1, where we vary only the total time
tQ over which we ramp down the trap depth.

8.3.1 Quench protocol 1: Basic KZ scaling and its breakdown
In QP1 we restrict tQ to values between 0.2 s and 3.5 s, for which we observe thatthe cooling curves are self-similar4 . We find that we always cross Tc = (70 ± 10) nK
at timetc = (0.72 ± 0.05) tQ ,(8.22)

and always have the same atom number (within ±20%) at the end of cooling. The
self-similarity of the measured cooling trajectories and the essentially constantevaporation efficiency indicate that for this range of tQ values the system is always
sufficiently thermalised for the temperature (as determined from the thermal wings
in ToF) to be well defined during the quench5 . Furthermore, from the definition of ,
we can linearise the cooling trajectory near tc and calculate τQ from the derivative
as follows:

d
1
=− ,
dt t≈tc
τQ(8.23)

4Note that we empirically find that for tQ < 0.2 s the evaporation is less efficient and the coolingtrajectories are no longer self-similar
5
Near Tc the mean free path for classical elastic collisions is about four times larger than the sizeof the box. Hence, while the evaporation takes place at the box walls, we can also safely assume that
T is uniform across the sample.

8.3

Direct observation of Kibble-Zurek freeze-out

131Fig. 8.7.: Quench protocols. (a) In QP1, the laser power is ramped down linearly over time tQ to
reduce the trap depth. In QP2 the laser power is ramped faster (at a slope characteristic of
tQ = 0.2 s) after tk & tc . (b) The self-similar QP1 cooling trajectories are shown in blue fortotal cooling time tQ = 0.2 s (upper panel) and 3.2 s (lower panel) (four experimental runs
per point; error bars indicate standard deviation in mean). We use polynomial fits to thedata (such as shown by the solid lines) to deduce tc and τQ . QP2 is shown in the lower
panel by the orange points, with the kink at tk = 0.85 tQ .

and due to the self similarity, we find that to good approximation6

τQ ∝ tQ .

(8.24)

It is worth briefly commenting on this result in light of the non-linearity of the coolingtrajectories in Fig. 8.7:
The definition of τQ for a non-linear quench
The KZ theory is built upon the assumption of a linear quench for which the derivativein Eq. 8.23 is constant and τQ is unambiguously defined. Experimentally, we do
not perform linear temperature quenches and for a quench which is not truly linear,
there is some uncertainty over exactly where the derivative should be evaluated(hence we write t ≈ tc in Eq. 8.23). According to the KZ hypothesis, the domain
size is determined by the system properties at the freeze-out point. Therefore, it ismost reasonable to define τQ from the derivative at the point, tc − t̂+ , given by the
˙solution to the (non-linearised) equation |ξ/ξ|= 1/τ .
In Appendix D, we use a simple numerical model for our non-linear quench toinvestigate whether evaluating τQ from the slope at the freeze time rather than at
the critical time affects the simple proportionality in 8.24 (see Fig. 8.8). The model
confirms two important results:
6

When the derivative is evaluated at exactly at t = tc , we find a constant of proportionality
τQ = (0.41 ± 0.02) tQ .

132

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Fig. 8.8.: Technical details of our non-linear quench trajectory. We parameterise the self-similar shapeof the QP1 cooling curves by the quadratic curve shown in blue. In a numerical model inAppendix D we calculate the freeze-out conditions (blue shaded region) and illustrate thetQ = 1 s case above (assuming F-Model critical exponents). To link this non-linear curve tothe linearised KZ theory, we can define τQ as the inverse slope of (t) evaluated at eithert = tc (gray) or t = tc − t̂ (red). See text for a discussion of these two scenarios.

• The KZ scaling law for the domain size at the critical point is only recoved
exactly if we use τQ defined by the slope of the quench trajectory at tc − t̂+ .
• There is a small non-linearity between the τQ evaluated from the slope at thefreeze time and at the critical time. A numerical power-law fit in both the MF
and F Model for characteristc parameters in our experiment gives τQ |t=tc −t̂+ ≈1.1(τQ |t=tc )0.94 , i.e. τQ |t=tc −t̂+ ∝ t0.94
Q . This non-linearity means that we must
be careful when replacing τQ with tQ when evaluating any exponents.
In summary, the numerical analysis tells us that if we were to ignore all complications
arising from our non-linear quench and evaluate the derivative in Eq. 8.24 at t = tc ,then we would recover a value of the exponent b which is 6% smaller than thetrue value. Since this correction is small (considerably smaller than the difference
between the values of b in the MF and the F-Model), we will proceed with theassumption of a linear quench (i.e. that the proportionality in Eq. 8.24 always holds).
Only once we have experimentally evaluated the exponent b under this assumption
will we attempt to account for any corrections arising from the non-linearity of ourcooling curve.
With this assumption in mind, we now inspect the size of the domains formed at theend of a QP1 quench:
QP1 Results
In Fig. 8.9(a) we plot `(= d) vs. tQ (∝ τQ ), for experiments using QP1. For tQ ≤ 1 s
we observe a slow power-law growth of ` with tQ , in good agreement with the

8.3

Direct observation of Kibble-Zurek freeze-out

133

Fig. 8.9.: Kibble-Zurek scaling and the freeze-out hypothesis. (a) Coherence length ` as a functionof tQ . Blue points correspond to QP1, error bars indicate fitting errors on `. The shaded
blue area shows power-law fits with 1/4 < b < 1/3 to the data with tQ ≤ tbr
Q = 1 s. The
horizontal dotted line indicates our instrumental resolution (arising from the finite Bragg
pulse time). (b) Coherence length ` measured following QP2, as a function of tk /tQ , fortQ = 3.2 s (orange), 0.7 s (green), and 0.3 s (purple) (four experimental runs per point; error
bars indicate standard deviation in mean). The shaded areas correspond to the essentiallyconstant ` (and its uncertainty) in the freeze-out period tk − tc < t̂. (For tQ < tbr
Q the systemnever unfreezes). The (average) ` values within these plateaux are shown in their respective
colours as diamonds in panel (b).

expected KZ scaling. However, for tQ > tbr
Q ≈ 1 s this scaling breaks down and `grows faster, quickly approaching the system size.
For quenches longer than tbr
Q , we suggest that the breakdown of the KZ scaling isdue to the system unfreezing and healing significantly during the quench and before
it is observed. We note from Eq. 8.22 that the time between crossing Tc and the end
νz/(1+νz)of cooling is tQ − tc ≈ 0.28 tQ ∝ tQ , while the KZ freeze-out time is t̂− ∝ tQ
(assuming Eq. 8.24). Due to the different scaling of these two times with tQ , it isalways possible to perform slow enough quenches such that tQ − tc exceeds t̂− forνz > 0 (see Fig. 8.10). Solving for the intersection t̂− ≈ 0.28 tbr
Q in Fig. 8.10, we
obtain an estimate of the absolute value of t̂− without a priori knowledge of τ0 or
λ0 .

t̂− ≈ 0.28 tbr
Q

tQ
tbr
Q

!νz/(1+νz).

(8.25)

Since νz = 1 holds at bothq the mean field and F Model level, we can say that for
both models t̂− ≈ 0.28 tQ tbr
Q . In the next section we will have an opportunity todirectly measure t̂− for comparison with this expression.

134

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Fig. 8.10.: Proposed mechanism for the breakdown of the KZ scaling in Fig. 8.9(b). We plot tQ − tc
and t̂− as a function of tQ and suggest that beyond the intersection point, tbr
Q (orange
shaded region), the system unfreezes and begins to heal during the QP1 quench. To
explore this region we must used QP2.

8.3.2 Quench Protocol 2: Directly observing the KZ freeze-out
Our second quench protocol consists of two steps as shown in Fig. 8.7. This protocol
allows us to directly observe the freezing of the system and to fully understand thebreakdown of the scaling law at tbr
Q in the QP1 results. Once this breakdown isunderstood, we can also use QP2 measurements to extend the range of tQ overwhich we can fit a power law. Using this extended range we will experimentally
distingush the MF and F Model critical exponents.
In QP2 we initially follow the QP1 trajectory for a given tQ , but then at a variable"kink" time tk & tc we accelerate the cooling. The last part of the trajectory always
corresponds to the final portion of our fastest, 0.2 s cooling trajectory. This way,even for tQ > tbr
Q we can complete the cooling and measure C1 before the system
has time to unfreeze.
In Fig. 8.9(b), the orange points show the QP2 measurements of ` for tQ = 3.2 s
and various values of the kink position tk /tQ . These data reveal two remarkablefacts:• For a broad range of tk values, ` is indeed constant (within errors), and thewidth of this plateau agrees with our estimate from Eq. 8.25, t̂− ≈ 0.5 s fortQ = 3.2 s, indicated by the horizontal arrow.
• The value of ` within the plateaued region falls in line with the KZ scaling law
in Fig. 8.9(a).
We also show analogous QP2 measurements for tQ = 0.7 s (green) and 0.3 s(purple); in these cases tQ < tbr
Q , so t̂− is longer than tQ − tc , the system never

8.3

Direct observation of Kibble-Zurek freeze-out

135

{ HΜmL

2.0

1.0

0.1

0.4
ΤQ HsL

1.6

Fig. 8.11.: Critical exponents of the interacting BEC transition. Orange circles and diamonds show
` values obtained using QP2, as in Fig. 8.9(b); the diamonds show the same three data
points as in Fig. 8.9(a). Blue circles show the same QP1 data with tQ ≤ 1 s as in Fig. 8.9(a).
The horizontal error bars indicate the systematic uncertainty in τQ based on evaluationsys.
of −()˙ −1 over the range t = tc − t̂+ → tc . We obtain b = 0.33±0.02±0.03 stat. (solid line), inagreement with the F model prediction b ≈ 1/3, corresponding to ν ≈ 2/3 and z = 3/2,
and excluding the mean-field value b = 1/4.

unfreezes, and thus the acceleration of the cooling has no effect on `. These
plateaux provide direct support for the KZ freeze-out hypothesis.

8.4 Kibble-Zurek scaling and universal criticalexponents
To accurately determine the KZ exponent b, we have made extensive measurements
following QP2, extracting ` from the plateaued regions of width min[t̂− , tQ − tc ], asin Fig. 8.9(b). In Fig. 8.11 we combine these data with the QP1 measurements
for tQ ≤ tbr
Q , and plot ` versus τQ . To construct this plot we must return to the
problem raised in section 8.3 of exactly where on the non-linear cooling trajectory
we should evaluate τQ = −()˙ −1 . There are two routes to proceed which we deal
with separately below:

Evaluate τQ at tc
The simplest analysis which ignores the non-linearity of the cooling trajectory is
to accept the proportionality τQ ∝ tQ and find b as the slope of a plot of log(`) vs.
log(tQ ). This gives b = 0.31 ± 0.03, which is in agreement with the F Model (b = 1/3)
and excludes the MF prediction (b = 1/4). Furthermore, note that this approachgives a lower bound on the value of b, and the numerical simulation in Appendix D
suggests that we should include a 6% systematic correction to this value (becauseτQ ∝ t0.94
Q ). Applying this correction (which is smaller than the statistical error on

136

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

b) hints at a value b = 0.33 which is in even better agreement with the F Model
prediction.
Evaluate τQ at tc − t̂+
The study above gave considerable support to the beyond mean-field F Model
exponents, however rather than accessing b directly, it only provided a tight lower
bound to b and relied on numerical simulations to indicate the true value of b. Ideally,
we would not rely on any simulations, but instead estimate t̂+ directly from theexperiment, and evaluate τQ at tc − t̂+ . Measuring t̂+ is experimentally challenging,
however we proceed with the following assumptions:• First we assume from Eq. 8.5 that t̂+ /t̂− = (X+ /X− )z/(1+νz) ≈ 0.4, where weassume F Model results for ν, z and X+ /X− in the hope of self-consistently
measuring b.
q• Next we assume that the value of t̂− = 0.28 tQ tbr
Q suggested in section 8.3.1
is a good estimate of t̂− from which we can evaluate t̂+ = 0.4 t̂− .
These assumptions are difficult to justify rigorously. In particular, the second assumption requires that the microscopic time and length scales τ0 and λ0 do notchange when a BEC is present. Nevertheless, we can follow this protocol withtbr
Q = 1 s to achieve an estimate b = 0.35 ± 0.03. This is again consistent with the FModel, and gives us a way to assess the systematic error in our value of b.
One could even combine the values of b for the two methods above to give b =sys.
0.33±0.02±0.03 stat. . It is reassuring to note that the systematic variation on b as we analyse
the data in different ways is not larger than the statistical uncertainty on any given
analysis, and is also significantly smaller than the difference between the F Modeland MF theory.

8.4.1 Dynamical critical exponent, zHaving observed excellent agreement with the KZ theory, we now discuss the implications of our measurements for the critical exponents of the interacting BEC
phase transition, which is in the same universality class as the λ-transition of 4 He.
While ν ≈ 0.67 has been measured in both liquid helium (see [170]) and atomic
gases [171], the dynamical exponent z has, to our knowledge, never been measured
before (see [163, 172]). Using the well-established ν = 0.67 and Eq. 8.6, we obtainsys.
z = 1.5±0.2±0.3 stat. . In contrast, MF theory does not provide a self-consistent interpresys.
tation of our results, since fixing ν = 1/2 yields an inconsistent z = 1.0±0.2±0.3 stat. .
Interestingly, if we instead fix νz = 1, which holds at both MF and F-model level,

8.4

Kibble-Zurek scaling and universal critical exponents

137

sys.
from Eq. 8.6 we obtain a slightly more precise z = 1.5±0.1±0.1 stat. and also recoversys.
ν = 0.67±0.04±0.06 stat. .

8.5 Conclusion
This chapter has presented an experiment which is the culmination of all the workin the second half of this thesis. We developed a versatile trapping technique inchapter 5, applied this to make a uniform system in chapter 6, introduced bragg
spectroscopy to benchmark the equilibrium ground state properties in chapter 7,
and now finally applied this in a dynamical study of the KZ freeze-out mechanismand measurement of the dynamical critical exponent for our universality class.
Note that our uniform trapping potential was essential for the clean observation of
the KZ mechanism presented here. In an inhomogeneous potential U (r), the criticaltemperature is a function of position. This means that  is position dependent and as
we decrease T , the critical point (x, t) = 0 is reached at different points at different
times. Condensation happens first at the point(s) r0 = arg min[U (r)], and then a"transition front" radiates out from this(these) point(s) at speed:

vF =

∂/∂t
∂/∂x

(8.26)

Independent phase domains will only form if this front travels faster than the speed vs
at which the choice of symmetry breaking phase can be communicated7 . Whereas ina uniform system, vF diverges (because ∂/∂x = 0 - i.e. the entire system becomes
critical at once), in an inhomogeneous system, we need to take the competition
between vF and vs into account. This means that we cannot directly observe theKZ freeze-out (only local regions of the system are frozen at any time), and the(local) size of the domains and is generally not given by any simple power law.
Instead, a harmonically trapped system is described the so-called inhomogeneous
KZ theory [165], which is only able to predict approximate scaling laws with b ≥ 1
holding near r0 . Hence, our uniform system was essential for the observation of thefundamental mechanism and consequences of KZ theory that we have described
here.

7

ˆ given by the local frozen quantities
Intuitively, vs can be bounded by the velocity scale vs < ξ/τ̂ξˆ and τ̂

138

Chapter 8

Critical Dynamics of Spontaneous Symmetry Breaking

Conclusions and Outlook

9

The experiments in Part I and Part II of this thesis were carried out on two separate
machines. The first machine is able to tune inter-atomic interactions and allowed us
to study dynamical effects of superheating and three-body loss in regimes of very
weak and very strong interactions respectively. The second machine is specialisedfor producing novel optical geometries, and specifically we created the worlds firstatomic Bose gas in a uniform potential. The chapters presented here reviewed aseries of equilibrium and dynamical experiments performed so-far in this uniformtrap.
In the future, we plan to combine the two key features of both machines in a third machine which will allow us to tune interactions and geometry on the same experiment.
Work has already started on this machine, and several proof-of-principle experiments
have been carried out on our existing machines to lay the ground-work for when thethird machine is fully operational. The uniform system in chapter 6 offers such arich range of experimental paths that our investigations in the immediate future willconcentrate on this simple geometry rather than expanding to more elaborate trap
shapes. There are four experiments which we envisage for the near future, which
are described in the sections below. The first two are extensions to chapters 7 and
8, and the last two involve small additions to our uniform trap geometry.

9.1 Thermodynamics with tuneable interactions
The first proposed experiment continues our strand of thermodynamic experimentsfrom chapter 7. In particular, we first wish to investigate the effect of interactions on
the the critical temperature (Tc ) for condensation.
In a harmonically trapped gas, Tc is shifted relative to the ideal gas value by amean-field effect arising from the inhomogeneity of the trap. It was recently possible
to also measure a more interesting Tc -shift arising from interaction-driven criticalcorrelations on top of this much larger mean-field shift [41] (see Eq. 2.11). Further
accurate measurement is not possible in a harmonic trap because the large meanfield effect dominates over any interesting physics. In a homogeneous gas, there
is no mean-field Tc -shift, and any Tc -shifts due to critical correlations become thehighest order effects [173]. This means that experiments on our uniform system

139

will give us an immediate handle on the fundamental critical phenomena driving
the beyond-mean field Tc shift, which has been the focus of theoretical debates formany years (see [26] for a historical summary).

9.2 Kibble-Zurek with tuneable interactions
In chapter 8, we observed behaviour of a thermally quenched Bose gas consistent
with the beyond mean-field critical exponents of the F Model. The reason for the
failure of mean-field is that fluctuations in the order parameter become as large as
the value of order parameter near the critical point. It is possible to derive a criterion(the so-called Ginzburg criterion) for the point at which critical fluctuations cause
the breakdown of mean-field theory as [171]:ξ > ξG = √λ2c128π 2 a(9.1)

where a is the s-wave scattering length. In our experiment in chapter 8, ξG ≈ 0.8,
and since the freeze-out values of ` are consistently higher than this value, ourobservation of beyond mean-field exponents is consistent with the Ginzburg criterion.
When we tune the interaction strength, a, in our new machine we will be able toincrease ξG and repeat our Kibble-Zurek measurements in this weakly interacting
gas to try to measure mean-field exponents in a regime where ξ < ξG .
Another interesting future study could focus on the dynamics of domain coarsening
after unfreezing on the ordered side, or on the variance and distribution of domain
sizes formed after the quench.

9.3 Interferometry
In Fig. 6.11 we saw how simple it is to apply our digital holographic methods tocreate trapped-atom interferometers. Atom interferometers can be used to detect
any external stimulus (e.g. a magnetic field gradient) which causes the phase of oneof the interfering clouds to evolve at a different rate to the other. Interfering two BECs
in uniform traps offers the following fundamental advantages over interferometersconstructed using locally harmonic potentials:
1. Our trap is dark, and therefore there is no dephasing between the trapped
condensates arising from a differential AC Stark effect caused by small trap
asymmetries.

140

Chapter 9Conclusions and Outlook

Fig. 9.1.: Trapped atom interferometry using our uniform system (preliminary). (a) In situ image ofa single condensate split into two halves using the three-wall potential illustrated in thegreen profile. (b) Typical ToF images showing interference fringes when the condensates
have been held in their separate halves for 1, 20 and 500 ms. We mark the locations of thefringe peaks after 1 ms hold time with vertical lines. We obtain the same pattern with highprobability after 20 ms hold time, but the fringe phase is randomised after 500 ms. (c) We
quantitatively extract the phase and amplitude of the fringes using a fit (red dotted line) toa cut through the fringes in the x direction (blue line). (d) Interferometric measurement of
decoherence time. We hold the gas in the split trap for 1, 20 or 500 ms and then observe
the interference in ToF. Repeating this many times, we can extract the distribution of thephase, φ, and amplitude, A, of the fringes and plot them on a polar plot. We find that for
hold times greater than ∼ 100 ms, the phase is fully randomised.

2. The peak atom density in uniform traps is low compared to a cloud of the same
number of atoms in locally harmonic traps. This allows us to put enough atoms
in our uniform interferometer to achieve a good signal-to-noise ratio, while
maintaining a low density such that dephasing from differential mean-field
shifts significantly reduced. Mean-field dephasing can be further reduced byturning off interatomic interactions.
Dephasing from the AC Stark shift and the mean-field shift can severely limit the
integration time of traditional trapped-atom interferometers [174, 175]. To indicate
the advantages of our homogeneous arrangement, we carried out the following
proof-of-principle experiment:
We create a uniform BEC with a small central wall running up the middle of the trap
(see Fig 9.1(a)). Initially, this wall is sufficiently low with respect to the chemical
potential of the BEC to allow a single, connected BEC to form in the entire trap.
We then ramp up the green laser power to raise the wall, and split the BEC in two.

9.3

Interferometry

141

Fig. 9.2.: Transport phenomena in an ultracold Bose gas (preliminary). (a) A cartoon of the opticalgeometry for creating a narrow channel for investigation of quantum transport properties.
The central green wall is split to create a channel which can be as narrow as the diffraction
limit. The atoms are shown in red, and for clarity, we suppress the appearance of the tube
beam. (b) An experimental image using the apparatus of chapter 6 to load atoms into thispotential.

Since the two BECs were formed from a single common ancestor, they are initiallyphase-locked with each other. Therefore if we immediately release the green trap,
we should always observe the same fringe pattern when they interfere. We candetect any dephasing of the two condensates after a hold time t as a randomisationof the fringe pattern in repeated realisations at the same hold time. Figure 9.1(d)shows the distribution of phases and amplitudes of the interference fringe pattern
after t = 1, 20 or 500 ms. We can see that the dephasing time is at least 20 ms in ourapparatus, and further experiments gave evidence that the dephasing time is longer
than ∼ 100 ms. This is similar to state-of-the-art trapped atom interferometers using
squeezed states or non-interacting gases which achieve dephasing times of a few
hundred milliseconds [49, 174].

9.3.1 Transport and thermomechanical phenomena
There has been a recent growth in interest in transport [76, 77, 176–178] and thermomechanical [179, 180] effects in ultracold gases. We note that our holographic
methods make it very simple to construct a geometry in which two uniform reservoirs
are connected by a thin channel (see Fig. 9.2). Using a magnetic field gradient, we
can load all the atoms into one reservoir, and then watch the atoms diffuse throughthe channel when we turn off the gradient. For our bosonic gas, we should see twodifferent transport effects: the thermal atoms should move ballistically through thechannel, while the BEC should show superfluid properties. We suggest that it may
be possible to tune the parameters of the channel (e.g. by changing the channel
width or inserting a tunnelling barrier) such that the superflow is much faster than theballistic themal diffusion. This will allow us to create a "superleak" which only allows

142

Chapter 9Conclusions and Outlooksuperfluid through. We will use this to explore thermomechanical effects in our Bosegas [179–181] similar to those observed in superfluid helium experiments [182].
Other transport effects could be observed by shaking our optical box (e.g. using
an oscillating magnetic field graident) and observing the superfluid response using
bragg spectroscopy.

9.3

Interferometry

143Appendices

145

Experimental Sequence Overview

A

During this thesis, we have not dwelt on the preparatory stages of laser- andevaporative-cooling required to reach condensation. Instead, we have concentrated on experiments performed when the cloud is already near degeneracy or
deeply condensed. It is somewhat humbling to note that all the techniques andexperiments presented in this thesis take place only in the final few moments of along experimental sequence. The bulk of the sequence was built up by previous
students and is summarised in their theses: [25, 26, 33] (Part I machine) and [39, 40](Part II machine). For specific details of the early experimental procedure, the reader
is directed towards these theses. Here we will just provide a brief overview of the
key concepts of the simpler Part II machine in a manageable two-page cartoon. It is
hoped that this overview will be a useful first roadmap for any future students tasked
with extending upon this work.

A.1 Rubidium-87 D2 LinesThroughout the cartoons on the following pages, we will illustrate the relevant lasertransitions on a skeleton of the 87 Rb D2 line. For clarity, labels are omitted from thisskeleton, and a labelled level spectrum is included in Fig. A.1 for reference.

Fig. A.1.: Hyperfine structure of the 87 Rb D2 lines. Data (reported in MHz) reproduced from [183].

147

20mm

7 G/cm

1. Magneto-optical-trap

68MHz

2. Optical Molasses

mF = -1
0
+1

B

x=0

B

B

x=0
B
x

For atoms moving the +x direction, the beam propagating in the –x direction is blueshifted towards resonance. This produces a velocity-dependent damping force whichcools the trapped atomic cloud.

For a stationary atom at x > 0, the |0,0 → |1,-1 (σ-) transition is closest to resonance.
We can arrange a restoring force by choosing circularly polarized light such thatphotons are preferentially scattered from the beam propagating in the –x direction.
This restoring force provides a position-dependent trapping potential.

F=0

F=1

v

The diagrams below illustrate the detuning of the lasers (red) from the atomic levels(black) for a stationary and moving atom:`

1. The static detuning, δ0, of the laser from the zero-field F=0 → F’=1 transition.
2. The Doppler-shift, k∙v, due to the atoms’ velocity.
3. The Zeeman-shift, gFmFμBB/ħ, of the mF=+1,0,-1 hyperfine levels.

1. Magneto-optical-trap: A combination of circularly polarized laser beams alongeach coordinate direction and a quadrupole magnetic field provide a position- andvelocity-dependent trapping force. In practice, we use lasers δ0=25MHz detuned
from the F=2 → F’=3 transition, but the principle is best understood by considering asystem with a simpler F=0 → F’=1 transition. The detuning of the laser light from thistransition has three contributions:

Laser Cooling: Capturing and cooling a 87Rb atoms from a background vapour.

25MHz

Energy5G

4. Magnetic Trap

100 G/cm

2. Optical Molasses: The quadrupole field is turned
off and the cloud is allowed to cool in the presenceof only the velocity-dependent damping force.
Without the magnetic field, additional, ‘sub-Doppler’cooling mechanisms (Sisyphus cooling) cansignificantly reduce the temperature of the cloud.

When cooling on the F=2 → F’=3 transition,
spontaneous emission to the F=1 ground state ispossible via higher order transitions. Atoms in the
F=1 level are brought back into F=2 using re-pumplight resonant with the F=1 → F’=2 transition.

With appropriately polarized beams in all 6
directions, the atoms are trapped and cooled in a
3D.

Evaporative cooling is used in later
steps to beat the recoil limit

2. Recoil limit: Absorption/emission ofphotons only affects the atom’s
momentum in units of ħk. This
discretisation gives a ‘recoil limit’ for
laser cooling kBT= (ħk)2/(2m) (~200nK
for 87Rb).

1. Doppler cooling limit: Spontaneousemission at a rate given by the inverse
excited state linewidth, Γ-1, heats at a
rate (ħk)2/(2mΓ). Balancing this with thelaser cooling damping rate (-F∙v ~ v∙v)
gives an equilibrium temperature of
kBT= ħΓ/2 (~150μK for 87Rb).

Laser cooling limits:

2. Magnetic trapping: The low-field-seeking |2,2 state is confined by a strong
quadrupole field.

1. Optical pumping: Puts atoms in the magnetically trappable |2,2 state by driving
the σ+ |F = 2,mF → |F’ = 2,mF+1 transition using circularly polarized lightpropagating along a quantization axis defined by a single coil. Atoms pumped into
the excited states spontaneously decay back to the ground states to be pumped
again until the population builds up in the dark |F = 2,mF = 2 state. Any atoms which
decay into the F = 1 manifold are re-pumped back into the cycling transition by repump lasers resonant with |F = 1 → |F’ = 2.

Magnetic Trapping: Confining the atoms in a strong quadrupole magnetic field
which splits the hyperfine levels ready for radio-frequency evaporation.

3. Optical Pumping

x=0

B

Atoms transferred toun-trapped statesB

xProfile before
thermalisation

Lower temperature
after thermalisation

Sweeping the radiation from 15 → 5 MHz
in 2 s, cools the cloud to 90 μK.

|2,-2

|2,-1

|2,0

|2,1

|2,2

`

Zeeman splitting of the F=2 manifold in
the quadrupole field allows us to address
atoms in different regions of the trap bytuning the frequency of radio-wave
radiation flooding the apparatus. Bytargeting the highest energy atoms (atlarge radius from the magnetic zero) andtransferring them from the trapped |2,2
state into un-trapped states, we allow theremaining atoms to re-thermalise at alower temperature. This is illustrated
schematically below:

Removing the highest energy atoms toallow the remaining atoms to rethermalise at a lower temperature.

Radio-Frequency Evaporation:

100 G/cm

5 W 15-5 MHz radiation

5. Radio-frequency Evaporation

Trappedatom density

EnergyMagnetic, optical + gravitational potential

time

Optical Evaporation: The ODT is ramped
down from 5W exponentially over 7s. Thisevaporatively cools the gas, achieving
condensation at 0.16W (2 μK trap depth,
90Hz radial frequency)

In the ODT, the atoms are confined along
the beam by the residual 7.4 G/cm
quadrupole field. Off axis this fieldprovides an approximately harmonic
potential with trapping frequency 30Hz.

z

Majorana spin-flips occur at thequadrupole magnetic zero where theatom’s spin (precessing at a vanishing
Larmor frequency) cannot adiabaticallyfollow the change in magnetic field
direction. These flips eject low energy
atoms dwelling near the magnetic zero.
To cool below the μK level, we transfer
the atoms into a single-beam (harmonic)
optical dipole trap (ODT) far red-detuned(1070nm) from all resonances andpositioned 60 μm (1 optical waist) below
the magnetic zero. Reducing thequadrupole field ‘pours’ the atoms into
the ODT:

Hybrid-trap & Optical Evaporation:

6. Hybrid-trap & Optical
Evaporation
15.276 G/cm

8. Uniform trap evaporation

See chapter 7

Either by Bragg-spectroscopy (pure BEC)
or time-of-flight (anti-gravity field lefton)

Query cloud properties via
momentum distribution

9. Bragg spectroscopy & Time-ofFlight

Trap loading: 532 nm shaped uniform
trap beams are ramped on as the ODT isramped down. To release atoms into thebox. A uniform bias field is added to thequadrupole field to create a uniform anti-

17.5 G

5G

Circularly polarized light drives the σ+|2, 2 → |3, 3 transition in the presenceof a 5G guide field. A camera captures
the shadow in the profile of the atoms
as they absorb photons.

Absorption Imaging

10. Absorption Imaging

See chapter 6

Optical evaporation: The green laser power isramped down exponentially over 2s toachieve condensation

gravity magnetic field gradient 15.276 G/cm.

Uniform-trap Transfer atoms into and condense in novel trap geometry

7. Uniform trap loadingRaster Holography

B

In Fig 5.7(b), we saw the Gerchberg-Saxton algorithm for generating raster holograms. The Gerchberg-Saxton algorithm calculates a phase hologram by linking
the known moduli in the SLM and trapping planes through simulation of the lightpropagation back and fourth between these planes. After each propagation, we
manually impose the modulus constraints while leaving the phase to converge on therequired solution. The convergence of this algorithm is problematic (see Fig. 5.8),
and in this section, we describe improvements to this computational method, buildingon the work of Pasienski and DeMarco [103]. In addition, we present experimental
results which indicate that the traps produced by our method are of sufficient quality
to integrate into cold atom experiments.

B.1 MRAF: Mixed-region amplitude freedom
To combat the fact that the Gerchberg-Saxton algorithm generally fails to converge
correctly, Pasienski and DeMarco devised the MRAF algorithm. The idea of MRAF
is to enhance convergence of the Gerchberg-Saxton algorithm in one region of
the trapping plane by giving up control of the remaining regions [103–106]. Moreprecisely, MRAF defines a "drawing" D in the trapping plane containing all the points
in which the desired potential is non-zero, and an additional "canvas" region, C,of zero potential around the drawing (see Fig. 5.8). The modulus constraint isthen manually imposed only inside C and D at each iterative step, thus reducing
the number of constraints the algorithm aims to satisfy. It is hoped that with theadditional freedom of amplitude outside C the algorithm will correctly converge onto
the desired diffraction pattern inside C and D. Meanwhile, a complex pattern offringes and speckles develop in the unconstrained region outside C,
In the original MRAF paper, region C is limited to a tight border around D (witha typical width of ≈ 1% of the trapping region length scales). Although C is afeatureless region of zero potential, it is an integral part of the trap since it defines
the zero to which all the pixels in D are referenced. For practical purposes, sucha small canvas can be a severe limitation because the fringes and speckles in theunconstrained region could greatly complicate the process of loading atoms into
the desired region, D. Therefore, our method aims to expand C to approach its
theoretical limit, which is set by Nyquist’s theorem to be 25% of the area of the151simulated trapping plane1 . In this large canvas regime, MRAF suffers the same
fate as the Gerchberg-Saxton algorithm: convergence stagnates to produce poor
solutions even after many iterations.

B.2 Optical vortices
The reason for the failure of the MRAF algorithm in the extended canvas regime isthe formation of large populations of optical vortices during the early iterations (approximately 1 vortex for every 10 pixels in Fig. B.2). Optical vortices are topologicalfeatures of the light field corresponding to a phase winding around a point of zerointensity. Large vortex populations are problematic because individual vortices are
difficult to eliminate in subsequent iterations of the algorithm. Unwinding a vortex
requires global disruption of the hologram, which cannot be achieved in late iterations of Gerchberg-Saxton-type algorithms which monotonically reduce the meansquare deviation from the target [184, 185]. Therefore any erroneous vortices are“frozen in” early on, and the algorithm gets stuck in a local optimum corresponding
to a particular vortex distribution.

B.2.1 Offset-MRAF
One possibility for handling vortices is by pairwise creation and annihilation, which
requires only local operations; computational algorithms which encourage such
pairwise operations are given in [186]. However, here we consider a much simpler
solution. Our method simply offsets the trapping pattern inside C and D by a uniform
intensity, |∆|2 , to remove all points of zero light intensity. This redefines the zeroof the potential but does not change the physics of the trap. The aim is to ensure
the phase of all points inside C and D is well defined for all iterations. If a pointof zero intensity is generated, then the phase becomes ambiguous, and we risk
forming a vortex in later iterations (see Fig. B.1). To ensure |∆|2 is sufficiently large
to prevent vortex formation, we perform a single iteration of the algorithm and countthe number of vortices produced. This first iteration starts from a parabolic phase
hologram which defocuses the light into a vortex-free patch of characteristic size
set by C. We then gradually increase |∆|2 until a single iteration from this starting
point does not introduce any vortices. We call this method of introducing an offset
to prevent vortex formation the “offset-MRAF" (OMRAF) algorithm.
1
The Nyquist argument proceeds as follows: An N ×N SLM plane can be fast-Fourier-transformed
to an N × N trapping plane with a diffraction limited spot size of exactly 1 pixel. The Nyquist limit
states that the diffraction limit should be reduced to half a pixel (the smallest feature in the trap is 1
pixel). This can be achieved by padding the computer-simulated SLM plane with zeros to increase its
size to 2N × 2N . This means that only 25% of the simulated SLM plane is covered by the physical
SLM, so we can at best control only 25% of the trapping plane.

152

Chapter B Raster Holography

Fig. B.1.: Choosing the optimal offset intensity. (a) Clarification of the role of the offset as a means
to eliminate vorticies and points of zero intensity which could develop into vortices. (b)The method for picking the optimal offset, ∆, measured in units of the peak amplitude in
the target pattern. We perform a single iteration of the algorithm, for different offsets, toproduce patterns such as those shown in the insets. We then choose the lowest value of ∆(indicated by the dashed line) which gives no vortices within C or D in this first iteration. (c)Improvement in the trap reconstruction associated with the offset.)

Fig. B.2.: Computational results for traps with a well defined zero potential background. We present
reconstructions of the test trap with MRAF and OMRAF methods. We also show detailsof the local reconstruction error (bottom insets) and the local phase (right insets). In thephase plots (obtained after 30 iterations) we use black to white to represent 0 to 2π and red
dots to mark vortex cores.

B.2 Optical vortices

153

For maximum light-usage efficiency, we want to minimise the proportion of theincident light which is steered into the featureless background, i.e. choose theminimal |∆| which gives satisfactory results. We find that for all the patterns wetested a suitable offset was |∆| ∼ 10−15% of the maximum amplitude in the trapping
plane (i.e. |∆|2 ∼ 1% of the maximum intensity). A computational simulation of thetrapping potential produced using the OMRAF algorithm is shown in Fig. B.2. Our
method reduces the number of vortices seen in C after 30 iterations by a factor of ∼ 20compared to MRAF in the large-canvas limit. This allows excellent convergence of
the algorithm, leaving only 4% RMS error between the desired trap and the OMRAF
result after 30 iterations and 1% RMS after 1000.

B.2.2 Efficiency
One compromise associated with the OMRAF method is a reduction in the efficiencyof light use. We define efficiency as the ratio of the integral of the trap intensity
above the background offset to the total integral across the trapping plane. Including
the non-zero ∆ causes a drop in efficiency by a factor of approximately 2 relative to
MRAF, from 43% to 24%, for the trap shape shown in Fig. B.2. Nevertheless, ourmethod remains more efficient than intensity-masking methods such as [187], which
report 3% efficiency for simple patterns. We calculate that for more complex patterns
such as our test trap shape, intensity masking would be less than 1% efficient.

B.3 Experimental realisation with laser light
We test our OMRAF algorithm experimentally by producing light patterns using 532
nm laser light in the arrangement shown in Fig. 5.2. Throughout this section, we
use the following techniques to improve the quality of our final traps:• We use the standard SLM-based Shack-Hartmann algorithm [188] to correct
for the low spatial frequency phase aberrations in the optical system andcharacterise our input laser beam.
• We use an active feedback algorithm [8] to optimise the final experimental
patterns. This feedback routine is essential to remove the “sinc envelope”
caused by the pixellation of the SLM, which would otherwise globally modulate
the intensity pattern.
• Finally, in practice, we find that when we try to produce experimental patterns
with C covering the maximum theoretically permitted area (25% of the trapping
plane), we cannot achieve patterns with less than 20% RMS fluctuations. This

154

Chapter B Raster Holography

is significantly improved in the results presented below by reducing C to cover
only 10% of the trapping plane.

Unfortunately, even with these precautions, we are not immediately able to generate
traps with the low RMS errors predicted by our simulations. Below we suggest afurther improvement which leads to traps of sufficient quality to consider integration
into cold atom experiments.

B.4 Fourier vs Helmholtz
In the previous section, we presented computational results generated under the
paraxial approximation in which the propagator mapping the light in the SLM planeto the trapping plane is given by a scaled Fourier transform. Using this approachin our experiments with an f = 200 mm lens, we are able to produce a trapping
pattern with RMS error of ≈ 11% (see Fig. B.3). As highlighted in Fig. B.3(b), themost prominent form of error is semi-regular fringing. We suggest that the source of
this error is the inadequacy of the paraxial approximation, and we use a numerical
Helmholtz solver to test this hypothesis.
The Helmholtz equation for light propagation in free space gives that the Fouriertransform of a light field, E(x; z), is modified by a phase factor Θ(ξ; z) as it propagatesin the +z direction [87]:n poF [E(r; 0); ξ] = F [E(r; z); ξ] exp ikz 1 − (λξ)2 ,
|
{z
}

(B.1)

Θ(ξ;z)

where F [g(r); ξ] denotes the Fourier transform (FT) of g, making the Fourier variablepairing r ↔ ξ explicit, and k = 2π/λ denotes the wave vector of the light. The
routine for propagation of E0 via the apparatus in Fig. 5.2 can then be conceptuallydecomposed into 3 stages:“Propagate FT from SLM to lens"“Apply Lens in Fourier space"“Propagate FT to trapping plane"Then:ψ1 (ξ) = F
h [E0 (r);i ξ] Θ (ξ; f )e (η)ψ2 (η) = ψ1 ~ L
ψ3 (η) = ψ2 (η)Θ(η; f )PHelmholtz [E0 (r); R] = F −1 [ψ3 (η); R] ,(B.2)

B.4 Fourier vs Helmholtz

155

Fig. B.3.: Experimental results with 532 nm laser light. (a) Experimental patterns based on four
different computational methods (for display we crop the trapping plane around region C).
We observe low quality traps with the MRAF algorithm, and we are unable to improvethem by adjusting the propagator (indeed, in this high RMS error regime, the additional
complexity of the Helmholtz propagator can be detrimental). However, incorporating aHelmholtz propagator into our OMRAF algorithm allows us to produce the test patternwith only 7% RMS error. (b) Illustrates the need for a beyond-paraxial propagator. The
left panel shows an experimental pattern based on OMRAF calculation under the paraxial
approximation. The right panel shows that similar fringing is reproduced when the same
paraxial kinoform is fed into a Helmholtz propagator to simulate the experiment. (c) To test
the universality of our method, we also show experimental OMRAF patterns cropped to theboundary of C for a uniform square trap and an annular BEC stirrer [8, 103].

where PHelmholtz represents the overall Helmholtz propagator, ~ represents convoe represents the FT of the phase pattern imprinted by the lens, which,lution, and L
after aberration correction, we assume to be: 2 
ikreL(ξ) ≈ F exp
;ξ .
2f

(B.3)

To show that the Helmholtz propagator is required, we use the OMRAF algorithm
under the Fourier approximation to produce a “Fourier-generated” kinoform, andthen input this kinoform into our numerical Helmholtz propagator to simulate theexperimental apparatus. This indeed produces the fringing pattern similar to thatobserved experimentally, which gives us confidence to continue with the Helmholtz
method (Fig. B.3(b)). We thus replace all instances of the Fourier propagator in the
OMRAF algorithm with a Helmholtz propagator. In this way, we are able to eliminate
the erroneous fringing, and reduce the RMS error of the experimental pattern to
7%. As shown in Fig. B.3, we can produce other trap shapes with similar fidelity.
Thus, even for traps with sharp edges and an extended canvas region, we achieve
RMS variation comparable to the 4% fluctuations previously seen only for simple

156

Chapter B Raster Holographysmooth traps with negligible canvas [8]. In [189], it has already been shown that 5%
RMS errors are sufficiently low for superfluid circuit and atomtronic applications. We
therefore believe that our methods show great promise for future applications.

B.4 Fourier vs Helmholtz

157

Graphics Card Acceleration ofComputations for Ultracold AtomExperiments

C

Computers play a central role in almost every aspect of a cold atom experiment. In
particular, there are two large computational jobs:• Data Analysis: The amount of data produced by the experimental machines is
intractable without a concerted computational effort• Physical Simulations: Often even "simple" calculations (e.g. determination of
the ToF profile after release from a uniform trap in Fig. 7.6) must be performed
numerically.
These applications generally require a large number of relatively simple operations
to be perfomed on a large data set. Such tasks are ideally suited to a graphicalprocessing-unit (GPU), which can be visualised as a large team of very basic
processors which all operate concurrently on separate parts of the data set (seeFig. C.1). In the sections below we will give one example from each two categories
above in which the use of a GPU has had a significant impact.

Fig. C.1.: Comparison of a computer’s central-processing-unit (CPU) and graphical-processing-unit(GPU). The CPU can be visualised as running a heavy-duty thread sequentially through thedata to be processed, whereas a GPU uses multiple light-weight threads running in parallelon different parts of the data.

.

159

C.1 Image processing: AnalysisGpUI
A typical project may require upwards of 105 experimental runs from the first planningstages to the final publication. This corresponds to approximately 1TB of raw datain the form of camera images which need to be processed. Image processing is not
only required in the final stages of data analysis: more often, image processing mustbe performed instantaneoustly while the machine is running such that an opticalcomponent or machine setting can be adjusted with near real-time feeback. For thisreason, we harness massively parallel GPU for rapid processing of experimental
images. In particular, we usually want to fit a theoretical model to the atomic profilein the images in order to extract atom numbers and temperatures. This requires
an efficient technique for non-linear least-squares fitting, as discussed in the next
section.

GPU Levenberg-Marquardt least squares fitting
A GPU based Levenberg-Marquardt non-linear least-squares solver library1 waswritten and embedded in a user friendly interface2 shown in Fig. C.2 (note that thesoftware name, AnalysisGpUI, is a play on the acronyms GPU and GUI (graphicaluser-interface)). The Levenberg-Marquardt algorithm finds the parameter set β for amodel function fi (β) which minimizes the sum of the square deviation from data yi
Pover all points, i: arg minβ ( i |yi − fi (β)|2 ). This algorithm proceeds by iteratively
solving for an improved parameter set β 0 = β + δ given by:

 T
J J + λ diag(JT J) δ = JT [y − f (β)](C.1)

where Jij = ∂fi /∂βj , diag(M) indicates the square matrix constructed from the
diagonal of M, and λ is a tuning parameter which is decreased when the computer
detects it is approaching convergence. Solving Eq. C.1 is not computationally demanding (the dimension of the matricies is equal to the number of fitting parametersin the set β). However, constructing the objects (particularly JT J) in this expression
from the experimental images can require a very large number (∼ 1010 ) of very basic
floating point operations per image. This is were we can profit from the parallel GPU
architecture.

1
2

160

Written in CUDA/C++
Written in C#/OpenGL

Chapter C Graphics Card Acceleration of Computations for Ultracold Atom Experiments

Fig. C.2.: AnalysisGpUI user interface.

.

C.1 Image processing: AnalysisGpUI161

Fig. C.3.: Performance analysis of our GPU algorithm. (a) Image processing time as a function ofimage size. The 0.1 s GPU initialisation time per image makes it slower than the CPU forvery small tasks, but (b) the GPU is up to ×16 faster than the CPU for large tasks in our
test rig.

.

GPU performance
In Fig. C.3, we compare the performance of our GPU implementation with thesame code running on a CPU (we also compare the speed to a previous-generation
MATLAB-based fitting software [26]). The hardware used for this comparision was:

CPU (Intel Core 2 Duo T9400)
GPU (NVIDIA GeForce 9600M GT)

Memory Bandwidth

Peak giga-FLOPS

3.2 GB/s
25.6 GB/s

20120

As expected, the GPU performs much better than the CPU for large problem sizes,
with a speedup of up to 16×. Since performing this benchmark test, we haveupgraded to an NVIDIA GTX 550Ti card which provides up to 64× speedup, andthe fitting process is now primarily limited by the process of loading the image filefrom the network.

C.2 Physical simulation: EvolveGPuE
Throughout this thesis, there have been several occasions when we have had toresort to numerical simulations to make a quantitative comparision with experiments(see e.g. Fig. 7.6 and 7.10). Here we will briefly outline how these simple simulationsof pure condensates can be quickly performed on a graphics card.

162

Chapter C Graphics Card Acceleration of Computations for Ultracold Atom Experiments

We consider a condensate described by the classical field ψ(x, t) in the GrossPitaevskii equation (GPE):



∂ψ
~2 2
2
i~
= −
∇ + U (r, t) + V0 |ψ| ψ∂t{z } |
{z
}
| 2m
ĤKin

(C.2)

Ĥ1

The formal solution to this equation is [190]:i∆t
ψ(r, t + ∆t) = exp −(ĤKin + Ĥ1 ) ψ(r, 0)
~






i∆t
i∆t
i∆t
= exp −
Ĥ1 exp −ĤKin exp −Ĥ1 ψ(r, t) + O(∆t3 )
2~
~
2~
{z
}|
{z
}|
{z
}
|
X̂

K̂

X̂

(C.3)

Where in the last line we will drop commutators scaled by ∼ ∆t3 and higher. Since X̂
is diagonal in position space (X̂f (x) = X(x)f (x)) and K̂ is diagonal in momentum
space (K̂f (p) = K(p)f (p)), we can perform appropriate (scaled) Fourier transforms
to write Eq. C.3 in a form which can be calculated on a computer:

ψ(x, t + ∆t) = X(x)×F −1 [K(p)×F [X(x)×ψ(x, t)]] ,(C.4)

where F denotes an appropriately scaled Fourier transform to map from position tomomentum space.
Note that if we choose an imaginary value for ∆t, Eq. C.3 decribes the decayof all excited components of ψ in imaginary time, and ψ(r, t) asymptotes to theground state of the system. Therefore, to simulate the time-of-flight dynamics ofan equilibrium system, we start with any ψ(x, 0) which has finite overlap with theground state and propagate in-trap in imaginary time until the system converges on
the ground state. Then we release the trap and propagate in real time.
The GPE on a GPU
We simulate the GPE with arbitrary U (r, t) using our program EvolveGPuE (a play on
the acronyms GPU and GPE) to perform the parallelisable operations highlighted inblue in Eq. C.4 on the GPU. Since we are often interested in watching the dynamical
evolution of our system (e.g. in ToF or as a response to a change of trap geometry),
EvolveGPuE, is purpose-built to display ψ(r, t) immediately as it is calculated (seeC.2 Physical simulation: EvolveGPuE

163

Fig. C.4: An example dynamical simulation.
A cylindrical obstacle is dragged at ≈1 mm/s through an anisotropic
harmonically trapped cloud withtrapping frequencies of approximately 60,14 and 9 Hz in the threecoordinate directions. The cloudcontains 5 × 105 87 Rb atoms.
Snapshots of the cloud are illustrated every 15 ms showing
the development of quantized vortices (seen as tubes of zero density) in the wake of the object.
This simulation is carried out on
a 256 × 256 × 256 grid, and witha time discretisation of ∆t =0.04 ms, we obtain a simulation
speed of 1.4 seconds per 1 ms ofevolution.

164

Chapter C Graphics Card Acceleration of Computations for Ultracold Atom Experiments

e.g. Fig. C.4). Since an NVIDA GTX 680 graphics card takes ∼ 0.06 s to calculate
the update t → t + ∆t on a spatial grid of 256 × 256 × 256 points, the frame rateof the calculation is fast enough to play as a continuous movie as the calculation
proceeds, and the user can interact with this movie by changing the camera angle toinspect the 3D cloud as it evolves (see real-time videos at [191,192]). To ensure thatthe dynamically updating display does not significantly slow down the calculation,
the calculation is done directly in part of the GPU’s memory already belonging to
the OpenGL display pipeline. This minimises the movement of the large array ofnumbers representing ψ in memory.
Note that a CPU based calculation is > 10 times slower than our GPU calculation,and the overhead of transferring this data to the GPU for display makes it impracticalto run a continuous movie of the CPU calculation in real time. The main limitationof our GPU solution is the limited graphical memory resources available. Highend consumer graphics cards have up to 4 GB of RAM which (once the memory
overheads of the fast-Fourier transform routine are subtracted) allows us to workwith grid dimensions up to 4853 (in this limit ψ ∼ 1 GB).
Volumetric display of ψ
The calculation of ψ is performed in 3D, but the results must be displayed on the
2D screen. This is achieved by sampling the volume occupied by ψ on ≈ 200
equally spaced planes perpendicular to the viewing direction. The value of |ψ|2 isinterpolated onto these planes and translated into a colour and transparency (denser
regions being more opaque). The semi-transparent stack of planes is then rendered
onto the screen one by one with the colours blending according to the transparencyof the added plane. This gives the illusion of a semi-transparent cloud in 3D space(see Fig. C.5).

C.2 Physical simulation: EvolveGPuE

165

Fig. C.5.: Volumetric rendering of ψ. (a) The volume occupied by ψ is sampled by many equally
spaced semi-transparent planes perpendicular to viewing direction (green - only 10 planes are shown).
These planes are rendered one by one to the screen with the final pixel values being the accumulated
colour and transparency. (b) When 200 planes are used, we see the illusion of a 3D cloud. The cloudshown here is the same as in Fig. 7.6 (50 ms ToF).

.

166

Chapter C Graphics Card Acceleration of Computations for Ultracold Atom Experiments

Numerical Simulation of
Kibble-Zurek Dynamics

D

Kibble-Zurek theory is deliberately very simple and very general, making only broadclaims about the exponents of the power laws governing the frozen properties ofa system as its dynamics slows near the critical point. To give definite numericalvalues to the microscopic length- and time-scales λ0 and τ0 is certainly not in thespirit of the original formulation, and to claim that the KZ theory should capture
anything but the most basic properties of the dynamical evolution of ξ near thecritical point is incorrect.
Nevertheless, in this appendix, we will break these conventions and try to directlycompare the KZ predictions to a simple numerical model of the system’s dynamics
near the critical point. Our motivation for performing this study is to assess theextent to which the non-linear form of our temperature quench, illustrated in Fig.
8.7, affects our interpretation of the experiment in chapter 8 where we evaluate
parameters based on a linear KZ theory.
Since this section is an abuse of the intended purpose of KZ theory, there will bemany questions which arise during the formulation of our dynamic model. These
questions are highlighted in blue below, and it must be remembered that our answers
to these questions are sensible but not rigorously justified. Although the assumptions
are uncontrolled, we include the full details of our model here because it reproduces
the KZ predictions remarkably well.

D.1 Dynamical evolution of ξ
We are fundamentally interested in the size of phase coherent domains in our BEC,
but critical phenomena are all phrased in terms of the correlation length ξ. Therefore,
our first question is:
Question 1: How does the correlation length on the approach to thecritical point relate to the size of the BEC domains on the ordered side?
We simply assume ξ(tc ) = d, and thus to evaluate d we must simulate the evolutionof ξ up to the critical point. This evolution is governed by the relaxation timescale,167τ , however there is some question over the exact form of the dynamical equation
linking ξ and τ :Question 2: What is the functional form linking ξ/ξ˙ and τ ?
Here we assume a simple form inspired by the nomenclature "relaxation time".
ξ−ξξ˙ = −
τ

eq

(D.1)

where ξ eq = λ0 X||−ν and X = X+ for  > 0 and X = X− for  < 0. In equilibrium,
the dynamical scaling hypothesis says that we can write down a "dispersion relation"
between τ eq and ξ eq

τ eq = τ0 (ξ eq /λ0 )z ∝ ||−νz(D.2)

Note that we were careful to include the critical amplitude ratio, X± , in the expressionfor ξ, but we leave it as an open question whether there is such a ratio for τ :Question 3: Is there a critical amplitude ratio associated with τ ?
When the system is far from equilibrium, it is not obvious that τ = τ eq . Instead, it ispossible that τ is given by an extension of the dispersion relation to non-equilibrium
scenarios:

τ dyn = τ0 (ξ/λ0 )z(D.3)

These two possible definitions of τ lead us to another question:
Question 4: Is the relaxation dynamics governed by the distance tothe critical point (i.e. by τ = τ eq ) or by the instantaneous ξ and thenon-equilibrium dispersion relation (i.e. by τ = τ dyn )?
We answer this question by noting that inserting the (finite) τ dyn into Eq. D.1 leads
˙ → ∞ at the critial point, which is not in the spirit of a "critical slowing down".
to ξ/ξ
Therefore, we believe that the choice τ = τ eq is more realistic. For interest we also
show calcuations with and τ = τ dyn in Fig. D.1.
The full numerical model for a quench with a quadratic non-linearity matching our
experimental cooling trajectory (see Fig. 8.8) is written explicitly below. For the168

Chapter D Numerical Simulation of Kibble-Zurek Dynamics

Fig. D.1.: Numerical simulations for the dynamics during a non-linear quench self similar to theexample shown in Fig. 8.8. We show separate simulations for tQ = 0.2 s, 1 s and 3.2 s. The orange
line shows ξ eq and the blue line shows the dynamically evaluated ξ based on τ eq (for interest, we also
show ξ evaluated base on τ dyn in gray). The vertical blue dashed line indicates tc and the horizontal
line marks ξ(tc ). The frozen region is shaded blue. Note that ξ eq (tc − t̂+ ) = ξ(tc ) as expected from
KZ theory.

purposes of illustration, we run the simulation with the F Model critical parameters.
We have checked that inserting general ν and z leads to consistent results.
Cooling trajectory



t−tc
c
 = − t−t
1
−
γ
tQ
τ0

tc = 0.72 tQ

τQ0 = 0.41 tQ

Dynamical equationsξ˙ = −(ξ − ξ eq )/τ eq

ξ eq = λ0 X||−ντ eq = τ0 (ξ eq /λ0 )zFreeze-outξ̇ eq (t̂± )ξ eq (t̂± )d = ξ(tc )Critical Exponentsν = 2/3

z = 3/2

Microscopics

λ0 = 0.7 µm

τ0 = 30 ms

Q= τ eq1(t̂ )±X+ = 1γ = 0.7

X+ /X− = 0.3

D.2 Simulation results
The simulation produces a number of reassuring conclusions for our non-linear
cooling curve which are briefly enumerated below:• The model with no free parameters captures the size d of the domains measured in the experiment remarkably well (see Fig. D.2)
• The model correctly predicts tbr ≈ 1 s (see Fig. D.1)
• ξ eq (tc − t̂+ ) is very close to the dynamical value ξ(tc ), which numerically justifies
the KZ hypothesis that the system’s correlation length as it crosses the criticalpoint is equal to the equilibrium correlation length at the freeze-out time.

D.2 Simulation results169

Fig. D.2.: The effect of calculating τQ at tc − t̂+ . When we run the simulation and plot ξ(tc ) vs. τQevaluated at tc , we obtain the gray line which is (i) not perfectly linear and (ii) gives a KZ exponent of
0.94ν/(1 + νz). If we instead evaluate τQ at tc − t̂+ , we obtain the blue line, which is (i) linear and (ii)gives b = ν/(1 + νz). To compare the absolute values of ξ(tc ) we plot the experimentaly measured
domain sizes as grey points. These points are shown "un-processed" (i.e. with τQ evaluated at tc ) forcomparison with the grey line. Note that here we used τ0 = τel = 30 ms and λ0 = λc = 70 µm, whichwere independently estimated. The fact that the absolute values of the experimentally measured `are close to our prediction with no free parameters is remarkable because there could be several
numerical factors of order 1 which we have omitted.

• For a linear quench (γ = 0) we recover b = ν/(1 + νz) to three digit precision• For γ = 0.7 (corresponding to our experiment) we only recover b = ν/(1 + νz)
when we evaluate τQ at tc − t̂+ . Evaluating τQ at tc gives a strict lower boundon b.
• A power law fit to τQ |tc −t̂+ vs. tQ gives τQ |tc −t̂+ ∝ t0.94
Q indicating that analysing
the data with τQ |tc ∝ tQ gives an underestimate of b by 6% (see Fig. D.2).

One surprising prediction from our simulation is the observation that ξ continuously
evolves during the "frozen" period. This is in contradiction with a literal interpretationof the KZ freezing hypothesis, and also in contradiction with the plateau observedin the experimentally measured value of d after tc (see Fig. 8.9). We suggestthat this contradiction could be due to the fact that the experiment measures thephase coherence length and not the correlation length. Although we assume thatthe coherence length is set by the correlation length at the critical point it does notnecessarily evolve in the same way as the correlation length. This can be particularly
seen in the long-time limit where the coherence length will increase as the system
unfreezes and cools further, while the correlation length of the fluctuations will
reduce. Since the evolution of the coherence length is not captured by our model, it
is surprising that the experimental breakdown time tbr = 1 s is still approximately
consistent with the quench time for which tc + t̂− = tQ in our simulation. The fact that170

Chapter D Numerical Simulation of Kibble-Zurek Dynamics

the simulation correctly predicts tbr gives us some reassurance that t̂− calculated
from Eq. 8.25 is a relevant unfreezing time for the KZ theory.
In conclusion, although there are still many open questions, this simulation produces
results which are reassuringly similar to the experimental values. Most importantly, itjustifies our analysis of the data when extracting critical exponents in chapter 8.

D.2 Simulation results

171

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