Exploring Interacting Bose Gases
in and out of Equilibrium

Christoph Eigen
Gonville & Caius College

This dissertation is submitted for the degree ofDoctor of Philosophy

August 2019
Cavendish Laboratory
Department of Physics

To the memory of my grandfather, Manfred.
. . . . . . . . . . . . . . .

Contents
Declaration of Authorshipi

Abstractiii

Acknowledgementsv

Acronyms and Global Variablesix

1

Introduction
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2

2

Theory of Bose Gases
2.1 The ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Theory of the condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Bogoliubov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Tan’s contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
6
8
9
13
15

3

Experimental Platform
3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Bose–Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Quasi-uniform potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Manipulating and probing samples . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Tuning interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Imaging & measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Radio-frequency spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
17
18
19
20
21
22
23
25

4

Turbulent-Cascade Fluxes
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Emergence of a turbulent cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Synthetic dissipation scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Probing the turbulent cascade fluxes . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Turbulent cascade atom-loss dynamics . . . . . . . . . . . . . . . . . . . .
4.4.2 Steady-state particle and energy flux . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Pre-steady-state cascade-front dynamics . . . . . . . . . . . . . . . . . . .
4.4.4 Unifying qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
32
37
39
39
41
42
43
45

5

Weak Collapse of a Bose–Einstein Condensate
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Qualitative picture and scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Critical interaction strength . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Collapse time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Atom loss and weak collapse theory . . . . . . . . . . . . . . . . . . . . . .

49
49
51
52
53
53

5.3

Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Critical interaction strength . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Collapse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Aftermath of the collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54
54
56
59
70

6

Moderately Strongly Interacting Bose Gases
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Quantum depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Bragg filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Quantum depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Quasi-particle excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Beyond Bogoliubov theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Spectroscopy of quasi-particle excitations . . . . . . . . . . . . . . . . . . .
6.3.3 Exploring the limits of validity of Feynman–Tan theory . . . . . . . . . . .
6.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71
71
73
74
78
79
80
81
83
84
87
88

7

Boses Gases Quenched to Unitarity
89
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1.1 Advantages of an optical box potential . . . . . . . . . . . . . . . . . . . . 92
7.1.2 Atom-loss scaling-law predictions . . . . . . . . . . . . . . . . . . . . . . . 93
7.1.3 Quench protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Degenerate Bose gases quenched to unitarity . . . . . . . . . . . . . . . . . . . . . 99
7.2.1 Universal loss dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2.2 Momentum distribution dynamics . . . . . . . . . . . . . . . . . . . . . . . 104
7.2.3 Molecular correlation dynamics . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3 Thermal Bose gases quenched to unitarity . . . . . . . . . . . . . . . . . . . . . . . 110
7.3.1 Momentum redistribution dynamics . . . . . . . . . . . . . . . . . . . . . . 110
7.3.2 Atom-loss and correlation dynamics . . . . . . . . . . . . . . . . . . . . . . 116
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8

Outlook

5.4

120

A Cross-calibration of the zero-crossing of a123

B Atom number control & stability

124

C Inverse-Abel transform

125

D Field control & stability
126
D.1 Mains oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D.2 Dynamical field control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.2.1 Primary field response time . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D.2.2 Quenches to unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Declaration of Authorship
I certify that this thesis is the result of my own work unless declared otherwise in the text. It is
not substantially the same, in whole or in part, as any that I have submitted, or, is concurrently
being submitted for a degree (or other qualification) at this or any other University (or similar
institution). This thesis does not exceed the world limit of sixty thousand words, excluding
figure captions, acknowledgements, and bibliography, as set out by the Faculty of Physics andChemistry.

Christoph Eigen,
August 2019

Cavendish Laboratory,
University of Cambridge,
J J Thomson Avenue,
Cambridge CB3 0HE,
United KingdomiAbstract
This thesis describes a collection of experiments that explore interacting ultracold Bose gases,
both in and out of equilibrium. Our experiments are performed using a gas of 39 K or 87 Rb confined in the uniform potential of an optical box trap, a novel testbed for quantum many-bodyphenomena. Our work focuses on weakly interacting non-equilibrium systems, moderately interacting systems that are still in equilibrium, and the unitary Bose gas which is both stronglyinteracting and eludes equilibrium.
We begin with studies of weakly-interacting gases far from equilibrium, which feature ties
to nonlinear wave phenomena. Highlights of our experiments include the direct measurementof turbulent-cascade fluxes, which (alongside realizing a tuneable dissipation scale) allow us todemonstrate the zeroth law of turbulence, and the first observation of weak collapse, a general
type of nonlinear wave collapse predicted over 40 years ago.
We then turn to moderately strong interactions, confronting existing theories of interacting
quantum fluids. One of our most important results is the first quantitative measurement of
the quantum depletion of a Bose–Einstein condensate, confirming a 70-year-old theory firstdeveloped to describe liquid helium.
The culmination of our work explores the unitary Bose gas, where interparticle interactions
are as strong as allowed by the laws of quantum mechanics. This strongly-correlated statepromises tantalizing possibilities, including emergent universal behavior set solely by the gas
density and novel forms of superfluidity. However, the strong interactions also lead to a complex interplay between coherent and dissipative dynamics. By disentangling these two processes, we have caught a glimpse of the promises that the unitary Bose gas holds. In particular,
we observe the emergence of universal behavior, and find that the gas features a well definedquasi-equilibrium state, with a non-zero condensed fraction.

iii

Acknowledgements
Expressing genuine gratitude to the people that one owes everything to is one of the mostdaunting challenges of writing a thesis, since one faces the inevitable failure of doing themjustice. Nevertheless, trying matters, so here goes.
First and foremost, I would like to thank Rob, who has been the kind of supervisor any
student can only dream of1 . From day one (at the start of my MPhil) he has been there to
guide my steps, and his patience and unerring support have been invaluable. It is unclear
what would have become of me without his inherent urge for directly confronting problems
and identifying issues at their root.
On equal footing, I would like to thank Zoran, who also took on (or rather retained) therole of a supervisor. His immeasurable2 enthusiasm for science, as well as his high standards(sometimes seemingly unattainable so) are outright infectious. Together Rob and Zoran formeda supervisor dream team, complimenting each other across the board but ultimately sharing
their love of science. I am deeply indebted to both Rob and Zoran for investing so much of
their time into not only my scientific prosperity but also my career.
Throughout the years I have had the pleasure to be mentored exclusively by brilliant postdocs, whose contributions to the work presented in this thesis are massive. Before I knew what
was going on, Alex had already taken me on board, teaching me the arts of sculpting3 while ona fast-track involuntary crash-course in how to rebuild the machine4 . His love for computersand physics (it’s a tough call) paved the way for success, and in particular his pioneering box
traps underpin this entire thesis. Nir offered continual advice and valuable insight at all times,
and was in charge of running BEC I. While his sense of humor often kept me on my toes5 , hebrought an overall badass scientist vibe to the lab, making it a stimulating and unique environment. Raphael was unarguably one of the most (openly) passionate people around, and both
his flair for science and his intrinsic drive to never stop left a profound impression. His ‘hostile
takeover’ (in a good way), implementing the Bragg setup and subsequently putting it to use,
1

I say that even though he left me to go to the other place.
I realize what I’m getting myself into.
3
Optical boxes, that is.
4This incident is no longer spoken of.
5
Passwords are useful it turns out.
2v

was extremely fruitful in many ways, and led to a large number of nights and weekends in thelab, which were however also regularly complemented by tennis breaks. While not directly involved in any of my work, Rich (who was leading BEC II at the time) was readily available foradvice, either during coffee break discussions or while enforcing key lab traits (such as looking
over ones shoulder while working and asking stimulating questions). All of them have since
moved on, and their departure marked a notable transition in the lab dynamics.
When Jake joined as a fellow PhD student, he swiftly transformed our early dabbles in theunitary regime with his savvy sense for electronics and problem solving. It has been a delight
exploring the unitary regime with him, and I owe a great deal to him. I also want to thank Jinyi,
who took over Nir’s role as the BEC I postdoc, keeping its future bright. He has been central toacquiring the data that BEC I has amassed.
There were a large number of summer and Part III students that have made an appearance
throughout the years; I would like to thank them all for their involvement (from small conveniences to large contributions), particularly emphasizing Adam and Sam.
I would also like to thank Timon, who as the newest BEC III postdoc has already imprinted
many of the stern Munich standards onto daily lab life. While we kind of threw Lena, our new
PhD student addition, in at the deep end, this seems to have paid off with significant progress.
Indeed, the current BEC III team is being remarkably productive in my writing absence, making
me eagerly anticipate my return.
I am particularly fond of a collaboration with Eric when he visited us for a sabbatical, instilling a refreshing and inspiring atmosphere to the lab, and providing profound insight into ourstudies of the unitary regime. I would also like to thank our theory collaborators, Kazuya and
Makoto, who performed the numerical simulations in our studies on turbulence.
The last person associated with BEC III who I would like to thank is Aziza, who even though
we never overlapped, I owe a great deal to as she orchestrated the majority of the building of
the machine at the heart of this thesis.
The physical segregation6 of office occupations for BEC II and BEC III has slowly drifted into
crisis. From back in the early days when things were still in order, I would also like to thankJay and Konrad (who deserted to the lab next door), who were also new recruits back when I
joined the group and who were central in ensuring that days spent in the lab were a delight. I
would also like to thank Milan and Nathaniel (Rob’s ‘students in crime’), as well as Max who
we unfortunately lost to Innsbruck; they all enhanced daily lab life.
Turning to the current BEC II inhabitants, I would like to thank Panos, Maciej, and Julian.
6

Overpopulation led to division into two offices.

vi

While faced with a bumpy start, their now operational 2D box experiment features optical potentials which are something to be jealous of. I have enjoyed their company and contributions
to daily lab life throughout, as well as social activities from football games to pub nights.
I also enjoyed the numerous aperiodic encounters with Konrad and Matteo, who started
their PhDs next door at the same time as I did, and with whom I share many good memories.
More generally, I would also like to thank Mete, Ulrich, Andreas, and Richard, alongside
all of their students and postdocs, for making the AMOP group an awesome, scientificallystimulating environment in which to pursue a PhD. I also extend thanks to all friends andpeers that have supported me or otherwise enhanced this journey.
Finally, I would like to turn to my family and those nearest my heart. First, I would like tothank my extended ‘science family’ from the Winter Seminar in Klosters. I have been fortunate to attend this unique conference since I was a child, and presenting our novel findings inthis setting has been priceless. I am thankful to everyone that attends and keeps this special
experience alive, but would like to particularly thank Michael, Markus, and Eberhard.
Finally, I cannot express enough gratitude to my parents Gerald and Andrea, as well as mybrother Julian. They have supported me on all fronts, and they are directly to ‘blame’ forinstilling my curiosity for science.
Last and quite a lot, I would like to thank Janine, without her constant support and unconditional compassion none of this would have been as enjoyable.

vii

Acronyms and Global Variables
BEC Bose–Einstein Condensate
rf Radio Frequency
ToF Time of Flight
DF Diffracted Fraction
TF Transferred FractionODT Optical Dipole Trap
SLM Spatial Light Modulator
OD Optical Density
QD Quantum DepletionkB - Boltzmann’s constant~ - reduced Plank’s constantm - atom massn - atomic density
nk - momentum distributiona - s-wave scattering length
T - temperatureE - energy per particle
L - length
R - radius
N - atom number
s.e.m. - standard error on the meanix

1 Introduction
Our world is inherently transient and interactions lie at the heart of its many-body nature.
While different areas of research present unique and novel problems at every stage of thefundamental ‘cascade’ linking them [1], there still exists a curious relationship between thestrange simplicity governing the fundamental laws of physics and the complex familiarity that
describes our everyday world [2].
Understanding quantum matter which is either far from equilibrium or strongly-interacting,lies at the heart of most fundamental problems in modern physics. Since its inception two
decades ago, when the first gaseous atomic Bose–Einstein condensates were created [3, 4], andfollowing the advent of degenerate atomic Fermi gases a decade later [5], the field of ultracold
atoms has established itself as an ideal setting for investigating quantum many-body phenomena, in the spirit of Feynman’s proposal of quantum simulation [6]. Key ingredients for thefield’s success are the pristine control of atom-trapping geometries, the ability to coherently
manipulate the atom’s internal states, the experimentally resolvable intrinsic timescales, andthe unique ability to tune the interparticle interactions using molecular resonances [7].
Over the last two decades, ultracold atom research has led to myriad achievements, whichare (at least in part) reviewed in [8–20]. Some of the major achievements can be attributed to theuse of optical lattices to mimic the solid-state setting, freezing-out of excitations to gain accessto the peculiarities of lower-dimensional physics, and artificial gauge potentials to simulate
behavior of arbitrarily charged particles in electromagnetic fields.
Our focus here is on Bose gases, which tend to form Bose–Einstein condensates at low temperatures, when the thermal wavelength becomes comparable to the interparticle separation.
This statistical second-order phase transition leads to a macroscopically occupied ground state,
and the emergence of the wave properties of matter. Such condensates are generally well understood when they are in equilibrium and the interparticle interactions are weak. However,if the gas is driven out of equilibrium or if it is strongly-interacting, things typically escalate,
swiftly becoming more complicated. Exploring these two regimes is at the heart of this thesis. We utilize ultracold Bose gases confined in a quasi-uniform box potential [21, 22], sculptedfrom repulsive (blue-detuned) laser light. Most of our experiments are performed using ul-

1

CHAPTER 1. INTRODUCTION

Experimental platform

turbulent-cascade fluxes

tuneable interactions

optical
box

Bres

trap
depth

0

ts (s)
cascadefront

magnetic
magneticfield,
field Bparticles

N. Navon et al.
Science 366, 382 (2019)

testbed for quantum many-body physics

Strong interactions

R. Lopes et al.

(a)�� (%)

50µm
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universal2 scaling laws

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FIG. 2. (color online) Bragg� spectroscopy, for n ⇡ 2.0⇥1012 cm 3 ,q = 1.7 krec , and a ⇡ 1000-�
a0 . (a) Typical absorption
image,
�� taken��� (%)

distances [22, 25]; this ‘factorisation’ of the effects of manybody correlations (captured by C) and the short-distancetwop
body physics was highlighted by Tan [26]. For na3 ⌧ 1,q
the contact density is C ⇡ (4⇡na)2 , and for our experimental
parameters |S(q) 1| < 0.03, so 1/S(q)QD 1 ⇡ 1 S(q).
This ‘Feynman-Tan’ (FT) approach thus gives the interaction
shift of the excitation resonance4⇡~na ⇣
⇡qa ⌘
!FT =
1
.
(3)
m
4

C. Eigen et al.
PRX 6, 041058 (2016)

Unitary Bose gas

quasiparticle energy�� (%)

quantum depletion

weak collapse

forcing

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prethermal dynamics
thold (μs)

ultracold
atoms

scattering length,ainteraction
strengthuniform potential

Non-equilibrium

���

(μ�)

C. Eigen et al.

For qa ! 0, !FT reduces to !B , but for increasing a (atPRL
190404
PRL 119, 250404 (2017)
fixed q) it back-bends
and119,changessign at a(2017)= 4/(⇡q) [see PRL 118, 210401 (2017)
Fig. 1(a)]. In Ref. [6] the largest value of a reached was 0.8/q
along the radial direction of the cylindricalbox trap, after the 2-ms
Δν (���)and back-bending was observed, but ! remained positive.
Bragg pulse and 20 ms of time of flight. The spherical halo arises
Let us also consider the dispersion relation, !(q) at fixedfrom the collisions between the stationary and diffracted clouds;
these collisions do not change the centre of mass of the atomic distria. The energy of the low-q phonons is above the free-particle
bution. (b) Bragg spectrum. Diffracted fraction (DF) as a function ofdispersion ( ! > 0) [23, 27], while according to Eq. (3) thethe frequency difference between the two Bragg beams, referenced
energy of particle-like excitations with q > 4/(⇡a) is below
to !0 , which was calibrated using a non-interacting cloud. The resoit ( ! < 0); finally, for q ! 1 the quasiparticle energy
nance is determined from a Gaussian fit to the data (solid line).
is expected to approach the free-particle dispersion from below ( ! ! 0 ) [22, 23]. As illustrated in Fig. 1(b), for alarge enough a the dispersion relation has an inflection point,taken after Bragg diffraction, and in Fig. 2(b) an example ofwhich is a precursor of the roton minimum that fully develops
a Bragg spectrum used to determine the resonance shift !.
only for extremely strong interactions [22, 23]. In Eq. (2) the
The diffracted fraction of atoms is determined from the centre
maximum in S(q) for fixed n and a, which is conceptuallyof mass of the atomic distribution [6, 8]. In all our measureassociated with the roton [22, 28],
ments we keep the maximal diffracted fraction to . 10%; thisp occurs at q = 8/(⇡a), independently 39
of n, and only for na3 ⇠ 1 this coincides withshould result in . 10% systematic errors in our interaction
1/3
the familiar result for liquid helium,
frequency shifts [5, 31].
p qroton ⇠ n .
In our experiments the regime na3 ⇠ 1 is not reachable
In Fig. 3(a) we plot ! versus a for two different combidue to significant losses on the timescale necessary to pernations of the BEC density n and excitation wavenumber q.
form high-resolution Bragg spectroscopy. Nevertheless, we
In both cases we observe good agreement with the prediction
reach the regime where interactions are strong enough to obof Eq. (3), without any adjustable parameters; for the lower nserve a dramatic departure from Bogoliubov theory and thewe reach higher a and clearly observe that ! changes sign.
precursors of roton physics.
Defining a dimensionless interaction frequency shift
Our setup is described in Ref. [29]. We produce quasimq
pure homogeneous 39 K BECs of N = (50 160) ⇥ 103
!,
(4)
↵⌘
atoms in a cylindrical optical box trap of variable radius,4⇡~n
R = (15 30) µm, and length, L = (30 50) µm. The BEC
theFT
prediction
of
Eq.
(3)
is
recastas:is produced in the lowest hyperfine state, which features a Fes⇣hbach resonance centred at 402.70(3) G [30]. The condensed⇡ ⌘↵FT = qa 1
qa ,(5)
fraction in our clouds is > 90% and we hold them in a trap of
4
depth ⇡ kB ⇥ 20 nK. By varying N , L, and R, we vary n inwhich is a universal function of qa only; with the same northe range (0.2 2.0) ⇥ 1012 cm 3 . The three-body loss ratemalisation the Bogoliubov theory gives ↵B = qa. Noteis / n2 a4 , so working
at such low n is favourable for increaspthat the normalisation in Eq. (4) also allows us to correct foring both qa and na3 . We prepare the BEC at a = 200 a0 ,the small (±10%) density variations between measurements
where a0 is the Bohr radius, and then ramp a in 50 ms to thetaken with different values of a and the same nominal n. In
value at which we perform the Bragg spectroscopy. For eachFig. 3(b) we show measurements of ↵ with three different
n we limit a to values for which the particle loss during the
combinations of n and q, which all fall onto the same uniwhole experiment is < 10%; note that in our trap the threeversal curve, in good agreement with the FT theory.
body recombination does not lead1to any observable heating.
By varying the angle between the Bragg laser beams we also
In Fig.
p3(b), for our most strongly interacting samples qa ⇡explore three different q values: 1.1, 1.7 and 2.0 krec , where
2.5 and na3 ⇡ 0.05. In the final part of the paper we explore
krec = 2⇡/ and = 767 nm. For all our parameters weeven stronger interactions and observe that the FT theory also
stay in the regime of particle-like excitations, with q⇠ values
breaks down. In Fig. 4(a) we show measurements of ! with
between 5 and 40.
n ⇡ 0.2 ⇥ 1012 cm 3 and q = 2 krec , for which we explore
In Fig. 2(a) we show an example of an absorption image
scattering lengths up to ⇡ 8⇥103 a0 , corresponding to qa ⇡ 7

100

+

30

�

10

-

0

5
10 15
� (μ�-�)C. Eigen et al.
Nature 563, 221 (2018)

Fig. 1.1 Thesis outline. Our experimental platform, which combines a uniform potential and the ability to tune theinterparticle contact interactions, establishes a novel testbed for quantum many-body physics. In the six
papers that form the basis of this thesis we explore systems that are either out-of-equilibrium or stronglyinteracting, or, in the case of the unitary Bose gas, both.

tracold

K gases in our second generation box-experiment [22]; this combines the benefits ofa uniform trapping potential with tuneable interparticle interactions [7], establishing a novel
testbed for fundamental many-body physics.

More generally, our experiments are part of a shift towards uniform box potentials [21–25].
These box-traps are an invaluable tool for extracting bulk properties of the gas, and are particularly promising in the context of non-equilibrium phenomena, where the otherwise nonuniform density imposed by more traditional harmonic traps can obscure or even alter the
resulting physics .

1.1 Outline
As outlined in Fig. 1.1, the work presented in this thesis closely follows the six papers [26–31],
which explore intricate many-body phenomena in systems that are either far-from-equilibriumor strongly-interacting, or, in the case of the resonantly interacting unitary Bose gas, both.
Following a brief introduction of the pertinent theory of Bose gases in Chapter 2, we begin
in Chapter 3 by giving an overview of the experimental platform used to create and manipu1

Note that for experiments performed in harmonic traps it has been possible to deal with the varying density inclever ways, even in some cases exploiting it as an advantage, broadly speaking achieved by assuming the local
density approximation to extract uniform system results.

2

CHAPTER 1. INTRODUCTION
late a 39 K Bose gas at nanokelvin temperatures. We highlight our key experimental features,
including the quasi-uniform box potential [21] and the ability to tune the interparticle interactions using magnetic Feshbach resonances [7]. In Chapters 4 and 5 we present experimentsthat investigate nonlinear wave phenomena in weakly-interacting gases. In Chapter 4 we drive
the cloud out of equilibrium and probe the fluxes underlying the emerging turbulent cascades,
while in Chapter 5 we study the effect of attractive interactions in a BEC, which leads to dramatic non-equilibrium behavior as the BEC collapses. Chapter 6 focuses on ‘moderately’ strong
interactions, where simple mean-field theories break and beyond-mean-field quantum correlation effects become important, but the experiments also remain tractable within the existing
theories. Chapter 7 explores the unitary Bose gas, where interactions between particles becomeas strong as theoretically possible and a complex interplay between coherent and dissipative
dynamics occurs. Finally, in Chapter 8 we briefly discuss future research avenues.

32 Theory of Bose Gases

Erst die Arbeit, dann das Spiel

Our goal in this chapter is to provide a minimal basis of the theory of Bose gases that wewill rely on throughout this thesis; a comprehensive treatment can be found in several places,
for example [8, 17], to which we refer the reader (and which we in part follow closely here).
Instead, for the core concepts of atomic physics we recommend e.g. [32].
If we consider two indistinguishable particles, the corresponding wave function under particle exchange (indicated by a subscript change) generically satisfies|ψ1,2 |2 = |ψ2,1 |2 .

(2.1)

For this relation to hold, we require ψ1,2 = |ψ| exp(iγ), where γ could in theory take any realvalue. In practice, all known particles fall into one of two categories, either γ = 0 or γ = π,defining bosons and fermions respectively1 . Fermions have half-integer spin while bosons,which we consider throughout this thesis, have integer spin. In the case of fermions, Eq. (2.1)
reveals the Pauli exclusion principle, while as we will see below bosons are more ‘sociable’.
As highlighted by the cartoon in Fig. 2.1, a Bose gas can typically be described by three
relevant lengthscales: the interparticle spacing n−1/3 (set by the gas density n), the thermal√
wavelength λ = h/ 2πmkB T (set by the temperature T ), and the s-wave scattering length a(capturing the two-body contact interactions)2 . In the following sections we will discover how
these lengthscales (or combinations thereof) reveal themselves.

1

An active field of research constitutes attempts to create so-called ‘anyonic’ quasi-particles in lower dimensions,where γ can take values other than 0 and π.
2
The finite size of the box can also play a role (e.g. when the interaction energy is comparable to the kinetic energy).
The van der Waals length avdw (which is ≈ 60 a0 for 39 K [33]) can also play a role when a . avdw . Additionally,in Bose gases the Efimov effect occurs [34], which formally requires the introduction of a three-body lengthscale
R3 (see e.g. [35, 36]), which can also become important.

5

CHAPTER 2. THEORY OF BOSE GASES

optical box

three relevant length scalesinterparticle spacing
thermal wavelengths-wavescattering length

n-1/3
λ
a

ultracold Bose gas
Fig. 2.1 Box-trapped Bose gases. A weakly interacting gas can typically be described by three characteristic
lengthscales:
the interparticle spacing n−1/3 , where n is the gas density, the thermal wavelength λ =√h/ 2πmkB T , where T is the gas temperature, and the s-wave scattering length a, which characterizes
the strength of the two-body contact interactions.

2.1 The ideal Bose gas
We begin our discussion entirely neglecting interactions. The equilibrium Bose distributionfunction isf (i ) =



∂Φ
∂µ



=

T,V

1

,
exp kiB−µ
T −1

(2.2)

where kB T is the thermal energy [37] (commonly defined by a temperature T and Boltzmann’sconstant kB ), i is the energy of a single-particle state, µ is the chemical potential, V is thevolume, and Φ = −kB T ln Z is the grand potential, where Z is the grand partition function.
In the case of non-interacting bosons, Eq. (2.2) gives the mean occupation of number of agiven state, and summing over all states simply yields the total particle numberN=X

f (i ) .

(2.3)

i

If the thermal energy is much larger than the energy difference between neighboring states(kB T  ∆i ), we can resort to the semi-classical approximation, treating the discrete energy
spectrum as a continuum, and replacing the sum over states with an integral over a densityof states g() (ignoring the quantum-mechanical ground-state energy). Within this approximation, g() encapsulates the thermodynamic behavior, and for all cases relevant here one finds
the form g() = Cα α−1 , where Cα is a constant and α is the thermodynamic scaling parameter.
From Eq. (2.2), we see that the need for positive occupation numbers defines a critical atom
number Nc (µ → 0 ), at which point the number of excited states saturates and the system
undergoes the statistical phase transition known as Bose–Einstein condensation [38, 39]. The

6

CHAPTER 2. THEORY OF BOSE GASES
critical atom number is given by
Nc =

Z ∞

g()f ()d = Cα ζ(α)Γ(α)(kB T )α ,

(2.4)

0

R∞

where ζ(α) is the Riemann function and Γ(α) =0xα−1 e−x dx (assuming <(α) > 0). For

N > Nc , since the excited states are saturated at Nc all of the remaining particles occupy theground state, forming a Bose–Einstein condensate, which can be treated separately3 .
We can rewrite Eq. (2.2) as an explicit function over phase space4
f (r, p) =1
exp[ (r,p)−µkB T ] − 1

,

(2.5)

where
(r, p) =p2+ U (r) .
2m

(2.6)

In addition to the free-particle dispersion relation we have included a potential energy term
U (r). We assume a general power-law form, which to simplify our discussion here we assume
to be spherically symmetric, that is U (r) = U0 rb . We can relate the form of this power-law
trapping potential directly to the density of states (captured by the thermodynamic parameterα), which yields
α=

3 3
+ .
2 b

(2.7)

For a uniform potential (b → ∞) α = 3/2 and for a harmonic trap (b = 2) α = 3.
Finally, we obtain the momentum and density distribution by (respectively) integrating Eq. (2.5)
over r or p: 3   

4π U0 − b31p2 f (p) =
Γg̃ 3µ−,
b kB Tb b kB T
2m
1nth (r) = 3 g̃ 3
λ 2





µ − U (r)
exp
kB T



,

(2.8)

(2.9)

where g̃` (x) is the polylogarithm of order ` and λ is the thermal wavelength.
Around the critical point λ is comparable to n−1/3 , which is intuitive as quantum effects
should become important when the wave packet size is comparable to the interparticle spacing.
Indeed, in an infinite uniform non-interacting gas the critical phase space density nc λ3 ≈ 2.6[see Eq. (2.9)].

3
4

In interacting systems this separation works to a first approximation.
This violates the Heisenberg uncertainty principle, and highlights the fact that it is a semi-classical approximation.

7

CHAPTER 2. THEORY OF BOSE GASES

2.2 Scattering theory
While the ideal gas description above lends some insight, atoms typically do interact witheach other. In general, one is faced with the daunting task of developing a full microscopic
description of the quantum-mechanical scattering problem. One of the most compelling aspects of ultracold atom experiments is that one can dramatically simplify this problem, andachieve conditions where the interactions between atoms can be described by a single parameter, known as the s-wave scattering length, a.
An extensive discussion of scattering theory, and the approximations required to formulate
such a simple description, can be found in many places (e.g. [8, 32, 40]); below we merely provide an outline.
Solving the Schrödinger equation for the relative motion of two particles (within the Born
approximation) yields the resultant scattering wave function

exp(ikr)
ψ(r) ≈ φk (r) +
r

z


f (k)

}|
{
mr
−
hφk’ (r)|Ṽ (r)|φk (r)i2π~2(2.10)

where mr is the reduced mass, φk (r) = exp(ik · r) and Ṽ (r) is the scattering potential. Physically, we interpret this as the sum of an incident plane wave and a spherically scattered wave
modulated by the scattering amplitude f (k).
The general form of Ṽ (r) can be rather complicated, however the short-range nature of thevan der Waals potential (falling faster than 1/r) facilitates progress. If we compare the range
of interaction rint with the gas’s relevant length scales (n−1/3 and λ), we find that for a dilutegas (rint  n−1/3 ) 5 , at degeneracy (n−1/3 ∼ λ), the angular momentum of the interacting
pair (approximated by ∼ hrint /λ), is small6 . This allows us to only consider the sphericallysymmetric l = 0 (s-wave) component [so that f (k) = f (k)], since the collisions lack the relative
momentum to significantly admix states with l > 0 into ψ(r).
Assuming that only elastic scattering events occur, the sole effect of the collision is to imparta phase-shift δ (bounded between ±π/2) on the l = 0 component, which defines the scattering
length a (in the limit k → 0) as

1
k cot δ = − ,a(2.11)The range of interaction is typically ∼ 100 a0 ≈ 5 nm [e.g. for 39 K the van der Waals radius rvdw ≈ 60 a0 ], while
for our typical densities n−1/3 ∼ 1 µm.
6Here we approximated the impact parameter bim of the interaction with rint , which overestimates the ratio since
bim < rint .
58

CHAPTER 2. THEORY OF BOSE GASES
and the scattering amplitude isf=ae2iδ − 1=−.
2ik
1 + ika(2.12)For bosons, the scattering cross-section can be obtained by integrating |f |2 over the solid angle,
also accounting for the bosonic factor of 2 (to ensure symmetric wave functions):σ=8π.
a−2 + k 2

(2.13)

In summary, we have found that Ṽ (r) (and more generally the entire scattering process),
however complicated it may be, is parametrized entirely by a. Consequently, given that a’pseudopotential’ reproduces the same a, its exact shape is unimportant, which is particularly
useful for theoretical considerations. Physically, this can be traced back to the fact that thecollision does not probe the short-distance behavior, and so we can simply reduce the entire
description into e.g. a contact interaction with a delta-function potential Ṽ (r) = gδ(r), whereg = 4π~2 a/m.
Finally, a remarkable feature of ultracold atom systems is that we can experimentally vary δ(and hence a) using molecular Feshbach resonances [7], by simply changing a magnetic field(see Section 3.2.1). From Eq. (2.11) we see that for |a| → ∞ the phase shift becomes maximal(|δ| = π/2), which means that the interactions are as strong as allowed by the laws of quantum mechanics; this is known as the unitary regime. As discussed in Section 3.2.1, Feshbachresonances directly give us access to this regime, as well as any other values of a along the way.

2.3 Theory of the condensate
In our discussion in Section 2.1 we considered a statistical description of the thermally excitedstates. We now turn to the Bose–Einstein condensate (BEC), a coherent macroscopic quantum
object, which we can describe by a macroscopic wave function

Ψ(r1 , r2 , ..., rN ) =NYφ(ri ),(2.14)

i=1

where the single particle wave functions are normalized [hφ|φi = 1].
Taking a step further, let us consider the general Hamiltonian of a Bose gas with two-body

9CHAPTER 2. THEORY OF BOSE GASES
contact interactions, which in second-quantized form is
Ĥ =

X
|

j

εj â†j âj +
{z

}

K̂

g X α † †Iijkl ai âj âk âl ,
2V
ijkl
|
{z
}

(2.15)

Iˆ

where K̂ and Iˆ are the kinetic and interaction energy operators (respectively) and we havemade use of an effective psuedopotential, characterized by g = 4π~2 a/m, neglecting the shortrange details of the two-body potential. Here â†j (âj ) is the creation (annihilation) operator for
R ∗ ∗
α =
the eigenstate αj , and Iijkl
αi αj αk αl d3 r, with wave functions normalized to unit volume.
The optimal basis choice for αi depends on the specific problem (see e.g. [41]), though we canalways rewrite the condensate wave function as a superposition of these eigenstates.
A convenient (and widely used, albeit slightly unphysical) choice is to assume periodic
boundary conditions (or an infinite system) and use momentum eigenstates7 . In this case the
Hamiltonian becomes
Ĥ =

X p2

2m

g X † †âp1 âp2 âp1 âp2 ,2Vâ†p âp +(2.16)

where the annihilation (creation) operators âp (â†p ) can be used to write the field operators as
plane waves
Ψ̂(r) =

X1âp √ eip·r/~ .
V
p(2.17)In the case of weak interactions (na3  1) it is insightful and often sufficient to resort to√a mean-field approximation, disregarding all operators with p 6= 0 and replacing â0 = N0 ;for simplicity, here we assume a pure zero-temperature gas where N = N0 8 . Within thisapproximation, we can readily identify the ground-state energy
E0 =

gN 2
,
2V

(2.18)and proceed along the typical lines to identify macroscopic thermodynamic quantities such as
the pressure
P =−

∂E0
= gn2 /2 ,∂V

(2.19)µ=∂E0= gn ,∂N(2.20)

chemical potential

7

While it is fundamentally unphysical, it does generally offer a good approximation assuming that kinetic energy
is negligible compared to interaction energy.
8
We turn to a description of both quantum and thermal depletion of the condensate in Section 2.4, while the moregeneral case of interacting partially condensed gases is beyond the scope of this thesis (see e.g. [17] as a starting
point).

10

CHAPTER 2. THEORY OF BOSE GASES
and the compressibility
∂n
1
1
=
=
,
∂P
gn
mc2
where c =q

(2.21)

gn
m is the speed of sound.

Time-independent Gross–Pitaevskii equation
Starting from Eq. (2.15), as long as a state (identified as the condensate) has a macroscopic
population much greater than all other states orthogonal to it, it is possible to show9 that thecondensate wave function Ψ is described by the time-independent Gross–Pitaevskii equation−~2 2∇ Ψ(r) + U (r)Ψ(r) + g|Ψ(r)|2 Ψ(r) = µΨ(r) ,2m(2.22)

which is a nonlinear Schrödinger equation with a cubic interaction term [and we have introduced a potential energy term U (r)]. Here µ = ∂E/∂N is the Hartree–Fock chemical potential
and the condensate density is n(r) = |Ψ(r)|2 . If we neglect the kinetic energy to make a connection to the infinite system results (known as the Thomas–Fermi limit), we find a densityn(r) =µ − U (r),g

(2.23)

which for a uniform box potential corresponds to a uniform condensate density inside the system volume V , gracefully decaying to zero10 at the trap edges to avoid an otherwise diverging
kinetic energy.
In experiments using quasi-homogeneous trapping potentials the density is also not perfectly
uniform due to the unavoidable smooth edges of the trapping potential, which can be describedusing a power-law potential of the form U (r) = U0 rb 11 . While it is typically sufficient to assume
a constant volume, it can be beneficial to assess the residual dependence of the effective volumeof the gas on its typical energy scale. To this end, we define an effective size reff using theb ∝ gn (within the Thomas–Fermi approximation). We then self-consistently
condition U0 reff
3 , such thatsolve for reff using n = N/Veff with Veff ∼ reff

reff ∝



gN
U0

 1

3+b

.

(2.24)

While this is only a rough estimate of the condensate size (as the prefactor and exact box shapematter), it is valuable in that given a measurement of the condensate size we can analytically9

See e.g. [8, 17, 41].
√
This change occurs over a distance set by the healing length ξ = 1/ 8πna .
11Here we assume spherical symmetry only for the simplicity of the argument.
1011CHAPTER 2. THEORY OF BOSE GASES
correct for slight size variations for experiments with different g, N , and U0 . A similar correction is obtained for a thermal gas, though as the temperature is independent of V one simply
1

has reff ∝ (kB T /U0 ) b [22].
Time-dependent Gross–Pitaevskii equation
For investigating the dynamics of the condensate, a natural extension to Eq. (2.22) is the timedependent Gross–Pitaevskii equation
i~

∂Ψ
~2 2
=−
∇ Ψ + U Ψ + g|Ψ|2 Ψ ,∂t
2m(2.25)

where under stationary conditions we require the time evolution of Ψ(r, t) to be set by e−iµt/~[i.e. demanding consistency with Eq. (2.22)]. It is important to recall that the GP equation is anonlinear Schrödinger equation with cubic nonlinearity, and above all else it is a wave equation. The GP equation is at the heart of our theoretical understanding weakly-interacting ofout-of-equilibrium Bose gases, and will be of utmost importance in Chapters 4 and 5. Curiously, it is possible to use Eq. (2.25) to arrive at equations of motion which are remarkably
similar to the Euler equations governing classical ideal fluid dynamics (i.e. in the absence ofviscosity and thermal conductivity).
The condensate satisfies the continuity equation for particle density12∂n+ ∇ · (nv) ,∂t(2.26)

where the velocity of the condensate is given byv=−
By writing Ψ =√i~ (Ψ∗ ∇Ψ − Ψ∇Ψ∗ )
.
2m
|Ψ|2

(2.27)

neiφ as a complex number in terms of its amplitude and phase, we identify
v=

~
∇φ ,
m

(2.28)

which shows us that the condensate dynamics satisfy potential flow, with a velocity potential~φ/m, and consequently highlights the profound physical trait that the motion of the condensate is irrotational [∇ × v = 0], apart topological defects in the form of vortex lines, whichfeature quantized circulation (in units ~/m) and a vanishing density along the vortex core (oc12This is relatively simple to show by multiplying Eq. (2.25) by Ψ∗ (r, t) and subtracting the complex conjugate fromthe resultant equation, further using n = |Ψ|2 and identifying v. Interestingly, it is also entirely independent of
the nonlinear term (as this is real and cancels).

12CHAPTER 2. THEORY OF BOSE GASES
curring in a regular manner over the healing length ξ).
Our next goal is to find an equation of motion for v. By inserting Ψ =√neiφ into Eq. (2.25),

retaining the real part (the imaginary part recovers the continuity equation), and taking thegradient, we get

m

∂v
= −∇
∂t



√
1
~2
√ ∇2 n + U
mv 2 + gn −
2
2m n



.

(2.29)

By introducing the pressure using the Gibbs–Duhem relation dP = ndµ, we finally arrive at∂v 1m+ ∇P = ∇
∂t
n






~2
mv 2
2√
√ ∇ n −∇
− ∇U ,
22m n(2.30)

which we can now directly compare to the classical Euler equation∂v 1m+ ∇P = mv × (∇ × v) − ∇
∂t
n



mv 2
2



− ∇U .

(2.31)Taking the restricted irrotational flow of our superfluids into account, the two equations
differ only in that the GP equation has an additional term on the r.h.s., known as the quantum
pressure term. Physically, this term originates from forces that arise due to spatial variations of
the magnitude of the condensate wave function, and it becomes important only when spatial
variations of the density occur on lengthscales shorter than the healing length.

2.4 Bogoliubov theory
In this section we give a brief overview of the pertinent aspects of Bogoliubov theory [42],which perturbatively extends our previous simple mean-field approximation to include thenext order corrections from Eq. (2.16), and forms the basis of our discussion in Chapter 6.
At the heart of Bogoliubov theory is the formulation of non-interacting quasi-particle excitations, which historically matched Landau’s original heuristic picture of interacting quantum
fluids. These quasi-particles feature the dispersion relation~2 k 2
~ω =
2m

r
1+

2
k2 ξ 2,(2.32)

which we plot as a function of k in Fig. 2.2 for 39 K at fixed ξ¯ = 1 µm. For large kξ, Eq. (2.32)
reduces to a quadratic dispersion relation and the excitations are particle-like, whereas at lowkξ the dispersion is linear in k and the excitations correspond to collective sound-waves.
To obtain this quasi-particle description from Eq. (2.16), one recasts the Hamiltonian into
quadratic form neglecting terms cubic and higher in âk 6= 0, which yields a description in

13

1.0

Δω/(2π) (Hz)ω/(2π) (kHz)CHAPTER 2. THEORY OF BOSE GASES

150
100
50

Δω

0
0

1

2
k k(1/ξ)
ξ

3

0.5

0
0

1

2

3

k kξ
(1/ξ)
Fig. 2.2 Bogoliubov dispersion relation. The solid black line shows the Bogoliubov dispersion relation ωB calculated for 39 K as a function of k [Eq. (2.32)], in units of 1/ξ, with fixed ξ¯ = 1 µm (so as to feature typicalexperimental values; for example a gas with n = 2 µm−3 and a = 400 a0 has ξ ≈ 1 µm). The dotted blueline shows√ the free-particle dispersion relation ω0 , and the red dashed line corresponds to ωp = ck, where¯ is the speed of sound. The inset shows the frequency shift from the free-particle dispersionc = ~/( 2 ξm)
2relation ∆ω = ωB − ω0 (dot-dashed line), which for large kξ¯ approaches a constant value ∆ωB = ~/(2mξ )(horizontal line).

terms of independent quasi-particles with energy  = ~ω, again having replaced k = 0 operators√
Pwith N and made use of particle number conservation â†0 â0 = N − k6=0 â†k âk . These newquasi-particles are bosonic in nature and described by annihilation (creation) operators b̂ (b̂† ),so thatĤ = E0 +X

(k)b̂†k b̂k ,(2.33)

k6=0

where the ground state energy is found to be
gN 2
E0 =
2V



128 √ 3√1+
na + ... ,15 π(2.34)

as first theoretically demonstrated by Lee, Huang, and Yang [43,44]13 , and recently experimentally confirmed by Navon et al. [45, 46].
This reduction of the many-body Hamiltonian to diagonal form is achieved using the Bogoliubov transformation∗âk = uk b̂k + v−kb̂†−k , â†k = u∗k b̂†k + v−k b̂−k ,(2.35)

where |uk |2 − |v−k |2 = 1 (to satisfy the bosonic commutation relations) and"1 + 1/(kξ)2p+1
2
1 + 2/(kξ)2

1
u2k =

13

This ∝#,#
1 + 1/(kξ)2p−1 .
2
1 + 2/(kξ)2

1
2
v−k
=

"

√
na3 correction to the mean-field energy is known as the LHY correction.

14(2.36)

CHAPTER 2. THEORY OF BOSE GASESThe number of quasi-particles is not conserved (µ = 0), and we assume a thermal population
Nquasi = hb̂†k b̂k i =1.
exp[/(kB T )] − 1(2.37)

This is to be contrasted to the number of particles in the condensate
N0 = N −X
k6=0

Vhâ†k âk i = N −
(2π)3

Z 

|uk |2 + |v−k |2|v−k | +exp[/(kB T )] − 1
2



dk .

(2.38)

Interestingly, the number of particles out of the condensate is non-zero even at T = 0; this is theinteraction-driven quantum depletion of the condensate, which arises entirely due to quantum
fluctuations. Performing the integral yields the famous prediction for the condensed fraction
8 √ 3
n0 /n = 1 − √
na .
3 π

(2.39)

Finally, it is insightful to further resolve the momentum distribution of the quantum depleted fraction, which is given by the Bogoliubov coefficient |v−k |2 (corresponding to pair-wise
excitations out of the condensate), and reads
1
nQD
k = 2

"

#
1 + 1/(kξ)2p−1 ,1 + 2/(kξ)2(2.40)

which is a distribution of typical width 1/ξ, and features power-law behavior at both low andhigh momenta:



nQD
k





1
1
.
, nQD
=
= √
k
4
4(kξ)
kξ→0
kξ→∞
2 2kξ

(2.41)

2.5 Tan’s contact
The two-body contact parameter C2 is the central quantity amid a powerful set of universal
relations that link the strength of short-range two-body correlations to the macroscopic thermodynamic behavior of a quantum many-body system. In particular, the contact connects
quantities such as the ground-state energy and the high-k behavior of the momentum distribution and spectroscopic responses. It was first uncovered by Tan [47–49] in the context of
ultracold Fermi gases, where it has been further explored theoretically [50–54] and verified
experimentally [55, 56] (see also [57–61]).
The description also extends to Bose gases [28, 36, 62–65], however the situation is complicated by Efimov physics which leads to three-body bound states [33, 34, 66] and three-bodycorrelations that cannot be inferred from knowledge of the pairwise ones [62, 63, 65]. Formally,

15

CHAPTER 2. THEORY OF BOSE GASES
universality is broken and requires the introduction of a second contact parameter, known as
the three-body contact C3 . However, away from unitarity coherent signs of the three-bodycontact are elusive, and one expects three-body physics to only slightly modify the scenario.
We restrict our discussion to two-body contact interactions, governed by the two-body contact
density C2 , to briefly illustrate the power of these universal relations.
The two-body contact is related to the ground-state energy E0 via a derivative with respect
to a 14 , known as the adiabatic sweep theorem
∂E0
~2
=
C2 ,
∂a
8πma2

(2.42)

which upon inserting E0 [Eq. (2.34)] gives
2C2 = (4πna)64 √ 31+ √na + ... .
3 π(2.43)

To give one concrete example, we can now simply obtain the high-k behavior of the momentum distribution

nk→∞ =

C2.
k4(2.44)

Indeed, this agrees with our Bogoliubov theory result [see Eq. (2.41)], which however only
captures C2 to mean-field level15 with C2 = (4πna)2 .

14

For comparison, the three-body contact is related to the ground-state energy via a derivative with respect to thethree-body parameter κ∗ (which is inversely proportional to the size of the Efimov state).
15
As typical for perturbation theory, the wave functions are ‘one step behind’ the energy.

16

3 Experimental Platform
In theory, there is no difference between theory and practice.
In practice, there is.
- anonymous -The experimental platform at the heart of this thesis is an ultracold gas of 39 K confined tothe quasi-uniform potential of an optical box trap. It combines key aspects of the two othermachines in our group: the general design, vacuum chamber, and atomic-species-specific apparatus are inspired by ‘BEC II’ (see [67–70]), while the quasi-uniform optical box trap closely
follows the set-up used in ‘BEC I’ (see [21, 71]). The aim of this chapter is to merely provide
an overview of our experimental platform and the tools at our disposal for manipulating andprobing ultracold samples, as the details of building and characterizing this machine are documented in two MPhil theses [22, 72] 1 .
We begin in Section 3.1 with a brief overview of how we produce quasi-pure Bose–Einsteincondensates in a quasi-uniform optical box potential, starting from a gas of 39 K at room temperature. In Section 3.2 we detail several key methods for probing and manipulating our ultracold
samples, including the versatile ability to tune the effective interparticle interaction strength bychanging an external magnetic field.

3.1 Sample preparation
Here we provide a brief overview of the initial stages of our experiment, describing how ourultracold 39 K samples are prepared; we refer the readers to [22, 72] for a more detailed accountof our machine (see also [68, 69, 71]). In Section 3.1.1 we give an overview of the experimental apparatus and provide an outline of the methods used to initially cool a gas of 39 K fromroom temperature to tens of nanokelvin. Section 3.1.2 illustrates how we sculpt our optical box
1

It should be noted that in Chapter 4 our experiments are performed on BEC I (using 87 Rb in a box), which weomit in our discussion here as the relevant experimental concepts are the same; for a more in-depth discussionof BEC I see [71, 73–75].

17

CHAPTER 3. EXPERIMENTAL PLATFORM

Fig. 3.1 Photograph of the main experimental apparatus. The final part of the experiment takes place within the‘science chamber’, a glass cell surrounded by the white cross-shaped Feshbach coil holder seen in the middle of the image.

trap and describes the loading of our pre-cooled atoms into this final trapping geometry, also
providing our benchmarks of its uniformity.

3.1.1 Bose–Einstein condensation
The vacuum system that comprises the backbone of our experimental apparatus consists of twochambers, connected by a narrow tube to allow for differential pumping (using two ion pumps
along the tube)2 . Our magneto-optical trap (MOT) is situated at one end, where each experiment begins by laser-cooling a sample of 39 K from background vapor3 . After initial trappingand cooling of the atoms in the MOT we use a gray molasses cooling protocol [76–78], whichenables sub-doppler laser cooling of the atoms to ∼ 10 µK. We then optically pump (using D1light) the atoms into the |F, mF i = |2, 2i hyperfine ground state before trapping them in a magnetic quadrupole trap and mechanically transporting ∼ 3 × 108 atoms to the ‘science cell’ at theother end. Figure 3.1 shows a photograph of our experimental apparatus, while Fig. 3.2 showsa detailed overview of the coils and laser beams in the region surrounding the science cell.
Once in the science cell, we proceed by ramping on an optical dipole trap (ODT), generatedusing a 20 W 1070 nm laser beam of ∼ 30 µm waist crossed with a recycled beam of waist∼ 100 µm. We subsequently switch off the magnetic trap to leave ∼ 10% of the atoms trappedin a purely optical trap. We perform a radio frequency Landau-Zener (LZ) [79, 80] sweep of the
2
3

We reach a pressure of ∼ 10−12 mbar in the science cell, manifesting itself as a one-body lifetime of > 100 s.
To maintain a steady background vapor gas of 39 K in the vacuum system we provide additional 39 K using getter
material that resides inside the vacuum chamber, and which we discharge daily by applying a current of 5.5 A
through the material for ≈ 80 s (see also [68]).

18

CHAPTER 3. EXPERIMENTAL PLATFORM(a)horizontal
imaging (y)(b)
quadrupole coil 1xfast coilpair 1
Feshbachcoil 1sheetsFeshbachcoil 2science cellyquadrupole
coil 1sheetsscience cell

fast coilpair 2z
quadrupole coil 2comp coil 1comp coil 3

tube

verticalimaging (z)

x

ODT

ODT

comp coil 2

Fig. 3.2 Schematic of the science cell surroundings. We highlight the essential laser beams and current carrying
coils used to trap and manipulate our atomic samples. (a) side view along -y and (b) top view. The repulsive
light used to form the cylindrical box potential (sheets and tube) is shown in green. The optical dipole trap
used for evaporation is shown in dark red, while the imaging light is shown in orange. The location of theatoms is indicated by the white circle. The Feshbach coils (blue) are used to create the uniform magnetic
field. The quadrupole coils (purple) initially trap and then mechanically transport the atoms to the sciencecell. Quadrupole coil 2 is used in conjunction with the three compensation coils (’comp coil’, brown) tomagnetically levitate the cloud. A pair of auxiliary ’fast coils’ (cyan) is positioned around each Feshbachcoil in order to minimize inductive coupling and allow fast switching of the magnetic field.

atoms from the |2, 2i state to the |1, 1i state, where a Feshbach resonance at a field of 402.70(3) G
occurs (see Section 3.2.1), enabling us to vary the scattering length of the atoms during theevaporative cooling process. This allows us to optimize the evaporation procedure, which
relies on scattering events to redistribute energy in order for cooling to occur. After ∼ 5 s ofevaporation, where the trap depth is decreased by a factor of ∼ 700, we form (harmonicallytrapped) BECs of up to ∼ 3 × 105 atoms, at condensed fractions of around 50% 4 .

3.1.2 Quasi-uniform potential
We then load the condensate into the cylindrical optical box potential formed by 532 nm laser
light, blue-detuned to ensure that the atoms feel a repulsive potential due to the dipole force(see e.g. [32]). The method used to sculpt the box potential directly follows the footsteps of [21],and is described in detail for our experimental set-up in [22]. In short, we begin by controllingand stabilizing the power of a near-gaussian beam coming directly from a commercial source5
using an acousto-optic modulator with a proportional-integral feedback control system. The
diffracted gaussian beam is then reflected off a single phase-only spatial light modulator (SLM)
in order to produce three spatially separated beams at a first focal plane: One hollow ring (the4Note that while lowering the trap depth further would result in more pure harmonically trapped samples, wefind that these parameters maximize the final atom number in the box.
5
We use a Laser Quantum Gem 532, which features a short coherence length (∼ 1 cm) so as to minimize interference effects.

19CHAPTER 3. EXPERIMENTAL PLATFORM
tube of the cylinder) and two thin sheets (its end caps). These beams are overlapped at aright angle and re-imaged to form a cylindrical box of light in which the atoms reside. We
ensure the focus and quality of our sculpted optical box by performing a wave-front correction
protocol [22], inspired by previous work [81] that exploits the tuneable nature of the SLM to
implement the Shack-Hartmann procedure [82]. To further ensure a uniform potential in thedark central region of the box we cancel gravity and the effect of any magnetic field gradientsto the ∼ 10−4 g0 level, where g0 is the gravitational acceleration. Our coils are also positioned toachieve low (negligible) residual magnetic field curvatures with associated frequencies ≈ 1 Hz
(trapping in a direction in the x-z plane and anti-trapping in the two orthogonal directions).
We achieve a quasi-uniform trapping potential, which is thermodynamically well describedby a r15(4) power-law. For most purposes in this thesis our potential can be well approximatedby a uniform cylindrical box of length L and radius R. The box dimensions are set by the SLM
parameters and are tuneable between experimental repetitions6 . We typically work with box
dimensions in the range (10 − 50) µm, for which we are able to ensure uniformity [22].
In order to form quasi-pure condensates in our box trap, we perform a final stage of evaporation where we lower the depth of the box trap to < kB × 50 nK. This typically produces
condensates of up to (2 − 3) × 105 , with condensed fractions of > 95% and a shot-to-shot standard deviation of around 3%. In order to reduce the atom number we simply increase thescattering length at the end of evaporation (before loading into the box) to make use of threebody recombination losses. This is beneficial as it is a self-normalizing process (i.e. one loses
more when starting with more); for details of our atom number control and stability see Section B. In order to prepare thermal samples we subsequently raise the trap depth again (whichsets the maximum temperature that can be reached due to evaporation) and violently oscillatea magnetic field gradient which ultimately leads to an equilibrium thermal gas at varying temperature and atom number (set by the shaking amplitude, trap depth, and initial atom number;see [22] for details)7 .

3.2 Manipulating and probing samples
Having reviewed how we prepare initially degenerate (or thermal) Bose gases in a quasiuniform box potential, we now turn to an overview of the tools used to manipulate and probe
our gases. In Section 3.2.1 we outline how Feshbach resonances allow us to vary the interaction
6

Our phase-only SLM (Hamamatsu X10468-04 with pixel size 20 µm and dimensions 800 pixels ×600 pixels) does
not allow for reliable dynamical changes on experimentally relevant timescales.
7
We use such a reheating protocol compared to directly loading a thermal cloud as this method avoids trapping
cold atoms outside of the box potential (e.g. in the tube), which are subsequently difficult to remove.

20

CHAPTER 3. EXPERIMENTAL PLATFORM

200

a (a0)

100
0
-100
-200

104
102
100
10-2
300

400

402
400
B (G)

500

Fig. 3.3 Controlling interparticle interactions using magnetic Feshbach resonances. We plot the magnetic Feshbach resonances for 39 K in the |1, 1i (red) and |1, 0i (blue) hyperfine state around 400 G. The approximate
resonance position is indicated by the vertical dotted lines, while the horizontal dashed line depicts the
background scattering length abg for |1, 1i. The open circles indicate the approximate zero-crossing of thetwo resonances. For the |1, 1i state B∞ = 402.70(3) G [65], ∆ = −52.25(4) G [26], and abg ≈ −29 a0 [83,84].
For the resonance in the |1, 0i state high-precision measurements are elusive, and we instead plot two independent theoretical predictions: solid - [83] and dashed - [84]. In the inset, we show a zoom-in of the region
around the |1, 1i resonance on a log plot. This reveals the remarkable feature that while the scattering length
in the |1, 1i state varies from ∼ 400 a0 to ∞, the |1, 0i state is essentially non-interacting (|a| < 10 a0 ), but
repulsive (a > 0).

strength by changing the magnetic field strength. Section 3.2.2 describes our absorption imaging protocol used to photograph (and ultimately destroy) the sample after every experimental
cycle. We also include a brief summary of two invaluable spectroscopic tools: radio-frequency
spectroscopy in Section 3.2.3 and Bragg spectroscopy in Section 3.2.4.

3.2.1 Tuning interactions
Owing to the existence of molecular Feshbach resonances [7], we are able to tune the effective
scattering length a by varying an external magnetic field. This remarkable feature makes ultracold atom experiments unparalleled in terms of studying the effect of interactions in quantum
many-body problems.
Feshbach resonances
Feshbach resonances occur due to resonant behavior when the energy of a pair of particles issimilar to that of a molecular bound state, in an energetically forbidden quantum state. Since
the energy depends linearly on the magnetic field strength (due to the Zeeman effect) this leads
to a second-order perturbation-theory-type dependence of a on B given bya(B) = abg 1 −

21


∆
,
B − B∞(3.1)

CHAPTER 3. EXPERIMENTAL PLATFORM
where abg is the background scattering length, B∞ the resonance position and ∆ the resonance width, defined as the distance in field between B∞ and B0 , where a(B0 ) ≡ 0 8 . In Fig. 3.3
we show a sketch of the Feshbach resonances for both |1, 1i and |1, 0i states of 39 K at around
400 G. We perform most of our experiments using spin-polarized samples in |1, 1i, though the|1, 0i state acts as an invaluable resort.
Field control
As we have seen the problem of tuning a has been reduced to a problem of controlling themagnetic field strength. We use a set of electromagnetic coils (see Fig. 3.2) to create a tuneable
magnetic field, that is essentially uniform apart from a magnetic field gradient used to cancel
the gravitational force that the atoms experience (for details see [22]). To vary B we simply
vary the current in the coils, while striving to maintain gravity (and other field gradient) compensation throughout. For our main coils these field variations are limited to & ms timescales.
For situations where faster field quenches are desired, we additionally utilize a set of auxiliary‘fast coils’ that allow rapid field quenches in a few µs. A detailed characterization as well as
additional technical details can be found in Section D.2 (see also [86]).

3.2.2 Imaging & measurement
Our experiments are performed in cycles, where for every experimental realization we begin
with a room temperature gas of 39 K and subsequently produce quasi-pure condensates within∼ 20 s 9 , as described above. At the end of each cycle, we record absorption images [32] of theatomic samples, providing a glimpse of its properties (such as atom number or kinetic energy
per particle), before ultimately destroying the sample.
We image the atoms at low field, first optically pumping them back into the |F = 2i state10

before resonantly imaging them using the |F = 2i → |F 0 = 3i transition along one of two directions [viewing either along the axis of the cylinder or in a perpendicular direction (from thetop)]. Our imaging is performed using low intensity, with I ≈ 0.1 Is , where Is = 1.75 mW/cm2
is the saturation intensity of the transition. The absorption imaging essentially integrates overthe line-of-sight, and so we obtain the two-dimensional column density after conversion from8

Feshbach resonances can also occur when using other methods to shift the energy levels with respect to eachother; for example optical potentials can be used to achieve similar feats (see [85] for an overview).
9
The typical duration of an experimental cycle ranges between 30 - 40 s, depending on the experiments performed.
10
Our pumping uses D1 light with σ + polarization, which pumps atoms to |F 0 = 2, m0F = 2i in the 42 P1/2 state.
The atoms subsequently decay back to the |F = 2i manifold, but not all of the atoms end up in the |2, 2i state (asignificant fraction also decays to |2, 1i). We have calibrated the resulting error in the measured atom number
by instead using an essentially 100% efficient LZ transfer back to the |2, 2i state. We find that the atom number
measured this way is 1.32(1) times that measured using the optical pumping protocol.

22CHAPTER 3. EXPERIMENTAL PLATFORM
(a)

(b)

(c)

(d)

0

OD

≥3

Fig. 3.4 Absorption images of homogeneous Bose–Einstein condensates trapped in a cylindrical optical box. (a,b)
In-situ images for horizontal and vertical imaging, respectively (the actual in-situ optical density is > 100
however our imaging saturates at about 3). Scale bar corresponds to 30 µm. (c,d) Time-of-flight absorption
images, allowing the clouds to expand in the presence of interactions. Image adapted from [22].

the measured optical density (where we assume an ideal imaging cross-section, taking into
account the pumping efficiency and the finite imaging intensity).
In Fig. 3.4 we show examples of typical absorption images of box-trapped quasi-pure homogeneous Bose–Einstein condensates along these two directions, both in-situ and following
ToF expansion in the presence of interactions. The in-situ images highlight the cylindrical box
shape, while in ToF (in the non-symmetric direction) the characteristic diamond-like shape withflattened edges clearly reveals the quantum nature of the gas [71, 87, 88].

3.2.3 Radio-frequency spectroscopy
In order to accurately tune the interparticle interaction strength we require precise control ofthe magnetic field that the atoms experience. Using ordinary field probes is not possible due tothe fact that the atoms reside in a vacuum chamber that cannot regularly be opened. However,
our atomic samples themselves are extremely sensitive to magnetic fields, and thus provide thebest magnetic field probe at our disposal.
An external magnetic field leads to a hyperfine splitting between the different internal states,
which can be calculated using the Breit–Rabi formula [89], and is plotted for 39 K in Fig. 3.5(a).
Outside of the Paschen–Back regime (high-B), the energy differences between the states are
field dependent, which allows for a direct probe of the magnetic field using radio-frequency(rf) photons. Here we confine our discussion to rf-driven transitions between the |1, 1i and the|1, 0i state, which are suitable for our field calibration but also have the additional advantage of23

CHAPTER 3. EXPERIMENTAL PLATFORM(a) (b) (c) (d)m(a)F, (b)(c) (d)ΔEHF (GHz)

1.0

2, 2
2, 1
2, 0
2,-1

0.5

F

1, 0
1.1

0.0
-0.5
-1.0
0

2,-2
1,-1
1, 0
1, 1

100

200 300
B (G)
(G)
B

(a) (b) (c) (d)

400

1

2

≥3

OD

(a) (b) (c) (d)TF (%)

N (104 )

0

500

8
4
0
0

1.0
1, 0

z

0.6
0.3
0

60
τt (µs)

120

98.72
98.75
98.78
ωrf /(2π) (MHz)Fig. 3.5 Radio-frequency spectroscopy. (a) Hyperfine structure in the 42 S1/2 ground state of 39 K as a function ofB, calculated using the Breit–Rabi formula. (b) Measured evolution of N in |1, 1i (blue) and |1, 0i (red) for
an on-resonance rf-pulse of variable pulse duration τ , exhibiting a coherent Rabi oscillation between thetwo states. (c) ToF absorption image following a π/2 pulse for a cloud at a magnetic field of 399.38(1) G[inferred from (d) and indicated as the red vertical line in (a)]. The difference in relative size of the clouds
clearly highlights the dramatic difference in interaction strength between the two states, and the two states
separate in space due to their slightly different magnetic moments at these intermediate field strengths(we use 45.08(3) ms ToF expansion). The scale bar corresponds to 50 µm. (d) Radio-frequency spectrum,plotting the transferred fraction (TF) versus applied rf-frequency ω̃rf /(2π). We extract a resonance frequency
ωres /(2π) = 98.7471(1) MHz, corresponding to a 2-mG statistical uncertainty in B.

enabling a fast interaction switch (see Section 3.2.1 and [29, 65, 90]). The splitting between |1, 1i
and |1, 0i grows monotonically in field, and for the range of fields of interest to us (350 - 450) G,
the sensitivity of our RF spectroscopy (set by the slope of the energy-level difference) ranges
between 63 and 38 kHz/G.
Throughout an rf-pulse the system evolves as a superposition of the two states, exhibiting
Rabi oscillations between the two states [32]. For a system initially spin-polarized in |1, 1i, thetransition probability to |1, 0i takes the form
P0 =

p
Ω2
2
sin
(
Ω2 + δ 2 τ /2) ,δ 2 + Ω2

(3.2)

where τ is the rf-pulse duration, δ = ωres − ω̃rf is the detuning of the applied rf-frequency ω̃rf
with respect to the resonance frequency ωres (set by the energy level splitting11 ), and Ω is theon-resonance Rabi frequency. In Fig. 3.5(b) we show a measurement of such an on-resonanceRabi oscillation with a π-pulse time τπ = 53.6(1) µs, corresponding to Ω/(2π) ≈ 9.3 kHz.
√
From Eq. (3.2) we see that the effective Rabi frequency Ωeff = Ω2 + δ 2 slightly increases off
11

While rf spectroscopy is also sensitive to differences in interaction energy between the clouds in the two states,
these shifts are typically . 100 Hz, corresponding to only a few-mG systematic shifts.

24

CHAPTER 3. EXPERIMENTAL PLATFORM
resonance, while the maximal transfer probability is suppressed. For our perturbation Ĥp =Ĥp0 cos(ω̃rf τ ), the Rabi frequency is given by the matrix element between the two states ~Ω =√h1, 1|Ĥp0 |1, 0i, which experimentally translates to a scaling Ω ∝ Prf , where Prf is the rf-power
supplied12 .
As illustrated by the ToF absorption image in Fig. 3.5(c), performing a ToF expansion realizes
a Stern–Gerlach experiment [92], as the two spin states separate in space due to their different
magnetic moments at these intermediate field strengths. We compensate gravity and any residual magnetic field gradients to within 2 × 10−4 g0 for the |1, 1i state, which leaves the |1, 0i statewith the ratio of magnetic moments as the residual acceleration. At a field of 399.38(1) G, as
used in Fig. 3.5(c), this corresponds to feeling a residual 3.6% of gravity, which for our ToF of
45.08(3) ms predicts a separation of 360(1) µm, in agreement with the measured displacementof 358(4) µm 13 . The fact that the two spin components separate in a clean manner without any
apparent collisions is in line with the relatively weak inter-state scattering length, which for therange of fields explored here is positive but below 10 a0 [84].
In Fig. 3.5(d) we show a typical example of a high-resolution rf-spectrum, used to calibrate
the magnetic fields at which we work. We achieve a precision on the single-mG level, which issmaller than typical absolute field drifts (of order ∼ 10 mG) over the course of a day. Note thatin order to achieve these precisions, we are required to trigger our measurements on the 50 Hz
alternating-current mains cycle, which feeds into our coils and leads to a 50 Hz (and multiples
thereof) modulation of the field that our coils produce, with a non-negligible amplitude of≈ 100 mG 14 . A detailed characterization of this oscillation, as well as further information
about our field control and stability can be found in Section D.1.

3.2.4 Bragg spectroscopy
In the context of ultracold atomic gases, Doppler-sensitive two-photon Bragg scattering [93,94]
has proven to be a versatile spectroscopic probe, and it has become a standard tool in the arsenal of many modern experiments. In contrast to the historical Bragg scattering of x-rays
off periodic atomic lattices [95, 96], here instead the atoms scatter off a periodic light potential (obtained by interfering two laser beams), which selectively imparts momentum to theatoms. In this section we provide an overview of the experimentally relevant aspects of Bragg
12

The rf antenna is formed by a single  3.5 mm wire wound into a circular  70 mm coil, designed to achieve
maximal emission signals at around 100 MHz. We use a variable capacitor to achieve impedance matching;see [91] for details.
13
We have separately calibrated the apparent pixel size spix using the mechanical transport track to controllably
move the cloud in a magnetic trap (and cross-calibrated both our imaging systems this way); this could otherwise also be achieved using the measurement presented here (or generally the free fall of a cloud under gravity).
For horizontal imaging we have spix = 2.59(3) µm, whereas for vertical imaging spix = 1.60(4) µm.
14This issue could be readily overcome by running the power supplies on batteries.

25

CHAPTER 3. EXPERIMENTAL PLATFORM(a) (b) (c) (d)EnergyEnergy/ℏ(a) (b) (c) (d)
Δ

y

|~ki
|~ki
i

ω1

θ2

Δω

δ

ℏq
ω1

|~ki
|~ki
ℏ(k+q)

Momentum

ℏq

θ3

ω2

ω2

|~ki
ℏk

x

θ1

Fig. 3.6 Bragg spectroscopy concepts. (a) Sketch of laser arrangement for performing Bragg spectroscopy (not toscale). The two Bragg beams couple the atomic states |~ki and |~(k + q)i via an intermediate state |ii.
The resonance condition is given by δ = 0, and depends on both the initial momentum and the dispersion
relation. (b) Top-view cartoon of the orientation of the two laser beams intersecting on our atomic sample(blue) confined in the cylindrical box trap (green). The cartoon is not to scale: in reality the box dimensions are . 50 µm and the beam widths are ≈ 1 mm. The beams form a moving optical lattice of period
2π/q and speed ∆ω̃/q (not shown). We highlight our most used laser configuration with θ2 = 133.6(1)◦ ,which approximately achieves an axial momentum kick. We make use of a third counter-propagating beam
overlapped with ω2 (so that θ3 = π), to easily obtain access to three distinct q.

spectroscopy; a detailed description of the theory of two-photon atom-light interactions can befound in many places (e.g. [32]), and the more technical aspects of our Bragg spectroscopy set
up are detailed in [91, 97].
The essence of Bragg scattering can be understood by simple means of energy and momentum conservation. As depicted in Fig. 3.6(a), let us consider an artificial three-level system
consisting of two momentum states (from the continuum) that are coupled by two laser beams
of frequency ω1 and ω2 via an intermediate state |ii (from which we are detuned by ∆). The
two beams essentially have the same wavelength (and corresponding recoil momentum krec )apart from a small frequency difference ∆ω̃ = ω1 − ω2 . The beams are intersected on the atoms
at an angle θ, so that the corresponding wave vector is q = k1 − k2 with its magnitude given
by

 
θ
q = 2 sinkrec .
2(3.3)

In a stimulated process the atom initially in momentum state |~ki can absorb a photon from

ω1 and coherently re-emit into ω2 , which imparts a momentum ~q to the atom. This process isresonant (δ = 0 so that ∆ω = ∆ω̃) when energy is conserved~∆ω =~2 q 2~2+ ∆Eint + k · q ,
2m
m

(3.4)

where the three terms correspond (left to right) to: the recoil energy acquired by a diffracted
atom, the difference in per-particle interaction energy between the two states (see e.g. Section 2.4), and the energy shift arising from the Doppler effect when the atom’s initial velocity

26

CHAPTER 3. EXPERIMENTAL PLATFORM
has a component along q.
During a Bragg pulse the two atomic states |ki and |k + qi in our artificial three-level system
are coherently coupled by the Bragg beams and thus undergo Rabi oscillations as a function of
the pulse duration τ , where the probability to be diffracted to |k + qi is
PB =

hp
Ω2
τi
2
2 + δ2sinΩ.
Ω2 + δ 2
2(3.5)

Here Ω is the two-photon Rabi frequency, which is related to the single-photon Rabi-frequenciesof the two individual transitions via Ω ' Ω1 Ω2 /(2∆) (provided |∆|  Ωi , with i ∈ {1, 2}, whichis satisfied throughout).
Experimentally, we use D2 laser light with a wavelength λD2 = 766.7 nm, so that krec =2π/λD2 ≈ 8.2/µm. The intensity of the two beams is kept roughly equal (to within . 5%), thebeams are ≈ 1 GHz red-detuned from the D2 transition, and have π̂ polarization. These parameters ensure that losses due to spontaneous emission are minimal; the expected fraction lost perπ-pulse is πΓ/|∆| ≈ 2%, where Γ/(2π) ≈ 6 MHz is the line width of the intermediate state. In

our experiments in Chapter 6 we make use of three angles {θ1 , θ2 , θ3 } = {66.4◦ , 113.6◦ , 180◦ }[see Fig. 3.6(b)], which gives us access to three different q with q ∈ {1.1, 1.7, 2.0} krec , and cor-

responding recoil velocities |vr | = ~|q|/m ∈ {14.6, 22.3, 26.7} µm/ms. In the absence of any interaction shifts the free-particle resonance frequencies are ∆ω/(2π) ∈ {10.43, 24.37, 34.81} kHz.
The two-photon Rabi frequency Ω is linked to the experimental parameters via
Γ2 I
,
4∆ Is

Ω=

(3.6)

which crucially is proportional to the beam intensity.
Equipped with our three-level system analysis let us now consider the full continuum of
momentum states that we experimentally probe. For the purposes here we restrict our discussion to the use of q2 , which is aligned with the axial length of our cylindrical box, so weeffectively probe the integrated one-dimensional momentum distribution of the cloud alongŷ. For for a homogeneous pure BEC the distribution is sinc2 -like with a Heisenberg-limited
momentum width ∼ ±~/L, corresponding to a spectral full-width-half-maximum (FWHM)
0.44 × 4πvr /L [87]. Our spectroscopic resolution is set by Ω, corresponding to a momentum
resolution of Ωm/q. To perform high-resolution Bragg spectroscopy we thus want to minimizeΩ and maximize τ , while keeping the diffracted fraction . 10% so that our experiments can beinterpreted within linear-response theory [17].
In Fig. 3.7(a) we show an example of a typical Bragg spectrum of a weakly-interacting sam-27

CHAPTER 3. EXPERIMENTAL PLATFORM

�� (��-�)(a) (b)�� (c) (d)(a) (b) (c) (d)

�

ℏq2

�
�
��

��
��
Δω/(2π)Δω/(�π) (kHz)
(���)

0

OD

≥3

Fig. 3.7 Bragg spectroscopy. (a) Bragg spectrum showing the recorded diffracted fraction (DF) as a function of
the frequency difference between the beams ∆ω̃/(2π). The spectroscopy is performed with a 1-ms Bragg
pulse with wave vector q2 on a weakly-interacting cloud with a ≈ 70 a0 and n ≈ 2.0 µm−3 , in a boxof length L ≈ 50 µm. We use a gaussian fit to extract the resonance position ∆ω/(2π) = 24.39(2) kHz
and full-width-half-maximum σFWHM = 1.13(4) kHz. (b) Absorption image of the Bragg diffraction of an
essentially non-interacting homogeneous sample after 15 ms time-of-flight showing the stationary (bottom)
and diffracted (top) clouds. The cloud is prepared as in (a) but for aesthetic purposes we perform a 64 µs
π-pulse to the |1, 0i state, before applying the Bragg pulse. Scale bar corresponds to 50 µm.

ple, where we are still sufficiently deep in the Thomas–Fermi regime so as to keep the density
quasi-uniform, but where interaction shifts are essentially negligible. We obtain a resonancefrequency ∆ω/(2π) = 24.39(2) kHz (extracted using a gaussian fit), which demonstrates thefact that the interaction energy is near-negligible (indeed we expect only a 20 Hz shift), andthe displacement following ToF is in line with vr . To obtain a Heisenberg-limited width (whichin this case would correspond to σFWHM ≈ 400 Hz) we would also need to avoid any Fourier
broadening and use pulse times τ & L/vr ≈ 2.2 ms, which equivalently ensures that a recoilingatom can traverse the box during the pulse (and leave assuming that the recoil energy is larger
than the trap depth, as is the case here)15 .

15

While measurements with τ & 4 ms do recover Heisenberg-limited widths to within < 10% (see also [87]), we
empirically found the best signal-to-noise (for fixed ≈ 100 images), and ultimately precision on ∆ω, by allowing
for slightly larger Ω but accepting slight Fourier broadening by using τ between 1-2 ms (see also Fig. 6.7).

28

4 Turbulent-Cascade Fluxes
Big whirls have little whirls
that feed on their velocity,
And little whirls have lesser whirls
and so on to viscosity- Lewis F. Richardson [98] -4.1 Introduction
A continuously driven fluid can become turbulent, exhibiting irregular fluctuations. This familiar yet complex phenomenon is observed in a myriad of physical systems ranging from
biology to astrophysics; examples include arterial blood [99], surface water waves [100–104],
clear air [105], optical beams [106–109], supernovae [110], and interplanetary media [111, 112].
While turbulence is truly ubiquitous, its multifaceted nature also makes it difficult to define
concisely. Turbulence generally describes a field that is in a state far from equilibrium, which
involves many interacting degrees of freedom, spans vastly different lengthscales, displays
chaotic properties, involves drive and dissipation, is statistically steady, and displays local
restoration of symmetries (isotropy and homogeneity)1 .
The study of turbulence is a long-standing endeavor, with early observations dating back to
da Vinci in the early 16th century [114] and first quantitative experiments being performed by
Reynolds in the late 19th century [115]. Throughout the years it has attracted considerable crossdisciplinary attention, however even today it remains a somewhat mysterious phenomenon.
A foundational breakthrough occurred in 1941, when Kolmogorov and Obukhov discovered a universal law describing the transfer of energy from large to small lengthscales in turbulent flows [116–118] 2 . In contrast to their extremely complex real-space dynamics, turbu1

Note that all of these properties are not necessarily fulfilled, a particularly famous example being intermittency
bursts, breaking the statistically time-invariant nature of the state (see e.g. [113]).
2
Note that in certain cases inverse cascades can also occur, where energy flows from small to large lengthscales
instead; two-dimensional fluids are a famous example in which this is the case [119–122].

29

CHAPTER 4. TURBULENT-CASCADE FLUXES

0

~1/kF

~1/kD

Onset ofsteadystate?

Time

td

Fig. 4.1 Cartoon of turbulent cascade dynamics. For illustration, we consider a compressible field in real space,
where from time ts = 0 onwards energy is injected at a large lengthscale (1/kF ). The nonlinear interactions then propagate the energy to ever smaller lengthscales (larger k), and once the excitations reach thedissipation lengthscale (1/kD ) at time td , a steady-state can be established. Figure from [27].

lent flows often feature comparatively simple Fourier-space dynamics, where scale-invariant
cascade-fluxes transport energy across very different lengthscales, sustaining scale-free steadystate behavior across the propagation lengthscales.
A typical scheme for generating turbulence is by injecting a disturbance into a field at a specific length scale. As shown in Fig. 4.1, the (in this case compressible) field initially at rest is continuously forced (at times t > 0) at large lengthscale 1/kF . Nonlinear interactions then promote
the excitations to ever smaller lengthscales (larger k) in relay fashion until they reach the dissipation lengthscale 1/kD (at time td ), at which point the field is fluctuating on all lengthscales
between 1/kF to 1/kD 3 . The interplay between driving and dissipation can, rather miraculously, lead to steady-state properties on these intermediate lengthscales known as the inertial
range. If such a steady-state is achieved, conservation of energy simply dictates that energy
is dissipated at kD at the same rate at which it is injected at kF . Crucially, within the inertial
range universal transport laws emerge, responsible for turbulent cascades in which conserved
quantities such as the energy or wave density obey k-space power-law distributions.
Even though not always understood from first principles, this simple picture has provento be remarkably universal, and such turbulent cascades have been observed in a plethora ofdifferent systems, including tidal channels [123], solar wind [124], interplanetary plasma [112],
liquid helium [125], dwarf galaxies [126], financial markets [127], and the hippocampus [128],
just to name a few. In comparison, the underlying scale-invariant cascade-fluxes are harder tomeasure as their extraction typically relies on non-trivial assumptions [129–132].
3

As a familiar classical example, consider water flowing around a pillar. The velocity field is perturbed on thescale of the pillar’s diameter. Now, further down the flow, large eddies interact with each other forming eversmaller eddies. This transfer of energy to smaller scales happens until the energy is ultimately dissipated at verysmall scales, where inertial forces are overcome by viscous ones.

30

CHAPTER 4. TURBULENT-CASCADE FLUXES
Turbulence in quantum (superfluid) systems features two profound differences as compared
to classical fluids. First, superfluids have no intrinsic dissipation scale, which alters the conventional picture of how dissipation can occur [133]. Secondly, superfluids exhibit highly restrictedflow properties, as the flow is irrotational apart from vortex lines with quantized circulation (inunits ~/m), in which the density must vanish along the core in a well-defined manner featuring a regular core structure (see Section 2.3 and e.g. [8, 17]). It is thus natural to decompose thekinetic energy of the fluid into two parts [134]: a compressible but irrotational one associatedwith wave-like excitations, and an incompressible one associated with the vortex dynamics.
The vortices present in quantum systems are topological defects; they cannot terminate within
the fluid and must either form loops or terminate on the walls of the container, and they cannot
simply decay4 . Vortex structures are responsible for peculiar dynamics at the heart of so-called
quantum turbulence [140–144] 5 , and more generally the absence of intrinsic dissipation andthe highly restricted flow raise fundamental questions about the character of turbulent cascades
in quantum fluids (see e.g. [142, 147]).
Over the last few decades in the emerging field of turbulence in quantum (superfluid) systems many experiments were performed on liquid helium, exploring both vortex [125,142,148–
150] and wave turbulence [151–153]. A particularly exciting observation was the existence of a
Kolmogorov energy spectrum ∝ k −5/3 [125] in incompressible-flow-dominated superfluid helium, highlighting universal parallels to classical-fluid turbulence [118]. Steady progress in thefield has been ensured by inventive techniques, such as the addition of tracer particles to visualize quantum vortices directly [154] and the ability to probe the vortex line length with second
sound attenuation [155]. However, unfortunately, liquid helium is a strongly-interacting fluid,
for which only approximate theoretical descriptions exist, rendering a first-principles understanding of the experiments intractable.
More recently, there has been a fast-growing interest in the possibility to study turbulencein the highly controllable setting of ultracold atomic gases [27, 156–168], which are more suitable for a fruitful interplay between experiment and theory. Experimentally, first qualitative
evidence for turbulence in ultracold atomic gases was observed in dramatically perturbed
harmonic traps [156], but quantitative comparisons with theory were hindered by the inhomogeneous density imposed by the harmonic trap. The advent of uniform-density quantum
gases [21–25] has proven to be a considerable advance in establishing quantum gases as plat4

Vortex lines can sustain helical deformations (known as Kelvin waves) [135], and interact with each other via
vortex reconnections [136, 137]. In both cases this leads to sound-wave emission and thus provide an energy
transfer mechanism between incompressible and compressible flows [138, 139].
5
Onsager uncovered the first theoretical links between turbulence and point vortex dynamics in 2D fluids [145],
and later Feynman envisioned tangles of vortices at the heart of quantum turbulence in 3D [146].

31

CHAPTER 4. TURBULENT-CASCADE FLUXES
forms for elaborate studies of quantum-fluid turbulence [27, 162, 166, 167]. A major reason forthis is that weakly-interacting zero-temperature Bose gases are well described by the classicalfield Gross–Pitaevskii (GP) equation, a nonlinear Schrödinger equation with cubic nonlinearity. This nonlinear wave equation is frequently used to model turbulence in quantum fluids [134, 169–173] and more generally constitutes a universal model of turbulence [169], withstriking similarities to the classical Euler equation (see Section 2.3).
In this chapter we present studies of a turbulent cascade in a box-trapped quantum gas[27,162], where we exploit unique features of ultracold atom systems to measure properties thatare otherwise not easily accessible. In Section 4.2 we provide the experimental context of earlier
work observing the emergence of a turbulent cascade in our box-trapped quantum gas [162],
which provides the starting point for our work studying the underlying cascade fluxes [27]. In
Section 4.3 we outline the pertinent theoretical framework in order to directly link dissipationin the gas to the cascade fluxes. Exploiting the synthetic dissipation scale in our gas, we study
the cascade-induced atom-loss dynamics in Section 4.4, which reveals both particle and energy
fluxes in steady state and provides insight as to how the cascade front propagates in momentum space before a steady-state has been established. Throughout we confront all our findings
with ab-initio simulations of the GP equation. While the complex far-from-equilibrium quantum dynamics that occurs in our system are not necessarily captured by this simple model, ourexperiments directly probe its regime of validity. Finally, we conclude in Section 4.5, outlining
avenues for future research.

4.2 Emergence of a turbulent cascade
We begin by reviewing our procedure for initiating a turbulent cascade in our box-trapped
quantum gas (following [162]), which sets the scene for our more recent studies on the underlying turbulent cascade fluxes [27]. Our experiments start with a quasi-pure Bose–Einsteincondensate of 87 Rb atoms in the uniform potential of a cylindrical optical box trap of radiusR ≈ 16 µm and length L ≈ 27 µm 6 . We prepare the atoms in the |F, mF i = |2, 2i hyperfine
ground state, where they are weakly-interacting with an s-wave scattering length a ≈ 100 a0 .
For our typical atom numbers of N ∼ 105 the healing length is ξ ∼ 1 µm and so the condensate
density is quasi-homogeneous7 . The corresponding chemical potential is µ ∼ kB × 2 nK, whichsets the gas’s natural energy scale.
The experiments in this chapter were performed using the first-generation box experiment [21], which uses 87 Rb
as opposed to 39 K (see [71, 73–75] for experimental details).
7
The homogeneity is also limited by the imaging resolution of our optical setup used to create the optical box trap;
for a characterization of the trapping potential see [21, 71].
632

CHAPTER 4. TURBULENT-CASCADE FLUXES(a) (b) (c) (d) (e) (f)(a) (b) (c) (d) (e) (f)(a) (b) (c) (d) (e) (f)

Force

xyz

xyz
xyz

Time

Fig. 4.2 Initiating the turbulent cascade. (a) Gross–Pitaevskii equation simulations of the shaking protocol used
to initiate a turbulent cascade in our box-trapped Bose gas. As one continuously drives the lowest-lying
axial mode, the system transitions from unidirectional sloshing to an isotropic turbulent state. The blue
shading indicates the gas density and the red lines indicate vortices. (b,c) Experimental absorption images
taken along x after 100 ms ToF expansion, with N ≈ 8 × 104 atoms. (b) Initial quasi-pure Bose–Einsteincondensate and (c) after shaking for 2 s at 8 Hz with amplitude F0 L/µ ≈ 1.2. The white dashed circle in(c) shows the size that would constitute an expansion energy of kB Tc /2, highlighting that E  kB Tc . Figure
adapted from [162] c Nature Publishing Group.

As illustrated by the simulations shown in Fig. 4.2, the turbulent cascade is initiated by resonantly driving the lowest-lying axial Bogoliubov sound mode of wavelength 2L (so kF = π/L)
using a spatially uniform force Fs (r, t) = F0 sin(ωs t)ŷ, where ŷ is the axial unit vector8 . Thesesimulations reveal that as one continuously excites the gas, the initially unidirectional sloshingtransforms into irregular, statistically isotropic motion over many ever smaller lengthscales9 .
Experimentally, electromagnetic coils provide the magnetic field gradient that creates a potential Vs (r) = F0 y and after resonantly [ωs /(2π) ∼ 10 Hz] exciting the condensate for several
seconds with an amplitude F0 ∼ µ/L the initially anisotropic, diamond-shaped ToF absorption images [Fig. 4.2(b)] also become statistically isotropic [Fig. 4.2(c)]. This provided the firstqualitative indication of a kinetic-energy-dominated turbulent state where phase coherence isdestroyed. It is important to distinguish this far-from-equilibrium state from an equilibrium
thermal state, which is also isotropic and would be achieved if the gas was heating to temperatures T > Tc . From the second-moment of the ToF images the total (mostly kinetic) energy per
particle was found to be E ≈ 0.12kB Tc (here Tc ≈ 50 nK), and so an equilibrium gas with thesame E would still be deeply degenerate (with a condensed fraction η ≈ 0.7). Moreover, as a
proof, if the forcing is stopped the phase coherence of the gas gradually re-establishes itself anda BEC with η ≈ 0.7 reforms [162]. These relaxation dynamics constitute a fascinating problem
in their own right, but here we will restrict ourselves to the establishment and the steady-state
properties of the turbulent state.
8

The fact that the excitation spectrum is discrete constrains how the excitations can propagate to larger k, and thefull microscopic picture of how this initially occurs is still an open problem.
9Note that the cylindrical symmetry of the (time-dependent) Hamiltonian is broken. In any real physical system
imperfections would always lead to such symmetry breaking. In the simulations the position of the numerical
grid provides the symmetry breaking instead.

33

vz (µm/s)

vvzyz (µm/s)

vz (µm/s)

CHAPTER 4. TURBULENT-CASCADE FLUXES

50

ΔU/kB = 0.3 nK

0
-50
100 0
0

0.4
τ (s)

ΔU/kB = 0.8
1.3 nK

0.4
τ (s)ΔU/kB = 0.8
3.6 nK

-100
200 0
0
-200
0

0.4
ττ (s)(s)0.8

Fig. 4.3 Nonlinear damping of the Bogoliubov sound mode. Axial velocity oscillations vy (τ ), following pulsed(∆t = 20 ms) excitations with three different amplitudes ∆U (see top right corner of the panels); here
N = 1.2(1) × 105 . We infer vy (τ ) from the center-of-mass displacement after ToF. The solid black lines are
exponentially decaying sinusoidal fits to the early-time data (τ < 0.25 s). With increasing ∆U the extracted
frequency ωres /(2π) ≈ 9.0 Hz remains essentially unchanged (to within ≈ 2%), while the initial damping
increases dramatically (mind the varying y-axis scale). At large ∆U the decay exhibits clear non-exponential
behavior. Figure adapted from [41].

Spectroscopic measurements elucidate how resonantly exciting the lowest-lying axial mode
eventually leads to a turbulent state [162]. For weak driving amplitudes (F0 L/µ  1) one
probes the linear response of the collective mode, allowing for precise measurements of theresonance frequency, which are found to be in good agreement with Bogoliubov theory (seealso [41]). Instead, for stronger drive amplitudes (F0 L/µ & 1) nonlinear behavior is observed,
where the response curves broaden, indicating that energy is leaking out of the mode. A particularly striking example of such nonlinear behavior can be directly observed in complementary’kick‘ measurements, where instead of driving the mode a potential difference ∆U is pulsed
across the box for a time ∆t = 20 ms (shorted compared to the period of the mode). The ensuing dynamics of the excited cloud are then observed as a function of the subsequent in-trap holdtime τ . As shown in Fig. 4.3 for three different ∆U , the cloud’s axial velocity vy (τ ) exhibits oscillations with τ . The extracted oscillation frequency ωres /(2π) ≈ 9.0 Hz is the same (to within≈ 2%) for all three ∆U , and matches the expected frequency from Bogoliubov theory [41],in line with complementary stroboscopic measurements [162]. The weak kick measurement(∆U = kB × 0.3 nK = 0.15µ) only features a subtle damping, which is well described for all τby an exponentially decaying sine fit to the early-time data, τ < 0.25 s (solid line). Instead, forlarger ∆U the initial damping is increasingly rapid, but a long-lived oscillation persists. Thisnonlinear decay is further accentuated by the same early-time fitting procedure, which fails to

34

γ

100

nk (µm3)

10-1

7
5
3

γ

CHAPTER 4. TURBULENT-CASCADE FLUXES

0

kL

2
ts (s)
ts (s)

7
5
3
4 0

1 2
F0 L/μ
ΔU/μ

10-2
10

-3

10-4

kD

∝ k-3 ln(k/k0 )-2/3∝ k-3 ln(k/k0 )-2/3
∝ k-γ

0.5

1
-1
kk (µm
(µm-1
))

2

4

Fig. 4.4 Emergence of a turbulent cascade. Momentum distribution nk (k) of a turbulent gas (on log-log scale); here
N = 7(1) × 104 , ∆U/µ = 1.1(1), ts = 4 s, ωs /(2π) = 9 Hz, and tToF = 100 ms. To obtain nk we average over
20 experimental repetitions and perform an azimuthal average, before finally performing an inverse-Abeltransform. The red vertical lines indicate our k-space resolution kL and the momentum cut-off introducedby the finite trap depth kD . In between kL and kD the momentum distribution is well characterized by apower-law nk ∝ k−γ with γ ≈ 3.5 (solid line). We also plot the weak-wave turbulence prediction [169]; thedotted line shows the simple nk ∝ k−3 , whereas the dashed line includes expected logarithmic corrections
from a forcing scale k0 , which we assume
R to be equal to kF (see text). The vertical gray line corresponds
to 1/ξ. The normalization is such that 4πk2 nk dk = 1. The insets show the extracted γ as a functionof shaking time ts (left, with fixed F0 L/µ ≈ 1) and amplitude F0 L/µ (right, with fixed ts = 4 s) in bothexperiments (blue) and simulations of the GPE (red), highlighting the robust steady-state nature of thecascade, characterized by γ0 ≈ 3.5 (dotted horizontal line). Data from [162] c Nature Publishing Group.

capture the data at long times (τ & 0.4 s). Our observations hint at a nonlinear many-body decay mechanism, and understanding the microscopic origin of this decay is an important task,
however here our focus is to simply exploit this feature as a route to turbulence.
In Fig. 4.4 we present an example of the turbulent state momentum distribution nk which
our gas attains. The isotropic ToF images correspond to the convolution of the in-trap momentum distribution with the initial trap size, integrated along the line of sight. We assume themomentum-space mapping ~kr = mr/tToF , where r is the radial distance to the cloud’s center,
which is a good approximation between kL and kD for our kinetic-energy-dominated gas10 ;here kL = mL/(2~tToF ) is our momentum-space resolution set by the initial size of the box and√
kD = 2mUD /~ is the high-k sink introduced by the finite trap depth UD .
To reconstruct nk from the line-of-sight integrated distribution presented in [162], we perform an inverse-Abel transform (see Section C). The distribution nk is well described by apower-law hnk i ≈ nk ∝ k −γ at intermediate k (between kL and kD ), where nk decreases byover two orders of magnitude. A fit to the data (solid line), which we use to characterize the
distribution, yields γ ≈ 3.5 11 . In the inset we map out the dependence of γ on ts and F0 L/µ,10

The validity of this mapping relies on the fact that the energy per particle E ≈ kB × 10 nK is significantly larger
than the interaction energy Eint ≈ kB × 1 nK. However, we do expect some contamination at lower k whereatoms are propelled to slightly larger k due to the conversion of interaction energy to kinetic energy in ToF.
11
For details concerning the extraction of γ see [162].

35

CHAPTER 4. TURBULENT-CASCADE FLUXES
at fixed F0 L/µ = 1 and ts = 4 s respectively, which shows that the gas indeed reaches a robust
steady-state power-law distribution (γ0 ≈ 3.5) under continuous forcing. These measurements
are complemented with numerical simulations of the GP equation, which give a consistent
steady-state γ0 ≈ 3.5 (see [27, 162] and Section 4.4.4 for additional details). It is not a given thatthe classical field approximation of the GP equation should hold in this far-from-equilibrium
scenario, especially at high k where the occupations become small, however the fact that it succeeds at describing the experiments suggests its validity. Finally, we also compare our results tothe analytical predictions of weak-wave turbulence theory of the 3D GP equation with particlelike excitations, 4-wave interactions, and a direct energy cascade [169]. A dimensional estimate
in this regime predicts nk ∝ k −γw , with γw = 3 (dotted line). It is however expected that due
to a weak non-locality of the interactions, the form of nk slightly deviates from this prediction
by a logarithmic factor. Such a logarithmic deviation has so far not been experimentally or numerically validated, though it is analytically expected to be of the form nk ∝ k −3 [ln(k/k0 )]−2/3 ,where k0 is a low-k cut off associated with the pumping scale [106, 169, 174].
The dashed line in Fig. 4.4 shows this prediction, fixing k0 = kF , which is barely distinguishable from a power-law with γ0 ≈ 3.5 across our k-range. However, it should be noted thatthe choice of k0 = kF is somewhat arbitrary (especially as kF is not isotropic [175]) as well as
the fact that the theory assumes an infinite system. A free fit to the data yields k0 ≈ 0.3/µm.
Instead, performing the same analysis on the nk obtained from our GP equation simulations(Section 4.4.4) yields an optimal k0 ≈ 0.7/µm. The analytical logarithmic correction to the naive
γw is also qualitatively in line with recent theoretical work which links the deviation from γw
to finite size effects, employing an approach based on the kinetic theory of non-thermal fixed
points [176].
While the data seem to support this logarithmic correction, we note that the analytical weakwave-turbulence theory neglects the role of vortices which are expected to be present in our
turbulent gas, and while in our simulations the compressible flow contribution to the energy
dominates over the incompressible one, vortices could have a residual effect [162]. Other interlinked factors that may play a role and which cannot be ruled out are the not necessarily weakinteractions [153, 169] and the increasing importance of quantum pressure in GP equation withincreasing k 12 . Finally, our GP equation simulations in Section 4.4.4 reveal self-similar behavior, which is at odds with such a logarithmic correction (albeit probing only a rather limited
k-range where one may be insensitive to this).
12

As described
2.3, the GP equation reduces to the classical Euler equation when the quantum pressure
 in Section√√
1
term m∇ ~2 /(2m n)∇2 n is negligible, which is the case in the absence of spatial variations of the wave
function on the order of the healing length [8].

36CHAPTER 4. TURBULENT-CASCADE FLUXES(a) (b) Real(c) (d)(e) (f) (a) (b) (c) (d) (e) (f)space

Momentum space
Cascadefront,
kcf (ts )

Trap
depth

xyz

xyz
xyz

L

ky

kx

kD

Forcing, kF

Particlests < td

ts > td

Fig. 4.5 Synthetic dissipation scale. (a) Our optical box trap (green) confines atoms (blue) in a cylindrical
trap with√a readily tuneable energy-depth UD . (b) In momentum space UD corresponds to kD = 2mUD /~ (green
circle). At ts = 0 the anisotropic forcing is initiated (dark blue disk elongated along ky ), which quickly
becomes isotropic as nonlinear interactions establish a cascade front kcf , which propagates to larger k. The
cascades front kcf reaches kD at ts = td and subsequently particles are continuously lost from the trap. Figure
adapted from [27].

4.3 Synthetic dissipation scale
Dissipation in turbulent flows is fundamental to the existence of a steady state under continuous energy injection. The emergence of a steady-state nk in our quantum gas suggests thatboth energy and particle fluxes through the inertial range are k-independent, however, it does
not directly reveal the fluxes. Here we study the role of dissipation in our gas, which offers awealth of information about the cascade dynamics, including direct access to the particle flux.
In conventional fluids the hydrodynamic (macroscopic) degrees of freedom are accessible,
but there are also ’hidden’ degrees of freedom (microscopic motion). Dissipation, which isgoverned by the viscosity ν of the fluid, occurs in the form of heating on lengthscales set by
the viscosity, where energy is transferred from the hydrodynamic degrees of freedom into themicroscopic ones. Conventional fluids are also typically thermally coupled to their surroundings, making it difficult to measure this (often minute) heating [177]. Moreover, ν is generally
not tuneable in an independent manner.
Our quantum gas offers direct access to the atomic momentum distribution, and hence all
microscopic degrees of freedom, and it is thermally isolated from its surroundings. When aspecific mode is damped it just means that energy is transferred to other modes, which we
also observe. Even though our superfluid lacks an intrinsic dissipation scale, the optical box[Fig. 4.5 (a)] has a finite trap depth UD which realizes a synthetic dissipation scale and acts as aparticle and energy sink. Atoms can simply escape from the trap when they attain sufficiently
high energies, and such atom loss is experimentally measurable to high precision (∼ 1%).
√

In momentum space [see Fig. 4.5 (b)], UD corresponds to a spherical shell of radius kD =2mUD /~, which is readily tuneable by simply varying the trapping laser power. The initially37CHAPTER 4. TURBULENT-CASCADE FLUXES
anisotropic forcing, that injects energy into the system at kF (indicated by the dark blue area
elongated along ky ), quickly becomes isotropic and establishes a cascade front which propagates through momentum space (at times ts < td ). When it reaches the dissipation scale kD attime td , atom loss occurs and a steady-state can be established as atoms are transported fromthe source to the sink.
To explore the experimental possibilities arising from this tuneable dissipation scale we turnto wave-turbulence theory. Within its framework, it is possible to derive a continuity equation from the equations of motion, including a source and a sink, that is local in momentum
space [169]:∂ ñk (k, t)= F (k, t) − D(k, t) − ∇k · Πn (k, t) .
∂t

(4.1)

Here F (k, t) and D(k, t) respectively correspond to the forcing and dissipation, ñk is norR
malized so that ñk dk = N , and ∇k · Πn describes the nonlinear interactions, where Πn is

the particle flux. In the absence of forcing and dissipation (F = D = 0) the steady-state solutions are thermodynamic equilibrium states with Πn = 0. Now, if F and D are nonzero but
localized in momentum space, then non-equilibrium steady-state solutions also exist, where ascale-independent particle flux is sustained between the source F and the sink D.
Assuming that the outflow is isotropic, we can integrate out all angular dependencies andconsider the total radial particle flux Πn (k) = 4πk 2 |Πn (k)|. Applying Eq. (4.1) in the inertial
range then yields 4πk 2 ∂ ñk /∂t = −∂Πn /∂k, and after integrating over k (up to kD ) we have∂N≡ −Πn (kD , t) .
∂t

(4.2)

As one may intuitively expect, in steady-state the particle flux through the momentum shellat kD is simply given by the rate at which atoms are lost from the trap. Moreover, as themomentum distribution in the inertial range is also stationary (see Fig. 4.4) this means that theradial particle flux is both k- and t-independent, such that Πn (kD , t) = Πn (k, t) = Πn .
The corresponding total radial energy flux ΠE (k, t), is also k- and t-independent in the inertial
range, and it reflects the rate of energy dissipation. To formulate a relationship between ΠE (k, t)
and Πn , let us consider the general case of weakly-interacting particles with the dispersion
relation ω(k), where the energy spectrum is E(k, t) = ~ω(k)nk (k, t); in our case the excitations
approach particle-like13 behavior ω(k) ∝ k 2 .
One may naively expect that ΠE (k) = ~ω(k) Πn (k), but this cannot be true as both ΠE andΠn are independent of k in steady state, while ω(k) is not. This nontrivial relationship between
13

The Bogoliubov dispersion relation ω = ~k2 /(2m)becomes particle-like at large kξ (see Section 2.4).

p√
1 + 2/(k2 ξ 2 ), where ξ = 1/ 8πna is the healing length,38

CHAPTER 4. TURBULENT-CASCADE FLUXES
the energy and particle fluxes stems from the fact that they are net fluxes, and the microscopic
interactions drive particles to both smaller and larger k, for k < kD . However, at kD there is no‘back-flow’ from the sink into the inertial range and the particles only flow one way, allowing
us to writeΠE (kD ) = ~ω(kD ) Πn (kD ) .

(4.3)

Furthermore, the existence of a steady state requires that ΠE = ~ω(kD ) Πn throughout the2 Π .
inertial range. For our dispersion relation ω(k) ∝ k 2 we have that ΠE ∝ kDn

A more formal approach to deriving Eq. (4.3) consists of multiplying Eq. (4.1) by ~ω(k) andinvoking the energy continuity equation to obtain∂ΠE (k, t)∂Πn (k, t)= ~ω(k)
∂k
∂k

(4.4)

in the inertial range. This equation is trivially satisfied for k < kD as the particle and energy
fluxes are k-independent in the inertial range and so both sides are zero, and no relation between ΠE (k) and Πn (k) is imposed. However, by integrating across the momentum-space shellkD , and assuming that ñk and all fluxes are zero for k > kD , one recovers Eq. (4.3).
In summary, we have shown that both the particle and energy fluxes underlying the turbulent cascade can be extracted from a simple time-resolved measurement of N (ts ) at different kD(keeping F0 fixed).

4.4 Probing the turbulent cascade fluxes
4.4.1 Turbulent cascade atom-loss dynamics
We now turn to our measurements of the atom-loss dynamics arising from the turbulent cascade. At ts = 0 we initiate the turbulent cascade using a fixed forcing amplitude F0 L =kB ×2.5 nK tuned to resonance [ωs /(2π) ≈ 9 Hz]14 . Our initial quasi-pure BEC has N ≈ 1.2×105 ,and we map out the cascade-induced loss Nloss as a function of ts for different trap depths15
2 . To accurately determine N
UD ∝ k D
loss we perform differential measurements in which we

subtract each measurement from a reference measurement with F0 = 0 but otherwise identicalparameters. This allows us to mitigate the effects of few-percent drifts in the initial N as well14

A weak dependence of ωres on UD exists due to imperfections in the trapping potential, and so for each UD wetune ωs to resonance [27].
15
The radial and axial trapping potentials are independently controlled, and we have performed several calibration
measurements to make them as equal as possible (including Bragg diffraction to give the atoms a directionspecific velocity and see if they leave the trap). Overall, these tests have suggested that various directions areequal to better than ∼ 20%, which is smaller than the range of UD that we explore and is no larger than thesystematic uncertainty in the average UD , averaged over directions.

39CHAPTER 4. TURBULENT-CASCADE FLUXES

��/�� (��)3 3
N lossloss(10(10
) )
Atom

60

24
39
133

40
20
0
0

2

4

6

ts (s)
Fig. 4.6 Turbulent cascade atom-loss dynamics. Measured atom loss Nloss due to the turbulent dynamics as afunction of shaking time ts , for three different trap depths UD (see legend). The initial atom number at
ts = 0 is N ≈ 1.2 × 105 . The error bars arise from averages over typically 50 repetitions, and they indicate
the error on the mean. The dashed lines show piece-wise linear fits that we use to extract the delay time td
and the subsequent initial loss rate (see text). The systematic error in UD values is 20%. Figure from [27].

as the additional one-body loss from collisions with the background gas16 .
In Fig. 4.6 we plot Nloss versus ts for three different UD , which reveals the expected qualitative
behavior. We find that the loss at short times is consistent with zero for all UD , in line with our
expectations that it takes time for the excitations to cascade through k-space and reach kD at
td . For t > td and relatively small total loss (< 30% of the initial N ≈ 1.2 × 105 ) Nloss is linear
in ts and so ∂Nloss /∂t is constant in time. Following Eq. (4.2), we identify this loss rate as thesteady-state particle flux Πn (kD ) = Πn in the inertial range (see Section 4.3). Instead, when thefractional loss is large and the low-k source of atoms is significantly depleted the loss rate is nolonger constant and the steady-state assumption also breaks down17 .
In order to extract both the loss onset delay time td and the particle flux Πn we performpiecewise linear fits (dashed lines). Qualitatively, we see that for increasing UD the delay time
td increases while the particle flux Πn decreases. In the following two subsections we willquantitatively study these two dependencies and compare our experiments to numerical simulations of the GP equation as well as analytical predictions of turbulent cascades. Finally, in
Section 4.4.4 we present a unifying picture of the cascade dynamics, which we further validateusing our simulations.

16

We assume that this additional loss (the one-body lifetime in this experiment is ≈ 14 s) does not substantiallyaffect the turbulent√ loss dynamics, which we have confirmed in GPE simulations.
17
Note that ωres ∝ N , and while the response curves are significantly broadened (see e.g. [162]) the system does
eventually shift out of resonance when N is significantly reduced.

40

CHAPTER 4. TURBULENT-CASCADE FLUXES

20

Πn (ms-1)

10
5

2

± 20%
20

50
U D/kB (nK)

100

200

Fig. 4.7 Steady-state particle flux. Log-log plot of the particle flux Πn versus trap depth UD ∝ kD2 for both experiment (open symbols) and simulation (solid circles). The colored points correspond to the values extracted from the data shown in Fig. 4.7 and the horizontal error bar shows the systematic uncertainty inexperimental values of UD . The solid line shows a power-law fit to the experimental data, which yields−1.05(8)−2.10(16)Πn ∝ UD∝ kD, providing a demonstration of the zeroth law of turbulence. Figure from [27].

4.4.2 Steady-state particle and energy flux
We begin by studying the steady-state particle flux, and in Fig. 4.7 we plot Πn versus UD on loglog scale. The linear dependence of the experimental data (open symbols) indicates power-law−1.05(8)behavior of Πn with UD , and a fit yields Πn ∝ UD

−2.10(16)∝ kD. Our experimental data arein agreement with numerical simulations based on the Gross–Pitaevskii equation (solid circles),in which we mimic the experimental protocol and Πn extraction without any free parameters(details are provided in [27]); a fit to the numerical data (omitted for visual clarity) yields a−1.04(1)consistent Πn ∝ UD

, although the numerical values lie systematically slightly above theexperimental ones.
Our results indicate that for our experimentally fixed F0 , in the limit kD → ∞ (vanishingly
2 Π resmall dissipation lengthscale) the particle flux vanishes while the energy flux ΠE ∝ kDnmains approximately constant, which is a hallmark of a direct energy cascade. This rather
counter-intuitive feature is a demonstration of the zeroth law of turbulence, which was first
formulated in the context of classical incompressible fluids, where in the limit of vanishing viscosity (analogous to our 1/kD → 0) it stipulates that for constant energy input the steady-state
rate of energy dissipation tends to a non-zero constant [118,178] 18 . Note that while energy conservation requires that in steady-state the energy input rate  must be equal to ΠE , this alone
is not sufficient to predict a constant ΠE for fixed F0 , as it is not a given that the rate at which18

In classical fluids the dissipation lengthscale generally depends on the viscosity ν of the fluid; as an example,
for incompressible turbulence the Kolmogorov dissipation lengthscale depends on the viscosity of the fluid asν 3/4 [118, 169].

41

CHAPTER 4. TURBULENT-CASCADE FLUXES

td (s)

3.0

1.0

± 20%
0.3
20

50
U DD/kBB (nK)

100

200

Fig. 4.8 Cascade front dynamics. Onset time for dissipation td versus UD on log-log scale for both experiment(open symbols) and simulation (solid circles). The colored points correspond to the data from Fig. 4.7
and the horizontal error bar shows the systematic uncertainty in experimental values of UD . A powerlaw fit, td ∝ UDβ , to the experimental data (solid line) gives β = 0.73(6). The numerical data shows thesame dependence with a slight systematic offset; a power-law fit to the numerical data yields a consistentβ = 0.71(1). Figure from [27].

the system absorbs energy is kD -independent19 . It is the constancy of ΠE inferred from Fig. 4.7(alongside energy conservation) that shows that the steady-state  is independent of kD acrossour experimentally accessible k-range.

4.4.3 Pre-steady-state cascade-front dynamics
We now turn to a time-resolved study of the initial stage of turbulence [179–181], where thecascade front is propagating through k-space and dissipation has not yet occurred. Our ability
to access this regime is rather unique, and in almost all past experimental studies of turbulence,
turbulent cascade spectra in k-space were obtained with single-point probes by translating time
variations into the spatial fluctuations required for k-space spectra measurements (through theso-called Taylor frozen turbulence hypothesis [182]). It has thus been difficult to experimentally
extract real-time information on the cascade-establishing dynamics20 . The fact that we observe
the initial dissipationless stage of turbulence (ts < td , see Fig. 4.6) together with our ability to
study the dependence of td on UD provides direct access to the dynamics of the cascade front.
In Fig. 4.8 we explore the dependence of td on UD , which we find to be well described by apower-law td ∝ UDβ , with β = 0.73(6) (solid line). Consequently, the dynamics of the cascade
1/(2β)

front are described by kcf (ts ) ∝ ts

. The results of our numerical simulations (solid circles)

exhibit a similar trend with UD but feature a small systematic offset. A fit to the numerical data
19

For example, in the familiar case of the damped harmonic oscillator, the steady-state energy-absorption ratedepends on both forcing and dissipation.
20
While our time-resolved measurements of nk yield qualitative insight into the pre-steady-state dynamics [162],
the low-k region (of interest at early times) is sufficiently contaminated by interactions in ToF to hinder a quantitative study.

42

log n klog n(k)log Πn Πℰ

CHAPTER 4. TURBULENT-CASCADE FLUXES

�����
�

kF

t1

�� ��
���

t2

t3
~k-2

log k

kD

kcf
ts < td

ts > td
log k
log k

Fig. 4.9 Momentum-space turbulent dynamics. Energy is continuously injected into the system at kF from ts = 0
onwards. The steady-state turbulent cascade distribution ñk is established in the wake of the cascade front
kcf (ts ) as it propagates towards kD (for ts < td ), reaching it at td . For the sake of our qualitative message, weshow an idealized sketch with an exaggerated ratio of kD /kF ; numerical simulations with our experimental
parameters, which support this qualitative picture, are shown in Fig. 4.10. The inset shows our consistentpicture for the evolution of both energy and particle fluxes, for three different times t1 (blue) < t2 (purple)< t3 (black), with t2 < td < t3 . The vertical arrows indicate the forcing and dissipation scales, as in themain panel. Figure from [27].

(omitted for visual clarity) yields a consistent β = 0.71(1).
In Section 4.4.4 we present a unifying picture of both pre-steady-state and steady-state turbulent dynamics, which offers insight into how the cascade forms as well as the value of β.
We furthermore verify this picture using the time evolution of the momentum distributions
extracted from our numerical simulations.

4.4.4 Unifying qualitative picture
In Fig. 4.9 we present our consistent picture of the Fourier-space dynamics. At ts = 0 we initiate the shaking, which continuously generates a surplus of atoms at kF in anisotropic fashion.
The nonlinear interactions establish an isotropic cascade front kcf (ts ) which subsequently propagates to larger k. As the cascade front propagates it forms the steady-state turbulent cascadeñk ∝ k −γ0 in its wake. Once kcf reaches kD (at time td ) dissipation occurs, and a steady-stateis established where the energy input rate  is equal to the energy flux ΠE (kD ). As the cascade front propagates (ts < td ), the instantaneous particle flux, which is k-independent fork < kcf (ts ) and zero for k > kcf (ts ), must equal the rate at which the population increases in the
2 dk = Π (k ) dt ,inertial range (to ensure atom number conservation), so that ñk (kcf ) 4πkcf
n cf
s
cf
−γ0
where ñk (kcf ) ∝ kcf. Moreover, to ensure energy conservation, the instantaneous energy-

injection rate  must equal the energy flux ΠE (kcf ) in the inertial range, so we analogously have
4−γ0kcf
dkcf ∝ ΠE (kcf ) dts .

We found experimentally that  does not depend on kD for ts > td (see Section 4.4.2). If we

43CHAPTER 4. TURBULENT-CASCADE FLUXESn(k)n k (µm3)(a) (b)5 (c) (d) (e) (f)
10

103

0.34
0.45
0.60
0.82
1.49
3.02

101

10-1

∝ k-3.5ts (s)

0.5

1

2
-1
kk (µm(µm-1))

4

8(t(ts /ts /trefref)a)ann(k)(µm3)3)k (µm(a) (b) (c)6 (d) (e) (f)
10
104
102
100
10-2
0.3

1
3
-b
-1
b
-1
(t
/t
)
k
(µm
)
(tss /trefk (µm )
ref )10Fig. 4.10 Momentum-space dynamics in numerical simulations. (a) Log-log plot of the calculated ñk versus k for
UD = kB × 130 nK and several ts (see legend), revealing how the power-law ñk ∼ k−γ0 , with γ0 ≈ 3.5,
develops in the wake of the cascade front. Once the cascade front reaches kD (indicated by the vertical
P gray2line) a steady-state is established over the inertial range. The profiles are normalized such thatk 4πk ñk δk = N , where δk = π/(20ξ) is the grid resolution in k space. (b) To highlight the selfsimilar nature of the pre-steady-state dynamics, we rescale the curves from (a) with ts . td ≈ 1.0 s, usingā = γ0 /(5 − γ0 ) = 2.33, b̄ = ā/γ0 = 0.67, and tref = 1 s (see text), collapsing our data onto a single curve.
Figure from [27].

extend this to the pre-steady-state dynamics and assume that  is also independent of kcf for−2
ts < td , we find that for k < kcf (ts ) the instantaneous particle flux Πn (ts ) ∝ kcf. As shown in

the inset of Fig. 4.9, this elegantly unifies our description of the fluxes for ts < td and ts > td ;the k-independent Πn is the same function of the high-k cutoff up to which the steady-statenk has been established (above which there is no back-flow), whether this is the instantaneous
kcf (ts ) < kD (for ts < td ) or kD (for ts > td ). Crucially, this picture also makes a concrete
prediction for β, which we can compare to our measured value β = 0.73(6) (see Fig. 4.8). With4−γ0a time-independent , we have kcf
dkcf ∝ dts , which for γ0 < 5 and kD  kF yields thepower-law prediction td ∝ UDβ , with β = (5 − γ0 )/2 ≈ 0.75(5), in good agreement with themeasured value.

44

CHAPTER 4. TURBULENT-CASCADE FLUXES
To further test this qualitative picture we compute the evolution of ñk during shaking using GP simulations including dissipation (see [27] for details). In Fig. 4.10(a) we show snapshots of ñk (ts ) for UD = kB × 130 nK, which support the qualitative picture put forward in
Fig. 4.9; the cascade front is more rounded here due to the moderate ratio of kD /kF ≈ 60 21 .
In order to highlight that ñk evolves in a self-similar way as the cascade front propagates
1/(2β)

through k-space with kcf (ts ) ∝ ts

, we rescale the data from Fig. 4.10(a) for t . td suchthat (ts /tref )ā ñk [(ts /tref )−b̄ k, ts ], with b̄ = 1/(2β) = 1/(5 − γ) = 0.67, ā = b̄γ = 2.33, and setting
arbitrarily tref = 1 s. This successfully collapses all the curves onto a single one, providing
strong support for this picture. Moreover, the particular values of ā and b̄ confirm our assumption that  is time-independent.
Interestingly, the condition for td to exhibit scaling behavior (γ < 5 and hence β > 0) is
closely linked to whether the steady-state ñk has infinite energy capacity, which also occurs
for γ < 5 and means that the cascade carries infinite energy for kD → ∞. In general it is expected that in infinite-capacity systems the cascade front propagates at a finite speed and thata Kolmogorov–Zakharov turbulence spectrum forms behind it [183], in line with our observations.

4.5 Conclusion
In this chapter we provided a consistent unifying picture of the wave-turbulent dynamics inour box-trapped quantum gas, finding good agreement with ab-initio simulations of the GP
equation throughout. More generally, our quantum gas provides an example of an essentially
stationary non-thermal state, and is of interest in the broader context of far-from-equilibrium
many-body quantum systems [31, 184, 185]. Our experimental platform is an excellent testbed
for future studies of turbulence, with many intriguing possibilities in sight. Below we present
several of these, starting with those that are more immediate (A) before turning to more generaltasks (B).

A1 - It would be interesting to perform dynamical experiments in which one suddenly varies
kD , further exploiting the tuneable nature of the dissipation in our gas. Such experiments could
be interesting in two different ways. First, they could provide a complementary way to measure the atom number in a particular k-space shell, which one could use to cross-validate the nk
measurements [162]. Secondly, one could study quenches between different turbulent states,
offering a glimpse at how the system transitions from one turbulent state to another with a
21

For the lowest kD that we explore kD /kF ≈ 23.

45

CHAPTER 4. TURBULENT-CASCADE FLUXES

G1

10.01 ( )

0.5

0
0

10

20
x (µm)

0.1 ( )

trelax (s)0.5 ( )

30

Fig. 4.11 Relaxation of the turbulent state. Following the preparation of a turbulent state, we halt the forcing at
trelax = 0 and study the relaxation dynamics. Here we show preliminary results of the extracted correlation
function G1 (x) = g1 (x)(1 − x/L) using a homodyne detection scheme [186], where L is the box lengthand g1 (x) ≡ <[g1 (r, r + xx̂)] is first-order two-point correlation normalized so that g1 (0) = 1; see [71, 186]
for additional details of the experimental protocol. The main panel shows G1 (x) for three different trelax ,where the coherence length increases as a function of trelax , from initially highly-suppressed long-rangeorder towards that of a pure BEC (dashed line). The inset shows the corresponding absorption images,
which qualitatively show how a BEC with the characteristic diamond-shape reforms.

new kD . In particular, such a new steady-state would require Πn to adjust across the entire
inertial range (kF < k < kD ), so that Πn is both kD -dependent, to obey the zeroth law, andk-independent for a given kD .

A2 - In nature energy injection always eventually stops, which inevitably leads to the decay
turbulence - its final stage. Experiments where we carefully study the relaxation of our turbulent gas back to a quasi-pure condensate are under way, and a glimpse of these are shown in
Fig. 4.11. We use a homodyne detection protocol [186] to measure the two-point correlation
function for different relaxation times, as phase coherence is re-established and a BEC reforms.
These experiments will pose as fundamental tests of both the decay of turbulence in quantum
gases [143, 187] and more generally the phase-ordering kinetics of Bose gases (see e.g. [188]).

A3 - Extending our studies on turbulence using our box-trapped 39 K gas will allow us to
explore the role of interactions in all three stages of turbulence: its establishment, steady-state
properties, and eventual decay.

B1 - An important future task is exploring the relationship between incompressible and compressible turbulent dynamics. A integral part of this endeavor is benchmarking different excitation protocols to find regimes where the gas reaches turbulent states in which incompressible
flow dominates and vortex tangles form. This is also coupled with designing probes that provide access to the quantities of interest from a theory perspective, such as the incompressible

46

CHAPTER 4. TURBULENT-CASCADE FLUXES
energy spectrum or the average vortex line length [142, 144, 159, 160, 168].

B2 - An exciting prospect is to study turbulence in homogeneous two-dimensional quantum
gases, where an inverse-energy cascade propagating from large to small scales is expected tooccur (see e.g. [122, 160]). Indeed, recent experiments have already detected signatures of such
dynamics [166, 167].

B3 - It would also be interesting to explore the long time behavior for different forcing amplitudes, to see how (or even if) the gas eventually does become thermal. Alternatively, for weakenough forcing relaxation processes could dominate the dynamics and prohibit ever reachinga turbulent state.

47

5 Weak Collapse of a Bose–Einstein
Condensate
Not so fatal attraction- Zoran Hadzibabic -5.1 Introduction
Even weak attractive interactions can lead to dramatic far-from-equilibrium behavior in BECs.
In the presence of attractive interactions, BECs are prone to collapse. Such an implosion is however opposed by the cloud’s kinetic energy, which stabilizes it. A metastable state is formed forsufficiently small interaction energies, however if the attractive forces overwhelm the dispersive ones, then collapse occurs. The collapse of a BEC is an example of wave collapse, a phenomenon ubiquitous in nature. Examples include white-caps on choppy water [189, 190], selffocusing light beams propagating in nonlinear optical media [191–193], the collapse of Langmuir waves in plasmas [194–199], and discontinuities formed in gas-dynamics [200]. It is also
thought to share parallels with the collapse and subsequent supernova explosion of stars at theend of their lifetime [201, 202]. Wave collapse entails the formation of a singularity in a finite
amount of time. However, in real physical systems, dissipative effects necessarily intervene
before this singularity is reached.

The unifying theoretical framework for understanding wave collapse is provided by thenonlinear Schrödinger equation. Theoretically, general forms of the nonlinearity are considered [190, 193], however the case of a cubic nonlinearity is of particular physical interest. It
describes both atomic BECs with s-wave two-body interactions, and nonlinear optical media
with a Kerr nonlinearity, and thus has been a theoretical focus [190, 193, 203–205]. In this case,the nature of the collapse is governed by the dimensionality of the system, which has impor-49

w e a ks tro n g

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

tim eFig. 5.1 Cartoon comparing strong and weak collapse. In the case of strong collapse, a finite part of the wave
(100% for simplicity) collapses into the singularity, whereas for weak collapse this fraction diminishes astime progresses and the singularity is approached. Figure from [26].

tant consequences for how effective collapse is as a nonlinear energy dissipation mechanism.
In 2D the collapse is classified as strong, while in 3D it is weak, the difference being that in strong
collapse the weight of the collapsing part of the wave function1 is constant and a finite amountof energy is trapped in the singularity, whereas for weak collapse the weight of the wave function that collapses into the singularity tends to zero as the collapse proceeds [203, 204] (see also
Fig. 5.1). This has the counter-intuitive consequence that in weak collapse less dissipation occurs for stronger attraction, as the collapse proceeds further before dissipative forces intervene.
Collapse does not occur in 1D for a cubic nonlinearity. Instead, stable solitons are formed,
where the dispersion (kinetic energy) exactly balances the nonlinear attraction (interaction energy).
The collapse of a BEC was first observed during the evaporative cooling of an ultracold gasof 7 Li [206,207]. More detailed studies were enabled by the use of a Feshbach resonance in 85 Rb,
where the interaction strength could be tuned to initiate the collapse [208, 209]. This confirmed
the expected condition for collapse N |ac | = kh aosc , arising from the balance of kinetic andinteraction energy. Here ac is the critical interaction strength, N the condensate atom number,aosc the harmonic oscillator length and kh a constant. Moreover, these quench experimentsallowed a detailed study of the collapse dynamics, which in turn led to the observation of a richphenomenology, including the formation of solitary waves in the aftermath of the collapse [210]and ‘jets’ of atoms emanating from the BEC during the collapse [209]. All previous experiments
studying the collapse of atomic BECs with contact interactions were performed in harmonic
traps [206–212]. While the critical point and the collapse dynamics were in general agreementwith theory [213–220], the observed atom loss was only seen to grow with |a| [210], and noevidence for the theoretically expected weak collapse was found.
1

For example, in the case of BECs the weight is the number of atoms in the collapsing wave function, whereas for
optical beams it is the power of the self-focusing beam.

50

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
Since then, collapse has also been studied in cases where the initial density distribution isroughly uniform locally, and a modulation instability seeds the collapse [221]. More exotic
collapse phenomena have also been studied in BECs. Collapse was found to occur in Bose–
Fermi mixtures [222–224], and the collapse of condensates with dipolar interactions has shown
clear signs of the d-wave nature of the interactions in the collapse remnants [225–227]. More recently, the beyond-mean-field quantum-fluctuations in such dipolar gases were used to oppose
the attraction, forming so-called quantum droplets [228–230], and transient regimes where supersolid properties emerge were observed [231–233]. Similar quantum droplets were later also
observed following the original proposal by Petrov [234], using a mixture of two Bose–Einsteincondensates [235–237].
In this chapter we study the collapse of a 39 K BEC confined in an optical box potential (seealso [26]). Our discussion is guided by three principal questions:(i) What is the condition for collapse?
(ii) What is the time to collapse?
(iii) What is aftermath of the collapse?

In Section 5.2 we introduce the relevant theoretical background and motivate a qualitative
picture to guide our discussion. Dimensional analysis allows us to hint at the universal scaling laws that should govern the collapse process. In Section 5.3 we present our experiments,
starting with equilibrium studies aimed at answering (i) before turning to quench experiments
to tackle (ii, iii); in our quench experiments we prepare a gas close to the critical point where
it is still stable, before rapidly tuning the interaction strength beyond the critical point andobserving the ensuing dynamics.

5.2 Qualitative picture and scaling laws
As a starting point for our discussion we consider the generalized Gross–Pitaevskii (GP) equation2 , modified to include three-body recombination events (a nonlinear dissipative loss mechanism). In a uniform potential, this is given by

i~
2

∂ψ
~2 2
~K3 4
=−
∇ ψ + g|ψ|2 ψ − i|ψ| ψ,∂t
2m
2

The Gross–Pitaevskii equation is the nonlinear Schödinger equation with a cubic nonlinearity.

51

(5.1)

tc

0

nn(t)/n(0)0 (t)/n0 (0)EnergyCHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

CollapseTimet
Condensate size, x
Fig. 5.2 Simple collapse model. The main panel shows the energy landscape as a function the condensate size x,
for different values of the interaction parameter α. For α ≥ 0 (black solid line) only one minimum exists,
and the BEC is stable. For any negative α, a new minimum at x = 0 appears, and the non-zero x solution
is metastable. For small negative |α| the condensate is protected from collapse (x → 0) by a kinetic energy
barrier (purple dashed line). At the critical interaction strength (α = αc , blue dot-dashed line) the energy
barrier disappears, rendering the condensate unstable. If α is suddenly quenched below αc (red dotted line)
the condensate collapses (x → 0) in a time tc . The inset shows a sketch of how the central density varies
after such a collapse is initiated.

where ~ is Planck’s constant, m is the atomic mass, and g = 4π~2 a/m the nonlinear couplingstrength (describes the contact interactions in a mean-field picture with an s-wave scattering
length, a). K3 is the three-body loss coefficient, which controls the importance of the dissipativeloss mechanism and is crucial for the aftermath of the collapse. Even though the potential
energy term does not feature, the walls of the box potential impose the boundary condition
that ψ → 0 at the edges. We proceed by re-writing Eq. (5.1) using the dimensionless variablesr̃ = r/L and t̃ = t/τ0 with τ0 = 2mL2 /~, where the L is the characteristic size of the box,
yieldingi∂ ψ̃= −∇2 ψ̃ + α|ψ̃|2 ψ̃ − iη|ψ̃|4 ψ̃,
∂ t̃

(5.2)

N 2 mK3
,
~L4

(5.3)

where
α=

8πaN
L

and η =and ψ̃ 2 is now normalized to unity rather than N . With Eq. (5.2) in hand, we turn to our threeprincipal questions.

5.2.1 Critical interaction strength
We begin with question (i) What is the condition for collapse? Collapse occurs due to thecompetition between kinetic energy (∝ 1/x2 ) and interaction energy (∝ α/x3 ), for a state ofcharacteristic size x. In Fig. 5.2 we show a sketch of the total energy as a function of x for
different representative values of α. In the case where the interactions are repulsive or zero52

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE(α ≥ 0), only one energy minimum exists and the BEC is stable. For attractive interactions(α < 0) a new global minimum emerges at x = 0. While the minimum at x = 0 represents
the singularity to which the BEC can collapse, we see that for small negative α a kinetic energy
barrier leads to a metastable solution at non-zero x. At αc this barrier disappears, leaving
nothing to stop the BEC from collapsing towards x = 0. This stability criterion is a staticparameter, and only depends on α and not on η 3 . The critical interaction strength ac thus
scales with atom number N and box size L as ac ∝ L/N [c.f. Eq. (5.3); ac = αc L/(8πN )].

5.2.2 Collapse time
Here we elaborate on the second question (ii) What is the time to collapse? After performing aquench of α to below αc , the condensate is free to collapse towards the singularity (x = 0). In
the absence of losses (η = 0) this occurs in a time tc , where the central density asymptoticallydiverges, as shown in the inset of Fig. 5.2. Such an increase in density however eventually
results in atom loss. As the losses are a higher-order nonlinear phenomenon, for relatively
small η their stabilizing effect is only encountered close to tc and the collapse time remains
essentially unchanged compared to the lossless (η = 0) case. Within this approximation, thedimensionless collapse time tc /τ0 is expected to only depend on α.

5.2.3 Atom loss and weak collapse theory
The answer to (iii) What is the aftermath of the collapse?, should be contained in Eq. (5.2), andhence it depends on both α and η. Moreover, the fraction of atoms lost, ∆N/N , should also justbe a function of α and η. The nature of the collapse event (i.e. whether it is strong or weak) will
dictate the dynamics around tc and determine how many atoms are lost in a collapse event.
Collapse has been extensively studied theoretically for η = 0, where it has been shown that thewave function adopts a self-similar form close to tc [203, 204]. The most pertinent predictionof weak collapse, which is expected in 3D, is that the collapsing self-similar solution contains adecreasing number (and fraction) of atoms as the singularity is approached. As atom loss only
occurs once the central density reaches ∼ |α|/η, this means that the stronger the attractive interactions, the further the collapse progresses towards tc and the higher the maximum density
reached. Combining these insights, counter-intuitively predicts that the stronger the attractive
interactions, the fewer atoms will be lost in a single collapse event4 .
3This still relies on the implicit assumption that η  1, so that three-body loss is negligible in the metastablecondensates and only becomes important if the BEC collapses.
4
It has been theoretically shown that when considering η 6= 0 for a cubic self-focusing term and a quintic nonlinearity the collapse is a critical case between weak collapse and the formation of a hotspot, where atom loss
occurs for an extended amount of time once the singularity is reached [238–240].

53

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

5.3 Experimental studies
5.3.1 Critical interaction strength
In order to measure the critical interaction strength we initially prepare a quasi-pure BEC with
N atoms, confined in a three-dimensional cylindrical box trap of linear size L ≈ 2R, at a scattering length of 4 a0 . Note that for the range of scattering lengths that we explore K3 is essentiallyconstant [241, 242], with the value 1.3(5) × 10−41 m6 s−1 [243] 5 . For the range of L and N ex-

plored throughout, this corresponds to small values of η (< 6 × 10−4 ), which ensures thatthree-body loss is negligible in our (meta)stable condensates. However, if collapse occurs, thensignificant loss occurs, providing the primary experimental signature of a collapse.
We slowly (over ≈ 1 s) ramp the scattering length to a final value a and hold in trap for an
additional ≈ 3 s before taking an absorption image after 80 ms time-of-flight (ToF) expansion
at 20 a0 . In Fig. 5.3(a) we show the number of atoms remaining, denoted by Nf , as a functionof a for a fixed N = 18.7(5) × 104 and L = 30(1) µm. A sharp drop in Nf is observed at awell defined a, clearly signifying ac . This is also accompanied by a dramatic deformation of
our clouds, as shown in Fig. 5.3(b). Fig. 5.3(c) shows the measurements of ac for L = 30(1) µm,
spanning over a factor of 20 in N . A striking linear dependence of ac on 1/N is observed, withthe gradient of a linear fit giving N ac = −4.88(5) µm 6 .
Furthermore, as shown in Fig. 5.3(d), by plotting N ac versus L, we extract |αc | = 4.0(1)(αc = 8πN ac /L). Numerically solving the GP equation for our cylindrical box using imaginary
time evolution to determine the ground state yields the critical point |αc | = 4.3. While thisdiscrepancy is larger than our statistical error bar, it can be readily explained by considering our
two main sources of systematic errors. Our calibration of the absolute atom number presents
a lower bound, and it could reasonably underestimate the actual atom number by up to 10%.
Our simulations also assume an ideal cylindrical box potential, whereas any imperfections in
the trapping potential could lead to small deviations of αc . In particular, we cancel gravity andany other stray magnetic field gradients so that the atoms experience a residual acceleration(in all directions) on the 10−4 g0 level [22], where g0 ≈ 9.84 m s−2 is the local gravitational

acceleration. This leads to a gravitational potential energy of L×4 kB pK (µm)−1 across the box,
which is equal to the ground state energy in a cylindrical box (with L = 2R) when L ≈ 40 µm.
56The indistinguishability of atoms in a BEC leads to a reduction of K3 by a factor of 3! compared to thermal gases.
By extrapolating ac to the 1/N → 0 limit, we are able to slightly refine the location in magnetic field at which thescattering length vanishes in the |1, 1i state. We refine the previously determined value of B0 = 350.4(1) G [244]to B0 = 350.45(3) G [26]. A third, independent measurement of the zero crossing (see Section A), based on
the failure of evaporation in a thermal Bose gas, is also consistent with this value of B0 . The remaining 30-mG
uncertainty in B0 corresponds to a systematic uncertainty in our a values of ≈ 0.02 a0 .

54

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE(a) (b) (c)
2 (d)

0000
00

0

1.0
1.0
1.0

-0.49 a0-0.54 a0(a) (b) (c) (d)-a-ac c(a(a00))(a) (b) (c) (d)
666
44
3
22

0.5
0.5
0.5
-a
(a(a0)))
-a
-a(a
00

-Nacc (µm)
(μm)

10
10
10
000

OD

N
(10444)))
NNff f(10(10(a) (b) (c) (d)
15
20
15

0.5
0.5
0.4
-4 )
1/
N
(10-4
-4
1/
1/ N
N (10
(10
))

1.0
1.0
0.8

6
3

2R=L
L

0
0

20
L (μm)(µm)
L

40

Fig. 5.3 Critical interaction strength. (a) Atom number, Nf , after a ramp to a final scattering length, a. The criticalinteraction strength ac is clearly visible by a sharp drop in Nf . (b) Absorption images of clouds produced
under identical conditions apart from a small 0.05 a0 change in a, clearly indicating the collapse event.
(c) Variation of ac with N for a cylindrical box with L ≈ 2R ≈ 30 µm. The linear fit confirms ac ∝1/N . (d) Variation of N ac with L, where the aspect ratio is fixed such that L = 2R (within experimental
uncertainties). The linear fit (whose intercept is consistent with zero) confirms the expected N ac ∝ L scaling
and its gradient gives the critical parameter αc = −4.0(1). Figure from [26].

However, the linear dependence of −N ac on L in Fig. 5.3(d) suggests that the condensate size is
only weakly perturbed by these not-so-small gradients, and that the uniform potential remains
a good approximation for describing our optical box trap.
To further investigate these effects experimentally, we explore how the critical condition forcollapse is altered in a tilted cylindrical box potential, which we achieve by purposefully applying a magnetic field gradient along the axis of the cylinder. To supply the field gradient,
we use the pair of compensation coils along the experimental y-axis (see Fig. 3.2). The atoms
experience a force given by the Zeeman interaction, which leads to an axial linear potential,
resulting in a ‘wedge’-like trapping potential as shown in Fig. 5.4(a). We use an axial field gradient corresponding to an acceleration gy = 4.5(2) × 10−3 g0 along y 7 . For our L ≈ 30 µm box
the resultant gravitational potential energy ≈ kB × 6.2 nK dwarfs the ground-state energy of
only ≈ kB × 0.2 nK and so it dominates the non-interacting Hamiltonian.
We now repeat the measurements presented in Fig. 5.3 for the tilted box potential. In Fig. 5.4(b)we show a dataset taken for fixed N = 9.2(4) × 104 . A sharp drop in N at well defined ac isalso observed here, exhibiting the same qualitative behavior as for the uniform box potential.
There is however a qualitative difference in the absorption images recorded [see Fig. 5.4(b)].
The uncollapsed metastable BEC is already significantly perturbed as compared to the uniform
case [c.f. Fig. 5.3(c)]. Once the collapse has occurred, the absorption images now also feature
7

We calibrate the field gradient by measuring the acceleration felt by the atoms as the current in the coil is varied.
We use both 70 ms and 100 ms ToF to guard against any initial position or velocity effects, which we find to be
negligible. We obtain gy = 1.70(1) I × 10−3 g0 , where I is the current in the coils (in ampere).

55

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

L

5
0
0.2

ac
0.4 0.6-a (a00)

3-0.54 a0N f (104)(a) (b) (c) (d) (e) (f)10N6-ac (a0)-0.46 a0-0.46 a0(a) U(b)(c) (d) (e) (f) (a) (b) (c) (d) (e)(a) (f)(b) (c) (d) (e) (f)
y

0
0

0.8

0.4
1/N
1/ N (10-4)

0.8

Fig. 5.4 Critical condition for collapse in a tilted box. (a) Sketch of both the tilted box potential (purple solid line)
and the uniform box potential (blue dotted line). (b) Typical measurement of Nf as a function of a for thetilted box potential; a sharp drop in Nf occurs at ac . (c) Absorption images of the clouds before (top) andafter (middle and bottom) collapse, taken with 100 ms ToF at aToF = 20 a0 . The clouds before collapse
already look perturbed, and once collapse has occurred they feature a myriad of density perturbations andhigh-contrast dips. (d) Extracted ac values versus 1/N , comparing wedge (purple) and uniform (blue) box
data under otherwise identical conditions.

myriad high contrast holes and perturbations.
In Fig. 5.4(c) we plot the measured ac for various different 1/N in our tilted box, and for convenient comparison we also include the data from our quasi-uniform box experiments. In bothcases we see linear behavior in 1/N , however the gradient of the tilted box potential is reducedby ≈ 50% compared to the quasi-uniform case. This change is to be expected as the gradientintroduces another term in the Hamiltonian, which reduces the effective condensate size. On
dimensional grounds one can predict that N ac ∝ ` holds true for any dominant lengthscale `involved. It would be interesting to test these scalings, however due to the anisotropic natureof the gradient it would also require an understanding of the dependence of αc on the aspectratio of the box, which we have fixed to 2R ≈ L throughout.

5.3.2 Collapse dynamics
Here we explore the dynamics of collapsing BECs and our experiments aim to answer thesecond question posed in Section 5.1: (ii) What is the time to collapse? To this end we performquench experiments. We begin by preparing a metastable BEC with N atoms in a box of linear
size L ≈ 2R, at a scattering length ai < 0, a small distance (≈ 0.3 a0 ) above ac 8 . We then quench
the interaction strength to a final value a < ac , before waiting for a variable hold time t. The
8This allows us to indiscriminately quench to final values of a > ac , providing a simultaneous measurement of acfor every set of N and L; this acts as a cross-check against magnetic field drifts that could otherwise compromise
our calibration of a. We checked that our results do not appreciably depend on the exact choice of ai .

56

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

10

55
6
00

10
10

100
100
tt (ms)(ms)(a) (b) (c) (d)

5
6

00
1000
1000

t < tctctc(ms)(ms)

tctc(ms)(ms)tc (ms)-3
ttc /τ
0 (10 -3)
c /τ0 (10 )

30

3

100
100
tt (ms)
(ms)

1000
1000

t ≫ tc

300
300

30
30

(a) (b) (c) (d)
300

10
10

t > tc(a) (b) (c) (d)
300
300

3
3
0
0

-2.19 a0-0.86 a0 12(104)4)
NNf f(10

NNf f(10(104)4)(a) (b)(c) (d)
10
10
12

1
1
|a| (a )|a| (a00)

2
2

30
30
3
3
0
0

1
1
|a| (a )|a| (a00)

2
2

L (µm) N (104 )16
6.3(1)
30
7.2(1)
30 11.4(3)30 15.9(4)30 20.3(5)41 12.3(1)

0.03

0.30
(α-α
(α-αc )/αc

3.00

Fig. 5.5 Collapse dynamics. (a) Final atom number Nf versus t, for two different a, where N = 11.4(3) × 104 andL = 30(1) µm. We extract the collapse time tc (green vertical bands) as the onset of atom loss. For small|a| (left) Nf suddenly jumps to a lower value at tc . For much larger |a| (right) the loss occurs graduallyin time until a final value is reached. (b) Example images at various hold times after a quench to below
ac ; here a = −1.02 a0 and aToF ≈ 20 a0 . (c) The extracted collapse time tc versus |a| for six series withdifferent N and L [see legend in (d)]. The shaded vertical bands correspond to ac . The difference in N
predominantly leads to an x-axis shift, related to a shift in ac . By changing L, drastically different collapse
times are observed, varying between 3 ms and 300 ms. (d) Universal collapse dynamics. Dimensionlesscollapse time tc /τ0 versus the dimensionless log-distance of α from αc . A striking coalescence of all the six
data series is seen. The solid line shows the results of lossless GP simulations without any free parameters
and the dashed line shows tc /τ0 ∝ [αc /(α − αc )]1/2 , which is predicted from a simplistic classical model(see text). Figure adapted from [26].

57CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
starting point for measuring t is defined as the half-way point between ai and a 9 . The collapse
process is finally halted by a jump in scattering length to aToF ≥ 0 a0 , before switching off thebox trap letting the cloud expand in 80 - 100 ms ToF.
In Fig. 5.5(a) we show typical time traces of Nf in two qualitatively different regimes. For
quenches just beyond the critical point (small |a − ac |) the atom number exhibits a sudden drop
to a stable lower value at tc . We interpret this as a single collapse event. Instead, for quenches
far beyond the critical point (large |a − ac |) Nf decays in a seemingly continuous manner before it ultimately stabilizes. This behavior, also seen in previous measurements of collapse inharmonically trapped BECs [209, 217], is understood to arise from a series of multiple, experimental unresolved, collapse events [214, 215, 245–251]. In this multiple-collapse regime we
associate tc with the onset of atom loss, which coincides with the time when the atom loss-rate
is maximal.
Fig. 5.5(b) shows a series of representative images observed throughout the collapse process.
For these measurements we use aToF ≈ 20 a0 , which leads to an additional expansion of theclouds by adding repulsive interaction energy at the start of ToF, which is then converted tokinetic energy during ToF. This avoids saturation of the images to allow for accurate counting of
the remaining atoms. Moreover, as the interaction energy increases quadratically with density,
this procedure also indirectly provides a measure of the in-situ density. The atom number
remains constant up to the collapse time, tc , but during the approach we see a clear swelling of
the clouds. This increase in expansion energy is interpreted as the shrinking of the BEC wave
function on route to collapse. Shortly after tc (within ≈ 10 ms), we observe collapse remnantsconsisting of two parts: a lower-energy central part, and a higher-energy shell, reminiscent of
the atom bursts generated in [209]. Similar shells are observed when imaging the samples froma perpendicular direction, indicating that they are spherical. From the size of these shells in
ToF we infer that they expand at a rate of ≈ 2 µm/ms. At longer times, more irregular patterns
are observed, which is consistent with the shell reflecting off the trap walls and interfering withthe central part of the cloud after ≈ 10 ms 10 .
It is already evident in the two examples shown in Fig. 5.5(a) that the time it takes for collapse
to occur decreases as the scattering length becomes more negative. We have repeated suchmeasurements for a range of different a, N , and L. In Fig. 5.5(c) we plot the extracted collapse
times tc for each series (fixed N and L) as a function of |a|. We see divergent behavior of tc
as a approaches ac (the vertical shaded bands indicate ac for each series). The predominant9

We define tq = t + δt as time when the quench is initiated. For the quenches used here we have measured
δt = 5(2) ms using RF spectroscopy, approximating the smooth change of a (varying from 20% to 80% in≈ 4 ms) as an instantaneous jump centered at the half-way point, which is only meaningful when δt/t is small.
10
A more in-depth discussion of the collapse remnants can be found in Section 5.3.3.

58

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
effect of increasing N is a negative x-axis shift, accompanied by a shift in ac . By changing L
for series with similar ac instead, we observe a drastic change in the observed collapse times,
which vary by over two orders of magnitude. Due to the divergent behavior around ac andfrom the expected scaling laws (see Section 5.2.2) predicted by the dimensionless generalized
GP equation [Eq. (5.2)] for small values of η (for our six data series η varies between 4 × 10−5

and 4 × 10−4 ), we plot tc /τ0 versus (a − ac )/ac ≡ (α − αc )/αc in Fig. 5.5(d) on log-log scale.
A remarkable coalescence of all six series is observed, supporting the universal scaling lawsthat Eq. (5.2) predicts for small η where the losses only occur very close to the critical point andthus do not appreciably alter tc . The solid line in Fig. 5.5 shows tc extracted from numericalsimulations of the lossless GP equation, without any free parameters. This reproduces a similar
dependence of tc on α, although the numerical values lie systematically slightly below theexperimental ones.
To provide some additional qualitative insight, we turn to the simple model from Section 5.2.
We consider the classical dynamics of a ball on the energy landscape shown in Fig. 5.2. The
acceleration of the ball is proportional to the gradient dE/dx. Since the ball is moving slowly
at the start, it is reasonable to assume that the initial change in x will dominate the time ittakes to reach x = 0 (collapse to occur). This yields a simple over-estimation of tc , where weassume that the acceleration remains constant at its initial value. As E depends linearly on α,
the initial acceleration is proportional to αc − α, and hence under these assumptions we predict√tc ∝ 1/ α − αc . This power-law prediction is shown by the dashed line in Fig. 5.5(d), and isin relatively good agreement with the data (in particular for small α − αc ) for such a simplistic
picture.

5.3.3 Aftermath of the collapse
To address question (iii) What is the aftermath of the collapse?, we further consider the quench
experiments from Section 5.3.2, now focusing on t > tc . The remnant atom number Nf provides
our primary experimental signature of the aftermath of the collapse, but we also examine thestructure of the collapse remnants.

Observation of weak collapse
We begin by considering the loss dynamics in a single quench experiment series with fixed
N and L. In Fig. 5.6(a) we plot the fractional atom loss ∆N/N versus t for four different a.
Here ∆N = N − Nf , where N is the initial pre-collapse atom number, and Nf the final timedependent one. For small |a − ac | (left panel), in the regime of single-collapse events where the59CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
atom loss occurs abruptly, we observe the surprising result that increasing |a| leads to less atom
loss, indicating weak collapse. Instead, for quenches to much larger |a − ac | (right panel), in themultiple-collapse regime where the atom loss occurs gradually once tc has been reached, theopposite trend is observed in the long-time limit, in line with previous experiments [209–211].
In order to understand the non-monotonic behavior between these two regimes, we plot thefractional atom loss ∆N/N versus ac /|a| for all hold times in Fig. 5.6(b). Our choice of x axis
captures the full range of a < ac , from a/ac ≈ 1 to a/ac → ∞. For t < tc , collapse has not
yet occurred, and thus our data is scattered around ∆N = 0. When collapse occurs (t > tc ),outcomes at non-zero ∆N/N are observed. As ac /|a| increases, ∆N smoothly decreases, inline with weak collapse theory. For ac /|a| < −0.6, in the regime of single-collapse events, weclearly see that the atom loss decreases with increasing ac /|a|, and extrapolates to zero as ac /|a|vanishes. This is the hallmark signature of weak collapse.
A linear extrapolation of ∆N/N (ac /|a|) gives a y-intercept of ∆N/N = −0.02(2) (dot-dashed(a) (b) (c) (d)-0.49 a0-0.67 a0

0.6
ΔN/N

ΔN/N
ΔN/N

0.6
0.3

0.3

0

0
10

(a) (b) 1.0(c) (d)ΔN/N
ΔN/N-1.71 a0-1.20 a0100
tt (ms)
(ms)

1000

10

100
tt (ms)
(ms)

1000

0.5

0
-1.0

-0.5
ac /|a|0

Fig. 5.6 Observation of weak collapse. (a) Fractional atom loss ∆N/N versus t for four different a with N =20.3(5) × 104 and L = 30(1) µm. For a close to ac (left panel) the fractional loss decreases for increasing|a|, whereas for much larger |a| (right panel) the opposite trend is seen in the long-time limit. (b) ∆N/N
versus ac /|a| for the entire data series from (a) (including all a < ac and t); the colors indicate the respective
series from (a). The points around ∆N/N = 0 show BECs that have not yet collapsed (t < tc ). For aclose to ac , in the regime of single collapse events, we see ∆N/N monotonically decrease with increasing|a|. Moreover, extrapolating to ac /|a| → 0 shows how ∆N/N vanishes (solid and dot-dashed black lines),
which is the unambiguous signature of weak-collapse. When ac /|a| > −0.6 the single-collapse atom loss
fails to stabilize the clouds (as too few atoms have been lost), and multiple collapse events occur; the purple
dashed line indicates the equilibrium stability requirement, ∆N/N = 1 − ac /a. Figure from [26].

60CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
black line in Fig. 5.6). Instead, fitting a power-law we obtain ∆N/N ∝ |a|−1.05(7) , which isshown by the (almost indistinguishable) solid black line.
For ac /|a| > −0.6, a deviation from the weak-collapse trend is observed, where the dwindling single-collapse atom loss can no longer stabilize the remnant, and additional collapse
events occur. The purple dashed line shows the equilibrium post-quench stability criterion∆N/N = 1 − ac /a, which defines (assuming equilibrium) the required atom loss in order tostabilize the cloud after the quench [Nc (a) = αc L/(8πa) = N ac /a]. The validity of this condition is not a given in the far-from-equilibrium setting following the first collapse event [210].
Nevertheless, we find that it gives a decent estimate for both the crossover between singleand multiple-collapse regimes (ac /|a| ≈ −0.6), and the long-time loss for large |a|. Note thatthe slight underestimation of the long-time loss at large |a| is consistent with the fact that any
added kinetic energy from the first collapse event will help stabilize the clouds.

Weak collapse scaling laws
We now turn to unravelling the scaling laws that describe the weak-collapse atom loss. We
perform a similar analysis as above for our five other quench experiment series with different
N and L (from Section 5.3.2). We restrict ourselves to only include those values of a for which
the first collapse event is clearly resolved. This predominantly includes cases where only asingle collapse event occurs, but also some where both single and double collapse events areresolved.
For our six series, we follow the same procedure as in Fig. 5.6(b) and perform power-law
fits of the form ∆N/N ∝ |a|−γ to extract γ. As shown in Fig. 5.7(a), we always obtain values
consistent with γ = 1; a weighted average over all six series gives γ̄ = 1.02(2).
Having experimentally uncovered the form ∆N/N = C/|a|, we now turn to the N and L
dependence of C. Plotting C versus N on a log-log plot [see Fig. 5.7(b)] reveals power-law
behavior with C ∝ N −0.51(2) , and no experimentally resolvable dependence on L; our two data
series with L ≈ 16 µm and L ≈ 41 µm lie on the same curve as the four with L ≈ 30 µm.
Hence we have experimentally deduced that the weak-collapse atom-loss scales as ∆N/N ∝√1/( N |a|). Re-writing this in terms of α and η [see Eq. (5.3)] yields ∆N/N ∝ η 1/4 /|α|. Whilethe determined weak-collapse scaling laws are independent of the linear size of the box L, notethat they may depend on its aspect ratio (we fix L ≈ 2R throughout); this could be interesting
to study in the future.
In Fig. 5.7(c) we show the fractional single-collapse atom loss for all our series versus η 1/4 /|α|,where each point in a series corresponds to a specific a, averaged over all t > tc . We find that61

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
L (µm) N (104 )C (a0)C (a0)γ(a) (b) (c) (d)
1.3
0.4
1.0
0.7
(a) (b) (c) (d)
0.4
0.3
0.2
0.2
5
6
(a) (b) (c) (d)

1012
N (10
(1044))
N

24

15

16
30
30
30
30
41

6.3(1)
7.2(1)
11.4(3)15.9(4)20.3(5)12.3(1)

ΔN/N

0.4

0.2

0
0

0.02
η 1/4/ |α|0.04

Fig. 5.7 Weak collapse scaling laws. (a) We fit a power-law of the form ∆N/N ∝ |a|−γ to the single-collapse
atom loss data for our six series (see legend). A weighted average yields γ̄ = 1.02(2). (b) Assuming thefunctional form ∆N/N = C/|a|, we get C ∝ N −0.51(2) (solid black line), without any dependence on L. (c)Universal weak-collapse atom loss behavior. By plotting the single-collapse data for different a, N and L
versus η 1/4 /α, we find that all our data lie on a single universal curve. Moreover, this clearly highlights how
the single weak-collapse atom loss vanishes in the limit of infinitely strong attractive interactions, |α| → ∞.
We obtain ∆N/N ≈ 13η 1/4 /α from a linear fit. Figure from [26].

all our data can be described by a single universal function ∆N/N ≈ 13η 1/4 /|α|. Theoreticallyunderstanding this scaling law remains an open problem; our measured scaling differs froma simple prediction based on scaling arguments within weak-collapse theory, which predicts
η 1/2 /|α|2 [204]. Moreover, as ∆N/N cannot exceed 100%, its linear dependence necessarily
breaks down for larger values of η 1/4 /|α|. In future it would be compelling to experimentally
explore the regime of strong dissipation by, for example, using a different Feshbach resonanceor atomic species.
In the same spirit, it is interesting to ask what experimental parameters maximize the range
over which single weak-collapse events are observable. The critical scattering length is given by
ac = αc L/(8πN ), and the equilibrium post-quench stability criterion reads Nsafe = αc L/(8πa).
Moreover, the experimentally extracted scaling of ∆N implies Nf = N − ∆N = N − c1 N 1/2 /a,where c1 is a constant. We expect that for more negative a single weak-collapse events occur
only until Nf exceeds Nsafe , as then not enough atoms are lost in the single collapse event, owing
to the diminishing weak-collapse atom loss for stronger attractive interactions. We denote the62CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
scattering length at which Nf = Nsafe as a = a∗ ; solving for a∗ yieldsa∗ = ac +

c1
.
N 1/2

(5.4)

Now, the relevant distance between a∗ and ac in log-space is given bya∗ − ac8πc1 N 1/2
=
,
ac
αc L

(5.5)

which suggests that range over which single-weak collapse events are resolvable in our experiments increases for increasing N and decreasing L 11 . This conclusion is also supported
experimentally, where the range over which clearly resolvable single collapse events occur isthe largest for series with large N and small L, as evidenced in Fig. 5.7(c) where these series
span the largest portion of the graph.

Crossover between single and double collapse
Having elucidated the scaling laws for single weak-collapse events, we now present experimental evidence for a gradual crossover between single- and double-collapse events as |a| isincreased. We start by considering a single series with L ≈ 30 µm and N = 11.4(3) × 104 . In
Fig. 5.8(a) we show the time evolution of Nf following a quench to various closely spaced abetween ac and a∗ (defined as Nf = Nsafe ), which reveals a striking bifurcation of the collapse
outcome. We observe two clearly resolved branches, which we interpret as the result of either
one (upper branch) or two (lower branch) collapse events; the accompanying histograms of Nfvalues accentuate this.
The existence of such a crossover regime, where the system moves between preferentiallycollapsing one or two times, strongly supports our interpretation that an increasing numberof sequential collapse events occur as |a| is increased, until for large |a − ac |, in the regime ofmultiple collapse, the different discrete branches become experimentally unresolved and theloss is seemingly continuous.
In Fig 5.8(b), we show the corresponding ∆N/N versus |a| on a log-log plot, from ac to a∗ .
Importantly, in the regime where double-collapse occurs, the single collapse branch still clearly
follows the weak-collapse scaling ∆N/N ∝ 1/|a|. Interestingly, the bifurcation regime does nothave the same extent in a for all our data series; it seems to occur more prevalently for small N
and large L. In these cases we also see hints at higher order bifurcation branches (though notstatistically unambiguous).
11

Note that a∗ − a increases for decreasing N .

63

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE-0.97 a0N f (104)12

N f (104)128

N f (104)612

N f (104)

12

10

100
1000
-0.99 a0t (ms)

10

100
1000
-1.02 a0t (ms)

10
10

100
1000
100
1000
-1.04 a0(ms)
tt (ms)

10

100
1000
-1.07 a0t (ms)

10100
tt (ms)
(ms)

6

46

CountsN f (104)(a) (b) (c) (d)
12

6

6

(a) (b) (c) (d)1000

N f (104)

0.5

ΔN/N

0.4

ac

0.3

0.2
0.8

0.9

1.0

|a| (a0)
Fig. 5.8 Transition from one to two collapse events. (a) Nf (t) for L ≈ 30 µm, N = 11.4(3) × 104 , and finely spaced
a in the range between ac and a∗ , accompanied by histograms of Nf values on the right. For t > tc , as
we increase |a| and the weak-collapse atom loss decreases, a second outcome branch in Nf emerges. The
two clearly resolved Nf branches correspond to one (upper branch) and two (lower branch) collapse events.
The double-collapse probability gradually increases with |a|. (b) ∆N/N versus |a| on a log-log plot. The
raw data are shown as transparent black circles, while the colored circles and diamonds, respectively, show
the average values for single- and double-collapse events. The error bars indicate the standard deviation.
The purple dashed line shows the equilibrium post-quench stability criterion, and the band indicates its
uncertainty. The solid black line shows the single-event weak collapse scaling law, ∆N/N ∝ 1/|α|, fit to thedata. Figure from [26].

64CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
L (µm) N (104 )ΔN (104)

10

16
30
30
30
30
41

3

6.3(1)
7.2(1)
11.4(3)15.9(4)20.3(5)12.3(1)10.5

1.0|a| (a0)

2.0

Fig. 5.9 Double-collapse scaling laws. We show ∆N versus |a| on a log-log plot for both single (open symbols)and double (solid symbols) collapse events. Plotting the data this way predictably fails to align the single,
weak collapse data. However, a striking coalescence of the double-collapse branch is observed. A weighted
power-law fit to the double-collapse data for fixed L yields ∆N ∝ a−1.50(4) (black dashed line). The data dosuggest a weak dependence on L (∼ L0.25(10) ) as evidenced by the larger (smaller) box data lying slightly
above (below) the fit.

Double-collapse scaling laws
We proceed by repeating the above analysis for all our six data series, empirically distinguishing both single- and double-collapse branches12 . In Fig. 5.9 we show the time-averaged ∆Nversus |a| on a log-log plot for both single- and double-collapse branches (open and solid symbols, respectively). Plotted this way, the single-collapse data [from Fig. 5.7(c)] do not fall onto auniversal curve, as expected since this form is not equivalent to the weak-collapse scaling law√∆N/N ∝ 1/( N a). On the contrary, and rather surprisingly, the data for the double-collapse
branch does fall onto a single curve. From a power-law fit to the double-collapse data with
fixed L (dashed black line) we obtain ∆N ∝ a−1.50(4) . Note that the data do suggest a weak dependence on L, and optimal coalescence is achieved with ∆N/N ∝ L0.25(10) /(N a3/2 ). It should
be noted that this scaling is no longer dimensionless (or solely captured by η and α), implyingthat our dimensional analysis from Eq. (5.2) is no longer adequate.
The mere existence of such a simple scaling for the double-collapse atom loss is alreadysurprising when one considers the far-from-equilibrium dynamics that occurs in a doublecollapse13 . In general, a particularly important problem is that the initial condition for thesecond collapse event (if truly sequential in nature) is relatively unclear, and numerical continuations of the GP equation beyond the singularity are difficult, demanding adaptive grids andother subtle assumptions on continuation through a singularity (see e.g. [193]). Fully under1213

Cases where this distinction is ambiguous are rare, and disregarding them does not affect the results.
As an example, there is numerical evidence that Eq. (5.2) presents a critical case between weak collapse and theformation of a hotspot [238–240], which could be related to the curious dynamics that we observe.

65CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
standing these scaling laws presents an exciting task for future work.

Structure of the collapse remnants
In the preceding sections we focused on the atom number of the collapse remnant, Nf , as ourprimary experimental signature. Here we extend our analysis to include the momentum distribution of the collapse remnants, which offers insight into the rich post-collapse dynamics.
To this end, we perform quench experiments with aToF ≈ 0 for otherwise identical parameters
as in Fig. 5.5(a) [where aToF ≈ 20 a0 ]. This avoids adding expansion energy in flight, allowing a more accurate measurement of the shape of the momentum distribution, but remains
complimentary to the aToF ≈ 20 a0 measurement as it does not faithfully measure Nf 14 .
We start by considering the single-collapse regime, and in Fig. 5.10 we show typical aToF ≈ 0
images of the collapse dynamics [see Fig. 5.5(b) for the aToF ≈ 20 a0 counterpart]. As tc is
approached, we observe a less-pronounced swelling of the clouds compared to the aToF ≈ 20 a0case, as we are not as sensitive to the diverging central density as the cloud shrinks. For times
directly after tc , such that tc < t < tr , we again observe isotropic shell-like structures (here tr
is the time it takes for the shell to reflect off the trap walls)15 . Interestingly, the central lowerenergy part of the collapse remnant is seemingly unperturbed initially [see also 5.5(b)].
For t > tr [Fig. 5.10(c)] we see structured anisotropic patterns, indicating intricate postcollapse dynamics16 . However, the general structure and orientation of the ordered structuresthat we observe is surprisingly reproducible. At later times (t  tr , about 100 ms after tr ),isotropic disordered structures are predominantly observed [Fig. 5.10(d)].
We propose the following qualitative picture that is consistent with the observed dynamics.
Following the first collapse event, a spherically symmetric shell of atoms is initially travelingoutwards. As the system has stabilized (Nf < Nsafe ), this shell simply propagates outwards
and eventually reaches the walls of the box, where it is reflected, as its energy (≈ 2 nK) is
still significantly lower than the trap depth (≈ 30 nK)17 . Experimental imperfections in thehomogeneity of trap depth, which we expect to be around 20%, then cause path-differences
between different parts of the shell as it is reflected, which in turn leads to a failure to refocus,
Our imaging is insensitive to high optical density (OD & 3) regions, where it underdetermines the occupations.
While the shell-like structures are most spectacularly observed in the single-collapse regime, they are also observed for much larger |a| close to (but above) tc .
16
Experimental variability in the initial conditions leads to scatter in tc from shot to shot, and as we only know theaverage value of tc the exact time since collapse has occurred is unknown in each realization. Consequently,
the trends described here are based on empirically assessing the occurrence probability of specific types of images. Implementing a weak measurement imaging technique (e.g. phase contrast imaging [252]) would allow
quantitative time-resolved measurements of the same cloud as it collapses.
17
We have checked that if the box trap is turned off 15 ms before tc , the collapse still occurs and shell-like structures
are observed predominantly (also for t > tr ).

14

15

66

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE(a)#1452 , t=3s(c)#1778 , t=56.7s#1648 , t=76.5s#1714 , t=200.3s#1574 , t=493.4s#1758 , t=32s(b)#1332 , t=37.2ms

tc < t < trt < tc#1664 , t=126.7s

t > tr(d)#1694 , t=997.7s

t ≫ trFig. 5.10 Single-collapse remnants. Typical absorption images taken with aToF ≈ 0 for clouds in the single-collapse
regime (low |a|) at different times t (a-d); see text. The images highlight the rich collapse dynamics before,
at long times, isotropic disordered structures eventually emerge.

generating ’tail-like’ structures. Subsequently, additional reflections occur, leading to peculiar
ordered structures, which only at long times, after numerous reflections, form an isotropic
disordered final state.
We now turn to the question: What is the nature of this final state? To address this question,
we proceed by extracting the final-state momentum distribution of the remnants. The distribution measured in ToF is the convolution of the in-trap momentum distribution with the initialsize of the trap, integrated along the line of sight. The mapping to momentum space is given by~kr = mr/tToF , where r is the radial distance from the center of the cloud. The initial size of thebox defines kL = mL/(2~tToF ), the smallest momentum which can be meaningfully measured,√and the trap depth UD introduces a high-k cut-off, and defines the maximum kD = 2mUD /~.
In between these two limits, we can safely posit that kr ≈ k. In order to reconstruct the 3D
momentum distribution ñk , we first azimuthally average the images (assuming isotropy andk = kr ), before performing inverse-Abel transforms (see Section C) and finally averaging overtime (for t  tc ). In this final step we assume that any relaxation is slow on this ∼ 1 s timescale,
which we will scrutinize experimentally in the following.
In Fig. 5.11 we show ñk averaged for 0.2 s < t < 1.8 s on a log-log plot (solid blue line) in the
single-collapse regime (a = −0.86 a0 ) for N = 11.4(3) × 104 and L = 30(1) µm. The blue bandshows the standard deviation of the time average, and the thin dotted orange and green lines
show, respectively, ñk at ≈ 0.2 s and ≈ 1.7 s, confirming our assumption that the final state
momentum distribution is stationary on these timescales. The inset shows the time-averaged

67

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
106

a = -0.86 a0(µm3)
n k (μm

105

q=

kL

3

104
kD

103
102

ΔN/N = 0.33(1)

0.1

0.2

0.5
k (μm
(µm-1)

1

2

Fig. 5.11 Remnant momentum distribution in the single-collapse regime. We show a log-log plot of ñk for a =−0.86 a0 , N = 11.4(3) × 104 , and L = 30(1) µm. The blue solid line shows the time average for 0.2 s <t < 1.8 s. The shaded band shows the standard deviation of the time average, while the dotted orange
and green lines show ñk at ≈ 0.2 s and ≈ 1.7 s, respectively. The vertical dotted lines indicate kL and kD ,corresponding to L = 30 µm and UD /kB = 30 nK. The black dashed line shows a decaying power-law withexponent q = 3, which ñk follows over a decade in k. The inset shows the time average of the absorption
images, revealing an isotropic distribution.

absorption image, further highlighting the isotropy of the collapse remnants. The momentum
distribution ñk exhibits a striking power-law dependence ñk ∝ k −q , with q ≈ 3 (black dashedline). It is curious that this power-law coincides with that expected for a turbulent cascade in
the weak-wave turbulence regime [169] (see also Section 4.2).
So far we only considered collapse remnants in the single-collapse regime, and we now
turn to the remnants in the regimes of multiple collapse. We start by considering the bifurcation regime, where both single- and double-collapse events can occur (see also Fig. 5.8). In
Fig. 5.12(a) we present typical aToF ≈ 0 images for t  tr at a = −1.04 a0 with N = 11.4(3) × 104
and L = 30(1) µm. The two collapse outcome branches, which feature different ∆N/N , are also
clearly resolved by a dramatic difference in the central low-energy part of the cloud. The images from the double-collapse branch (left), where more atoms are lost, are visually similar to
the remnants in the single-collapse regime at lower |a| (see Fig. 5.11). Instead, for the singlecollapse branch (right), where only one collapse event occurs and less atoms are lost, the images
now look strikingly different, with a completely disturbed central core.
In Fig. 5.12(b) we show ñk in the bifurcation regime (a = −1.04 a0 ), where we separately
average the single- and double-collapse branches for t > 0.2 s. A clear difference in ñk is also
observed between the two branches, where the single-collapse branch features a significant increase in occupation of momentum states at around 0.5(2)/µm. In both cases the distribution
for k & 1/µm still remains consistent with the decaying q = 3 power-law dependence (dashedblack line). The insets, showing time-averaged images, reiterate the striking difference between

68CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE

Double#1392 , t=1695.1ms

(a)

(b)

Single#1394 , t=1225.9ms#1264 , t=267.5ms#1054 , t=238.2ms

ΔN/N = 0.45(1)106

ΔN/N = 0.29(1)a = -1.04 a0(µm3)
n k (μm105kL

ΔN/N = 0.29(1)

104

kD

103
102
(c)

106

ΔN/N = 0.45(1)

0.1

0.2

0.5
k (μm-1)

1

2
a = -2.48 a0(µm3)
n k (μm

105
kL
104

q=

3

kD

103
102

ΔN/N = 0.65(1)

0.1

0.2

0.5
k (μm
(µm-1)

1

2

Fig. 5.12 Remnant momentum distributions in the regimes of multiple collapse. Here N = 11.4(3) × 104 andL = 30(1) µm. (a) In the bifurcation regime (a = −1.04 a0 ) where either single- or double-collapse events
can occur, the absorption images also clearly exhibit a dramatic visual difference between single- anddouble-collapse events. (b-c) Momentum distribution ñk for (a) a = −1.04 a0 in the bifurcation regime(separately averaging single and double collapse outcomes) and (b) a = −2.48 a0 in the multiple collapse
regime. The solid curves are the long-time (t > 0.2 s) averaged ñk and the bands show the standard
deviation. The black dashed line shows the power-law ñk ∝ k−q , with q = 3, and the vertical dotted lines
indicate kL and kD (see text). The insets show the corresponding averaged images, highlighting dramatic
visual differences between regimes.

69

CHAPTER 5. WEAK COLLAPSE OF A BOSE–EINSTEIN CONDENSATE
the two outcome branches. Physically, in cases where a second collapse occurs, the situation
is similar to the single-collapse regime, but with a non-equilibrium initial condition. However,the fact that being prone to collapse a second time profoundly affects the post-collapse dynamics, even if the second collapse event does not occur, is strong evidence that the two collapse
events are sequential in time.
In Fig. 5.12(c) we show the long-time averaged ñk for the same N and L, but at large |a|(a = −2.48 a0 ) in the multiple-collapse regime, where atom loss has occurred in seemingly continuous fashion through multiple (experimentally unresolvable) collapse events. Here the distribution resembles that from the single-collapse branch in the bifurcation regime of Fig. 5.12(b),and it remains consistent with a power-law distribution ñk ∝ k −q , with q = 3 for 0.7 < k < kD .

5.4 Conclusion
In summary, we performed a detailed study of the collapse of a box-trapped BEC. We havesuccessfully answered the three questions that we set out to explore. Most importantly, we
provided first experimental evidence of weak collapse in any physical system, demonstrating
the associated scaling laws that dictate the weak-collapse atom loss. While our experiments
provide concrete benchmarks for the theory, they also pose many new questions and puzzles
for future research.
A specific, particularly intriguing example is the loss of phase information following a collapse event [193, 253], which is thought to be a general property of collapse. After a singularity
occurs, the solution of the nonlinear Schrödinger equation is only determined up to a phasefactor of exp(iθ). In real physical systems, where the singularity is only approached and notentirely reached due to dissipative loss mechanisms that enter, this is thought to translate to
the fact that the phase is ‘almost’ lost, in the sense that the final phase becomes highly sensitive
to small changes in the initial conditions. It is therefore thought that interactions between separate post-collapse components result in a chaotic final state [193] (see also [254]). This property
was recently tested in the collapse of optical beams [255], where wave function phase scrambling is presented as an explanation for the chaotic and turbulent behavior that is observed in
the aftermath of collapse events.
However, in their case the collapse is strong, and it is unclear how these notions extend to
weak collapse theory, where the collapse is only partial in that only a part of the wave collapses
towards the singularity (see Section 5.2.3). Nevertheless, turbulence at low nonlinearity levels
is generally thought to correspond to a set of waves whose phases are close to random [169],
which is consistent with the notion of phase loss and our observations.

70

6 Moderately Strongly Interacting Bose Gases
Therefore love moderately: long love doth so;
Too swift arrives as tardy as too slow.
Romeo & Juliet- William Shakespeare -6.1 Introduction
Superfluidity is one of the most dramatic manifestations of coherent quantum many-body phenomena. Following its monumental discovery in liquid 4 He in 1937 [256–258], London [259]
had the intuition to conceptually link this fascinating behavior to Bose–Einstein condensation,
which to fair accuracy allowed him to predict the lambda temperature of liquid 4 He. Tisza
shortly afterwards followed suit when using the notion of Bose–Einstein condensation to formulate his two-fluid model [260], which qualitatively explained the superfluid fountain effectand predicted the existence of two sound velocities. However, these early links did not go
uncontested, and most prominently include Landau’s criticism that an ideal-gas picture of thestrongly-interacting liquid 4 He state is fundamentally inadequate. Landau had formulated
a phenomenological description of superfluids based on mixtures of weakly interacting elementary excitations (phonons and rotons) [261], which even though not rooted in microscopic
theory, did correspond to the later measured excitation spectrum of superfluid 4 He.
In 1947 Bogoliubov developed a microscopic theory of the interacting Bose gas based on a
perturbative treatment of excitations above a macroscopically occupied condensate [42], arriving at an effective quadratic Hamiltonian valid for contact interactions governed by an interaction parameter na3  1 (see Section 2.4). This constituted a breakthrough in the understandingof Landau’s excitation spectrum, providing a direct link between Bose–Einstein condensationand superfluid behavior, while also reproducing the linear dispersion at low momenta. However any comparison to liquid helium remained only qualitative, owing to the inherently strong71CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
and poorly understood interactions in superfluid 4 He, which bring it far out of the theory’s
regime of validity1 .
Bogoliubov theory has become a cornerstone of our modern understanding of interacting
quantum fluids. The advent of laboratory synthesized gaseous atomic Bose–Einstein condensates [3, 4] which offer truly weak interactions (na3  1) brought experiments directly into
Bogoliubov theory’s regime of validity, and allowed many of its aspects to be directly experimentally tested (see e.g. [8,17,267,268] and references therein). By exploiting magnetic Feshbachresonances [7] (see Section 3.2.1), gaseous atomic BECs offer readily tuneable contact interactions characterized by an s-wave scattering length a, providing a flexible setting for pushingtowards the strongly interacting regime na3 ∼ 1, where such experiments have started to probe
the richer physics of interacting Bose fluids [10, 18]. However, increasing a also dramaticallyenhances inelastic three-body collisions, which makes experiments on strongly interacting bulk
BECs [28–31, 46, 64, 269–271] difficult and still scarce2 .
More generally, the connection between condensation and superfluidity, as well as superconductivity (a superfluid of electrons), remains an active research topic; see e.g. [12, 272].
In this chapter we present our first studies of strongly-interacting homogeneous Bose gases.
Here we focus on ’moderately‘ strong interactions, where simple mean-field theories break
down and one can observe beyond-mean-field quantum correlation effects, but the gas remains in equilibrium and the experiments remain tractable within the existing theories. Our
experiments provide steps towards the long-term goal of bridging the gap between weaklyinteracting BECs and the richer phenomena observed in liquid helium, in a regime where theinteraction parameter na3 is no longer  1.
We divide our discussion into two sections focusing on the ground state (Section 6.2, seealso [29]) and the elementary excitations (Section 6.3, see also [28]). We note that many of thepertinent aspects of Bogoliubov theory, alongside our primary experimental tool for probing
moderately strongly interacting samples, Bragg diffraction, have already been introduced in
Chapter 3, to which we refer the reader as we will assume knowledge of these throughout.

1

Bogoliubov’s theory was later extended by Lee, Haung, and Yang to hard-sphere bosons [44], and field theoretical formulations extended the predictions to the regime of strong interactions, showing that the linear low-k
dispersion was a robust general feature [262, 263]. A vast amount of theoretical effort has been dedicated to theunderstanding of liquid 4 He (see e.g. [264–266] and references therein).
2
In both mass-balanced Fermi gases near Feshbach resonances, and Bose and Fermi gases in optical lattices, theregime of strong correlations is readily reached experimentally owing to the suppression of three-body losses.

72

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES

6.2 Quantum depletion
The central goal of this section is to experimentally address the 80-year old question - Howmuch of a superfluid is really a Bose–Einstein condensate? Bogoliubov’s 1947 theory [42] describing
the zero-temperature properties of weakly-interacting (na3  1) Bose gases provides a tentative answer to this question. Within this theory, the condensate excitations are expressed as asuperposition of pairs of counter-propagating single-particle states, and quantum fluctuations
ensure that even at zero temperature there exists a non-zero occupation of such excitations (seeSection 2.4). This leads to the coherent interaction-driven depletion of the condensate, andpredicts the remaining condensed fraction√
nBEC /n = 1 − γ na3 ,

(6.1)

√
with γ = 8/(3 π) ≈ 1.5.
While Bogoliubov first developed this theory in order to describe the superfluid state of 4 He,
it was never directly applicable to it. Indeed, due to inherently strong interparticle interactions
in superfluid 4 He, while 100% of the system is superfluid only about 10% of the atoms are
actually in the condensate [264, 273]. The remaining atoms are coherently expelled from thecondensate and spread over a large range of momenta.
Numerically, diffusion Monte Carlo simulations have rigorously tested the regime of validityof Eq. (6.1), and found it to be quantitatively valid for na3 . 10−3 [274]. The advent of gaseous
atomic BECs offered new hope in experimentally confirming Eq. (6.1), as the interparticle interactions are well described by s-wave contact interactions and the achievable values of na3 fall
directly into the theory’s regime of validity.
Experimentally, however, an orthogonal challenge presented itself: the predicted depleted
fraction nQD /n is typically below3 1%, making it rather difficult to measure. In harmonicallytrapped ultracold atomic gases, characteristic signs of quantum depletion were observed byeither enhancing the role of interactions using optical lattices [275] (see also [276, 277]), or by
performing high-resolution studies of the expansion of a weakly-interacting gas [278]. Nevertheless, only semiquantitative comparison to theory has been possible, due to complications
associated with the addition of the lattice, the inhomogeneous gas densities involved, and theinterpretation of the expansion experiments [279]. Recent experiments in the non-equilibrium
driven-dissipative setting of polariton condensates [280] have also observed characteristic sig3

For typical parameters a = 100 a0 and n = 1019 m−3 , we have na3 ≈ 1.5 × 10−6 and Eq. (6.1) predicts a depleted
fraction of 0.2%.

73

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
natures of quantum depletion [281].
Here we experimentally create a textbook setting in order to quantitatively test and verify Bogoliubov’s theory of quantum depletion. Using our experimental platform, outlined in
Chapter 3, we produce homogeneous 39 K condensates with a readily tuneable scattering lengtha. We prepare our clouds in relatively large cylindrical boxes, with R ≈ 32 µm, L ≈ 50 µm inorder to achieve low-density quasi-uniform condensates with n = 3.5×1017 m−3 , as this allows√
us to reach larger values of na3 while enduring the same three-body loss rate Ṅ /N ∝ n2 a4(ignoring the modulation due to Efimov physics [33, 34, 66]).
We turn to Bragg spectroscopy [93, 94], a coherent momentum-selective two-photon process,in order to access the underlying momentum distribution in spectroscopic fashion (see Section 3.2.4 for an overview of this technique and details of our experimental setup).
As we show below, the large separation of momentum scales between the Bose–Einsteincondensate and the quantum depletion allows us to employ a BEC filtering technique [282],
which in turn allows us to spatially separate the BEC from the high-momentum componentsof the gas. This significantly simplifies the experiments and, to within a good approximation,
conceptually reveals the condensed fraction of the cloud in a single experimental run.

6.2.1 Bragg filtering
Our goal is to spatially separate the BEC from its depleted counterpart, in order to directly
measure the condensed fraction and test Eq. (6.1). As Bragg spectroscopy probes the onedimensional momentum distribution of the cloud n̆1D
k , we begin by taking a closer look at thetheoretically expected zero-temperature 1D momentum distribution of both the BEC, n̆BEC , andthe quantum depletion, n̆QD , for realistic experimental parameters.
The BEC distribution n̆BEC (k) features a Heisenberg-limited width ∝ 1/L [87] and exponentially suppressed tails. A top-hat BEC wave function ψ(y), of extent L, would give n̆BEC ∝sinc2 (kL/2), however the unphysical sharp edges in real space give unphysical high-k momentum tails ∝ 1/k 2 . In reality, the wave function is rounded off (over the healing length ξ) near
the walls and so the momentum distribution has exponentially suppressed high-k tails. In the√for |y| < L/2 [8], which we use to compute n̆BEC (k)limit ξ  L, we have ψ(y) ∝ tanh L/2−|y|2ξnumerically.

On the other hand, the quantum depletion n̆QD (k) has a width ∝ 1/ξ and polynomial tails[8,64,270,278]. In order to compute n̆QD (k) we assume that any finite-size effects are negligible
and use the standard textbook prediction [8, 17] for the three-dimensional QD in an infinite
system (see Section 2.4 for additional details). Integrating Eq. (2.40) over the two orthogonal74

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASESpn�˘ 1Dk���na3 = 0.1
L/⇠ ⇡ 40

���

����

L
����
�

���

-�/ξ

�/ξ

�

���

n�˘ 1Dk�pna3 = 0.2
L/⇠ ⇡ 50

���

����

����
�

���

�/ξ

-�/ξ

�

�/ξ

�/ξ

�
Fig. 6.1 Momentum distribution of a zero-temperature homogeneous Bose gas. Integrated 1D momentum distributions n̆1D
k for a gas of density n and size L at two different interaction strengths a. We normalize the
1D
distributions such that n̆1D
k (0) = 1 [setting γ = 0 in Eq. (6.1)]. The total n̆k (k) consists of two parts: thelow-k BEC (blue) with a Heisenberg-limited width ∝ 1/L, and a broad quantum-depletion pedestal (orange) with characteristic width set by the inverse healing length 1/ξ (see text). The low-k√ distribution is(to good approximation) the same as for a pure BEC just rescaled by a factor ∼ 1 − γ na3 (see dashedlines). The insets on the right accentuate that n̆QD (k)  n̆BEC (k) at large k. The cartoons on the left depict
the underlying microscopic picture of the coherent excitations out of the condensate (blue), which occur asa superposition of pairs of counter-propagating particles.
√ Note that while we use experimentally relevant
values of L/ξ, we do use slightly exaggerated values of na3 for visual clarity. Figure from [29].

directions givespn̆QD (k) ∝ a 1 + ξ 2 k 2 − ξ 2 k 2 (ξ 2 k 2 + 2) .

(6.2)

√
We normalize the r.h.s. of Eq. (6.2) to γ na3 , and the numerical BEC momentum distribution√
to 1 − γ na3 . In Fig. 6.1 we show examples of the expected n̆1D
k (k) = n̆BEC (k) + n̆QD (k), where
we scale both distributions so that in the absence of quantum depletion n̆1D
k (0) = 1.
As long as L/ξ  1, separation of scales is ensured and n̆QD (k) extends over a much wider
range of momenta than n̆BEC (k). This in turn allows a Bragg pulse with two-photon Rabi
frequency Ω to be tuned to resonance with k = 0 to selectively diffract the BEC, leaving mostof the high-k QD tails behind. However, for a non-infinite L/ξ the choice of Ω is non-trivial,
as both the imperfect diffraction of the BEC and the partial diffraction of the QD introduce
opposite systematic effects4 . As shown in Fig. 6.2, we find satisfactory results with Ω = 2π ×
4

Naively one might choose Ω so that a π-pulse has a Fourier width equivalent to the central (zero-to-zero) widthof the BEC spectral line [set by 4π~q/(mL) ≈ 2π × 900 Hz for our q = 1.7krec ≈ 13.7/µm considered here].
However, it is beneficial to capture more of the sinc2 -tails to avoid leaving behind a significant fraction of the
BEC (see [29] for a detailed discussion).

75

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
���

���

����
����
�π

���
-��π

˘
�n ��
QDn�˘ BEC
���

����

�

����

�π

����

��π

��

-��

�

��

�ξ

Fig. 6.2 Bragg filtering efficiency. Expected zero-temperature 1D momentum distributions of both √n̆BEC (left) andn̆QD (right) for typical experimental parameters: n = 3.5 × 1017 m−3 , L = 50 µm, and na3 ≈ 0.04.
The shaded regions indicate the fraction of the distribution that a Bragg π-pulse centered on k = 0 with
Ω = 1.8 kHz diffracts (see Section 3.2.4). Integrating yields a predicted 96% diffraction of the BEC, whileleaving behind 71% of the QD, and to a relatively good approximation achieves the desired filtering. Notethe difference in x-axis between the two panels; here L/ξ ≈ 60. Figure from [29].

1.8 kHz for representative experimental parameters. In the following, we will first assume thatthe filtering works perfectly, before later assessing the impact of this systematic effect, as wellas those arising from the small non-zero temperature of our samples.
Finally, using qξ  1 permits the separation of diffracted and non-diffracted portions of thecloud in time-of-flight expansion, as it ensures that the momentum kick received by a diffracted
atom, ~q, is much larger than the QD momentum spread. Combining the above requirements,
we arrive at the necessary separation of scales 1/L  1/ξ  q. For our experiments we achieve
L/ξ > 30 and qξ > 12, allowing us to directly employ this filtering technique.
Our experimental protocol is as follows. We start by preparing a weakly interacting BEC√
at a ≈ 200 a0 , such that na3 < 10−3 . The condensate is quasi-pure, supported by the factthat we do not discern a thermal fraction in time-of-flight expansion. Our clouds are initiallyprepared in a trap of depth U0 ≈ kB × 20 nK, but before tuning a we adiabatically increase
U0 by a factor of 5 in order to ensure that the interaction energy is smaller than the trap depth[U0  ~2 /(2mξ 2 )].
We then adiabatically (in 150 − 250 ms) increase a to a value between 700 − 3000 a0 , andmeasure the diffracted fraction (DF) as a function of τ . We limit ourselves to values of a which
fulfill two requirements: (i) three-body losses are kept to below 10%, and (ii) when reducing aback to 200 a0 no signs of heating are apparent.
In order to avoid scattering between atoms as they are diffracted out of the cloud, which
would jeopardize our filtering protocol, before the Bragg pulse we rapidly turn off the interactions (in ≈ 60 µs), using an RF π-pulse to transfer the atoms to the essentially non-interacting|F = 1, mF = 0i state (see Section 3.2.1). This sudden turn off of the interactions before probing the gas greatly simplifies the interpretation as it eliminates any final-state interaction effects, ensures that the diffracted and non-diffracted components separate without undergoing76

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
��� (c) (d) (e) (f)
a (b)
(a)
���

��
��

zy

���
���

���
���
���
���

���
���

���
���

(��
��))
ττ ((ms)

��

(b)
b ���
���

a

���
���

3000 a0
3000

t
����
����

����

700 a0
700

����
����

ττ ((ms)��))
Fig. 6.3 Bragg filtering and reversible interaction tuning of the condensed fraction. (a) Diffracted fraction (DF) as
a function of the Bragg pulse duration, τ , for Rabi frequency Ω = 2π × 1.8 kHz and a ≈ 3000 a0 . The reddiamonds correspond to data extracted from the absorption images in the background, which comprise of
the spatially separated stationary (bottom) and diffracted (top) parts of the cloud. (b) Zooming in aroundthe maximum diffracted fraction η (where τ is close to π/Ω), at three points in a round-trip protocol (seeinset): at 700 a0 (blue circles), after increasing a from 700 a0 to 3000 a0 in 80 ms (orange diamonds), andafter returning to 700 a0 in another 80 ms (open green circles). Each data point corresponds to the averageof typically 10 repetitions (error bars indicate s.e.m.). The solid lines show the quadratic fits [η − c1 (τ − τ0 )2 ]used to extract η. Figure from [29].

collisions, and suddenly freezes the momentum distribution before we probe it using Bragg
diffraction [283].
Following the Bragg pulse of duration τ , we wait for 10 ms, which allows the diffracted andnon-diffracted portions of the gas to separate by ≈ 220 µm, and ensures that the individualclouds do not significantly expand beyond the initial size of the box. Finally, we observe thegas by taking an absorption image along the z-axis (perpendicular to y).
In Fig. 6.3(a) we show an example of the measured diffracted fraction (DF) as a function of
the Bragg pulse duration τ at a ≈ 3000 a0 , for our chosen Ω = 2π × 1.8 kHz (see also [29]). Representative absorption images at various stages throughout this Rabi oscillation (red diamonds)
are shown in the background. The images depict the clearly separated stationary (bottom) anddiffracted (top) clouds.
We will initially assume that our Bragg pulse perfectly filters out the condensate from thehigh-k components of the gas, and that the condensed fraction of the cloud is simply given by
the maximal diffracted fraction, η (which is observed for τ = π/Ω ≈ 0.28 ms). The measured η77CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
is slightly below unity, with a small portion of the cloud visibly left behind (see central image).
While this is expected from quantum depletion, it could in practice also occur due to other reasons, including experimental imperfections or an inevitably nonzero gas temperature (whichleads to thermal depletion).
To demonstrate that we are in fact observing quantum depletion, our experiments are performed in differential fashion - that is, we study how η varies with a, while keeping the otherexperimental parameters the same. Moreover, we verify that the initial tuning of η with a is adiabatically reversible, which excludes the possibility that the gas’ condensed fraction is reduced
due to non-adiabatic heating or losses. In Fig. 6.3(b), where we focus on DF at τ ≈ π/Ω, we
exemplify such adiabatic reversible tuning by performing three (sequential) protocols, as depicted in the inset: for a cloud initially prepared at 700 a0 , following a ramp of a to 3000 a0 , andafter reducing a back to 700 a0 (completing the round-trip). We find that η indeed is reduced asa is increased, and crucially that this effect is fully reversible (within statistical uncertainties).
We have verified such reversible tuning for our entire experimental range of a values.

6.2.2 Quantum depletion
We explore the variation of η with the interaction parameter√na3 in Fig. 6.4. Within our as-

sumptions, this directly tests Eq. (6.1). We observe the expected linear dependence of η on√√na3 , and from a linear fit to the data, of the form η0 (1 − γ na3 ), we obtain an offset η0 close
to unity and a slope γ = 1.5(2), in good overall agreement with Eq. (6.1).
Our next step consists of scrutinizing our assumptions. In the inset of Fig. 6.4 we summarize
our numerical assessment of the systematic effects arising from imperfections of the filtering
protocol both due to the non-infinite L/ξ (see Fig. 6.2), and due to a small nonzero initial temperature T . The magnitude of both of the systematic effects is of order . 20% and they partiallycancel; note that here we only provide a brief account of these calculations, whereas they canbe found in full detail in the supplementary information of [29].
The dashed line in the inset of Fig. 6.4 shows the simulated dependence of η on√

na3 forT = 0 and our experimental parameters (n, L, and Ω). As already indicated in Fig. 6.2, any noninfinite Ω slightly reduces η0 as the tails of the BEC are not fully captured, which is in part why
η0 deviates from unity. Moreover, while the quantum depletion is spread over a large range ofmomenta, for non-infinite L/ξ we inevitably also diffract a portion of the quantum depletion,
which has the effect of reducing the apparent γ; a linear fit to the dashed line (omitted forclarity) gives γ0 ≈ 1.2.
The small systematic difference between our data and this T = 0 simulation can be explained78

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES

����

����

η

����
����

����

����

����
����

p����
na3

����

Fig. 6.4 Measurement of the quantum depletion of an interacting BEC. Plot of the maximal√ diffracted fraction η,
which we associate with the condensed fraction, versus the interaction parameter na3 . The solid line isa linear fit to the data, from which we obtain the offset η0 = 0.954(5) and slope γ = 1.5(2). The verticalerror bars show fitting errors (see Fig. 6.3), while the horizontal ones reflect both the ≈ 10% uncertainty inn and the uncertainty in the Feshbach resonance position at 402.70(3) G [65]. The inset presents a detailed
analysis of the systematic effects, including the non-infinite L/ξ and a small finite T . The dashed line shows
numerical simulations of the expected η based on our Bragg filtering protocol for T = 0. The orange shaded
region extends the simulations to include the effect of initial temperatures (at a = 200 a0 ) between 3.5 to
5 nK (top to bottom). Figure from [29].

by assuming a small nonzero initial temperature. At a finite temperature, thermal depletion occurs, arising from the presence of thermal excitations (phonons); this generally reduces η as the
majority of the participating atoms are also not diffracted by the Bragg pulse tuned to k = 0.
Moreover, even if a is increased adiabatically, the amount of thermal depletion increases, because changing a modifies both the dispersion relation and the particle content of the thermally
populated low-k excitations [8, 29]. If we consider our experimental protocol in which we initially prepare a gas at 200 a0 and small T > 0 before adiabatically increasing a, then due to thisadiabatic heating the finite initial temperature would not only lead to a reduction of η by a con√stant offset (independent of na3 ) but it would also slightly increase the apparent γ. Indeed,by taking this into account in our numerical simulations we obtain the orange shaded band for
an initial T between 3.5 and 5 nK, which is in quantitative agreement with our data. This low
initial T is also in line with the fact that we do not discern the corresponding thermal fractionsof . 10% in ToF at 200 a0 , and it is reasonable for our trap depth U0 ≈ 20 nK. It is worth not√ing that due to these effects η( na3 ) is also not expected to be perfectly linear, however this isnegligible on the scale of our experimental errors.

6.2.3 Conclusion
In conclusion, our experiments quantitatively confirm Bogoliubov’s theory of the quantum
depletion of a Bose–Einstein condensate. This theory forms the backbone of our understanding79CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
of interacting quantum fluids. In the future, it would be interesting to perform experiments√
which access even larger values of na3 , while also maintaining the stringent requirementson adiabaticity, heating, and losses. Deviations from Bogoliubov theory are expected to be√observable at our level of statistical precision already for values of na3 about twice larger[274] 5 . Our measurements also reveal a remarkable sensitivity to the temperature of the clouds,
and we envision that the Bragg filtering protocol, as originally conceived [282], could be used
to achieve high precision thermometry in cases where the thermal fraction is too low to bediscernible in ToF. Finally, it would also be interesting to use Bragg spectroscopy to measure
the full 1D momentum distribution, and thus experimentally study the form of n̆QD (k).

6.3 Quasi-particle excitations
Elementary excitations of a many-body system provide a window into the system’s groundstate, and they determine how the system responds to external perturbations. They are consequently at the heart of a range of macroscopic many-body phenomena, such as superfluidity [256–258]. A powerful tool for probing elementary excitations is Bragg spectroscopy,
a coherent two-photon process which provides direct access to the excitation energy ~ω ata well defined wavenumber q [55, 87, 93, 269, 284–286] (see Section 3.2.4). Previous experiments on weakly-interacting Bose gases have used Bragg spectroscopy to broadly confirm
Bogoliubov theory, both in harmonically trapped gases relying on the local density approximation [284, 285], and more directly in homogeneous box-trapped condensates [87].
In strongly interacting atomic BECs much richer physics is expected (see [18] for a recent
review), including phenomena traditionally associated with superfluid 4 He, such as the roton
minimum in the excitation spectrum [287–289] (see also [264–266] and references therein). A
deviation from the Bogoliubov spectrum was observed in Bragg spectroscopy of large-q excitations in a harmonically trapped 85 Rb BEC [269], and has inspired various theoretical interpretations [18, 269, 290–294], with no consensus or complete quantitative agreement with theexperiments being reached so far6 .
In this section we present our results of the Bragg spectroscopy of a moderately strongly√
Additional measurements that explore up to na3 ≈ 0.08 by abandoning our stringent adiabaticity criteria
can be found in [29]. While having to be taken with a pinch of salt, they do tentatively suggest a deviationfrom Bogoliubov theory of the same sign and roughly the same magnitude as predicted by the Monte Carlosimulations [274].
6
An inflection point in the dispersion relation has been observed in the one-dimensional regime in harmonicallytrapped gases [295], however arising purely from confinement-induced effects. Recent experiments using dipolar quantum gases have also observed a roton minimum in the excitation spectrum using oblate trap geometries [296], though here the origin of the minimum links back to the attraction between dipoles that reduce theenergy cost for out-of-plane excitations.
580

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES

(a)

Δω

(b)

ω
a1

0

a2

�

�/(π�)

�0Fig. 6.5 Predictions for the excitation resonances. (a) Interaction shift ∆ω for particle-like excitations with fixedq. The dashed line shows the Bogoliubov prediction (valid only for small qa), while the solid one shows
the Feynman–Tan prediction. (b) Sketches of the dispersion relation ω(q). The solid lines correspond totwo distinctly different scattering lengths (a2 > a1 ), following [297]. The dotted line corresponds to thefree-particle dispersion relation ω0 . Figure from [28].

interacting homogeneous BEC, focusing on large-q excitations. We significantly extend previously achieved values of a, thus confronting modern theories of interacting Bose gases.

6.3.1 Beyond Bogoliubov theory
Following Bogoliubov’s quasi-particle description (see Section 2.4), for qξ  1 the excitations
are particle-like and the predicted interaction shift from the free-particle dispersion relation∆ω = ω−ω0 , where ω0 = ~q 2 /(2m) is ∆ωB = 4π~na/m, which is q-independent and linear in na[see Fig. 6.5]. For a more qualitative understanding of this shift, consider that the atoms in thecondensate feel a chemical potential µ = 4π~2 na/m, while the diffracted (high-k) atoms haveequal direct and exchange interactions with the condensate, amounting to 2µ, the difference of
which recovers ~∆ωB .
In order for the prediction ∆ωB to be valid, one requires that: (i) the excitations are particle√
like (qξ  1), (ii) the gas is weakly-interacting ( na3  1), and (iii) the two-body scattering
amplitude is k-independent (q  1/a), for which one can neglect the short-range two-particle
correlations at distances r . a.

Assuming condition (ii) holds and√na3  1, the Feynman energy relation [298, 299] relates

ω0 to the excitation resonance via the static structure factor Sq , i.e. ω = ω0 /Sq . For short-range
correlations and assuming qξ  1, Sq is given by
C2
Sq = 1 +
8n



1
4
−
q πaq 2



,

(6.3)

where term in the brackets arises from two-body correlations at short-distances, and C2 is thetwo-body contact density (see Section 2.5), which captures the effects of many-body correlations [17, 294] and can be seen as a measure of the probability of finding two particles at thesame point. Note that in general the possibility of such ‘factorization’ of two-body and many-81

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES

Sq

1.10

9000 a07000 a05000 a0
3000 a0

1.05

1.00
0

1

2

3

q/krecFig. 6.6 Emergence of a peak in the structure factor. We plot the predicted
Sq within the FT formalism for a gas√
3 ranging between 0.03 and 0.15, revealing
density n = 0.2 × 1018 m−3 at various a (seelegend),withna√
the emergence of a clear peak structure as na3 increases. For all plotted values qξ & 2. The dashed line
indicates the peak position for continuously varying a [see Eq. (6.3)].

body physics was first highlighted by Tan [47], and so we coin this the ‘Feynman–Tan’ (FT)
formalism. By noting that for our experimental parameters |Sq −1| < 0.03, where we have used√the contact density C2 ≈ (4πna)2 valid for low na3 , we can approximate 1/Sq − 1 ≈ 1 − Sq .
Within this approach the prediction for the interaction shift from ω0 is then given by∆ωFT =πqa 
4π~na 
1−
.
m
4

(6.4)

In the limit of small qa, ∆ωFT approaches the expected Bogoliubov result ∆ωB [dashed line in
Fig. 6.5(a)], however for increasing a at fixed q it back-bends, changing sign at a = 4/(πq) [solidline in Fig. 6.5(a)]. In contrast, for phonon-like excitations (qξ  1) ∆ω remains positive for alla [297]. As depicted in Fig. 6.5(b), this implies an inflection point in the dispersion relation[ω(q) at fixed a]. Such an inflection point is a precursor of the roton minimum, which fully√
develops only for very strong interactions ( na3 ∼ 1) [294, 297]. The static structure factor
Sq (at fixed n and a) reaches a maximum at q = 8/(πa) [see Eq.(6.3)], independent of n, as
plotted for relevant experimental parameters in Fig. 6.6. While this maximum has been linked
to the roton minimum [294, 300], the liquid helium scaling qroton ∼ n1/3 is only recovered for√na3 ∼ 1.
It is useful to define a dimensionless interaction frequency shift
α≡

mq
∆ω ,
4π~n

(6.5)

in order to recast the FT prediction [Eq. (6.4)] asπ αFT = qa 1 − qa ,
4
82

(6.6)

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES

(a)
(�)
�
�
q�
�~
�
�
�
�
�
�
-�

(b) �� �(�)�� (%)(%)
��(%)
�� (%)50µm
��µ�

�� (%)

�� (%)

�� �
�
�
�
�� �
�
-�-�
� �
-�
�� �
�
Δω
/
�π
(���)
Δω
/
(
�
π)
(
���
)
�
Δω
π) ((���
��Δω // ((�
�π)
���))
OD
�
�
�
Δν (���) BEC.
�
Fig. 6.7 Bragg spectroscopy of an interacting
Here n ≈ 2.0 × 1012 cm−3 , q = 1.7 krec (along the ŷ axis), and� (taken along ẑ) following a 2-ms Bragg pulse, and subsequenta ≈ 1000 a0 . (a) Typical absorption image-�� occurs due to� collisions between the stationary
15 ms ToF. It should be noted that the scattering
halo, whichΔν(���)and diffracted cloud, does not alter the centre-of-mass of the atomic distribution. (b) Bragg spectrum for aninteracting cloud, obtained by the varying frequency difference between the two Bragg beams (referenced
to ω0 , which we calibrated using a non-interacting cloud) and recording the diffracted fraction for each(obtained from the centre-of-mass). We extract the resonance position ∆ω from a gaussian fit to the data(solid line). The typical statistical error on each data point is ∼ 1%. Figure from [28].

which is a universal function of qa. Within this formulation the Bogoliubov result reads αB =qa. In addition to its aesthetic qualities, this form is experimentally useful as it allows plotting
the results with different n on the same universal curve.

6.3.2 Spectroscopy of quasi-particle excitations
We now turn to our experiments performing Bragg spectroscopy of interacting homogeneous
Bose–Einstein condensates. As described in Section 3.1, we produce 39 K BECs in the lowest
hyperfine state in order to access the Feshbach resonance at 402.70(3) G. For our purposes here,
we prepare quasi-pure homogeneous BECs of N = (50 − 160) × 103 atoms, in a cylindricalbox of variable length L = (30 − 50) µm and radius R = (15 − 30) µm; the corresponding gas

densities n range between 0.2 and 2.0 × 1018 m−3 . It is again beneficial to work at low densities,√
as one can reach larger values of qa and na3 for the same three-body loss rate Ṅ /N ∝ n2 a4(ignoring the modulation due to Efimov physics [33, 34, 66]). Throughout we use a trap depth
U0 ≈ kB × 20 nK, and the condensed fraction is > 90% (see Section 6.2).
We always start by preparing the BEC at 200 a0 , before ramping (in 50 ms 7 ) the scattering
length to its final value a at which we perform high-resolution Bragg spectroscopy8 . We limit
the maximal a that we explore for each n so that the total particle loss during the experimentis < 10%. A significant advantage of using a homogeneous sample is that three-body loss
does not intrinsically lead to significant heating [301]; this is in stark contrast to harmonic traps
where ‘anti-evaporative’ heating occurs as atoms with below-average potential energy (at the7

We have checked that the Bragg spectra are not broadened for these ramp speeds, whereas for faster ramps wedo observe significant broadening.
8
Our Bragg pulses are 2-ms in duration, which minimizes any Fourier broadening and ensures a near-Heisenberglimited width [87] when the sample is non-interacting.

83

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
trap center) are preferentially lost [302].
We choose three different Bragg angles θ to achieve three different values of q/krec : 1.1, 1.7,
and 2.0, where krec = 2π/λ and λ = 766.7 nm (see Section 3.2.4 for an overview of our Bragg
set-up). For our experimental range of parameters we always stay in the regime of particle-like
excitations: qξ varies between 5 and 40.
In Fig. 6.7(a) we present a typical absorption image following a 2-ms Bragg pulse and subsequent 20 ms ToF. At moderately strong interactions, the images feature a clear halo, whichis spherical in shape and occurs due to collisions between the diffracted cloud as it leaves
passing through the stationary BEC (see also [303, 304]). Importantly, these collisions do notalter the centre of mass of the atomic distribution, which we use to infer the diffracted fraction(DF) [269, 286]. We keep the maximum DF . 10% in order to keep the experiments within thelinear response regime9 .
We map out DF as a function of the frequency difference between the two Bragg beams ω̃; inFig. 6.7(b) we show a typical example of such a Bragg spectrum, with ω̃ referenced to ω0 10 . We
extract the resonance shift from the free-particle dispersion ∆ω using a gaussian fit to the data.
In Fig. 6.8 we plot the measured ∆ω, recast as the dimensionless α [see Eq. (6.5)], as a functionof qa for three different combinations of density n and excitation wavenumber q 11 , which all
fall onto a single universal curve. For qa & 0.5 we observe back-bending and a clear deviationfrom the Bogoliubov theory prediction (dashed line)12 . Increasing qa further shows how α
becomes clearly negative. We find excellent agreement with the Feynman–Tan (FT) prediction[solid line, see Eq. (6.6)], without any free parameters.

6.3.3 Exploring the limits of validity of Feynman–Tan theory
While we observe good agreement between our experiments and the FT prediction for qa . 2.5,
we note that around qa = 2.5 the validity of this theory is questionable, and the apparent
agreement could be fortuitous. For these data the interaction parameter is already sizeable; the
Lee-Huang-Yang (LHY) prediction for the next-order correction to C2 is of the order 50% [43,44, 62] and beyond-LHY corrections [305, 306] could even be significant. The Feynman relation√is also expected to be quantitatively reliable only for na3  0.1 [297, 307].
√
We now consider even larger values of qa and na3 in order to test the limits of validity of the
The measured shift ∆ω reduces for larger DF, approaching 0 (ω = ω0 ) for a π-pulse. The requirement DF . 10%
should keep the resulting systematic relative suppression of ∆ω to . 10% [285].
10
We separately measure ω0 using a non-interacting sample (see Section 3.2.4).
11
Note that the normalization of Eq. (6.5) allows us to take into account small ±10% variations in density for thedata taken with the same nominal n and varying a.
12Previous Bragg spectroscopy experiments on harmonically trapped strongly interacting 85 Rb samples [269]
achieved maximal values of a that correspond to 0.8/q, and while back-bending was observed ∆ω remained
positive, consistent with FT theory.
9

84

Δ

-���

-���

CHAPTER 6.� MODERATELY
BOSE GASES� STRONGLY� INTERACTING�� (��� ��)

(b)

α

�

-�

(µm12-3cm) -3) qq/k(knn (10recrec)
1.1
1.1
2.0
2.0
1.7
1.7
2.0
0.8
0.8
2.0

-�
�

�

�

��
Fig. 6.8 Breakdown of the Bogoliubov approximation and observation of negative frequency shifts. Dimensionless frequency shift α versus qa for three different combinations of n and q (see legend). The solid line shows
the FT prediction αFT [see Eq. (6.6)] with no adjustable parameters, whereas the dashed line shows the Bogoliubov prediction αB = qa valid for small qa. The vertical error bars show statistical fitting errors andthe horizontal error bars reflect the uncertainty in a arising from uncertainty in the position of the Feshbachresonance. Figure adapted from [28].

FT prediction. In Fig. 6.8 we saw that, as anticipated, by decreasing the density (from 2.0 µm−3
to 0.8 µm−3 ) we were able to reach significantly larger values of a (up to ≈ 3 × 103 a0 ) while
adhering to our loss-requirement of . 10%. Here we use even lower densities n = 0.2 µm−3

(while also ensuring that our cloud remains quasi-homogeneous) to allow even larger values√of a ≈ 8 × 103 a0 , corresponding to na3 ≈ 0.1, and with q = 2krec we have qa ≈ 7.
To explore deviations from FT theory we combine data taken using many different combi√
nations of {n, q, a}, grouping them into sets with (roughly) equal na3 , but varying values of
qa. This allows us to attempt to disentangle the ‘culprit’ responsible any deviations, since if√
one simply increases a at fixed q and n then one simultaneously increases both qa and na3 ,thus making it difficult to identify which of the two dimensionless parameters is (primarily)
responsible for any deviation13 .
In Fig. 6.9 we plot the deviation of the dimensionless frequency shift from FT theory α − αFT√
versus qa, where the different symbols (see legend) correspond to different na3 . For ourvalues of qa & 3 we see a clear deviation from the FT prediction, which overestimates the√(negative) shift. As we always observe comparable shifts at the same qa and varying na3 ,we conclude that (at least for the parameter range explored here) the breakdown of FT theory√
occurs for qa & 3, independent of na3 .
The deviation from FT theory that we observe for qa & 3 can be explained by turning to a
recent calculation based on the Wilson operator product expansion (OPE) [294] (see also [262]),13

While increasing a at fixed q and n also reduces qξ (which could lead to√the breakdown of our assumption ofparticle-like excitations), we still have qξ ≈ 11 for our largest qa and na3 , which is larger than our lowestqξ ≈ 5 from Fig. 6.8, which we found to display universal behavior, and so we rule this out.

85

Δ
-���
-���

CHAPTER 6.�� MODERATELYSTRONGLY��INTERACTINGBOSE GASES������ (��
(���� ����))

(b)
��
��
αα

��

- αα��
αα ��

-�
-�
-�
-�
��

��

��

��
��

p

na3 (%)

2.5
4.3
6.4
9.3

��
��

��

��

��
��
Fig. 6.9 Deviation from the Feynman–Tan prediction. We plot the deviationof the dimensionless frequency shift√
α from the FT theory versus qa, for four different values of na3 , reaching up to 9.3% (see legend). As
described in the text, the dashed line shows the OPE prediction with C2 = (4πna)2 and no adjustableparameters. The dot-dashed line shows the OPE prediction, self-consistently including LHY corrections
to C2 . The inset shows a comparison of the FT (solid) and OPE (dashed) predictions at low qa. Figure
from [28].

which predicts a frequency shift


~C2
2
∆ωOPE =
−1 ,
4πmna 1 + (qa/2)2(6.7)

which assuming the low-na3 result C2 = (4πna)2 , and after being recast into dimensionlessform [see Eq. (6.5)] becomes
αOPE = qa




2
−1 .
1 + (qa/2)2(6.8)

In Fig. 6.9 the dashed black line depicts αOPE − αFT , and we find that this prediction captures

the data well for qa & 3. While Eq. (6.8) is also a universal function of qa, the theory allows forthe self-consistent inclusion of beyond-mean-field corrections to C2 . To test this we write thetwo-body contact density including the Lee-Huang-Yang (LHY) correction [43, 44, 62]:2

C2 = (4πna)64 √ 31+ √na + ... ,
3 π

(6.9)

√
and calculate the LHY-corrected α̃OPE for our largest na3 = 0.093 (α̃OPE now depends on√
both qa and na3 ). The dot-dashed line in Fig. 6.9 shows this prediction, upon subtracting theuniversal αFT . Interestingly, this does appear to provide slightly better agreement with the data,
but this observation is not statistically conclusive (see also [64]). It should be noted howeverthat the LHY correction changes C2 by about a factor of 2, and so at this stage beyond-LHY
corrections, which also depend on the van der Waals length, could be significant [305, 306].
In the inset of Fig. 6.9 we highlight that while we find success for the OPE prediction at large

86CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
qa, it is experimentally excluded for qa . 3, where it predicts a zero crossing of the frequency
shift at qa = 2 instead of qa = 4/π. As observed experimentally, and also as theoretically described in [294], the two theories are complementary. In fact, while both FT and OPE theories
address the short-distance behavior of the density response function, the main difference between the two lies in the type of excitation: FT theory assumes a single particle excitation, while
OPE assumes high-momentum pair excitations. Theoretically, while OPE theory is predicted
to be exact at q → ∞, for any finite q instead there exists a regime at small values of |Sq − 1| atwhich FT theory still holds.
Finally, it is curious to note that if αOPE were valid below qa = 1.57 > π/4, then the magnitude of the frequency shift would be larger than the FT prediction, which formally gives anupper bound to the frequency shift due to the sum rules on which it is based [17].

6.3.4 Conclusion
In conclusion, we have studied the quasiparticle excitations in a moderately strongly interacting homogeneous BEC. Our experiments push far beyond the regime of validity of Bogoliubov√
theory. For a range of interaction strengths qa . 3 (and small na3 ) our experiments arequantitatively captured in the framework of the Feynman energy relation, taking into accountshort-range two-particle correlations, in the spirit first introduced by Tan [47]. As predicted bythis theory, we observe that the shift of the excitation resonance from the free-particle energychanges sign from positive to negative, and confirm that this sign change occurs at qa ≈ 4/π.
For qa & 3 we find that this theory also breaks down, suggesting the need for more sophisticated theoretical approaches. In particular, a recent such approach, based on the Wilson
operator product expansion, shows good agreement with our data for qa & 3, however fails
to capture the low-qa behavior. Unifying the different theoretical descriptions of quasiparticle
resonances in all interaction regimes remains an open theoretical challenge.
Our experiments focus on the regime of particle-like excitations (qξ  1), however in the
future it would be interesting to map out the full dispersion relation by experimentally probing lower values of qξ. This can be achieved by reducing the angle between the Bragg beams
θ, or through parametric excitation via the modulation of the scattering length [308] or density [309]. This should allow the experimental observation of the expected inflection point in
the dispersion relation at large a, a precursor to the roton excitation.
In the opposite limit, we already exhaust the maximal q = 2krec achieved with a maximum
angle of θ = π. However, it would be interesting to perform high-precision Bragg spectroscopyusing a higher-order Bragg process: a four-photon process could effectively give access to twice87

CHAPTER 6. MODERATELY STRONGLY INTERACTING BOSE GASES
the maximum q, thus significantly extending our accessible range of qa.
Finally, a high precision study of the shape of the Bragg spectra (to first order characterized
by their width) would provide additional valuable information [294], and is anticipated as afruitful study for the future. Indeed, here box-traps are particularly suitable as they avoid
additional broadening due to the inhomogeneous density present in harmonic traps and allow
for Heisenberg-limited spectrum widths [87].

6.4 Conclusion
In conclusion, in our first studies of ‘moderately’ strongly-interacting gases we explored regimes
where simple mean-field theories break down and one can observe beyond-mean-field quantum correlation effects, but the experiments also remain tractable within the existing theories.
In Section 6.2 we presented the first quantitative measurement of the quantum depletion of a
Bose–Einstein condensate, confirming Bogoliubov’s 70-year-old theory that describes the microscopic origin of how atoms are expelled from the condensate due to interactions. In Section 6.3 we mapped out the energy of particle-like excitations using two-photon Bragg spectroscopy, observing dramatic deviations from Bogoliubov theory and confronting modern theories of interacting quantum fluids.
We already provided detailed summaries and outlined future goals within each section, andso here we will constrain ourselves to a few general remarks. In the current state of affairs, wehave more or less reached an impasse in terms of the maximal na3 that we can explore while
adhering to our stringent loss, heating, and adiabaticity requirements. In Chapter 7 we willventure down the same path as taken in the pioneering experiment [270], where the solution is
to forsake any hope of preparing an equilibrium state adiabatically, and instead turn to quench
experiments, allowing access to the unitary regime (na3  1) though also finding oneself far
from equilibrium.

88

7 Boses Gases Quenched to Unitarity
When you come to a fork in the road, take it.
- Yogi Berra -7.1 Introduction
Strongly-correlated states of matter, from novel superconductors to the quark-gluon plasma,exhibit some of the most fascinating phenomena in nature. Understanding such systems,
and especially their dynamics far from equilibrium, poses a fundamental challenge in modern physics.
Ultracold atomic gases act as remarkable quantum simulators for these problems, due totheir tuneable interparticle interactions and experimentally resolvable intrinsic timescales. In
particular, the ability to exploit magnetic Feshbach resonances [7] and tune a → ∞ offers accessto the unitary regime, where the interactions between particles are as strong as theoreticallypossible. In this strongly-correlated regime the microscopic details of a particular system arethought to become less important, so that connections between seemingly disparate quantum
many-body systems can be drawn. In particular, in unitary ultracold atomic gases the underlying physical processes inevitably no longer depend on the diverging a, and instead are thought
to display new types of universal behavior [11, 12, 16, 18, 310, 311].
Studies of the unitary Fermi gas (see e.g. [11, 12, 16] and references therein), which most
famously allowed the exploration of the crossover between Bose–Einstein condensation andBardeen–Cooper–Schrieffer superconductivity [5, 312–316], have been among the highlights ofultracold-atom research for the past 15 years.
The unitary Bose gas holds as many and as tantalizing promises as its Fermi counterpart,
including novel forms of superfluidity [317–319] and emergent universal behavior set solely
by the gas density [310, 311]. However, in Bose gases the strong interactions also lead todramatically enhanced particle loss and associated heating; this establishes a complex inter-89n-1/3 /λincreasing degeneracy

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

unitary thermal
Bose gas

thermalBose gas{n-1/3 , λ}{n-1/3 , λ, a}1?

beyond mean fieldmean-fieldBEC{n-1/3 , a}{n-1/3 }1

n-1/3 /aincreasing interactions
Fig. 7.1 Sketch of the equilibrium phase diagram for strongly-interacting Bose gases. We are not able to adiabatically prepare unitary gases to directly explore bottom left corner of this phase diagram. Instead, we
perform quench experiments to enter the unitary regime, starting with equilibrium degenerate (blue region)or thermal (red region) Bose gases at large n−1/3 /a.

play between the coherent and the dissipative dynamics, and makes the study of the unitary
Bose gas an inherently dynamical, non-equilibrium problem. Experimental studies of unitary
Bose gases are only recently emerging [46, 65, 90, 270, 271, 320–322], with steady theoreticalprogress [18, 319, 323–354], establishing an exciting research frontier.
While our principal aim is uncovering the nature of the unitary Bose gas, this stronglyinteracting far-from-equilibrium scenario also fits into recently emerging theoretical paradigm
concerning general properties of equilibration and thermalization in quantum many-body systems far from equilibrium (see e.g. [176, 355–358]). These theories, aimed at unravelling theorganizing principles of non-equilibrium quantum systems, have predicted elegant universal
scaling behavior, akin to the quasi-equilibrium behavior near phase transitions between different states of matter.
As a starting point for our discussion, it is insightful to consider a cartoon sketch of theequilibrium phase diagram of strongly interacting Bose gases (Fig. 7.1) within the universality
hypothesis. A weakly interacting thermal Bose has three relevant lengthscales: the interparticle√
spacing n−1/3 , the thermal wavelength λ ∝ 1/ T and the s-wave scattering length a. At low
temperatures, when our gas forms a quasi-pure Bose–Einstein condensate, λ drops out of the
problem we are left with just n−1/3 and a. Here the gas is described well by mean-field theory
assuming a single macroscopic wave function, as long as the interactions are relatively weak.
As one starts tuning to stronger interactions, one begins to explore beyond-mean-field effects,
such as those discussed in Chapter 6; for a sense of scale there we reached n−1/3 /a ≈ 70 withn−1/3 /λ . 0.4.

90

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

Fig. 7.2 Borromean rings act as a symbolic representation of three-body Efimov states. In this picture the threerings are bound together, but if one of them is removed the other two separate as well. The quantum
mechanical analogue occurs when three bosons interact, in a situation where two-body interactions do notsupport a two-body bound state (as is the case at unitarity). In this case Efimov predicted a spectrum ofbound three-body states [34].

In thermal unitary gases the diverging a is capped by λ. While one still has two lengthscales,
in many cases their dynamics are coupled, and one can still predict universal behavior; forexample, one expects essentially universal behavior of the loss dynamics, as experimentallyconfirmed using harmonically trapped gases [320–322].
In interaction-dominated degenerate gases, we are left with only a single lengthscale, theinterparticle spacing n−1/3 , which sets the natural momentum, energy, and time scales:~kn = ~ 6π 2 n

1/3, En =~2 kn2~, and tn =
.
2m
En

(7.1)

While these ‘Fermi’ scales are relevant to both Fermi and Bose gases, in Bose gases the implied universal behavior can be broken by Efimov physics [33–35, 64–66, 271, 348, 359–365]; theEfimov effect [34], a quantum few-body effect first discussed in nuclear physics in 1970, is aquantum-mechanical analogue of the Borromean rings (see Fig. 7.2), where even though there
is no two-body bound state, novel three-body bound states, known as Efimov states, still exist.
The Feshbach dimer molecular state, which is responsible for the resonance, is of size a witha binding energy Eb ≈ ~2 /(ma2 ), and so becomes unbound as a → ∞. The infinite series of
excited Efimov trimer states, each of a size 22.7 times larger than the previous one, can introduce new lengthscales into the problem, and are thought to profoundly affect the many-body
behavior (see e.g. [319, 341, 346, 366]).
First experimental forays into the unitary regime date back to [46], where the authors obtained a lower bound for the equation of state µ = 0.44(8)En , in line with theoretical estimates [310, 323–325, 327, 332, 334, 343, 367, 368] that predict a ground state energy ranging from
0.39 to 1.75 En (though note that some are variational upper bounds). Indeed, it is natural toexpect the ground state to have an energy ∼ En .
In the pioneering experiment [270], the authors embraced the non-equilibrium nature of the91CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
problem, and performed quench experiments to reach the unitary regime, entirely forsaking
equilibrium. The gas was allowed to then evolve at unitarity for a variable time, before quenching back to low a to observe the samples. Rather remarkably, this revealed that the absolutepost-quench occupation of high-momentum components reached a steady-state, while the total atom number decayed continuously. This provided first experimental evidence that thelossless processes are somewhat faster than the recombination induced ones, and raised fundamental questions about the extent to which the gas has well-defined equilibrium properties.
Overall, these degenerate-gas dynamics displayed universal behavior, consistent with tn as the
only characteristic timescale [270, 271].
However, many-body interferometry of a thermal unitary Bose gas has revealed novel threebody correlations [65], and Efimov trimers have been observed following a ramp out of theunitary regime [271].
In this chapter we study the dynamics of homogeneous Bose gases quenched to unitarity,
focusing on initially degenerate gases in Section 7.2 and thermal gases in Section 7.3; note thatcore parts of our discussion closely follow [30, 31].
Before turning to our experiments we first highlight the advantages of performing unitarygas experiments using homogeneous Bose gases (Section 7.1.1), detail the pertinent atom-loss
scaling law predictions (Section 7.1.2), and outline our quench protocol used to probe the unitary regime (Section 7.1.3).

7.1.1 Advantages of an optical box potential
All previous unitary Bose gas experiments were performed with harmonically trapped gases,
and their interpretation relies on knowledge of the inhomogeneous density profiles. For degenerate gases, the density profile is known prior to the quench to unitarity, and therefore also justafter it, but the subsequent evolution is complicated by strong interactions and inhomogeneous
losses.
Here we prepare our gases in the quasi-uniform potential provided by a cylindrical opticalbox trap [21], and so they are essentially homogeneous (see Section 3.1.2). Crucially, the density
in our gas is always closely related to the total atom number, which sets quantities such as En
and tn , which are global variables1 . We can therefore assume that (to a good approximation) thevolume remains constant throughout an experiment, which allows us to quantitatively study
the full evolution of a cloud as it decays and heats.
For the experiments in this chapter we always use the same optics (spatial light modulator
1

In experiments the density is never perfectly uniform due to edge effects, but for our box trap we assess that for
more than 80% of the atoms the local En is within 10% of the trap average value.

92

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
parameters) to create the quasi-uniform trapping potential, which has an aspect ratio is 2R/L ≈0.63 and a volume V ≈ 3 × 10−14 m3 . While it is reasonable to assume the same constant V for
all our data, varying the laser power (to vary the trap depth U0 ) and the typical energy scaleof the gas (kB T for thermal gases and µ ∝ N a for condensates) does lead to variations in theeffective initial sample volumes for different experiments, which we can readily account for. In
order to assess these initial volume differences we first measure a nominal volume2 , and thencorrect the volumes of other samples assuming a power-law potential of the form ∼ r15(4) [22](see also Section 2.3).
Another valuable benefit of using a homogeneous trap is that it eliminates ’anti-evaporative’heating present in harmonic traps, where atoms at the trap center with below-average energy
are preferentially lost [301,302,369]. However, to avoid heating from the loss processes one still
has to ensure that the loss products leave the box without undergoing secondary collisions3 .

7.1.2 Atom-loss scaling-law predictions
Here we present theoretical predictions of the atom-loss scaling laws across the different interaction and temperature regimes (see Fig. 7.1). In our experiments three-body recombination is
the leading order loss process, as one-body loss4 is usually negligible (with a lifetime > 102 s)
and two-body inelastic collisions are absent as our gas is prepared in the lowest hyperfine state.
The per-particle three-body loss rate generally takes the formṄ /N = −g3 L3 hn2 i ,(7.2)

where g3 is the zero-distance three-body correlation function, L3 is the three-body loss coefficient, and h...i denotes a trap average. For a thermal gas boson bunching gives g3 = 3!, which
is suppressed in a weakly-interacting zero-temperature Bose–Einstein condensate [370–373].
While for na3 → 0 we have g3 = 1, the beyond-mean-field LHY correction to g3 (which arises
from the depletion of the condensate) is
64 √ 3g3 ≈ 1 + √
na .
π
2

(7.3)

Our in-situ analysis used to extract V is described in detail in [22]. In short, we perform partial re-pump absorption imaging to measure the in-situ cloud shape. To extract the effective trap dimensions we convolve theexpected line density distributions with a gaussian point spread function to account for both finite imaging resolution and edge effects. For our thermal gases we measure V ≈ 2.7 × 10−14 m3 for a cloud at T ≈ 140 nK in
a trap of depth U0 /kB ≈ 3 µK. For our degenerate gases we measure V ≈ 3.7 × 10−14 m3 for N ≈ 1.2 × 105 at
a ≈ 400 a0 in a trap of depth U0 /kB ≈ 50 nK.
3
To fulfill this requirement one requires a mean-free path 1/(8πna2 ) large compared to the box dimensions (toavoid secondary collisions) and a trap depth U0  Eb [assuming that three-body recombination results in theformation of an atom and a weakly bound dimer, which (respectively) gain 2/3 and 1/3 of Eb .]4
Some one-body loss is always present due to e.g. collisions with the background gas in the vacuum chamber.

93

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY√
Interestingly, this correction has a rather large prefactor (64/ π ≈ 36) and so for na3 = 10−5
we already expect a ≈ 10% increase of g3 5 .
For a homogeneous gas, the density is constant across the trap and the size (set by the trap
volume V ) is independent of time (i.e. while losses occur and N decreases), which dramaticallysimplifies matters. In particular, we simply have hn2 i = n2 = N 2 /V 2 . Equation (7.2) thengenerally takes the power-law formṄ /N = −c0 N γ ,(7.4)

where c0 is a constant. Integration explicitly yields the atom-loss dynamics

−1/γ
−γ
N (t) = c0 γ t + N0
,

(7.5)

where N0 is the initial (t = 0) atom number and at long times we expect power-law behavior
N (t → ∞) = (c0 γt)−1/γ .

(7.6)

In the following we consider the different temperature and interaction regimes to deduce theexplicit form of Eq. (7.4) expected in each regime (i.e. finding γ).

Weakly interacting regime
In degenerate gases away from unitarity6 , on average (ignoring the additional log-periodic
variation with a that arises from Efimov physics [33,35,360]) we have L3 ∝ ~a4 /m [302,374], sothat γ = 2. More generally, for a loss process involving i particles, one expects (on dimensional
grounds, neglecting Efimov physics)Ṅ /N ∝ ~/m a3i−5 ni−1 ,(7.7)

where γ = i − 1 is an integer [375, 376].
In the following we will consider how these atom-loss scaling laws are modified in the unitary regime, when a → ∞.
5

The large prefactor makes studies of g3 a promising candidate for exploring beyond-LHY corrections. In particular, for realistic experimental parameters a = 3000 a0 and n = 5 µm−3 the prediction from Eq. (7.3) exceeds 6(the thermal gas g3 ), which is impossible, and one would already expect strong deviations when the correction
is of order 1.
6Here we assume that we are still in the regime where a is large compared to avdw ≈ 60 a0 (for 39 K).

94

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

Thermal unitary regime
In the thermal unitary regime the diverging a is capped by the thermal wavelength λ, which isa statistical measure of the inverse relative atomic momenta. We therefore have that (averaging
over a thermal distribution) L3 ∝ ~λ4 /m, and generally Ṅ /N ∝ −hn2 i/T 2 . More quantitatively(up to negligible corrections), the loss coefficient is√√
3 3~ 46 3π 2 ~5
L3 = ζ3,λ = ζ3 32m
m (kB T )2

(7.8)

∗

where ζ3 = (1 − e−4η ) is a non-universal prefactor set by Efimov physics [320–322, 376–378],
and we have omitted the theoretically expected weak modulation log-periodic in T 7 . The
Efimov width parameter η ∗ defines the lifetime of a deeply bound Efimov trimer ~/(η ∗ ET ),∗where ET is the energy of the trimer state. The physical origin of ζ3 is that a fraction e−4η ofthree-body collisions do not result in atom loss but instead form a transient trimer that decays
back into free atoms which are not lost from the trap. This form of L3 for thermal unitary
Bose gases was verified experimentally using thermal unitary Bose gases prepared in harmonic
traps [320–322].
In our homogeneous gases the fractional loss rate simplifies to√Ṅ
36 3π 2 ~5 N 2= −ζ3 3 2 2 2 .
N
m kB V T(7.9)

In contrast to the weakly-interacting case, a thermal unitary gas intrinsically heats while undergoing three-body loss because of the unitarity limited scattering cross-section σ = 8π/(k 2 +a−2 ) ≈ 8π/k 2 , which leads to atoms with lower relative momenta having a higher unitarity limited loss rate and being preferentially lost [320, 322, 379]. While we now have two lengthscales,
n−1/3 and λ, their dynamics are coupled and we can consider the heating caused by loss andhence reduce Eq. (7.9) into the form of Eq. (7.4) to obtain the relevant γ. Following [322, 379],
we write out the total energy before (l.h.s.) and after (r.h.s.) a loss event

αN kB T = α(N + dN )(T + dT ) − (α − δ3 )T dN ,(7.10)

separating the terms corresponding to atoms that remain in the trap (first r.h.s. term) andthose that are lost (second r.h.s. term). Here α = E/(kB T ) = 3/2 is the thermodynamic
parameter (assuming nλ3  1) and δ3 = 2/3 corresponds to the excess energy per parti7

While theoretically expected, such a log-periodic modulation in T has not been resolved experimentally. Both the∗rather weak relative oscillation amplitude ≈ 0.022e−2η (a few % for typical η ∗ ) and extremely slow variation
with T [a period corresponds to a factor of e4π/s0 ≈ 3 × 105 (s0 = 1.00624 for identical bosons)] make this aparticularly challenging pursuit for identical bosons [322, 379].

95

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
cle from the three-body loss event, associated with the k-dependence of the recombination
events [320, 322, 379] 8 . Simplifying and rearranging Eq. (7.10) yields
dT
δ3 dN
4 dN
dE
=−
=−
=
,
T
α N
9 N
E

(7.11)

which upon integration gives T ∝ N −4/9 . Putting the ingredients together, we predict theloss rate scaling law γ = 2 + 2 × 4/9 = 26/9. We also obtain the energy dynamics by simply

replacing N (E) in Eq. (7.9), which retains the power-law form and reads Ė/E ∝ E −13/2 , resulting in the same general form as Eq. (7.5) after integration. We thus find the expected long-time
power-law behavior E(t → ∞) ∝ t2/13 .
Degenerate unitary regime
In the degenerate unitary regime, within the universality hypothesis [311], there is only one
lengthscale so we expect that a is limited at unitarity by n−1/3 = (V /N )1/3 , which directlypredicts the scaling law γ = 2/3. Equivalently, one can simply argue that since tn is the only
timescale in the problem we requireṄ /N = −A/tn ,(7.12)

where A is a constant. Such a scaling was recently observed in the initial loss rate of a harmonically trapped 85 Rb gas quenched to unitarity [271] over two orders of magnitude in average initial density hn0 i, with A ≈ 0.18. We however do expect that A depends on non-universal
Efimov physics through ζ3 (as in the thermal unitary regime), and an additional log-periodic
modulation as function of n is also expected [366, 380] 9 .
While our derivation of Eq. (7.12) was based on a 3-body loss process, we note that in a
degenerate gas higher-order loss processes cannot be ruled out. However, by considering any
i-body loss event [see Eq. (7.7)] one actually recovers the same scaling law (as na3 ∼ 1), andso all loss processes scale in the same way (set by ∼ tn ). This recapitulates our key message inthat this is a many-body loss rate, set by the only timescale in the problem ∼ tn .
This universality reveals a more serious issue, as it raises the question whether such a gas
can ever exist in equilibrium. Indeed, if one considers the ‘quality factor’ of the system, that is
8

Here we assume that the trap depth U0  kB T , so that evaporation is negligible. In [320, 322] the authors tuned
U0 & kB T in order to use evaporative cooling to compensate the loss-induced heating and achieve an essentiallyconstant T .
9The expected oscillation period corresponds to a change of density by a factor of e3π/s0 ≈ 104 (s0 = 1.00624
for identical bosons). The expected relative amplitude for 85 Rb is ∼ 20% [366], and the data from [271, 380],
spanning about half of the expected log-period, do show a suppression at the predicted minimum position.
However, the results are inconclusive owing to systematic uncertainties in the calibration of hn0 i [380].

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CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
the product of the energy of the many-body state (∼ En ) and its lifetime (∼ tn ) then we are left
with just a number, which is a priori unknown and not necessarily larger than 1 (required for awell-defined state to exist).
We can make some quantitative progress by taking a step back and asking whether a thermal
unitary Bose gas can remain in equilibrium under three-body loss [321]. We need to compare
the per-particle loss rate [Eq. (7.9)] to the two-body elastic collision rate nσv, where on averagep
σ = 8πλ2 and v is the relative velocity between two thermal atoms 4 kB T /(πm). The ratio of
the two rates is then√
16 2 1√,9ζ3 3 nλ3(7.13)

which is a function of η ∗ and phase space density nλ3 . Using η ∗ ≈ 0.09 measured for our 39 K
state [321] this reduces to ≈ 5/(nλ3 ), which shows us that we do not expect to be able to adiabatically prepare an equilibrium unitary state for deeply degenerate gases. Our experiments
confirm this in a sobering manner. If we use our normal field ramps to tune a → ∞ on ∼ 10 mstimescales then initially pure BECs become thermal, losing a large fraction of atoms during theramp, with the degenerate unitary regime eluding us.
To overcome this problem, we forsake our dreams of adiabatically preparing a degenerate
unitary Bose gas, and instead turn to quench experiments, where we rapidly tune a → ∞ toobserve the ensuing dynamics.

7.1.3 Quench protocol
Here we outline our experimental protocol used to quench both initially degenerate and thermal Bose gases into the unitary regime. We prepare homogeneous Bose gases in the quasiuniform potential of a cylindrical optical box trap with volume V ≈ 3 × 10−14 m3 and aspectratio 2R/L ≈ 0.63 (see Section 7.1.1).
We begin our experiments in the weakly interacting regime, with initial gas parameters
n0 a3i < 10−3 . In Fig. 7.4 we introduce our quench protocol, which we illustrate for an initiallydegenerate gas, while highlighting differences to our thermal gas experiments which enjoy several simplifications. In our degenerate gas experiments we use initial atom numbers N0 in therange (1−22)×104 , with corresponding initial densities n0 in the range (0.4−5.2) µm−3 , so thattn0 varies between (150 − 27) µs. However, most of our experiments are performed with four
initial N0 = {48, 98, 143, 214} × 103 , with s.e.m. < 1% 10 . In order to reach interaction param10

While our statistical uncertainty in N is small (which allows us to detect relative differences in N to high precision) do note that there is an overall systematic uncertainty in N of ≈ 10%).

97

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

quench to unitarity~2 µs

na3 < 10-3

quench backevolve for time
thold at unitarity

nk

N

1/ℛ

na3 < 10-3

a→∞Fig. 7.3 Quench Protocol. A quasi-pure BEC is rapidly quenched to unitarity, the gas evolves under unitary interactions for a time thold , before ramping back to low a at a variable ramp-out rate R = −dB/dt. Immediately
allowing the atoms to expand in ToF at aToF ∼ 0 gives us access to post-quench momentum distribution
nk . Instead, by holding the cloud at low a in an essentially closed environment for 40 ms and subsequently
performing ToF at aToF ≈ 103 a0 we are able to spread out the optically dense low-k atoms sufficiently tofaithfully measure the atom number N (see text). The circles in the cartoons depict atoms, and their sizes
the interaction strength, limited at unitarity by the interparticle spacing.

eters n0 |a|3 > 1 (entering the unitary degenerate regime) for our full range of n0 , we need totune B to within δB ∼ 60 mG of the Feshbach resonance at B∞ = 402.70(3) G (see Section D for
details of our field calibration at unitarity). The requirement for reaching the thermal unitary−1/3

regime (a/λ > 1) is somewhat relaxed, since as λ < n0

we automatically satisfy a/λ > 1 if

n0 |a|3 > 1.
To enter the unitary regime, we rapidly (within 2 µs  tn0 ) quench the magnetic field to B∞
using fast auxiliary coils strategically placed to avoid coupling to the Feshbach coils. We areable to perform field quenches of up to ∆B ≈ 3.6 G ≈ 60 δB, corresponding to a minimum
ai = 390 a0 .
Our quasi-pure condensates are prepared in a U0 /kB ≈ 50 nK deep trap, however to avoid
the loss of atoms due to spilling or evaporation in the unitary regime (as En0  U0 ) we raise thetrap depth to Uh /kB ≈ 2 µK  En0 at unitarity11 . We hold the gas at unitarity as it evolves for avariable time thold , before returning to low a in order to observe the samples; we use 6 − 32 ms
time-of-flight (ToF) expansion and our imaging protocol detects only free atoms.
The functional form of our B field ramp out of the unitary regime is exponential (see Section D), and since we always ramp B to more than 10 δB away from B∞ the ramp is essentiallylinear in and near the unitary regime, where its form is most relevant [381]. We therefore characterize different ramps by their initial ramp-out rate R = −dB/dt = ∆B/τQ , where ∆B is thechange in field from B∞ 12 and τQ is the ramp’s exponential time constant, which we can vary
between (1 − 10) µs.
Conceptually, an infinitely fast ramp-out should project the resonantly interacting cloud onto
free atom states. This avoids the formation of an atom-molecule mixture [271] and allows a11This occurs rapidly (. 10 µs) alongside the quench of B. We have studied the consequences of such a trapdepth quench at low a (without quenching a → ∞), and find that the dynamics it induces occur on > 100 µs
timescales, leading to an increase in E which at all times is < 10% of that due to a quench to unitarity.
12
The initial and final distance from B∞ are the same as our current electronics do not offer separate control.

98CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
study of the gas’s post-quench properties, including the atom number N , energy per particle
E, and momentum distribution nk . Instead, ramping away from unitarity slowly and creating
an atom-molecule mixture allows us to study the dynamics of correlations in the unitary Bose
gas13 .
If following the quench protocol we immediately release the cloud in ToF14 , then our absorption images give the post-quench line-of-sight integrated momentum distribution (convolved
with the initial size). This allows us to obtain rich momentum- and time-resolved information,
which is at the heart of Section 7.2.2. However, alone, these measurements have a shortcomingin that the low-k region of the cloud, which at short thold contains the majority of the atoms, is
optically dense and so does not permit an accurate measure of N (see Fig. 7.3).
Our solution to this is that following the quench back to low a we introduce a 40 ms wait
time at ≈ 200 a0 , which allows the gas to rethermalize, spreading the BEC to larger momenta
while maintaining an essentially closed system15 . Note that for these measurements we also
increase the scattering length for ToF (to aToF ≈ 103 a0 ) to aid the expansion16 .
Our thermal gas measurements are simpler in several regards. Here we keep the trap depth
fixed throughout, and prepare all gases in the same deep trap with U0 = Uh ≈ kB × 3 µK. Thereis also no dramatic separation of momentum scales following the quench, so the immediaterelease nk measurements also faithfully reveal N and the energy per particle E.

7.2 Degenerate Bose gases quenched to unitarity
Here we present an overview of the typical atom-loss, correlation, and energy dynamics that
we observe using our quench protocol.
We begin by investigating the role of the sweep out of the unitary regime (see Fig. 7.4).
Our goal here is to disentangle the two reasons that the observed atom number Nobs candecrease with thold : (i) due to losses that occur at unitarity, and (ii) because the ramp-out
creates a mixture of atoms and (Feshbach and/or Efimov) molecules [271], while we only
observe the free atoms. The molecular fraction of the gas following the ramp-out is governed by both the correlations that have developed at unitarity, and the ramp-out rate R =−dB/dt [271, 353, 366, 381–383]. In the limit of an infinitely fast ramp-out we expect that weare no longer sensitive to the many-body state at unitarity and that we essentially project the13

As we will see in Section 7.2, these measurements also show that our technically limited fastest ramp-out rate
1/R = 0.3 µs/G underestimates the actual atom number by . 10%.
14
Upon releasing the gas we simultaneously (within about 3 ms) also turn off the interactions (a → 0).
15
If we vary Uh by ±20% and repeat our experiments at various thold (up to 2 ms) we obtain the same Nobs withinerror bars, and in cases where the BEC is small (or absent) both measurements give consistent Nobs .
16
We checked that this does not cause (detectable) atom loss using quasi-pure BECs prepared at low a.

99

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
version 25(a) (b) (c) (d)(a) (b) (c) (d)
6

1/a
�/�

�
�obs
N
(104)
��� (��

1/ai
00(a) (b) (c) (d)
10

Time
time

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4
11/R
/ ℛ(μs/G)(µs/G)

6

4
�N���(10(��)�)

����
����
5

2
10

100
(µs)
hold (μs)
tthold1000

Fig. 7.4 Atom-loss and correlation dynamics of a degenerate Bose gas quenched to unitarity. (a) Quench protocol.
A quasi-pure BEC is quenched to unitarity, held there for a variable time thold , and finally ramped back to
the weakly-interacting regime at a variable ramp-out rate R = −dB/dt. An infinitely fast ramp (1/R → 0)
would project the resonantly interacting cloud onto free-atom states, while a finite-rate ramp-out creates a
mixture of atoms and molecules. (b) The open circles show the observed atom number Nobs as a function of
inverse ramp-out rate 1/R, for N0 = 98 × 103 and thold = 80 µs. An exponential fit to the data (solid line)
gives an exponential constant 2.2(3) µs/G. The solid squares show Nobs if we perform a second magneticfield pulse to dissociate the molecules. (c) The evolution of Nobs as a function of thold on a log-log plot for
our fastest ramp-out (0.3 µs/G, circles) and a much slower one (5.8 µs/G, diamonds). The fast ramp-out
reveals the on-resonance atom loss, while the difference between the two curves unveils the correlation−9/26
dynamics in the unitary gas. At long times we observe power-law behavior Nobs ∝ thold (dashed line),corresponding to the loss expected for a thermal unitary Bose gas (see Section 7.1.2). The statistical error
bars are smaller than the point sizes. Figure adapted from [30].

resonantly interacting cloud onto free-atom states, such that Nobs = N .
We study the dependence of Nobs on 1/R for N0 = 98 × 103 and a fixed thold = 80 µs [opencircles in Fig. 7.4(b)], where we have varied 1/R by varying both τQ between (1−10) µs and ∆B
between (3.6−1.8) G. We find that the data are well captured by an exponential fit, characteristicof a Landau-Zener process [382, 383]. We extract an exponential constant 2.2(3) µs/G, and by
extrapolating 1/R → 0 we assess that our technically-limited fastest ramp-out 1/R = 0.3 µs/G

measures the true N to within . 10%; we corroborate this with additional measurements for
our full range of N0 at several thold (see Section 7.2.3). For the N0 and thold considered here
we see that our slowest ramp-outs reduce Nobs dramatically, by up to 40%. To experimentally
verify that this reduction is due to the creation of an atom-molecule mixture, we apply a second
brief (8 µs) pulse back to B∞ after the first trip to resonance (with slow ramp-out), in order tobreak up the molecules, finding that we can make most of the missing atoms reappear [see100CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY� (���)��(a) (b) (c) (d)�E/kB (nK)(a) (b) (c) (d)
��

�

��
thold (µs)

���

300
100
30
10

100 1000(µs)
hold (μs)
ttholdFig. 7.5 Energy dynamics of a degenerate Bose gas quenched to unitarity. We use a BEC with N0 = 214 × 103
and probe the gas using our fastest ramp-out (1/R = 0.3 µs/G), where Nobs ≈ N . (a) Representative absorption images taken along the cylinder axis, after 12 ms ToF expansion at weak interactions for different
on-resonance hold times thold . (b) Extracted kinetic energy per particle E as a function of thold . For comparison, the interaction energy per particle after the ramp-out is < kB ×20 nK. The solid line shows a power-law
with exponent 2/13, as predicted for the long-time behavior of a thermal unitary Bose gas, where E ∝ T(see Section 7.1.2). Figure from [30].

solid squares in Fig. 7.4(b)]17 .
In Fig. 7.4(c) we plot Nobs versus thold on log-log scale for both our fastest ramp-out (0.3 µs/G)
and a much slower one (5.8 µs/G). We see that at short thold the difference between the two
curves is small, highlighting that it takes time for the system to develop the correlations thatallow us to create an atom-molecule mixture. We also see that the two curves overlap again
at long times, indicating that the gas eventually becomes uncorrelated. Indeed, at long times−3.0(2)(thold & 1 ms) we observe power-law behavior with Nobs ∝ thold, which is in agreement withthe scaling predicted for a thermal unitary Bose gas N ∝ t−9/26 (see Section 7.1.2), indicatingthat degeneracy is ultimately lost.
In Fig. 7.5(a) we show typical absorption images at different thold , which illustrate how initially there is a dramatic population of high-k momentum states with the central low-k region
still present, before at much longer times the central low-k population disappears, in line withthe fact that as the samples decay and heat, they ultimately become thermal. In Fig. 7.5(b) weshow a log-log plot of evolution of the extracted kinetic energy per particle E(thold ) 18 . We seethat E monotonically increases with time, and at long times we find excellent agreement withthe predicted recombination heating in the thermal unitary regime E ∝ t2/13 , assuming thatE ∝ T (see Section 7.1.2).
As we have seen above, eventually, the condensate inevitably vanishes, but does the gas
attain a strongly-correlated quasi-equilibrium steady state before degeneracy is lost? If so,
what is the nature of this state? In the following we will explore the post-quench dynamics
17

Reminiscent of the experiments in [384], we observe an oscillation of Nobs as a function of the time spent at low abetween the two magnetic field pulses, and we tweak this time to maximize Nobs . As in [384], we find that thefrequency of the oscillation is set by the binding energy of the Feshbach dimer state at the intermediate low a (towithin our uncertainty in a). We found no discernible dependencies of ν on n or thold (varying n by more than afactor of 3 and exploring up to ∼ 4 tn0 ). We also observe an oscillation of E at the same frequency.
18
In extracting E we empirically suppress noise at very high-k using a gaussian fit.

101

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

103-dN/dt (µs-1)

γ=�/�
102

γ=��/�

10
2

5

10

20

� (���)Fig. 7.6 Atom-loss scaling laws. We show the numerically differentiated loss curve Nobs (thold ) for N0 = 214 × 103(thold monotonically decreases from right to left, as losses reduce N ). We observe both the degenerate(γ = 2/3, solid line) and thermal (γ = 26/9, dashed line) scaling laws predicted for the decay of unitary
Bose gases in the two temperature regimes. The crossover between the two regimes occurs at a well definedatom number which we denote Nc (where the lines cross). Figure from [30].

in further detail, seeking answers to these questions. To study the atom-loss dynamics (andassociated energy and momentum distribution dynamics) [Sections 7.2.1 and 7.2.2] we use ourfastest ramp-out and assume that this faithfully measures N (i.e. defining N = Nobs for ourlargest R). Instead, in order to map out the correlation dynamics [Section 7.2.3] we consider∆N , defined as the difference between N and Nobs measured using a much slower ramp-out[e.g. the difference between the two curves in Fig 7.4(c)].

7.2.1 Universal loss dynamics
To determine the instantaneous loss rate we numerically differentiate the atom-loss curves
N (thold ) 19 , providing an elegant way to directly test Eq. (7.4). In Fig. 7.6 we show the extracted Ṅ versus N on log-log scale for N0 = 214 × 103 , corresponding to n0 = 5.1 µm−3 . For
large N (short thold ) we clearly observe degenerate-gas behavior with γ = 2/3, while for small
N (long thold ) we instead observe the thermal-gas scaling γ = 26/9 (see Section 7.1.2). The
crossover between these two regimes occurs at a well-defined Nc and corresponding tc .
We now extend our analysis to include two additional data series with different initial n0 20 .
In Fig. 7.7(a) we show a log-log plot of the relative loss rate Ṅ /N versus n for our three different
n0 . We see that for large n/n0 (short thold ) all curves follow the same γ = 2/3 degenerate gas
scaling law (solid line). As n reduces, each data series individually bends away in similar
19

To numerically compute the derivative, for each experimental repetition of the full N (thold ) curve (of which typically 30 are measured), we select the ith and (i + δi)th point to calculate the instantaneous loss rate and average
N for each pair [typically δi = 3]. Note that we rely on the dense logarithmic spacing of our data in thold . We repeat this procedure for all i and all our repetitions before logarithmically binning our data in the relevant x-axis(e.g. N in this case).
20
In contrast to [30], here we do take into account slight variations in the initial volume (see Section 7.1.3).

102

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

5.1
2.7
1.6


-1-1 )dn/dt
1/n(µs(µs-N /N)n0 (µm-3)(a) (b) (c) (d)
10-2

10-3

10-4

1.0
-3
n
n (µm
(µm-3))

0.3
0.3

1.0
�/��n
E/E

(a) (b) (c) (d)
1.00

3.0

0.5

�t�c/�
/t�n�0

0
10 1
3
1

2 4 8
n0 (µm-3)

ΓN c /N0(a) (b) (c)1.0(d)

0.3

0.10

0.01
1

2 4 8
-3
n
(µm-3
n00 (µm))

3.0

Fig. 7.7 Universal crossover from a degenerate to a thermal unitary gas. (a) Log-log plot of −Ṅ /N versus n forthree different n0 (see legend). The same γ = 2/3 law, Ṅ /N = −0.28/tn (solid line), captures the degenerate
gas data (large n/n0 ) for all n0 . The dashed lines show the expected thermal unitary scaling law γ = 26/9.
(b) For our three n0 the crossover between the two regimes occurs at almost the same N/N0 and thold /tn0 ;averaging we get Nc = 0.43(4) N0 and tc = 4.0(3) tn0 . (c) Log-log plot of the dimensionless per-particle
loss rate Γ versus E/En , which collapses our data onto a single universal curve. The solid line corresponds
to the degenerate unitary gas Γ = 0.28, and the dashed one shows Γ ∝ (En /E)2 , expected for a thermal
unitary gas. The crossover between the two regimes occurs at Ec = 1.7(2)En . Figure adapted from [30].

fashion, separately following the thermal unitary scaling law γ = 26/9 (dashed lines).
Focusing on the degenerate gas regime, we extract the degenerate gas scaling law coefficient
A = 0.28(3) [defining Ṅ /N = −A/tn , see Section 7.1.2]. We assess that due to the < 10%
difference between Nobs and the actual N , we may overestimate A by up to 0.04. It is interesting
to compare our results to the 85 Rb data [271], from which we extract a slightly lower A ≈ 0.18.

This difference is consistent [328] with the difference in the Efimov width parameters: η ∗ =0.09(4) for our |1, 1i 39 K state [321] and η ∗ ≈ 0.057(2) for the |2, −2i 85 Rb state [64], though aquantitative comparison will require more precise and accurate measurements of η ∗ .

We now set out to answer the question: When does the crossover from degenerate gas (γ =2/3) to thermal-gas (γ = 26/9) behavior occur? In Fig. 7.7(b) we map out the atom number Nc
and time tc at which the crossover occurs for the different n0 . We find that, within errors, thecrossover always occurs at the same fraction of N0 and the same thold expressed in units of tn0 .
Averaging our data gives Nc = 0.43(4) N0 and tc = 4.0(3) tn0 .
To understand why this crossover occurs, we proceed by relating the change in γ to thegrowth of the dimensionless E/En . We define the dimensionless per-particle loss rate
Γ = −tn Ṅ /N ,103

(7.14)

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
so that in the degenerate regime we simply have Γ = A, and in the thermal regime Γ ∝N 4/3 /T 2 ∝ (En /E)2 [using Ṅ /N ∝ −N 2 /T 2 and assuming E ∝ T , see Section 7.1.2].
As shown in Fig. 7.7(c), we find that plotting Γ versus E/En collapses all our data onto asingle universal curve, with the crossover between degenerate and thermal gas behavior at
Ec = 1.7(2) En . In comparison, for an ideal Bose gas in equilibrium this energy would berather high, corresponding to T ≈ 3 Tc , where Tc is the BEC critical temperature. However, ina unitary gas we expect the energy to be of order En even at zero temperature [324, 329, 332].
We note that these measurements do not disentangle the contributions to the initial growthof E/En (at thold < tc ) due to heating and due to the development of the interaction-induced
correlations that coherently broaden the momentum distribution [270, 324, 333, 341, 342].

7.2.2 Momentum distribution dynamics
Having established a comprehensive picture of the loss dynamics, we now turn to a momentumand time-resolved analysis of the post-quench dynamics, which allows us to isolate the effectsof the initial coherent dynamics. We use our fast ramp-outs that minimize molecule production [see Fig. 7.8(a)] to measure the momentum distribution nk . To reconstruct nk from our absorption images (which measure the line-of-sight integrated momentum distribution) we first
average each image azimuthally and then average over the experimental repetitions (of which
typically about 20 are taken). We then perform an inverse Abel transform (see Section C), beforeR
finally normalizing by the total atom number N , such that 4πk 2 nk dk = 1 within experimen-

tal errors. Note that due to the initial cloud size and the non-infinite ToF, our measurements of
nk are not quantitatively reliable for k < 2 µm−1 .
In Fig. 7.8(b) we show the full nk (k) for fixed n0 = 5.1 µm−3 at several values of thold . Our key
experimental observation is illustrated in Fig. 7.8(c), where we finely resolve the time-evolutionof nk for individual k states21 . Plotted this way, we discern separate stages in their evolution:
following a rapid initial growth, nk approaches an intermediate (quasi-)steady-state, before
heating takes over at long times (mind the logarithmic x-axis). While the timescales for thedifferent processes are all of order tn0 , they remain distinguishable. We determine such time
separation for k/kn0 & 0.8, and for each k in this range we identify the plateau occupation nk(dashed lines). We then use sigmoidal fits (solid lines) to extract the characteristic time τ (k) for
the rapid initial growth, which we define as the half-way rise time, i.e. nk (k, τ (k)) = nk (k)/2.
A crucial feature of Fig. 7.8(c) is that the curves are not aligned in time and the dynamics are21

To provide a better sense of scale we have plotted these dynamics in dimensionless form: nk kn3 0 versus thold /tn0 ,but note that this scaling affects all curves in the same way as n0 is fixed here, while it will become importantlater when we compare different n0 .

104

(a) (b) (c) (d)
-1

1/ai

1/�

1/a

(a) (b) (c) (d)nk (μm(µm33)

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

0
0
Time
Time

(a) (b) (c) (d)0.2

thold (µs)

t hold (µs)
0
12
28
56
166

10

10-3
10-5
0

5
10
-1
k (μm
(µm-1
))

15

nk (1.0k n0 )nk (1.6k n0 ) × 15

nk k3n30

nk (2.0k n0 ) × 100

0.1

0.0
0.3

1
tthold/thold /tnn0

3

Fig. 7.8 Momentum-resolved dynamics of a degenerate Bose gas quenched to unitarity. (a) Quench protocol,
highlighting that we use our fastest ramp-out (1/R = 0.3 µs/G) to minimize the conversion of atoms intomolecules. (b) Momentum distributions for various thold (see legend); here n0 = 5.1 µm−3 , corresponding
to kn0 = 6.7 µm−1 and tn0 = 27 µs. (c) Populations of individual k states exhibit a fast initial growth,saturation at (quasi-)steady-state values nk (dashed lines), and long-time heating. We use sigmoidal (inlog-space) fits of the form nk (1 − 1/[1 + (thold /τ )1/δτ ]) (solid lines) to extract the half-way rise times τ (k).
Figure from [31].

clearly k-dependent. In particular, nk (kn0 ) reaches its steady-state value only after nk (2kn0 )already shows signs of heating. This observation explains why previous measurements could
not quantitatively separate lossless and recombination dynamics by considering all momentum
states at the same thold . As a concrete example, if one considers the kinetic energy per particleR
E = ε dk, where ε = ~2 /(2m) × 4πk 4 nk , then these rich momentum-dependent dynamics are
washed out by the sum over k [see e.g. Fig. 7.5(b)].

We proceed by separately extracting nk for different k and piecing together the functionnk (k). This does not reveal the momentum distribution for any specific thold , but instead allows
us to infer what the steady-state nk (k) would be in the absence of losses and heating. Our
curves now include information arising from different thold , and so we just use tn0 and kn0 set
by the initial density n0 ; note that for our longest τ we do observe ≈ 20% particle loss. It should
also be noted that we assume that at early times (thold ∼ tn0 ) all nonzero-k states are primarily
fed from the macroscopically occupied condensate [see Fig. 7.8(b)].
In Fig. 7.9 we plot the dimensionless τ /tn0 and nk kn3 0 as a function of the dimensionlessk/kn0 , for three different n0 . By expressing all quantities in such dimensionless form we findthat within experimental errors our data points fall onto universal curves.
Across our experimentally accessible range of momenta the universal τ /tn0 is consistent with105

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY(a) (b) (c) (d)(a) (b) (c) (d)
10-1

1.0
-3

3

nnkk kkn03

1.6
2.7
5.1

ττ/t
/tnn0

nn0 (µm )

0.3

0.1

∝ (k n0 /k)2∝ (k n0 /k)2
1

10-2
10-3
10-4

2

3

0

k/k
k/knn0

1

2

3

k/k
k/knn0Fig. 7.9 Universal post-quench dynamics and steady-state momentum distribution in the degenerate Bose gas.
We show (a) τ (k), and (b) the steady-state nk (k), for three different n0 (see legend). By expressing all
quantities in dimensionless form, using the natural Fermi scales tn0 and kn0 , we find that all our data fall
onto universal curves. The solid line in (b) is an exponential fit to the data, nk kn3 = 1.53 exp[−3.62 k/kn ].
Figure from [31].

the scaling ∝ kn0 /k at low k and ∝ (kn0 /k)2 at high k [see Fig. 7.9(a)]. Such scaling was qualitatively predicted for the emergence of a prethermal steady state [327, 328, 330, 335, 342] (seealso [354]). Within this picture, immediately after the quench the excitations resemble Bogoliubov modes in a weakly interacting BEC, with phonon- and particle-like character at respectively low and high momenta, but with the usual mean-field energy replaced by an energy∼ En . Consequently, the speed of sound is ∼ ~kn /m and the crossover between phonon- andparticle-like regimes is at k ∼ kn . The predicted τ (k) is determined by the dephasing time,
which is approximately equal to the inverse of the excitation energy.
The universal nk curve is more of a surprise, and presents a new theoretical challenge. Empirically, we find that over three orders of magnitude in nk kn3 our data are well described by an exponential A exp[−Bk/kn ], with A = 1.53(5) and B = 3.62(2). Interestingly, if we take this funcR
tional form at face value it implies a finite condensed fraction of η = 1 − 4πk 2 nk dk = 19(4)%.
Our experiments do not feature the asymptotic form nk ∼ 1/k 4 expected at very high k [47] 22 ,but even if nk were to change to this slower-decaying function right outside of our experimental range this would change η by less than 3%. While our estimate of η is close to the predictions
for a prethermal state in [335, 342], we note that this functional form was not predicted so far,and it could require explicit consideration of our final quench back to low a. That been said,
qualitative similarities (up to ≈ kn ) to theoretical calculations are present [331, 341, 352].
It is also interesting to use the exponential nk to compute the kinetic energy per particleR
E = ε dk, defining the prethermal state spectral energy density ε = ~2 /(2m) × 4πk 4 nk . We

obtain E = 0.74(4)En0 , which highlights that the prethermal state is strongly correlated, but
22

Efimov physics is thought to modify the 1/k4 tail with log-periodic modulations, and several theoretical interpretations (e.g. [333, 336]) of the pioneering 85 Rb experiment [270] compared such equilibrium unitary predictions
to the data. However, it is not clear that such a tail would survive a quench back to low a, and generally it has
been numerically observed that the contact parameter (characterizing the high-k momentum tail) can exhibit
large oscillations following a quench [385], and more generally that low momenta remain frozen while highmomenta evolve adiabatically [283].

106

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

1/ℛ (µs/G)� �
�� (���)Δ� (���)ΔN (104)(a) (b) (c) (d)
4

2

����
���
���

0
10

100
(µs)
thold (μs)1000(a) (b) (c) (d)0.4 0.4

ΔN/N

0.2
0
0

2

4

0.2

0
0.1

1
thold /tn0

10Fig. 7.10 Correlation dynamics of a degenerate Bose gas quenched to unitarity. (a) We plot ∆N versus thold forthree different N0 and the same slow ramp-out 1/R = 5.8 µs/G [note the logarithmic x axis]. (b) ∆N/N
versus the dimensionless time thold /tn0 , for different N0 and 1/R; see legend in (a). The shaded region indicates the crossover to the thermal regime at tc = 4.0(3) tn0 . The inset shows a zoom-in on the degenerate
regime on linear scale, highlighting that the correlations reach a (quasi-)steady state well before tc . Figure
from [30].

for quantitative purposes one should bear in mind the caveat that we do not observe the veryhigh-k tails experimentally.

7.2.3 Molecular correlation dynamics
We now turn to the dynamics of correlations at unitarity, which are indirectly revealed byslowly ramping the magnetic field away from resonance and creating an atom-molecule mixture. We study ∆N , the reduction in Nobs due to the slow ramp-out, assuming that N = Nobs
for our fastest ramp-out with 1/R = 0.3 µs/G.
In Fig. 7.10(a) we plot ∆N versus thold for three different N0 , using the same slow ramp-out
with 1/R = 5.8 µs/G. In Fig. 7.10(b) we show a rescaled version, where we plot the fractional
conversion efficiency ∆N/N versus the dimensionless thold /tn0 23 , where we have also included
23

We use the initial tn0 to normalize thold , as the relevance of tn is not clear in the thermal regime. In the degenerate
regime, one can readily relate tn0 to tn , by simply rewriting the degenerate-gas loss rate [Eq. (7.12)] as ṫn = 2A/3,
so that tn /tn0 = 1 + 2A/3 × thold /tn0 .

107

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
an additional data series with twice the ramp-out speed. Plotting the data this way achieves
horizontal alignment of all our curves, and apart from their heights they all look essentially thesame. This shows us that correlation dynamics are universally set by the initial gas density.
We can qualitatively understand the non-monotonic shape of the curves as arising from thecompetition of two effects. At short times following the quench to unitarity, the gas is uncorrelated as it takes time for the correlations to develop (on a timescale set by tn0 ). Instead, at longtimes the system becomes uncorrelated again as it heats up and the phase space density drops
dramatically (see also [36, 383]).
We highlight the early-time dynamics (on linear timescale) in the inset of Fig. 7.10(b), whichshows how ∆N/N becomes essentially constant well before tc (where the crossover to the thermal unitary regime occurs, see Section 7.2.1). This suggests that the correlations in the systemreach a quasi-equilibrium state while the gas is still degenerate, which in our homogeneous
system implies a global (quasi-)equilibrium.
In the following we present an attempt at experimentally disentangling how the steadystate conversion efficiency ∆N/N depends on R and N0 . Based on recent experiments with85 Rb [271], which observed that the post-quench molecular gas contains Efimov trimers in thefirst excited state, it is likely that our atom-molecule mixture also includes such Efimov trimers(instead of simply consisting of Feshbach dimers). The presence of these trimers could breakuniversality and lead to a nontrivial dependence of ∆N/N on N0 , as the on-resonance size of
this state is of order 1 µm [33,35,348,362,363,386], which could set a scale that separates ‘small’and ‘large’ densities [346, 347, 349, 366].
To provide a baseline for our expectations, we begin by considering the theoretically expected dependencies of the Feshbach dimer production efficiency. Following [381] (see also
[353, 380]), we introduce a simple model to describe the ramp-out from the unitary regime,
which as we will see predicts that the Feshbach dimer conversion efficiency is a universal function of N0 /R (see also [383]). We split the ramp into two regimes, a initial sudden diabatic
one and subsequent slower adiabatic one. To estimate the scattering length a∗ at which thecrossover between the two regimes occurs, we assume that the ramp-out remains diabatic up
to the point when Eb /~ = Ėb /Eb , after which the process is assumed to be become fully adiabatic, molecular conversion is halted, and one simply follows the final energy states. Assumingthat we are close to the Feshbach resonance (B − B∞  ∆), we substitute the Feshbach dimer
binding energy Eb ≈ −~2 /(ma2 ) and solve Ėb = Eb2 /~ to obtaina∗ =~abg ∆
2mR

108

1/3

.

(7.15)

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

fit results

ΔN/N

0.4

0.2

0
0.0

N 0 (103 )

1/ℛ (µs/G)

48(1)
98(1)
143(2)

2.7(1)
5.8(1)

0.5
N0/ℛ (s/G)(ΔN/N)mβm (s/G)

0.38(1)0.11(1)0.42(2)0.19(2)0.49(3)0.34(4)0.32(1)0.07(1)0.45(1)0.15(1)

1.0

Fig. 7.11 Molecular conversion efficiency in the degenerate post-quench quasi-equilibrium state. We study themolecular correlation dynamics in the quasi-steady state (using thold = 2.7(1)tn0 ), and plot ∆N/N versus
N0 /R for three fixed N0 varying R and two fixed R varying N0 (see legend). Plotted this way, we would
expect our data to collapse onto a single universal curve if we were producing only Feshbach dimers (seetext). To determine ∆N/N we measure the reduced Nobs due to the slower ramp-outs but compute N from
N0 using the adjusted A = 0.24, which ensures that an extrapolation to 1/R → 0 recovers ∆N/N = 0. We
fit the data with a saturating exponential (∆N/N )m (1 − exp[−N0 /(Rβm )]) (see table), finding evidence
that universality is broken.

Within this picture, our ramp projects the initial many-body state onto the molecular state atscattering length a∗ . For larger R one achieves smaller a∗ , and consequently a more localized
molecular state. Assuming that universality holds, the original many-body state at unitarity−1/3

has a size set by the interparticle spacing n0. To assess the conversion efficiency, the next stepwould be to compute the overlap integral between these two states. However, for our purposes
here we simply resort to dimensional arguments and posit that the conversion efficiency is a−1/3function of the ratio of sizes a∗ /n0, i.e. a function of N0 /R.

In Fig. 7.11 we present a summary of all our measurements of ∆N/N , which we always
extract at the same fixed thold /tn0 = 2.7(1). As outlined in the internal legend, we vary N0 /R
in two different ways. For three of our data series (circles) we keep N0 fixed and vary 1/R byvarying both τQ and ∆B [as in Fig. 7.4(b)]. For the other two series, we instead fix 1/R and vary
N0 . We find that a saturating exponential of height (∆N/N )m and 1/e rise-constant βm fits all
series well individually. However, we find that our data do not collapse onto a universal curvewhen plotted in this way. This is perhaps most clearly revealed by the individual exponential
fits to the different series, which give inconsistent values of (∆N/N )m and βm .
Our data thus indicates that there is more to the story than just Feshbach dimer formation,in line with [271] and the expectation that our atom-molecule mixture contains Efimov trimers.
Quantitatively understanding the composition of the atom-molecule mixtures, as well as thefull ∆N/N curves, remain interesting problems for future work, which have already attracted
significant theoretical attention [353,366]. Experimentally, a clear limitation is that we currently

109

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
cannot achieve large values of N0 /R for which ∆N/N truly saturates (as our maximal τQ is limited to 10 µs). Additionally, gaining a better understanding of the expected few-body physics
for 39 K will be beneficial to pin-point the expected densities at which deviations to universal
behavior are expected (see e.g. [366]).
A more general remark is that our measurements currently infer the presence of molecules by
the absence of atoms, and in the future we intend on moving towards the direct observation of
the molecules, which following [271] also offers the possibility of distinguishing trimers fromdimers by studying the decay dynamics of the molecular gas at low a.

7.3 Thermal Bose gases quenched to unitarity
In this section we explore the dynamics of thermal Bose gases quenched to unitarity, and whilethese experiments do offer some significant simplifications compared to their degenerate counterparts, they also come with additional surprises. We begin in Section 7.3.1 by studying theearly-time post-quench redistribution dynamics in a momentum-resolved manner. The dynamic and thermodynamic properties generically depend on both n and T , but we find thatthey can still be expressed in terms of universal dimensionless functions. In Section 7.3.2 weturn to atom-loss and correlation dynamics, where we primarily focus on longer times after thequench, when three-body recombination dominates the gas dynamics.

7.3.1 Momentum redistribution dynamics
A simplification for studying unitary thermal gases compared to their degenerate counterparts
comes from the fact that in a thermal gas three-body recombination is slowed down more thanthe lossless dynamics [320, 321, 326]. As shown in Fig. 7.12(a) for a thermal gas with n0 =5.6 µm−3 and T = 150 nK 24 , simply looking at the energy per particle E(thold ) on log-log scale
now reveals two separate stages in the post-quench evolution: a rapid initial growth where E
more than doubles, followed by heating on a significantly longer timescale.
The behavior at long times agrees with the expected energy-growth due to recombination
heating in a thermal unitary Bose gas [320] (see Section 7.1.2), while the shape of the curve
resembles those obtained for the evolution of individual k states in our measurements using
initially degenerate samples [see Fig. 7.8(c)]. All the above reinforces our general interpretation of the two-step post-quench dynamics, for both degenerate and thermal gases. Here wewill focus on the early-time dynamics [e.g. corresponding to thold . 100 µs in Fig. 7.12(a)].
In Fig. 7.12(b) we show how the momentum distribution nk is essentially identical for 60 and
24Here, and throughout this section, T is the initial temperature (before the quench to unitarity).

110

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
126 µs, which demonstrates how on this timescale a true steady-state is established for all k (asopposed to the prethermal (quasi-)steady-state that we observe for degenerate gases).
In thermal gases it is natural to look at the redistribution of particles in k-space, as even√prior to the quench to unitarity nk is significant for k . 1/λ, where λ = h/ 2πmkB T is theinitial thermal wavelength. We thus consider the change δnk (k) with respect to our initial
distribution measured at thold = 0, and the corresponding change in the spectral energy density
ε = ~2 /(2m) 4πk 4 nk . A complication arising in thermal unitary Bose gases is that they feature
both n−1/3 and λ as potentially relevant lengthscales (and any combination of the two), and it
is a priori unknown whether the dynamic and thermodynamic properties can be expressed interms of dimensionless universal functions.
In Fig. 7.12(c) we show a plot of the time-resolved population changes in different sphericalk-space shells, 4πk 2 δnk . We find a special momentum k0 (dashed line) at which the population
remains essentially constant, separating the momenta that are depleted (blue) from those thatgrow (red). In Fig. 7.12(d) we plot vertical cuts through Fig. 7.12(c), for k < k0 , k = k0 , andk > k0 . We use sigmoidal fits (solid lines) to extract τ (k) away from k0 where the populations
either diminish (k < k0 ) or grow (k > k0 ). Close to k0 , where the population changes are small,
we find that nk initially slightly decreases before subsequently increasing again, behavior towhich we cannot assign a single timescale.
In Fig. 7.12(e,f) we (respectively) plot the extracted τ (k) and steady-state δε(k), for two difR
ferent combinations of n0 and T . The δε(k) curve resolves the energy growth ∆E = δε dk,arising from the redistribution of particles from k < k0 to k > k0 . The dispersive shape ofτ (k) was not anticipated, and invites further theoretical work. We do note a qualitative similarity to our degenerate gas data for k > k0 when δε is positive, where we find that τ smoothly
decreases for increasing k.

Universal laws dictating the early-time momentum redistribution dynamics
We proceed by empirically investigating whether the τ (k) and δε(k) curves (see Fig. 7.12) can bescaled into universal dimensionless functions. We have taken such curves for 15 combinationsof n0 and T , with corresponding initial phase-space densities n0 λ3 between 0.2 and 2.
In order to rescale the data horizontally, and thus find the natural scale of k we use k0 [definedby the zero-crossing of δε(k)] for guidance. In Fig. 7.13 we plot k0 versus T on log-log scale forall our 15 series (see legend), observing power-law behavior; a fit yields k0 ∝ T 0.53(2) (dashedline). We thus deduce that the natural scale of k0 is 1/λ, independent of n0 . Indeed, as shown in
the inset, if we consider series at similar fixed T ≈ 0.2 µK but varying n0 we see that k0 remains

111CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

thold (µ s)

0.4
0.2

10-3
10

-4

10

-5

0

-

80
40
0
0

-1

k (µ m )(a) (b) (c) (d)t hold (µ s)
0
60
126

5
10
-1
k (μm
(µ m-1
))

0

(e)

15

0.1

k (µ m-1 )
9.4
6.0
4.4

0.0
-0.1
10

30
thold
µs)
hold ((μs)

100

(f)

δε/k (nK µm)
δε/kBB (nK μm)µm33)
nk ((μm

100 1000
tholdµs)
hold ((μs)

44πkπ k22δnδ nkk (μm)
(µ m)

10

(a) (b) (c) (d)+τ ( µs)τ (μs)(a) (b) (c) (d)0.6

4π k 2δ nk

E/k (µ K)
E/kBB (μK)(a) (b) (c) (d)

5
10
15
-1
k (μm
(µ m-1
))

30
15
0
0

5
10
15
-1
k (μm
(µ m-1))

Fig. 7.12 Momentum-resolved dynamics of a thermal Bose gas quenched to unitarity. (a) Log-log plot of the
kinetic energy per particle E versus thold , exhibiting a rapid initial growth at thold . 100 µs, and significant
heating only for thold  100 µs; the black line shows the prediction for recombination heating. Here, andin (b-d), the initial density and gas temperature are n0 = 5.6 µm−3 and T = 150 nK. (b) Momentum
distributions nk (k) for different hold times thold at unitarity. The initial redistribution of particles from low
to high momenta (indicated by the dotted arrow) essentially occurs within 60 µs, and nk is almost identicalat 126 µs. (c) Time-resolved population changes in different k-space shells, 4πk2 δnk ; the population in
k0 = 6.0 µm−1 (dashed line) remains essentially unchanged. (d) Vertical cuts through (c), above, at, andbelow k0 . The solid lines show the sigmoidal (in log-space) fits of the form δnk (1 − 1/[1 + (thold /τ )1/δτ ])used to extract the half-way time τ at each k. (e,f) Here we map out the momentum dependence of theextracted parameters for our data series with n0 = 5.6 µm−3 and T = 150 nK (blue circles), and an
additional one with n0 = 1.3 µm−3 and T = 70 nK (red diamonds). (e) Half-way time τ as a function ofk. (f) Change in spectral-energy density (between the initial, pre-quench state, and the post-quench steadystate), δε(k) ∝ k4 δnk (k). Figure from [31].

essentially constant. We proceed by fitting the data with a fixed k0 ∝ T 1/2 ∝ 1/λ scaling,

and obtain k0 = 4.40(2)/λ (solid line). Rather curiously, we find that rewriting ~2 k02 /(2m) =1.54(1)kB T yields an energy associated with k0 which is very close to the average kinetic energyof a monatomic ideal gas (3/2 kB T ).
We now turn to the full τ (k) and δε(k) curves for our 15 combinations of n and T , introducing
1/λ as the natural scale for k. In Fig. 7.14(a) we plot τ (k) as a function of kλ on log-log scale,and in Fig. 7.14(d) we plot δε(k)/λ versus kλ, ensuring that the area under each curve remains
unchanged [i.e. conserving ∆E(n0 , T )]. In both cases we find successful horizontal alignmentof the curves, further establishing that the natural scale of k is 1/λ.
Our next question, which poses more of a challenge, is whether these two families of curves
can be vertically scaled to each collapse onto a single universal curve. To address this we takea heuristic approach, introducing a scaling time ts (n0 , T ) and energy Es (n0 , T ). Our conjectureis that both ts and Es take power-law form, namely that ts ∼ tαn0t tβλt , where tλ = ~/(kB T ), andsimilarly Es ∼ Enα0E (kB T )βE . We then simply search for the values of α and β that yield thebest data collapse. In our analysis we treat α and β as independent, however in the absence of
any other relevant scales one expects αt + βt = αE + βE = 1 for dimensional reasons. To assessthe degree of data collapse we use a single figure of merit σ, which is obtained by calculating
the standard deviation of our data for all n0 and T at fixed kλ, before summing over kλ. We
normalize by σ0 , corresponding to the value obtained for no scaling (α = β = 0).

112

7

k0 (µm-1)

6

k0 (µm-1)

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
nn0 (µm-3 ) T (nK)

8

0.7
1.1
1.2
1.3
1.8
2.0
2.3
2.7
3.2
3.8
4.2
5.0
5.1
5.6
6.7

4
0
0

3
6
-3
n
(µm
)
-3
0
n0 (µm )

5

4
80

120
T (nK)

200
160
100
70
80
190
100
170
100
190
150
190
190
150
200

200

Fig. 7.13 Momentum-scale governing the redistribution dynamics. We show a log-log plot of the extracted k0[defined such that δε(k0 ) ≡ 0, see Fig. 7.12(f)] as a function of the initial gas temperature T , showcasingour 15 different combinations of n0 and T (see legend). We find power-law behavior, and a fit to the data
gives k0 ∝ T 0.53(3) (dashed line), suggesting that k0 ∝ 1/λ, independent of the initial gas density n0 .
Fixing the power-law dependence and refitting gives k0 = 4.40(2)/λ (solid line), which defines the natural
momentum scale of the redistribution dynamics. The inset shows k0 versus n0 for a subset of the data forwhich n0 varies but T ≈ 0.2 µK is approximately constant, highlighting the lack of n0 dependence.

In Figs. 7.14(b) and (e), respectively, we plot σ/σ0 for τ and δε/λ. Focusing on the temporal
scaling first [Fig. 7.14(b)], we find the minimum in σ around αt = βt = 1/2, which suggests√√that ts = tn0 tλ . In Fig. 7.14(c) we plot τ / tn0 tλ versus kλ and observe that, within experimental scatter, all our data fall onto a single universal curve. Our intuitive interpretation ofthis scaling is that because the particles in a thermal gas do not overlap, they must first meet inorder to start feeling the unitarity interactions after the quench. Their ‘meeting time’, up to a
√
−1/3
−1/3
and
numerical factor, is tn tλ ∼ n0 λ m/~ [i.e. the ratio of initial interparticle distance n0

characteristic thermal speed ∝ ~/(mλ)].

A final tentative remark regarding Fig. 7.14(c), is that it seems that at large k we approach
power-law behavior τ /(tn0 tλ )1/2 = c̃/(kλ) (dashed line), where c̃ is a constant. Curiously,√
one can simply rewrite this scaling as τ /tn0 = c̃/(2 π) kn0 /k, where λ fully drops out. Moreover, this scaling is the same as observed for low-k in our degenerate gas data, where we hadτ /tn0 ≈ 0.98 kn0 /k [see dashed line in Fig. 7.9(a)]. Indeed, the dashed line shown in Fig. 7.14(c)√corresponds to setting c̃ = 0.98×2 π, extracted from the degenerate gas data. Surprisingly, we
find that this quantitatively captures the thermal data at high k (within experimental scatter).
We currently do not understand why these two different momentum regimes in thermal anddegenerate gases would behave in the same way, however we also cannot rule out that it is justa coincidence.
Turning to the spectral energy density [Fig. 7.14(e)], we instead find the optimal αE ≈ 1 andβE ≈ 0, which rather surprisingly implies that Es = En0 . In Fig. 7.14(f) we set out to test
this scaling, plotting δε/(λEn0 ) versus kλ, and find that it indeed collapses all our data onto a113

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

0
2

5

10

0.4
0

kk λ
λ

1

(�) (�) (�) (�) (�) (�)

8

12

0.5
0

1

αE

0.2

δε/(λ
δε/(λ Enn)0))

τ/(tnt0ntλtλ)1/2
τ/

4

kk λ
λ

αt

1.0
0

0
0

1.5

1

25

βE

0.8

βt

ττ ((μs)
µs)

30

200
160
10
100
70
80
(a)
190 (b) (c) (d)
2.0
100
170
100
1.0
190
150
0.5
190
190
150
0.2
200

(�) (�) (�) (�) (�) (�)

50

σ/ σ0

0.7
1.1
1.2
1.3
1.8
2.0
2.3
2.7
3.2
3.8
4.2
5.0
5.1
5.6
6.7

1.2

1

σ/ σ0

nn0 (µm ) T (nK)(�) (�) (�) (�) (�) (�)(a) (b) (c) (d)100δε/(λBkλ)(nK)B ) (nK)δε/(k(a) (b) (c) (d)
-3

0.1

0.0

2

5
kλ

0

10

4

8

12

kλ

Fig. 7.14 Universal dynamic and thermodynamic functions dictating the evolution of thermal Bose gases
quenched to unitarity. (a, d) By plotting τ and δε/λ as a function kλ we horizontally align our curves
for 15 different combinations of n0 and T (see legend). (b, e) Supposing that the characteristic timescalet βt
for the dynamics takes the form ts ∼ tα
n0 tλ , we find the best data collapse (captured by small σ/σ0 )√for αt ≈ βt ≈ 1/2, implying that ts = tn0 tλ (see text for details). Similarly, for the energy scale
αE
Es ∼ En0 (kB T )βE we obtain αE ≈ 1 and βE ≈ 0, suggesting that Es = En0 . (c, f) To within exper√
imental scatter, we find that the dimensionless τ / tn0 tλ and δε/(λEn0 ) are universal functions of thedimensionless kλ. Figure from [31].

universal curve. It is quite remarkable that while δε(k) depends on both n0 and T , its integral∆E ends up being independent of T .
Due to the lack of T dependence, in order for both thermal and degenerate gas regimes toconnect gracefully we would require that ∆E/En0 is the same in both cases, inviting a comparison to our degenerate gas data from Section 7.2.2. Bearing the caveat in mind that weonly observe the distributions up to kλ ≈ 12 and in general do not observe their very high-k
tails, from our thermal-gas data in Fig. 7.14(f) we estimate ∆E/En0 = 0.7(1). In comparison, taking the exponential nk kn3 from our degenerate gas data in Fig. 7.9(b) gives a consistent
E/En0 = 0.74(4).
It is insightful to compare the universal δ to the expected very-high-k equilibrium behaviorof (k) at unitarity, which is expected to be governed by the unitarity limited two- and three√body contact densities C2 = 32πn2 λ3 and C3 ≈ 3 3s0 n3 λ4 [65, 333]; in this case we would
expect the very-high k behavior to be given by!
√16
ω3 (k) 3 3s0 3
1/3= 7/3 2/3 n λ 1 +nλ,λEnkλ 32ππ 6(7.16)

where the log-periodic function ω3 (k) ≈ 89.3 sin [2s0 ln(k/κ∗ ) − 1.34] with s0 = 1.00624 andκ∗ is the three-body parameter. We see that this is not a universal function of kλ, but rather
depends on the phase-space density of the gas. Looking past the log-periodic modulation (andsafely neglecting the initial low-a contact density in a thermal gas), we find that for our range of114

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

30

k = k020

k < k010-0.1
30
thold (µs)

100δε/kB (nK µm)0.0k > k0τ (µs)

4πk2δnk (µm)

0.3(1) (µs/G)
2.8(1) (µs/G)

10(a) (b) (c) (d)40(a) (b) (c) (d)(a) (b) (c) (d)0.1 1/ℛ (µs/G)

1/ℛ (µs/G)
0.3(1) (µs/G)
2.8(1) (µs/G)

25k (µm-1)

1020

1/ℛ (µs/G)
0.3(1) (µs/G)
2.8(1) (µs/G)

0
0

4

8
12
k (µm-1)

16

Fig. 7.15 Effects of the ramp-out rate on the initial redistribution dynamics. We compare two different R for our
momentum- and time-resolved measurements of the initial redistribution dynamics of a thermal Bose gas
quenched to unitarity. In both cases we prepare the cloud with the same n0 = 5.6µm−3 and T = 150 nK.
We show a direct comparison of the results for the two different R, for (a) the population dynamics ofthree characteristic k-space shells, and (b,c) the extracted half-way rise time τ (k) and change in spectralenergy density δε, respectively.

phase-space densities this prediction sits far above our data (the prefactor evaluates to ≈ 0.34).
This suggests that our quench protocol is not sensitive to the contact at unitarity, most probably
due to the final quench back to low a (though we also do not resolve very large values of kλ).
Do the redistribution dynamics depend on R?
In the discussion above we solely focused on data taken with our largest R, which should approximately project the many-body wave function onto free-particle states and faithfully measure nk . However, as discussed for degenerate gases in Section 7.2.3, we expect that the a∗ one
reaches before the system can adiabatically follow the ramp-out does depend (albeit nontrivially) on properties of the sample. This suggests that while we did keep R fixed, the physically
relevant parameter may vary between the different series. It is therefore a worthwhile question
to ask how our measurements depend on R.
To investigate this, we focus on a single data series; we use n0 = 5.6 µm−3 and T = 150 nK,
as this is the series with the largest phase-space density and so most conducive to creatinga significant fraction of molecules [383]. We repeat our full momentum- and time-resolved
measurement with identical initial conditions (within 1 s.e.m.) apart from the use of a 10 times
slower ramp-out, achieved by increasing the time-constant of our ramp-out τQ . A comparison
between the two measurements is shown in Fig. 7.15, where we note that to normalize nk
we use the individually observed atom number Nobs , which as we will see in Section 7.3.2 isreduced by up to ∼ 20% for our slow ramp-out.
In Fig. 7.15(a) we study the population dynamics for three characteristic k-space shells, plotting 4πk 2 δnk versus thold for our two R. We see that data for the two different R essentially lie
on top of each other, with only small deviations.
In Figs. 7.15(b,c) we compare the extracted τ and δε versus k, respectively. For τ (k) [Fig. 7.15(b)]we observe small but statistically significant differences of the extracted τ for k < k0 and

115

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY

0.3
0.2
0.1
0

8

(a) (b) (c) (d)1.0
1/ℛ (µs/G)
0.3(1) (µs/G)
2.8(1) (µs/G)
0.3
3
30
1/2
tthold /(tn tλ ) 1/2
hold /(tn00 tλ )nobs /n0

N obs (104)ΔN/N(a) (b) (c) (d)16

1/ℛ (µs/G)
0.3(1) (µs/G)
2.8(1) (µs/G)

4
0.1

0.5

0.2

1

10

100

1

2

thold /(tn0 tλ)1/2

3

E/E0

Fig. 7.16 Atom-loss, correlation, and energy dynamics of a thermal Bose gas quenched to unitarity. (a) Log-log√
plot of Nobs as a function of thold (normalized by tn0 tλ ) for our fastest ramp-out (0.3 µs/G) and a muchslower one (2.8 µs/G). In the inset we show the the molecular conversion efficiency ∆N/N , where weuse the long-time behavior (which agrees with the expected recombination heating loss rate) to calculate
N (thold ) (dotted line in the main panel). (b) We plot nobs /n0 versus E/E0 on log-log scale for our two R.
The dashed line corresponds to the prediction for three-body recombination heating, which matches ourdata at long times (large E/E0 ).

around k0 , while the two agree well for k > 1.5k0 . For the slow ramp-out we are able toextract τ for values significantly closer to k0 , as the region in k-space where the populationshows a non-monotonic (first decreasing before subsequently increasing) time dependence, issmaller. Instead looking at δε [Fig. 7.15(c)], we see that the two curves are essentially identicalapart from at very large k, where we observe a suppression of δε for the slow ramp-out.

7.3.2 Atom-loss and correlation dynamics
We now turn to the atom-loss, correlation, and energy dynamics of a thermal Bose gas quenched
to unitarity. We consider the same data series as in Section 7.3.1, with n0 = 5.6 µm−3 andT = 150 nK, comparing the two R that differ by a factor of 10.
In Fig. 7.16(a) we plot the evolution of Nobs for our two different R, where we have normal√
ized thold by the characteristic initial meeting time tn0 tλ ≈ 36 µs [see Fig. 7.14(c)]. Similar
to our degenerate gas data, we see that the two curves converge at both short and long times,suggesting that it takes some time for correlations to develop before ultimately the phase-spacedensity drops sufficiently for molecule formation to become negligible [383]. At long times,−9/26

we observe the expected power-law behavior Nobs ∝ thold[solid red line in Fig. 7.16(a), seealso Section 7.1.2]. Assuming that the same three-body loss extends to short times we can use
Eq. (7.5) to compute the expected loss dynamics (dotted red line) given our N0 . Interestingly,
for both our R we observe more loss than expected from our calculation25 . For our slow ramp25

Note that recent interferometric measurements of the three-body contact density C3 in a thermal unitary Bosegas [65] observed a relatively slow evolution of C3 following the quench to unitarity, approaching its expected
theoretical equilibrium value [333] in 100 µs ∼ 5tλ . Since it is expected that C3 ∝ L3 [36, 333], it is not clear
whether we expect any traditional three-body loss immediately after the quench.

116

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
out we observe a dramatic suppression of Nobs , compared to both the fast-ramp data and our
calculation based on the long-time behavior. This is further highlighted in the inset, where
we study the evolution of ∆N/N , which we compute using the calculated loss curve to define√
N 26 . The conversion efficiency reaches a broad maximum at thold / tn0 tλ ≈ 1. We note thatthe significant molecule production can be attributed to the rather large phase-space density of
the gas (n0 λ3 ≈ 2) [383], and ∆N/N is significantly suppressed for samples with lower n0 λ3 .
At this point it is insightful to recall the two-step dynamics of E, and the associated redistribution dynamics that we presented in Section 7.3.1. We found that the change in energy was set√
solely by En , while the timescale for the changes was set by the ‘meeting time’ ( tn0 tλ ), with all√
dynamics essentially saturating after thold / tn0 tλ ≈ 2. We also found that these redistribution
dynamics are remarkably insensitive to R, while the observed early-time atom loss depends
sensitively on R.
In Fig. 7.16(b) we compare the decrease in nobs /n0 to the increase in E/E0 27 . While we doobserve loss at early times, visually extrapolating our fast ramp-out data to early times suggeststhat the loss that we observe does not lead to traditional three-body recombination heating, andin the limit of 1/R → 0 it looks as if we are approaching an essentially lossless increase in E.

Instead, at long times we observe the anticipated n ∝ E −9/4 [see Eq. (7.11) in Section 7.1.2] forrecombination heating.
Attaining a full understanding of the loss dynamics at early times is an important endeavor−1/3

for the future, which will require meticulously disentangling the effects of {n0, λ, R, andthold }. It should be noted that while for reference we consider the dynamics to be a function of√√tn0 tλ , it is not necessarily true that the peak position of ∆N/N is set by tn0 tλ ; indeed it has
been theoretically shown that the dynamics of the two-body contact density C2 are set by tλ ina thermal unitary Bose gas [65].
From here on we will focus only on our fast ramp-out (1/R = 0.3 µs/G) data (which minimizes molecular conversion) and examine the observed atom-loss dynamics for different initialconditions (recall that we took data for 15 different n0 and T ). To study the loss dynamics wecompute the dimensionless loss rate√  
18 3 En 2Ṅ,Γ = −tn = ζ3 2
N
π
E
26

(7.17)

This assumption is not necessarily correct, but plotting the data this way allows us to compare curves of ∆N/N
for both our R, and our previous definition (setting Nobs = N for our largest R) simply consists of subtracting
the two (aligned) curves from each other.
27
Here we have applied a dynamical correction to the volume as the gas heats, assuming the power-law shape of
our trap (see also Section 7.1.1). For a sense of scale, within this model a change of E by a factor of 3 leads to achange in volume by 33/15 ≈ 24%.

117

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY(a) (b) (c) (d)1.0(a) (b) (c) (d)

0.1

0.01

nn0 (µm-3 )
0.7
2.0
3.8
5.0
5.1
6.7

1

Γ

Γ

0.1

0.01
2

5

2

E/En

3

4

5

6

nn0 (µm-3 ) T (nK)
0.7
200
1.1
160
1.2
100
1.3
70
1.8
80
2.0
190
2.3
100
2.7
170
3.2
100
3.8
190
4.2
150
5.0
190
5.1
190
5.6
150
6.7
200

E/En

Fig. 7.17 Loss dynamics of thermal Bose gases quenched to unitarity. (a) Dimensionless loss rate Γ versus E/En
for data with varying n0 but essentially constant initial T ≈ 200 nK. The solid line shows the expected
result for three-body recombination in a thermal unitary gas Γ ∝ (En /E)2 , in excellent agreement with√√the long-time data [thold > 2 tn0 tλ , filled symbols]. At early times [thold < 2 tn0 tλ , open symbols], we
observe enhanced three-body loss compared to the long-time extrapolation, and a peculiar dependence
on n0 , with the lowest phase-space density gases showing the most dramatic deviations from the long√time behavior. (b) Dimensionless loss rate Γ versus E/En , focusing only on long times (thold > 2 tn0 tλ )after the quench when the gas is essentially in equilibrium. Our data is described well by Γ ∝ (En /E)2
[solid line]. The dotted line corresponds to our degenerate gas result from Fig. 7.7, and while the absolutevalues differ, the two measurements are in agreement taking into account systematic uncertainties in box
volume; the red shaded region depicts a ±10% uncertainty in the linear box size. The dashed line shows
the previously measured loss rate from [321], and the blue shaded region its systematic uncertainty.
∗

to facilitate comparison with our degenerate gas data, where we recall that ζ3 = (1 − e−4η )and η ∗ is the Efimov width parameter (see Section 7.1.2).

In Fig. 7.17(a) we plot Γ versus E/En on log-log scale comparing data series with approximately constant T ≈ 0.2 µK but varying n0 , at all times after the quench. At long times√
when the gas has reached an equilibrium state thold > 2 tn0 tλ (filled symbols) we observe
excellent agreement between the data which follow the expected Γ ∝ (En /E)−2 scaling expected for three-body recombination heating of a thermal-unitary Bose gas. At early times√(thold < 2 tn0 tλ , open symbols) we observe somewhat enhanced losses, with a rather peculiarfeature that low phase-space density data deviate most dramatically. We currently do not understand the reason for this behavior, but it could be an indication for Efimov physics breaking
universality.
√
In Fig. 7.17(b) we turn our attention exclusively to the long-time behavior (thold > 2 tn0 tλ ),and plot the extracted Γ versus E/En for all our 15 data series (see legend). We find that thiscollapses all our data onto a single universal curve, in agreement with the anticipated scalinglaw Γ ∝ (En /E)2 . The solid line shows the best fit value to the data, which would suggest that

ζ3 ≈ 0.13, and a remarkably low corresponding η ∗ ≈ 0.03. However, our ∼ ±10% systematic
uncertainties in the linear box dimension lead to relatively large systematic uncertainties in ζ3(red band). Our measurement is consistent with our previous result for initially degenerategases that have heated to become thermal (dotted line, see Section 7.2), which feature similarsystematic uncertainties. Overall, our measurements are in good agreement with previous118

CHAPTER 7. BOSES GASES QUENCHED TO UNITARITY
measurement of the thermal unitarity-limited loss rate coefficient ζ3 for our 39 K state, whichfound η ∗ = 0.09(4) [321] (dashed line, and corresponding blue shaded region).

7.4 Conclusion
In conclusion, we have performed a detailed study of both degenerate and thermal Bose gases
quenched to unitarity. Our experiments establish a comprehensive view of the prethermal dynamics and thermodynamics at both low and high temperatures, providing benchmarks for thetheory, and uncovering new conceptual puzzles. They reveal remarkably robust universal behavior, which would be interesting to further test by performing the same quench experiments
using a far-from-equilibrium initial state, such as a turbulent gas.
Focusing particularly on the degenerate unitary Bose gas (where our understanding is stillin its infancy), our experiments unveiled three key facts that make further exploration of thissystem promising:
1. The post-quench dynamics reveal that the gas, even though not in true thermal equilibrium,
does attain a well-defined quasi-equilibrium (prethermal) state.
2. The quench-induced correlation energy is on the order of En , indicating that this is astrongly-correlated state.
3. The prethermal unitary gas has a non-zero condensed fraction, of about 20%, suggesting thatit may be a novel kind of superfluid.
The reason that such a superfluid state may be truly unique is that Efimov physics is thought
to play a fundamental role in the gas’s behavior. Experiments using unitary thermal Bose
gases have recently revealed few-body signatures of such novel three-body correlations [65].
A major experimental goal is finding out whether these states also have a profound impact
on the many-body state at low temperature, where theorists have predicted the formation of amolecular superfluid state consisting of Efimov trimers [319]. In the future we hope to devise
spectroscopic and interferometric measurements that reveal and probe this exciting new stateof matter.

119

8 Outlook
Libiamo, libiamo ne’lieti calici
La Traviata- Giuseppe Verdi -Throughout our four experimental chapters we delved into fundamental quantum manybody physics, exploring interacting Bose gases both in and out of equilibrium. Our experiments operate at the interface of theory and experiment, providing invaluable input for theoretical discourse, in the true spirit of quantum simulation. We have reaped the benefits of ourquasi-uniform optical box potential as well as our ability to precisely control the interparticle
interactions. Having already provided detailed summaries of the experiments and our more
immediate future goals within each chapter, here we take a step back and consider a few possible future research avenues, which leverage the key advantages of our experimental platform.

Critical dynamics. It would be interesting to investigate how interactions affect the critical behavior near the (second-order) Bose–Einstein condensation phase transition. In general, close
to such a phase transition the details of the short-range physics become unimportant and all
systems can be classified into just a few categories known as universality classes, according
to their generic properties such as symmetries, dimensionality, and type of interaction [387].
This was previously studied in our group using 87 Rb in a box [186] (see also [388]), and alongside two other experiments [23,389] comprehensively explored and validated the Kibble-Zurek
mechanism [390, 391], obtaining unprecedented access to the critical exponents of the BEC
phase transition [392]. We intend on extending these experiments, making use of our readily
tuneable interaction strength. A particularly intriguing issue is that the ideal and interacting
Bose gas are in different universality classes. Experimentally, we are able to continuously tune
the interaction strength from interacting to non-interacting, raising the question: can one actually change the universality class of a system, and if so, how does this change occur?

120

CHAPTER 8. OUTLOOK
Critical temperature shift. In a homogeneous Bose gas, quantum correlations close to the critical point are believed to aid condensation. While mostly accepted today, this theoretical consensus emerged after an over 50-year-long debate (see e.g. [393–396]), following the pioneering work of Lee and Yang [43, 397]. The beyond-mean-field contribution has been observed
indirectly in harmonically trapped Bose gases [398], but a direct experimental observation remains absent. Our experimental platform offers a promising setting to observe this interactioninduced shift directly, which is expected to scale as∆Tca≈ 1.8 .
0
Tc
λ

(8.1)

Observing such (typically few %) shifts will require high-precision differential measurements
and perhaps even improvements to the uniformity of our quasi-uniform potential (if the relevant quantity for comparison is the shift ∆Tc , rather than Tc itself).

Space-borne homogeneous Bose–Einstein condensates with tuneable interactions. A major limitation in our experiments is the need to compensate the gravitational force to ensure a homogeneous trapping potential. A natural (albeit technically non-trivial) solution to this is to performthe experiments in outer space, which enjoys a low-gravity environment. The first BEC wasrecently created in space [399], with its sights on precision interferometry. Letting our imagination flow, here we outline a few particularly exciting prospects for using our experimental
platform in such a low-gravity environment (entirely neglecting technical challenges). It would
significantly relax the inherent ‘size-limit’ of our boxes (to ensure sample homogeneity). This
would in turn allow for vast experimentally accessible density ranges (in particular towards
lower densities). Such low densities would be invaluable in reducing three-body recombination compared to the effect of beyond-mean-field correlations (recall Chapter 6). To name afew particularly exciting prospects, it would be a remarkable setting for studying quantum
turbulence, spin mixtures, and the unitary regime.

121

A Cross-calibration of the zero-crossing of a
The measurements in Section 5.3.1 used the collapse of a Bose–Einstein condensate to measurea(B0 ), the zero crossing of the Feshbach resonance at B∞ = 402.70(3) G [65] in the 39 K |1, 1i
state. To measure B0 , we extrapolated the critical point for collapse to the 1/N → 0 limit, obtaining B0 = 350.45(3) G, where we calibrated the absolute field strength using rf-spectroscopy.
This result was found to be in good agreement with previous measurements [83, 243, 244], andcorresponds to a resonance width ∆ = −52.25(4) G.
Here we present an independent measurement of the zero crossing using the evaporation ofa thermal Bose gas. We initially prepare an equilibrium thermal cloud of N ≈ 1×105 atoms at atemperature of ≈ 200 nK at ≈ 200 a0 . We then ramp the scattering length to a final value a over
200 ms before lowering the trap depth by a factor of ≈ 6 in 3 s and subsequently waiting for an
additional 27 s. We finally image the cloud after 30 ms ToF expansion. In Fig. A.1 we plot thecloud’s energy per particle E versus a, where we assume our previous calibration of the Feshbach resonance to convert to a. A clear symmetric peak in E is observed, which arises from thefailure of the gas to thermalize during evaporation as scattering events become too infrequent.
The peak is centered at −0.05(8) a0 , thus cross-validating our previous measurement.

320

335

B (G)
(G)
B
350

365

E/kB (nK)

100
80
60
40
-10

-5

0
a/a
a (a00)5

10

Fig. A.1 Cross-calibration of the zero crossing of a in the lowest hyperfine state. We plot the energy per particle
of a thermal cloud after evaporation at a variable scattering length a (see text for details). A clear peakis observed around a ≈ 0 as scattering events become too rare for the gas to thermalize. The solid line
is a gaussian fit to the data, which gives the center position −0.05(8) a0 (red vertical band), consistent
with zero and thus cross-validating the previous independent calibration using the collapse of a BEC (seeSection 5.3.1).

123

B Atom number control & stability
The data supporting our results typically consist of tens of thousands of experimental repetitions spread over months. Hence, a particularly important aspect is the control and stability of
the initial atom number N0 in our experiments, which we provide benchmarks for below.
To reduce the otherwise maximal achievable N0 [which varies between (2 - 3) × 105 depending on how well previous cooling/preparation steps work] we use three-body losses at theend of evaporation in the ODT (before loading into the box). Experimentally, we vary the B
field (and hence a) at the end of evaporation using the control voltage VFeshODT to vary N0 [seeFig. B.1(a)]. The exact value of VFeshODT needed to produce the same average N0 , does varyslightly from day to day (as previous cooling steps may be more or less efficient), but this isreadily adjusted for by tweaking VFeshODT .
In Fig. B.1(b) we plot N0 as a function of time during continuous operation on three separate
days1 , intending to achieve the same N0 ≈ 9.8 × 104 . While the repetition rate here was 31.7 s,
we only sample N0 twice every 98 repetitions (0.86 hours) 2 . We see only minimal drifts inN0 over time, and averaging gives {9.6, 9.8, 9.8} × 104 (for the three series), with a standard
deviation of 0.3 × 104 (≈ 3%). We are typically able to continuously run for up to ∼ 50 hours
before we need to replenish our MOT using the getters (though this varies based on the desired
N0 ; lower/higher N0 are stable for longer/shorter durations).

(a) (b)30 (c) (d)(a) (b) (c) (d)N0 (104 )N0 (104 )

10
10
3
1
3.92 3.93 3.94 3.95 3.96
VFeshODT (V)

5

0

0

10
tdat (hours)

20

Fig. B.1 Atom number control and stability. (a) Quasi-pure homogeneous BEC atom number N0 versus control
voltage VFeshODT , which sets a at the end of evaporation (in the ODT). (b) N0 as a function of time duringcontinuous operation (N0 is sampled only in a subset of the data series which started at tdat = 0, see text).
1
2

We restrict ourselves to three for visual clarity.
These data correspond to a subset of those in which we explore the unitary regime (see Chapter 7). Out of thefull data series, consisting of ∼ 2000 images each, we have removed . 0.5 % in post-selection, corresponding
primarily to blank images (where it is clear that one of the preparation steps was erroneous).

124

C Inverse-Abel transform
We use an inverse-Abel transform to reconstruct the three-dimensional momentum distribution ñk from the line-of-sight integrated density profile n2D that our absorption images provide.
In the cases where the distribution is spherically symmetric (as satisfied for our uses), we canconsider the azimuthally averaged ñ2D (k̄) = F (k̄) without the loss of information. Formally,
for a function f (k), which drops to zero faster than 1/k, the inverse-Abel transform isf (k) = −

1
π

Z ∞
k

dF
dk̄
p
.
dk̄ k̄ 2 − k 2(C.1)

We implement this procedure numerically, converting the integral to a sum (as the measured
F (k̄) is discretized, with k̄ i ∈ {k̄ 1 , k̄ 2 , ..., k̄ m−1 , k̄ m }), and numerically differentiating F (k̄). We
also separate out the first term in the sum (at k) and analytically approximate its contribution
to tame the otherwise potentially divergent behavior (in the presence of statistical noise).

125

D Field control & stability
Here we give an overview of our field stability and control, with particular emphasis on ourfield quenches into the unitary regime (relevant for Chapter 7).

D.1 Mains oscillation
The coils used to create the magnetic field that the atoms experience (see Fig. 3.2) are driven
by power-supplies connected to the three-phase 50 Hz alternating-current mains. This unfortunately feeds into the fields that they produce, and leads to a modulation of B as a functionof time (with 20 ms overall period)1 . The variation in field is of order 100 mG, which is significantly larger than other technical limitations [e.g. typical field drifts over the course of a dayof order ∼ 10 mG, or the variation across the cloud from the field gradient 6.855 G/cm used tocompensate gravity (corresponding to ≈ 34 mG across 50 µm)].
While typically negligible at low a [where (i) the percentage modulation in a is small, and(ii) the cloud’s intrinsic timescales are much longer so that this averages out], it is a major issue
upon approaching the Feshbach resonance as well as for field-sensitive spectroscopy at all a 2 .
As a starting point for tackling this problem, we externally trigger on the mains cycle, which
synchronizes our experimental clock to it (by waiting for it in the sequence). This solves themain aspect of the spectroscopy problem, as it hampers the shot-to-shot field variations arisingfrom the oscillation over the course of the short (. 100 µs) pulses. We can now map out
the oscillation in detail, by varying the time ttrig following the trigger and performing highresolution rf-spectroscopy to measure B (see Section 3.2.3). In Fig. D.1 we plot B, normalized
by its average over the full range of data, revealing a striking oscillation. We fit the data with afunction of the form

10X

cj sin(2π ν0 j ttrig + φj ) ,

(D.1)

j=1

with ν0 = 50 Hz and where cj and φj are fitting parameters. This captures the data excellently, confirming the higher-multiple-of-ν0 frequency components in the oscillation (it is also
1
2

A solution to this would be to run the coils on batteries, which we are contemplating as an upgrade.
Two prime examples are rf-spectroscopy (see Section 3.2.3) and molecular dimer spectroscopy, where the binding
energy Eb ≈ −~2 /(ma2 ) (see e.g. [365]).

126

APPENDIX D. FIELD CONTROL & STABILITY

100(G)
BB--BB(mG)

50
0
-50
-100
0

5

10
ttrig
(ms)
t (ms)

15

20

Fig. D.1 Mains oscillation. We plot the measured field variation (B around its mean B) as a function of time ttrig
following an external trigger to synchronize with the mains cycle (see text). We identify a 2 ms window
following ttrig = 13.2 ms (dashed line), over which the field changes by . 30 mG. Field-sensitive experiments, in particular those quenching into the unitary regime (see Chapter 7), are always performed in thisregion. For experiments that are longer, we inadvertently sample the oscillation.

consistent with a Fourier analysis). The multiple-frequency components most likely arise due
to coupling between our different power-supplies used to control B, but the exact form is notimportant here. We identify a 2 ms window following ttrig = 13.2 ms where the field changes
by . 30 mG. We use this trigger at ttrig = 13.2 ms in all experiments that do not inevitably
average over this oscillation [i.e. where the critically sensitive part of the experiment occurs in 20 ms]3 .

D.2 Dynamical field control
Throughout this thesis we rely on dynamical control of the magnetic field, and consequently
a. The response time of the Feshbach coils (in conjunction with the entourage of coils striving
to maintain gravity compensation, from here on collectively referred to as primary coils) islimited to a few milliseconds, but is still typically suitable for our experiments. However, forour experiments at unitarity, this is far too slow, and we instead rely on a set of auxiliary ‘fast’coils to perform rapid field quenches (on µs timescales). In the following we first characterize
the response time of our slower primary field ramps, before detailing and benchmarking ourquenches to unitarity.

3This curve was measured continuously over a ≈ 12 hour period; each point corresponds to the center frequency
of a 31-point rf-spectrum, and the data were taken in randomized order). It therefore also acts as a measureof our field stability, which is typically better than ±10 mG over the course of a day (weekends and nights arebest), and a long warm-up period (> 3 hours) is typically necessary. We have also re-measured this curve several
times (months apart) and never observed a statistically significant difference.

127

APPENDIX D. FIELD CONTROL & STABILITY

D.2.1 Primary field response time
A prevalent situation in which we wish to push the primary field response time to the limit isfor turning off interactions during ToF4 . Such primary coil ramps are also at the heart of ourquench experiments exploring collapse dynamics (see Chapter 5).
In Fig. D.2 we plot B (measured using rf-spectroscopy) following a quench (initiated at t = 0)
using our primary coils. The example we provide corresponds to a typical measurement inwhich we wish to rapidly turn off the interactions for ToF. The blue circles depict the result if
we simply dial in an instantaneous quench (setting the control voltage Vc to its final value, seeblue line in inset). We obtain a slight improvement in the response time by programming in an‘overshoot’ (red circles, see also inset). The magnitude of such controlled field quenches (hererather large at ≈ 50 G) does not significantly affect the response time, though we do achievesignificantly faster response times when entirely turning off the coils (. 1 ms).

0

375

400
0

1
2
t (ms)t (ms)

350
0

5

tt (ms)(ms)

40

a (a0 )Vc (V)

5

a/a0

B (G)
(G)

400

10
4
0

10

Fig. D.2 Field response following a quench using our primary coils. The quench is initiated at t = 0 and the field
is measured using rf-spectroscopy as a function of t for two quench protocols depicted in the inset anddescribed in the text.

D.2.2 Quenches to unitarity
To reach the unitary regime we use a set of strategically placed auxiliary coils (a.k.a.‘fast coils’)that avoid significant inductive coupling to the Feshbach coils, thus minimizing eddy currents
and allowing for field switching on µs timescales. The auxiliary coils are powered by a Elektroautomatik PS9360-30 1U supply, and for switching we use an H-bridge circuit (see Fig. D.3),
which was designed, built, and tested by Jake Glidden [86]; here we only summarize key features. Using the H-bridge we can isolate the coil from the supply for switch-off, so that theenergy stored in the magnetic field can be dissipated over the the sum of the auxiliary coil’sinternal resistance and R2 . The on/off configurations of the H-bridge form a simple RL circuit,for which we expect exponential transient behavior, characterized by a rise time τin and a fall
4

This for example allows for more accurate measurements of the momentum distribution by avoiding contamination from the conversion of interaction to kinetic energy in ToF. Note that we can also achieve this more rapidly(≈ 60 µs) using an rf spin-flip to |1, 0i, though this is not always convenient or possible.

128

APPENDIX D. FIELD CONTROL & STABILITY

R1

AH

BH

Supply

R2

L

AL

BL

Fig. D.3 Schematic of the H-bridge circuit used for rapid field switching. We highlight only the core componentsof the circuit, which are the power supply, two resistors (R1 and R2 ), four n-channel IXFN48N50 MOSFETS(denoted AH, AL, BH, and BL), and our set of auxiliary coils (denoted L). The circuit is driven using a pairof half-bridge FAN7380 drivers; figure from [86], where additional details are available.

time τQ . Moreover, keeping the sum Rtot = R1 + R2 fixed allows us to fix τin while varying τQ .
The field probes that we currently have access to do not allow for a detailed characterizationof the response times5 , and instead we characterize the ramp speeds using the current in coil.
We however do measure the field using rf-spectroscopy at unitarity (once it has settled)6 .
As exemplified in Fig. D.4 (for two different initial ramp-out rates R = ∆B/τQ ), the measured current Iq shows an exponential rise/fall as a function of t (we show examples of atypical quench into the unitary regime with thold = 20 µs)7 . The two different τQ are achieved
using a fixed Rtot = 10 Ω, using either R2 = Rtot or R2 = 0. This gives access to values of τQ
in the range (1-10) µs; Table D.1 summarizes the measured τQ for different combinations of R1
and R2 .
4

2

0

0
0

10

20

30

tt (µs)(µs)

40

ΔB (G)8

ΔB (G)(A)
IIqq (A)1650

Fig. D.4 Current response of the fast coils used for quenches to unitarity. Here we use an on-resonance hold time
thold = 20 µs, and calibrate the field jump magnitude ∆B using rf-spectroscopy at unitarity (the conversion
as a function of time assumes proportionality with Iq ).
5

We intend on implementing two-photon Raman spectroscopy between |1, 1i and |1, 0i, in order to both provide
a faster field probe and allow for rapid spin-flips (competitive with µs time scales).
6
For this purpose we typically work with low densities (≈ 0.3 µm−3 ) to mitigate any risk of significant potential
rf-resonance shifts of order En /~.
7
The low duty cycle (. 10−4 ) ensures that the coils do not burn; we also include a fuse in the circuit to avoid
disasters (it works).

129

APPENDIX D. FIELD CONTROL & STABILITY

Table D.1: Resulting τQ for different combinations of R1 and R2 , with Rtot = 10 Ω to maintain τin ≈ 1 µs.
The uncertainty in τQ is ≈ 0.2 µs.

R1 (Ω)

R2 (Ω)τQ (µs)

10.0
9.0
7.4
6.6
0.0

0.0
1.0
2.6
3.4
10.0

10.2
4.5
3.0
2.3
1.2

We use rf-spectroscopy to calibrate the field at unitarity, an example of which is shown in
Fig. D.5 as a function of thold (using a 30 µs rf pulse). We see that within the experimentally
accessible range of times8 the field is essentially constant at B∞ (within its uncertainty).
For our field calibration at unitarity we measure the field at early times (in the first 50 µs)
by performing rf-spectroscopy at unitarity with low-density samples. While our primary coils
are used to set the field before the quench, we do slightly tweak ∆B (by up to ≈ 30 mG) forfinal adjustments to ensure we always start within δB ≈ 20 mG of B∞ before starting longdata series. We always measure the field again after the run has finished (and occasionallyalso in-between) and retain only those data where subsequent field measurements lie withinδB < 40 mG of B∞ .

(a) (b) (c) (d)

402.8

0.4

B (G)

TF (%)(a) (b) (c) (d)0.2
098.88 98.90 98.92

402.7
402.6
0

50

100

150

thold (µs)

ωrf /(2π) (MHz)Frequency(MHz)

Fig. D.5 Field stability at unitarity. (a) Radio-frequency spectra showing the transferred fraction of clouds at various on-resonance thold , using a 30 µs rf pulse. (b) Measured field as a function of thold at unitarity, confirming that the field is constant to within 30 mG over > 150 µs, the most critical time frame in our experiments.

8

We cannot currently accurately measure much long times as the clouds decay and heat significantly, diminishing
our signal which relies on Stern–Gerlach separation of the spin components. State-selective imaging would
improve this (though we have no reason to expect any significant changes for thold < 2 ms).

130

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