Thermodynamics ofultracold 39K atomic
Bose gases withtuneable interactions

A dissertation submitted to the
University of Cambridge for the
degree of Doctor of Philosophy

Naaman Tammuz
Cavendish Laboratory, November 2011

Cavendish Laboratory© 2011 Naaman Tammuz

Cavendish Laboratory
University of Cambridge
19, J J Thomson Avenue
Cambridge CB3 0HE
UK

Abstract

This thesis can conceptually be divided into two sections. In the first, consistingof chapters 2 and 3, I describe the design and construction of an experimental
system for the production of ultracold Bose condensed gases of the isotopes 39 K
and 87 Rb. We work with 39 K primarily due to the existence of a very broad
Feshbach scattering resonance in its absolute ground state which enables precise
and wide tuning of the s-wave scattering length. In order to cool 39 K sufficiently
it is sympathetically cooled by 87 Rb while held in a magnetic trapping potential,
after which both species are transferred to a purely optical trap. The 87 Rb isremoved from the trap and the remaining 39 K is evaporated further by graduallowering of the trap depth. We produce quasi-pure 39 K condensates containing
over 4 × 105 atoms. The same apparatus can also produce 87 Rb condensates ofover 8 × 105 atoms.
The second part of the thesis, consisting of the chapters 4 and 5, describes experiments aimed at addressing the influence of interparticle interactions on thethermodynamics of ultracold Bose gases. Firstly, we investigate the saturationof excited states as the driving mechanism behind Bose-Einstein condensation,in accordance with Einstein’s original description. We find that real, interacting
gases deviate more strongly from this picture than predicted by simple mean-field
theories. Extrapolation to the non-interacting limit, however, allows us to recover
Einstein’s picture. Secondly, we investigate the effect of interaction strength on
the shift of the condensation critical point from its ideal-gas value. We reveal
for the first time the beyond-mean-field corrections to this quantity, in qualitative agreement with the most recent theoretical predictions, and also observe
non-equilibrium effects in the regime of very weak and very strong interactions.

iDeclaration

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

Cambridge, November 2011iii

Acknowledgements

It is a pleasure to thank the many people with whom I have had the good fortuneof sharing the last four years and who have directly and indirectly contributed
to the work described in this thesis. First and foremost, I would like to thankmy supervisor, Zoran Hadzibabic, for giving me the precious opportunity to join
the group at its inception, despite my minimal experimental background andrelatively fresh arrival to the field of cold-atom physics. The freedom given to
my colleagues and me during these early days of the project always made us feeltrusted, and Zoran did a good job of concealing any fears he might well have
harboured. His steady and optimistic hand at the group’s helm over the years
has helped us overcome many practical and conceptual obstacles and has madeour work more rewarding as a result.
There is a great deal of satisfaction and enjoyment to be had from working towards a common goal as part of a team, and I have been very fortunate in the
teammates with whom I have had the chance to spend time, both in the lab andout. It requires a delicate balance of personalities to produce a group that isboth fun and productive and I feel lucky to have been part of just such a group.
Thank you from the bottom of my heart to Rob Smith and Robbie Campbell
for all the shared experiences, for the good sense of camaraderie, for their humour and their relaxedness. The same thanks go to Scott Beattie and Stuart
Moulder, who arrived two years into our work and immediately became an indispensable source of aid and entertainment. The newest generation of students,
Igor Gotlibovych,Tobias Schmidutz, Alex Gaunt and Richard Fletcher are all
talented physicists and great individuals and it is very comforting to know thatboth of the atom-machines are in such capable and skilled hands. Thanks also to
Jun-Young Kim and Philip Richman, who have been part of the team along theway. I would also like to greatly thank Rob Smith, Alex Gaunt, Richard Fletcher
and Scott Beattie again for their reading of the manuscript of this thesis and forv

vi
their helpful comments and suggestions.
Throughout my PhD we were fortunate to have as guests in our group a steadystream of talented undergraduate students, whose work greatly contributed tothe overall progress of the lab and whose presence brought to the lab different
perspectives and a breath of fresh air. In particular, thanks go to Alexandre
Dareau, Nils Engelson, Mehdi Lallouache, and Diana Aude. Alexandre for the funtime spent designing our imaging system, dissecting projectors and playing with
SLMs, Nils for his work on our optical dipole trap (amongst other things), Mehdi
for his work on our fiberport cluster and Diana for her work on the microwave
evaporation system. We also had the distinctly enjoyable privilege of a 3-monthvisit to our group by Prof. Jean Dalibard from the ENS in Paris. Any timespent in Jean’s company is happy, well spent time and I am extremely gratefulto have been able to benefit from his presence for such an extended period. I amvery grateful for the good neighbourly relations we enjoyed with the friendly andhard-working members of Michael Köhl’s hybrid BEC-ion-trapping lab and Fermi
gas lab, who were always ready and willing to share from both their experiential
and material supplies, as well as their footballing prowess. Our other colleagues
in the AMOP group, from the teams of Mete Atatüre and Richard Phillips, werealso a source of help and encouragement. I wish them all success and happinessin all their future endeavours.
We also occasionally had to venture outside the lab, and when we did so we were
lucky enough to find ourselves in the friendly, safe and competent hands of thestaff of the Cavendish laboratory. Our first port of call for anything not directly
related to a cold atom is Pam, the most wonderful group administrator in theworld, who was always ready to help with any issue and often went to extra lengths
to ensure things got done right. A PhD devoted in large part to the constructionof an experiment from scratch, should not and can not be considered complete
without abundant thanks to the technical staff who make such a labour-intensive
task manageable. Special thanks go to the head of the student workshop, Nigel
Palfrey, with whom I spent many days milling, lathing, drilling and shooting
the breeze. Thanks also to Peter Norman and Chris Summerfield from the main
workshop for their expert and friendly handling of any machining request wethrew at them. Thank you to the staff of the Cavendish stores for providing uswith all the wires, tubes, pens, lab books, jubilee clips, resistors, capacitors andop-amps we could have dreamt of and a big thank you to Luis and his team in
the big-ass burrito van for making Thursday lunchtimes extra special.

vii
Thank you to my friends. Orr, Aharon and Adam, Allon and Roohia, Lail,
Roi, Jesper, Safi, and everyone else who is dear to my heart - thank you foreverything that has been and that is yet to come. Thank you to my family. My
unbelievably supportive and caring parents, my beautiful and gifted sister, myloving and cheerful grandparents, my cool relatives across the pond(s) and Gretha
and Victor, who have sent so much goodness and wisdom my way.
Last but most, all day and every day, thank you to Petra, who makes my heart
explode.

Contents

1 Introduction
1.1
1.2
1.3
1.4

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature, density and length scales . . . . . . . . . . . . . . .
Why 39 K? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Theoretical background
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Laser cooling and trapping . . . . . . . . . . . . . . . . . . . . . .
Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . . .
Evaporative and sympathetic cooling . . . . . . . . . . . . . . . .
Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . .
Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . .
Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . .
Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
4
6

9
10
25
26
30
37
39
44
48

3 Design, assembly and performance of our 39 K-87 Rb
apparatus
53
3.1
3.2
3.3
3.4
3.5
3.6
3.7

Sequence overview . . . . . . . . . . . . . . . . . . . . . . . . . .
Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . . .
Quadrupole and compensation coils . . . . . . . . . . . . . . . . .
Optical pumping and magnetic capture . . . . . . . . . . . . . . .
Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54
55
69
75
77
80
81

ix

x

Contents
3.8 QUIC trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Crossed dipole trap (CDT) . . . . . . . . . . . . . . . . . . . . . .
3.11 Feshbach field and Feshbach resonance . . . . . . . . . . . . . . .
3.12 Onset of BEC in 39 K . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 Sequence summary . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 Imaging setup and procedure . . . . . . . . . . . . . . . . . . . .
3.15 Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 Timing control . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
89
94
100
108
108
108
115
119

4 The role of saturation in Bose-Einstein condensation 123
4.1
4.2
4.3
4.4
4.5
4.6
4.7

Introduction and motivation . . . . . . . . . . . . . . . . . . . . .
Experimental procedure for investigating saturation . . . . . . . .
Observation of non-saturation . . . . . . . . . . . . . . . . . . . .
Varying interaction strength and temperature . . . . . . . . . . .
Mean-field theories of interacting gases . . . . . . . . . . . . . . .
Comparing experiment and theory . . . . . . . . . . . . . . . . . .
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . .

124
126
129
130
131
140
146

5 Interaction-dependence of the BEC transition temperature
151
5.1
5.2
5.3
5.4
5.5
5.6

Introduction and theory . . . . . . . . . . . . . . . . . . . . . . .
Previous experimental studies . . . . . . . . . . . . . . . . . . . .
Experimental method . . . . . . . . . . . . . . . . . . . . . . . . .
Extracting the Tc shift . . . . . . . . . . . . . . . . . . . . . . . .
Non-equilibrium measurements . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152
158
161
164
166
169

6 Summary and Outlook

171

Appendices177

A 87 Rb and 39 K properties

177

B A compact system for the rapid production of 87 Rb
BECs181
B.1 Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B.2 Lasers and coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B.3 Optical Plug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Contents

xi

B.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

C Using Spatial Light Modulators for versatile creation of optical potentials
193
C.1
C.2
C.3
C.4
C.5
C.6
C.7

Spatial Light Modulators . . . . . . . . . . . . . . . . . . . . . . .
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .
Trial 1: Laguerre-Gauss modes . . . . . . . . . . . . . . . . . . . .
Trial 2: 2D confinement and the 2D-3D crossover . . . . . . . . .
Trial 3: Arbitrary potentials . . . . . . . . . . . . . . . . . . . . .
Home-made SLM . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current status and future plans . . . . . . . . . . . . . . . . . . .

194
195
197
200
201
203
205

D MOT and science cell properties

207

E Magnetic fields produced by current-carrying coils

211

E.1 Fields from circular coils . . . . . . . . . . . . . . . . . . . . . . . 211
E.2 Fields from rectangular coils . . . . . . . . . . . . . . . . . . . . . 214

F A tightly confining potential for studies of twodimensional Bose gases
217
F.1
F.2
F.3
F.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The 0-π phaseplate . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating the trapping frequency and depth . . . . . . . . . . .
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .

217
219
221
222

“In the eighteenth century it was often convenient to regard man as aclockwork automaton. In the nineteenth century, with Newtonian physics
pretty well assimilated and a lot of work in thermodynamics going on, man
was looked on as a heat engine, about 40 per cent efficient. Now in thetwentieth century, with nuclear and subatomic physics a going thing, man
had become something which absorbs X-rays, gamma rays and neutrons.”

Thomas Pynchon (b. 1937), V.

1
Introduction“The general public has long been divided
into two parts; those who think that sciencecan do anything and those who are afraid it
will.”
Thomas Pynchon, Mason & Dixon

1.1 BackgroundThe basic dichotomy of the elementary particles of nature by now requires
little retelling: fermions, the fundamental constituents of matter possessing
half-integer spin, are prohibited from simultaneously occupying the same quantum state while bosons - elementary or composite particles possessing integer
spin - are sociable and prefer to aggregate in the same state. In this thesis weconcern ourselves only with the latter class of particles and can encapsulate theaims of the reported experiments as exploring the consequences of inter-particle
interactions on different aspects of their gregarious behaviour.

12Introduction

Central to the understanding of the properties of bosonic systems1 is the conceptof the Bose-Einstein condensation (BEC) phase transition, which occurs when the
wave-like nature of indistinguishable particles begins to dominate their behaviour.
In this “quantum-degenerate” regime, a macroscopic fraction of the particles occupies the confining potential’s single-particle ground state. The BEC transition
plays a key role in the explanation of experimental signatures observed in many
disparate systems. Manifestations of this bosonic quantum degeneracy include
the behaviour of Cooper pairs in superconductors (1911) [1], the superfluid properties of 4 He (1938) [2, 3, 4, 5, 6] and 3 He (1972) [7, 8, 9], the condensation ofmagnons in antiferromagnets (2000) and ferrimagnets (2006) [10, 11], fermionicpair condensates in dilute Fermi gases (2004) [12] and condensation of microcavity
polaritons (2006) [13, 14].
Perhaps the most striking and clear demonstration of this phenomenon, however,has been provided by the field of ultracold atomic Bose gases. This burgeoning field with its newly-developed experimental techniques, recognized by the
Physics Nobel prizes of 1997, 2001 and 2005 [15, 16, 17], has provided the meansfor cooling trapped atomic ensembles down to unprecedented temperatures - on
the order of tens of nanokelvin - allowing us to perform extremely sensitive measurements on the quantum nature of these versatile and highly-controllable systems. Magnetic trapping of neutral atoms was first reported in [18] and the first
demonstration of the magneto-optical trap was reported two years later [19]. The
concurrent introduction and development of evaporative cooling [20, 21, 22] leadto the first experimental observations of BEC in dilute atomic Bose gases in 1995
[23, 24, 25] and quantum degeneracy in fermi gases a few years later [26, 27]. To
date, fifteen atomic species have been condensed in ultracold atom groups aroundthe world (see table 1.1).
1995
Rb [23]
2001
41
K [32]
200984
Sr [38, 39]87

1995
Na [24]
2003
133
Cs [33]
2010
86
Sr [40]23

1995
Li [25]
2003
174
Yb [34]
2011164
Dy [41]7

1998
H [28]
2005
52
Cr [35]2000
Rb [29]2007
39
K [36]

85

4

2001
He [30, 31]
200940
Ca [37]∗

Table 1.1: Atomic species to have been condensed at the time of writing.
1

An even number of constituent elementary fermions (electrons, protons, neutrons) makes
an atom bosonic, while an odd number make it a fermion.

1.2 Temperature, density and length scales3

Our understanding of these systems’ properties has grown in parallel with ourexperimental ability to manipulate their external and internal degrees of freedom.
Novel magnetic and optical trapping methods enable samples to be loaded into
periodic, lower-dimensional and dynamically reconfigureable potentials, while thediscovery of Feshbach scattering resonances [42] has provided the ability to tune
the strength and nature of interatomic interactions. This has enabled investigation of the delicate interplay between interparticle interactions and externalpotentials in determining the thermodynamic behaviour of these systems. Experiment and theory continue to yield valuable insights into the underlying manybody physical effects as well as opening the way to new technological applicationsof these systems in fields such as high-precision metrology and quantum information processing. Reviews of the history of this field can be found in [43, 44] anda discussion of more recent experimental and theoretical results can be found in[45, 46].
The experiments described in this thesis investigate the effects of interactions
on the thermodynamic behaviour of a Bose gas, both well within the regime ofquantum degeneracy (chapter 4) as well as on its threshold (chapter 5). Below
is a summary of the relevant temperature, density and length scales as well as a
discussion of our motivations for working with 39 K. This is intended to serve as
an overview, with the technical elements introduced here to be explained more
rigorously in chapters 2 and 3.

1.2 Temperature, density and length scales
Broadly speaking, the effects of quantum statistics become significant when thenumber of thermally accessible states becomes comparable to the number of particles in the system, such that the average occupancy of a state below the system’s thermal energy becomes of order one. In terms of macroscopic parameters,
namely temperature and number density, the onset of quantum degeneracy occurs
when the characteristic spatial extension of a particle, as given by its thermal de
Broglie wavelengthh(1.1)
λdB = √
2πmkB T
becomes comparable to the mean inter-particle spacing ` ∼ n−1/3 , where n is theparticle number density, h is Planck’s constant, m is the mass, kB is Boltzmann’sconstant and T is the temperature.

4

Introduction

This is quantified by the phase-space density of the gas
ρ = nλ3dB(1.2)

which measures the number of particles within a ‘de Broglie volume’, λ3dB .
BEC therefore occurs when ρ ∼ 1. A property shared by all cold-atomexperiments is the extreme diluteness of the trapped samples, which is requiredin order to keep them in their metastable, weakly-interacting, gaseous phase.
The weak-interactions also make the system more amenable to theoreticaltreatment, which is otherwise complicated by the strong interactions present
in solids or quantum liquids such as the helium superfluids. Densities on theorder of 1013 cm−3 - roughly ten orders of magnitude below those of a typicalsolid - are required in order to avoid prohibitive molecule formation rates via
three-body recombination. Table 1.2 shows some typical densities for comparison.

nucleuswhite dwarf

lead

water

air

BEC

1038 cm−31030 cm−31023 cm−31022 cm−3

1019 cm−3

1013 − 1015 cm−3

Table 1.2: Typical number densities of some materials in comparison to those at
the centre of a BEC.

As an example, a cloud of 106 87 Rb atoms held in a harmonic potential of frequency ω/2π = 100Hz at T = 100µK has a phase-space density of ρ ∼ 10−7 anda thermal radius of RT ≈ 50µm. The same cloud at T = 1µK has ρ ≈ 0.1 andRT ≈ 15µm. The BEC transition temperature for the above cloud occurs aroundTc ≈ 450nK. Typical trapped samples contain between 103 and 107 atoms andhave an experimental trap lifetime in the range 1-100s. Section 2.7 discusses the
BEC transition in more detail.

1.3 Why 39K?
A relative newcomer to the condensate family, 39 K was condensed for the first timein 2007 [36] in the LENS1 group in Florence following several years of pioneeringstudies by this and other groups into the collisional properties of this isotope andits suitability for BEC experiments [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,60]. The fermionic isotope 40 K was first cooled to degeneracy in 1999 by the JILA
1

European Laboratory for Non-Linear Spectroscopy

1.3 Why 39 K?

5group [26], while the other bosonic potassium isotope, 41 K, was first condensed at
LENS in 2001 [32]. Amongst other findings, it was demonstrated that 87 Rb waswell suited to act as a refrigerant for sympathetic cooling of 39 K and 41 K [32, 59].
The existence and properties of multiple Feshbach scattering resonances in 39 K
were predicted and observed [60] and several interspecies 39 K-87 Rb and 41 K-87 Rb
Feshbach resonances have also been found [61, 58, 36, 62].
39

K is an attractive species to work with1 . The existence of other bosonic andfermionic isotopes, both of which have also been cooled to degeneracy, allow experiments to be performed on ultracold Bose-Bose and Bose-Fermi mixtures inwhich the two species possess almost identical masses2 [65]. 39 K’s several broad
Feshbach resonances, located at moderate magnetic fields, allow fine tuning of thes-wave scattering length over a large range, from strongly attractive to strongly
repulsive. The ability to cancel out interactions altogether enables precision atominterferometry experiments to be performed as well as studies on phenomena suchas Anderson localization [66, 67]. 39 K’s negative background scattering length,
which constitutes a significant hurdle on the route to BEC, can be utilized inexperiments on the effects of attractive interactions [25, 68, 69, 70, 71]. From atechnical point of view, its well-understood level structure, high natural abundance (93.26%) and the availability of commercial laser systems at the relevant
wavelengths also add to its appeal.
Working with this isotope poses several difficulties, however. Chief amongst theseare (1) the inefficiency of conventional laser cooling mechanisms due to 39 K’s unresolved excited-state hyperfine structure, (2) the relative inefficiency of evaporative cooling due to the small background scattering length and (3) the instabilityof 39 K BEC resulting from the attractive nature of this background interaction
strength. The first of these is overcome by careful choice of laser frequencies, thesecond by performing sympathetic cooling with 87 Rb and the third by harnessing a Feshbach resonance to change the interaction strength from attractive torepulsive. These problems and the measures for overcoming them are describedin detail in chapters 2 and 3.

1
2

No pun intended.
The only other species-pairs for which this has been achieved are 7 Li-6 Li [63] and 84 Sr-87 Sr
[64].

6

Introduction

1.4 Thesis outline
The remainder of this thesis is structured as follows:• Chapter 2 provides some theoretical background to the various techniques
and phenomena utilized in the production and manipulation of ultracold
atomic gases.
• Chapter 3 describes the design, assembly and performance of our apparatus
for producing ultracold dual-species 87 Rb-39 K clouds.
• Chapter 4 discusses experiments in which a condensed 39 K gas with tuneable interaction strength was used to investigate the role of excited-state
saturation as the driving mechanism behind the Bose-Einstein condensationtransition in a non-interacting gas.
• Chapter 5 describes our measurements of the interaction-shift in the BEC
critical point. We observe the hitherto unseen beyond-mean-field effects on
the shift in the critical temperature in addition to non-equilibrium effects
in the regimes of very-weak and very-strong interaction strengths.
• In chapter 6 we summarize our results and mention possible directions for
future work.
Appendicies
The appendices at the end of this thesis include additional information that did
not find a place in the main text but that may prove useful to others working on
similar experiments. This includes
1. A summary of the relevant optical and physical properties of both 87 Rb and39
K (appendix A).
2. A description of a compact system for rapid production of 87 Rb BECs assembled to provide a convenient platform for proof-of-principle experiments
and testing of components (appendix B).
3. Experimentation with the use of a Spatial Light Modulator (SLM) for generation of Laguerre-Gauss beams, arbitrary optical potentials and dynamic
control of trapping potentials (appendix C).
4. Information on the dimensions and anti-reflection (AR) coatings of the
MOT chamber and the science cell (appendix D).

1.4 Thesis outline

7

5. The formulae used for calculation of the magnetic fields generated by our
various magnet coils (appendix E).
6. Preliminary results for the design and testing of a setup for producing
tightly-confining optical potentials for future studies of two-dimensional
Bose gases with tuneable interactions (appendix F).

8Introduction

Publications arising from this work
• The results described in chapter 3 are summarized in:
1. Efficient Production of Large 39 K Bose-Einstein Condensates.
R. L. D. Campbell, R. P. Smith, N. Tammuz, S. Beattie, S. Moulder, and Z.
Hadzibabic, Phys. Rev. A 82, 063611 (2010)• The results of chapter 4 on the role of saturation in BEC were published in:
2. Can a Bose gas be saturated?
N. Tammuz, R. P. Smith, R. L. D. Campbell, S. Beattie, S. Moulder, J. Dalibard
and Z. Hadzibabic, Phys. Rev. Lett. 106, 230401 (2011)• Our measurements of the interaction shift in the BEC critical point, as described in chapter 5, appear in:
3. Effects of Interactions on the Critical Temperature of a Trapped Bose Gas.
R. P. Smith, R. L. D. Campbell, N. Tammuz, and Z. Hadzibabic, Phys. Rev.
Lett. 106, 250403 (2011)• Analysis of beyond-mean-field contributions to the condensed fraction were
published in:
4. Condensed Fraction of an Atomic Bose Gas Induced by Critical Correlations.
R. P. Smith, N. Tammuz, R. L. D. Campbell, M. Holzmann, and Z. Hadzibabic,
Phys. Rev. Lett. 107, 190403 (2011)

2
Theoretical background“Why should things be easy to understand?”Thomas Pynchon, quoted in Playboy (March 1977)

Abstract
This chapter summarizes some of the theory underlying the various stages in the
production and manipulation of our ultracold 39 K and 87 Rb gases. In the interestsof conceptual clarity and convenient browsing, the chapter is structured so as toroughly correspond to the progression of the experimental procedure used to create
the degenerate and quasi-degenerate clouds which constitute the starting point
for our investigations. We discuss the principles behind (i) laser cooling andtrapping, (ii) magnetic trapping, (iii) evaporative cooling, (iv) optical trapping,(v) Feshbach scattering resonances, (vi) the properties of thermal and condensed
atomic clouds and (v) absorption imaging.

910

Theoretical background

2.1 Laser cooling and trapping
Laser cooling and magneto-optical trapping have established themselves as thestarting point of virtually every single ultracold atom experiment conducted today. The use of laser light for cooling and trapping atoms enjoys a rich anddistinguished history, including the award of the 1997 Nobel prize in Physics[72, 73, 74] for development of the related experimental methods and theoreticalunderstanding. However, the mechanical effects of radiation on matter particles
had been hypothesized and investigated long before the first proposals and experiments harnessing these effects to cool atoms were carried out in the mid 1970’s.
Table 2.1 presents a chronology of milestones related to these theoretical andexperimental studies.
Year

Name

1619

Keppler1873

Maxwell

1876

Bartoli

1901

Lebedev

1901-3

Nichols &
Hull

1909

Einstein

1933

Frisch1975
1975

Hänsch &Schawlow
Wineland &Dehmelt

Contribution
In 1619 treatise, “De Cometis”, hypothesises that the
deflection of comet tails away from the sun is a consequence of pressure exerted by solar rays.
In “Treatise on Electricity and Magnetism”[75],showed that radiation pressure is equal to energy density of electromagnetic wave.
Derived,from arguments based on 2nd law of thermodynamics, radiation pressure equal to that calculatedby Maxwell [76, 77].
First measurement of light pressure on a solid bodyusing container of water and torsional balance [78].
Use the torsional Nichols radiometer to measure radiation pressure [79, 80].
In [81], showed that Planck’s black-body formula implied that light quanta also carried quantized momentum p = h/λ.
Observed defection of a beam of sodium atoms by lightresonant with a sodium energy level transition [82].
Proposed using laser for cooling of neutral atoms [83].
Proposed laser cooling of trapped ions [84].

2.1 Laser cooling and trapping

1978

AshkinNeuhauser
et al.
Andreyev /1981-2 Phillips &Metcalf
Chu,1997 Phillips &C.-Tannoudji
1978

11

Proposed “a method of stably trapping, cooling, andmanipulating atoms on a continuous-wave basis...using
resonance radiation pressure forces” [85].
Laser cooling of trapped ions to a few millikelvin [86].

Cooling of Na beams with laser light [87, 88]Nobel prize “for development of methods to cool andtrap atoms with laser light.” [74, 72, 73]

Table 2.1: Milestones in the understanding of the mechanical effects of light onmatter.

The fields of atomic spectroscopy and atom clocks have also benefited hugely fromthe reduced Doppler shifts and increased interaction times afforded by magnetooptical traps (MOTs), in which many different atomic species can be cooled withrelative ease to temperatures on the order of 100µK. The physics governing laser
cooling and the operation of MOTs has been presented and scrutinized in many
excellent books [89, 90, 91], review articles [72, 73, 74, 92], research papers andtheses. Here we provide a brief review of the main operating principles responsible
for the laser cooling and trapping of atoms from a background vapour in ourdouble-species 39 K-87 Rb MOT. The basic theory is discussed in the context of a
two-level system and in subsection 2.1.4 we discuss how well this approximationis realised in practice for our atomic species.

2.1.1 Atom-light interactions
The total force acting on atoms as a results of their interaction with an electromagnetic field is the result of three distinct processes taking place during theinteraction. These are the absorption, spontaneous emission and stimulated emission of photons from the radiation field. The contribution of the first two of theseis usually called the scattering or radiation force and the dispersive force arising from the third process is called the dipole force. These two components areoften treated separately and in ultracold atom experiments typically play very
distinct roles in the cooling and manipulation of trapped clouds. Laser cooling12

Theoretical backgroundand trapping constitute the first stage of the experimental sequence and rely on
the scattering force, as described in 2.1.2 and 2.1.3. Optical trapping using the
dipole force is utilized during the final part of the experiment, where its tight confinement and state-independent nature make it convenient for use in conjunction
with evaporative cooling and Feshbach resonances, respectively. Here we outlinea derivation, along the lines presented in [89], of the force resulting from theinteraction between a two-level system and a plane-wave electric field along one
spatial dimension, z. For an atom, this force arises from the interaction between
the field and the dipole moment, −er, it induces in the atom. The Hamiltonian
for the energy of a dipole in an electric field is
H = −e E(r, t) · r(2.1)

where E(r, t) is the electric field operator and er is the dipole moment operator.
The force along the z-direction can be found using the Ehrenfest theorem asFz = −

∂H
∂z




=e

∂
(E(r, t) · r)
∂z


(2.2)

Using the dipole approximation, which assumes a constant value of the electricfield across the dipole (i.e. λ  |r|), we can exchange the order of the derivative
and the expectation value such thatFz = e∂(hE(r, t) · ri)
∂z

(2.3)

Since the atom has no dipole moment when in an energy eigenstate (i.e. hg|r|gi =he|r|ei = 0), the dipole operator can be written as r = reg |eihg| + r∗eg |gihe| =reg ρeg + r∗eg ρ∗eg , where ρeg is the off-diagonal element of the density matrix ρ =|ΨihΨ|. The electric field operator is given by E(r, t) = E0 cos(ωt − k · r) and thedefinition of the Rabi frequency isΩ≡−eE0
he|r|gi~ | {z }

(2.4)

≡X12

where we have defined X12 ≡ he|r|gi. The force of equation 2.3 can then be
written as


∂Ω ∗
∂Ω∗
(2.5)
Fz = ~
ρ +
ρeg
∂z eg
∂z

2.1 Laser cooling and trapping

13

where in obtaining equation 2.5, the rotating-wave approximation (RWA), whichassumes a near resonant field such that δ ≡ ωL − ω0  ωL + ω0 , has been used.
Splitting the derivative of the Rabi frequency into its real and imaginary parts
according to∂Ω= (qr + iqi )Ω(2.6)∂z
reduces equation 2.5 to
Fz = ~qr (Ωρ∗eg + Ω∗ ρeg ) + i~qi (Ωρ∗12 − Ω∗ ρ12 )
~s
=
1+s



1
−δqr + Γqi
2


(2.7)

where δ = ω − ω0 is the detuning of the field from the transition frequency andin going from the first to the second lines of 2.7 we have inserted the expressionfor ρeg obtained from the steady-state solutions of the optical Bloch equations(OBE):
iΩ
ρ12 =
(2.8)
2(Γ/2 − iδ)(1 + s)
in which the saturation parameter, s, is given by
s≡
with
s0 ≡

2Ω2
I
=
2
Γ
IS

s0
1 + (2δ/Γ)2

and

(2.9)

IS ≡

πhc
3λ30 τ

(2.10)

with c the speed of light, λ0 the resonance wavelength and τ = Γ−1 the transition
lifetime. The above expression for the saturation intensity, IS , can be foundusing Einstein’s A and B coefficients and the intensity, I is related to the peakfield value E0 via: I = 0 cE02 /2. Within the RWA, the field can be written as
E(r, t) = E0 exp(−i(ωt − k · r)). Inserting this expression into equation 2.6 andusing the definition 2.4, we obtain∂Ωe∂E0 −i(ωt−k·r)
e
= X12
e
+ i kz X12 e−i(ωt−k·r)
∂z
~
∂z
~
=

1 ∂s0
Ω + i |{z}k Ω
2s0 ∂z| {z }
qi(2.11)

qr

where we have also made use of equation 2.10. Inserting these values of qr and qi

14

Theoretical backgroundinto equation 2.7, yields the total force, which can be generalized to three spatial
dimensions by replacing the partial derivative with a gradient. This results inthe following expression for the total force
Scattering rate: Rscatt =Γρ22

z}|{∇(I/IS )
I/IS
~δ
Γ
F=− ·
+ ~k ·
2 1 + I/IS + (2δ/Γ)2
2 1 + I/IS + (2δ/Γ)2
|
{z
} |
{z
}
Scattering Force: ∝ I

Dipole force: ∝ ∇I

= Fdip + Fscatt(2.12)

The first term on the right-hand side of equation 2.12 is the dipole force. It is
proportional to the gradient of the field intensity and is the result of the refractionof light by the atom. For large detunings, such that |δ|  Γ, the dipole force
reduces to the derivative of the light shift due to the a.c. Stark shift obtainedusing time-dependent perturbation theory
Fdip = −∇Udip ' −∇

~Ω2
4δ


(2.13)In a non-homogeneous field, the difference in intensity across the atom results ina net force, whose direction is dictated by the field’s detuning from the atomic
resonance, as can be seen in equation 2.12: a red-detuned field (δ < 0) gives riseto an ‘attractive’ force in the direction of highest intensity, while a blue-detuned
field (δ > 0) ‘repels’ atoms towards the region of lowest intensity. The dipole
force vanishes on resonance (δ = 0).
The second term in equation 2.12 is the scattering force. It arises from absorptionof photons from the incident field and its form can equally be derived by considering that the force is equal to the rate of change of momentum. The momentum
imparted by a single photon of wavevector k is p = ~k and the rate at whichsuch photons are absorbed is given by Rscatt = Γρ22 where ρ22 is the excited-state
population, given by ρ22 = (1 − w)/2, with w the third component of the Blochvector whose steady state value for a two-level system is1
 


u
Ωδ
1
 


v  = 2 ΩΓ/2 δ + Ω2 /2 + Γ2 /4
δ 2 + Γ2 /4w
1

this result assumes the RWA.

(2.14)

2.1 Laser cooling and trapping

15

The force is simply equal to the momentum transfer times the scattering rate:
Fscatt = ~kRscatt , which reproduces the second term in equation 2.12.
The following two subsections discuss the use of the scattering force for laser cooling and trapping of atoms. Broadly speaking, trapping in position-space requires
a spatially-dependent restoring force and trapping in momentum-space requires
a velocity-dependent restoring force. In a MOT, these are generated by engineering the space- and velocity-dependence of quasi-resonant photon scatteringfrom an incident laser beam. Below we first discuss the case of a purely velocitydependent force (‘Doppler cooling’), followed by the case of combined velocityand spatially-dependent force (‘magneto-optical trapping’).

2.1.2 Doppler cooling
As discussed above, laser cooling relies on the momentum imparted to the atom
during repeated stimulated-absorption/spontaneous emission cycles. Each absorption event results in the transfer of the photon’s linear momentum, ~k, energy ~ω and angular momentum ~, to the atom, together with its promotion toa higher-lying energy state. The recoil from the subsequent spontaneous photon
emission occurs isotropically and hence averages to zero over many absorptionemission cycles. As a result, a net force acts on the atom in the direction of thelight propagation and with a magnitude given by the single-photon momentum
recoil, ~k, multiplied by the photon scattering rate, Rscatt , as given by equation
2.12. The dipole force is negligible compared to the scattering force in this casesince relatively small detunings are used, δ ∼ Γ and the characteristic distanceover which the beam intensity varies is large compared with the wavelength (i.e.
kΓI  δ ∇I). The velocity-dependence of this force arises from the Doppler effectwhich causes an atom in motion with velocity v to experience a laser detuning
given by k · v. The effect, which has also given its name to this cooling method,
therefore results in a combined detuning of±δDopp= δ ∓ kv

(2.15)

when the atoms are moving towards or away from the incident laser beam withvelocity v, respectively. In order to see how the scattering force of equation 2.12
can be used to cool atoms, consider now two counter-propagating laser beams
of intensity I and frequency ω. Each of these beams exerts a force on an atom
moving with velocity v according to equation 2.12 using the detunings of equation

16

Theoretical background

2.15, resulting in a total force given by
FTOT (v) = F+ + F−
Γ I
= ~k
2 IS

"

1

1

2 −
2
+
−
1 + I/IS + 2δDopp/Γ
1 + I/IS + 2δDopp/Γ#(2.16)

where the relative minus sign comes from the counterpropagating wavevectors.
To see the velocity dependence more clearly, we consider the small-velocity limitv  δ/k and expand equation 2.16 to first order in v:
FTOT (v) ≈4~k 2 (I/IS ) (2δ/Γ)v ≡ −βv[1 + I/IS + (2δ/Γ)2 ]2(2.17)

When the laser is red-detuned (δ < 0), this has the form of a viscous damping
force, acting to oppose the atom’s motion and hence slowing it down, with theanalogy of movement through a viscous medium leading to the adoption of thename optical molasses for this beam configuration. Figure 2.1 shows the form of
60

Force/mass (103 ms-2)

50
40
30
20
10
0
-10
-20
-30
-40
-50
-60

-20

-10

0

Velocity (ms )
-1

10

20

Figure 2.1: Acceleration due to scattering force in a 1D MOT plotted against
atomic velocity. Curves are plotted for 87 Rb with I/IS = 1 and for different
laser detunings. From the innermost to the outermost curves, the detunings are:-0.1Γ,-0.3Γ,-0.5Γ,-Γ,-2Γ,-3Γ. The red dashed lines show the separate componentsof the force arising from each of the two counter-propagating beams for the case ofδ = −Γ.

2.1 Laser cooling and trapping

17

this force as a function of atom velocity for various values of the (zero-velocity)
beam detuning, δ. The molasses laser configuration can be extended in orderto provide a viscous force along all three spatial directions by adding two morecounter-propagating beam pairs, orthogonal to each other and to the first pair(i.e. six beams in total).

The Doppler limit
The Doppler limit represents the point at which Doppler cooling is balanced by
the heating resulting from the discrete recoil events associated with the absorption
and spontaneous emission from the laser beams. The light field of a single beamtransfers an energy of 2~ωr to an atom with each scattering event, where ~ωr =~2 k 2 /2m is the recoil energy, and does so at a rate of 2Rscatt , with the factor of two
accounting for the two counter-propagating beams and assuming no saturation of
the transition (i.e. I/IS  1). The heating rate is therefore 4~ωr Rscatt . Equating
this to the cooling rate FTOT · vx , with FTOT given by equation 2.16, gives thesteady-state kinetic energy~Γ1 2
mvx = −
2
8



Γ
2δ
+
Γ
2δ


(2.18)

Equating this to 12 kB T and minimizing with respect to the detuning, δ, yields the
Doppler temperature~ΓTD =(2.19)
2kB
obtained when δ = −Γ/2. The Doppler temperatures for the transitions usedin the cooling of 87 Rb and 39 K (see subsection 2.1.4) are 146µK and 145µK,
prespectively, corresponding to velocities, vD = kB TD /m, of 12cm/s and 18cm/s.
Sub-Doppler cooling and the recoil limit
Temperatures below the Doppler limit can be obtained in optical molasses [93],most notably via the mechanism known as Sisyphus cooling [94, 95, 96, 97]. SubDoppler cooling relies on the existence of magnetic sublevels in the atomic groundstate1 , which were omitted from the discussion above. Figure 2.2 illustrates thismechanism in a simplified system consisting of a J = 1/2 ground state and a
J 0 = 3/2 excited state.
1

For example the mF sublevels of the hyperfine states - see subsection 2.1.4.

18

Theoretical backgroundJ’=3/2!

MJ=+1/2!
J=1/2!
MJ=-1/2!

!
!

!

!

+

!

!

+

!+
x!
lin!

0

!
8

lin!

!
4

3!
8

lin!

!
2

5!
8

lin!

3!
4

7!
8

lin!

!

Figure 2.2: Illustration of the polarization-gradient (Sisyphus) cooling mechanismin the lin ⊥ lin configuration. An atom in the MJ = +1/2 state at x = 0 moves
along its potential curve until it reaches x = λ/4, at which the polarization isσ− circularly polarized. The atom is excited to the |J 0 = 3/2, MJ = −1/2i stateand decays preferentially down to the |J = 1/2, MJ = −1/2i state via spontaneousemission before the atom has moved a significant distance. The atom in the MJ =−1/2 state finds itself in a potential minimum and loses energy as it scales thepotential landscape until it gets pumped back into MJ = 1/2 at x = λ/2, where
the polarization is σ + -polarized.

In brief, the periodic polarization gradients that arise from the interference ofcounter-propagating MOT beams of different polarizations1 cause a spatiallyperiodic modulation of the energies of the magnetic sublevels. This periodic
potential is out of phase for different sublevels, with the maxima of one level
coinciding with the minima of another. There follows a process of continuous
back-and-forth optical pumping between the sublevels, with atoms at a potential peak absorbing a circularly-polarized photon and spontaneously emitting ahigher-energy photon in order to decay to a lower-lying sublevel. The atom inthis lower sublevel then continues moving until it reaches the next maximum in
its potential and gets transferred back to the original state, which is now thelower-lying of the two. The name given to this mechanism refers to the fact thatthe atom continually finds itself ascending the potential landscape, only to be
1

Most examples consider two beams linearly-polarized along orthogonal directions (the lin⊥ lin configuration) but the effect is also present for two circularly-polarized beams of
opposing handedness (the σ+ − σ− configuration).

2.1 Laser cooling and trapping

19

quasi-instantaneously transferred to a neighbouring potential minimum in orderto repeat the process.
The limit of this mechanism is reached when the atom’s energy loss in beingtransferred to the potential minimum, proportional to I/ |δ|1 , is balanced by therecoil energy it acquires during spontaneous emission. The lowest temperatureachievable in Sisyphus cooling is thus reached when this potential depth, which
is proportional to the light intensity (U0 ∝ I/|δ|), becomes equal to the kinetic
energy imparted to an atom by the spontaneous emission of a photon, the socalled recoil energy, giving(~k)2(2.20)
Trec =
2mkB
The recoil temperatures for 87 Rb and 39 K are 180nK and 415nK, respectively.
2
The corresponding recoil velocities are simply found from kB Trec = 12 mvrec
andare equal to 5.88mm/s and 1.34cm/s, respectively, although in practice the lightintensity cannot be reduced indefinitely and attainable temperatures are limited
to several tens of recoil temperatures [98].
Since the velocity capture range of the sub-Doppler cooling mechanisms is significantly narrower than that of Doppler cooling, laser cooling sequences are usuallydivided into two distinct stages. A typical laser cooling sequence begins with a
Doppler cooling stage, during which the detuning is on the order of the linewidth(δ ∼ −Γ) and the intensity is on the order of IS . This provides a strong radiation
force for initial cooling and brings the temperature into the capture range for subDoppler cooling. The detuning is then taken even further from resonance and theintensity is decreased in order to obtain a temperature below the Doppler limit.
This strategy requires some modification in the case of a manifold of excitedstates to which transitions may occur during optical molasses, and furthermore
is very sensitive to the proximity of these transitions to each other [99]. This isdiscussed in the context of 39 K in subsection 2.1.4 below while chapter 3 describes
the sequence we employ in further detail.

2.1.3 Magneto-optical trapping
Although optical molasses can cool the atomic cloud, they cannot prevent the
diffusion of atoms from the region in which the beams overlap and hence do notconstitute a trap, for which a spatially-dependent restoring force is required. In
1

See sections 2.1.1 and 2.5 for discussion of optical potentials.

20

Theoretical backgrounda magneto-optical trap (MOT), this spatial dependence is established by making
the laser detuning a function of the atom’s position. This is accomplished by
exploiting the Zeeman shifts of the atom’s different magnetic sublevels in conjunction with the optical selection rules.
|B|!

E!

|B|!

mJ=1!
0!
-1!

!

!+

!!

J=1!

!!
x!

J=0!

Figure 2.3: Illustration of the operating principle of a MOT in a J = 0, 1 system.
A magnetic field gradient in conjunction with circularly polarized beams adds aspatial dependence to the scattering force used in the molasses configuration.

Figure 2.3 illustrates the MOT mechanism in a one-dimensional two-level system
consisting of a J = 0 ground state and a J = 1 excited state, split into its three
Zeeman sublevels, mJ = −1, 0, +1, by a linearly varying magnetic field. The
two counterpropagating beams are circularly polarized in opposite senses andred-detuned from the atomic resonance frequency. The circular polarization ofeach beam is chosen such that the only transition it can induce (∆mJ = ±1 forσ + /σ − , respectively) is brought closer to resonance by the Zeeman shift wheneveran atom moves away from the trap centre towards the beam. Helpfully, thisZeeman shift simultaneously ensures that the transition induced by the beam of
opposite circular polarization is shifted further away from resonance and hence
does not accelerate the atom along its direction of motion. Figure 2.4 shows themost common MOT configuration, consisting of the three orthogonal, counterpropagating optical molasses beam pairs, intersecting midway between a pair ofanti-Helmholtz magnetic coils. The coils produce a magnetic field with a zero at
the point midway along their mutual axis and increasing linearly with position
in all directions such that its value is given by B = B 0 (x, y, −2z), where B 0
is the field gradient. See appendix E for a more detailed description of the field
produced by this configuration. The total detuning of each beam from the relevant2.1 Laser cooling and trapping

21

transition is now given by±δMOT= δ ∓ k · v ± (ge me − gg mg )µB B 0x ≡ δ ∓ k · v ± αx~(2.21)

where the third term is the detuning due to the Zeeman shift, U = −µ · B,
between the two states (see section 2.3 below) and µ is the atom’s magnetic
moment (recall: µB h ∼=1.4 MHz/G). The subscripts ‘g’ and ‘e’ represent thelower and higher states in the transition, respectively.

!+

!+

!!

!+

!!

+z!
+y!
+x!

!!

Figure 2.4: Relation between the currents generating the quadrupole field andthe beam polarizations in a MOT. The black arrows indicate current direction andthe notation on each beam indicates the type of transition it drives in the trapped
atoms (σ +/− ). For this configuration, all the beams in the x and y directions areleft-circularly polarized when viewed along their propagation direction (i.e. theirpolarization vector rotates anti-clockwise), whereas the beams in the z direction
are right-circularly polarized.

The total scattering force now acquires a position-dependence in addition to
the Doppler cooling velocity-dependence F (v) → F (x, v). Repeating the lowvelocity expansion of equation 2.16 but using the detunings of equation 2.21, andalso assuming a weak magnetic field, i.e., v  δ/k and B  ~δ/µB yields
4~k (I/IS ) (2δ/Γ)F (x, v) = 2 (kv + αx) ≡ −βv − κx
1 + I/IS + (2δ/Γ)2(2.22)

22

Theoretical backgroundwhere the spring constant κ = αβ/k. The atom’s equation of motion is therefore2ẍ + γMOT ẋ + ωMOT
x=0

(2.23)

This is the equation of a damped harmonic oscillator in which γMOT = β/mp
κ/m are the damping coefficient and effective oscillation freand ωMOT =quency, respectively, with m the atomic mass. For typical MOT parameters(δ = −Γ, I/IS = 1, B 0 = 10Gcm−1 ), oscillation frequencies are on the order ofa few kHz while damping rates are on the order of tens of kHz. The damping√
ratio ζ = β/(2 mκ) is therefore much greater than one and the motion is overdamped. Any displacement of the atom within the MOT will (exponentially)
decay towards the trap centre without oscillating [100, 101].
We note that in general, the presence of the MOT’s magnetic field gradient greatly
complicates the mechanisms normally responsible for sub-Doppler cooling due tothe differential Zeeman shifts imparted to each of the magnetic sublevels. This canbe qualitatively be viewed as resulting from the competition between the Larmor
precession about the external field, and the optical pumping process which tends
to align the atomic dipole with the orientation of the electromagnetic field [102].

2.1.4 Laser cooling of 39 K and 87 Rb
The basic theory of laser cooling is formulated for a two-level system, and while
real atoms invariably possess a more complicated electronic level structure, thesingle valence electron of the alkali metals gives rise to a relatively simple level
hierarchy which makes this family particularly convenient cooling targets. Indeed only recently have non-alkali atomic species such as Calcium [103], Magnesium [104], Strontium [103, 105], Argon [106], Krypton [106], Ytterbium [107],
Dysprosium [108, 109] and others been used in laser cooling and degenerate-gas
experiments.
Figure 2.5 shows part of the the hyperfine energy level structure for the bosonicspecies used in our experiments, 87 Rb and 39 K. More specifically, we focus on theD2 transition between the |L = 0, J = 1/2i ground state and the |L = 1, J = 3/2i
excited state manifolds. The natural linewidths of the D2 lines in both 87 Rb and39
K are approximately Γ = 2π × 6MHz, i.e. an excited state lifetime of 26ns.
In both species, the ground state is split by the hyperfine interaction into |F = 1i
and |F = 2i hyperfine states and the excited state (52 P3/2 for 87 Rb and 42 P3/2 for2.1 Laser cooling and trapping

23
F’=3!

87Rb!

(gF=2/3)!

(gF=2/3)!

21.2MHz!

267MHz!

52P3/2!

F’=3!

39K!

F’=2!

42P3/2!
F’=2!

(gF=2/3)!

157MHz!

(gF=2/3)!

9.4MHz!

F’=1!

F’=1!

(gF=2/3)!

72MHz!

(gF=2/3)!

3.3MHz!

F’=0!

F’=0!

F=2!

F=2!

(gF=1/2)!

52S1/2!

(gF=1/2)!

42S1/2!

6.834GHz!

F=1!

(gF=-1/2)!

repump!

repump!

cooling!

766.7nm!

cooling!

780.24nm!

462MHz!

F=1!

(gF=-1/2)!

Figure 2.5: Hyperfine structure of the D2 -line states in 87 Rb and 39 K. The total
angular momentum of each state, F = I + J, is shown together with its Landég-factor, gF . Red (blue) arrows indicate the cooling (repump) frequencies. The
precise values of the detunings used in our experiments are given in table 3.2.

K) into |F 0 = 1, 2, 3i states. In our dual-species MOT, the role of the two-levelsystem for both species is played by the |F = 2i and |F 0 = 3i hyperfine states,
and it is the transition |F = 2i → |F 0 = 3i that constitutes the cycling or cooling transition used during laser cooling. The saturation intensities for the D2
transitions in 87 Rb and 39 K are 1.67 mW cm−2 [110] and 1.75 mW cm−2 [111],
respectively. Chapter 3 contains a detailed description of our experimental parameters, while appendix A contains a more detailed energy-level diagram for thetwo species.

39

The selection rule ∆F = 0, ±1 implies that decay from |F 0 = 3i can only occur tothe |F = 2i state. However, the finite linewidth of all the hyperfine levels together
with the required detuning of the cooling beam mean that occasional excitationof atoms occurs from the |F = 2i to the |F 0 = 2i state. Decay from |F 0 = 2i to|F = 1i is then allowed and as all the atoms gradually accumulate in the |F = 1i
state they are removed from the cycling transition and are no longer cooled. This

24

Theoretical backgroundpumping is countered by the so-called repump beam, which is resonantly tuned
to the |F = 1i → |F 0 = 2i transition. Atoms ‘repumped’ to |F 0 = 2i then haveanother chance at decaying to |F = 2i and hence back onto the cooling cycle.
The cooling and repump transition are indicated in figure 2.5 as red and blue
arrows, respectively.
The situation in 39 K is modified by the narrow, 34MHz, overall splitting of theexcited-state hyperfine manifold [49]. This unresolved level structure has two
main consequences for the laser cooling process:1. Firstly, the unresolved excited-state gives rise to much stronger pumping
into the |F = 1i ground state since the probability of |F = 2i → |F 0 = 1, 2itransitions is greatly increased. This requires the use of repump light ofsimilar intensity to the cooling light in order to prevent accumulation of
the atoms in the |F = 1i state. This pumping is so frequent that the contribution of the repump beam to the cooling process becomes comparable to that of the cooling beam. The functional distinction between thetwo is effectively blurred and the use of their original names is simply amatter of convention. In addition, use of the same detunings as used forother species (δ ∼ Γ), results in heating due to the blue-detuning of thebeams from the nearby hyperfine states. For this reason, it has been found[47, 112, 49, 48, 113, 59] that the largest capture velocity is obtained withboth the cooling and repump beams red-detuned from the entire excitedstate manifold, as shown in figure 2.5.
2. Although this large detuning of both beams yields a large capture velocity1 ,it does not produce an effective sub-Doppler cooling force [49] and undernormal operating conditions, the temperatures achieved during optical molasses are on the order of 1mK. In general, sub-Doppler cooling mechanisms
are only efficient when the excited-state hyperfine splitting is much larger
than the natural linewidth Γ, as in Rb, Na and Cs, or when it is smaller
than Γ, as in Sr [114, 105]. When it is on the order of Γ, as is the case for39
K and 41 K, sub-Doppler cooling is impeded by heating forces or resonant
photon reabsorption [99]. It has, however, been shown both theoretically[49, 99] and experimentally [115, 116] that sub-Doppler cooling of theseisotopes is made possible by a suitable modification of the usual coolingstrategy. Both [115] and [116] employ an initial Doppler cooling stage withboth beams detuned from the entire excited-state hyperfine manifold. This
1

Found experimentally in [47, 112] to exceed 30m/s.

2.2 Optical pumping

25

is followed by a molasses stage that in general involves a significant reduction in the cooling beam’s detuning and intensity, together with an evenlarger reduction in the detuning and intensity of the repump beam. With
their respective strategies, [115] and [116] report temperatures of 34µK and
25 ± 3µK, respectively, for their 39 K clouds.

2.2 Optical pumping
At the end of the MOT and molasses stages, the atoms are distributed between
the various |F = 2i Zeeman sublevels - magnetically trappable and untrappable
alike - and must be transferred into the |2, 2i state prior to application of thequadrupole field in order to trap the highest possible atom number. In orderto accomplish this, a uniform magnetic guide field is initially applied in order todefine a quantization axis. It is important to ensure that this quantization field is
not strong enough to take the magnetic sublevels out of resonance with the lasers.
We use a coil pair aligned along the propagation direction of the pumping beamin order to produce a field of several Gauss at the location of the atoms. The
atoms are then illuminated with a beam resonant with the |F = 2i → |F 0 = 2itransition and σ + -circularly-polarized with respect to the quantization axis. The|F = 1i → |F 0 = 2i repump light also remains on during the first half of thisprocess.
Optical pumping relies on the fact that transitions driven by σ + -polarized lightmust satisfy ∆mF = +1. i.e., in our case |2, mF i → |2, mF + 1i. From thisexcited F 0 = 2 state the atom can spontaneously decay to one of the groundstates while obeying ∆mF = 0, ±1, and with a probability determined by the appropriate Clebsch-Gordan coefficients. The presence of the repump light ensuresthat any atoms decaying to the F = 1 ground state are transferred back into
F 0 = 2 and kept on the pumping cycle. After several absorption/emission cycles,
the atoms accumulate in the |2, 2i state, labelled ‘stretched’ since mF is at its
maximal value and ‘dark’ since, for lack of an |mF + 1i state, it can no longer
interact with the pumping light.
Transfer of all the atoms into the |2, 2i state is not only important for maximizing the number of initially trapped atoms in the magnetic trap, but also forlimiting subsequent trap-loss collisions which reduce numbers, and cause heating.
These processes are even more critical in our case since we rely on interspecies
87
Rb-39 K collisions for the sympathetic cooling of 39 K. If a significant fraction of

26

Theoretical backgroundthese collisions are inelastic, sympathetic cooling will not be efficient enough tosufficiently increase the 39 K phase-space density and could prevent condensationof the sample. Indeed, extra effort is made during evaporative cooling to actively
remove any |2, 1i atoms, whether initially present or subsequently created, fromthe trap (see section 3.9).

The inelastic collisions most likely to cause losses from our trap are so called
hyperfine-changing collisions (HCC) of the type Rb |2, 1i + K |2, 2i → Rb |1, 1i +K |2, 2i+KE , in which the 87 Rb ground state hyperfine energy, 6.8GHz, is released
and divided between the kinetic energies of the two emerging atoms. The |1, 1i
atom is anti-trapped and the |2, 2i atom is also ejected if its final kinetic energy
exceeds the trap depth. The same process with the species reversed can also occur
but presents a lesser threat since (a) there are far fewer 39 K atoms present and(b) the 39 K ground state hyperfine splitting, 462MHz, is much smaller than thatof 87 Rb (although still larger than the trap depth). The process |2, 1i + |2, 1i →|2, 2i + |2, 0i can also occur, leading to further losses. By contrast, atoms instretched states do not experience such spin-exchange collisions since the total
angular momentum projection ΣmF , must be conserved in these collisions.

In addition to maximizing the number of trapped atoms and suppressing trap
losses and heating, atoms in the |2, 2i state experience the tightest magnetic
confinement and hence minimal contact with the transfer tube walls as well as
the highest elastic collision rate for efficient evaporative cooling.

2.3 Magnetic trapping
Magnetic traps are used in order to provide a conservative potential which does
not rely on photon absorption and hence enables atoms to ultimately be cooled totemperatures below the recoil limit. Our atoms posses a non-zero magnetic dipole
moment and hence experience a force in the presence of an external magnetic field
gradient, which can therefore be used to generate a suitable trapping potential.
We capture our atoms in a purely magnetic trap generated by our quadrupole
coils prior to transporting them to the science cell.

2.3 Magnetic trapping27

2.3.1 Energy shift in an external magnetic field
The energy shift experienced by an atom due to a static external magnetic field,
B, is given byµB(gS Sz + gL Lz + gI Iz ) Bz
(2.24)
HB =~where Bz is the component of the field along the z (quantization) axis, (Sz , Lz , Iz )are the operators for the z-components of the spin, orbital and nuclear angularmomenta, respectively, and (gS , gL , gI ) are the corresponding Landé g-factors. If
this energy shift is small compared to the hyperfine splitting
Hhfs = Ahfs I · J⇒

1Ehfs = Ahfs [F (F + 1) − I(I + 1) − J(J + 1)] (2.25)
2arising from the coupling between the electronic and nuclear angular momenta,
then F = I + J and mf are good quantum numbers. Ahfs is the hyperfine
structure constant, equal to Ahfs = h · 3.417GHz and h · 230.86MHz for 87 Rb and39
K, respectively. Equation 2.24 can then be written as
HB = µB gF Fz Bz(2.26)

where gF is the relevant Landé g-factor. The resulting Zeeman energy shift issimply linear in the field strength
U (r)|F,mF i = −µ · B (r) = µB gF mF Bz(2.27)

The force on an atom in such a potential is thenF (r) = −∇U (r) = −µB gF mF ∇Bz(2.28)

From equation 2.27, it follows that any states for which gF mF > 0 can lower theirpotential energy by moving to regions of lower magnetic field. These states aremagnetically trappable in a local field minimum and are hence known as low-fieldseeking states. The Landé factors of the ground states of both 87 Rb and 39 K aregF =1 = −1/2 and gF =2 = 1/2, and hence the trappable |F, mF i states of both
species are |1, −1i, |2, 1i and |2, 2i.
When the external-field and hyperfine shifts are comparable, however, the full
Hamiltonian H = Hhfs + HB has to be diagonalized. An analytical result exists
for the energy shift of the states belonging to the fine-structure ground state

28

Theoretical background

manifold (i.e. L = 0, J = 1/2) in the form of the Breit-Rabi formula [117]
∆Ehfs
∆Ehfs
E|J=1/2mJ ImI i = −+ gI µB mB ±2 (2I + 1)
2

1/2

4mx
2
(2.29)
1++x
2I + 1In this formula ∆Ehfs = Ahfs (I + 1/2) is the hyperfine splitting, m = mI ± mJ =mI ± 1/2, B = |B| is the magnitude of the magnetic field and x is given byx=(gJ − gI ) µB B
∆Ehfs

(2.30)For both 87 Rb and 39 K, I = 3/2. The energy shifts for the D-line ground stateof 39 K are shown in figure 2.6.
0.06

F,mF

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

0.04

E\kB

0.02
0.00
-0.02
2,-2
1,-1
1, 0
1, 1

-0.04
0

100

200

300

B[G]

400

500

600

Figure 2.6: Hyperfine structure in the 42 S1/2 ground state of 39 K in a magnetic
field, calculated using the Breit-Rabi formula 2.29. States are labeled with their
low-field quantum numbers |F, mF i. The dashed line shows the maximal field,
82.4G, at which the |1, −1i state can be magnetically trapped. The maximal trap
depth for atoms in this state is roughly 1.48mK.

We make use of the Breit-Rabi formula, equation 2.29, in our experiments forthe calibration of the magnetic field used to access the Feshbach resonance at402.5G. This is done via RF spectroscopy on the transition between two magnetic
sublevels in 87 Rb, the precise separation of which is obtained using the Breit-Rabi
formula. Our procedure for doing this is described in subsection 3.11.3.
The depth of a magnetic trap is given by U = µ∆B, where ∆B is the difference2.3 Magnetic trapping

29

between the field maximum and its minimal value. This depth can be expressedin units of temperature as U = µB gF mF ∆B/kB . For example, neglecting gravitational effects, 87 Rb and 39 K atoms in the |2, 2i state experience a trap of depth
U = µB ∆B/kB ∼= (0.7mK/G) × ∆B. Chapter 3 contains the details of themagnetic trap configurations used during the transport and sympathetic coolingstages of our experiments.

2.3.2 Non-adiabatic losses from magnetic traps
The atom’s magnetic dipole moment precesses about the local external magnetic
field at the Larmor frequency, ωL = µB/~, and as long as the external field direction as experienced by the atom changes slowly with respect to ωL the moment
will adiabatically follow the field direction an hence remain in its trappable state
relative to the instantaneous magnetic field. This criterion can be written as
1 |v · ∇B|
1ωL
B(2.31)

i.e. that the change in field direction experienced by the atom during one precession about the field is much smaller than 2π. However from criterion 2.31, we
notice an important problem: In regions of very low magnetic field, the adiabaticity criterion can be violated. Atoms of sufficient temperature will experience too
rapidly changing a field and will transition to a different mF state, most likely an
untrapped one (gF mF < 0), and will hence be lost from the trap. These transitions are known as Majorana spin flips [118, 119] and especially plague traps withmagnetic field zeros, at which atoms of arbitrarily low velocity are transferred tountrapped states and lost from the trap. Since the magnetic zero is also theplace where the coldest atoms congregate, Majorana spin flips also cause acuteheating of the atomic cloud and ultimately prevent the sample from reaching thetemperatures required for entering the quantum degenerate regime [120, 121].
The quadrupole magnetic field produced by our MOT coils contains a magnetic
zero at its origin and is therefore susceptible to such losses, rendering it unsuitable for sustained evaporative cooling. In order to address this problem, after
transport of the atoms to the science cell in the quadrupole field the trap is modified so as to remove the zero-field region. This trap configuration consists of aharmonic potential whose minimum is offset from zero. More details about thisQUIC (Quadrupole-Ioffe configuration) trap can be found in section 3.8.

30Theoretical background

2.4 Evaporative and sympathetic cooling
2.4.1 Evaporative coolingLaser cooling as described in section 2.1 produces atom samples with temperatures below the Doppler limit but still significantly above the recoil limit. Typical
MOT phase-space densities are on the order of ρ ∼ 10−6 and it was soon realisedthat a novel cooling technique was required in order to achieve the higher phasespace densities (ρ ∼ 1) required for the onset of BEC. This led to the proposalof evaporative cooling [20] as a means of raising the phase-space density and wasa key milestone on the route to the first experimental realizations of BEC. The
first use of evaporative cooling was for cooling hydrogen atoms in 1988 [21].

f(E)

f(E)

f(E)

Evaporative cooling relies on the selective removal from the trap of atoms withhigher-than-average kinetic energy. Rethermalization of the gas via elastic twobody collisions, leaves the remaining atoms at a lower equilibrium temperature(see figure 2.7). The gradual removal of atoms with ever decreasing energies
enables an arbitrary reduction of the temperature, limited in principle only by
the initial number of trapped atoms. The current lowest reported temperature
obtained via evaporative cooling is 450 ± 80pK, achieved by the MIT group [122].

T1

T2<T1
E

E

E

Figure 2.7: Principle of evaporative cooling. Lowering the trap depth removes
atoms from the high-energy tail of the thermal distribution characterised by temperature T1 (bottom left), after which the remaining gas re-equilibrates via elastic
collisions at a lower temperature, T2 < T1 (bottom right).

2.4 Evaporative and sympathetic cooling31

2.4.2 RF and microwave evaporation
The most common technique for selective removal of high-energy trapped atoms
relies on the spatially-varying Zeeman shift experienced by the atoms in themagnetic trap. Atoms with higher kinetic energies will be located further out
in the magnetic potential according to 21 mv¯2 ≈ gF µB mF B. High-energy atoms
can therefore be selectively transferred to untrapped states by the application ofresonant oscillating magnetic field. The most commonly used transitions used in
evaporative cooling are the RF transitions between the magnetic sublevels of thehyperfine state in which the atoms are confined.

E!

mF!

F, mF ;n2!

2, 2;n2, !2;n + 4

1!

2,1;n

0!

2, !1;n + 2

F=2!

-1!
2, !2;n-2!

!! HF = 6.834GHz!! MW = !! HF + 3!! RFX Axis

1!
0!

F=1!

-1!

r!
Figure 2.8: Evaporative cooling in a harmonic potential. Radio-frequency radiation tuned to the ∆mF = 1 resonence frequency induces a series of four singlephoton transitions to the untrapped |2, −2i state. Within the dressed atom picture,
The asymptotic state|2, 2; ni (where n is the number of photons) is adiabaticallyconnected to the untrapped |2, 2; n + 4i state by an effective four-photon transition. The hyperfine evaporation transition |2, 2i → |1, 1i is indicated by the orange
arrow. The dashed orange arrow shows the route through which |1, 1i atoms canreach the |2, 1i state.

32

Theoretical background

These ∆mF = 1 transitions occur on a surface of constant magnetic field definedby the condition~ωRF = µB gF B
(RF evaporation)(2.32)For 87 Rb and 39 K atoms in the |2, 2i state this radiation, also referred to as an RF
knife, drives sequential single-photon transitions between the Zeeman sublevels,
ultimately transferring the absorbing atoms into the untrapped mF = 0 and theanti-trapped mF = −1, −2 states which are removed from the trap. This process
amounts to truncating the trap depth at the knife energy, which is normally
expressed as a multiple, η, of the cloud’s instantaneous temperature i.e.,~ωevap = ηkB T(2.33)

where η is known as the evaporation parameter. The process is illustrated, withinboth the diabatic and dressed-state [95] pictures, in figure 2.8. For 87 Rb and 39 K
atoms in the |2, 2i state for which gF = 1/2, the resonance frequency for such a∆mF = 1 transition corresponds toµB gF ∆mF
νRF=
= 1.4 MHz/GB
h(2.34)

This method is problematic, however, for use in our sympathetic cooling stage
since due to the almost identical potential experienced by the two species, as many39
K as 87 Rb atoms would be removed from the trap. In order to circumvent thisproblem, we employ evaporative cooling on the ground-state hyperfine transitionof 87 Rb. The transition frequency of νevap ≈ 6.834GHz gives this approach thename microwave (MW) evaporation. We induce transitions between the trapped|2, 2i state and the antitrapped |1, 1i state in 87 Rb by applying radiation at afrequency given by~ωMW = ~ωHF + 3~ωRF(MW evaporation)(2.35)In terms of temperature, the height of the evaporating knife in both cases is thenνRFkB=
= 0.01 MHz/µK
T
2h

(RF)

νMW
3kB
=
= 0.03 MHz/µK
T
2h

(MW)

(2.36)In experiments, an initial frequency is chosen for the knife which is subsequently

2.4 Evaporative and sympathetic cooling33

ramped down in order to ensure continual cooling of the sample. This procedure
is known as forced evaporative cooling. During microwave evaporation, atoms
transferred into the |1, 1i state can be transferred back up into into the trapped|2, 1i state by the evaporation radiation on their way out of the trap (see figure2.8). These |2, 1i atoms lead to heating and atom loss via spin-exchange collisions
and hence need to be removed during evaporation. For this reason, during ourhyperfine evaporation sequence we continuously sweep a second field in order totransfer any residual |2, 1i atoms to the untrapped |1, 0i state. See section 3.9
for more details.

Evaporation efficiency
Since this form of cooling relies on the removal of atoms, the overall efficiency ofthe process is best characterized by the gain in phase-space density, ρ, comparedto the change in atom number, N . The most commonly used figure of merit for
the evaporation efficiency is the logarithmic derivative, measuring the relative
fractional changes in these quantitiesγ=−d ln ρ
dρ/ρ
=−dN/N
d ln N(2.37)In the absence of any trap losses due to inelastic collisions the optimal rate forevaporative cooling, in the sense of yielding the largest atom number at the BEC
transition temperature, is set by the initial temperature and by the elastic collisionrate, Γel = 1/τel , which is responsible for restoring the cloud to its equilibrium
thermal distribution. However, in practice atoms are constantly lost from thetrap due to background and three-body collisions at a rate Γloss = 1/τloss andthis acts to oppose the rise in phase-space density caused by cooling. Therefore
optimal evaporative cooling, in attempting to maximize the value of γ, relies on
a trade-off between efficiency, as determined by the value of η, and cooling speed,
as determined by the rate at which the knife is swept through the cloud.
As a limiting example, in the absence of inelastic losses η could be chosen to bearbitrarily large and the sweep rate arbitrarily slow, such that after an arbitrarily
large time a single atom containing an arbitrarily large fraction of the cloud’s
kinetic energy would be evaporated from the trap, cooling the remaining atoms
to a corresponding arbitrarily low temperature. It is therefore apparent that the
number to be maximized for efficient evaporation is R = τloss /τel , the ratio of34

Theoretical background‘good’ elastic collisions to ‘bad’ inelastic collisions (i.e. the number of elastic
collisions per trapping time). This is simply stating that the atoms should reequilibrate as fast as possible compared to the rate at which they are being lostfrom the trap.
In our experiments the initial value of η is usually chosen to be around 7-8, bothfor evaporation of 87 Rb in the QUIC trap (see section 3.8) as well as for the
optical evaporation of 39 K in the CDT (see section 2.5). The evaporation speed
is then empirically found by maximizing the phase-space density at the end of
the evaporation sweep.
Runaway regime
A desirable regime to enter during evaporative cooling is that of so-called runaway
evaporation, in which the elastic collision rate
τel = σnv̄(2.38)

increases as the temperature is decreased. Here σ is the scattering cross-section,n is the density and v̄ is the mean thermal velocity. This implies that the increasein density due to the reduction in temperature more than compensates for theconcomitant reduction in thermal velocity and atom number. Within a simple
model1 , the scaling of temperature with number can be expressed as T ∝ N β
with β = (η − 3)/3. Similarly, The volume scales as V ∝ N α with α = (η − 3)/2.
Hence the scattering rate above is found to scale asσnv̄ ∝ N 1−β(2.39)

which implies that runaway cooling requires β > 1 or η > 6. By contrast, ina uniform system the decrease in temperature leads to an overall decrease in
the elastic scattering rate and hence to a drop in the evaporation efficiency. In
general, the dependence of the density in a power-law trap with exponent ν isgiven by−Ar ν
(2.40)
n(r) = n0 e kB T
which leads to a temperature dependence of n0 ∝ T −3/ν . This implies that in a
linear potential (ν = 1) the runaway regime is entered for a lower value of theratio R and evaporative cooling is hence more efficient. See [123, 22, 124] for
1

Found by solving: 3kB (N + dN )(T + dT ) = 3N kB T + ηkB T dN

2.4 Evaporative and sympathetic cooling35

a more detailed discussion. This benefit of the linear potential is harnessed inour experiments with an optically-plugged quadrupole trap, as described in appendix B. Another experimental constraint on the evaporative cooling procedureis the requirement that the finite frequency-width of the evaporation knife, due
to power-broadening, magnetic field noise, etc., be smaller that the temperatureof the remaining atoms. Careful field stabilization and choice of radiation power
throughout the evaporation ramp can mitigate these effects [125, 126, 127, 128].

2.4.3 Sympathetic cooling
Introduction and principles
There are circumstances, however, in which attempting to perform direct evaporative cooling on an atomic species is undesirable. This is the case, for example,
when the scattering length is very small or when one is working with a lowabundance species and wishes to avoid the heavy atom losses inherent to evaporative cooling1 . For fermionic species at low temperatures, s-wave collisions are
forbidden by the Pauli exclusion principle and therefore these species also require
the use of an indirect evaporation procedure [129, 26, 27, 130, 131].
The aim of sympathetic cooling is the cooling of a ‘target’ gas, in thermal contact with a ‘buffer’ gas, by forced evaporative cooling of the buffer gas. The
temperature of the target atoms is reduced through their thermalization withthe cold buffer-gas reservoir. Since the cooling of both species depends on theremoval of only one species from the trap, for a given initial buffer gas number
there is a natural tradeoff between the final size of the cooled load and the lowest
achievable temperature. The background triplet scattering length of 39 K is small
and negative, with a value of abg = −33a0 [132, 60], where a0 is the Bohr radius, a0 ≈ 0.53Å. Furthermore, the Ramsauer-Townsend minimum in the s-wave
scattering cross-section in 39 K occurs at a temperature of T ∼ 320µK, where contributions from higher partial waves are still small (σl ∝ T 2l , [133]). These facts,
together with the modest MOT numbers compared to the temperatures achievedby laser cooling, make sympathetic cooling of 39 K the most promising approachfor cooling this species2 . The effectiveness of 87 Rb as a buffer gas for 39 K has
1

2

An exception to this is the recent direct evaporative cooling of the low-abundance species
84
Sr and 86 Sr [38, 39, 40], made possible by the existence of a metastable state in which alarge number of atoms could be accumulated during laser cooling.
Despite these difficulties, the Florence group has recently reported on the condensation of39
K without a sympathetic cooling stage.

36

Theoretical backgroundalready been demonstrated in a series of experiments [59, 36]. Furthermore, theuse of an 8.5G-wide inter-species Feshbach resonance located at 317.9G in the|1, 1i states of 87 Rb and 39 K enables the enhancement of the 39 K-87 Rb scatteringcross section from its background value of aKRb = 36a0 to values on the orderof aKRb ≈ 150a0 and hence increase the efficiency of sympathetic cooling, as iscarried out in [36].

A model of sympathetic cooling
A simple and analytically soluble model for sympathetic cooling has been proposed and used by the Aspect group in 2001 [134] in order to better understandthe results of their experiments on simultaneous production of BECs in samplesof 87 Rb containing atoms in the |2, 2i and |1, −1i states. In these experiments,
the |1, −1i state (the buffer gas) was evaporated directly, since due to its smaller
magnetic moment its spatial extent in the magnetic trap was larger, while the|2, 2i component (the target gas) was cooled to degeneracy sympathetically via
elastic collisions with the |1, −1i atoms. A similar experiment with the same components had already been performed in the Wieman group in 1997 [129]. Below
we review this model and in chapter 3 we apply it to the results of our 87 Rb-39 K
sympathetic cooling measurements.
In addition to the discretization of what is actually the continuous process of
atom removal and thermal re-equilibration, this model makes some further assumptions.
1. Both species are assumed to always be thermalized with themselves andwith each other i.e., the thermalization rate is much higher than the coolingrate.
2. The gas is described by classical statistics (i.e., the classical equipartition
theorem, allocating 21 kB T to each degree of freedom, is valid)
3. The number of target atoms, N2 is taken to be constant.
4. Atoms escaping from the trap neither collide nor exchange energy with theremaining atoms.
Initially, the total energy of N1 buffer and N2 target atoms in a harmonic trap is
E = 3 (N1 + N2 ) kB T

(2.41)

2.5 Optical trapping

37

Assuming a number, dN1 , of atoms is evaporatively removed from the trap withan energy cutoff of ηkB T , the corresponding (negative) energy change is
dE = dN1 (η + κ) kB T

(2.42)

where κ ∈ [0, 1] is a function of η and of the effective dimension of the surface on
which evaporation takes place [22]. After rethermalization at a new temperature,T + dT , the total energy is
E + dE = 3 (N1 + dN1 + N2 ) kB (T + dT )(2.43)

Substituting in Eqns. 2.41 and 2.42, and keeping only first-order terms yields
dN1
dT
=α
T
N1 + N2(2.44)

with α = (η + κ) /3 − 1. Assuming evaporation at constant η, we get
T = Tmin

N1
+1
N2

α


with

Tmin = Tinit

N2
N1initα(2.45)

where Tinit and N1init are the initial values of the respective quantities and N2
has been neglected with respect to N1init in the expression for Tmin . This lastapproximation is valid since in order to reduce the temperature by several ordersof magnitude, the initial fraction of target atoms in the gas has to be small. In
our experiments its value is no larger than 1/10.

2.5 Optical trapping
In this section we briefly discuss the optical dipole force, introduced in section
2.1, which provides the dominant contribution to the total optical force when thelight field is far-detuned from resonance and the scattering force is weak. A nonmagnetic trapping potential is required for experiments utilizing Feshbach resonances, especially in our case since the |1, 1i state in which our resonance exists
is high-field seeking and therefore magnetically untrapped. Feshbach resonances
involve the application of a uniform magnetic field, as discussed in the following section, and are therefore incompatible with non-uniform magnetic trapping
potentials. In addition, the tight confinement provided by optical traps enables
more efficient evaporative cooling, which we employ following the sympathetic
cooling stage of our 39 K production sequence, as described in chapter 3.

38

Theoretical backgroundAs mentioned previously, the dipole force arises from the interaction of the atom’s
induced dipole with the intensity gradient of the light field. This is a conservative
force and hence can be related to a potential which can be used for trappingand manipulation of atoms. This can be seen more clearly by integrating theexpression for the dipole force in equation 2.12, which we repeat here for clarity:Z
Udip (r) = −Fdip · dr with Fdip = −Udip (r) =∇(I/IS )~δ
·
2 1 + I/IS + (2δ/Γ)2(2.46)~δln(1 + I/IS + (2δ/Γ)2 ) + C12=

~δln(1 + (I/IS + 1)/(2δ/Γ)2 ) + C2
2≈

~δ Γ2 I2 (2δ)2 IS=

3πc2 Γ
I(r)2ω03 δ

((δ/Γ)2  I/IS )(2.47)

where ω0 is the transition frequency. In the above, we have assumed a large
detuning and low intensity, (δ/Γ)2  I/IS , have neglected any constant factors
along the way and have used the relation IS = πhc/3λ3 τ of equation 2.10. Within
the same large-detuning approximation, the scattering rate from equation 2.12 isgiven by 2
3πc2 Γ
Rscatt (r) =I(r)(2.48)
2~ω03 δA semi-classical model can also be used to derive the optical dipole potential,
as is done for example in [135], which also contains a more exhaustive review
of optical trapping of neutral atoms. By using the expression for the complex
atomic polarizability, derived using the classical Lorentz oscillator model [136],
the resulting expressions are


Γ
Γ
3πc2
Udip (r) = − 3+I (r)2ω0 ω0 − ω ω0 + ω 3 
2
3πc2 ω
ΓΓ
Rscatt (r) =+I (r)2~ω03 ω0
ω0 − ω ω0 + ω(2.49)

which reduce to equations 2.47 and 2.48 when the field is tuned closer to resonance2.6 Feshbach resonances

39

such that |δ|  ω0 and the second terms in the above equations can be neglected.
Hence red-detuned light (δ < 0) gives rise to negative potentials, attracting atoms
to intensity maxima, while blue-detuned light (δ > 0) attracts atoms to intensity minima. Atoms can therefore be trapped in a suitably configured intensity
distribution. Since the potential depth scales as I/δ while the scattering rate
scales as I/δ 2 , dipole traps typically use high intensities and large detunings tokeep scattering as low as possible while providing a sufficiently deep potential.
A common way to create such a confining potential is in the intersection regionof two non-parallel red-detuned beams, as is done in our experiments, or using
a single red-detuned beam and relying on the finite Rayleigh range to provide
the axial confinement [137, 138]. Periodic optical lattices can also be generated
by the standing-wave interference pattern of pairs of counter-propagating red- or
blue-detuned beams [139, 135].

2.6 Feshbach resonances
As discussed in chapter 1, one of the main motivations for our use of 39 K is theprecise tuning of the interparticle interaction strength over a broad range affordedby the existence of a wide and conveniently-located Feshbach scattering resonancein the atoms’ absolute ground state. This resonance gives us access to a large
range of interaction strengths, from strongly repulsive to strongly attractive andincluding the special case of an ideal, non-interacting, gas. This external control
over the scattering length requires only the application of a suitable homogeneous
magnetic field to the atoms, as discussed below.
The following provides a description of the rudiments of scattering theory andFeshbach resonances, and is intended only to introduce the relevant quantities
and parameters which will be referred to throughout the rest of this thesis, aswell as providing references for further reading.

2.6.1 Scattering theory
When an incident particle with momentum k, in the form of a plane wave ψinc =eik·r , scatters from a spherically symmetric potential, V (r), the asymptotic formof the overall wavefunction, ψk (r) at a large distance compared to the effective

40

Theoretical background

range of the potential, is given by [140]ψk (r) ∼ eik·z + f (k, θ)eikrr(2.50)

where θ is the angle between the momentum of the incoming plane wave and thatof the scattered wave. f (k, θ) is the amplitude of the outgoing spherical wave andis called the scattering amplitude. At low energies, only partial waves with zeroangular momentum (l = 0) contribute to the outgoing wavefunction. This can beunderstood qualitatively as higher partial waves of angular momentum quantum
number l not possessing sufficient energy to cross the centrifugal energy barrier,~2 l (l + 1) / (2mr r2 ), where r is the interparticle distance, and simply being reflected from the scattering potential without sampling the short-range potential
V (r). In this limit the scattering amplitude sheds its angular dependence andapproaches a constant, −a, called the scattering length. Hence in the k → 0 limitof vanishingly-low-energy collisions, the wavefunction reduces toψ (r) = 1 −ar(2.51)and the scattering length can therefore be associated with the intercept of theasymptotic wavefunction ψ along the radial axis. Comparing equation 2.51 withthe solution of the radial Schrödinger equation allows us to relate the scattering
length, a, to the phase shift δ0 accumulated, in the same low-energy limit, by theasymptotic scattered wavefunction during the time it spends inside the effective
range of the scattering potential and gives
tan δ0k→0k

a = − lim

(2.52)The scattering cross-section, σ, can be obtained from the scattering amplitude
using
Z 1
dσ
2
= |f (θ)|⇒σ = 2π|f (θ)|2 d (cos θ)(2.53)dΩ−1which in our low-energy limit with f (θ) = −a gives
σ = 8πa2(2.54)

where a factor of 2 has been added to the result of the integration due to therequisite symmetrization of the bosonic wavefunction [140].

2.6 Feshbach resonances

41

2.6.2 Feshbach resonances
In scattering theory, open channels and closed channels refer to the collectionof quantum states in which particles are allowed or forbidden, respectively, toemerge from a scattering event by energy conservation. If Etot is the total energy
1−2of the incoming particles, defined by Etot = Eψ1 + Eψ2 + Ekin
, where Eψ1 /ψ2 are
1−2
the internal energies of the particles in states ψ1 /ψ2 and Ekin is their relative
kinetic energy, then any channel with energy E 6 Etot is called an open channel
and any channel with E > Etot is called a closed channel. A closed channel caninvolve, for example, two particles that are in different internal states to those of
the incoming particles, and the difference in energy between the two molecular
scattering potentials is then determined by the difference in the internal energiesof the two states.
A Feshbach resonance (see [42], and references therein) occurs when the energy ofan incoming open elastic channel1 is magnetically tuned so as to coincide with thatof a bound (molecular) state of an energetically closed channel. This tuning canbe accomplished when the two channels possess different total magnetic moments
and therefore experience a differential Zeeman shift in the presence of an external
magnetic field. The difference between the energy of the incoming state and thatof the closest bound state is thenE − E0 = ∆µ (B − B0 )(2.55)

where ∆µ is the difference between the magnetic moments of the two channels
and E0 and B0 are the energy and magnetic field positions of the resonance. This
scenario is illustrated in figure 2.9.
In the vicinity of a Feshbach resonance, the dependence of the s-wave scattering
length on the external magnetic field is conventionally described bya (B) = abg∆
1−
B − B0(2.56)

where abg is the background scattering length far from the resonance and the parameter ∆ is the resonance width, defined as the distance between the resonancecentre and the field, Bzc , at which the scattering length vanishes - the so called
1i.e. one in which the initial and final relative kinetic energies are equal.

42

Theoretical background

Energy!

closed-channel"
bound states!

closed channel!

Asymptotic open-channel energy!
open channel!

Interparticle"
distance!

Figure 2.9: A Feshbach scattering resonance occurs when the energy of two particles in an open collisional channel coincides with that of a bound state in a closed
molecular channel. The distance between the two molecular potentials can tuned
by an external magnetic field.

zero-crossing 1∆ ≡ Bzc − B0 ∝1
abg ∆µ(2.57)

Equation 2.56 can be intuitively understood as the result of second order perturbation theory, with the second order process describing incoming particles
coupling to an intermediate closed channel which then decays into two particles
exiting in an open channel. This intuitive understanding, together with equations 2.55, 2.56 and 2.57 can also be used to see that the positive scattering
length which arises when the energy of the incoming channel is higher than thatof the closed-channel bound state, leads to a repulsion between the states (similarly to the optical potential of section 2.5) and that a negative scattering length
correspondingly leads to an interparticle attraction when the incoming energy islower than that of the bound state.
The variation of the scattering length with magnetic field described by equation
2.56 is shown in figure 2.10 for the position and width of the resonance used inour experiments - 402.5G and 52G, respectively.
It is worth noting that in the vicinity of the zero-crossing i.e. B ≈ Bzc , the1i.e. the width characterizes the field range over which the resonance dominates the contributions of other states to the scattering length.

2.6 Feshbach resonances

43

100

resonance
B0=402.5

80

Scattering length [a0]

60

a>0

40

zero-crossing
Bzc=350.5

20
0
-20
-40

a<0

-60
-80
-100

0

100

200

300

400

B[G]

500

600

700

800

Figure 2.10: Variation of the scattering length with magnetic field in the vicinityof the 52G-wide Feshbach resonance at 402.5G used in our experiments.

scattering length is given approximately bya (B) ≈abg
(B − Bzc )∆(2.58)

Therefore when precise control of the scattering length near the zero-crossing isdesired for experiments on extremely weakly-interacting gases, a resonance with asmall background scattering length and a large resonance width is advantageous.
This is precisely the case in 39 K whose −33a0 background scattering length and
52G-wide resonance make it a natural candidate for experiments such as thoseinvestigating Anderson localization [67, 141, 142] and interferometric experiments
wishing to minimize the effects of interactions [143].
The Feshbach resonances present in the |F = 1i ground state of 39 K, as well as
several inter-species 87 Rb-39 K resonances, have been discovered and investigatedin a series of experiments by the group at LENS [59, 60, 36, 58, 62, 61]. The
techniques described in the preceding sections can be used in order to lower thetrapped cloud’s temperature into the regime of quantum degeneracy, in which
the effects of quantum statistics must be accounted for in describing the system’s
thermodynamic behaviour. The following section outlines perhaps the most wellknown manifestation of these effects in the field of ultracold atoms - the BoseEinstein condensation transition.

44

Theoretical background

2.7 Bose-Einstein Condensation
The theory of Bose-Einstein condensation in ideal and interacting gases has been
expounded, discussed and dissected in the pages of a huge collection of literature,including books [123, 144], review articles [45, 145], research papers and theses.
Below we very briefly review some of the main results, most of which will be
discussed in further detail in chapters 4 and 5 in the context of our experimentson the thermodynamics of 39 K bose gases and are given here in order to introduce
the main quantities of interest and their origins.

2.7.1 Non-interacting gas
The mean occupancy, f (p ), of a single-particle state of energy p in an ideal Bosegas in thermodynamic equilibrium at temperature T is given by the Bose-Einstein
distribution function
1(2.59)f (p ) = (p −µ)/k T
Be−1
where kB is the Boltzmann constant and µ is the chemical potential, enforcing
the conservation of particle number. The total atom number is therefore givenby the normalization condition
Ntot =Xf (p )

(2.60)

p

Replacement of the summation by an integration does not correctly account forthe occupancy of the ground state and therefore the total atom number is divided
into a ground-state contribution in addition to the excited state number
Z ∞Ntot = N0 + Nex = N0 +f ()g()d(2.61)

03/2 √
where g () = 4πV2 2m
 is the 3D density of states1 with V the system volume~2
and N0 is the ground-state occupancy. Within the semi-classical approximationand in the presence of an external potential, V (r), the single-particle energies
are given by p (r) = p2 /2m + V (r). With the substitution x = p2 /2mkB T , theintegral in equation 2.61 for the number of atoms in excited states can be written
1

We assume all particles are in a single spin state.

2.7 Bose-Einstein Condensation

45

as
Nex
2
nex (r) =
=√ 3
V
πλdB

Z ∞
0

x1/2
dx
z −1 ex − 1(2.62)

where nex (r) = Nex /V is the excited-state (‘thermal’) density distribution, theparameter z(r) is defined as z(r) = e[µ−V (r)]/kB T and λdB is the thermal de Broglie
wavelength, given by
s
2π~2
mkB T

λdB =

(2.63)The result of the integral in equation 2.62 isnex (r) =g3/2 (z (r))λ3dB

(2.64)

where the polylogarithm g3/2 is given bygγ (z) =

∞
X
zn
n=1

(2.65)

nγ

Therefore, armed with the functional form of the external potential, the total
atom number and the temperature, equation 2.62 can be solved for the excitedstate spatial density distribution.

Uniform system
In an infinite uniform system, V = 0, and since for an ideal gas µ 6 0, the valueof z (r) cannot exceed unity and hence the value of g3/2 (z (r)) cannot exceed
g3/2 (1) = ζ (3/2) = 2.612. This bound on the excited-state atom density signals
the onset of condensation, with any additional particles present in the systemconstrained to occupy the ground state. The temperature at which this occurs
can be found by inserting z (r) = 1 into equation 2.64 and solving for T .
2π~2
Tc =
mkB



n
g3/2 (1)2/3≈ 3.31~2 n2/3
mkB(uniform, ideal gas)

(2.66)

This is the temperature at which the number of atoms in excited states is equal
to the total atom number, and using equations 2.66 and 2.61, the condensate

46

Theoretical backgroundfraction at temperatures below Tc may be obtained as
N0
=1−
Ntot



T
Tc

3/2
(uniform, ideal gas)

(2.67)

In summary, for a uniform system, the onset of BEC occurs when the density reaches the critical value ncrit = 2.612/λ3dB i.e. the phase-space densityρ ≡ ncrit λ3dB = 2.612.
Harmonic potential
For a three-dimensional harmonic potential of the form
1
V (r) = m ωx2 x2 + ωy2 y 2 + ωz2 z 22(2.68)

1/3
1
2
the density of states is given by g () = 2(~ω̄)
the geo3  , with ω̄ = (ωx ωy ωz )metric mean of the trapping frequencies. Performing the integral of equation 2.64
over r and setting µ = 01 , yields the equivalent expression to 2.66 for the criticaltemperature for saturation of the excited state atom number.

~ω̄
Tc =
kB



N
ζ (3)

1/3
= 0.94~ω̄ 1/3NkB(harmonic potential)(2.69)

P∞ −αis the Riemann zeta function. This can equally bewhere ζ (α) =n=1 nexpressed as a critical number for a given temperatureNc = ζ (3)

kB T
~ω̄

3


= 1.202

kB T
~ω̄

3
(harmonic potential)(2.70)

Using equation 2.69 the expression for the condensed fraction in an ideal harmonically confined gas is given by
N0
=1−
Ntot



T
Tc

3
(harmonic potential)(2.71)

Below Tc , one has µ = 02 and equation 2.64 for the density distribution of the
1

2

Condensation occurs in general when the chemical potential reaches its maximal value,
namely the lowest value of the potential energy min(V (r)). For the 3D harmonic potenP3
tial considered here, this corresponds to the zero-point energy 12 ~ i=1 ωi which can beneglected at temperatures when kB T  ~ω̄Neglecting the zero-point energy - see previous footnote

2.7 Bose-Einstein Condensation

47

excited-state atoms reduces tonex (r) =g3/2 e

g3/2 e−V (r)/k T
B−mω 2 r2 /2kB T

=

λ3dB

λ3dB


(harmonic potential)(2.72)The density profile of the condensate wavefunction in the harmonic potential isn0 (r) = |φ0 (r)|2 = mω̄ 3/2
π~

"  
 2  2 #
2
x
y
z
exp −
−
−
ax
ay
ax

(2.73)

where φ0 (r) is the single-particle ground state wavefunction,
whose widths areq~
given by the corresponding oscillator lengths ai = mωi .

2.7.2 Weakly-interacting gas
Interactions between the atoms modify the above results and will be discussed
in more detail in chapters 4 and 5. Here we only discuss some of the propertiesof the condensate.

The Gross-Pitaevskii equation
At the low energies where s-wave scattering dominates and collisions can be described as contact interactions, the effective interaction strength is given byg=

4π~2 am(2.74)

where a is the scattering length described in section 2.6, N is the atom number
and m is the mass. The equilibrium state of the condensate is governed by thetime-independent Gross-Pitaevskii (GP) equation


~2 2
2
−
∇ + V (r) + g |ψ| ψ = µψ2m(2.75)

where ψ = ψ (r) is the many-body wavefunction and the non-linear third termon the left-hand side accounts for the mean-field interaction energy produced byatom ensemble. The density and atom number are thenZ2n (r) = |ψ (r)|⇒N = |ψ (r)|2 dr(2.76)

48

Theoretical backgroundA corresponding mean-field description of the non-condensed component will bepresented in chapter 4.

The Thomas-Fermi approximation
The Gross-Pitaevskii equation, 2.75, has a simple solution which is valid forsufficiently large clouds, such that the kinetic energy term is much smaller than
the interaction energy term and can be neglected in the GP equation. In thisapproximation, the atom density can therefore be found as

V (r) + g |ψ|2 ψ = µψ⇒n (r) = |ψ (r)|2 =µ − V (r)g

(2.77)

Hence the boundary of the cloud is given by V (r) = µ, which for the harmonic
potential of equation 2.68 gives a so-called Thomas-Fermi radius of
Ri2 =

2µ
,
mωi2

i = x, y, z

(2.78)

The chemical potential, µ is found from the normalization condition of equation
2.76 to be2/5~ω̄ 15N aµ=(2.79)
2āhoq~where āho =is the harmonic oscillator length associated with the meanmω̄trapping frequency, ω̄. The condensate density therefore has the form of an
inverted parabola


x2
y2
z2
n (r) = |ψ| = n0 1 − 2 − 2 − 2
Rx Ry Rz
2

(2.80)

See section 3.15 for a description of how these functions are used to extract
physical parameters from images of the atomic cloud.

2.8 Absorption imaging
In most ultracold atom experiments, information about the system is extracted
from images of the atomic density distribution, with the vast majority of experiments utilizing optical techniques in order to measure this spatially-dependent
quantity. Several optical imaging schemes exist, each with its own advantages
and drawbacks, lending themselves to imaging under different experimental con-2.8 Absorption imaging

49

ditions. In general, different techniques rely on either the absorptive or dispersive
components of the atom-light interaction, which predominantly either attenuate
or introduce a phase shift of the incident light field. The relative contribution ofeach of these components is determined by the density of the cloud and the detuning of the probe light. Below is a brief description of the most commonly-used
imaging technique - absorption imaging.
Absorption imaging relies on the attenuation of an incident beam as a resultof being partially absorbed by the atomic sample, with the shadow cast by thecloud in the plane of the atoms being imaged onto a charge-coupled device (CCD)
by an imaging system1 . According to the Beer-Lambert law, described below,
the spatially varying intensity profile recorded on the CCD is a function of thecloud’s spatially varying optical density, which is itself directly proportional tothe density distribution of the cloud, integrated along the propagation axis of
the imaging beam. Physical quantities are then inferred from the form of thisdensity profile (see section 3.15 for a description of our analysis procedure). The
dynamic range of this technique is limited to imaging optical densities no higher
than roughly 4, since achieving detectable transmission at these densities requiresa large detuning, which would also lead to a significant dispersive contribution
to the interaction, causing spreading of the refracted light beyond the collection
range of the optics and leading to a a false absorption signal [145]. In order
to avoid this problem, dense clouds are allowed to expand during time-of-flight(TOF) such that their optical densities are sufficiently reduced to obtain a strong
enough signal-to-noise ratio. This is especially important for condensed clouds,
whose in-situ optical densities are typically on the order of a few hundred.

2.8.1 Theory of absorption imaging
For a monochromatic beam of intensity I (x, y), propagating through a cloud ofdensity n, the change in beam intensity, ∆I upon passing a distance dz is given by
the product of the photon energy, the scattering rate and the number of particles
per unit area, ndz
dI= −~ωRscatt n = −σnI
(2.81)dz
1c.f. Plato’s Allegory of the Cave.

50

Theoretical backgroundwhere in the second equality the scattering rate, σ is given by
≡σ

0
z}|{
3λ2
1
σ=
2π 1 + I/IS + (2δ/Γ)2(2.82)

obtained from the scattering rate in equation 2.12 and the saturation intensity,
Isat of equation 2.10. Integrating equation 2.81, gives an exponential decay of theintensity as the beam propagates through the cloudZ

I (x, y) = I0 (x, y) exp −σn (x, y, z) dz ≡ I0 (x, y) e−OD

(2.83)

where the optical density, OD, is defined byZ
OD (x, y) = σ|n (x, y, z) dz = − ln{z}I (x, y)I0 (x, y)(2.84)

column density, n̄(x,y)

Hence imaging the intensity distribution gives us direct access to the singlyintegrated number density, also known as the column density, n̄ (x, y), of thecloud, from which physical properties such as atom number and temperature
can be extracted (see section 3.15). The total atom number can be found using
equation 2.84
Z
Z Z
1
3
N = n (r) d r =OD (x, y) dxdy
(2.85)σIn practice, stray light impinging on the CCD even in the absence of the imaging
beam can bias the measured optical density, and so a background ‘dark’ image isalso recorded and subtracted from each of the intensity profiles on the right-handside of equation 2.84 prior to division. Equation 2.84 therefore becomesOD (x, y) = − ln

I (x, y) − Idark (x, y)
I0 (x, y) − Idark (x, y)
(2.86)

To summarise, an image with the atoms present is captured (yielding I(x, y))followed by an image in the absence of the atoms but in the presence of theimaging beam (yielding I0 (x, y)). Finally, either one or two background images
are taken in the absence of both atoms and imaging light (yielding Idark (x, y)).
Section 3.14 describes our absorption imaging setup and procedure, while section

2.8 Absorption imaging

51

3.15 discusses the methods by which we extract the temperature of the cloud, as
well as the number of atoms in the condensate and in the thermal component,from these absorption images.

3
Design, assembly andperformance of our 39K-87Rb
apparatus“Keep cool but care.”Abstract

Thomas Pynchon, V.

This chapter describes the design and construction of our apparatus for producing
ultracold 39 K and 87 Rb clouds, as well as the experimental procedure we utilize inorder to cool the samples to quantum degeneracy and the results of the various
stages within this process. It is intended to serve as a manual or field-guide ofsorts for any student tasked with the assembly of a similar experimental setup, or
indeed maintenance and expansion of the one described here. Care has been taken
to try and motivate the choice of various aspects of the experimental apparatus andsequence, not only in order to reflect the large portion of my and my colleagues’
work that has gone into the system’s design and construction, but also in the hopethat others will benefit from this information as I have benefited from reading aboutothers’ experiences and insights.

53

Design, assembly and performance of our 39 K-87 Rb apparatus

54

3.1 Sequence overview
Mirroring the structure of chapter 2, the sections of this chapter trace the progression of a typical experimental sequence culminating in a condensed cloud ofover 4 × 105 39 K (or 8 × 105 87 Rb) atoms confined in a purely optical trap. The
general outline of such a sequence is as follows:
The experiment takes place within an ultra-high vacuum (UHV) chamber (Section
3.2). Potassium and Rubidium atoms are released from heated dispensers andthe desired isotopes, 39 K and 87 Rb, are jointly cooled and confined in a magnetooptical trap (MOT) formed by three pairs of dual-frequency, counter-propagating
frequency-locked laser beams (Section 3.3) and a quadrupole magnetic field (Section 3.4). Both species are then optically pumped into magnetically-trappablestates and transferred into a purely magnetic quadrupole trap (Section 3.6) for
transport (Section 3.7) to the ‘science cell’ located in the higher-vacuum regionof the vacuum chamber in preparation for the subsequent evaporative coolingstage. Figure 3.1 shows a diagram of the vacuum chamber, translation stage and
PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT
quadrupole coil mount.
1!

1!

3!

7!
8!

9!
4!

6!

5!

Figure 3.1: Diagram of the vacuum chamber, translation stage and quadrupole
coil support. Visible are (1) the ion pumps, (2) turbo pump, (3) atom source
module, (4) MOT cell and quadrupole coils, (5) quadrupole coil mounting arm, (6)translation stage, (7) Ti-Sub pump, (8) science cell and (9) six-way cube.

In the science cell, the linear magnetic potential generated by the quadrupole coils
PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCTis converted into a harmonic potential with a non-zero field minimum (Section
3.8) in order to avoid the spin-flip losses which occur in zero-field regions. TherePRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCTPRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT2!

3.2 Vacuum system

55

follows a stage of sympathetic evaporative cooling, in which 87 Rb atoms withabove-average kinetic energy are forcibly ejected from the magnetic trap, leaving
behind colder atoms whose return to thermal equilibrium leads both to coolingof the 87 Rb atoms and to efficient cooling of the 39 K atoms with which they
are in contact (Section 3.9). In preparation for the application of a magnetic
Feshbach field, both species are transferred into a crossed optical dipole trap
(CDT) formed by a far-red-detuned IR laser (Section 3.10). The 87 Rb is thenremoved from the CDT by a resonant light pulse and the 39 K transferred into
its |F = 1, mF = 1i ground state, in collisions of which a broad, convenientlylocated Feshbach resonance exists. The homogenous Feshbach field is then appliedin order to bring the inter-particle scattering length to a large, positive value(Section 3.11). This is followed by an optical evaporation stage during which
the CDT power is gradually lowered. The 39 K is thus evaporatively cooled to thedesired temperature and its phase-space density increased, if desired, through thecondensation critical point. In order to extract information about the sample,
all magnetic and optical fields are extinguished and the cloud is illuminated witha resonant light pulse, in the presence of a magnetic guide field, after several
milliseconds of free expansion. An absorption image of the illuminated cloud isrecorded on a CCD camera (Section 3.14) and physical information extracted
from it by means of a least-squares fit to a theoretical absorption profile (Section
3.15). We begin, therefore, with a description of our vacuum system.

3.2 Vacuum system
3.2.1 Background
The vacuum chamber to which the atoms are confined - and inside which all our
experiments take place - is one of the key components of any cold-atom apparatus.
Collisions with room-temperature background atoms cause heating of the species
under study and invariably transfer sufficient energy to cause the ejection of atoms
from the trapping potential. The rate of these ‘one-body’ background collisions
depends on the ambient pressure and is minimized by ensuring as high a vacuum
as possible in the trap region.
Cloud lifetimes on the order of several minutes are desirable in order to provide
sufficient time for efficient evaporative cooling and to ensure that experiments
can be performed in thermodynamic equilibrium. Such lifetimes typically require
an ultra-high vacuum (UHV) in the region of 10−11 mbar. In addition, since we56Design, assembly and performance of our 39 K-87 Rb apparatus

wish our dual-species MOT to be fully loaded from a background vapour of 87 Rb
and 39 K in a few seconds, we require a higher pressure in the MOT region. A
pressure on the order of 10−9 mbar is high enough to provide rapid MOT loading,while being low enough to ensure that a sufficient number of atoms survive thetransport to the lower-pressure region without being lost to background collisions
on the way. This pressure difference between the two stages of the experimentlends itself to the idea of spatially separating the regions in which the MOT andevaporation steps take place in order to guarantee optimal conditions for eachprocess [146].
The design of the vacuum chamber also dictates the amount of access available to the science cell. Access is required for the various trapping and imaging
beams with their associated optics, as well as for the placement of the numerous
magnetic coils used in the generation of the magneticic trapping potentials, theFeshbach field, compensation fields and required field gradients. In this context
too, separation of the MOT and science cell regions is advantageous, as it allows
us to greatly increase access to the atomic sample. This section will describe thedesign, implementation and performance of our vacuum system, which utilises a
double-differential configuration to achieve a pressure ratio on the order of 103 between the MOT and science cell regions of the system. We initially outline some
guiding principles and define some useful quantities to be used in the design ofvacuum chambers with specific desired characteristics.

3.2.2 Pressure regime
In designing our vacuum system, it is important to remember that at the pressures of interest, the gas inside the chamber is well within the molecular flow
regime, in which the mean free path of a molecule is much longer than the characteristic dimension of the apparatus [147]. This is quantified by the so-called
Knudsen number Kn = λ/D, where λ is the mean free path and D is the characteristic dimension, which for a tube would be its diameter. As an example, from
simple kinetic theory, for pressures on the order of P = 10−10 mbar and at room
temperature, the number density n = p/kB T is around 2 · 1012 m−3 , implying amean free path λ = (nσ)−1 of over 4km for nitrogen molecules (whose diameteris around 3.7Å). The ratio of molecule-chamber to molecule-molecule collisions
is typically given by a few times the Knudsen number and so at these pressuresand temperatures, molecule collisions occur almost exclusively with the chamber walls and not with each other. In the molecular flow regime (Kn  1), the3.2 Vacuum system

57

conductance of a tube whose length is not much longer than its diameter, to an
accuracy of a few percent, is given by [147]:
12.4D3 /L −1
Ctube =ls
1 + 4D/3L(3.1)

where D and L are the tube diameter and length in cm, respectively. This expression is obtained by combining the conductance of a tube whose length is muchlarger than its diameter, with that of an aperture appropriate to the entrance
area of the tube.

3.2.3 Differential pumping
In order to achieve a pressure ratio of two or three orders of magnitude, we makeuse of a two-stage differential pumping setup. The basic principle of differential
pumping is illustrated in figure 3.2a. Two regions of the vacuum chamber, oneof which is connected to a pump of speed S, are separated by a low-conduction
section of conductance C1 1 .

a)!

b)!

Pump!
S!
(P≈0)!

Pump!
S2!
(P≈0)!

S*!

Pump!
S1!
(P≈0)!
S2*!

CP!

CP2!
P1!

P0!
C1!

S1*!
CP1!

P2!

P1!
C2!

P0!
C1!

Figure 3.2: Differential vacuum schematic. a) Single differential pumping stage.
A chamber connected to a vacuum pump of speed S is connected to another, higherpressure, chamber via a low-conductance section, C1 . The conductance of the section connecting the pump to the first chamber is denoted by CP and the pressuresin the low- and high-pressure chambers are denoted by P1 and P0 , respectively. b)
Two-stage differential pumping simply adds a second differential stage to the setup
described in (a).

We can estimate the pressure in different regions of the chamber using the usefulanalogy between the chamber parameters and those of electrical circuits obeying
1

Note: The units of both conductance, C, and pumping speed, S, are ls−1 .

58Design, assembly and performance of our 39 K-87 Rb apparatus

Ohm’s law, V = I · R. The throughput, Q, related to the particle flow rate by
dN/dt = Q/kB T , is analogous to the electrical current, I. The pressure, P, isanalogous to the voltage, V, and the tube conductance, C, is analogous to theelectrical conductance, G = 1/R. Pumps are modelled as regions of zero pressure
and pumping speed S. It is important to remember to combine the pump’s speedwith the conductance, CP , of the tube leading to the pump in order to obtain theeffective pumping speed, S∗ according to 1/S ∗ = 1/S + 1/CP .
The vacuum ‘circuit’ shown in figure 3.2a can then easily be seen to consist of a‘pressure divider’, and the ratio of pressures in the first and second chambers isgiven by:
1/S ∗
C1
C1
P1
=
=
≈ ∗
∗
∗
P0
1/S + 1/C1
C1 + S
S

where

C1  S ∗(3.2)

The generalization to a second differential stage, as shown schematically in figure
3.2b is given by:


C2
C1 C2
P2
1
C2
C1
·
≈ ∗ ∗
=
· P1 ·
=
∗
∗
∗
∗
∗
P0
P0
C2 + S2
C1 + S1 + (C2 S2 )/(S2 + C2 ) C2 + S2
S1 S2
(3.3)
where the last approximation is valid when C1 , C2  S1∗ , S2∗ . In addition tothese considerations, other important principles to follow are the minimizationof internal surface area to reduce outgassing, the use of high-conductance tubes
near pump intakes and the use of low-outgassing materials such as stainless steel
and titanium.

3.2.4 Our chamber
The schematic layout of our vacuum system is shown in figure 3.3. The MOT
and science cells are connected by a transfer tube 65cm in length, with an inner diameter (ID) of 10mm. Two ion pumps are located 19cm and 49cm along
this tube, providing the double-differential described above. The tubes connecting the ion pumps to the transfer tube are 38mm in diameter near the transfer
tube and 63mm in diameter at the pump entrance. They are both 30cm long,with the one nearest the science cell incorporating a vertically-mounted titaniumsublimation (Ti-Sub) pump. Located on the transfer tube, halfway between thetwo ion pumps, is a small six-way cube incorporated in order to allow viewing

3.2 Vacuum system

59of the cloud during transport for diagnostic purposes and/or to allow for a preliminary evaporative cooling step. The conductances of the components labelled
in figure 3.3 are listed in Table 3.1. In the absence of any pressure source inthe science cell region we expect P2 = P3 , and using equation 3.3 we obtain an
estimate for the pressure differential of P0 /P2 ≈ 1600. In practice, we observea pressure difference of around 300 estimated from the ratio of MOT and magnetic trap lifetimes. We attribute this discrepancy to outgassing from the vacuum
components and helium diffusion through the science cell walls.

Pump!
S2!
(P≈0)!

Pump!
S1!
(P≈0)!
S2*!

CP2!

S1*!
MOT cell!

CP1!

Science cell!
P3!

P2!
C3!

P1!

P0!
C1!

C2!

Figure 3.3: Schematic of our double-differential vacuum chamber, showing the
MOT cell, science cell, transfer tube and pumps. The labels P0−3 indicate thepressure at different positions and C1−3,P 1,P 2 indicate conductances.

Part
C1
C2
C3
CP 1 ,CP 2
S1 , S2S∗1 , S∗2

Length [mm]Diameter [mm]

126
60
300
160
210
90
0
300

10
16
10
16
63
38
63
–

Conductance[ls−1 ]
0.72
0.4
2.51
33.16
60
21.36

Table 3.1: Dimensions and conductances of the main segments in our vacuum
chamber. Refer for figure 3.3 for the location of each part.

60

Design, assembly and performance of our 39 K-87 Rb apparatus

3.2.5 Pumps and gauges
There is no one type of vacuum pump capable of establishing the required 10−14
drop between atmospheric pressure and our ultra-high vacuum and hence a sequence of pumps is used during the chamber’s evacuation to reach the desiredpressure range.
Roughing and turbo pumps
After initial assembly of the vacuum chamber, an all-metal right-angle valve
located near the MOT cell is used to connect a turbo-molecular pump1 , with
its exhaust venting into the intake of an oil-free scroll pump2 . In addition
to serving as a backing pump for the turbo pump, which cannot vent toatmospheric pressure, this scroll pump is the first to get turned on andbrings the chamber down to a pressure of around 5 × 10−2 mbar in a matterof seconds. The oil-free design of the scroll pump ensures that no liquid
finds its way out via the turbo pump into the chamber. The turbo pumpis then activated and left to pump on the chamber for the duration of thesystem’s bakeout (see section 3.2.9).
Ion pumps
During the cooling phase after the bakeout, when the pressure has dropped
below ∼ 10−6 mbar, two ion pumps3 are turned on. These pumps act as getters for any atoms that enter the region between their cathode and anodeand hence act as pumps on the chamber. The ion pumps, after bakeout,
bring the pressure in the chamber down to below 10−11 mbar and are operated continuously throughout the several-year lifetime of the apparatus
in order to continuously remove from the system any excess atoms released
from our atom sources, outgassed impurity atoms from the metal components of the chamber and any helium migrating through the walls of thequartz science cell.
Since these pumps rely on large magnetic fields, it is in general desirable
to distance them from the main experimental region in order to minimize
the effects of stray fields, which in turn implies a reduction of their effective
pumping speed. Additional magnetic shielding4 made from nickel-coatedlow-carbon steel has also been fitted to both ion pumps to reduce the stray1234œrlikon Leybold TURBOVAC TW 70 H
Leybold SC5D
Gamma Vacuum, TiTan 75S
Gamma Vacuum, 75S-6S

3.2 Vacuum system

61

magnetic field near the science cell. According to the manufacturer’s specifications, at the location of the atoms in the science cell this results in a
reduction of the stray field from a value of 0.282G to 0.1G. The pumps usedin our experiment have an effective pumping speed of between 60-70 ls−1 atpressures between 10−10 -10−6 mbar, respectively, They are located no closer
than 30cm from the science cell, have a quoted lifetime of 50,000 hours at
a pressure of 10−6 mbar and a maximal bakeout temperature of 250℃.
Titanium sublimation pump
In addition, a titanium sublimation (Ti-Sub) pump1 is mounted near thescience cell and is occasionally activated in order to enhance the vacuum.
Titanium sublimation pumps are getter pumps often used in addition to ion
pumps to remove molecules such as H2 , N2 , O2 , CO2 and H2 O. A current
on the order of 50A is used to sublimate a thin layer of titanium onto an
inner surface of the chamber, which then reacts with active gas molecules
to form low vapour-pressure compounds that are bound to the film. The
pump is mounted so as to provide as large an area as possible on which todeposit the titanium film. which is replenished once it has saturated. In
our case, the pump is only activated for 1.5 minutes at 47A roughly once
every six months. Since this pump produces no magnetic fields it can beplaced in close proximity to the experimental region and hence maximize
its pumping speed.
Bayard-Alpert gauge
Although the ultimate benchmark for our vacuum is the lifetime of ourtrapped cloud, we are able to obtain estimates of the pressure from a
Bayard-Alpert ionisation gauge2 mounted in close proximity to the sciencecell (see figure 3.6). In addition, the ion current readout from the ion
pumps can be used to estimate the pressure according to the approximate
formula: Pressure = 10−3 × Current, with pressure and current in millibars
and amperes, respectively.

3.2.6 MOT cell
The region of the vacuum chamber inside which the MOT will be formed has tofulfil several criteria. Firstly, it must be able to accommodate the three orthogonal
1
2

Varian, TiSub Cartridge
VACOM, BAIS sensor

62

Design, assembly and performance of our 39 K-87 Rb apparatus

pairs of counter-propagating cooling/trapping laser beams (see section 3.4). The
diameter of these beams defines the capture volume of the MOT and so in order
to collect large atom numbers, large beams and windows are necessary. Secondly,
there are several considerations when choosing the separation between the coilpair used to generate the quadrupole magnetic field, required for operation of
the MOT as well as for trapping of the atoms during transport to the sciencecell. These considerations are related to the size of the coils used, which in turn
determines the spatial extension of the quadrupole field and hence the trapping
volume. In addition to a large trapping volume, the ability to produce large field
gradients at the atoms’ location is important for ensuring that as many atoms as
possible are safely transported to the science cell and are not lost en route in the
narrow tube connecting the two regions. Large field gradients are also requiredin order to maximize the efficiency of evaporative cooling (see section 2.4). We
would hence like the smallest coils that provide sufficient MOT trapping volume,and that can still be spaced such as to maximize the field gradient they produce
at a given current. The maximal field gradient at the centre of a single coil pairis obtained when the coil separation is equal to the coil radius, as described inappendix E. The MOT cell must also provide a path to the atomic sources aswell as a means of collecting the MOT fluorescence which serves as a probe of
the number of trapped atoms.

Figure 3.4: a) Image of the MOT chamber showing the viewports, transfer tube
(left) and atom source module (background). b) Science cell mounted on vacuum
chamber prior to the addition of the various surrounding beams and magnet coils.

With these considerations in mind, a custom non-magnetic, stainless steel(316LN) MOT cell was designed and manufactured1 . The cell consists of a cylindrical chamber of 110mm diameter and 44mm height, incorporating eight 23/4”
1

Fabricated by Kurt J. Lesker Company

3.2 Vacuum system

63(DN35CF) viewports, a further 23/4” (DN35CF) flange for mounting of the atom
source module, and a smaller 11/3” (DN16CF) flange for connecting the differential vacuum tube. The viewports are made from Kodial glass, mounted in 316LN
flanges, and are AR coated on both sides with a coating optimized for 767nm
and 780nm. The viewports are distanced from the central part of the chamberin order to allow vertical compensation coils to be mounted around the verticalflanges as well as to allow the quadrupole coil support, mounted on the translation stage, to hold the coils as close as possible to the cell itself while having
sufficient clearance to translate them to the other end of the chamber. Figure
3.4a shows the assembled MOT cell prior to chamber bakeout.

3.2.7 Science cellIn the interests of maximizing optical access and versatility, and minimizing any
unwanted magnetic fields caused by eddy currents, we have chosen to use a science cell consisting of a rectangular quartz cell1 . This is manufactured fromhighly polished, optical quality Spectrosil® Quartz in a fusing process utilizing
no intermediate bonding materials, and is annealed after assembly to remove
any residual strain. Spectrosil® Quartz (also known as Far UV Quartz) has
good transmission properties in the range 190 - 2700nm. Prior to assembly, abroadband anti-reflection (AR) coating is applied to the external walls of the cellto ensure high transmission at the wavelengths which are to be used for imaging of both species (767 and 780nm) as well as trapping and manipulation of
the ultracold cloud (532nm and 800-1100nm). The cell’s outer dimensions are
30×30×110mm and the cell wall thickness is 5mm. This thickness was chosen
so as to avoid permeation of helium into the cell, while allowing a sufficient inner volume for trapping and imaging purposes, as well as avoiding any opticalbirefringence that might be caused by mechanical stress to the cell wall during
evacuation of the chamber. The outer dimensions where chosen so as to allow
placing of trapping and Feshbach coils in close proximity to the atoms in order toallow sufficiently large fields and steep gradients to be applied. Figure 3.4b shows
the mounted science cell during chamber assembly and appendix D contains amore detailed diagram of the cell in addition to reflectance spectra for the AR
coating at both normal and 45° incidence.
The quartz cell is connected to the rest of the vacuum chamber by means of a1

Manufactured by Starna Scientific Ltd.

64

Design, assembly and performance of our 39 K-87 Rb apparatus

graded glass-to-metal seal which is required in order to overcome the difference incoefficients of thermal expansion between the steel and quartz components of thechamber during bakeout as well as during normal operation. The DN16CF flange
attached to the glass-to-metal seal is made from non-magnetic 316LN stainlesssteel in order to minimize eddy currents near the trap centre.

3.2.8 Atom sources
The vapours of 87 Rb and 39 K from which our MOT is loaded are released into
the MOT cell from a set of commercial vapour sources1 . In total, eight sources
are currently installed in our vacuum chamber, four sources on each of two 8-pin
electrical feedthroughs2 . The feedthroughs are connected to two perpendicular
arms of a T-shaped tube, the third arm of which is attached directly to the MOT
cell. After installation inside the vacuum chamber, the sources are thermally
activated during the bakeout process (which also expedites the removal of theIndium used to seal the sources) and subsequent release of atoms is affected byrunning a current of between 4-8A through the desired source. In order to accommodate experiments involving any of the commonly used Rb or K isotopes, wehave installed sources of four different compositions: The bosonic 85 Rb and 87 Rb
isotopes can be released either from one of two natural-abundance Rb sources(72% 85 Rb, 28% 87 Rb, 50mg total per source3 ) or from one of two 87 Rb-enrichedsources (98% 87 Rb, 50mg total per source4 ). We also have two types of K sources
installed in the system: two natural-abundance sources containing 93% of thebosonic 39 K, 0.012% of the fermionic 40 K and 6.7% of the bosonic 41 K (60mg
total per source5 ), and two 40 K-enriched (10%) sources (25mg total per source6 ).
Figure 3.5 shows a schematic of the source module and an image of one of thetwo feedthroughs on which our sources are mounted.
We have learnt two valuable lessons regarding the use and mounting of this typeof vapour source. The first is that mounting of these tube-shaped containers
with the tube opening facing down is highly undesirable since the alkali metal
can partially or fully detach from the walls of the tube. In the best case this only
1
2
3
4
5
6

Alvatec, Alvasources®
Kurt J. Lesker, EFT0084033
AS-3-Rb-50-V
AS-3-Rb87(98%)-50-V
AS-3-K-60-V
AS-3-K40 (10%)-25-V - The 40 K-enriched sources were produced by Alvatec, using material
processed by Technical Glass Inc. (since renamed Precision Glassblowing).

3.2 Vacuum system

65

Figure 3.5: a) Schematic of the commercial atom sources used in our system. The
source is connected to an electrical feedthrough via the contacting flaps. Prior tobakeout the source material is kept in a pure Argon atmosphere, which is release
when the source’s indium seal is melted. b) Image of the sources mounted on theelectrical feedthrough in the system.

reduces the sublimation rate of the material due to the decreased contact withthe heated walls, and in the worst case can lead to the material falling out of
the source. Since the material in the sources is actually an alloy of the specificalkali metal (K or Rb) with Bismuth (Bi) and has a very low vapour pressure,
this does not change the pressure in the chamber although it does, naturally,
render the source useless. Secondly, we have found that during the activation of
any source a significant amount of material is deposited on the surfaces of othernearby sources, even though their openings face the same direction. This resultsin small but noticeable amounts of Rb being released from K sources and viceversa. To preclude this it is recommended to mount all sources of each element,
rather than a combination, on a single feedthrough.
For the experiments described in this thesis, an enriched 87 Rb source was typically
fired for 1.5mins at 5A at the start of every day and for ∼ 30s during eachexperimental cycle in the course of measurements, with a 39 K source being fired
once, typically for 1.5mins at 6A, every several weeks.

3.2.9 Assembly and bakeout
Assembly
In order to achieve pressures in the range < 10−11 mbar, it is crucial to remove as
many of the impurity atoms adsorbed onto the interior surfaces of the chamber aswell as those trapped inside the bulk of the metal vacuum components during theirproduction. Even very small amounts of contamination from skin oils, grease,

Design, assembly and performance of our 39 K-87 Rb apparatus

66

lint and water vapour can prevent or hugely delay the achievement of a sufficientvacuum and so all contact with the vacuum components during chamber assemblymust be carried out using clean latex gloves to prevent any fingerprints being left
on inner surfaces. Prior to assembly, every component is thoroughly cleaned using
acetone and rinsed with methanol, with any suitably small components also being
cleaned beforehand in an ultrasonic bath1 . It is important to work quickly inorder to minimize exposure of interior surfaces to air in order to limit the amountof water vapour condensing on them. All vacuum connections are made using
ConFlat flanges, non-reusable copper gaskets and silver-plated vacuum bolts.
Figure 3.6 shows the assembled vacuum system prior to baking.

Figure 3.6: Assembled vacuum system with turbo pump connected prior to bakeout. The (vertical) Ti-Sub pump and the pressure gauge feedthrough pins arevisible on the left, near the science cell.

Bakeout
Following assembly, the entire chamber undergoes a baking stage during which its
temperature is gradually elevated to around 200℃. It is held at this temperature
for two weeks while continuously being pumped by the turbo and ion pumps. The
elevated temperature (exponentially) accelerates the outgassing of impurity atoms
trapped inside the metal components during their production as well the traces1

Langford Electronics, Sonomatic 375

3.2 Vacuum system67

of water and other organic hydrocarbons adsorbed onto the chamber surfaces
during assembly. These released atoms are then removed from the system by thepumps. It is important to bake the entire system as any unbaked areas will act
as condensing surfaces for desorbed atoms which will then be redistributed after
cool-down.
In preparation for the bakeout, the entire system is wrapped in resistive heatingtapes1 which are connected to mains-voltage variacs2 in order to control their temperature. Thermocouples are then placed against the chamber walls in several
strategically-chosen locations in order to provide local probes of the temperature. This is important in order to ensure the relatively slow (not exceeding
2-3℃/minute) and uniform heating required in order to avoid large temperature
gradients which could damage the vacuum due to differential expansion of thematerials making up the chamber. The chamber is then wrapped in several layersof aluminium foil for insulation and the voltage across the heating tapes gradually
increased. The temperature of the thermocouples is measured using a 16-channel
thermocouple monitor3 and is monitored and logged using a LabView VI communicating with the thermocouple monitor over GPIB. The variac voltages areadjusted in order to ensure smooth, uniform and gradual heating of the system toits final temperature. Figure 3.7 shows the system during preparation for baking
and during the baking itself.

Figure 3.7: The vacuum chamber before (a) and during (b) bakeout.

The most sensitive parts of the system during baking are the ion pumps, sciencecell and viewports. Above 250℃ the ion pumps’ permanent magnets start losing
magnetisation. The maximal bakeout temperature of the quartz science cell is12
3

Tyco Thermal Controls, isopad TeMS2, 40W/m, max. 260℃.
Clairtronic, 10551
Stanford Research Systems (SRS), SR630

Design, assembly and performance of our 39 K-87 Rb apparatus

68280-300℃, while that of the Kodial viewports is 350℃. The glass-to-metal sealon the science cell is especially sensitive to temperature gradients. The ionisation
gauge is also activated during the bakeout - once the pressure drops into its
operation range (P < 5 × 10−2 ) - in order to efficiently remove impurities released
during its initial degassing. It can be baked to 250℃ while still activated (400℃
while disconnected).
The process is reversed to bring the system back to room temperature at a pressure in the region < 10−11 mbar, with the turbo and roughing pumps being valved
off when the pressure is between 10−8 −10−9 mbar, leaving only the ion pumps active. The measurement limit of our ionization gauge is ∼ 3 × 10−11 mbar 1 . To
illustrate the exponential dependence of the pressure on the chamber temperature, figure 3.8 shows a plot of the logarithm of the pressure, as measured by
the ionization gauge, against the inverse temperature, measured by a thermocouple close to the gauge, during system cool-down. The (approximate) linearity ofthis plot is consistent with the Clausius-Clapeyron relation, dP/dT = L/(T ∆V ),where P is the pressure, T the temperature, L the latent heat and ∆V the volume
change.
-14

ln(P

gauge
ln(P Gauge
)

)

-16
-18
-20
-22
-24
0.00200.0022 0.0024

0.00260.00280.0030 0.0032

0.0034-1-1 )
1/TGauge
1/T
[K(K
]

Figure 3.8: Natural logarithm of the pressure, ln(Pgauge ), during chamber bakeout, plotted against the inverse temperature, 1/T. The pressure spikes around T≈ 150℃ and 200℃ are due to activation of the Ti-Sub pump during the bakeout.

1

This limit is set by the pump’s so-called ‘x-ray limit’ at which the pressure-dependention current becomes smaller than the pressure-independent electron current caused by thephotoelectric effect3.3 Laser System

69

Note regarding vacuum leaks:

As an additional note, it is worth mentioning that at some point during thesystem’s day-to-day operation, a small leak was introduced by a collision
between the translating quadrupole coil mount and one of the horizontallymounted flanges on the MOT cell. The leak was opened at the joint between
the MOT cell and the transfer tube and was detected by spraying the region
with helium and observing the sharp rise in system pressure. Rather thanre-bake the chamber, we opted to try and seal the leak using a purposemade silicon resin sealant1 . The sealant is applied directly to the leak andrequires several days to fully cure at room temperature. This successfully
restored UHV pressure and has not presented any problems in the several
years since its application, with typical 1/e lifetimes of 300s being achieved
for thermal clouds in the science cell.

3.3 Laser System
The light used for laser cooling, optical pumping (see section 3.6) and imaging (see section 3.14) of the atom cloud is required to address specific atomic
transitions and hence must possess a narrow linewidth compared with both the
Doppler width and the splitting between the hyperfine states involved. Laser
linewidths are typically stabilized to below the linewidth of the relevant atomic
transition, which for the cooling transitions of 87 Rb and 39 K are (2π×) 6.07MHz
and 6.04MHz, respectively. Linewidths on the order of 1MHz and below are easily achieved by commercially available stabilized diode lasers. Our laser system
consists of a combination of external-cavity-stabilized diode lasers (ECDL) andtapered amplifier (TA) modules. The ECDLs emit light with a linewidth below
1MHz, a mode-hop-free tuning range typically between 20-50GHz and powers
in the range 80-150mW, while the TA chips are designed to amplify the powerof the diode lasers to around 1W while preserving their spectral characteristics.
An overview of the laser system is shown in figure 3.11 in addition to the moredetailed description below. We also make use of far-off-resonant light to generate the optical dipole potential used in our optical trap (section 3.10), as well
as for our experiments with optically-plugged magnetic traps (appendix B) andtwo-dimensional confinement (appendix F).
1

Kurt J. Lesker, KL-5-5B

Design, assembly and performance of our 39 K-87 Rb apparatus

70

3.3.1 Saturated absorption and locking scheme
Despite their narrow linewidths and relative stability, ECDL lasers are not immune from unacceptable long-term frequency drifts. In order to preclude thesedrifts and to ensure that the lasers are tuned precisely to the desired frequency,
all our diode lasers are actively locked to atomic transitions in a vapour cell containing a sample of the desired atomic species (Rb or K). The locking scheme
employs the well-known technique of saturated-absorption spectroscopy [90, 148]in order to obtain Doppler-free absorption spectra from which an appropriate
error signal can be derived and fed back to the laser in order to stabilize its
frequency. Figures 3.9 and 3.10 show the saturated absorption spectra obtainedfor 87 Rb and 39 K, respectively, using the beam configurations illustrated in figure
3.11 and described in the following two subsections.

Photodiode signal!

424MHz!

a)!

b)!

5!
1!

4!

2!
3!

6!

-

c)!

=

157MHz!

267MHz!

Figure 3.9: Saturated-absorption spectroscopy on the |F = 2i → |F 0 i transitionon the D2 line of 87 Rb. (a) Absorption spectrum showing saturated-absorption
peaks superposed on the background Doppler spectrum. The peaks labelled 1,
3 and 6 are the |F = 2i → |F 0 = 1, 2, 3i transitions, respectively, and the peaks
labelled 2, 4 and 5 are the |F = 2i → |F 0 = 1/2i, |F = 2i → |F 0 = 1/3i and|F = 2i → |F 0 = 2/3i crossover peaks, respectively. Figure (b) shows the Doppler
spectrum in the absence of saturated absorption and (c) the signal resulting fromsubtraction of the latter from the former. The relative positions of the peaks canbe deduced from the spacings indicated in figure (c), with crossover peaks locatedhalfway between the participating transitions.

3.3.2 Rb lasers
The 87 Rb cooling light is derived from an integrated master-oscillator-poweramplifier (MOPA) unit1 , providing ≈800mW of 780nm light at its main output.
The unit’s rear auxiliary output provides several mW in total, part of which is1

Toptica, TA100

3.3 Laser System

71
0.005

Photodiode signal (a.u.)

2!
0.000

1!

3!

-0.005-0.010-0.015

0.01

0.02

0.03

0.04

Time (s)Figure 3.10: Saturated Absorption spectroscopy on the D2 transition of 39 K. The
figure shows the large |F = 1i → |F 0 i / |F = 2i → |F 0 i crossover dip to which welock our 39 K master laser. Shown are the saturated absorption curves at vapour
cell temperatures of 19.8℃ (orange), 35℃ (green), 40℃ (blue), 45℃ (red) and 50℃(black).

sent to a saturated-absorption locking setup and part of which is used for imaging light. The saturated-absorption setup contains a double-pass AOM driven at
approximately 98MHz and is used to offset-lock the laser frequency to 2×98 =196MHz above 1 the |F = 2i → |F 0 = 2, 3i crossover peak which is located approximately 133MHz below the |F = 2i → |F 0 = 3i cooling transition i.e. 63MHz blue
detuned. All subsequent frequency tuning of the cooling, imaging and pumping
beams is carried out with this double-pass AOM in order to preserve the beamalignment. The imaging beam is then brought into resonance by another singlepass AOM operating at around 67MHz, while the cooling beam is brought toa detuning of approximately -3Γ by another single-pass AOM operating near
83MHz. The light used for optical pumping (see section 2.3) is split off after thisAOM and is passed through another single-pass AOM at 200MHz in order tobe able to bring it into resonance with the |F = 2i → |F 0 = 2i transition during
optical pumping.
The 87 Rb repump light is generated by a separate ECDL2 , locked via a separate
saturated-absorption setup to the |F = 1i → |F 0 = 1, 2i crossover peak, 78.5MHz
below the |F = 1i → |F 0 = 2i repump transition and brought into resonance bya single-pass AOM operating at approximately 78MHz.
1
2

i.e. at a higher frequency
Toptica, DL100

λ/4!

150!

LP!

λ/2!
λ/4!

λ/2!

λ/2!

400!

+78!

-87!

λ/2!

λ/2!

λ/2!

87Rb repump!

39K TA!

Iris!

Mirror!

AOM!

λ/4!

750!

25!

75!

λ/2!

λ/4!

-98!

λ/2!

150!

150!

400!

λ/2!

-67!

λ/2!

Horiz.!
Imaging!

λ/2 @ 767nm!
λ/2!

Cooling!

Pumping!

imaging and !
pumping!
λ/2!

87Rb cooling,"

PM Fiber coupler!

87Rb TA!

-87!

150!

λ/2!

75!

0!
+4

125!

-200!

50!

λ/4!

125!

imaging and pumping!

87Rb !
Repump!

Shutter!

39K cooling, repump,"

Lens!

Polarizing beamsplitter cube!

Photodiode!

Wave plate!

Anamorphic prism pair!

100!

λ/2!

100!

-200!

λ/2!

λ/2!

λ/2!

Vert.!
Imaging!

72
Design, assembly and performance of our 39 K-87 Rb apparatus

Rb cell!

-83!

DL100 780nm!

Rb cell!

K cell!

DLPro 767nm!

Figure 3.11: Overview of the laser system used to generate the cooling, repump,imaging and pumping beams used in the experiment. See text for a detailed description.

3.3 Laser System

73

3.3.3 K lasers
All the 39 K beams are derived from a single ECDL1 , providing roughly 80mW
of 767nm light, with the cooling and repump beams seeding a TA unit2 with amaximal output power of approximately 1.5W. Light is split off from the ECDL
output and a double-pass AOM operating at 87MHz is used to offset-lock thelaser 2×87 = 174MHz below3 the |F = 1, 2i → |F 0 i crossover dip, which is itself
231MHz red (blue) detuned from the |F = 2i (|F = 1i) states i.e. the laser gets
locked approximately 57MHz red of the |F = 1i → |F 0 i transition. The light not
used for locking is again split into two beams, with one traversing a double-pass
200MHz AOM to generate the cooling/imaging/pumping beam and the otherencountering a single-pass 40MHz AOM to bring this repump beam closer toits final frequency. The cooling and repump beams are them overlapped on apolarizing beamsplitter cube (PBS) and the overlapped, orthogonally polarized
beams are then rotated with a halfwave plate in order to tune the balance between
the cooling and repump power seeding the TA. In order to ensure perfect spatial
overlap of the MOT beams for both 39 K and 87 Rb, we overlap the combined 39 K
cooling/repump beam with the 87 Rb cooling beam. The two beams are spatiallyoverlapped using a PBS cube and their polarizations are aligned prior to fibre
coupling using a dichroic waveplate4 , designed to act as a half-wave plate at
767nm and a full-wave plate for 780nm. All coupling is done into polarizationmaintaining (PM) single-mode fibres. The 39 K imaging and pumping beams are
likewise overlapped with their 87 Rb counterparts before being coupled and sent
to the vacuum system, while the fibres containing the repump and cooling lightare sent to the fiberport cluster located near the vacuum chamber.

3.3.4 Shutters
Complete extinction of laser beams requires the use of opaque mechanical shutterssince AOMs always exhibit small amounts of leakage light into the diffracted
beams. We use Thorlabs5 and Sunex6 shutters, whose characteristic opening andshutting times are on the order of 1-2ms. The vibrations caused by the Thorlabs
1
2
3
4
5
6

Toptica, DLPro
Sacher Lasertechnik, TEC-400
i.e. at a higher frequency than the |F = 2i → |F 0 i transition.
LENS-Optics GmbH, W2M25-767&780
Thorlabs, SH05
Sunex Inc., SHT934

74

Design, assembly and performance of our 39 K-87 Rb apparatus

shutters can temporarily perturb the locking of nearby lasers and are reducedby mounting them on sorbothane shock-absorbing posts. In addition we havealso made use of home-made shutters, produced from voice-coil actuators foundin generic hard disk drives. The design follows that of the Melbourne group1[149, 150].

3.3.5 AOMs and drivers
Fine adjustment of laser frequencies, on the order of . 1MHz, is accomplishedusing acousto-optic modulators (AOM). These devices rely on Bragg-like scattering of an incident laser beam from a travelling refractive-index grating, generated
by an RF signal in the range f ∼20-400MHz, to shift both the frequency andthe direction of the outgoing beam and hence can be used to control the beam’s
detuning as well as acting as high-speed switches, respectively. Most AOMs allow frequency shifts over a range of ±20% of their centre frequencies with thedeflection speed of the models used in our experiments2 depending on the beamwidth roughly as 15ns/100µm.

Figure 3.12: (a) Dual-channel AOM driver interior. Visible for each channel fromright to left are the VCOs3 , attenuators4 , switches5 and amplifiers6 . Also visible are
the power supplies and fans (top) and frequency counters (bottom). Two amplifiersof different gains are used in this driver in order to drive AOMs requiring differing
amounts of drive power without risking damage due to excess power. (b) Driver
front panel.
1
2
3
4
5
6

http://optics.ph.unimelb.edu.au/atomopt/shutter/shutter.html
Crystal Technology Inc., 3080-125, 3110-110, 3200-115 and AA Opto-Electronic
Mini-Circuits, ZOS-50, ZOS-100, ZOS-200, ZOS-300
Mini-Circuits, ZX73-2500-S+
Mini-Circuits, ZYSWA-2-50DR
Delta RF Technology, LA2-1-525-30 and Mini-Circuits, ZHL-3A, ZHL-32A

3.4 Magneto-optical trap (MOT)75

The electronics required to drive an AOM consist of a voltage-controlled oscillator(VCO), a voltage-controlled attenuator and an amplifier which is required in orderto generate the ∼1W of RF power required to drive most commercial AOMs attheir maximal diffraction efficiency of 80-90%1 . Figure 3.12 shows the internallayout of one of the drivers constructed for use in our experiment.

3.3.6 Fiberport cluster
In order to divide the cooling and repump light into the six separate beams
required for operation of our MOT, we have made use of a commercial fiberport
cluster 2 . This is a compact, modular assembly of fibre couplers, half-wave plates
and polarizing beam splitters (PBS) designed to accept two beams at different
input ports and to allow their relative intensities to be adjusted at each of six
output ports by adjusting the alignment of the six half-wave plates. All inputs
and outputs are coupled into polarization-maintaining single-mode optical fibres
and the light intensity at both inputs can be monitored from built-in photodiodes
to which ∼1% of each input beam is diverted. When calculating the powers at
each of the six outputs, it is important to take account of the imperfections in
the PBS cubes, which typically reflect ∼5% of the p-polarized light together with
∼100% of the s-polarized component.
The triple-wavelength beam containing the 39 K cooling/repump frequencies andthe 87 Rb cooling light is sent into the first fiberport-cluster input while the 87 Rb
repump light is sent into the second input. Due to the PBS cube imperfections,perfect balancing of all the power from one input, i.e. ∼ 17% at each output,
results in the second input being distributed between the outputs with values
between 7-25% of the total power at this input. Since the precise balancing ofthe (relatively weak) 87 Rb repump light is not crucial to the operation of the
MOT, we take care to balance the triple-wavelength ‘cooling’ beam outputs, andallow the Rb repump to remain unbalanced.

3.4 Magneto-optical trap (MOT)The preceding sections described our vacuum system and laser setup, both ofwhich play a role in multiple stages of the experiment. We now turn to describing
1
2

As opposed to switching speed, maximal diffraction efficiency grows with beam diameter.
Shäfter + Kirchhoff GmbH, Fiber Port Cluster 2-to-6

76

Design, assembly and performance of our 39 K-87 Rb apparatus

in more detail the individual stages of the experimental sequence, from collection
and cooling in the dual-species MOT via sympathetic cooling in the science celland ending with evaporative cooling in the optical dipole trap and time-of-flight
imaging.

3.4.1 MOT laser parameters
As mentioned in section 2.1.4, the unresolved excited-state hyperfine manifold in39
K leads to strong pumping of atoms into the |F = 1i state (requiring comparable‘cooling’ and ‘repump’ intensities) and potentially to strong heating due the bluedetuning from nearby states (requiring detuning of both cooling and repump
beams from the entire excited-state manifold).
For 39 K during the MOT phase, we use detunings of -7.6Γ and -4.6Γ from the 39 K
|F = 2i → |F 0 = 3i and |F = 1i → |F 0 = 2i transitions, respectively. However,in order to achieve lower 39 K temperatures prior to magnetic trapping, during
the last 40 ms of the MOT stage, the two detunings are reduced to -1.5Γ and-4.1Γ, respectively, while the repumping intensity is also slightly reduced. This
results in lower temperatures of several hundred µK without an appreciable drop
in atom number [49]. For 87 Rb during typical MOT operation, the cooling lightis detuned by -3.1Γ from the cycling transition, while the repump light is close
to resonance with the |F = 1i → |F 0 = 2i transition. Under these conditions,the atom numbers in our single-species 87 Rb and 39 K MOTs saturate at roughly
5 × 109 and 1 × 109 , respectively, after ∼ 10s of loading. Simultaneous trappingof 87 Rb and 39 K results in a roughly 15% reduction in 87 Rb number and a factorof ∼3 reduction in the 39 K number due to light-assisted interspecies collisions.

3.4.2 MOT field gradient
In optimizing the size of the 87 Rb and 39 K MOTs individually, we have found
field gradients of 9.6G/cm and 8G/cm1 , respectively, to yield the largest MOTsfor our laser parameters. However, when both species are trapped together, theirrelative abundances also influence the optimal conditions and we have empiricallyfound a weaker gradient of 6.4G/cm to provide the best results in terms of final
39
K condensate size.
1

Along the ‘strong’ axis, connecting the coil centres.

3.5 Quadrupole and compensation coils

77

The MOT stage is followed by 6ms of optical molasses. During this step the87
Rb cooling light is taken further away from resonance to -4.9Γ , the 39 K coolingbeam detuning is left at -1.5Γ and the 39 K repump detuning is increased to -4.1Γ.
Table 3.2 summarizes the laser and magnetic field parameters during the MOT
and molasses stages.
Laser parameters
Beam87

Rb Cooling87
Rb Repump
39
K Cooling
39
K Repump
87

Rb Cooling87
Rb Repump
39
K Cooling
39
K Repump

Detuning [Γ]Total power (at atoms) [mW]
MOT (duration: ∼ 20s)
-3.1
130
0
10
-7.6 (-1.5 during final 40ms)
155-4.6 (-4.1 during final 40ms)
155
Molasses (duration: 6ms)
-4.9
60
0
10
-1.5
110
-4.1
10
Beam diameter (1/e2 ) [cm]∼3
Field gradient (strong) [G/cm]
6.4 (9.6 during final 40ms)
Atom number (approx.)

5 × 109 (87 Rb-only)

1 × 109 (39 K-only)

4 × 109 /3 × 108 (87 Rb-39 K)

Table 3.2: Summary of MOT and molasses parameters. Detunings for the 87 Rb
beams refer to the |F = 2i → |F 0 = 3i and |F = 1i → |F 0 = 2i transitions for thecooling and repump beams, respectively. Detunings for the 39 K beams refer tothe entire excited-state hyperfine manifold, |F = 2i → |F 0 = 3i and |F = 1i →|F 0 = 2i, respectively.

3.5 Quadrupole and compensation coils
Here we describe the electromagnetic coils used to establish the quadrupole field
for our MOT as well as the compensation coils used for nulling stray fields in the
MOT region.

78

Design, assembly and performance of our 39 K-87 Rb apparatus

3.5.1 Quadrupole coils
In addition to providing the magnetic field for our MOT, our main quadrupole
coils are also responsible for confining the atoms during transport to the sciencecell and during the initial evaporative cooling stage, as well as generating thequadrupole component of our quadrupole-Ioffe configuration (QUIC) magnetic
trap, in which sympathetic cooling of 39 K by 87 Rb takes place.
The coils are wound from insulated, hollow, circular copper tubing (OD: 4mm,
ID: 2.8mm) and consist of 40 turns (4×10) with an inner diameter of 51mm andan outer diameter of 135mm. They are mounted 100mm apart (at their closest
point), as close as possible to anti-Helmholtz configuration in order to maximize
their gradient (see appendix E), and provide a field gradient of 160G/cm along thevertical (strong) axis when run at a current of 200A. The wound coils are coatedin protective epoxy and mounted in a heavy-duty aluminium support designed tofirmly hold the coils, enable their mounting onto the translation stage, eliminate
mechanical vibrations due to water flow and magnetic forces, and ensure sufficient
optical access both at the MOT and science cell ends of the apparatus (see figure3.13). A narrow slit is machined into the brackets supporting the coils themselvesin order to avoid eddy currents being formed near the location of the atoms when
the coils are turned off.

Figure 3.13: (a) Quadrupole coil pair mounted on the translation stage and (b)MOT compensation coils mounted on the MOT cell flanges. The red pipes carrycurrent and cooling water to the quadrupole coils.

The two coils are connected in series and their current is provided by a 6kW(30V, 200A) DC power supply1 down 35mm2 -gauge cables, connected to the coils
1

Delta Elektronika, SM 30-200

3.5 Quadrupole and compensation coils

79

via copper adapters designed to also allow easy connection of water-cooling hoses
to each coil. The current is controlled remotely by an analog signal to the power
supply and is switched digitally using a high-voltage IGBT1 .

3.5.2 Coil safety
The IGBT driver circuit is also responsible for switching off the current in theevent of the coils overheating or a drop in the cooling water flow. A thermistor2
with a threshold temperature of 60℃ is mounted on each quadrupole coil anda third thermistor is mounted to the large copper heatsink to which the IGBT
is bolted. A varistor is connected in parallel with the IGBT in order to shuntaway any current from transient voltage spikes that would otherwise damage the
IGBT. In addition, a flow switch3 near the coils is used to ensure the flow does
not drop below 0.5lmin−1 . These fours signals must all be normal in order for
current to be allowed to pass through the coils.

3.5.3 Compensation coils
The various sub-Doppler cooling mechanisms at work - and hence the lowest
temperatures attainable - during optical molasses are sensitive to both the direction and magnitude of any magnetic fields present [90, 151]. Transfer of the
maximal number of atoms, with the highest possible phase-space density, into
the purely magnetic trap further requires the MOT to have a regular shape, its
centre to overlap with that of the magnetic trap and its free-space expansion
to be isotropic and determined only by the cloud temperature. Furthermore, awell-defined quantization axis is required during the optical pumping stage prior
to magnetic trapping (see section 2.3).
Meeting these criteria requires the cancellation of any stray magnetic fields at thelocation of the atoms, arising from the earth’s magnetic field or nearby instrumentation4 . Field compensation is accomplished by three orthogonal coil pairs
mounted directly onto the flanges of the MOT cell through which the cooling andrepump beams are directed (see figure 3.13b). The pair providing the verticalcompensation field is mounted around the top and bottom flanges of the MOT
1234SEMIKRON, SEMiX 402GB066HDs
EPCOS, B59901D0060A040
Gentech International, FS-10
Ion pumps in particular produce significant static magnetic fields

Design, assembly and performance of our 39 K-87 Rb apparatus

80cell. They consist of 9 turns (3×3) of 1mm-diameter enamelled copper and havean inner diameter of 90mm. The other two pairs consist of 40 turns (5×8) of
1mm-diameter enamelled copper wire with an inner diameter of 72mm. Eachhorizontal pair produces a field of roughly 0.6G/A at its centre and the verticalpair generates roughly 1.7G/A. The currents to these coils are provided by a
3-output, low-ripple power supply1 .
Despite not being mounted precisely in Helmholtz configuration due to spaceconstraints, the coil separation was chosen so as to achieve as uniform a fieldin the trap centre as possible, with the non-uniformity estimated numericallyto be around -0.012GA−1 cm−2 for the horizontal pairs and -0.002GA−1 cm−2 forthe vertical pair. The independently-controlled currents in each coil pair were
adjusted in order to optimize the number and temperature of atoms transferred
into the quadrupole magnetic trap.

Note
For a description of the Feshbach, compensation and guide-field coils
mounted about the science cell, see section 3.11.

3.6 Optical pumping and magnetic capture
Our optical pumping and magnetic capture stages proceed as follows:• At the end of the 6ms molasses stage, all cooling beams are extinguishedand both repump beams are left on for a further 1ms in order to transfer
all atoms into their respective |F = 2i manifolds. The guide field is thengiven 800µs to ramp up before both pumping beams are switched on for200µs. The repump beams are only on for the first 100µs.
• Immediately after the optical pumping stage described above, all lights are
extinguished and the quadrupole coils are abruptly switched on at 40 G/cm
(strong axis) in order to capture the atoms, and then linearly ramped up to
160 G/cm in 50ms. The capture efficiency is typically ∼80% for 87 Rb and∼65% for 39 K.
1

Hameg Instruments, HM7042-5

3.7 Transport

81

3.7 Transport
3.7.1 Background - moving vs. stationary coils
Once the atoms have been optically pumped and magnetically trapped, theymust be transported over a distance of roughly 75cm to their final position within
the science cell. There exist several experimental approaches to the problem oftransporting laser-cooled atoms from one region of an experimental apparatus tothe other. This transport should transfer the maximal number of atoms whileremaining as adiabatic as possible. Some of the schemes that have been developed
and previously demonstrated include:• Use of moving red-detuned optical tweezers [152, 153]
• Use of resonant ‘push beams’ in conjunction with a 2D ‘atomic funnel’ MOT
to create a slow atom beam [154, 155, 156, 157, 57, 158, 159]
• Use of a chain of static magnetic coils in which coordinated control of thecurrents allows the creation of a moving magnetic trap [146, 160]
• Use of a single magnetic coil pair, translated by means of a mechanicalstage and confining the atoms magnetically for the duration of the transport[161, 162, 163, 164].
Given the vibrational stability issues encountered in translation of the lenses usedin moving optical tweezers [152, 165], the need for multiple additional laser beams
and magnetic coils associated with the ‘push beam’ method, and the many coils
and sensitive current optimization required for the static coil scheme, we opted
to implement a moving coil system in our experiment.
Advantages This scheme has the advantages of conceptual simplicity, ease of
setup and optimization and the use of only a single coil pair, significantly
reducing the construction, mounting, alignment and water-cooling burdens
compared to the static coil method.
Disadvantages The main disadvantages of the moving coil method are the needfor care to minimize vibrations due to the coil translation and the extra
volume swept out by the translation stage and any coil mounts, which
would normally be available for mounting components.
Both of these disadvantages can be ameliorated through prudent design of thevacuum chamber geometry, the coil mount and the transport sequence. The

82

Design, assembly and performance of our 39 K-87 Rb apparatus

high-precision translation stages and control/feedback electronics commerciallyavailable today allow for great accuracy and repeatability, and we have found thismethod to perform very well in our experiments.

3.7.2 Track details
The translation stage selected for our transport system1 is capable of 900mm oftravel, is fitted with an optical linear encoder with 0.5µm resolution2 , a 6000rpm
brushless servo motor3 and is controlled via a servo drive/controller4 communicating with our control PC.
An important selection criterion is the use of a ball-screw-based stage instead of alinear-motor-based stage to ensure that any magnetic field generated by the track
is localized as far as possible from the science cell. The magnets in a linear-motortranslation stage are located along the whole length of the track as well as on
the moving stage itself, whereas a ball-screw stage relies on a motor located at
its far end to rotate a precision screw which translates the stage along the track.
We position the motor at the MOT-cell end of the chamber, as far as possible
from the science cell. We have also found it crucial to to ‘de-energize’ the track(i.e., shut down the motor) immediately after the transport stage to eliminate
electronic RF noise emitted by the motor which was observed to severely reduce
the number of trapped atoms.
The quadrupole coils are mounted to the translation stage via an aluminium supporting ‘arm’ (see figure 3.13b), which also provides a channel for the electricaland water connections to the coil tubing. These are held within a flexible, articulated cable housing5 one end of which is anchored to the optical table and theother end of which is free to follow the track motion, keeping the water hoses andelectrical cables out of any beam paths while providing strain relief and sufficient
bend radius.
1
2
3
4
5

Parker 406XR series. Full part number: 406.900XRSD5.H14L13.C10.M29.E3.B1.R1.P1,
from which the various features and specifications can be extracted.
Renishaw, RGH24Z30F00A
Part number: SMH82.6003.8.14.21D65.2
Compax3, C3 S063 V2 F10.I12.T11
igus, E-Chain System® , 2500.09.125.0

3.8 QUIC trap83

3.7.3 Transport parameters and efficiency
The 741.21mm transfer takes approximately 1.7s and the parameters used are
given in Table 3.3.
Travel distancejerk

acceleration

velocity

741.21 mm

50,000 mm/s3

5,000 mm/s2

500 mm/s

Table 3.3: Transport parameters. Jerk is the time-derivative of the acceleration,and the acceleration and velocity given represent peak values.

We typically transfer well over 50% of the atoms from the MOT to the science cell,with the transfer efficiency for 39 K lower than for 87 Rb since its higher temperatureleads to removal from the trap via contact with the transfer tube walls. The
temperature of these ‘evaporated’ atoms can be estimated by equating the spatial
extent of the cloud, rcloud ≈ kB T /µB B 0 , with the transfer tube radius, rtube =0.5cm. For our gradient of B 0 ≈ 80G/cm along the weak trap direction, thisgives a temperature of Tevap ≈ 2.7mK.

3.8 QUIC trap
Our 39 K cloud must ultimately be transferred into a purely optical trap prior tocondensation in order to access the desired Feshbach resonance in the magnetically anti-trapped |1, 1i state. However, the 39 K sample must be cooled prior tothis transfer in order to ensure the maximal initial atom number in the optical
trap. This is accomplished via sympathetic cooling with 87 Rb and our precise
cooling targets are described in section 3.9. With 87 Rb alone it is possible toevaporate directly in the quadrupole trap in the presence of an optical ‘dimple’potential [166] offset from the trap centre, and in this way produce opticallytrapped condensates of 5 × 105 87 Rb atoms with less than 10s of evaporation.
However, 39 K cannot be effectively cooled sympathetically in similarly short evaporation sweeps due to the relatively small K-Rb interspecies background scattering length aK-Rb ' 36a0 [58], where a0 is the Bohr radius a0 ≈ 0.53Å, compared
to aRb ' 99a0 [167] for 87 Rb.
As mentioned in section 2.3, the region of very low magnetic field at thecentre of the quadrupole trap restores the degeneracy of the Zeeman states andallows adiabatic transitions to untrapped states which are rapidly lost fromthe trapping potential. Long evaporation sweeps in the quadrupole trap areDesign, assembly and performance of our 39 K-87 Rb apparatus

84

therefore prohibited due to loss of 39 K atoms via such Majorana flips as well
as via inelastic collisions with |2, 1i 87 Rb atoms likewise created by Majoranatransitions. Here too, several schemes have been proposed and demonstrated inorder to address the presence of these Majorana spin flips by ensuring that theatoms do not experience a region of zero magnetic field. In our system, we havechosen to implement a so-called Quadrupole-Ioffe configuration (QUIC) trap,a Ioffe-Pritchard-type trap first realised by Esslinger et. al. in 1998 [168] andsubsequently used by many other groups. This configuration offsets the trap
minimum to a positive value while converting the linear quadrupole trap to aharmonic trapping potential.

135mm!
51mm!

62mm!

100mm!

+z!
+y!

56mm!

152mm!

+x!

Figure 3.14: Schematic of the QUIC trap coil configuration in our system. The
quadrupole coil pair in conjunction with the Ioffe coil generate the offset harmonic
potential, with the rectangular anti-bias coil pair opposing the field along the axisof the Ioffe coil. See table 3.4 for a summary of the coil parameters.

3.8.1 Trap configuration
The QUIC configuration consists of a pair of coils producing a quadrupole field
together with a third coil whose axis is oriented perpendicular to that of thequadrupole coils as shown in figure 3.14. This third coil is known as the Ioffe coiland is responsible for producing the offset harmonic confinement when its field issuperposed with the quadrupole field.

3.8 QUIC trap85

In addition to the Ioffe coil, we make use of an additional ‘anti-bias’ Helmholtz
coil pair, also shown in figure 3.14, to provide a uniform field in the opposite
direction to that created by the Ioffe coil along its axis. This anti-bias field isused in order to provide better control over the trapping potential during loadingof the QUIC trap and the subsequent transfer into the optical trap.

3.8.2 Pros and cons
The quadrupole field used in our QUIC trap is generated by the same quadrupole
coil pair used in our MOT and transport stages. Since the atoms are alreadyconfined in this quadrupole field upon arrival in the science cell, conversion into
the QUIC trap is relatively straightforward and efficient, consisting of simply
ramping up the Ioffe and anti-bias coils so as to retain the largest number ofatoms possible in the resulting harmonic potential. Other advantages of the QUIC
configuration include its simplicity, the relatively large optical access afforded by
the reduced number of coils compared to other configurations and the good modematching that can be achieved between this type of trap and the gaussian intensity
profile of the optical dipole trap due to the harmonic nature of the confinement(as opposed, for example, to the optically plugged trap discussed in appendix B).
Disadvantages of this configuration are the weaker confinement compared to thequadrupole trap, which makes evaporative cooling less efficient as mentioned in
section 2.4, as well as the fact that during its operation the centre of the QUIC
trapping potential is displaced from the quadrupole trap centre in the direction of
the Ioffe coil, potentially reducing optical access to the cloud. We overcome thisproblem by evaporating in the displaced QUIC before ramping up the appropriate
currents in order to temporarily restore the cloud to the centre of the cell prior
to loading of the optical trap.

3.8.3 Trapping potential
The QUIC trap fields are similar to those produced by a standard Ioffe-Pritchard
configuration near the trap minimum [169], consisting of a harmonic potential
with a non-zero offset. To obtain an estimate for the trapping frequencies, we
can use the near-origin approximation for the field produced by a single coil1 to
1

See equation E.3 in appendix E

Design, assembly and performance of our 39 K-87 Rb apparatus

86

write the field produced by the Ioffe coil as
Ioffe
0
00
Bax
= B0 + Bax
x + Bax(x2 + (y 2 + z 2 )/2)p
1 0 p 2
Ioffe
00
Brad
= Bax
y + z 2 − Bax
x y2 + z2
(3.4)
2
p
0
where we have written the radial coordinate as ρ = y 2 + z 2 . Bax
is the ax00
ial field gradient, Bax is the axial field curvature and we have used the coordinate system of figure 3.14. This field is superposed with that of the quadrupole0
trap Bquad = B 0 (x, y, −2z) and the axial gradient Bax
is chosen to cancel the0quadrupole gradient B along the Ioffe coil axis at the quadrupole centre (i.e.
0= −B 0 ). Combining the two fields and keeping only terms to second orderBaxin position yields the following for the total field|Btot | =q
B2Ioffe + B2quad

100 2≈ B0 + Bax
x +
2


3 0 2
B
2
B0

!
00
− Bax
ρ2

(3.5)

where we have reinserted the radial coordinate,ρ. The trapping potential hence
has the form
2 2
2
U = gF mF µB B ≈ U0 + (1/2)mωax
x + (1/2)mωrad
ρ2

(3.6)

with the trapping frequencies given by
r
ωax =

00
2gF mF µB Baxmrωrad =

gF mF µB
m


3 0 2
B
2
B0

!1/2
00
− Bax

(3.7)

For typical values of the bias field, quadrupole gradient and axial curvature used00
in our sequence (B0 = 5.3G, B 0 = 64Gcm−1 and Bax= 120Gcm−2 , respectively),
this predicts trapping frequencies of ωax = 2π×21(14)Hz and ω⊥ = 2π×76(51)Hz,
respectively. Since the radial frequency is usually significantly higher than theaxial frequency, atoms in the QUIC trap occupy an elongated, cigar-shaped potential. Figure 3.15a shows the magnetic field profile along the axial direction as
the current in the Ioffe coil is increased in 5A increments with the quadrupole
gradient held at 80G/cm. In addition to the primary minimum moving towards
the Ioffe coil, a second field minimum appears outside the science cell and the two
minima approach each other until eventually joining to form an offset harmonic3.8 QUIC trap87

potential. There is therefore a risk during the Ioffe coil ramp-up, that atoms will
collide with the cell wall on their way to the secondary minimum. The intuitiverole of the anti-bias field is to ensure that the primary minimum is deep enoughduring the Ioffe-coil ramp-up such that when atoms do start spilling into the secondary minimum it is already within the science cell and the atoms are not lost
via collisions with the cell wall, as illustrated for our experimental parameters infigure 3.15b.
b)

200

160

160

80

120

Ioffe coil

Field [G]

200

120

Ioffe coil

Field [G]

a)

80
40

40

-2

-1.5

-1

-0.5

0

z [cm]

0.5

1

1.5

-2

-1.5

-1

-0.5

0

z [cm]

0.5

1

1.5

Figure 3.15: Field profile along the Ioffe coil axial direction (a) without and (b)with the anti-bias field contribution. The red dashed line shows the position of the
Ioffe coil and the shaded regions show the locations of the science cell walls. In(a), the Ioffe current is increased (light to dark, 0-25A) in 5A increments with thequadrupole gradient held at 80G/cm. Atoms spilling into the secondary minimum
collide with the cell wall. Figure (b) shows the profile at various points (light todark) through our actual experimental sequence, with the anti-bias field, directedalong −z, ensuring that atoms entering the secondary minimum do not collide withthe cell wall.

Figure 3.16 shows images of the cloud in the QUIC trap taken perpendicular to
the Ioffe coil axis (i.e. vertically in the lab frame) and showing a sequence with
progressively increasing anti-bias field. As the anti-bias field is increased, with
the Ioffe field remaining unchanged, the initially-tight axial confinement graduallyrelaxes until the dual minima can clearly be seen.

3.8.4 Ioffe coil and trap loading
Our Ioffe coil is located between the two quadrupole coils, at a distance of 2mm
from the outer wall of the science cell along its long edge (see figure 3.14). The
coil consists of 18 turns (3×6, axial×radial) of 1mm-diameter enamelled copper

88

Design, assembly and performance of our 39 K-87 Rb apparatus

Figure 3.16: Finding the anti-bias voltage in the compressed QUIC trap to provide
maximal compression without creating a magnetic field zero. The trapped cloud isimaged perpendicular to the trap axis and the Ioffe coil is located at the bottomof the image. From top left to bottom right: 22.77G, 23.16G, 23.82G, 24.49G,
25.15G, 25.81G, 26.47G and 27.14G

wire, and has a clear inner diameter of 10mm. It is mounted on a water-cooled
copper block with a clearance hole through which both the axial imaging and the2D confinement beams can access the cell.
Conversion of the quadrupole trap to a QUIC configuration as efficiently as possible requires coordination between the currents in the quadrupole, Ioffe andanti-bias coils. The transfer sequence lasts 5s in total and is shown in figure 3.17,
with the axial field value at various points during the transfer shown in figure
3.15.
With the quadrupole coils at 200A (80G/cm, weak axis), the Ioffe coil is ramped
up over 500ms to a current of 24A while the anti-bias coil pair is simultaneously
ramped up to 19.3A (28G) in order to counter the movement of the trap minimum
towards the cell wall caused by the Ioffe coil field. The current in the quadrupole
coils is then reduced to 160A (64G/cm, weak axis) over 250ms, after which theanti-bias current is ramped down to 3.4A (5G) over 500ms and finally, 250ms
later, the Ioffe coil current is gradually ramped down to its final value of 19.7A
over 3.5s, during which the trap centre moves gently towards the cell wall, causing
some removal of the hottest atoms from the trap.
With this procedure 4 × 108 87 Rb and up to 4 × 107 39 K atoms are loaded into

Ioffe Current [A]Quad gradient(weak) [G/cm]

3.9 Sympathetic cooling

89

80

64

24.4
19.7

Antibias Field [G]

0

1.5

28.1

5

0
0

0.5

1

0.75 1.25

2

3Time [s]45

6

Figure 3.17: QUIC transfer sequence. Currents and field values of the quadrupole,
Ioffe and anti-bias coils during conversion of the quadrupole field into the QUIC
configuration.

the QUIC trap and these final currents result in trapping frequencies of ωz /2π =14 (21)Hz axially and ωr /2π = 43 (64)Hz radially with respect to the Ioffe coilaxis for 87 Rb (39 K) in the |2, 2i ground state, in the presence of the 5G bias
field. In addition to numerical simulation, trapping frequencies in the QUIC trap
were determined by measuring the differential gravitational sag, ∆z ≡ zRb − zK ,between the trapped 87 Rb and 39 K clouds, from which the frequencies can begK
obtained using ∆z = 1 − mmRb.
ω2
Rb

3.9 Sympathetic cooling
Our sympathetic cooling target is to bring as large a number of 39 K atoms as
possible to a temperature of approximately 5µK. This goal is motivated by two
factors:
1. As discussed in section 3.10, this temperature coincides with the optimal
temperature found for loading the largest possible 39 K samples into ourDesign, assembly and performance of our 39 K-87 Rb apparatus

90
optical trap.

2. In our QUIC trap, the difference between the gravitational sags, g/ωr2 , for87
Rb and 39 K is 73µm. This corresponds to the (vertical) thermal radiifor the two species, RTRb ≈ RTK , at a temperature of roughly 4µK, wherep
RT = kB T / (mωr2 ). Below this temperature, the reduced spatial overlap
between the species renders sympathetic cooling inefficient.
The forced evaporation of the 87 Rb buffer gas is carried out by inducing the|F = 2, mF = 2i → |1, 1i transition between the |2, 2i state of our trapped atoms,
and the anti-trapped |1, 1i state residing in the lower hyperfine (F = 1) groundstate (see figure 3.18). These two states are separated by the ground state hyperfine splitting of 6.835 GHz, in addition to their relative Zeeman shifts due tothe presence of the magnetic trap, which in this case gives a total splitting of:fM W = fHF + 3µB B. We evaporate on this ‘microwave’ transition rather thanthe more closely-spaced Zeeman transitions in order to avoid cutting directly into
the 39 K cloud. The 39 K atoms are cooled via their elastic collisions with the cold87
Rb reservoir.
6909.68MHz!

mF!
2!

a)!
E!

b)!

1!
0!
-1!

!! HF

!! evap

QUIC!
Trap!
loading!

-2!

r!
-1!

F=1!

6873.68MHz!

6879.68MHz!

F=2!

0!
1!

2s!

6s!

τ=14s!

87Rb evaporation!

6847.24MHz!

56s!

c)!
10-turn"
helix!

SMA"
connector!
ground plate!

Figure 3.18: Hyperfine evaporation of 87 Rb. (a) 87 Rb atoms in the |2, 2i state
are transferred to |1, 1i using a microwave-frequency sweep. The 5G field offset
at the QUIC trap centre allows us to also selectively remove residual |2, 1i atoms
using a second, simultaneous sweep as described in the text. (b) The sweep sequence used for hyperfine evaporation includes a 2s preliminary evaporation in thequadrupole trap, a 6s interval during ramp-up of the QUIC trap and a 56s evaporation sweep with a time constant of 14s. (c) The helix antenna used for theevaporation (modified from [170]).

3.9 Sympathetic cooling91

The sympathetic cooling sequence proceeds as follows: In order to transfer as
many 39 K atoms into the QUIC trap as possible, we perform a short sympathetic
cooling stage while still in the pure quadrupole trap, ramping the microwave
frequency from 6909.68MHz to 6879.68MHZ linearly in 2s. The knife is then liftedto a high frequency during the 6s loading of the QUIC trap. Once our 87 Rb-39 K
mixture has been loaded into the QUIC trap, we employ a 56s exponential sweepof the frequency from 6873.68 to 6847.24MHz with a 14s time constant. To obtainthe starting frequency, we first find the ‘bottom of the trap’ i.e., the frequency
at which all atoms are just evaporated. Equation 2.36 then allows us to pick
an initial temperature from which to commence evaporation. The long sweep isrequired in order to allow for thermalization between 39 K and 87 Rb atoms due tothe small value of aK-Rb , whereas in 87 Rb-only experiments we observe an almost
identical efficiency with a shorter evaporation time of 32s. In addition, during
the evaporation time, we continually sweep a separate microwave source between
6838 and 6844MHz. The 5G field offset at the minimum of our QUIC trap allows
this sweep to remove any |2, 1i 87 Rb atoms without affecting the atoms in the|2, 2i state.

Hardware
The microwave radiation is delivered to the atoms from an antenna driven by afunction generator1 via an amplifier2 . In order to achieve efficient evaporation,
it is desirable for as much of the power delivered to the antenna to be radiatedin order to maximize the magnetic field strength at the location of the atoms.
Our antenna consists of a ten-turn helix with an overall length of 10cm, woundfrom 1mm copper wire around a teflon tube of wall thickness 0.4mm and innerdiameter 14mm. The helix is connected to a 30mm-diameter copper ground plate
fitted with an SMA connector and is impedance matched to the voltage source by
deforming the first 1/4 turn closest to the ground plate. The output power of our
25dB-gain amplifier is limited to 38dBm (∼6.3W), with the added directivity3of the antenna, Dpeak ≈ 4.5 (DdBi = 10 log[D] ≈ 6.5), providing sufficient powerat the atoms to perform efficient state transfer. Figure 3.18b shows the helix
antenna and figure 3.18c shows the antenna’s directivity measured for both linear
polarization components. See Igor Gotlibovych’s project report for more details
12
3

Agilent, N5183A MXG
Microwave Amplifiers Ltd., AM53-6.7-7-25-38
Antenna directivity is defined as the ratio of the power density radiated in the directionof strongest emission to that of an ideal isotropic radiator radiating the same total power.

92

Design, assembly and performance of our 39 K-87 Rb apparatus

on the setup and characterization of the microwave system [170]. The radiation
used for removal of |2, 1i 87 Rb atoms during the evaporation sweep is derived
from a sweep generator1 and is added to the main evaporation frequency on a
power combiner2 before being sent to the amplifier and helix antenna.

3.9.1 Sympathetic cooling results
Figure 3.19 shows absorption images3 of both 39 K and 87 Rb taken at various
points during the sympathetic cooling sequence.
39K!

87Rb!

T ≈ 200µK!
N ≈ 25 x 106!

T ≈ 120µK!
N ≈ 400 x 106!

T ≈ 5µK!
N ≈ 14 x 106!

T ≈ 4µK!
N ≈ 60 x 106!

Figure 3.19: Sequence of images taken at the start, middle and end of a typicalsympathetic cooling sequence. While 87 Rb cools from 120 to 4µK, losing roughlya factor of seven in atom number, 39 K is cooled from 200 to 5µK with less than afactor of two drop in number.

Figure 3.20 shows the evolution of the phase-space density of both the 87 Rb andthe 39 K components against their respective numbers during the sympathetic
cooling stage in the QUIC trap.
We can characterize the efficiency of our sympathetic cooling using the evaporation parameter, γ, defined in the discussion of evaporative cooling is section
2.4. For 87 Rb alone, we observe very efficient cooling with γ = 3.6. For 39 K we
observe an extremely steep rise in phase-space density with only small atom loss,
indicating the effectiveness of our sympathetic cooling. The temperature of the39
K cloud is reduced from 200 to 5 µK while the atom number is reduced by less1
2
3

Agilent/HP, 83622A synthesized sweeper
Mini-Circuits, ZX10-2-71-S+
See section 3.14.

3.9 Sympathetic cooling

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

93

Phase space density

87
87

39

10

7

Rb alone

Rb with 39K

K with 87Rb

10

8

Atom number

Figure 3.20: Evolution of phase-space density with atom number during sympathetic cooling. The solid black line shows 39 K cooling with less than a factor of two
drop in number. The dotted black line shows the data for 87 Rb in the presence of
39 K and the dashed black line shows 87 Rb alone. The dashed red line shows theresult of a fit to the sympathetic cooling model of [134], described in section 2.4.

than a factor of two, from ∼ 25 × 106 to ∼ 14 × 106 . The trap lifetime would
account for a roughly 20% reduction in NK over the evaporation period, so theobserved loss shows evidence for some inelastic processes. By analogy with conventional evaporative cooling we can express the efficiency of this process with
an effective γsymp ≈ 20.
Using the expression for the peak phase-space density of a non-degenerate cloudof N atoms, at temperature T, in a 3D harmonic potential of mean frequency ω̄ρ0 = n0 λ3dB = N



~ω
kB T

3
(3.8)

in conjunction with equation 2.451 allows us to model the evolution of the 87 Rb
phase-space density with (decreasing) atom number and compare it to our experimental results. A fit to the data is shown in figure 3.20. This fit has assumed
a constant 39 K number, taken to be the mean of the initial and final values, andyields the result α = 1.47 ± 0.03 for the evaporation parameter α = (η + κ) /3 − 1
defined in this model. Assuming a constant value of η = 7 throughout the evaporation, results in the prediction κ = 0.41. Despite its simplicity, the model of1

init
init αReminder: TRb = Tmin (NRb /NK + 1)α with Tmin = TRb(NK /NRb)

94

Design, assembly and performance of our 39 K-87 Rb apparatus

Temperature [µK][134] is seen to provide a good fit to the data. Figure 3.21 shows the temperatureof the 39 K cloud as a function of the number of 39 K atoms remaining at the endof the evaporative cooling stage. The temperature can be seen to approach thefinal temperature reached by 87 Rb in the absence of 39 K.

6
5
4
3
0

5

10

15

20 x 106

Number of K atoms after sympathetic cooling39

Figure 3.21: 39 K temperature as a function of the number of 39 K atoms remaining
at the end of the sympathetic cooling.

3.10 Crossed dipole trap (CDT)The final cooling stage is performed in a crossed optical dipole trap (CDT) after
transfer of the atoms from the QUIC trap. The light for our crossed optical dipole
trap is generated by a 10W, single-mode, linearly polarized, CW, Ytterbium fibre
laser1 , with a wavelength of λ = 1070nm and a linewidth specified at δλ ≈ 1.5nm
The laser is fitted with a collimator to produce a beam with a 1/e2 diameter of
5mm and a 1/e2 divergence of 0.3mrad. The red-detuning of the beam from theD2 transition for both species produces an attractive potential, with atoms drawn
to the intensity maximum at the beam centre.
Our trap is formed by a single beam arranged in a ‘bow-tie’ configuration asshown in figure 3.22.
This configuration, as opposed to two independently-controlled beams, was chosen in order to maximise the amount of trapping power available. The beam isinitially telescoped down to a 1mm diameter by a lens pair in order to enable
it to pass through a 110MHz AOM2 , selected primarily for its large active aperture. The beam then has its polarization cleaned up by a polarizing beamsplitter
1
2

IPG Photonics, YLM-10-LP-SC
Crystal Technology Inc., 3110-197

3.10 Crossed dipole trap (CDT)

95

300mm!
300mm!

From laser"
output!
250mm!

periscope!

125mm!

To beam dump"
and photodiode!

-25mm!

PBS!
-50mm!
100mm!

Figure 3.22: CDT optics layout. The beam diameter is initially reduced by a 1:5
telescope before passing through an AOM and PBS cube. It is then expanded andfocussed at the atoms by another lens pair. After passing through the cell, thebeam is collimated, refocussed and sent back to cross itself orthogonally, before
being sent to a beam dump and monitor photodiode. Distances are not to scale.

cube, before being expanded and refocused at the centre of the science cell by
a concave-convex lens pair (f = -50/+100mm) mounted in a rigid cage system.
This lens configuration allows simple adjustment of the beam waist between 100
and 160µm. After passing horizontally through the science cell, the beam is collimated, reflected and focused back to cross itself orthogonally with the same
waist. A maximum power of 7.5W per beam reaches the atoms, although wenormally limit this to 7W to ensure power stability.
The beam has a Gaussian intensity profile and so the combined potential taking
into account both beams as well as gravity, and neglecting the Rayleigh range,πω 2which for our 140µm waist is zR = λ 0 ≈ 6cm, can be written as
Beam along ‘y’
Beam along ‘x’
}|
{z
}|
{


gravity22
y 2 +z 2−2 x +z
−2
z}|{
2
2
w0
w0
Utot (x, y, z) = −U0 e
−U0 e
+ mgz

z


 2


 2

x + z2y + z2≈ −U0 1 − 2− U0 1 − 2
+ mgz
w02
w02

(3.9)
= −2U0 1 − (x2 + y 2 + 2z 2 )/w02

96

Design, assembly and performance of our 39 K-87 Rb apparatus

where U0 is the peak potential created by a single beam and w0 is the beam waist.
The potential can thus be approximated, for small displacements (x, y  ω0 ), as
harmonic, with the trapping frequency along the z-direction larger by a factor of√
2 than those along the x and y directions. Further discussion of the trap characteristics, including trap-depth, frequency, gravitational sag and anharmonicity
simulations, can be found in the doctoral thesis of Robert Campbell.

Intensity stabilization
The optical trapping potential can be stabilized, and its depth varied over twoorders of magnitude, by means of a servo loop consisting of the AOM throughwhich the beam is initially passed, a photodiode1 and a proportional-integralderivative (PID) controller, whose output is fed back to the AOM in order tostabilize the beam intensity in proportion to a reference voltage provided by our
computer-controlled analog voltage sources (see section 3.16). The photodiode isilluminated by a small (but constant) fraction of the total CDT power, picked off
the main beam by a glass microscope slide after both its traversals of the sciencecell.

3.10.1 Transfer into the CDT
In preparation for transfer of the atoms into the CDT, the QUIC trap is translated
to the centre of the science cell as follows: The quadrupole gradient is reduced
linearly from 128G/cm to 80G/cm over 1s, while the current in the Ioffe coil issimultaneously ramped up from 19.7A to 28A. During the second half of thisramp, the anti-bias field is linearly increased from 5G up to 22.8G.
In order to load the CDT from the QUIC trap, the power in the CDT beam islinearly ramped up to 6.8W in 1s, followed by a 0.5s settling time. Over thefollowing second, the quadrupole gradient is reduced to 48G/cm, the anti-bias
field to 3.72G and the Ioffe coil current to 15.6A. The quadrupole and Ioffe coils
are then abruptly switched off, leaving the atoms confined solely by the CDT,
whose depth at this point is ∼30µK for 39 K and ∼35µK for 87 Rb. The anti-bias
field remains on in preparation for the removal of the 87 Rb from the trap (seebelow).
1

Thorlabs, DET36A/M

3.10 Crossed dipole trap (CDT)

97

Figure 3.23 shows the number of 39 K atoms transferred into the CDT as afunction of their final number in the QUIC trap, together with the temperatures
before and after the transfer. For samples with temperatures of up to 6µK in
the QUIC trap, equivalent to an evaporation parameter of η ∼ 5, we observe
a constant transfer efficiency of ∼60%. This constant factor is due to theimperfect spatial overlap between the two trapping potentials and is limited by
the geometries of the two traps, despite the large (140µm) CDT waist being
beneficial. For the QUIC trap frequencies specified above for 39 K, the radial andaxial diameters of the cloud at 5µK are 2RT ≈ 160 and 500µm, respectively,pusing RT = kB T /mω 2 .

5.9

9 x 106

39

K Number in CDT

8

(5.2)

5.6

7

(6.2)

(4.1)

6

6.8

5
5.0

4
3
2

(3.1µK)

1
0

(3.5)

4.9µK

0

5

39

10

15

K Number in QUIC

20 x 106

Figure 3.23: Transfer efficiency from QUIC to CDT. Plot shows number of 39 K
atoms initially in the CDT as a function of the final number in the QUIC trap,
together with the temperatures before (in brackets) and after the transfer.

For comparison, the loading efficiency was measured for a 100-µm beam waist andwas found to be limited to ∼20%, saturating at around a third of the available
CDT power. We cannot currently increase our beam waist without decreasing
the trap depth since we are operating at our maximum available laser power. We
typically load 8 × 106 39 K atoms into the CDT at a temperature of 6µK, fromthe 13 × 106 atoms remaining in the QUIC trap.

Design, assembly and performance of our 39 K-87 Rb apparatus

98

3.10.2 Optical evaporation strategy
With our atoms in the CDT, we are faced with the decision of how to proceed withcooling of the 39 K atoms. Since the potential is now state-independent, reductionof the trap depth is accomplished by reducing the CDT beam intensity. In orderfor sympathetic cooling to be effective in the CDT, the total trap depth for 87 Rb
has to be lower than that for 39 K so that the former can be removed without loss of
the latter. The relative trap depths for 87 Rb and 39 K depend on the competition
between two factors: (1) the strength of the electric dipole potential for the two
species and (2) the deformation of the potential by gravity. Both effects are larger
for 87 Rb, the former due to the smaller detuning of the CDT beam from the 87 Rb
D line and the latter because of its higher mass. As a result, at our initial laser
intensities the effect of gravity is negligible and atoms are preferentially lost along
the CDT beams such that the trap depth is higher for Rb. Only at sufficiently
low intensities does gravity distort the trapping potential enough for the Rb
trap depth to fall below that of K, making sympathetic cooling viable. See thecorresponding chapter of Robert Campbell’s thesis for a more detailed discussionof the optical trap depth. We therefore choose to switch to direct evaporation of
39
K and immediately after transfer of both species into the CDT we remove any
remaining 87 Rb atoms from the trap with a 2ms pulse of imaging light in order toprevent any spin-exchange collisions during the subsequent state transfer of 39 K
into the |1, 1i state, as described below.

3.10.3 State transfer
In order to prevent collapse of the 39 K cloud at low temperatures due to its attractive interactions [68, 69, 171], as well as to increase the evaporative coolingefficiency, we tune the scattering length to a positive and large value by meansof a magnetic Feshbach resonance1 . The Feshbach resonance on which our experiments rely occurs in |1, 1i + |1, 1i collisions of 39 K and so in preparation for
the application of the uniform Feshbach magnetic field, we transfer the entire
population from the |2, 2i to the |1, 1i state. The transfer is accomplished by a
35ms Landau-Zener sweep of the magnetic bias field from 3.72G to 3.14G in thepresence of 469.3MHz radiation, after all 87 Rb has been removed from the trap.
The 469MHz signal is generated by the same function generator used to pro1

See section 2.6

3.10 Crossed dipole trap (CDT)99

duce our 6.8GHz 87 Rb-evaporation radiation; its 5dBm output is controlled by a
voltage-controlled RF switch1 and amplified by a broadband RF amplifier2 to a
power of approximately 4W (36dBm) before being sent to a single-loop antennaof 36mm diameter, mounted horizontally less than 20mm from the location of theatoms. The amount of radiated power, measured by means of a small pick-up coil
was found to be hugely increased by the insertion of a tuneable 1-3pF capacitor3
in series with the antenna, whose primary role is the cancellation of the dominant
inductive reactance of the antenna, Xloop = ωLloop , with the inductance, Lloop ,given by [172, 173]

  
8a
−2
(3.10)
Lloop = µ0 a lnbwhere a = 18mm and b = 0.5mm are the loop and wire radii, respectively4 .
The use of such ‘tuning capacitors’ for impedance matching of loop antennas iswell known [172, 173]. The resonating capacitance is found by equating the tworeactances: (ωC)−1 = ωLloop , which for our parameters gives C = 1.4pF and does
not account for any ohmic losses due mainly to the skin effect in the wire. The
circumference of our antenna is C≈11.3cm, i.e. C ≈ λ/6 at 469MHz, classifying
it as a small-to-intermediate size loop.
Direct optical evaporation of 39 K proceeds, after the state transfer and ramp-up
of the Feshbach field, via an exponential sweep of the beam intensity over 6s fromits initial value of 6.7W per beam down to the desired final depth, which depends
on the desired final temperature or condensed fraction.

3.10.4 Trap frequencies
In order to measure the CDT trapping frequencies, we excite an oscillation of
the trapped cloud and extract its centre-of-mass position as a function of time.
We prepare a small, fully condensed cloud by performing an optical evaporation
sweep to an almost negligible trap depth (a few nK) and then abruptly ramping
up the trapping beam power to a final depth of 2.6µK in a two-piece linear rampof total duration 110ms, as shown in figure 3.24a.
The centre-of-mass vertical coordinate of each image was found from a fit to a12
34

Mini-Circuits, ZYSWA-2-50DR
Delta RF Technology, LA2-1-525-30
Vishay BCF280905215, 1-3.5pF, 300VDC voltage rating, can withstand 600V for 1 minuteExpression 3.10 for the inductance is valid for b  a i.e. when the wire radius is muchsmaller than the loop radius.

Design, assembly and performance of our 39 K-87 Rb apparatus

100

a)!

2.6µK!

b)!

730
720

Pixel

710

eva

por

3nK!

300nK!

700
690

atio
n!

680

100ms! 10ms!

240ms!

670
0

10

20

30

40

Hold time (ms)Figure 3.24: Measuring the CDT trapping frequency. (a) following optical evaporation, the CDT intensity profile is ramped up over 110ms and (b) the centre-ofmass position of the cloud is measured after a variable oscillation time and fit to asine function to extract the trap frequency.

pure Thomas-Fermi profile. Figure 3.24b shows the oscillation in the verticaldirection, together with a best-fit sinusoid which yields a frequency along thez-axis (vertical in lab frame) of 91.8 ± 1.0Hz. A numerical simulation of the CDT
trapping potential, including the effects of gravity, yields a predicted frequencyof ωz = 93.9Hz for this trap depth assuming no misalignment between the CDT
beams, in good agreement with the data.

3.11 Feshbach field and Feshbach resonanceWith the 39 K atoms transferred to the |1, 1i state, we can now access the Feshbach resonance at 402.5G in order to tune the s-wave scattering length to thedesired value. As discussed in chapter 2, this is accomplished by applicationof a homogeneous magnetic field and this section describes the coils used, theirmounting about the science cell, their field calibration and our measurements
used for locating the position of the resonance and the zero-crossing.

3.11.1 Feshbach coils
The Feshbach field is generated by a Helmholtz coil pair, with each coil consistingof 14 turns (2×7, axial×radial) wound from 4mm outer-diameter (OD) copper
tubing (ID: 2.8mm), coated in extra-thin heat-shrink material and having an
inner diameter of 70mm. The coils are mounted 37mm apart at their closest
point and produce a field of approximately 540G at the point midway between

3.11 Feshbach field and Feshbach resonance101

them when operated at their maximal current of 200A. These parameters werechosen in order to provide access to the 402.5G-resonance while keeping the sizeof the coils sufficiently small so as to be able to place them in close proximity
to atoms, within the space between the science cell and the quadrupole coils.
The field curvature at the location of the atoms is numerically estimated to be
7 × 10−4 GA−1 cm−2 .

3.11.2 Science cell coil mount
The Feshbach coil pair, as well as 5 auxiliary Helmholtz coil pairs (two axial,
two transverse and one vertical, with respect to the science cell’s long axis1 , theRF-evaporation antenna and the 462MHz hyperfine state-transfer antenna aremounted in their respective positions around the science cell by a single mounting
structure, designed to hold all the above components within the restricted space
between the science cell and the moving quadrupole coils.

Figure 3.25: The mount constructed for supporting the Feshbach, compensation,
anti-bias and TOP trap coils. An additional mount (not shown) is attached so as
to fit within the upper coil’s inner diameter and supports the meniscus constituting
the first component of the multi-element objective used for vertical imaging (seesection 3.14) at a distance of approximately 15mm from the science cell’s outerwall.

The auxiliary coils are used for providing the uniform compensation and bias
fields required during different stages of the experimental sequence such as QUIC
1

See figure 3.6.

Design, assembly and performance of our 39 K-87 Rb apparatus

102

trap loading, state transfer and imaging. The mount is constructed from Polyoxymethylene1 , a thermoplastic material possessing high stiffness, dimensional
stability and machinability, often used as a replacement for metals in mechanicalparts. This was used in order to avoid any eddy currents arising from the rapid
switching of the Feshbach, anti-bias and Ioffe coils during the experimental sequence. Figure 3.25 shows the coil mount before and after its insertion into theapparatus, and the specifications of all the coils mounted in the vicinity of thecell are summarized in table 3.4.

Name

ID
[mm]

Turns

Wire  Spacing Field
[mm]
[mm]
[G/A]

Feshbach

70

2×7

4/2.8

37

2.6

Feshbachresonance

62×152

8

0.71

56

1.14

Horizontal
imaging guide

74×149

8

0.71

56

1.14

—

190

3×7

1

70

2.28

Verticalimaging guide

62×152

10

1

56

1.46

QUIC anti-bias/
double-shuttering

74×149

10

1

56

1.43

—

10

3×6

1

NA

-

QUIC trap

36

1

1

NA

-

Evaporative
cooling

36

1

1

NA

-

15

10 (helix)

1

NA

-

Compensation(transverse field)
Compensation(axial field)
Compensation(vertical field)
TOP(transverse field)
TOP(axial field)
IoffeRF
antenna
462MHzantenna
Microwave
antennaRole

39

K statetransfer

87
Rbevaporation

Table 3.4: Summary of the various coils mounted in the vicinity of the science celland their properties.

1

Also known as acetal, or by one of its trade names, Delrin.

3.11 Feshbach field and Feshbach resonance103

3.11.3 Feshbach field calibration
We perform RF spectroscopy on the |2, 2i → |2, 1i transition of 87 Rb in order tocalibrate the field produced by the Feshbach coils. This is carried out by applying
RF radiation to a cold, |2, 2i 87 Rb cloud with a fixed frequency corresponding
to the splitting between the |2, 2i and |2, 1i states in the presence of a 402.5G
field. The splitting is found from the Breit-Rabi formula of equation 2.29 tobe ν402.5G = 249.444MHz, and this radiation is applied using the same function
generator and loop antenna used for 39 K state transfer (section 3.10.3). The
voltage controlling the current in the Feshbach coils is then swept linearly in
200ms over sequential overlapping intervals with a width of 0.004V and centred
at increments of 0.002V, until we observe transfer of the entire cloud to the |2, 1i
state (see figure 3.26).
a)!

b)!

c)!

d)!

e)!

Figure 3.26: A typical sequence of images used for calibration of the Feshbach
field. The atoms are irradiated with a frequency corresponding to the energy
splitting at 402.5G and the control voltage is swept over a series of overlapping
intervals: (a) 3.800V → 3.804V, (b) 3.802V → 3.806V, (c) 3.804V → 3.808V, (d)3.806V → 3.810V, (e) 3.808V → 3.812V. Transfer of atoms from the |2, 2i state(lower cloud) to the |2, 1i state (upper cloud) is monitored. In this case, transfer
occurs somewhere in the overlap between intervals (c) and (d) i.e. at a voltage of
3.807V, which therefore corresponds to a field of 402.5G.

In order to distinguish between the two states during imaging, a Stern-Gerlach
field is applied in order to spatially split the two components during time offlight. The Stern-Gerlach field is applied by jumping the quadrupole coil current
to 46A over 1-2ms in order to generate a gradient of ≈20G during the 14ms
time of flight, and results in a larger deflection for the |2, 2i component. Bothcomponents are simultaneously imaged by our σ + -polarized |F = 2i → |F 0 = 3ilight, with the Feshbach field being switched off 5ms prior to the imaging pulse. In
the current setup, we have found a control voltage value of 3.807V to correspondto a field of 402.5G, and we use this procedure to periodically ensure that thiscalibration has not drifted. Below we describe the experiments carried out inorder to characterize the resonance, namely location of the resonance centre andthe zero-crossing field.

104

Design, assembly and performance of our 39 K-87 Rb apparatus

3.11.4 Detecting the resonance
The presence of Feshbach resonances is most commonly detected by monitoring
atom losses in the cloud as the scattering length is tuned across the resonance.
In the vicinity of a Feshbach resonance, the scattering length rises sharply, as
described by equation 2.56, leading to an increase in the rate of inelastic threebody collisions. The three-body coefficient, K3 , for these collisions, defined byṅ = −K3 n3 , is expected to rise sharply with the s-wave scattering length and for
low-energy collisions at large scattering length, has been predicted to scale with
its fourth power, K3 ∝ ~a4 /m [174]. Experimental results consistent with thisprediction have been reported [175]. The divergence of a on resonance therefore
causes a correspondingly sharp rise in the inelastic collision rate. These threebody collisions result in the formation of a diatomic molecule, with the binding
energy released in the process being removed by the third colliding atom. Since
the final energy of this atom usually exceeds the trap depth, this process results in
atom loss from the trap. Furthermore, since this three-body recombination takes
place preferentially in the denser, cooler region of the cloud, the ejected atom
removes an energy below the mean thermal energy and hence causes heating of
the sample. To see this more quantitatively, we can consider the mean potential
energy associated with a thermal atom at temperature, T, trapped in a harmonic
potential and participating in such a three-body process
R
U n3 dV1= kB ThU i = R 3(3.11)
2n dVwith U = mω 2 r2 /2 and n = n0 exp [−mω 2 r2 /2kb T ]. This is less than the 32 kB T
mean potential energy in the gas and hence each atom lost due to three-bodyrecombination releases an energy of kB T into the system.
Observation of these sharp loss and heating features is therefore a useful signatureof the resonance and can be used to locate its centre. We perform this Feshbach
spectroscopy on our trapped 39 K atoms using the following procedure:
1. The atoms are held in the |1, 1i state in the optical trap at a temperatureof around 1.2µK.
2. The Feshbach field is taken to a value of approximately 403.9G and held at
this value for 10ms. This is done in order to try and eliminate the common
asymmetry in the loss profile that results from the field having to pass

3.11 Feshbach field and Feshbach resonance105

through the resonance centre on its way to values that lie beyond it, andhence suffering excess losses compared to field values on the same side of
the resonance as the initial field value. We therefore choose an initial fieldvalue as close as possible to the resonance centre.
3. The field is then swept to a final value in 20ms and held at this value for
100ms. This hold time is chosen so as to allow for a significant maximal
drop in atom number at the resonance, while not losing enough atoms so
as to blur the resonance peak as a result of reducing the density enough forthe losses to ‘saturate’.
4. The cloud is released from the trap and absorption imaged after an 8ms
time-of-flight. Upon release, the field value is taken close to the zero-crossing
at 350G in order to prevent any additional losses during TOF.
5. The atom number and temperature of the cloud are extracted and averaged
over 5 repetitions of the above sequence.
The results of this sequence are shown in figure 3.27. The atom number is seen
to drop sharply by a factor of two while the temperature concurrently rises byapproximately 25% at a field value very near 402.5G. The agreement of thisvalue with the results reported by other groups [60, 176] give us confidence inour field calibration and experimental procedure. Note that in figure 3.27 wealso observe additional losses and heating on the attractive (high-field) side of
the resonance centre, compared to those on the repulsive side. This is likely
due to the asymmetry in the three-body loss coefficient K3 on either side of theresonance [176, 177], with the a < 0 value of K3 being around 100 times larger
than its a > 0 counterpart just 0.3G from the resonance centre1 .

3.11.5 Detecting the zero-crossing
Two procedures have been used by various groups in attempting to detect thelocation of the zero-crossing for a given resonance.
1. The vanishing of the scattering cross-section causes a drop in the efficiencyof evaporative cooling due to the divergence of the rethermalization time(see section 2.4). This can be detected as a peak in the final temperature achieved during the evaporative cooling stage and, in the case of free,1

see figure 1c of [176]Design, assembly and performance of our 39 K-87 Rb apparatus

106

0.32
0.3

a)!

Atom number (106)

0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
1.6
1.55

b)!

7HPSHUDWXUH .

1.5
1.45
1.4
1.35
1.3
1.25
1.2
398 399 400 401 402 403 404 405 406 407 408 409 410 411)HVKEDFKÀHOG *Figure 3.27: Detection of the resonance centre. The location of the resonancemanifests itself as a sharp drop in the atom number and a corresponding rise in
the temperature due to the sharp rise in the rate of inelastic three-body collisions.
The resonance centre was found at 402.5G, in excellent agreement with previous
results [60, 176].

unforced evaporation, as a peak in the number of atoms remaining in thetrap [178, 179]. Similarly, in heteronuclear systems the zero-crossing ofinter-species resonances is found by monitoring the efficiency of sympathetic cooling as reflected in the final temperature and atom number of
the target gas, which both take on their peak values at the zero-crossing[61, 180, 181].
2. Alternatively, the zero-crossing can be located by observation of the onset ofcollapse of the condensate. This manifests itself in an abrupt drop in atom
number when the condensate becomes unstable. This instability occurs at
a negative value of the scattering length when the atom number increases

3.11 Feshbach field and Feshbach resonance107

sufficiently such that the self-attractive forces due to the negative scattering
length outweigh the repulsive force due to the zero-point energy of the atoms
in the harmonic trap [68, 69, 182, 171].
We have searched for the zero-crossing using the latter method. A 39 K condensate
is held in the CDT for three seconds, during the last 100ms of which the Feshbach
field is ramped from an initial value of around 390G to a value in the predictedvicinity of the zero-crossing at 350G in 50ms and held at this value for a further
50ms. The cloud is then imaged after a 15ms TOF. Figure 3.28 shows a sequenceof images taken as the Feshbach field approaches and then passes the zero-crossing
from above. After an initial, coarser, search the range of final values was narrowed
and the onset for the fragmentation of the condensate was found to appear ata field of between 350.2G and 349.7G. We therefore place the zero-crossing ata field of 350.0 ± 0.2G, giving a resonance width of ∆ = 52.5G, in very good
agreement with theoretical predictions and other experimental results [60, 36].

353.1G!

348.9G!

collapse!

351.0G!

350.6G!

350.2G!

349.7G!

349.3G!

Figure 3.28: BECs during coarse- and fine-tuning of the field near the zerocrossing

A more accurate estimate could perhaps be obtained by utilizing the first of thetwo techniques mentioned above and measuring the final temperature of the 39 K
cloud after the direct evaporative cooling stage in the CDT.

Design, assembly and performance of our 39 K-87 Rb apparatus

1083.12 Onset of BEC in 39K

Line density [103 μm-1]!

Evaporative cooling in the CDT is used to take the 39 K sample across the BEC
critical point. Figure 3.29 shows a sequence of images taken at various points
during evaporative cooling in the CDT, along with their density profile fits as
described in section 3.15 below.

2.5!
2!
1.5!
1!
0.5!
0!

-200

0

200!

Position [μm]!

-200

0

200!

Position [μm]!

-200

0

200!

Position [μm]!

-200

0

200!

Position [μm]!

Figure 3.29: Optical evaporation of 39 K across Tc . The leftmost image is taken
above Tc and in the following images the condensed fractions are 0, ≈20%, ≈40%and ≈80%, respectively, with the condensate number growing from approximately
1.5 × 105 to 4.2 × 105 as the temperature decreases from 170nK to 50nK. The
bottom row shows the theoretical fits to the data as described in section 3.15.

3.13 Sequence summary
Figure 3.30 shows an overview of a typical experimental sequence for producing
condensed clouds of 39 K in the CDT, including a plot of the evolution of boththe 87 Rb and 39 Rb peak number densities as a function of their temperature
throughout the sequence.

3.14 Imaging setup and procedure
This section describes the optical setups we use in order to capture absorption
images, both horizontally and vertically in the lab frame, following release of
the cloud from the magnetic or optical trapping potential. We also describe ourdouble-shuttering imaging procedure and the fitting procedure used to extract
physical information from the resulting absorption images.

3.14 Imaging setup and procedure

109♯! Stage!

duration!

1.! MOT!

NK / NRb!

T!

15s!

2x108 / 4x109!

1.5mK / 300μK!

2.! Optical Molasses!

6ms!

--------!

600μK / 50μK!

3.! Optical pumping!

200μs!

--------!

--------!

4.! Magnetic capture!

50ms!

1.3x108 / 3.2x109!

--------!

5.! Transport!

2.7s!

--------!

--------!

6.! Transfer to QUIC!

6s!

2.5x107 / 4x108! 200μK / 100μK!

7.! Sympathetic cooling by 87Rb! 56s!

1.4x107 / 6x107!

5μK / 4μK!

8.! Transfer to CDT!

2s!

8x106 / 3x107!

6μK / 5μK!

9.!

Removal of 87Rb!

2ms!

-------!

--------!

10.!

39K state transfer!

35ms!

-------!

--------!

5ms!

-------!

--------!

6s!

4x105 (39K)!

~400nK!

20ms (+180μs)!

4x105 (39K)!

~400nK!

11.! Feshbach field ramp-up!
12.!

Optical evaporation of 39K!

0 !4

3

!

0 !8

3

=1

dB

!

3

n

7-9!

=1

dB

BEC!

12-13!

n

1014

=1

00

13.! TOF (+Imaging)!

=1
0 !12

!

dB

n

3-6!

3

7,8,10,11!

!

=1

0 !14

dB

n

3

1!

!

dB

1010

2!

n

Central density (cm-3)

1012

3-6!
2!

1!

108

UHV!
background!

106

104

10-8

10-6

10-4

10-2
Temperature (K)

100

102

104

Figure 3.30: Summary of the experimental sequence for cooling 39 K to quantum
degeneracy. The table shows the duration of each step together with the associatedatom numbers and temperatures and the plot shows the evolution of both species’
peak number densities against their temperatures. Labels on the plot correspondto rows in the table and lines of constant phase-space density are indicated.

110

Design, assembly and performance of our 39 K-87 Rb apparatus

3.14.1 Horizontal imaging system
Our horizontal imaging system is an adaptation of the compound-lens design
proposed by Alt [183]. The motivation behind this design is to construct a highnumerical aperture objective using cheap, off-the-shelf optical components byusing the aberrations present in each component to cancel each other, as well as
the aberrations introduced by the science cell wall, and hence achieve diffractionlimited performance with a working distance large enough so as not to obstruct
the paths of the CDT beams. The original constraints presented in [183] weresimilar to those of our system in the horizontal direction and the only change
was the use of a standard meniscus as the first element instead of the custom
meniscus used by Alt. The recalculation of the lens spacings and the assembly
and testing of the objective were carried out by Alexandre Dareau using the
OSLO optimization package, and the detailed results can be found in his project
report [184].
In brief, the surface of the first meniscus in the objective is located ≈27mm
from the science cell wall along the optical axis, and the CCD surface is located≈111mm from the final surface of the focussing achromatic doublet. These spacings represent only initial guesses. The objective is mounted on a translation
stage and the distances are optimized empirically so as to focus on the atomic
cloud since it may not be situated precisely in the centre of the cell.

Horizontal-imaging magnification
The magnification of the horizontal imaging system is determined by measuring
the centre-of-mass (CoM) position of a dropped cloud at various times after its
release from the trapping potential, and fitting its trajectory to that expected forfree fall under gravity, h = 12 M gt2 . Our knowledge of the camera’s pixel size andthe fall time allows us to obtain the effective pixel size, i.e. the magnification. For
the horizontal objective, we obtain a magnification of Mhoriz = 2.8

3.14.2 Vertical Imaging system
The vertical imaging system was also designed by Alexandre Dareau andfull details are available in his report. The main additional constraint on
this system was the need to leave room for the translating quadrupole coils

3.14 Imaging setup and procedure111

To camera è!

j!
i!

h!
g!

Quadrupole !
Coil w/mount!

f!
Feshbach coil!
Camera!
atoms!

e!d! c!b!a!

Figure 3.31: Overview of the horizontal and vertical imaging systems around thescience cell.

Label

Description

Horizontal objectivea1” meniscus, f=100mma→b
2.53mmb
1” plano-convex lens, f=75mmb→c
2.80mmc
1” bi-convex lens, f=100mmc→d12.64mmd1” plano-concave lens, f=-75mm
d→e
arb.
e
1” achromatic doublet, f=100mm
e → CCD≈110mmPart number
LE1234-B
LA1608-B
LB1676-B
LC1582-B
AC254-100-B

Table 3.5: Descriptions, separations and part numbers for the components of ourhorizontal imaging system. Labels correspond to those shown in figure 3.31 andunless otherwise stated, part numbers are from Thorlabs. See Alexandre Dareau’s
project report for further details.

Design, assembly and performance of our 39 K-87 Rb apparatus

112

as shown in figure 3.31, with the imaging beam passing through the coilcentre. In order to cover as large a solid angle as possible, 2” optics wereused for this system, and a custom meniscus was mounted in close proximity to the science cell in order to further increase the effective numerical aperture.

Label

Description

Part number

Vertical objective
25mm Custom meniscus,fLens-Optics GmbH.
R1 = 26.06mm, R2 = −78.16mm
f→g
≈54mm
g
2” plano-convex lens, f=150mm
LA1417g→h
2.53mmh2” bi-convex lens, f=100mm
LB1630h→i
2.80mmi2” plano-concave lens, f=-75mm
LC1315i→j≈150mmj2” achromatic doublet, f=200mm
AC508-200-B
j → CCD≈200mmTable 3.6: Descriptions, separations and part numbers for the components of ourvertical imaging system. Labels correspond to those shown in figure 3.31 and unlessotherwise stated, part numbers are from Thorlabs. See Alexandre Dareau’s project
report for further details.

Vertical magnification

We obtain the magnification of the vertical imaging system by simultaneous vertical and horizontal imaging of a cloud released from the CDT after roughly 20ms
time-of-flight. Since in this time, any anisotropy due to the trapping potential
will have decayed away we can assume that the vertical and horizontal sizes of
the cloud are equal. Our knowledge of the horizontal-imaging magnification (seeabove) and the pixel size therefore allow us to infer the vertical-imaging magnification. We obtain a magnification in this direction of Mvert = 5.2 when using adoublet of focal length 200mm as the final element.

3.14 Imaging setup and procedure

113

3.14.3 Cameras and double shuttering
The camera model currently used in both the horizontal and vertical imaging
systems1 utilizes a so-called ‘interline’ CCD. The advantage of this type of sensor from our perspective is that it enables two separate images to be captured invery rapid succession, with inter-frame times as low as 5µs. This double-shutteringfeature, often used in particle image velocimetry (PIV) applications, allows us toeffectively remove the effects of beam inhomogeneity from our final OD profile
since any of the inevitable inhomogeneities in the imaging laser beam will nothave time to fluctuate between the images taken with and without the atomic
cloud - the I and I0 of equation 2.86, respectively - and will hence be efficiently
divided out when extracting the optical density. The primary disadvantage of thistype of sensor is its relatively modest quantum efficiency at the imaging wavelengths, approximately 25%, compared to more sophisticated back-illuminatedand electron-multiplying (EMCCD) cameras, in which efficiencies of > 95% areachievable. The camera is controlled via a custom graphical user interface (GUI)running within MATLAB, that automatically generates the final OD image(s)for each experimental run and also allows browsing and basic analysis of images
in real time.

Double-shuttering procedure
Since double-shuttering relies on the use of a masked region of the CCD for
temporary storage of the first exposure while the second exposure is recorded,
the duration of the second exposure must be at least as long as the time required
for the camera to read out the first exposure from behind the mask (≈80ms).
Therefore, since a double-exposure pair with very different exposure times isused to capture I and I0 , an identically exposed pair must be used to capture thedark background images for each of the two (see equation 2.86). Therefore two
double-exposures are used to record four images, with the third serving as thedark image for the first and the fourth serving as the dark image for the second.

3.14.4 Imaging procedure
Our 39 K imaging sequence proceeds as follows:
1

PCO imaging, pixelfly QE, double shutter, pixel size: 6.45µm, sensor: Sony ICX285AL

114

Design, assembly and performance of our 39 K-87 Rb apparatus

1. The Feshbach field is switched off 5ms prior to the end of time-of-flight.
2. Since our 39 K atoms are held in the |1, 1i state, while we wish to image on
the cycling |F = 2i → |F 0 = 3i transition, immediately prior to imaging all
the atoms are pumped back into the |2, 2i state by a 100µs pulse of repump
light.
3. The atoms are then illuminated with an 80µs pulse of circularly polarized,
low-intensity light (I/IS ≈ 0.2), resonant with the |F = 2i → |F 0 = 3itransition, and an image of the intensity in the plane of the atoms is recordedby the camera1 . During the imaging pulse, a quantization axis is defined byapplying a guide field of several Gauss along the propagation direction of
the imaging beam using the transverse compensation coils (see table 3.4).
4. A second imaging pulse is applied 60µs after the end of the first in order toimage the unattenuated beam profile, I0 (x, y). During this 60µs interval,
the vertical compensation field is rapidly pulsed to a few (∼ 10) Gauss inorder to detune the atoms from the imaging light and hence make theminvisible to the imaging beam.
5. Approximately 600ms later, a pair of background dark images is captured,with the CDT being briefly pulsed beforehand at the correct moment inorder to replicate more precisely the lighting conditions under which thefirst two images were taken.

3.14.5 Double-species imaging
We have also successfully used double-shuttering to obtain simultaneous images of
the 87 Rb and 39 K clouds. Three double-exposures are taken, with the first, thirdand fifth images constituting I, I0 and Idark for the first species, respectively, andthe second, fourth and sixth providing the same for the second species.

3.14.6 Cross section calibration
The inferred atom number depends on the value used for the absorption crosssection, σ, of equation 2.54 . Assuming the ideal theoretical resonant cross-section
3λ, typically underestifor a two-level system with transition wavelength λ, σ = 2π
mates the atom number. This is mainly due to imperfections in the polarization1

typical camera exposure time: 110µs

3.15 Image analysis

115

of the imaging beam and is identified in our experiments by comparing the criticalatom number, Nc , in a weakly-interacting gas with the theoretical prediction given
by Hartree-Fock mean-field theory (see chapter 4). In calculating the theoreticalprediction we also take into account the effects of the estimated trap anharmonicity and finite system size [46], which are less than 3% and 1%, respectively. For
87
Rb we observed a reduction in the cross-section by a factor of 1.5±0.3. For 39 K,
the experimental value of Nc was initially extracted from a series taken at 135a0 ,giving a cross-section reduction factor of 1.9±0.3. More careful measurements onvery weakly-interacting clouds as part of our investigation of the interaction-shiftof the critical point yielded a factor of 1.5.

3.15 Image analysis
As discussed above, at the end of a typical experimental run four images are captured using two double-shutter exposures, and the optical density profile OD(x, y)
is generated using equation 2.86. In order to extract quantitative informationfrom these optical density profiles, we fit them to theoretical predictions of thedensity profile after the appropriate time-of-flight, in which the temperature, thethermal atom number and the condensed atom number are directly related to thefitting parameters, as described below. For purely thermal, uncondensed clouds
far from the critical point, the absorption image is fitted with a Gaussian profileof the form:
ODtherm (x, y) = ODtherm (0)e−(x−x0 )2(y−y0 )2
−
2
2
2σx
2σy

(T  Tc )(3.12)

where x0 and y0 are the coordinates of the distribution peak, σx and σy are thewidths and OD(0) is the peak optical density. For thermal clouds near or below
the critical temperature, the fitting function used is the singly-integrated versionof equation 2.72 for the Bose-enhanced distribution:22(x−x0 )
(y−y0 )−−
ODtherm (0)222σyODtherm (x, y) =g2 (e 2σx)g2 (1)(T ≤ Tc )(3.13)

At high temperatures, as well as far from the cloud centre, the dilogarithmic
profile of equation 3.13 approaches the Gaussian distribution of equation 3.12.
Fully-condensed clouds are likewise fitted with the singly-integrated version of

Design, assembly and performance of our 39 K-87 Rb apparatus

116
equation 2.80

!3/2
(x − x0 )2 (y − y0 )2ODcond (x, y) = ODcond (0) max 1 −
−
,0
R2x
R2y

(3.14)

For partially condensed clouds, a bimodal fit is used to differentiate between
atoms belonging to the condensed and thermal components and the fitting function is simply the sum of equations 3.13 and 3.14. In addition, the fitting routine
also allows for gradients in the x and y directions, as well as a global offset. The
most general fitting function therefore has the form

OD = A1 · g2 (e|−(y−A5 )2
(x−A4 )2
−
2A2 2
2A3 2

{z

thermal component!3/2(x − A9 )2 (y − A10 )2) + A6 · max 1 −
−
,0
}
A7 2
A8 2
{z
}
|
BEC component+ A11 · x + A12 · y +|{z}gradients

A13
|{z}

(3.15)

global offset

3.15.1 Triple-pass fitting procedure
Fitting all thirteen of these parameters simultaneously is clearly computationallyexpensive and inevitably returns unreliable results. Therefore we make use of
an automated three-pass fitting procedure, designed streamline the process of
extracting all the required parameters while providing much more reliable andaccurate values [185, 186]. Our ‘triple-pass’ fitting sequence is as follows:
1. The first pass performs a bimodal fit with a global offset but no gradients
i.e. A1 − A10 , A13 of equation 3.15. The (relatively crude) results of this fit
will serve as initial guesses in subsequent fits.
2. The second pass performs a g2 -only fit, including a global offset andgradients, in order to find the thermal component i.e. A1 → A5 , A11 → A13
T F (1)of equation 3.15. In this fit, a region of radius αRx,y , where α is a userT F (1)chosen factor (typically 1.1) and Rx,y is the BEC radius found in the firstpass, centred on the BEC-centre found in the first pass, is excluded from thedata prior to fitting. This is done in order to prevent the BEC peak from
artificially reducing the width found by the thermal-only fit. The results ofthis fit are used to determine the thermal atom number.

3.15 Image analysis

117

3. The third pass performs a TF-only fit i.e. A1 −A5 of equation 3.15. This isdone by performing a full bimodal fit, including gradients and global offset,
but holding all the g2 , gradient and offset parameters at the values obtained
for them in the second pass and allowing only the five TF parameters tovary. The results of this fit are used to determine the condensate number.
4. Finally, a number of ‘excluded-region’ g2 -only fits are performed, excluding from the data regions with user-selected radii, typically chosen to be
multiples (e.g. 0.8, 0.9, 1, 1.1 and 1) of the thermal radius, centred on the
BEC centre found in the third pass. The results of these fits are used todetermine the temperature of the cloud and its error.

3.15.2 Number and temperature extraction
The temperature along the i-th direction (i = x/y) is extracted from the measured width of the cloud in pixels, σi , after t-ms time-of-flight, according to [145]Ti =m(σi · apix )2m(σi · apix )2
ω2· 6 i 22 ≈kB
10 + ωi t
kB t2(3.16)

where apix is the effective pixel size found from the imaging magnification, ωi isthe angular trapping frequency in the i direction, m is the atom mass, kB is theBoltzmann constant and the temperature is given in µK. The factor of 106 in the
denominator is introduced since t is taken in ms. The second approximation isvalid for long times-of-flight, when t is much larger than the in-trap oscillation
period.
The thermal atom number, in millions, is given by
Ntherm = 1.202

2π
σ



· A1 · σx · σy · a2pix × 10−18(3.17)

where σ is the absorption cross-section and A1 is the amplitude (i.e. A1 of equation 3.15). The condensed atom number, also in millions, is obtained using
Ncond =

2π
· A6 · Rx · Ry · a2pix × 10−18
5σ(3.18)

where A6 is the amplitude (i.e. A6 of equation 3.15) and Rx/y are the ThomasFermi widths in pixels (i.e. A7/8 of equation 3.15).

118

Design, assembly and performance of our 39 K-87 Rb apparatus

3.15.3 AnalysisGUIA user-friendly application and graphical user interface GUI written in MATLAB
was written for performing the fitting routines. The GUI is used for preview ofimages in order to enable scrutinization and elimination from batch fits, configuration of the desired fitting routine, display of the fit results superposed on
the image data, tabulated display of the fit results, numerical assessment of the
fit quality, extraction of physical parameters and export of data for later use.
The GUI also permits the selection of a smaller region of the image (region-ofinterest) for fitting in order to reduce computational cost, as well as allowing forthe optional input of initial guesses for the various fitting iterations, the input oftrapping frequencies and TOFs for number and temperature extraction and the
input of the desired radii to be used in the excluded-region fits described above.
Figure 3.32 shows a selection of screenshots from the application.

Figure 3.32: Analysis GUI screenshots showing the windows used for (a) image
preview, (b) visual image analysis and statistics and (c) image fitting.

3.16 Timing control

119

3.16 Timing control
3.16.1 Background
In order to control the power supplies, AOMs, shutters, function generators,
translation stage and other components of our experiment, analog and digital
control voltages as well as serial communications are required. Furthermore,experiments of the type reported in this thesis, with certain stages requiring hightemporal resolution, on the order of tens of µs and with overall durations on
the order of a minute, would not be feasible without the sophisticated electronic
devices that have become commercially available in the last few decades. Thissection briefly reviews the hardware and software components of our experimental
control system.

3.16.2 I/O devices
Digital and analog signals for our experiments are generated by a series of PXIbased input/output (I/O) devices. These cards are mounted in a PXI chassis1
communicating in real time with a PC2 running the control software. We currently have 64 digital channels3 and 16 analog channels4 available, with extra
channels easily added by purchasing addition output cards. In addition to thedigital and analog channels, communication with instruments via RS-232 andGPIB is enabled via a 4-channel RS-232 PCI card5 installed in the control PC anda USB-based GPIB controller6 . RS-232 and GPIB communications are used, forexample, in remotely configuring the outputs of the various function generators,
configuring and controlling the translation stage and logging the measurementsof the thermocouple monitor used during the chamber bakeout.
We note here that compared to the so-called hardware timing of the digital andanalog cards, which use a timing signal independent to that generated by the PC
operating system, GPIB and RS232 are software timed protocols whose output
timing is determined by the operating system and hence is subject to a muchlarger jitter. There is additional uncertainty on these timings since the receiving
1
2
3
4
5
6

National Instruments, PXIe-1062Q
2.40GHz, Intel Core™2 Duo processor, Windows XP, 4Gb RAM
National Instruments, PXI-6533, PXI-6534
National Instruments, 2 × PXI-6713
Brainboxes Intashield, IS-400
National Instruments, GPIB-USB-HSDesign, assembly and performance of our 39 K-87 Rb apparatus

120

Control PC!

Timing data!

Control software!

FPGA!
board!

Timing signal!
Sequence data!

RS-232!

GPIB!

Chassis and!
I/O devices!

Breakout + isolation!
Digital!
signals!

Analogue!
signals!

Instruments!

Figure 3.33: Schematic of the control hierarchy in the experiment. The Word
Generator software communicates with the NI hardware to output digital and analog signals. These are interfaced to the rest of the experiment via optical andcapacitive isolation circuits in order to avoid damage to the NI hardware fromvoltage spike in the apparatus.

devices also require a few ms in order to receive and output the command. We
have found that the typical GPIB command strings used in our experiment, forexample to change the frequency and time constant of an evaporation sweep,
should not be instructed to be sent to the device less than roughly 100ms apartin order to ensure a reliable response.

3.16.3 Word Generator
In addition to allowing the user to easily explore different experimental procedures, well-designed control software enables swift debugging and optimization ofexperimental sequences and control over a wide range of devices. In our experiments we have made use of the Cicero Word Generator control package1 , authoredby Aviv Keshet at the MIT Center for Ultracold Atoms2 and generously made12

http://web.mit.edu/~akeshet/www/Cicero/
http://cuaweb.mit.edu/

3.16 Timing control

121

available under the terms of the GNU General Public License1 . This software2
allows users to visually configure complex sequences and includes many timeand labour-saving features. In addition the software includes a variable timebasefeature which allows for the creation of sequences of much greater duration athigh temporal resolution by freeing the user from the constraint of a fixed timing
resolution and instead calculating a custom timing signal that only updates output values when the experimental sequence requires their value to be changed.
This feature can also be used in conjunction with an additional low-cost FPGA
board3 to generate high-temporal-resolution sequences of effectively infinite duration, and we make use of this functionality in our sequence design. Figure 3.33
shows the path of the control signal from the control software, via the PC andPXI chassis to the breakout panels and finally the target instrument.

3.16.4 Digital and analog isolation
In order to protect the control hardware from voltage spikes that can occur within
the various instruments, as well as attempting to reduce the presence of groundloops, all digital signals are isolated from the output cards by means of an optocoupler4 circuit prior to breakout of the signals to BNC connectors. Analog
signals can also be isolated by means of an isolation amplifier5 circuit after theirbreakout into BNC connectors6 . The two types of isolation boards are shown infigure 3.34, together with their respective single-channel circuit diagrams.

1
2
3
4
5
6

http://www.gnu.org/licenses/
Can be downloaded directly at: http://sourceforge.net/projects/cicerowordgener/
files/
Opal Kelly Inc., XEM3001
Fairchild Semiconductor, 6N138
Burr-Brown, ISO124P
Breakout panel: National Instruments, BNC-2110

Design, assembly and performance of our 39 K-87 Rb apparatus

122

1nF!

ISO124!

a)!
Vin!

V+!

4.7kΩ!
V+!
V-!

gnd!

V-!

9.1kΩ!
220pF!

b)!

6N138!
1!

8!

2!

7!

3!

6!

1kΩ!

50Ω!

-!
+!
V+!

V-!

Vout!

5V!
Vin!

100nF! 2.2µF!

470Ω!

+5!
+V! 0V!
IF0505S!
0V!

Vout!
100μF!

4!

5!

4.7kΩ!
gnd!

gnd!

Figure 3.34: (a) Four-channel analog isolation module and (b) eight-channel optocoupler board for digital isolation (DC-DC converter). The optocoupler boards
were constructed and tested by Alex Gaunt.

4
The role of saturation in
Bose-Einstein condensation“Proverbs for Paranoids:
1. You may never get to touch the Master, but
you can tickle his creatures.”Thomas Pynchon, Gravity’s Rainbow

Abstract
In this chapter, we describe the first experiments carried out following our condensation of 39 K. By harnessing our ability to tune the interaction strength, wehave been able to test the validity of Einstein’s textbook picture of BEC as arisingfrom saturation of the excited states of the system, as introduced in chapter 2.
We observe strong deviation from the saturation picture. In the small-condensate
limit, our results agree with a mean-field description of the interactions, while
for larger condensates these theories fail to capture the observed behaviour.
However, extrapolation of our measurements to the non-interacting limit allows
us to recover Einstein’s picture in a non-interacting gas.

123

124

The role of saturation in Bose-Einstein condensation

4.1 Introduction and motivation
Chapter 2 introduced the predictions for the onset of condensation in a noninteracting Bose gas, but also pointed out that interactions are expected to modify
these predictions. This chapter presents the results of our investigations aimed ataddressing the question of whether excited-state saturation is indeed the driving
mechanism behind Bose-Einstein condensation, as originally envisaged by Einstein [187], and our attempts at elucidating the role of interactions in the BEC
phase transition.

We recall that in the textbook theory of the non-interacting Bose gas, purely(id)statistical arguments set an upper bound Nc on the number of atoms NT occupying the excited states of the system (see chapter 2, [187], [188],[123] and [144]).
In the case of the ideal, harmonically-confined Bose gas, the prediction for thecritical value of NT at which this saturation occurs is given by equation 2.70 [189].
NT 6 Nc(id) = ζ(3)



kB T
~ω̄

3
(4.1)

where T is the temperature, ω̄ is the geometric mean of the trapping frequenciesand ζ is the Riemann function, with ζ(3) ≈ 1.202. The condensation observedin weakly-interacting harmonically-trapped atomic Bose gases [23, 24, 182] isgenerally believed to constitute a faithful representation of this statistical phasetransition. Figure 4.1 illustrates the saturation prediction and shows a plot of thenumber of thermal (red line) and condensed (blue line) atoms as a function of thetotal atom number. We emphasize that the term ‘saturation’ in this context refers
to the halt in the growth of the thermal component, predicted in the ideal-gas
case to coincide with the onset of condensation.

The idea of systematically scrutinizing the role of this saturation mechanismin the BEC transition followed on from an observation made in the course ofexperiments on the critical point of a two-dimensional Bose gas in the ENS group(see footnote 31 of [190]). In these experiments, the linear slope of the curve
dN0 /dNtot above (the blue line of figure 4.1) above the critical point was found to
take the value 0.7, clearly deviating from the unity slope expected for a saturatedgas.

4.1 Introduction and motivation

125

300

Saturated gas

200

BE
C

NT , N0 (thousands)

Nc

100

0

0

100

200

300

400

Ntot (thousands)

Figure 4.1: Saturated gas. The red and blue lines show the predicted thermal
atom number, NT , and condensed atom number N0 , respectively, as a function of
the total atom number Ntot = N0 + NT for a saturated gas at fixed temperature.
The critical point, Nc , is indicated by the vertical and horizontal dashed lines.

Chapter outline
This chapter proceeds as follows:• We first describe the experimental procedure we employ in order to probe
the saturation, or lack thereof, of our trapped gas (sections 4.2 and 4.3).
• We present the effects of varying both the interaction strength and thetemperature on the degree of deviation from the behaviour of a saturatedgas (section 4.4).
• In order to disentangle the roles of interaction strength and temperature
in Bose-Einstein condensation, we present two mean-field theories - thePopov and self-consistent Hartree-Fock approximations - and compare our
non-saturation measurements to their predictions (section 4.5).
• Our results demonstrate that above a certain condensate size, these meanfield theories can no longer reproduce the experimental observations and weadopt a more heuristic approach in order to quantify the lack of saturationin this regime (section 4.6).
• We use the results obtained using this approach to extrapolate to the limitof a non-interacting gas and show that in this limit the saturation picture

126

The role of saturation in Bose-Einstein condensation
originally described by Einstein is recovered within our experimental errors(section 4.6.1).
• We then summarize and discuss our results in the context of previouslyperformed related experiments and the predictions of beyond-mean-field
theories (section 4.7).

4.2 Experimental procedure for investigating
saturation
In order to investigate the condensation process, we perform conceptually simple
experiments in which the temperature of the gas is kept constant while the atom
number is varied. We choose to vary the atom number by allowing the trappedsample to gradually decay as a result of three-body recombination, scatteringof photons from the trapping laser beams and collisions with background atoms
in the vacuum chamber. Temperature variations are kept to a minimum bystabilizing the power in the trapping lasers in order to fix the depth of the opticalpotential.
The experiment begins with a partially condensed gas of 39 K atoms in the|F, mF i = |1, 1i state, prepared as described in chapter 3, at a scattering lengthof a = 135a0 . Following preparation of the sample, the optical trap depth israised slightly by ramping up the power in the trapping beams over 500ms inorder to prevent further evaporation and maintain a steady temperature for theremainder of the experiment. The mean trapping frequencies for data taken at
different temperatures varies between ω̄ = 60 − 80Hz. Once the final CDT power
is reached, the Feshbach field is ramped to its final value and the cloud is held
for up to several tens of seconds while the atom number decays.
Figure 4.2 shows the relevant part of the experimental sequence. The orange andgreen lines show the value of the CDT power and Feshbach field, respectively, as afunction of time. We perform measurements over a range of interaction strengths
and at various temperatures. The temperature of a series is set by the final
depth of the optical trap and the scattering length is set by the final value of the
Feshbach field. We vary the scattering length over roughly an order of magnitude,
a = 40a0 − 356a0 , and the temperature over the range T = 115 − 284nK for thedifferent series. The dashed lines in figure 4.2 indicate the range of values taken
on by the Feshbach field and CDT power for these measurements.

4.2 Experimental procedure for investigating saturation

127

hold time

CDT power (W)2

CDT power

380

Feshbach field (G)400

Feshbach field

1
360
0

3

6

9

time (s)1215Figure 4.2: Experimental protocol. Orange and green lines show the value of
the CDT power and the Feshbach field respectively, as a function of time. The
atoms are cooled by an exponential ramp-down of the trap depth, with their final
temperature being set by the final trap depth and the scattering length being held
constant at a = 135a0 during the ramp-down. The CDT power is then raisedslightly in order to maintain a steady temperature during the ensuing hold timeand once the final CDT power is reached the Feshbach field is ramped to its final
value. Dashed lines delimit the ranges used for the two parameters in the different
experimental series.

While the total atom number decays due to inelastic and background collisions,elastic collisions among the trapped atoms ensure equilibrium redistributionof particles between the condensate and the thermal gas. Since the densitydependent three-body recombination is expected to take place preferentially inthe cooler, denser region of the trap, heating of the sample might be expected.
Heating can also arise from light scattering in the optical trap. However any suchheating is at least partially compensated by residual evaporation taking place inthe finite-depth trap. For all the data presented in this chapter, the thermalization rate is much higher than the atom-loss rate and hence the details of the loss
mechanism do not affect the quasi-static thermodynamic properties of the gas.

Ensuring thermal equilibrium
In general, a system with continuous dissipation can never be at true thermodynamic equilibrium. The proximity to equilibrium for atomic gases depends on thedimensionless parameter, γel τ , measuring the relative rates of elastic and inelastic

128

The role of saturation in Bose-Einstein condensation

processes. Here, γel is the elastic collision rate and τ is the timescale characterizing losses arising from inelastic processes, often taken to be the 1/e-lifetime of
the cloud. Equilibration is usually taken to require 3 elastic collisions per particle[191] implying γel τ  1. Following [192], the mean elastic collision rate for each
atom (γel ) is given by
σ0(4.2)
γel = n(0)vth2
with the central density, n(0), the thermal velocity, vth and the scattering crosssection, σ0 , given by1
n(0) = N ω̄3m
2πkB T

3/2

r
vth =

8kB T
πm

and

σ0 = 8πa2(4.3)

For example, a cloud of 4×105 39 K atoms at T = 200nK and a scattering length of
a = 135a0 in an ω̄ = 80Hz trap has a mean elastic scattering time of 1/γel ≈ 13ms.
A basic condition for establishing global equilibrium in our system is that thecharacteristic inelastic-loss time, τ , should be larger than the period of the trapping potential, i.e. τ > 1/ω̄. The period of our optical trap in these experiments
varies between 2 − 2.7ms and we ensure that this global equilibrium criterion is
fulfilled for all the data reported below. Occasionally, data series will fulfil thelatter criterion without satisfying the former. For low scattering lengths this occurs due to the drop in the elastic collision rate and for high scattering lengths
due to the drop in the inelastic-loss time. For this reason we excluded from
our analysis series taken at scattering lengths and temperatures in which unreliable distortions of the data, such as large temperature variations, w ere ascribedto the onset of non-equilibrium effects. For example, for data series taken at
T ≈ 180nK, all series below roughly a = 60a0 were excluded from the analysis.
Chapter 5 contains a more detailed discussion of these equilibrium criteria in thecontext of our measurements of the critical point of our trapped gas, where theprecision with which we want to measure the critical number Nc prescribes thevalue of γel τ for which our sample is effectively at equilibrium.
Extracting numbers and temperature
For each hold time, we extract the condensate number N0 and the thermal number
NT from a triple-pass bimodal fit, as described in section 3.15, after 18ms of free
1

Similar values of n(0) (smaller than 10% difference) are obtained for our experimental
parameters when using the ‘Bose-enhanced’ expression, n(0) = ζ(3/2)/λ3dB .

4.3 Observation of non-saturation

129

Ntot (thousands)!

time-of-flight expansion from the optical trap. Figure 4.3 shows the total number,
Ntot , and temperature, T , versus hold time for an experimental series taken ata scattering length of a = 135a0 and a temperature of 177nK. The temperature
for Ntot > Nc shows no drift and has a standard deviation of 3nK. We assign anominal temperature, T ∗ , to the entire series. T ∗ is taken to be the experimentallymeasured temperature in the N0 → 0 limit, i.e. coinciding with the first non-zerocondensate number.
400!

200!

T (μK)!

0!
200!

100!

0!

0!

20!

40!

60!

Hold time (s)!

Figure 4.3: Total atom number and temperature against hold time for an experimental sequence taken with a = 135a0 and T = 177nK. The temperature showsno drift for Ntot > Nc and has a standard deviation of 3nK. The atom number isfitted with a double-exponential to guide the eye.

It is worth pointing out that harmonically trapped gases are well-suited for performing such experiments, which require reliable and high-precision distinction
between the condensed and non-condensed components. This is due to the fact
that, contrary to uniformly confined Bose-condensed systems such as liquid helium, condensation in harmonic traps manifests itself in coordinate space in addition to momentum space, causing the condensate to appear as a distinct peakin the atomic density distributions recorded after time-of-flight in experiments.
This simplifies the extraction of occupation numbers from such distributions (seee.g. [46, 123]).

4.3 Observation of non-saturation
Figure 4.4 is a plot of the thermal and condensate numbers against the total atom
number for the experimental series shown in figure 4.3. The total atom number

130

The role of saturation in Bose-Einstein condensation

is obtained by a direct summation over the density distribution rather than being
extracted from a fit to the data, and the relation Ntot = N0 + NT is found to besatisfied for all data points to within 0.5%. The data has been overlain on theideal-gas saturation predictions of figure 4.1.

Nc
Saturated gas

C

200

100

0

BE

NT , N0 (thousands)

300

0

100

200

300

400

Ntot (thousands)

Figure 4.4: Non-saturation in a weakly-interacting Bose gas. The thermal (red)and condensed (blue) atom numbers are plotted against the total atom number forthe data series of figure 4.3 (a = 135a0 , T = 177nK). Strong deviation from theideal-gas saturation prediction is observed, with approximately half of the atoms
added above the critical number being accommodated in the condensate.

The deviation of the experimental data from the ideal-gas prediction is striking.
In this example, as the total number is increased from the observed critical value,
Nc ≈ 200, 000, to 450,000, only half of the additional atoms accumulate in thecondensate.

4.4 Varying interaction strength andtemperature
As mentioned above, we repeat this procedure for a range of scattering lengths
and temperatures, ensuring that for the chosen values rethermalization is rapid
enough compared to the trap loss rate to maintain thermodynamic equilibrium.
Figure 4.5 shows the results of 18 such experimental series, focussing on the region
Ntot > Nc in which the condensate is present, and plotting N0 against Ntot − Nc .
The solid line shows the prediction for a fully-saturated thermal component:

4.5 Mean-field theories of interacting gases

131

N0 = Ntot − Nc . The deviation of the data from this prediction is clearly seen in
all the series and grows with both a and T .

Ta,

N0 (thousands)

100

50

0

0

100

200

300

122nK, 40a0
126nK, 62a0
115nK, 135a0
121nK, 221a0
157nK, 93a0152nK, 135a0
184nK, 62a0
181nK, 93a0
177nK, 135a0
171nK, 221a0
188nK, 135a0
186nK, 184a0
181nK, 274a0
233nK, 135a0
260nK, 135a0251nK, 221a0
267nK, 274a0
284nK, 356a0

400

Ntot -Nc (thousands)

Figure 4.5: Deviation from the saturation prediction for a range of scattering
lengths and temperatures. Shown is N0 vs. Ntot − Nc for 18 experimental series,
with the values of the scattering length (a = 40 − 356a0 ) and the temperature(T = 115 − 284nK) encoded in the colour of the data points (see legend). The solid
black line is the saturated-gas prediction: N0 = Ntot − Nc .

4.5 Mean-field theories of interacting gases
In this section, we briefly review the two schemes most commonly used for taking
into account inter-atomic interactions - the Popov and self-consistent HartreeFock approximations. These are both mean-field theories which avoid the difficulty of solving the full many-body Schrödinger equation for an interacting system
by reducing the many-body problem to a one-body problem via the introductionof an appropriate mean field potential generated by all the other particles. In
general, the effects of interactions are only expected to become sizeable in thepresence of a condensate, since only then does the peak density become sufficiently high for the interactions to significantly modify the system’s properties.
We begin by tracing the derivation of the Hartree-Fock-Bogoliubov (HFB) mean
field equations. The main approximation used in the derivation of these equations

132

The role of saturation in Bose-Einstein condensation(beyond the assumption of a contact interaction between the atoms and use of
the Bogoliubov approximation further along) is the introduction of mean fields
to replace what would otherwise be products of pairs of quantum field operatorsin the Hamiltonian. In this way, terms cubic and quartic in the field operators
are reduced to linear and quadratic terms. In addition to simplifying the resulting equations for the condensate wavefunction and the excitation spectrum of
the system, this scheme forms a starting point for the Popov and self-consistent
Hartree-Fock approximations. The outlined derivation below roughly follows that
described in several sources (for example [193], [194], [195], [196], [144] and [123]).
Its main purpose is to introduce the mean-field description as well as to highlightthe difference between the Popov and self-consistent Hartree-Fock approximations.

4.5.1 Hartree-Fock-Bogoliubov mean-field theory
Our starting point is the second-quantized many-body Hamiltonian for a Bosegas with contact interactions held in an external potential Vext (r) [123, 144]:

Z
Ĥ =


Z
g
~2 ∇2
+ Vext (r) Ψ̂(r) dr +Ψ̂† (r)Ψ̂† (r)Ψ̂(r)Ψ̂(r) drΨ̂ (r) −2m
2†

where Ψ̂† (r) and Ψ̂(r) are the creation and annihilation field operators, respectively, and in accordance with the Born approximation the microscopic interaction potential has been replaced with an effective potential characterized by the
coupling constant g = 4π~2 a/m introduced in equation 2.74. The equation of
motion for the field Ψ̂(r, t) in the Heisenberg representation is obtained from thisHamiltonian as 2 2∂ Ψ̂(r, t)~∇i~= [Ψ̂(r, t), Ĥ] = −+ Vext (r) Ψ̂(r, t) + g Ψ̃† (r, t)Ψ̃(r, t)Ψ̃(r, t)
∂t
2m
(4.4)
where we have made use of the bosonic commutation relations[Ψ̂(r), Ψ̂† (r0 )] = δ(r0 − r),[Ψ̂(r), Ψ̂(r0 )] = [Ψ̂† (r), Ψ̂† (r0 )] = 0(4.5)

The non-condensed, time-varying component of the field can be separated fromthe stationary condensed component by writing the total field as the sum of a

4.5 Mean-field theories of interacting gases

133

condensate wavefunction and an operator describing the fluctuations of the fieldΨ̂(r, t) = Φ(r, t) + ψ̃(r, t)| {z }| {z }
condensate(4.6)

fluctuations

where the Bogoliubov approximation involves the assumption that since theground state is macroscopically occupied, the number of non-condensed atoms
is significantly smaller than the ground-state occupation number1 . This form forthe total field is then inserted into the equation of motion 4.4. The self-consistentmean field approximation consists of the linearization of the interaction term cubic in the fluctuation field by reducing it to a set of interactions of a single fieldwith the mean field produced by the other two fields according toψ̃ † (r, t)ψ̃(r, t)ψ̃(r, t) ' 2hψ̃ † (r)ψ̃(r)iψ̃(r, t) + hψ̃(r)ψ̃(r)iψ̃ † (r, t)(4.7)

This yields an equation of motion whose stationary solution provides the equivalent of the Gross-Pitaevskii equation for the spatially-varying condensate wavefunction, Φ(r)2



~2 ∇2
+ Vext (r) − µ + g [n0 (r) + 2nT (r)] Φ(r) + g m̃(r)Φ∗ (r) = 0
−
|
{z
}
2m

(4.8)

‘anomalous’ term

where the chemical potential, µ, has been introduced since the condensate wavefunction evolves in time according to Φ(r, t) = Φ(r) exp(−iµt/~) and we haveintroduced the condensate, thermal and ‘anomalous’ densities, n0 (r), nT (r) andm̃(r), respectively3
n0 (r) = |Φ(r)|2nT (r) = hψ̃ † (r)ψ̃(r)im̃(r) = hψ̃(r)ψ̃(r)i
1

2

3

(4.9)

†
The Bogoliubov approximation consists of replacing both the creation√ (â0 ) and annihilation(â0 ) operators for the zero-momentum state by the real number N0 , where N0 is thecondensate occupation number.
The full expansion for the cubic field-operator term Ψ̂† (r, t)Ψ̂(r, t)Ψ̂(r, t) within themean-field approximation is given by: Ψ̂† (r, t)Ψ̂(r, t)Ψ̂(r, t) ' |Φ(r)|2 Φ(r) + 2[|Φ(r)|2 +hψ̃ † (r)ψ̃(r)i]ψ̃(r, t) + [Φ2 (r) + hψ̃(r)ψ̃(r)i]ψ̃ † (r, t) + 2Φ(r)ψ̃ † (r, t)ψ̃(r, t) + Φ∗ (r)ψ̃(r, t)ψ̃(r, t)This notation is roughly consistent with that of [194].

134

The role of saturation in Bose-Einstein condensation

We also note that terms linear in ψ̃(r, t) have vanished from the original expansion when taking the time average since the fluctuations are assumed to obeyhψ̃(r, t)i = hψ̃(r)i = 0. In the T → 0 limit, when the thermal fraction can beneglected, equation 4.8 reduces to the Gross-Pitaevskii equation introduced inchapter 2 for the weakly-interacting gas.
We now turn our attention to obtaining the excitation spectrum in a similar manner. Substituting the field decomposition of equation 4.6 into the Hamiltonian ofequation 4.4 and repeating the mean-field approximation for the resulting cubic
and quartic terms in ψ̃(r, t) and ψ̃ † (r, t) results in a Hamiltonian which can be
diagonalized by the linear transformationψ̃(r) =X(ui (r)α̂i − vi∗ (r)α̂i† )iXψ̃ † (r) =(u∗i (r)α̂i† − vi (r)α̂i )(4.10)iwhere α̂i† and α̂i are the creation and annihilation operators of the i-th elementary
excitation, respectively. Collecting the coefficients of α̂i† and α̂i in the expanded
Hamiltonian results in two coupled equations for the parameters ui (r) and vi (r)- the so-called Hartree-Fock-Bogoliubov (HFB) equations

~2 ∇2
−
+ Vext (r) − µ + 2gn(r) − i ui (r) − gm(r)vi (r) = 02m(4.11)

~2 ∇2+ Vext (r) − µ + 2gn(r) + i vi (r) − gm∗ (r)ui (r) = 0−2m(4.12)



with the total density, n(r), and the anomalous term, m(r), defined byn(r) = hΨ̂† (r)Ψ̂(r)i = |Φ(r)|2 + hψ̃ † (r)ψ̃(r)i = nc (r) + nT (r)(4.13)

m(r) = hΨ̂(r)Ψ̂(r)i = Φ2 (r) + hψ̃(r)ψ̃(r)i = Φ2 (r) + m̃(r)(4.14)

4.5.2 Popov approximation
The Popov approximation [197, 194, 195, 198, 123, 144] consists of neglecting
the anomalous density m̃(r) in equations 4.8, 4.11 and 4.12. This eliminates theanomalous density term in equation 4.8 and replaces m(r) and m∗ (r) in equations

4.5 Mean-field theories of interacting gases

135

4.11 and 4.12 by n0 (r), where we have used the Bogoliubov approximation to
replace Φ2 (r) ' n0 (see equations 4.13 and 4.14).
The HFB equations therefore reduce to the Popov equations

 2 2
~∇
+ Vext (r) − µ + 2gn(r) − i ui (r) − gn0 (r)vi (r) = 0−2m(4.15)

 2 2

~∇
−
+ Vext (r) − µ + 2gn(r) + i vi (r) − gn0 (r)ui (r) = 02m(4.16)In the semi-classical approximation, the kinetic energy operators, −~2 ∇2 /2m, inequations 4.15 and 4.16 can be replaced by p2 /2m and the excitation energies are
obtained as1/2(r, p) = (p2 /2m + 2gn(r) + Vext (r) − µ)2 − (gn0 (r))2(4.17)The density of non-condensed atoms in this approximation is then given byZnT (r) =1
dp p2 /2m + 2gn(r) + Vext (r) − µ3(r,p)/kBT − 1(2π~)
(r, p)e(4.18)

where (r, p) is given by equation 4.17 and the factor multiplying the excitation
distribution function is the ratio of the particle energy in the Hartree-Fock approximation (see below) to that in the Popov approximation. This corresponds
to the number of non-condensed particles associated to a given excitation andensures that we are effectively summing particles and not excitations when calculating the particle density, nT (r).
The condensate density, n0 (r) = |Φ(r)|2 is found by solving equation 4.8 with the‘anomalous’ term involving m̃(r) omitted, namely



~2 ∇2
−
+ Vext (r) − µ + g [n0 (r) + 2nT (r)] Φ(r) = 02m(4.19)

within the Thomas-Fermi approximation, the first term of equation 4.19 can beneglected and the chemical potential is then given byµ = Vext (r) + g [n0 (r) + 2nT (r)](4.20)

The four equations, 4.17, 4.18, 4.19 and 4.20 must be solved self-consistently in

136

The role of saturation in Bose-Einstein condensation

order to find the thermal and condensate densities.
3.5

nT , n0 , ntot (1011 cm-3)

3.0
2.5
2.0
1.5
1.0
0.5
0.0
-40

-30

-20

-10

0

r(µm)

10

20

30

40

Figure 4.6: Popov density profiles. Cuts through the thermal (red) and condensate(blue) in-situ spatial density distributions for a cloud with Ntot = 105 atoms at T =150nK. The dimple in the thermal cloud is a result of repulsion by the condensate.
The black dashed line shows the total density distribution, ntot = nT + n0 .

We note that at very low temperatures, when the non-condensed fraction nT
can be neglected, the Popov spectrum of equation 4.17 reduces to the usual
Bogoliubov quasi-particle excitation spectrum (see for example [199, 123]) and,as we have already mentioned, equation 4.19 for the condensate wavefunction
reduces to the usual Gross-Pitaevskii equation introduced in chapter 2. At higher
temperatures, when the condensed fraction n0 (r) can be neglected, the Popovspectrum coincides with the Hartree-Fock spectrum (see below).

4.5.3 Self-consistent Hartree-Fock approximation
A further simplification of the full Hartree-Fock-Bogoliubov equations is the socalled self-consistent Hartree-Fock approximation, which is widely accepted asproviding the simplest scheme for addressing the effects of atomic interactions ina Bose gas.
Mathematically, the approximation consists of again neglecting m̃(r) in equations
4.8, 4.11 and 4.12 and in addition neglecting any of the collective effects arisingin the HFB and Popov approximations. The latter approximation amounts toneglecting the parameter vi (r) in equations 4.10 and hence in equations 4.11 and4.5 Mean-field theories of interacting gases

137

4.11, which therefore yield the simplified, single-particle, excitation spectrum
HF (r, p) =p2
+ Vext (r) + 2gn(r)2m(4.21)

with n(r) = nc (r) + nT (r) as in equation 4.13. Within this approximation, thecondensate density is still found from equation 4.19 and the thermal density nT (r)is now obtained usingZ
HF
Tn (r) =dp13
((r,p)−µ)/k
HF
BT − 1(2π~) e(4.22)

Physically, this approximation corresponds to treating the thermal component as
a gas of ‘non-interacting’ atoms moving in a self-consistently determined meanfield potential given by
Veff (r) = Vext (r) + 2gn(r) = Vext (r) + 2g[n0 (r) + nT (r)](4.23)

Two further simplifications can be made as a consequence of the relative diluteness of the thermal component compared to the condensate:
1. Firstly, one can neglect any influence of the thermal component on thespatial distribution of the condensate wave function1 . Setting nT (r) = 0 in
equation 4.19 and applying the Thomas-Fermi (TF) approximation2 gives
the usual TF profile for the condensate
n0 (r) = max[(µ0 − Vext (r))/g, 0]
where the normalization conditionsity yields the chemical potential
~ω̄
µ0 =
2

R

(4.24)

n0 (r)dr = N0 on the condensate den-


2/5
a
15N0
aho

(4.25)

Note that this expression is strictly equal to the actual chemical potential,µ, only in the zero-temperature limit. More generally, µ0 provides the1
2

We note that in [186], a small deviation from this approximation was observed in the formof a ∼ 4% compression of the condensate length from its predicted TF size.
Use of the Thomas-Fermi approximation is valid as long as N  aho /a (i.e. N0  100 to1000 for our parameters)

138

The role of saturation in Bose-Einstein condensation
2/5

characteristic energy scale for the interactions1 and its scaling with N0
will inform the analysis of our experimental data (see section 4.6 below).

2. Secondly, we can neglect the mean-field energy, 2gnT (r), due to the thermal
component itself, so that the effective potential experienced by the thermal
atoms is then given by
Veff (r) = Vext (r) + 2gn0 (r) = |Vext (r) − µ0 | + µ0

(4.26)

Figure 4.7 shows an illustration of the potential, Veff (r), experienced by the thermal atoms in this approximation, which has the form of a 3D ‘Mexican hat’. We
note that under these approximations, the Hartree-Fock theory reduces to theso-called ’semi-ideal’ model, [200, 201, 202]. This model has been used in theanalysis of several previous experiments, as discussed in section 4.7 at the end of
this chapter.

Figure 4.7: Illustration of the “Mexican hat” potential of equation 4.26 experienced by the thermal atoms as a result of their interaction with the condensate at
the centre of the trap and the external harmonic potential.

In neglecting the contribution of the thermal atoms to the mean-field potential,
the second approximation also amounts to neglecting the mean-field shift of thecritical temperature due to interactions. This shift will be discussed in moredetail in chapter 5 where our observation of the mean-field Tc shift, in addition to observation of beyond-mean-field effects on the critical temperature, arepresented.
1

In the zero-temperature Thomas-Fermi limit, for example, the chemical potential is related
to the interaction energy per particle according to Eint /N = (2/7)µ0 .

4.5 Mean-field theories of interacting gases

139

Using the above expression for Veff (r), the number of thermal atoms at a given
temperature in the presence of a condensate of N0 atoms can be estimated byintegrating equation 4.22, i.e.
1NT =(2π~)3


 2

−1
Z 
1
p
exp
+ Veff (r) − µ − 1
d3 p d3 r
kB T 2m(4.27)

Introducing the dimensionless variables
R=

r
RT

α0 =

µ
kB T

(4.28)

pwith RT = kB T /mω 2 , we can express the thermal atom number after performing the momentum integral in equation 4.27 asNT = 4π

RT
λT

3 Z ∞√

2

R2 g3/2 (e−R /2+α0 )dR(4.29)

2α0

Making the substitution X 2 = R2 − 2α0 , the above integral becomesNT = 4π

≈ 4π

= 4π

RT
λT

3 Z ∞
2(X 2 + 2α0 )1/2 g3/2 (e−X /2 )XdXRTλT

3 Z ∞RTλT

3 Z ∞02

X 2 (1 + α0 /X 2 )g3/2 (e−X /2 )dX|0−X 2 /2

2

X g3/2 (e
{z
√
ζ(3)


=

kB T
~ω

(RTF  RT )

0

π
2


)dX +4π
}

RT
λT

3

Z ∞
α0
|0

2

g3/2 (e−X /2 )dX
{z
}
√
ζ(2)

π
2

3
[ζ(3) + α0 ζ(2)](4.30)In going from the first to the second row of equation 4.30 we have made theassumption that the condensate size is small compared to the thermal radius,
RTF  RT , and hence that Xα02  1. Using equation 4.1 for the ideal gas critical(id)number, Nc , we are left with
NT
ζ(2)
µ
=1+
α0 ≈ 1 + 1.37α0 = 1 + 1.37
0
Nc
ζ(3)
kB T

(4.31)

It has in fact been shown [203, 196, 46] that the description of the thermodynamic
behaviour of an interacting Bose gas in the thermodynamic limit for a given

140

The role of saturation in Bose-Einstein condensation

atom number, trap geometry, scattering length, etc., is fixed by the values of(id)
two dimensionless parameters: the reduced temperature, t = T /Tc , and theinteraction parameter, η defined as
2/5

1/6 a
η≡
' 1.57 N
(id)
aho
kB Tc
µ

(4.32)

where µ is the zero-temperature chemical potential in the Thomas-Fermi limit1 .
In the experiments described in this chapter, η, which quantifies the relative
importance of interactions, spans the range η = 0.21 → 0.61. A related dimensionless interaction parameter is given by, a/λ0 , the ratio of the scattering
length to the thermal wavelength evaluated at the ideal-gas critical temperature.
This parameter will be used in chapter 5 since most theories for the shift in Tc
are formulated in terms of expansions in this quantity. η and a/λ0 are related
via η ≈ 2.3 · (a/λ0 )2/5 and hence in the current experiments, we have probed therange a/λ0 = 2.7×10−3 → 3.6×10−2 . Physically, the origin of the non-saturation
predicted in this Hartree-Fock scheme can therefore be understood as resulting
directly from the geometry of the effective ‘Mexican hat’ potential allowing thethermal component to occupy a larger volume, which grows with increasing N0 .

4.6 Comparing experiment and theory
We initially compare our measurements to the more analytically-tractable predictions of Hartree-Fock theory above. Guided by the scaling implied by equations
2/5
4.25 and 4.31, we plot the thermal atom number, NT , as a function of N0 . Figure 4.8 shows this plot for the a = 135a0 , T = 177nK series shown in figures 4.3
and 4.4. From equations 4.25 and 4.31 we define the ‘non-saturation slope’ SHF :2/5

SHF = dNT /d(N0 ) = 1.37X(4.33)

where X is an interaction parameter, defined by
2 2/5

X = ξT awhere1ξ = ζ (3) 152/5
2



kB
~ω̄

2

−2/5

aho

(4.34)The blue line in figure 4.8 corresponds to this prediction, with the y-axis intercept
fixed by the measured value of Nc . As can be seen in figure 4.8, the Hartree-Focknon-saturation slope agrees well with the data in the regime of small condensates
1

This has the same functional form as equation 4.25, with the replacement N0 → N .

4.6 Comparing experiment and theory141

NT (thousands)

320

280

240

200
0

20

40

N0

60

2/5

80

100

Figure 4.8: Quantifying the lack of saturation. The thermal number, NT , is2/5
plotted against N0 for the series shown in figures 4.3 and 4.4. The horizontal
dotted line is the saturation prediction NT = Nc . The blue line is the meanfield Hartree-Fock result of equation 4.31 for a harmonically trapped gas for theappropriate scattering length (SHF = 699). The orange line is the prediction of thePopov approximation for the experimental parameters used. The solid red line isa linear fit to the data in the large-condensate regime, as discussed below, and theblack line is a second-order polynomial fit to guide the eye.

with N0 . 104 , corresponding to µ0 /kB T . 0.1, whereas the data deviates fromthe HF prediction for larger values of N0 . On figure 4.8 we have also plotted
the prediction of the Popov theory for the relevant experimental parameters. In
the small-condensate regime, the Popov result is almost indistinguishable fromthe Hartree-Fock prediction and its departure from the linear HF curve for larger
condensates still fails to describe the observed growth of the thermal component.
In order to more quantitatively test the predictions of equation 4.31, we took
several series at different scattering lengths (a = 56 − 274a0 ) and temperatures(T = 177 − 317nK), focussing on the regime of very small N0 . The method
we use to reliably detect and measure small condensates involves turning off theinteraction strength during time-of-flight and performing a direct summation of
the optical density. It is described in more detail in chapter 5 in the context of
our investigations into the interaction-shift of the transition temperature Tc . For
2/5
each of these series, we fit the initial non-saturation slope S0 = dNT /d(N0 ) forN0 → 0, and compare the result with the prediction SHF = 1.37X. Figure 4.9

142

The role of saturation in Bose-Einstein condensationshows the results of this comparison, in which experiment and theory agree towithin a few percent.
Since, as mentioned above, two independent parameters, t and η are requiredin order to fully capture the mean-field behaviour, we do not expect our simplified non-saturation slope S0 to scale exactly with the interaction parameter, X.
Despite this, the agreement with our single-parameter theory is good.

S0

2000

1000

0

0

400

800

X

1200

1600

Figure 4.9: Agreement of the initial non-saturation slope, S0 with the predictionof the ‘semi-ideal’ Hartree-Fock theory. The blue line shows the prediction ofequation 4.31. The vertical error bars are statistical, with the < 10% systematic
uncertainty on NT and N0 corresponding to a < 6% uncertainty in S0 . Horizontal
error bars include the 3Hz uncertainty in the trapping frequencies ω̄/2π and (for39 K) the 0.1G uncertainty in the position of the Feshbach resonance.

The agreement of figure 4.9 with HF theory is the first main quantitative result of
this investigation. From it we can deduce that the initial non-saturation slope S0
would indeed vanish in the non-interacting limit, in which µ0 → 0 for any valueof N0 .
Figure 4.10a shows the predictions of the Popov approximation for the thermaland condensate numbers for different values of the scattering length, at a temperature of T = 200nK, while figure 4.10b shows the data of figures 4.3 and 4.4
compared to the Popov model for the same experimental parameters. It is apparent from this plot that the mean-field Popov model does not fully capture theextent of the non-saturation observed in our clouds.
In order to quantitatively study the non-saturation effects taking place at larger
N0 , where the data is no longer well described by the Hartree-Fock theory, we
adopt a more heuristic approach. Although the observed increase of NT with
2/5
N0 is not perfectly linear, over a broad experimentally relevant range it can be
2/5
well quantified by a coarse-grained slope S = ∆[NT ]/∆[N0 ], as indicated by thesolid red line in figure 4.8. In order to treat data taken at different values of X,

4.6 Comparing experiment and theory

NT , N0 , (thousands)

a=4

250

00a 0

0a
a=20 0300

a=135a 0

250

NT , N0 , (thousands)

a)!

300

a=40a0

200
a=5a0
150

100
50

0

143

b)!
Popov

200

model

Saturated gas

150

100

50

0

50

100

150

200

250

300

Ntot (thousands)

350

400

450

500

0

0

50

100

150

200

250

300

350

400

450

500

Ntot (thousands)

Figure 4.10: The Popov model. Figure (a) shows the predictions of the Popovapproximation for different scattering lengths (a = 5, 40, 135, 200, 400a0 ) at a temperature of T = 200nK. Figure (b) compares the experimental non-saturation data
to the corresponding Popov prediction, which fails to fully capture the degree of
non-saturation.

i.e. different values of a and T , equally, for each experimental series we consider
the same range of values of µ0 /kB T , from 0.1 to 0.3. Note that this range spans
more than an order of magnitude of N0 values and encompasses the bulk of thedata shown in figure 4.5.
It is worth pointing out that the data shown in figure 4.8 can equally be describedby a second-order polynomial fit, instead of using the two slopes, S0 and S.
However, such a fit relies on precise measurements at very low N0 , which areoften not available. This is the reason we have chosen to use a single empiricallydefined parameter S for describing the experiments, valid over the typical range ofN0 values. Figure 4.11 shows the obtained slopes S(a, T ) for the 18 experimental
series of figure 4.5. Within experimental error, all data points fall onto a straightline when plotted against the dimensionless variable, X = ξT 2 a2/5 , supporting
the assumption that µ0 /kB T can still used as the relevant interaction parameter.
In order to further demonstrate the universality of our results and the validity of
our approach, figure 4.11 also shows data taken with a different atomic species,
87
Rb. This species was held in the |F, mF i = |2, 2i state and its scattering lengthof a = 99a0 is not tuneable. Two series were taken with this species, at T = 175nK
and at T = 203nK. These two data points lie well within the spread of the 39 K
data. The contributions of temperature drifts and finite-size effects to the data
must also be considered. We compensate for the small temperature drifts as
described below, while finite-size effects are shown to have no influence on the
exhibited degree of non-saturation.

144

The role of saturation in Bose-Einstein condensation
5000
4000

S

3000
2000
1000
0

0

400

800

X

1200

1600

Figure 4.11: Saturation in the non-interacting limit. The large-N0 slopes, S,
for the 18 data series of figure 4.5 and two additional 87 Rb series (black points),
plotted against the dimensionless interaction variable, X, of equation 4.34. An
unconstrained linear fit to the data gives an intercept S(0) = −20 ± 100, consistent
with complete saturation in the ideal-gas limit. See the caption of figure 4.9 for an
explanation of the error bars.

Correcting for temperature drifts
In our experimental series, we occasionally encounter temperature drifts of up toa few percent over the course of the series. We analytically correct for these drifts
by scaling every (NT ,N0 ) pair according to the scaling suggested by Hartree-Focktheory. We scale each experimental series by its nominal temperature, T ∗ . This
2/5
corresponds to plotting NT (T ∗ /T 0 )3 against N0 (T ∗ /T 0 ) in figure 4.8, where T 0
is the temperature assigned to each data point. T 0 is found by first performing
an exponential fit to a plot of measured temperature, Tm , against image index1 .
The value of this fit gives each image a temperature, Ti . We then perform alinear fit to the plot of Tm /Ti against N0 , yielding a factor, fN0 , for each image2 .
T 0 is then given by: T 0 = Ti ∗ fN0 . In plain words, this correction corresponds
to scaling the thermal and condensate atom numbers of each data point to therelative values they would take on, according to the Hartree-Fock theory, were
they at the nominal temperature of the series they belong to. This correction is
1
2

This is to take account of temperature drifts resulting from changes in the lab environment
over the course of the day
This takes into account the drift of temperature with decreasing condensate number.

4.6 Comparing experiment and theory

145

valid to first order, even though the HF theory is imperfect, since the temperaturecorrections involved are small.
Finite-size effects(id)

The first correction to the ideal-gas prediction, N0 /Ntot = 1 − (T /Tc )3 , due tothe effect of finite particle number has been found analytically by studying thelarge-N limit of the expression for the total number of particles at temperature
T [204, 205, 206, 207]
N=

∞
X
i=0

N (i ) =∞X

1e(i −µ)/kB T − 1
i=0(4.35)The result is given byN03ωho ζ(2) 2 −1/3
= 1 − t3 −tNN2ω̄(ζ(3))2/3|{z}(4.36)

finite-size correction(id)where ωho is the arithmetic mean of the trap frequencies and t = T /Tc is
the reduced temperature introduced above. This expression has been shown to
be indistinguishable from the exact value obtained using equation 4.35 alreadyfor N = 1000 [206]. As an example, for t = 0.75 and an atom number of
2 × 105 , with typical trapping frequencies, ω/2π = (65, 65, 84)Hz, the condensed
fraction is reduced by roughly 1.9% from its ideal gas value of 58%. However,(id)inserting expression 2.69 for Tc shows that the shift in the absolute number
N0 does not depend on the total number N and is instead given by δN0 =−(kB T /~ω̄)2 (3/2)(ωho /ω̄)ζ(2). At a given temperature, this is simply equivalent
to a uniform shift in the critical number, which in figure 4.8, corresponds toa uniform vertical shift of the data. This does not affect the extracted nonsaturation slopes and hence leaves the results of this chapter unchanged. Chapter
5 discusses finite-size effects on the shift of the critical temperature which must(id)
be taken into account when measuring the interaction-shift of Tc .

4.6.1 Extrapolation to the non-interacting limit
The saturation prediction of equation 4.1 for an ideal gas (a = 0) corresponds
to S = 0 at the origin of the graph in figure 4.11. It is important to distinguish146

The role of saturation in Bose-Einstein condensation

between the two ways in which the interaction variable, X ∝ T 2 a2/5 , can approachzero. In the T → 0 limit, NT vanishes for any value of N0 and so trivially S = 0.
It is therefore important for us to demonstrate that the slope S depends only on
the combination T 2 a2/5 , allowing us to obtain its value in the a → 0 limit for
any fixed, non-zero value of T . The solid red line in figure 4.11 shows the resultof this extrapolation by means of an unconstrained linear fit to the data. The
resulting value of the intercept, S(0) = −20±100, together with the demonstrated
agreement with Hartree-Fock theory in the small-N0 limit, confirm the concept ofa saturated Bose gas over a broad range of experimentally relevant parameters.

4.7 Discussion and Conclusions
The experiments described in this chapter consisted of probing the dependenceof the condensed fraction on the total atom number at a constant temperature.
This contrasts with the numerous studies, dating from some of the earliest experimental investigations of BEC to the present day, which have focussed on
measurement of the condensed fraction as a function of temperature and hence
have never explicitly and systematically addressed the issue of non-saturation of
the thermal component.
Nevertheless, these experiments drew attention to the need for a proper treatmentof the effect of interactions on Bose-Einstein condensation by providing clear
evidence of the inadequacy of ideal-gas theory. Below is a brief chronologicallyordered summary of those results most directly related to the experiments carriedout in our group and described in this chapter and in chapter 5.
Already within the few years following the first experimental realization of BEC
in dilute atomic gases, several experiments [208, 209, 210] compared the predictions of ideal-gas theory with their experimental findings. Figure 4.12 shows thechronological progression towards more precise condensed-fraction measurements,
revealing ever more clearly the deviations from the ideal-gas picture.
The first experiment to clearly demonstrate non-ideal behaviour was [209] (seefigure 4.12b), whose results also served as a helpful benchmark for subsequentdevelopment and testing of mean-field models for interacting clouds (see e.g.
[201, 123]).
Later studies [211, 212] also observed deviation from ideal-gas theory and insteadfound agreement with the ‘semi-ideal’ theory [201, 200]. In [213], the authors explicitly investigated the mutual interactions between the condensed and thermal

4.7 Discussion and Conclusions

a)!

b)!

147

c)!

Figure 4.12: The results of [208] (figure (a)), [209] (figure (b)) and [186] (figure(c))
for the condensed fraction, N/N0 , as a function of the reduced temperature, T /Tc0 .
The increasing precision of the measurements has enabled more detailed comparison
with mean-field models.

components. This experiment focussed on the narrow region at the interface between the condensate and the thermal cloud, which was expected to be sensitive
to these interactions. It was demonstrated that a dynamical model for the cloud’s
expansion based on the HFB theory presented in section 4.5 provided a good fit
to the experimental data, which clearly displayed a repulsion of the thermal cloudby the condensate in images taken after time-of-flight (see Fig. 2 of [213]).
A more comprehensive study was performed in [186], which measured the condensed fraction, expansion energy and equilibrium shape of a partially-condensed
87
Rb cloud. The use of Bragg scattering enabled the detection of larger condensed
fractions and accurate fitting of the thermal component at lower temperatures
than any previous study. The resulting condensed-fraction curves showed clear
deviation from the ideal-gas prediction (see figure 4.12c) and were compared toboth the semi-ideal model [201, 200], including finite-size effects, and to the selfconsistent Hartree-Fock theory described in section 4.5. The semi-ideal model
failed to reproduce the results (as in our experiments) although the self-consistent
Hartree-Fock theory yielded good agreement within experimental errors. The
authors also investigated the effect of the thermal cloud on the condensate bystudying its axial length as a function of temperature. The axial length is expected to grow very slowly due to the large aspect ratio of the trapping potential(ωz /ωr ≈ 0.02) and hence to closely reflect its in-situ value. A compression of thecondensate by the thermal cloud is observed. The semi-ideal model, which explicitly neglects such effects, unsurprisingly fails to fully account for this compression
whereas the Hartree-Fock model provides good agreement with the data. This
result again illustrates the over-simplicity of the semi-ideal model under typical148

The role of saturation in Bose-Einstein condensation

experimental parameters, as also shown by our results.
Furthermore, experiments described in [214] explicitly report the breakdown ofHartree-Fock mean-field theory in relatively small (Ntot ≈ 104 ), nearly-1D cloudsof 87 Rb. The authors observe the appearance of a quasi-condensate in the centreof the trap and show that Hartree-Fock theory fails to predict its existence, whichis consequently attributed to interaction-induced correlations between the atoms
as opposed to saturation of the excited states.
Two studies bearing conceptual similarities to the experiments reported in thischapter have been published in the last few years. In [215], condensation wasstudied in a spinor gas in which atoms were gradually transferred from two condensed spin components into an initially-unpopulated spin state. Non-saturationof the growing thermal cloud was not reported. However, since this component
experiences a potential modified by interactions with the already existing condensates, it is probable that non-saturation effects were merely obscured by thecomplicated dynamics of the system. The second study reports on the realizationof a BEC in a photon gas [216]. In this system, despite the absence of a directinteraction between the photons, their thermalization via collisions with the dyemolecules within the microcavity to which they are confined leads to a secondorder interaction between the light particles. The non-saturation arising fromthis effective interaction is visible in figure 2 of [216] as a clear increase of thearea under the ‘thermal’ part of the spatial intensity distribution as the numberof photons in the system is increased (at room temperature).
To this long line of investigations, the results of this chapter add the demonstration that under typical experimental circumstances, atomic Bose gases show
strong deviation from the saturation picture. In addition, this deviation is not
fully reproduced by the most commonly-used mean field theories. Our experimental control over the interparticle interaction strength, however, has enabled
us to extrapolate our results to the non-interacting limit and to provide thefirst direct evidence for excited-state saturation as the driving mechanism behind
Bose-Einstein condensation, confirming the purely statistical nature of the BEC
phase-transition as originally described by Einstein. We expect these results tobe generic to experiments on harmonically trapped 3D Bose gases with (relativelyweak) short-range s-wave interactions.
There exist several possible avenues for future research :1. We have so far measured only global properties (total numbers). In the4.7 Discussion and Conclusions

149

future it would be interesting to study saturation and other thermodynamic behaviour at the local level, by probing local thermal and condensate
densities. This would enable us to infer the effective equation of state of
the bulk system, as proposed in [217] and demonstrated, for example, in[218, 219, 220, 221].
2. The ability to probe local densities would also allow investigation of theeffects of different interaction potentials, such as very strong or long-range
interactions, which might influence saturation at the local level.
3. Trap geometry influences the behaviour of the thermal gas. The excludedvolume argument for non-saturation suggests that the effect grows with thedimensionality of the system and so it would be interesting to explore lowerdimensional as well as disordered geometries. In this context, Appendix F
describes our progress towards the implementation of a tightly-confining
optical potential for studies of two-dimensional systems.

5Interaction-dependence of the
BEC transition temperature“. . . I tell you there is no such message, no such home – only the millions of last
moments . . . nothing more. Our history is an aggregate of last moments.”
Thomas Pynchon, Gravity’s Rainbow

Abstract
While chapter 4 was concerned with the effects of interparticle interactions on
trapped gases below the critical temperature, in the presence of a finite condensed
fraction, this chapter describes experiments on the effects of these interactions
on the value of the transition temperature itself, i.e. in the limit of vanishingcondensate fraction. We observe for the first time the beyond-mean-field (bMF)
corrections to Tc in a harmonically-trapped gas and also observe non-equilibrium
effects on the apparent critical temperature in the regime of both very weak andvery strong interactions.

151

152

Interaction-dependence of the BEC transition temperature

5.1 Introduction and theory
As discussed in section 2.7, the critical temperature, Tc , is defined as the temperature at which the condensed fraction in the gas vanishes, N0 → 0 and is oneof the most fundamental quantities in the study of the BEC phase transition.
While clear predictions exist for the critical temperature of a non-interacting gas,
the effects of (repulsive) interparticle interactions on the condensation temperature have been the subject of theoretical debate for over fifty years, since thepioneering work of Lee and Yang [222, 223] on a homogeneous gas of hard-corebosons.
Despite the success of mean-field theories in predicting a broad range of properties such as the spatial distribution of the dilute gas in a harmonic trap, its
low-energy excitation spectrum and its small-amplitude oscillations, correlations
between the atoms due to their interactions are also expected to play an important role in many phenomena. These include the properties of superfluid
vortices, decoherence, quantum depletion, and others. Correlations also provide
corrections to the predicted mean-field values of thermodynamic quantities suchas the energy per particle and the transition temperature. Indeed, the former ofthese was recently measured in an experiment carried out in the ENS group in
Paris using a gas of 7 Li [220], while observation of the latter is the subject of thischapter, which is structured as follows:• Following a short summary of the theoretical predictions for the shift of
Tc with interaction strength in both a uniform and harmonically-trappedgas, we briefly review previous experimental measurements of the transition
temperature in trapped interacting clouds.
• We then describe our experimental sequence and analysis procedure formeasuring the shift of the critical temperature over a wide range of interaction strengths, and present our results for those measurements believed
to be at thermodynamic equilibrium.
• Finally we discuss the stringent requirements for performing equilibrium Tc
measurements, which are violated in the regimes of either very weak or verystrong interactions, and discuss their physical origins and significance.
• We conclude and discuss directions for future study.

5.1 Introduction and theory

153

5.1.1 Tc -shift in a uniform system
We recall that in an ideal, homogeneous bose gas, the transition temperature isgiven by
3/2

2π~2
n
(id)
(5.1)
Tc =
ζ(3/2)kB mwhich corresponds to the condition that the phase-space density reach its criticalvalue nλ30 = ζ(3/2) ' 2.612, as discussed in section 2.7. The fractional shift ofthis quantity by the inclusion of interactions is defined by
∆Tc
(id)
Tc

≡

Tc
(id)

Tc

−1

(5.2)

and is usually calculated as a function of the diluteness- or gas parameterδ ≡ an1/3(5.3)

i.e. the effect of interactions can be increased by either raising the atomic density,n, or by increasing the scattering length, a, for example by accessing a Feshbach
scattering resonance.
At the mean-field (MF) level, the shift of the uniform-system critical temperaturevanishes identically. Within this approximation, the interaction energy is given
by Uint = 2gn, where g is the interaction strength and n the density. A particle
with momentum p within this approximation therefore has an energy (p) =p2 /2m + 2gn. The result of this energy shift is a shift in the critical value ofthe chemical potential, µc , but not in that of the transition temperature. This iseasily seen by looking at the modification of the equation of state from that of an
ideal gas, arising from use of this expression for the energy (c.f. equation 2.64)
nλ3T = g3/2 (eβµ )(p)→(p)+2gn⇒

nλ3T = g3/2 eβ(µ−2gn)



(5.4)

where β = 1/kB T . At the transition, namely the point at which the argumentof g3/2 goes to 1, the criterion nλ3T = ζ(3/2) remains unchanged. The chemical
potential now has to reach a value higher by 2gn than its previous maximal
value, but the value of Tc is unaffected. In general, any constant shift in the
single-particle energies cannot change the critical temperature of the gas. A
correct calculation of the Tc shift therefore requires taking into consideration thecorrelations which develop near the critical point and are not captured within

154Interaction-dependence of the BEC transition temperaturemean-field theory.
This is a challenging task and only in the first few years of this century has thecommunity converged on a consensus for the functional dependence of the Tc shift
on the gas parameter. For an overview of these developments, see e.g. [224, 225,
226, 227]. Table 5.1 summarizes the evolution of the theoretical predictions since
the middle of the 20th century.
Year

Author(s)Prediction

1953

Feynman [228]∆Tc < 01958

Lee & Yang [223]

1960

Glassgold, Kaufman & Watson [229]1964

Huang [230]1971

Fetter & Walecka [231]1982

Toyoda [232]1992

Stoof [233]∆Tc
1/3(id) ∼ aρ
Tc∆Tc
1/3 1/2)(id) ∼ (aρTc∆Tc
1/3 1/2) , (a > 0)(id) ∼ (aρTc∆Tc(id) < 0Tc∆Tc
1/3 1/2)(id) ≈ −3.5(aρ
Tc
∆Tc
1/3
+ O(aρ1/3 )(id) = caρ
Tc

1996

Bijlsma & Stoof [234]c=4.661997

Grüter, Ceperley & Laloë [235]c=0.341999

Holzmann, Grüter & Laloë [236]

c=0.7

1999

Holzmann & Krauth [237]

c=2.3

1999

Baym, Blaizot, Holzmann, Laloë & Vautherin [238]

c=2.9

1999

Huang [239]2000

Reppy et. al. [240]2001

Arnold & Moore [224]c=1.32

2001

Kashurnikov, Prokofev & Svistunov [241]c=1.292003

Kleinert [242]c=1.142004

Kastening [243]c=1.272004

Nho & Landau [244]c=1.322010

Betz & Ueltschi [245]c=-2.33∆Tc
1/3 1/2)(id) ≈ 3.5(aρ
Tc
1

c=5.1

Table 5.1: Summary of the various functional dependencies and numerical coefficients predicted (and measured: [240]) for the lowest-order shift in the criticaltemperature of the homogeneous dilute Bose gas.

1

It was pointed out in [241] that the estimation of the interparticle scattering length in thisstudy was not correct.

5.1 Introduction and theory

155

It is now generally accepted that in a dilute Bose gas the Tc shift is, to first order,
linear in the gas parameter ([234, 238, 237, 240, 224, 241, 246, 225, 226, 227]), withrepulsive interactions believed to enhance condensation, i.e. shift the transition
temperature above its ideal-gas value1 , with the coefficient having a value ofapproximately c ≈ 1.3 [224, 241]:∆Tc≈ 1.3an1/3 ≈ 1.8
(id)

Tc

a
λ0

(5.5)

In the above expression, we have introduced the alternative interaction parametera/λ0 , with a the scattering length and λ0 the thermal wavelength at the ideal-gas
critical temperature. Conversion between the two parameters is done using theusual relationship nλ30 = ζ(3/2).

5.1.2 Tc -shift in a non-uniform system
The problem of the interaction shift of Tc in experiments on ultracold atomic
gases is made even more complex by the fact that these systems are produced inharmonic traps. For a harmonically trapped gas, the condensation temperature
is defined for a given atom number N , rather than for a given density due to theinhomogeneity of the cloud.
As mentioned previously, the ideal gas prediction for the transition temperature
in a trap of geometric-mean frequency ω̄ is~ω̄
Tc(id) =
kB



N
ζ(3)

1/3
(5.6)

where ζ(3) ≈ 1.202. This criterion corresponds to a phase-space density in thetrap centre equal to the uniform-system critical value i.e. n(0)λ3T = ζ(3/2). Theinteraction shift of the critical point can therefore be expressed either as a shift
in the critical temperature for a given atom number, ∆Tc (N ), for the purposesof comparison with theoretical literature, or as a shift in the critical number at
a given temperature, ∆Nc (T ), for easier visualization (as in figures 5.1 and 5.8,
below) and more closely related to our experimental procedure for determining
the critical point, described in section 5.3.
1

Namely, condensation is predicted to occur at a phase space density below the ideal gas
critical value of ζ(3/2) ≈ 2.612.

156

Interaction-dependence of the BEC transition temperature

Two opposing effects compete in determining the overall shift of the critical point
in a trapped gas: similarly to the homogeneous gas and in the spirit of thelocal density approximation (LDA), correlations near the critical point should
act to reduce the critical density i.e. raise the critical temperature, as implied byequation 5.5. However in conjunction with this, the mean-field repulsion between
the atoms broadens their spatial distribution and hence reduces their central
density, causing a reduction in the critical temperature. For weak interactions,the latter effect dominates the former, giving rise to a positive value of ∆Nc , orequivalently a negative value of ∆Tc . Figure 5.1 illustrates these two competing
effects.
2.612
3

n

V(r)

Figure 5.1: Opposing effects of interactions on the critical point of a Bose gas
in a harmonic potential V (r). Compared to an ideal gas (dotted blue line) withthe same Tc , repulsive interactions reduce the critical density, but also broaden thedensity distribution (solid red line). Mean-field theory (dashed line) captures only
the latter effect, and predicts an increase of the critical atom number Nc at fixed
temperature T , equivalent to a decrease of Tc at fixed N .

The negative Tc shift due to the broadening of the spatial profile can be calculated
within mean-field theory. This calculation was carried out in [198] and yields
∆Tc
(id)
Tc

≈ −1.33a 1/6
aN ≈ −3.426
aho
λ0

(5.7)

p
where aho is the trap oscillator length aho = ~/mω̄. This prediction corresponds
to the dashed-line spatial profile in figure 5.1 and we note that this result neglects
finite-size effects as well as (by definition) the correlation effects mentioned above.
The dominance of the negative MF shift of Tc over the positive beyond-MF one
goes beyond the difference in numerical pre-factors in equations 5.5 and 5.7. In
contrast to the homogeneous case, at the condensation point in a non-uniform
system only the central region of the cloud is close to criticality. The effect of
critical correlations on Tc is therefore expected to be reduced by a factor ∼ (a/λ0 )3compared to the uniform case, corresponding to the ratio of the volume of the5.1 Introduction and theory157

critical region to that of the thermal cloud. The MF result of equation 5.7 should
therefore be exact at first order in a/λ0 .
There have been several attempts to theoretically combine the effects of MF
repulsion and beyond-MF correlations on Tc for a harmonically trapped gas [247,248, 249, 250, 251], but no consensus has yet been reached beyond the expectationthat the additional beyond-MF shift should be positive.
 calculations, for  Recent
2ain addition to theexample, predict a logarithmic term of the form ln λa0λ0
quadratic term, such that the full functional form (neglecting finite-size effects)
is given by   2

a
∆Tc
a
a
00
0
(5.8)
≈ c1 + c2 + c2 ln
(id)
λ0
λ0
λ0
Tc

with c1 = −3.426032, c02 = −32πζ(2)= −45.856623 and c002 = −155 [249].
3ζ(3)Table 5.2 summarizes the Tc -shift predictions discussed in the preceding sections.

Uniform
(id)

Ideal

Tc

MF

=



n
ζ(3/2)

∆Tc
(id)
Tc

3/2

Harmonic
2π~2
kB m

(id)

Tc

∆Tc

=0

(id)
Tc

∆Tc
Interacting

∆Tc
bMF

(id)
Tc

≈ 1.3an1/3
≈ 1.8

a
λ0

= k~ωB

(id)
Tc

= c1



N
ζ(3)

1/3

= −3.426

a
λ0

a
λ0

  2aa
000+ c2 ln
+ c2
λ0
λ0
 3
a
+O
λ0

Table 5.2: Summary of the predicted critical temperatures and their interactioninduced shifts in uniform and harmonic trapping potentials.

158Interaction-dependence of the BEC transition temperature

5.2 Previous experimental studies
Previous measurements of the Tc shift have been performed for a/λ0 ranging
from 0.007 [252] to 0.024 [253]. The results of these measurements are consistent
with the MF prediction of equation 5.7 within experimental errors and could notdiscern the effects of critical correlations. Below is a brief summary of a selectionof these studies and their findings.
1. Ref. [209] reports the results of early experiments carried out using 87 Rb in
a trap with (ωx , ωy , ωz ) = 2π × (132, 132, 373)Hz. The condensed fraction
as a function of temperature is compared to the ideal-gas prediction
N0T= 1 − ( (id)
)3 , including finite-size effects, as shown in figure 5.2.
N
Tc

Figure 5.2: Condensed-fraction against reduced temperature, reproduced from
[209]. Condensation occurs with 40,000 atoms at a temperature of 280nK.

A least-squares fit to the form NN0 = 1−( TTc )3 yields a value of Tc = 0.94(5)Tc0
for the critical temperature, with finite-size effects estimated to account fora shift of 3% from the ideal-gas value and interaction effects speculated toaccount for the remaining shift.
2. Ref. [210] reports on experiments using 87 Rb in a cigar-shaped trap with(ωr , ωz ) = 2π × (11.7, 33.1)Hz. Condensation occurs with N ≈ 1.5 × 106
and T ≈ 430nK and the results are compared to the ideal-gas prediction,
as shown in figure 5.3.
Although a least-squares fit to the condensed fraction plot yielded Tc =(0.96±0.15)Tc0 for the critical temperature, in these experiments the thermaland condensed atom numbers are determined from gaussian fits to bothcomponents and the overall accuracy in the atom number is estimated at
20%. The accuracy in the determination of the temperature is estimated at

5.2 Previous experimental studies

159

Figure 5.3: Condensed-fraction against reduced temperature, reproduced from[210].

15% and therefore no clear departure of the critical temperature from its
ideal-gas value is observed within experimental error.
3. Ref. [253] reports on a systematic study of the interaction shift in a 87 Rbgas. The interaction parameter is varied by changing the atom number
loaded into the trap and the critical number is obtained by monitoring theappearance of a condensed component.

Figure 5.4: Tc -shift as a function of number, Tc (N ), as reported in [253].

Ideal-gas behaviour is excluded by over two standard deviations (see figure5.4) and the data agree with the mean-field prediction of equation 5.7.
Writing their results in the form ∆Tc /Tc0 = αN 1/6 , the authors obtain
α = −0.009(1)+0.002−0.001 , compared to the prediction α ≈ −0.007 of equation
5.7. Within experimental sensitivity, no evidence is seen for the predictedcondensate-enhancing effects of beyond-mean-field critical fluctuations.
The authors of [253] conclude that observation of beyond-mean-field effects
should be made possible by: (1) improved thermometry, with an accuracy of
1% or better, (2) enhancement of the interaction shift, possibly through use

160Interaction-dependence of the BEC transition temperatureof a Feshbach resonance and (3) local measurement of the critical density
near the trap centre, which should be most sensitive to critical fluctuations.
The results reported in this chapter are precisely an implementation of thefirst two of these measures. Furthermore, a separate study performed inour group [254] has used measurements of the finite condensed fraction
already present at the mean-field critical point to confirm the ‘condensateboosting’ effect of critical fluctuations and the expected scaling behaviour ofthe condensed fraction with interaction strength. This study also provides
a useful connection to the uniform-potential Tc -shift via the local density
approximation (LDA). However, discussion of these results is beyond thescope of this thesis.

4. Ref. [252] reports on experiments performing in-situ phase-contrast imaging
of trapped 23 Na clouds. Best agreement with the measured spatial profiles
is provided by the Popov mean-field approximation (see section 4.5). The
chemical potential extracted using this model is used to obtain an estimateof ∆Tc /Tc0 = −0.027±0.001 for the critical temperature, in good agreementwith the prediction of equation 5.7, ∆Tc /Tc0 ≈ −0.025.
5. The interaction shift in a uniform system was measured in a dilute sample
of 4 He adsorbed in a porous glass (vycor) [240]. The gas parameter an1/3
was varied over the range 0.03 → 0.075 (corresponding to a/λ0 = 0.021 →0.054) by changing the density of adsorbed 4 He and Tc was measured via
the onset of superfluidity in the sample (see figure 5.5).

Figure 5.5: Critical temperature Tc as a function of density for the 4 H-vycor
system. The onset of superfluidity is measured by observing the change in periodof a torsional oscillator to which the sample is mounted. Figure from [240].

These measurements confirmed the positive sign of the shift, in qualitative5.3 Experimental method

161

agreement with equation 5.5, but found a coefficient of 5.1 ± 0.9 for the
linear coefficient, closer to the value calculated in [233, 234].
Figure 5.6 shows the mean-field prediction of equation 5.7, together with theexperimental results reported in [209, 253] and [252].
0.00

Ensher et al. (1996)
Gerbier at al. (2004)
Meppelink et al. (2010)∆Tc/Tc0 (%)

-0.02
-0.04
-0.06
-0.08

MF theory

-0.10
-0.12
-0.14
0.00

0.01

0.02

a/λ0

0.03

0.04

Figure 5.6: The mean-field prediction of equation 5.7 is shown (blue line), together
with the measurements of [252] (green triangle), [209] (red point) and [253] (interval
between dashed lines)

5.3 Experimental method
The experimental procedure we use to probe the critical point in our interacting
gas is almost identical to that used in the non-saturation studies described inchapter 4. These measurements likewise begin with a partially-condensed 39 K gas,
prepared in its |F, mF i = |1, 1i internal state in our crossed optical dipole trap.
For the experiments described in this chapter, the geometric mean of the trap
frequencies is in the range ω̄ = 2π × (75 → 85)Hz corresponding to measurementswith atom numbers N ≈ (2 → 8) × 105 .
The critical point is measured by fixing the optical trap depth and allowing thetotal atom number to decay towards Nc via inelastic processes. Here too werely on elastic collisions to redistribute atoms between the condensed and thermal components and on the fixed trap depth to maintain an essentially constanttemperature (see figure 4.3).
We explore the range 0.001 < a/λ0 < 0.06 by exploiting the 402.5G Feshbachresonance in the |1, 1i state. The cloud is initially prepared at a scattering lengthof a = 135a0 , as described in chapter 3, which is then adjusted to the desired

162Interaction-dependence of the BEC transition temperature

value by ramping the Feshbach field to the appropriate strength over a few ms.
The cloud is held for a variable hold time before being released from the trap andimaged after a 19ms time-of-flight. The total hold time required for the atom
number to drop below Nc depends on the specific value of a being measured andthe temperature, ranging between 5 → 18s.
In addition to extending the a/λ0 range over previous experiments, use of the
Feshbach resonance allows us to eliminate several key sources of statistical andsystematic errors, greatly improving both the precision and accuracy of our ∆Tcmeasurements. More specifically, turning off interactions during TOF allows us
to detect smaller condensed fractions, and performing a low-a reference measurement for each interaction strength allows us to isolate the interaction shift andeliminate all a-independent systematic errors (see below).

1) Removal of interactions during time-of-flight
In order to home in on the critical point we use the Feshbach resonance to turn off
the interactions during time-of-flight. To accomplish this we quickly (in . 2ms)
ramp the Feshbach field to the zero-crossing at 350G immediately after release of
the gas from the trap. This minimizes the expansion of small condensates during
TOF and allows us to reliably detect condensed fractions as small as ∼ 10−3 (seefigure 5.7).

Figure 5.7: (a) An absorption image of a cloudwith 450 000 atoms and a
0.14 % condensed fraction.
We show an azimuthally averaged cut through the column density for illustrative
purposes, while the full 2D
distribution is used for fitting. The gas was preparedat a large scattering length,a = 274 a0 , but the interactions were turned off in TOF.

Once imaged, we determine small condensed numbers N0 by performing a direct
summation over the density distribution in the central part of the image, after5.3 Experimental method

163

subtracting a fit to the smooth thermal background. For N0 > 2000 this agrees
with a standard Thomas-Fermi fit, but is more reliable for smaller condensates.

2) Reference measurements at low a
For each measurement series at a given a and λ0 , we concurrently take a reference
measurement with a different a, same ω̄ and very similar N , hence very similar
λ0 . This is done by alternating experimental runs at the two different scattering
lengths. Specifically, for the reference point we choose a small a such that a/λ0 ≈0.005. Measurements at this value of the interaction parameter are still taken
to be in thermal equilibrium, as discussed in section 5.5. With this procedure,
we thus directly access the small Tc shift due only to the difference in a/λ0 ,and essentially eliminate all a-independent systematic errors that usually affectabsolute measurements of Tc (N, ω̄, a). These include uncertainties in the absolutecalibration of N and ω̄, as well as the additional Tc shifts due to finite-size effects[46] and the small anharmonicity of the trapping potential [255]. Figures 5.8aand 5.8b show the relationship between the condensed (N0 ) and thermal (NT )atom number near the critical point and illustrate the differential measurementfor one of our experimental series.

6

56a0

500

274a0

NT (thousands)

N0 (thousands)

8

4

400

2
0
350

274a0

450

400

450

NT (thousands)

500

350

56a0
0

10

20

N

2/5
0

30

Figure 5.8: Determination of the critical point and differential interaction shift.
(a) Condensed (N0 ) versus thermal (NT ) atom number for two concurrently takenseries with a = 56a0 (blue circles) and a = 274a0 (black squares). Note that all
points correspond to condensed fractions below 2%. The data is scaled to thesame temperature (T = 240nK) and shows the shift of the critical point in theform ∆Nc (T ). The solid point corresponds to the image shown in figure 5.7. Solidlines show the extrapolation to N0 = 0, necessary to accurately determine Nc . (b)2/5
NT is plotted versus N0 for the same data as in (a), showing more clearly the
extrapolation procedure.

164

Interaction-dependence of the BEC transition temperature

Systematic errors
In general, interactions increase the kinetic energy of atoms during TOF, resulting
in an a-dependent error in the measured temperature which is not cancelled out by
our differential measurements. To minimize this error we extract the temperaturefrom a fit to only the wings of the thermal distribution, excluding the central
thermal radius from the fit as described in section 3.15 (see also [186, 185]). The
removal of interactions during TOF using the Feshbach resonance also serves
to reduce this error, but the reduction of a is gradual over ≈ 2ms. We havemeasured the difference between the (apparent) temperatures with and ‘without’interactions during TOF to be approximately linear in a/λ0 , and about 4% for
a = 400a0 and λ0 ≈ 104 a0 . By varying the time at which we turn off a, weestimate our residual error to be 1−2% at a/λ0 = 0.04. This estimate is supported
by numerical simulations which reproduce the experiments quite well.
Additionally, interactions modify the initial in-trap momentum distribution. This
reduces the apparent T due to the positive chemical potential preferentially enhancing the population of low-energy states. We numerically estimate this effectto also be approximately linear in a/λ0 , and about 1 − 2% at a/λ0 = 0.04. Fortuitously, the in-trap and in-TOF effects partially cancel, resulting in a net error(id)in ∆Tc /Tc of at most ±1 % at a/λ0 = 0.04.

5.4 Extracting the Tc shift
Extrapolating to find Nc
The critical temperature of an experimental series is found by measuring thetemperature at the critical point, namely when Ntot = Nc = N (N0 → 0). In
order to precisely locate Nc , it is therefore essential to carefully extrapolate NT
to the N0 → 0 limit in order to accurately determine Nc . As can be seen in figure
5.8a, the rise of N0 with NT is not simply vertical due to the non-saturation of
the thermal component in a partially condensed gas (c.f. chapter 4). Since ourmeasurements take place at small N0 , we perform the extrapolation using the
Hartree-Fock non-saturation prediction described in section 4.6
2/5

NT = Nc + SHF N0(5.9)

with the non-saturation slope SHF (T, ω̄, a) calculated with no free parameters5.4 Extracting the Tc shift

165

using equations 4.33 and 4.34. This extrapolation is shown in figure 5.8b. The
shot-to-shot temperature fluctuations near Nc are at the 1% level, while the T
drift over the entire series is < 10%. This drift is analytically compensated,as described in chapter 4, by applying the mean-field scaling, which amounts
2/5
to plotting NT (Tc /T )3 against N0 (Tc /T ), as is done for the data presented inchapter 4.

Calculation of ∆Tc
The Tc shift for each series is obtained by comparing the measured value of Tc
with that obtained from its corresponding low-a reference series. If the two series
possessed identical Nc values, ∆Tc would simply be given by the difference intheir respective transition temperatures. In order to correct for the small (few%) difference in Nc , we apply the ideal-gas scaling Tc ∝ N 1/3 to the reference
series temperature. i.e. Tcref∗ = Tcref (Ncm /Ncref )1/3 , where (Ncref , Tcref ) and (Ncm , Tcm )are the critical number and temperature for the reference and ‘measured’ series,
respectively. In plain words, Tcref∗ is what the critical temperature of the reference
series would be if it had the same value of Nc as the measured series. The final
expression for the Tc shift is therefore given by
(id)

∆Tc

Tcm − Tc

Tc

Tc

=
(id)

(id)

(Ncm )(5.10)(id)where the value of Tc is obtained from Tcref∗ using the mean-field shift, i.e.
(id)
Tc = Tcref∗ /(1 − 3.426 λa0 ). We shift our results to ensure that our a/λ0 ≈ 0.005
reference point lies on the mean-field prediction, although assuming the ideal(id)gas value of Tc for a/λ0 = 0.005 would not affect our results within error
bars. Figure 5.9 summarizes our equilibrium measurements of the interaction(id)shift ∆Tc /Tc . We took data with a range of atom numbers, N ≈ (2 → 8) × 105(id)(corresponding to Tc ≈ 180−330 nK), in order to verify that our results depend
only on the interaction parameter a/λ0 .
The dashed blue line shows the MF result of equation 5.7, which agrees very
well with the data for a/λ0 . 0.01. For larger a/λ0 we observe a clear deviationfrom this prediction. All data points are fitted well by a second-order polyno(id)mial (solid red line), ∆Tc /Tc = b1 (a/λ0 ) + b2 (a/λ0 )2 , with b1 = −3.5 ± 0.3
and b2 = 46 ± 5. Calibration of the y-axis is done by ensuring that this fit ex(id)trapolates to ∆Tc /Tc = 0 for a/λ0 → 0. Within our error bars, the predicted166Interaction-dependence of the BEC transition temperature0∆Tc/Tc0 (%)

-2

-4

-6

-8
0.00

Mean-field
prediction
0.01

0.02

a/λ0

0.03

0.04

Figure 5.9: Interaction shift of the critical temperature. Data points were takenwith N ≈ 2 × 105 (blue circles), 4 × 105 (black squares), and 8 × 105 (red triangles)(id)
atoms. The dashed line is the mean-field result ∆Tc /Tc = −3.426 a/λ0 . The solidline shows a second-order polynomial fit to the data (see text). Vertical error bars
show standard statistical errors. Horizontal error bars reflect the 0.1 G uncertainty
in the position of the Feshbach resonance.

small logarithmic corrections of equation 5.8 are not discernible. The value ofb1 is in excellent agreement with the MF prediction of −3.426 [198]. The valueof b2 strongly excludes zero, and its sign is consistent with the expected effectof beyond-MF critical correlations. These measurements provide the first clear
observation of beyond-MF effects on the transition temperature of a harmonicallytrapped gas.

5.5 Non-equilibrium measurements
In this section we discuss the requirements on the relationship between the holdtime in the CDT, thold , the elastic scattering rate, γel and the relevant atomnumber decay time τ for the measurements of Tc to faithfully reflect the equilibrium properties of the gas. In general, a system with continuous dissipation can
only be ‘close to’ thermodynamic equilibrium. For an atomic gas, the proximity
to equilibrium depends on the dimensionless parameter γel τ , which measures therelative rates of elastic and inelastic processes.

5.5 Non-equilibrium measurements

167

Following [192], we make use of the following expressions in calculating γel :γel = n(0)vthn(0) = N ω̄
r
vth =

3



σ0
2

m
2πkB T

8kB T
πm

3/2

(5.11)

where n(0) is the central density, vth the mean thermal velocity and σ0 the scattering cross-section. In practice the elastic scattering rate γel required for equilibrium measurements also depends on the precision with which measurements
are made. In these experiments, we measure Nc to roughly 1%, and so we requirethat the gas continuously re-equilibrate on a timescale τ corresponding to only
a 1% atom loss. This implies a value of γel roughly 100 times higher than would
be naively assumed by taking the 1/e lifetime as the relevant timescale for decayof the cloud.
Equilibration is usually considered to take about 3 collisions per particle [191].
With this in mind, for all the measurements shown in figure 5.9 we made sure thatthe number of collisions occurring on the timescale τ fulfilled γel τ > 5. All ourdata also satisfy the condition thold > τ > 1/ω̄, necessary for global equilibrium
to be established.
It is interesting to observe what happens if we violate these stringent equilibrium
criteria. In figure 5.10a we show measurements with N ≈ 4 × 105 atoms (λ0 ≈104 a0 ), extending beyond the equilibrium region shown in figure 5.9.
We still show only measurements satisfying thold > τ > 1/ω̄ and γel thold > 5,so that there is nominally enough time for global equilibrium to be established.
However if γel τ is not large enough, the elastic collisions cannot ‘keep up’ withthe continuously present dissipation. The resulting non-equilibrium effects canthus not be eliminated by simply extending thold , but are an intrinsic property ofthe system. In figure 5.10b we plot γel τ , based on γel calculated from equation
5.11 and τ measured near the critical point for each series. Individually, γel ≈0.7 − 1000 s−1 and τ ≈ 2ms − 1s vary vastly as a function of a (γel increasing andτ decreasing), but the breakdown of equilibrium appears to occur at very similarvalues of γel τ in the low- and high-a limit.
Since non-equilibrium phenomena necessarily depend on additional factors such

168

Interaction-dependence of the BEC transition temperature∆Tc/Tc0 (%)

4
0

(a)
Equilibrium region

-4
-8
(b)

3

el

1

el /

2

10

1
0.00

0.01

0.02

0.03

a/λ0

0.04

0.05

0.06

0

(id)

Figure 5.10: Non-equilibrium effects. (a) ∆Tc /Tc
for N ≈ 4 × 105 atoms isdetermined following the procedure which assumes equilibrium (as in figure 5.8). At
both very low and very high a the apparent Tc deviates from the equilibrium curve.
(b) Equilibrium criteria: γel τ (solid squares) is the number of elastic collisions per
particle during 1% atom-loss; γel /ω̄ = 1 (open circles) marks the onset of thehydrodynamic regime.

as the initial conditions, we do not expect our quantitative results to be universal
and instead discuss only the qualitative trends in our data. In the small-a limit weobserve a smooth rapid rise of the apparent Tc above the equilibrium curve (and(id)
hence above Tc for a → 0). We can qualitatively understand this effect withina simple picture. In this regime, losses are most likely dominated by inelastic
collisions with background particles (‘one-body processes’) which equally affect
N0 and NT . The net effect of the equilibrating elastic collisions would therefore
be to transfer atoms from the condensate to the thermal cloud. However thedissipation rate is too high compared to γel , and so N0 remains non-zero evenafter the total atom number drops below the equilibrium critical value Nc (i.e.
the measured Tc is above the equilibrium value). Although strictly speaking T
is not defined out of equilibrium, the absolute value of the observed effect issufficiently small that an equilibrium distribution function fits the data very welland provides a good measure of the energy content of the cloud.

5.6 Conclusions

169

Our measurements in the large-a limit suggest that the initial breakdown of equilibrium again results in condensates surviving above the equilibrium Tc . However
the physics in this regime is much richer, with several potentially competing effects requiring further investigation. For example, three-body decay affects N0
and NT differently, the thermal component is far from saturation (chapter 4), andthe gas also enters the hydrodynamic regime, γel /ω̄ > 1 (see figure 5.10(b).

5.6 Conclusions
In conclusion, we have performed high-precision studies of the effects of interactions on Bose-Einstein condensation of a trapped atomic gas. In the regime
where equilibrium measurements are possible, our most important observation isthe clear deviation from mean-field behaviour for sufficiently strong interactions.
The additional positive shift of the critical temperature is a clear signature of
the condensation-enhancing effect of critical fluctuations. These measurements
should provide motivation and guidance for further theoretical studies of thisdifficult problem. We have also studied non-equilibrium condensation phenomena, for both very weak and very strong interactions. Further study of theseeffects should prove useful for understanding condensation in intrinsically out-ofequilibrium systems, such as polariton gases.
The interplay between trap geometry and interaction strength is also a promising
direction for future research on both the shift in the critical point and the out-ofequilibrium behaviour of the system. Indeed, preparations are already underway
in our group for generation of a two-dimensional trapping potential for exploring
the effects of reduced-dimensionality on the thermodynamic behaviour of ourgases. Preliminary designs and tests of the setup for generating these potentials
optically are described in appendices C and F.

6
Summary and Outlook“What goes around may come around, but it neverends up exactly the same place, you ever notice? Likea record on a turntable, all it takes is one groove’s
difference and the universe can be on into a whole’nother song.”
Thomas Pynchon, Inherent Vice

It is my hope that this thesis has served two main purposes. The first was toprovide relevant and useful information regarding the design and construction ofan apparatus for reliable production of ultracold rubidium and potassium gases.
During the design stage, we explicitly endeavoured to assemble an apparatus thatwas flexible, versatile and simple, while compromising as little on performanceas possible. Plentiful optical access, the implementation of both magnetic andoptical trapping potentials, the inclusion of fermionic 40 K sources, a reliable andstraightforward transport mechanism and an abundance of coils for productionof custom magnetic field configurations, should all contribute to the ability ofthis setup to serve as a platform for many successful experiments in the future. I
hope that other students, both within our group and out, find this record of ourexperiences to be a helpful resource in their own studies.

171

172

Summary and Outlook

We have used this system to produce condensed clouds of both 39 K and 87 Rb the second group in the world to do so. A conveniently-located, broad Feshbachresonance in the ground state of 39 K makes this species one of the most attractive for systematic studies into the effects of interactions on the thermodynamic
behaviour of cold gases. The second purpose of this thesis was the descriptionof two experiments, carried out using condensed clouds of 39 K and fundamentally reliant on this precise control over the atoms’ interaction strength. The
first of these experiments scrutinized Einstein’s original saturation picture as thedriving mechanism behind BEC. The deviation from this idealized picture due
to interatomic interactions was found to be very strong, and could only be captured quantitatively by a mean-field treatment in the regime of small condensate
number. In the larger-condensate regime a more heuristic approach was used toquantify the lack of saturation and by extrapolation of our results to the noninteracting limit we were able to recover complete saturation in the ideal-gas limit.
This provided the first direct experimental evidence for excited-state saturation
as the basic mechanism behind BEC, confirming the purely statistical nature ofthis phase-transition as originally described by Einstein.
The second experiment described in this thesis made use of similar experimental
techniques to those developed in the saturation study to perform high-precision
measurements of the shift in the critical temperature induced by interparticle
interactions. In addition to confirming the predictions of mean-field theory in theregime of weak interactions, we have provided the first observation of the effects
of interparticle correlations, arising in the vicinity of the transition, on the criticaltemperature. These beyond-mean field corrections are in qualitative agreementwith recent theoretical predictions and our measurements should serve as a useful
guide for further theoretical efforts. As part of this study, we have also observedsome intrinsic non-equilibrium behaviour of our trapped clouds in the regimes ofboth very weak and very strong interactions.
The technique shared by both of these experiments is their reliance on the gradualdecay in atom number resulting from inelastic processes and background collisions. Trap-loss measurements have long been used to study collisional dynamics
of trapped clouds in magneto-optical traps [256, 257, 258, 259, 260, 55] as well
as of condensed clouds [261]. To these studies the current work adds the probing of thermodynamic quantities such as the condensed fraction and the criticaltemperature. There can be no doubt that many more insights are waiting to be
extracted from such conceptually straightforward yet subtle experiments.

173
Many open questions remain, predominantly related to exploring the effects ofinterparticle interactions on the behaviour of these systems under different circumstances. These include, but are not limited to• Confinement in potentials of reduced dimensionality.
• Confinement in periodic potentials.
• In-situ measurements of thermodynamic quantities.
• Novel interaction regimes, such as very strong interactions and long-range
interaction potentials.
• Confinement in the presence of impurities and/or disorder.
• Trapping of Bose-Bose and Bose-Fermi mixtures in the presence of tuneableinterspecies and intraspecies interactions.
• Manipulation using artificial optically-generated gauge potentials.
All in all, degenerate quantum gases promise to yield many physical insights andapplications for years to come by providing an almost ideal test-bed in which
to investigate many open questions in condensed matter physics, materials science, fundamental quantum mechanics and even cosmology [262, 263], as well
as applications in fields ranging from metrology to quantum communication andquantum information processing.

Appendices

175

A

87Rb and 39K properties

General and optical properties of 87 Rb and 39 K. The 87 Rb data is taken from[110] and the 39 K data from [111], both of which contain references to the original
data sources.
Property
Natural abundance
Mass
Nuclear spin (I)Atomic number (Z)Melting point
Boiling point
Vapor pressure at 293 K
Density at 293 K

87

Rb value
27.83(2) %86.909180520(15) amu
3/2
37
39.31 ℃
688 ℃
4.0 × 10−7 mbar
1.53 g/cm3

39

K value
93.2581(44) %38.96370668(20) amu
3/2
1963.65 ℃(336.8 K)
774.0 ℃(1047.15 K)
1.3 × 10−8 mbar
0.862 g/cm3

Table A.1: General properties of 87 Rb and 39 K. Values taken from references [110]and [111], respectively.

177

87

178
PropertyFrequency
Wavelength
Wavenumber
Lifetime
Natural linewidth
Recoil velocity
Recoil temperature
Doppler temperature
Saturation intensity

symbol
ν
λ
k/2π
τ
Γ/2π
vrec
Trec
TD
Is

Rb and 39 K properties

D1 -line value
377.1074635(4) THz
794.9788509(8) nm
12578.950985(13) cm−1
27.70(4) ns
5.746(8) MHz
0.57754 cm/s
348.66 nK
137 µKD2 -line value384.2304844685(62) THz
780.241209686(13) nm
12816.54938993(21) cm−1
26.24(4) ns
6.065(9) MHz
0.58845 cm/s
361.96 nK
146 µK1.669(2)mW/cm2

Table A.2: D1 - and D2 -line optical properties of 87 Rb. Values taken from reference[110].

PropertyFrequency
Wavelength
Wavenumber
Lifetime
Natural linewidth
Recoil velocity
Recoil temperature
Doppler temperature
Saturation intensity

symbol
ν
λ
k/2π
τ
Γ/2π
vrec
Trec
TD
Is

D1 -line value389.286058716(62) THz
770.108385049(123) nm
12985.1851928(21) cm−1
26.37(5) ns
6.03(1) MHz
1.329825973(7) cm/s
0.41436702 µK
145 µKD2 -line value
391.01617003(12) THz
766.700921822(24) nm
13042.8954964(4) cm−1
26.37(5) ns
6.035(11) MHz
1.335736144(7) cm/s
0.41805837 µK
145 µK
1.75mW/cm2

Table A.3: D1 - and D2 -line optical properties of 39 K. Values taken from reference[111].

179

5 DATA TABLES

gF = 2/3(0.93 MHz/G)
193.7408(46) MHz
72.9113(32) MHz

23

F=3

266.650(9) MHz

5 2 P3/2
gF = 2/3(0.93 MHz/G)

229.8518(56) MHz

F=2

156.947(7) MHz
302.0738(88) MHz

5 DATA TABLES

72.218(4) MHz

gF = 2/3(0.93 MHz/G)

F=1
24

F=0

gF = 1/6(0.23 MHz/G)

5 DATA TABLES

780.241 209 686(13) nm
384.230 484 468 5(62) THz
2
816.549 389 93(21) cm-1
5 P1/2 121.589
049 439(58) eV

306.246(11) MHz

gF = 1/6(0.23 MHz/G)
gF = -1/6(0.23 MHz/G)

306.246(11) MHz2

5 P1/2F=2
F=1816.656(30) MHz
510.410(19) MHz

2.563 005 979 089 11(4) GHz2

24

816.656(30) MHz
510.410(19) MHz

5 S1/2F=2794.978 850 9(8) nm
377.107 463 5(4) THz
12 578.950 985(13) cm-1
1.559 590 99(6) eV

gF = 1/2(0.70 MHz/G)
gF = -1/6(0.23 MHz/G)

F=2
F=1

6.834 682 610 904 29(9) GHz

4.271 676 631 815 19(6) GHz
794.978 850 9(8) nm
377.107 463 5(4) THz
12 578.950 985(13) cm-1
1.559 590 99(6) eV

gF = -1/2(-0.70 MHz/G)

F=1

F=2

87

gF = 1/2(0.70 MHz/G)Rb D2 transition hyperfine structure, with frequency splittings between
the hyperfine energy levels.

Figure 2:
The excited-state values are taken from [6], and the ground-state values are from [16]. The approximate Landé
2.563 005 979 089 11(4) GHz
gF -factors for each level are also given,
with the corresponding Zeeman splittings between adjacent magnetic
sublevels.

5 2S1/2

gF = 1/2
6.834 682 610 904 29(9) GHz(0.70 MHz/G)

F=2

2.563 005 979 089 11(4) GHz
4.271 676 631 815 19(6) GHz

5 2S1/2
6.834 682 610 904 29(9) GHz
gF = -1/2(-0.70 MHz/G)

F=1

4.271 676 631 815 19(6) GHz
87

Figure 3: A.1:
Rb D87transition
hyperfinestructure,with frequency
splittings
between theenergy levels.
1 Rb
FigureD-linehyperfinestructure.
The LandégF -factors
forhyperfine
each level
The excited-state values are taken from [7], and the ground-state values are from [16]. The approximate Landéarealso given,withsplittings
betweenadjacent
magneticF=1
g -factors
for eachleveltheare corresponding
also given, with Zeeman
the corresponding
Zeemansplittingsg = -1/2between adjacent magnetic(-0.70MHz/G)
sublevels.
Valuestaken
fromreference[110]
sublevels.
F

F

Figure 3: 87 Rb D1 transition hyperfine structure, with frequency splittings between the hyperfine energy levels.
The excited-state values are taken from [7], and the ground-state values are from [16]. The approximate LandégF -factors for each level are also given, with the corresponding Zeeman splittings between adjacent magnetic

87

180
5 DATA TABLES

Rb and 39 K properties

gF = 2/3(0.93 MHz/G)
193.7408(46)14.4 MHzMHz
72.9113(32)MHz
6.7 MHz23

F=3266.650(9)MHz
21.1 MHz

P3/2
5422P
3/2
gF = 2/3(0.93 MHz/G)

229.8518(56)16.1 MHzMHz

F=2

9.4 MHz156.947(7)MHz
302.0738(88)19.4 MHzMHz

5 DATA TABLES

72.218(4)MHz
3.3 MHzgF = 2/3(0.93 MHz/G)

F=1
24

F=0

gF = 1/6(0.23 MHz/G)

5 DATA TABLES

780.241209nm
686(13) nm
766.701
384.230
484 THz468 5(62) THz
391.0162
816.549 389 93(21)cm-113042.894
cm-1
5 P1/2 12
1.589 049 439(58) eV

306.246(11) MHz

510.410(19) MHz

gF = 1/6(0.23 MHz/G)
gF = -1/6(0.23 MHz/G)

20.8 MHz306.246(11)
MHz

22

54 P
P1/2
1/2

F=2
F=1

816.656(30)MHz
55.5 MHz34.7 MHz
510.410(19)MHz

2.563 005 979 089 11(4) GHz2

24

816.656(30) MHz

1.619 eV

5 S1/2F=2794.978
850 9(8)nm
nm
770.108
377.107463 5(4)THz
THz
389.286-1
12 578.950 985(13) cm
12985.192 cm-1
1.559 590 99(6) eV

gF = 1/2(0.70 MHz/G)
gF = -1/6(0.23 MHz/G)

F=2
F=1

6.834 682 610 904 29(9) GHz

1.612 eV

4.271 676 631 815 19(6) GHz
794.978 850 9(8) nm
377.107 463 5(4) THz
12 578.950 985(13) cm-1
1.559 590 99(6) eV

gF = -1/2(-0.70 MHz/G)

F=1

F=2

87

gF = 1/2(0.70 MHz/G)Rb D2 transition hyperfine structure, with frequency splittings between
the hyperfine energy levels.

Figure 2:
The excited-state values are taken from [6], and the ground-state values are from [16]. The approximate Landé
2.563 005173.1
979 089
11(4) GHzMHz
gF -factors for each level are also given,withthecorresponding Zeeman splittings between adjacent magnetic
sublevels.

S1/2
5422S
1/2

gF = 1/2
6.834 682461.7
610 904
29(9) GHzMHz
(0.70 MHz/G)

F=2

2.563 005 979 089 11(4) GHz
4.271 676288.6
631 81519(6) GHzMHz

5 2S1/2
6.834 682 610 904 29(9) GHz
gF = -1/2(-0.70 MHz/G)

F=1

4.271 676 631 815 19(6) GHz
87

Figure 3: A.2:
Rb D39structure,with frequency
splittings
between theenergy levels.
1 transition
FigureK D-linehyperfine
hyperfinestructure.
The LandégF -factors
forhyperfine
each level
The excited-state values are taken from [7], and the ground-state values are from [16]. The approximate Landéarealso given,withsplittings
betweenadjacent
magneticF=1
g -factors
for eachleveltheare corresponding
also given, with Zeeman
the corresponding
Zeemansplittingsg = -1/2between adjacent magnetic(-0.70MHz/G)
sublevels.
Valuestaken
fromreference[111].
sublevels.
F

F

Figure 3: 87 Rb D1 transition hyperfine structure, with frequency splittings between the hyperfine energy levels.
The excited-state values are taken from [7], and the ground-state values are from [16]. The approximate LandégF -factors for each level are also given, with the corresponding Zeeman splittings between adjacent magnetic

B
A compact system for the rapid
production of 87Rb BECs

During the first year of our work, in parallel with the design and assembly of thelaser, imaging and control systems, we constructed a single-chamber apparatus
intended to serve as a platform on which to test the performance of these various
components . Initially intended to be used solely for the production of a MOT,
it was realised soon after this goal was achieved that condensation of 87 Rb in thissetup was a possibility. On the 24th Feb. 2009, we observed our first 87 Rb BEC in
this system. Condensation was achieved by evaporative cooling in an opticallyplugged quadrupole trap. This type of hybrid trap, utilizing both optical andmagnetic forces, was used in the first observation of BEC in the MIT group in 1995
[24] and has since been used successfully in several other groups [264, 265, 266].
Several other compact machines for producing degenerate Bose gases have been
demonstrated [267, 268, 269] and a commercial venture [270] has even been established to capitalize on the simplicity, reduced cost and versatility of such systems.

181

A compact system for the rapid production of 87 Rb BECs

182

Below is a brief description of the setup and the sequence used in order to produce
ultracold clouds in this relatively simple machine.

B.1 Vacuum chamber
The compact vacuum chamber consists of a circular quartz cell1 , fitted with seven1” AR-coated quartz windows around its perimeter and two 2” AR-coated quartz
windows on its top and bottom. This cell is connected to a stainless steel chamber
via a 15cm long glass tube of 15mm diameter. The vacuum chamber is continuously pumped by a 45ls−1 ion pump 2 and can be connected to a turbo pump3
via an all-metal right-angle valve4 during bakeout. A Bayard-Alpert ionization
gauge5 is fitted to the chamber for monitoring the pressure and a source module
consisting of four atom sources6 mounted to an eight-pin electrical feedthrough7
is also fitted. Figure B.1 shows a top-view schematic of the chamber.

1!
6!

7!

2!
Imaging!

4!
MOT!

3!

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCTPRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCTPRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

8!

5!

Figure B.1: Schematic of the small-system vacuum chamber. (1) Ion pump, (2)viewport for imaging/pumping/plug beams, (3) turbo pump, (4) right-angle valve,(5) atom source module, (6) ionization gauge and (7) glass MOT cell. Also shown
are the MOT and imaging beams, the quadrupole coils and the CCD camera used
for absorption imaging (8).
PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

Figure B.2 shows images of the bare assembled system, the various magnetic coils
1
2
3
4
5
6
7

Triad Technology, Inc.
œrlikon Leybold IZ50œrlikon Leybold TURBOVAC TW 70 H
VAT, Series 540 DN40
Varian, Inc., UHV-24p
Alvatec, 2 × AS-3-K-30-V and 2 × AS-3-Rb-50-V
Kurt J. Lesker, EFT0084033

B.1 Vacuum chamber

183

Figure B.2: (a) The bare system after assembly and bakeout, (b) the magnetic
coils mounted around the cell and (c) a top view showing the directions of the
MOT and imaging beams together with the optical cmponents.

184

A compact system for the rapid production of 87 Rb BECs

mounted about the glass cell and a top view of the apparatus indicating the beampaths of the MOT and imaging light.

B.1.1 Bakeout

Temperature [°C]

Pressure [10-7 mbar]

Bakeout of the chamber proceeded in a similar manner to that of the main chamber, described in section 3.2.9, with the chamber being wrapped in heater tapeand the applied heating current controlled using variacs. Temepratures at various locations were monitored using thermocouples and logged using a LabView
VI. Figure B.3 shows an overview of the system pressure (black line), as wellas the temperatures measured at various locations, during the first six days ofthe bakeout procedure. Clearly visible are the pressure spikes that accompany
the raising of the temperature, followed by a gradual drop in the pressure - atconstant temperature - as the released impurities are pumped from the system.

Time [days]

Figure B.3: Overview of the first six days of the bakeout process. The pressure asmeasured by the ionization gauge is shown in black and the temperatures at various
locations along the chamber are shown by the coloured data points. Temperaturerises are accompanied by pressure rises and subsequent decays as released impurities
are removed by the pump.

B.2 Lasers and coils
The cooling, repump, pumping and imaging beams for cooling 87 Rb in this chamber are generated by the same laser system used for the main experiment anddescribed in section 3.3. Similarly to the main experiment, once on the vacuum
table, the cooling and repump beams are spatially overlapped before being splitB.2 Lasers and coils

185

into the six beams required for the 3D MOT. Figure B.4 shows the home-madefiberport cluster assembled for this purpose using a custom-machined baseplate
together with commercially available miniature beamsplitter-cubes and mounts,
half-wave plates, polarizers, fibre couplers, photodiodes and ‘tweaker’ plates for
precision alignment1 . One input is used for the cooling beam and the second forthe repumper. Initial alignment and testing of the fiberport cluster was carriedout by a visiting student, Mehdi Lallouache.
a)!

Input 1!
Photodiode!

b)!

λ/2!

λ/2!

Output 3!

Photodiode!

λ/2!

λ/2!

Output 1!

Output 4!

λ/2!

λ/2!

Input 2!

Output 5!

Output 6!

Output 2!

Figure B.4: (a) Image and (b) schematic of the home-made fiberport cluster foroverlap and splitting of the 87 Rb cooling and repump beams.

Each beam is collimated to a diameter of approximately 1” and passed througha quarter-wave plate to produce the appropriate circular polarization. Two 2”
mirrors are used to steer the beams through the cell. A total of 84mW of coolinglight is divided equally between the six beams, with 10mW of repump light being
divided less evenly due to imperfections in the fiberport cluster. In additiona diode-pumped solid-state (DPSS) 532nm laser2 , capable of outputting 18W
of CW power, is used to provide a beam to be used for optical plugging of thequadrupole magnetic trap. The quadrupole magnetic field used both for operationof the MOT and for magnetic trapping is generated by a pair of coils mounted inanti-Helmholtz configuration above and below the cell. Each coil consists of 36
turns (6 × 6) of insulated copper tubing (OD: 4mm, ID: 2.8mm), encapsulated inan epoxy resin and wound with an inner diameter of 53mm, an outer diameterof 103mm and a height of 25mm. Cooling water is circulated within the copper
tubing. The coils are mounted 30mm apart and are powered by a power supply3
1
2
3

Thorlabs, Inc., Fiber Optomechanics range
Coherent, Inc. Verdi V18
Delta Electronika SM15-200D

186

A compact system for the rapid production of 87 Rb BECs

capable of providing up to 200A at 15.5V, corresponding to a field gradient of
400Gcm−1 along the strong (vertical) axis. Three smaller coil pairs mounted
in approximately Helmholtz configuration, provide compensation fields and asingle auxiliary coil is used to provide a guide field directed along the glass tube
connecting the cell to the rest of the chamber.

B.3 Optical Plug
Optical plugging relies on the use of a repulsive (blue-detuned) laser beam torepel atoms from the central region of a quadrupole magnetic trap in which they
are susceptible to Majorana spin flips. An advantage of this method is that thelinearity of the quadrupole trap is preserved and hence evaporative cooling ismore efficient than in a harmonic potential. This is crucial in our case, since
the single-chamber design already limits the lifetime of our trapped clouds toseveral seconds. Our plug light is generated by a pumped Ti:Sapph laser1 , whose
frequency is tuned in the range νplug = 740 → 770nm. The beam is sent tothe vacuum chamber in a polarization maintaining fibre, and upon exiting fromthe fibre is focussed at the centre of the glass cell by a single, f = 150mm, 2”
achromatic doublet2 . The beam path is shown in figure B.5.

Figure B.5: Diagram of the system showing the propagation path of the plugbeam.

As can be seen, the plug propagates along the weak quadrupole axis and hence
breaks the rotational symmetry of the trap. The beam waist is chosen to belarger than the estimated hole region in order to avoid sensitivity to alignment
12

Coherent, Inc., MBR-110
Thorlabs, AC508-150-B-ML

B.3 Optical Plug187

instabilities, while still providing a high enough barrier to incident atoms to keep
as many as possible away from the hole region. In our setup, we use a plug
waist of wplug ≈ 27µm and a peak barrier height of up to 750µK. The additionof the optical plug potential to the magnetic quadrupole field creates an overall
potential containing two minima on either side of the plug beam, as illustrated
in figure B.6.
800
700

Potential (µK)

600
500
400
300
200
100
0
-200

-100

0

100

200

Position (µm)

Figure B.6: Optically plugged quadrupole potential plotted against distance fromthe trap centre, for a strong-axis field gradient of 200Gcm−1 . The plug barrier
shown is for a waist of 27µm and a peak height of 725µK.

Assuming propagation along the y-direction and neglecting the Rayleigh rangeof zR = πw02 /λ ≈ 4mm for our 27µm waist as well as gravity, the total potential
is given by
UTOT = Umag + Uplugp222= µB gF mF B 0 x2 + y 2 + 4z 2 + U0 e−(x +z )/wplug(B.1)

where B 0 is the magnetic field gradient and wplug is the plug waist. Approximating
the potential minimum as an anisotropic oscillator, the trapping frequencies are
given by [24]sωx = ωy

4x20
− 1,
2
wplug

r
ωy =

µB gF mF B 0
,
2mx0

ωz =

√
3ωx

(B.2)

where x0 is the distance between the trap centre and the trap minimum. We have

188

A compact system for the rapid production of 87 Rb BECs

found that displacing the plug slightly to one side of the trap centre yields larger
condensate numbers due to the formation of a single trap minimum, similarly tothe results reported in [265].

B.3.1 Light-Induced Atomic Disorption (LIAD)
Use of a single-chamber system conflicts with the opposing pressure requirements
required for the MOT and evaporation stages, as discussed in section 3.2. To address this limitation, the technique of Light-Induced Atomic Desorption (LIAD)
was pursued with the aim of obtaining a higher pressure of the target species during MOT loading and a sharp decrease in the pressure prior to the start of evaporation. Photon-stimulated desorption was first demonstrated by A. Gozzini et al.
in 1993 [271] with the name light-induced atomic desorption being given to thephenomenon a year later [272]. LIAD makes use of incoherent short-wavelength(UV) light to desorb atoms from cell walls, analogously to the photoelectric effectfor electrons. This increases the vapour pressure in the cell and can hence yieldlarger MOT numbers. Re-adsorption of atoms once the light is extinguished thenlowers the pressure and hence increases the lifetime of the trapped atoms.
In recent years a number of groups have made use of this technique in cold-atomexperiments in order to achieve larger, longer-lived and faster loading MOTs
[273, 274, 275, 276, 277, 278, 279, 280, 281]. These experiments have explored thewavelength and intensity dependence of the desorption rate, as well as the effects
of different adsorbing substrates, coatings and atomic species on the desorption
dynamics. Figure B.7 shows our UV source and the layout of its control circuit.
a)!

TTL!

b)!

relay!

18-19.5V!

12V"
0.1A!

RT "
18-19V"
thermistor! <1A!

Fan!
2kΩ!

UV array!

FET!

68Ω!

Figure B.7: (a) UV source mounted on a copper heatsink fitted with a fan andthermistor1 . (b) Control circuit for the UV source and fan. A relay2 allows TTL
switching of the light, while a FET3 is used to turn off the source in the event ofoverheating (T > 70℃).

B.4 Performance

189

The source4 we use in our setup dissipates 18W of power and radiates 1250mW ofpower with a central wavelength of 374nm (FWHM: 12nm). It is clamped using
thermally conducting paste to a 40x40x15mm finned copper heatsink fitted witha cooling fan5 . A thermistor6 is mounted in close proximity to the UV source andis connected to a safety circuit in order to ensure the source is turned off in theevent of overheating (T > 70℃).
Somewhat disappointingly, we do not see the full benefits of LIAD in our system.
This may be due to the material from which our cell is made or the nature of
the AR coating on its viewports, although the exact reason remains unknown.
We have found however, that extended illumination with the source (≥30mins)
allows us to double the final MOT size and therefore the source is now employed
as an alternative mechanism to firing the conventional atom sources, which can
impair the immediate functionality of the system.
Interestingly, when testing the effect of the UV source on our main 39 K-87 Rb
system MOT chamber, we noticed a reduction in the MOT size of approximately
50% over ∼ 3s when the UV was turned on. Turning the UV off caused the MOT
to start growing immediately, reaching its former size after ∼15s. At the time thiseffect was attributed to the release of impurity atoms from the chamber walls.
However, similar observations reported by a group at the National University ofSingapore [282] were attributed to a different cause. Their MOT diminished insize only when the UV source was directed at the trapping region, and actually
grew in size when the source was directed at a cell wall away from the MOT
region. They therefore suspected that the depletion was due to the ionizationof neutral 87 Rb from the excited 5P3/2 state to singly charged 87 Rb, a transition
requiring radiation at 484nm or shorter.

B.4 Performance
The RF radiation required for evaporative cooling in the optically plugged trap
is provided by a small 4-turn coil placed near the vertical cell window and driven
1
2
3
4
5
6

EPCOS, B59901D0060A040
Tyco Electronics, Axicom IM 23
Toshiba, 2SK3869
Enfis, UNO Tag (UVA)
BiSonic, SP401012M, 12V, 0.09A
EPCOS, B59901D0060A040

190

A compact system for the rapid production of 87 Rb BECs

by a function generator1 via an RF amplifier2 outputting up to 2W. Figure B.8
shows sample images taken during alignment of the plug beam with respect tothe 87 Rb cloud, following an evaporation sweep from roughly 15MHz to 1MHz.

Figure B.8: Alignment of the optical plug. Images taken during the alignment
process show the enhancement of atom number resulting from correct central alignment of the plug. In these images, the plug beam is left on during the short, 4ms,time-of-flight.

A typical experimental sequence proceeds as follows:
1. A several-second MOT stage is followed by 2.5ms of optical molasses and a
20µs optical pumping stage to transfer atoms into the |2, 2i state.
2. The polarized sample is then magnetically captured by abruptly turning on
the quadrupole field with a gradient of 64Gcm−1 , which is then ramped up
to 200Gcm−1 over the following 500ms.
3. At this point the plug beam is ramped up to expel atoms from the trap
centre.
4. There follows a 300ms ‘settling’ hold in the quadrupole trap during which
a 10MHz RF knife is applied. A first evaporation stage is then carried out,
ramping the RF linearly from 10MHz to 2MHz over 3s in the 200Gcm−1
trap.
1
2

Agilent 33250A
Delta RF Inc., LA2-1-525-30

B.4 Performance191

5. In order to reduce trap loss due to three-body collisions and scatteringfrom the plug beam, we then decompress the quadrupole field to 80Gcm−1
in 200ms.
6. Following decompression the evaporation sweep is resumed, commencing
from a lower frequency of 1.4MHz and sweeping logarithmically to the desired final frequency in 0.5s.
Using this protocol, we have been able to obtain condensates containing over 105
atoms. Figure B.9 shows a trapped cloud imaged at various stages during theevaporation sweep. The two distinct potential minima are clearly visible. Figure
B.10 shows a bimodal fit to a partially condensed (28%) cloud, prepared as
described above. Figure B.11 shows a sequence of images taken at 1ms intervals
in the range 3-12ms after release from the trap.

Figure B.9: Evaporation sequence in the plugged quadrupole trap. Two distinct
minima are clearly visible and further evaporation will result in population of only
one well due to slight asymmetries in the overall potential.

a)!

800

Line Density (µm-1)

700

b)!

600
500
400
300
200
100
0
-250 -200 -150 -100

-50

0
50
Position (µm)

100

150

200

250

Figure B.10: Bimodal fit to a partially-condensed 87 Rb cloud. Fit carried out
according to the triple-pass procedure described in section 3.15. This example
possesses a condensed fraction of N0 /N ≈ 28% and a temperature of T ≈ 500nK,
as obtained from a fit to the thermal wings.

Further details can be found in the first-year reports of Stuart Moulder, RobertCampbell, Igor Gotlibovych and Tobias Schmidutz.

192

A compact system for the rapid production of 87 Rb BECs

Figure B.11: Atoms in freefall. After evaporation to a trap depth of 145kHz asequence of images was captured with the time-of-flight varied in 1ms intervals in
the range 3-12ms.

C
Using Spatial Light Modulatorsfor versatile creation of opticalpotentials

Manipulation of atoms for performing novel and high-precision experiments relies
on our ability to generate the desired static or dynamic potential landscapes. As
discussed in this thesis, optical trapping of neutral particles is a widely-usedtechnique which has enabled confinement and manipulation of large numbersof particles as well as single atoms. The most common optical trap geometryis that produced by a Gaussian laser mode which provides an approximately
harmonic trapping potential, as utilized in the experiments described in thisthesis (see section 3.10). Periodic optical lattices with tuneable parameters canalso be formed by interference of non-copropagating beams [135]. More recently,
diffusive optical elements have been used to project optical speckle potentials
onto trapped clouds for studies into the effects of disordered potentials [283, 67,

193

194

Using Spatial Light Modulators for versatile creation of optical potentials

284, 285, 142]. In general, it is clear that the ability to produce a diverse range of
optical potentials can provide access to many interesting experimental regimes.

C.1 Spatial Light Modulators
An emerging tool for creating highly-customisable optical potentials is the Spatial Light Modulator (SLM). The operating principle of an SLM is the ability
to control (modulate) the phase and/or amplitude at different points within the
wavefront of an incident beam. Knowing the subsequent propagation path of
this modulated field enables production of a tailored field configuration at thelocation of the sample simply by a suitable choice of phase and amplitude modulation in the SLM plane. The modulation itself is accomplished by reflecting ortransmitting the beam through an array of independently-controlled pixels. The
most common SLM implementations are based on liquid crystal displays (LCD),
although some purely intensity-modulating SLMs rely on a digital micromirror
device (DMD) consisting of an array of deformable micromirrors to deflect lightfrom the desired pixel and hence control the intensity attenuation. In an LCDbased SLM, phase control is provided by tuning the birefringence of each pixel,
while intensity control is accomplished by rotating the beam polarization at eachpixel in conjunction with polarizing filters, similarly to a conventional LCD projector. The SLM is controlled from a PC via a conventional video (DVI) interface.
After the SLM, a lens is typically used to generate a far-field image of the beamat the location of the atomic cloud.
The nature of this device naturally makes it much more flexible than the use ofstatic optical elements, such as phaseplates and masks, for the creation of arbitrary potentials and in addition opens up the possibility of dynamic manipulationof the atoms by real-time changes to the phase-pattern displayed on the SLM.
The rate at which the pattern can be updated varies from device to device basedon the method used to induce the phase-shift, but for the most common types(LCD) is on the order of 12Hz. SLMs based on ferroelectric liquid crystals canachieve frame rates on the order of 1kHz.
In order to explore the feasibility of using an SLM for generating optical potentials in future experiments, we conducted a trial using a commercial SLM1loaned from its manufacturer. Below, we briefly describe the experimental setup
used for testing the SLM, as well as the three classes of optical potential whose1

Hamamatsu Photonics, LCOS-SLM X10468-04

C.2 Experimental setup

195

production was trialled during these initial experiments with the SLM. These
were the Laguerre-Gauss (LG) modes, the variably spaced sheets (‘lids’) and thearbitrarily-shaped potential.

C.2 Experimental setup
The setup used for testing of our SLM is shown in figure C.1. After emerging froma single-mode polarization-maintaining fibre1 , the 532nm beam is collimated to a
1/e2 -diameter of ≈ 6mm by an f = 45mm lens2 and reflected from two steering
mirrors before being reflected from the SLM surface, passing through a 200mm
lens3 and impinging on a CCD camera4 . The final lens is located 200mm fromboth the SLM and the camera and all reflection angles are kept to a minimum.

Figure C.1: SLM trial setup. Shown are the fibre output collimator (bottom left),steering mirrors, SLM head, Fourier lens and CCD camera. The light is derived
from our 532nm diode-pumped solid-state (DPSS) laser.

Phase patterns, also known as kinoforms, are sent to the device in the form ofa 600 × 800 array of 8-bit values (0 → 255), corresponding to a phase shift ofapproximately 0 → 2π at the desired wavelength5 . The field incident on the
SLM acquires this phase shift and is therefore multiplied by the factor, P (x, y) =
1
2
3
4
5

Thorlabs, PM460-HP
Thorlabs, AC254-045-A-ML
Thorlabs, LA1708-A
Thorlabs, DCC1545M CMOS
In practice the precise pixel value corresponding to a 2π phase shift is found empirically.

196

Using Spatial Light Modulators for versatile creation of optical potentials

eiφ(x,y) , where φ(x, y) is the imprinted phase pattern, i.e.
incidentz }| {
E(r, zSLM ) = E(r, 0) e|iφ(x,y){z }(SLM plane)(C.1)

imprinted

The field then passes through the so-called Fourier lens and is focussed in its focal
plane. In order to find the final form of the field in the Fourier lens focal plane,
we make use of the following:
1. Propagation of the Fourier transform of the field, Ẽ, from z0 to z is accomplished by multiplication with a propagation phase factor:√
2Ẽ(ξ, z) = Ẽ(ξ, z0 )eik(z−z0 ) 1−(λξ)(C.2)

2. Within the thin-lens approximation, the effect of an ideal lens of focal length
f is simply to multiply the field by the phase factor,
2

E(r, z) → E(r, z)e−ikr /2f .

(C.3)

3. The field in the lens focal plane can be found using the Fresnel diffraction
formula:Zeikz ikr02 /2z
200E(r , z) =e
E(r, 0)eikr /2z e−ikr·r /z d2 r
(C.4)
iλz
Using the above we find that the field at the focal plane after the lens, z = f , isrelated, up to a global phase factor, to the field at a distance d before the lens,z = −d by
02
i kr(1− fd ) Z
2fe00
E(r , f ) =E(r, −d) eikr·r /f d2 r
(C.5)
iλf
In our experimental setup we have zSLM = −d = −f . The phase factor preceding
the above integral therefore vanishes, reducing the expression to a regular Fouriertransform.
Z
1
0
0
E(r , f ) =E(r, zSLM ) eikr·r /f d2 r
(focal plane)
(C.6)
iλf
with E(r, zSLM ) the field of equation C.1, including the imprinted phase pattern.

C.3 Trial 1: Laguerre-Gauss modes

197

C.3 Trial 1: Laguerre-Gauss modes
A uniform trapping potential can be generated by replacing the harmonic confinement provided by a Gaussian beam with a potential consisting of a flat bottom
with sharp outer walls. Laguerre-Gauss (LG) modes are the solutions to theparaxial Helmholtz equation1 in the case of cylindrical symmetry, and such a=0flat-bottomed potential can be provided by the annular LGl6p=0
modes, where land p are the azimuthal and radial mode indices, respectively. LGl0 beams canbe generated by imprinting a phase winding of the form
φ(x, y) = lθ(C.7)

onto an incident Gaussian beam, where θ is the polar angle in the xy plane andl is an integer, often called the charge or winding number. Beams generatedusing this method have attracted much attention recently since their photons
each possess L = l~ of orbital angular momentum which can be transferred tothe particles with which they interact [286, 287, 288]. Therefore, in addition
to enabling studies of gases confined in uniform potentials, the ability to easily
and dynamically change the charge of the LG beams also lends itself to use inexperiments studying the rotational properties of BECs such as vortex nucleation[289] and superfluidity measurements (see [290] and subsection C.7 below).
Figure C.2 shows some example kinoforms used for imprinting such phase windings, together with their resulting intensity profiles, which can be obtained asfollows: For an incident Gaussian beam of wavelength λ and waist ω, passing
through the ‘phaseplate’ of equation C.7, the field at the focus of a subsequent
Fourier lens of focal length f is given by equation C.6 using cylindrical coordinates√

I0
E(r , θ ) =
λf
0

0

Z

0

2

2

drdθ r eikr·r /f e−r /ω eilθ

(C.8)

where the field immediately before the phaseplate is taken to be E(r, θ) =√ −r2 /ω2
I0 e
. The Fourier term can be decomposed in terms of Bessel functions of
the first kind, Jm (z), using the identitye

iz cos φ=∞X

im Jm (z) e−imφ

(C.9)

m=−∞
1

Helmholtz equation: ∇2 ψ + k 2 ψ = 0. The paraxial (small-angle) approximation neglects
second-order variations of amplitude along direction of propagation.

198

Using Spatial Light Modulators for versatile creation of optical potentials

Figure C.2: Radial cross-section through the intensity distribution for LGl0 beams
of winding number l = 1, 2, 3, 6, 10. Shown on the right are the phase patterns used
for the l = 1, 6 and 10 beams.

and hence
ikr·r0 /fe

ikrr0 cos(θ−θ0 )/f

=e

=

∞
X

0

im Jm (krr0 /f )e−im(θ−θ )(C.10)

m=−∞

Inserting equation C.10 into equation C.8 and noting that only the term for whichl = m will survive the integral over θ, results in the following expression for the
field
Z ∞
p
2
0 0
l
ilθ0
E(r , θ ) = i I0 e
du u Jl (2ur0 /ω 0 ) e−u(C.11)
0where the substitution u = r/ω has been made and ω 0 = λf /πω is the waist
at the lens focus. The intensity is obtained as I(r0 , θ0 ) = |E(r0 , θ0 )|2 . Near the
optical axis, for r0  lω 0 , the Bessel function’s small-argument approximationJl (2ur0 /ω 0 ) ∼ (ur0 ω 0 )l /l! can be used to obtain 0 2lr
I(r ) ∝ Io
ω0
0

for

r0  lω 0(C.12)√
It can be shown [291] that for large radii, the intensity falls of as I(r0 ) ∝ I0 l2 r0−4 ,with the radial dependence, I(r0 ) ∝ 1/r04 , independent of the charge, l. The l2 -

C.3 Trial 1: Laguerre-Gauss modes

199

dependence is responsible for transferring large fractions of the beam power into
the wings and explains the large difference between the peak heights in figure
C.2. Figure C.3 shows LG beams recorded using the setup of figure C.1 using
the winding numbers l = 1, 3, 5, 11, 13, 15, 17, 19, 25, 30, 35, 45. We note that theimages of figure C.3 were taken after correcting for aberrations in the optical pathusing the Shack-Hartmann algorithm [292] and applying an annular aperture to
the SLM pattern which suppresses the fringes normally present outside the main
LG ring.

FigureC.3:
Measured
profiles
of
beams
with
l
=
1, 3, 5, 11, 13, 15, 17, 19, 25, 30, 35, 45 (top-left to bottom-right).
The field ofview in each image is d = 0.78 × 0.78mm (i.e. 150 × 150 pixels). The colorscale ofeach image is scaled to the image’s maximal and minimal values.

200

Using Spatial Light Modulators for versatile creation of optical potentials

C.4 Trial 2: 2D confinement and the 2D-3D
crossoverPhase gradients are used to produce a uniform translation of the field in theimage plane, as can be easily understood by observing that insertion of the factor
P (x, y) = e±ipx into equation C.6 is equivalent to a straightforward translation
by ±p in the Fourier domain. These gradients can of course be superposed totranslate the field in both directions, as illustrated in figure C.4.

Figure C.4: ‘Translation gradients’ are used to uniformly translate the field profile in the image plane. Horizontal (left) gradients are superposed with verticalgradients (middle) to translate diagonally. The higher the gradient, the larger thetranslation.

Such gradients are commonly used in order to separate the modulated lightdiffracted from the SLM from the light undergoing specular reflection from its
surface, which contributes an unwanted signal in the image plane. However, with
only a slight modification, similar gradients can also be used to provide a usefulpotential for exploring the 2D-3D crossover regime. Figure C.5 shows a phase pattern consisting of opposing linear gradients centred on the horizontal axis. This
phase pattern produces an intensity profile consisting of two vertically-spaced
thin ‘light sheets’ (aka lids), whose spacing grows with the phase gradient.
a) 2.5

b)

φ/2π
φ/2π

2.0

1.5

1.0

0.5

0.0
-1.0

-0.5

0.0

y/y
x/x00

0.5

1.0

Figure C.5: ‘Splitting gradients’ can be used to produce two thin light sheetsfor exploring the transition between the 2D and 3D confinement regimes. In thisexample, the gradient is p = 5π/y0C.5 Trial 3: Arbitrary potentials

201

The formation of this profile can be intuitively understood by considering illumination of the SLM by a plane wave. The phase pattern of figure C.5 has theforme−ipy if y ∈ [−y , 0]0P (x, y) =(C.13)
eipy
if y ∈ [0, y0 ]where y0 is the distance from the horizontal axis to the top of the SLM. The
amplitude in the image plane therefore takes the form
0

0

Z 0

Z y0

ei(p+k)y dy(C.14)0 i(p+k)y y0ei(−p+k)ye∝+i(−p + k) −y0i(p + k) 0

(C.15)eE(x , y ) ∝−y0i(−p+k)y

dy +
0



h
h
y0
y0
y0 i
y0 i
∝ e−i(−p+k) 2 sinc((−p + k) ) + ei(p+k) 2 sinc((p + k) ) (C.16)
22i.e. a sum of two sinc functions. Assuming a Gaussian incident beam would leadto a convolution of a gaussian with each of the sinc peaks. For large enough y0
and p, the cross term arising from taking the modulus of the amplitude can beneglected, producing two separated peaks.
The possibility of dynamically tightening the confinement into the 2D regime,
may also be useful for gathering as high a number of atoms as possible into
the trap, compared with current loading procedures, which simply turn on the
final two-dimensional potential in its final form, incurring the loss of those atoms
beyond its reach. We intend to explore this technique in future experiments.
Figure C.6 shows a series of intensity profiles, taken with varying values of the‘splitting gradient’.

C.5 Trial 3: Arbitrary potentials
It is clear that the ability to generate arbitrarily-shaped optical potentials would
make available a hugely versatile toolbox for performing a wide range of experiments in micron-scale custom-designed systems. Several algorithms exist for performing the phase-retrieval task required for producing arbitrary field profiles, the202

Using Spatial Light Modulators for versatile creation of optical potentials

Figure C.6: Production of variably-spaced light sheets for studies of the 2D-3D
crossover. The splitting gradient, p, was measured in terms of the number of SLM
pixels per 2π phase shift. From top-left to bottom-right the images correspond to
p = 200, 190, ..., 10. Note that in these images, a cylindrical lens profile has also
been added to the SLM in order to obtain the elongated aspect ratio shown.

most well-known of which is the Gerchberg-Saxton (GS) algorithm [293], which
has been widely documented, implemented and investigated. The GS algorithm
is an iterative algorithm for finding the phase pattern required to transform one
intensity distribution into another, when both intensity distributions are known.
Although the GS algorithm is in wide use, is has been shown that its convergence is imperfect and often results in the production of numerous optical vortices in the resulting intensity profile. Following our initial use of the GS algorithm for making arbitrary profiles (see figure C.7), a project student (Alex
Gaunt) conducted a theoretical, numerical and experimental investigation into
these limitations and developed a method based on an extension of the mixedregion-amplitude-freedom (MRAF) algorithm of [294, 295, 296] for producing
much smoother arbitrary potentials from which these optical vortices were alsoC.6 Home-made SLM

203practically eliminated. In addition, two further measures where implemented to
improved the profile smoothness:
1. The Shack-Hartmann aberration correction algorithm [292] was used in order to compensate for aberrations present in the SLM panel and opticalpath.
2. A feedback algorithm was implemented to empirically compensate for irregularities in the final image by proportionally adjusting the target intensity
profile fed to the phase-retrieval algorithm in response to the image recordedby the CCD camera.
Figure C.7 shows a rudimentary example of the improvements afforded by thesetechniques. For further details, see Alex Gaunt’s project report [297].

b)!

c)!

d)!

e)!

OMRAF (no feedback)!

OMRAF (w/feedback)!

a)!

Target image!

GS (no feedback)!

GS (w/feedback)!

Figure C.7: Comparison of the Gerchberg-Saxton algorithm and the OMRAF
algorithm developed by Alex Gaunt for producing smooth arbitrary potentials andremoving optical vortices. The field of view in each image is approximately 1.6 ×
1.6mm. (a) Target image and images produced by (b) GS algorithm with nofeedback, (c) GS algorithm with feedback, (d) OMRAF algorithm without feedbackand (e) OMRAF algorithm with feedback.

Further applications involving the dynamic manipulation of optical potentials
include mechanical stirring of condensed clouds [296] and the implementation ofa lens with dynamically varying focal length for translation or compression of
optical dipole potentials1 .

C.6 Home-made SLM
We have also briefly explored the possibility of constructing a home-made SLM
using the LCD panel and control electronics found inside a simple commercial
1

Note: the phase profile of a lens of focal length f is a quadratic: φ(r) = −kr2 /2f

204

Using Spatial Light Modulators for versatile creation of optical potentials

projector1 . The main motivation for this was the prospect of achieving comparable results to a commercial phase-only SLM for a roughly two-order-of-magnitude
magnitude reduction in cost. Commercial LCD panels have already been use asintensity modulators to control the intensity of an array of optical dipole traps
for manipulating a cold 85 Rb sample [298]. Figure C.8 shows the bare LCD panel
connected to the control board, after the safety circuit controlling the projector’shigh-power light source was bypassed.

Figure C.8: (a) Home-made SLM using LCD panel and control electronics fromcommercial projector. (b) close-up of LCD panel between crossed polarizers showing the displayed (l = 3) phase pattern.

The main drawbacks of this approach, aside from the lack of robustness/rigidityof the setup, are:
• The transmissive nature of the device significantly reduces the diffraction
efficiency. The commercial SLM operates reflectively and incorporates adielectric mirror to maximize reflection over the desired wavelength range.
• The spaces between the panel’s pixels give rise to a modulating envelope
to the entire image and degrade its smoothness. The commercial SLM
utilizes an LCOS panel, which operates in reflection and whose pixels are
intrinsically much more closely spaced, reducing the effects of pixellation.
• Use of twisted nematic liquid crystals as opposed to parallel liquid crystals.
Twisted nematics rotate the beam’s polarization which has an unknown
effect of the quality of the final image, relying as it does on interference of thebeams emerging from each pixel. The commercial SLM utilizes a parallelaligned nematic LCD which leaves the beam’s polarization unaffected.
1

Saville Audio Visual, TMX-1700XL. Cost: £150.

C.7 Current status and future plans

205

C.7 Current status and future plans
We are currently making use of a commercial SLM in experiments investigating
the proliferation and non-equilibrium behaviour of vortices nucleated in a 87 Rb
condensate using the LGl0 beams described above. The techniques under development may also find application in direct measurements of the superfluid fractionof ultracold gases, as proposed in [290].
In addition, plans exist for a series of experiments on uniformly-confined clouds inboth two and three dimensions, including the crossover between the two regimes.
Finally, another attempt at in-house construction of an SLM is also underway.
We hope to improve on our previous efforts by making use of a commerciallyavailable LCOS panel instead of the conventional LCD display mentioned above
in order to improve efficiency and reduce pixelation effects.

D
MOT and science cell properties• Figure D.1 shows top and side views of the 316LN stainless steel MOT
cell, as well a diagram of the quartz science cell with the dimensions inmillimetres.
• Figure D.2 shows the single-surface reflectance spectrum at normal (0°)
incidence (solid line).
• Figure D.3 shows the single-surface reflectance spectrum at 45° incidence.
The upper, long-dashed line is the spectrum for s-polarized light and thelower, short-dashed line is the spectrum for p-polarized light. The solid line
shows the mean for the two polarizations.
• Figure D.4 shows the transmittance spectrum at 45° incidence for an entire
cell, with only the external surfaces coated. The upper, short-dashed lineis the spectrum for p-polarized light and the lower, long-dashed line is the
spectrum for s-polarized light. The solid line shows the mean for the twopolarizations.

207

208

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

110.00

34.00

70.00

MOT and science cell properties
32.30

30.40

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

a)!

b)!

110.0032.30

c)!

Graded seal! Quartz!

DN16 flange!

70.00

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

34.00

44.00

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

34!

20!15!

7.6!

30.40

≈80!

30!20!

100!
110!

Figure D.1: Top (a) and side view (b) views of the MOT cell and a diagram ofthe science cell (c), including relevant dimensions (in mm).
44.00

91.40

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

Figure D.2: Science cell BBAR coating. Single-surface reflectance spectrum atnormal (0°) incidence (solid line). The dashed line shows an alternative coatingoffered by the manufacturer. Data provided by the manufacturer, Starna Scientific
Ltd.

91.40

209

Figure D.3: Science cell BBAR coating. Single-surface reflectance spectrum atnormal 45° incidence. Upper (long-dashed) line shows spectrum for s-polarized
light and lower (short-dashed) line shows spectrum for p-polarized light. Solid line
shows mean of both polarizations. Data provided by the manufacturer, Starna
Scientific Ltd.

Transmittance (%)

100
90
80
70
60
500

600

700

800

900

1000

1100

Wavelength (nm)Figure D.4: Science cell BBAR coating. Transmittance through entire sciencecell at 45° incidence. Only external surfaces are coated. Upper (short-dashed) line
shows spectrum for p-polarized light and lower (long-dashed) line shows spectrum
for s-polarized light. Solid line shows mean of both polarizations.Data provided by
the manufacturer, Starna Scientific Ltd.

E
Magnetic fields produced bycurrent-carrying coils

E.1 Fields from circular coils
The axial and radial field components generated by loop of current I with radiusR, displaced from the origin by a distance D (as illustrated in figure E.1a) are:µI
1R2 − ρ2 − (z − D)2
2
2
p
Bz =
K(k ) +E(k )2π (R + ρ)2 + (z − D)2(R − ρ)2 + (z − D)2


µI 1
z−D
R2 + ρ2 − (z − D)2
2
2
p
Bρ =
−K(k ) +E(k )2π ρ (R + ρ)2 + (z − D)2(R − ρ)2 + (z − D)2
(E.1)
where
k2 =

4Rρ
(R + ρ)2 + (z − D)2

(E.2)

211

212

Magnetic fields produced by current-carrying coils

and K(k 2 ) and E(k 2 ) are the complete elliptic integrals of the first and second
kind, respectively1 .

ẑ
a)!

ẑ

b)!

R

R
I

D

Bz ( !, z)DI!

( !, z)B! ( !, z)

D
z

!

I

ŷ

I!

ŷ

x̂

I!

x̂

Figure E.1: Single-loop and coil-pair configuration. Figure (a) shows a thin loopof radius R and current I, mounted a distance D from the origin. Figure (b) showsa coil pair mounted a distance d = 2D apart and operated in Helmholtz (blackarrows) and Anti-Helmholtz (red arrows) configuration.

E.1.1 Near-origin approximation for circular current loops
Near the origin (ρ = 0, z = 0), the power series expansion for the field due to asingle coil (see figure E.1a) to second order in ρ and z is:
Bz = µI

1R2
3DR2
3 R2 (4D2 − R2 ) 2
+
µI
z
+
µI
(z − ρ2 /2) + ...
2 (D2 + R2 )3/2
2 (D2 + R2 )5/2
4 (D2 + R2 )7/2
|
{z
}
|
{z
}
axial field gradientaxial curvature

DR2
3 R2 (4D2 − R2 )3zρ + ...
Bρ = − µIρ−µI
4 (D2 + R2 )5/2
4 (D2 + R2 )7/2
|
{z
}

(E.3)

radial field gradientE.1.2 Circular coil pairs
Figure E.1b shows a coil pair, operated in Helmholtz (black arrows) or AntiHelmholtz (red arrows) configuration. The coils are a distance d = 2D apart.
1

These functions are implemented in Mathematica as EllipticK[m] and EllipticE[m].

E.1 Fields from circular coils

213• For the Helmholtz configuration, the equations for the field near the origin
to third order are
Bz = µI

R2
3 R2 (4D2 − R2 ) 2+µI(z − ρ2 /2) + ...
(D2 + R2 )3/2
2 (D2 + R2 )7/2

Bρ = −µI

3 R2 (4D2 − R2 )zρ + ...
2 (D2 + R2 )7/2

(E.4)

The next terms in this expansion are only fourth order in z and ρ. In theideal Helmholtz case, when R = 2D (i.e. coil spacing = coil radius), thesecond term in Bz and the first term in Bρ vanish, leaving the uniform field8
Bz = µI √
+ ...
5 5R

(E.5)

Bρ = 0 + ...

(E.6)• For the Anti-Helmholtz configuration, the equations for the field near theorigin to third order are
15 R2 (4D2 − 3R2 ) 2
DR2
z + µI(4z − 6ρ2 z) + ...
Bz = µI · 3 ·
2
2
5/2
2
2
9/2
(D + R )24 (D + R )Bρ = −µI

3DR2
15 R2 (4D2 − 3R2 ) 3ρ+µI(ρ − 4ρz 4 ) + ... (E.7)
2 (D2 + R2 )5/2
16 (D2 + R2 )9/2

In this case, when R = 2D the field gradient is maximized and is given by
dBz
48
dBρ
= µI √
=2
2
dz
dρ
25 5R

(E.8)

It is also worth noting that the third order terms in both directions in bothp
directions vanish when R = 4/3D.

Approximating the coil pairs in our experiments as stacked single pairs of increasing radius and separation we obtain a good estimate to the total field bysumming the individual contribution of each pair according to the above equations using the appropriate spacings and radii. This was used in the design ofour quadrupole, MOT compensation and Feshbach coils.

214

Magnetic fields produced by current-carrying coils

E.2 Fields from rectangular coils
Below are the expressions for the magnetic field produced by a thin rectangularloop of current, I lying in the xy plane1 . These were used in the design of the coil
mount holding the rectangular Helmholtz coils mounted around the science celland described in section 3.11. Figure E.2 shows the geometry under consideration.

Figure E.2: Geometry for a single rectangular loop of wire with side dimensions
2 × a1 by 2 × b1 lying in the x-y plane. The field is evaluated at point P (x, y, z).

The field is obtained from the vector potential using
Bx = −

∂Ay
∂z

By =

∂Ax
∂z

Bz =

∂Ay ∂Ax
−
∂x
∂y

(E.9)

The components of the vector potential areµ0 I(r1 + a1 + x) (r3 − a1 + x)
Ax =
ln
·
4π
(r2 − a1 + x) (r4 + a1 + x)
µ0 I
(r2 + b1 + y) (r4 − b1 + y)
Ay =
ln
·
4π
(r3 − b1 + y) (r1 + b1 + y)(E.10)

where r1 , r2 , r3 and r4 are the distances from the corners of the loop to the point
P (x, y, z) at which the field will be calculated (see figure E.2). The components
1

Results reproduced from [299]E.2 Fields from rectangular coils

215

of the field at P (x, y, z) are then4 
µ0 I X (−1)α+1 zBx =4π α=1 rα [rα + dα ](E.11)

4 µ0 I X(−1)α+1 zBy =4π α=1 rα [rα + (−1)α+1 Cα ](E.12)

4 µ0 I X(−1)α dαCαBz =−4π α=1 rα [rα + (−1)α Cα ] rα [rα + dα ](E.13)

where
C1 = −C4 = a1 + x

d1 = d2 = y + b1

C2 = −C3 = a1 − x

d3 = d4 = y − b1

(E.14)

r1 =p(a1 + x)2 + (y + b1 )2 + z 2

r2 =

p

r3 =

p
(a1 − x)2 + (y − b1 )2 + z 2

r4 =p(a1 + x)2 + (y − b1 )2 + z 2 (E.15)(a1 − x)2 + (y + b1 )2 + z 2

Figure E.3 shows a screenshot from the Mathematica notebook used to obtainthe total field and curvature of the rectangular Helmholtz coils used in our experiment. The predictions of this simulation are in excellent agreement with themeasured field values reported in table 3.4.

Figure E.3: Mathematica notebook. Includes a diagram of the specified coils (topleft), a plot of the field contours in the y-z plane (top right) and Bz field profiles
along the y and z axes, with the other two coordinates equal to zero (bottom rightand left, respectively).

F
A tightly confining potential forstudies of two-dimensional Bosegases

F.1 Introduction
A system’s dimensionality plays a crucial role in determining its properties andthe nature of its phase transitions. In general, ordered states are more robust in
higher dimensions, while the effects of thermal and quantum fluctuations tend todestroy the ordering of the system in lower dimensions (see [300] and references
therein).
The case of two dimensions is especially interesting since it displays ‘marginal’
behaviour: Although, as described by the Mermin-Wagner theorem, thermal fluctuations preclude true long-range order at any finite temperature in an infinite

217

218

A tightly confining potential for studies of two-dimensional Bose gases

ideal system, superfluidity is still possible in an interacting infinite system atfinite temperature. The mechanism by which this superfluidity is established isnamed the Berezinskii-Kosterlitz-Thouless (BKT) transition, after those who first
proposed its existence [301, 302].
In order to enter the 2D regime in the context of experiments on ultracold atoms
a highly-anisotropic trap is required, possessing one spatial direction that is far
more tightly-confining than the remaining two. The tightly-confining directionis ‘frozen out’ when the energy spacing, ~ω, between the ground state and firstexcited state is much larger than both kB T and the mean-field interaction energy
gn, where g is the coupling constant and n the particle density. Both of thesequantities are typically on the order of a few kHz.
Our ability to fine-tune the interaction strength in our 39 K clouds opens theway for interesting experiments on the interplay between interactions and dimensionality in this system. The interaction strength in the 2D regime can becharacterized by a dimensionless coupling constant, g̃, related to the 3D couplingstrength, g via~2(F.1)
g = g̃
mwhere m is the atomic mass. For a harmonically-confined gas, this can be rewritten as√ag̃ = 8π(harmonic trap)
(F.2)
aho
p
where a is the scattering length and aho = ~/mω̄ is the oscillator length in thetightly-confining direction. As an example, assuming a trap frequency of 4kHz
(see section F.3 below), the range of interaction strengths used in the experiments
reported in this thesis (a ≈ 40 − 400a0 ), should allow us to span the interval g̃ =0.04 − 0.4. This compares to typical values of g̃ = 0.13 in the Paris experiments[190, 303] and g̃ = 0.02 in the NIST experiments of [304].
Furthermore, using the holographic techniques described in appendix C, novel
geometries can be created for containing these two-dimensional ultracold clouds.
These include, but are not limited to, uniform trapping potentials and nonsimply-connected geometries such as toroidal traps. With these longer-term goals
in mind, preliminary work was carried out on generating an optical potential possessing tight confinement along a single direction for studying two-dimensional
clouds.
Several experimental methods have been employed in order to achieve 2D andF.2 The 0-π phaseplate219

quasi-2D confinement of trapped clouds:• A single red-detuned gaussian beam was used in conjunction with a tightlyfocussing cylindrical lens to produce an attractive sheet trap. This was usedin the first production of a gas in the quasi-2D regime [305] as well as in
the experiments at NIST [304].
• Two interfering gaussian beams of wavelength λ, either perfectly counterpropagating or with a small angle, θ between them, can be used to produce astack of adjacent 2D clouds separated by λ/2 or λ/(2 sin(θ/2)), respectively[306, 307, 308, 309, 310, 311, 312].
• Atoms can be tightly-confined against an evanescent wave in close proximity
to a glass surface, with in-plane confinement being provided by an additional
optical trap or magnetic field gradient [313, 314].
• 2D confinement has also been implemented using rf-dressed magnetic traps[315, 316, 317].
• Another technique involves the use of blue-detuned light in conjunction
with a phaseplate to produce a potential consisting of two repulsive sheets,with the atoms confined in the dark plane and radial confinement provided
by an additional optical or magnetic potential[318, 319, 320]
We have chosen to implement the last of these schemes following recent encouraging results from the ENS group. The principle behind generating the potential
is identical to that described for spatial light modulators (SLMs) in appendix
C, with the SLM surface replaced by a static optical element. These plates are
made by spatially tailoring the thickness of an anti-reflection coating in order toproduce the desired phase shift at the design wavelength. Below we describe thismethod in further detail and describe the results of our preliminary tests.

F.2 The 0-π phaseplate
The most convenient phase-pattern to produce a suitable tightly-confining potential is the 0-π plate, consisting of a π phase-shift for Y < 0 and no phase shift
for Y > 0.

0 if y < 0
P (x, y) =(F.3)
π if y > 0This pattern produces a Hermite-Gaussian TEM01 -like mode at the focus of the

220

A tightly confining potential for studies of two-dimensional Bose gases

Fourier lens, which as usual is chosen to coincide with the position of the atoms.
Figure F.1 shows the phase pattern, resulting intensity profile and a comparison
to the unshifted Gaussian beam.
1.0

a)"

b)"

c)"
0.8

0"
I/I

0
I/I
0"

0.6

y"

π"

0.4

0.2

x"

0.0
-6

-4

-2

0

y/ω
y/
ωy y"

2

4

6

Figure F.1: Generating a 2D potential using a 0−π phaseplate. (a) The imprinted
phase pattern, (b) the resulting intensity profile for a Gaussian input beam and(c) cross-sections through the intensity profiles of the shifted and unshifted beams,
normalized to the peak intensity of the unshifted beam.

The field in the focal plane is given by (see equation C.6)
1
E(r , f ) =
iλf
0

Z ∞

Z ∞

dy −

dx
−∞

Z 0

0



0

dy E(r, 0) eikr·r /f

(F.4)

−∞

p
where r = x2 + y 2 , f is the Fourier lens focal length, E(r, 0) is the Gaussian
field incident on the phaseplate and vertical centre of the phaseplate is at y = 0.
For a Gaussian input beam of power P and waists ωx , ωysE(r, 0) =

 2
 2
2P
x
y
exp − 2 exp − 2πωx ωy
ωx
ωy

(F.5)

the integral of equation F.4 can be evaluated analytically to gives

 02 
 02 
 0
2P
x
y
y
exp − 0 2 exp − 0 2 erfi
E(r , f ) =00πωx ωy
ωx
ωy
ωy0
|
{z
}
{z
}
|
0

Focussed Gaussian(F.6)

modulation0
where ωx,y
= λf /πωx,y is the usual focussed Gaussian waist and the imaginary
error function, erfi(z), is related to the error function by erfi(z) = −ierf(iz),√ Rzwith erf(z) = 2/ π 0 exp(−t2 )dt. This has the form of a focussed Gaussian
modulated by an error function and is shown in figure F.1c. Since the error

F.3 Estimating the trapping frequency and depth221function is an odd function, the intensity profile vanishes along y = 0. This
property is useful since it ensures the minimal photon scattering rate for theatoms confined at the centre of the potential. This is also an advantage of using
blue- as opposed to red-detuned light. Furthermore, since blue-detuned light
can be focussed down to smaller waists, it also enables the production of more
tightly-confining traps than those using red-detuned light.

F.3 Estimating the trapping frequency anddepth
The trapping frequency near the intensity minimum can be estimated by using√
the first-order approximation erfi(z) ≈ (2/ π)z + O(z 3 ). Inserting this into theexpression for the intensity I(r0 , f ) = |E(r0 , f )|2 and using expression 2.47 for the
potential produced by a far-detuned beam of detuning ∆Udip (r) =3πc2 Γ
I (r)2ω03 ∆(F.7)

gives a potential in the y-direction of the form Udip (y) = 12 mΩ2 y 2 with the trapping frequency
24c2 ΓP(F.8)Ω2 =πmω03 ωy0 3 ωx0 ∆with Γ the natural linewidth, c the speed of light, m the atomic mass and ω0
the transition frequency. Plugging in the constants for 39 K and 532nm light gives0Ω2 = αP/ωy3 ωx0 , with α = 2.47 × 10−11 m4 /Js. Hence for a beam with transverse
waist ωx0 = 150µm and axial waist ωy0 = 5µm, this results in a vertical trapping√
frequency of Ωy = 2π × 5.77 × P kHz, with the beam power P given in Watts.
By setting dUdip /dy = 0, the distance, d, between the intensity maxima is foundto be dy = 1.8483 ωy0 . Using this result, the trap depth is found to be 0.37 times
the height of the unshifted Gaussian beam, i.e.
T =∆U
0.37 · αI00.37 · α2P
=
=
kB
kB
πωy0 ωx0 kB(F.9)

with α ≈ 8.1 × 10−37 Jm2 /W for both 39 K and 87 Rb. For the 5µm and 150µm
waists above, this amounts to a depth of roughly 18µK/W.

222

A tightly confining potential for studies of two-dimensional Bose gases

F.4 Experimental setup
Figure F.2 shows a schematic of the experimental setup designed for use in ourapparatus. At the time of writing of this thesis this hardware has not yet been
installed on the system but has undergone preliminary testing to ensure its suitability.
0-π "phase-plate"cylindrical #lens"
f=500mm"

f=45mm"dichroic#
beamsplitter"

imaging#
beam"from fibre"

f=15.3mm"
collimator"

Figure F.2: Illustration of our setup for producing the 2D confining potential.
In addition to being collimated and focussed to the desired waists, the trapping
beam passes through the 0-π phaseplate. The trapping beam is overlapped with
an imaging beam to aid alignment.

The main advantage of our configuration compared to those utilized by the Oxford
and Paris groups lies in its simplicity, consisting in essence of three lenses andthe 0-π phaseplate. By contrast, the Oxford setup contains seven lenses and thephaseplate, while the Paris setup uses four lenses in addition to the phaseplate.
Our system is comparable in simplicity to that used by the Austin group, although
they also include an aperture in the optical path. In addition, due to spatial
limitations, the Paris group are constrained to use 2-inch optics along the entire
beam path to avoid clipping of their 7mm-diameter beam. We are fortunate
in being able to use a 45mm lens as the Fourier lens (compared to 100mm and
160mm for the Paris and Oxford groups, respectively) and hence achieve an equal
spot size with a beam of less than half the diameter, allowing us to use only 1-inch
optics. Figure F.3 shows a top and side view of the beam trajectory.
The trapping beam emerges from a polarization-maintaining (PM) single-mode
fibre1 and is collimated to a diameter of dinit = 3mm by an f = 15.3mm spherical
1

Timbercon, Inc., PM-4630-0420-0420-008M

F.4 Experimental setup223

d≈2 x 1.5mm"d≈2 x 135μm!
zR≈10.8cm"≈120mm"top!
view"
f1=15.3mm"

fx=500mm"

0-π plate"

f2=45mm"

545mm"

45mm"

d≈2 x 1.5mm"d≈2 x 5μm!
zR≈150μm"

side!
view"
f1=15.3mm"

fy = ∞"

0-π plate"

f2=45mm"Figure F.3: Configuration for producing a tightly-confining optical potential. The
blue-detuned (532nm) beam is collimated, and then telescoped transversely to a
waist of ωx0 = 135µm while being focussed vertically to a waist of around ωy0 = 5µm.
On the way, it passes through the 0-π phaseplate which gives rise to the TEM01 -likemode in the focal plane.

lens1 . It then passes through an f = 500mm cylindrical lens2 which focusses thebeam along the transverse (horizontal) direction and acts as the first lens of a
two-lens telescope. The beam then passes through the 0-π phaseplate, is reflectedfrom a dichroic beamsplitter3 and is focussed on the atoms by the f = 45mm
spherical Fourier lens4 . This lens collimates the beam transversely to a 1/e2 waist of ωx0 = 135µm5 and focusses the beam axially to a waist of ωy0 = 5µm. The
dichroic beamsplitter is chosen to reflect 532nm light while transmitting imaging
light for both 87 Rb and 39 K in order to facilitate alignment of the 2D trap withthe trapped atoms. These focal length values were chosen to provide sufficiently
tight confinement, while possessing a Rayleigh range which was large enoughto ensure a uniform potential (matching the CDT waist) without wasting beam
power unnecessarily. Care was taken to match the transverse waist to the axial
Rayleigh range in order to ensure a uniform aspect ratio for the trapped cloud.
The phaseplate is mounted in a rotation mount6 to allow for precise alignment of
the trapping plane and this rotation mount is mounted onto an XY translation
stage7 to enable precise alignment of the phaseplate centre line with the trapping
1
2
3
4
5
6
7

Thorlabs, C260TME-A
Thorlabs, LJ1144RM-A
Semrock Inc., LM01-552-25
Thorlabs, AC254-045-A-ML
Note: Changing the transverse waist simply requires changing the cylindrical lens focal
length.
Thorlabs, CRM1
Thorlabs, ST1XY-D224

A tightly confining potential for studies of two-dimensional Bose gases

beam. The f = 45mm Fourier lens is mounted within a precision translation
stage1 to allow translation of the focal plane along the beam propagation axis
for overlapping with the atoms. Since after passing through the phaseplate thebeam is no longer Gaussian and will not retain its form during propagation, we
endeavour to place the Fourier lens as close as possible to the phaseplate. To thisend, the phaseplate, dichroic beamsplitter and Fourier lens, with their respective
translation stages, are all mounted within a right-angle kinematic cage mount 2
to ensure their proximity and alignment. After passing through the science cell,the trapping beam is separated from the imaging beam by another beamsplitter3
and sent to a beam dump in order to avoid damage to the camera sensor.

Updated design
Subsequent to the design of this system, the waist of the CDT was increased from
100µm to 140µm in order to increase the number of loaded atoms. Consequently,
the choice of focal lengths in the 2D setup should be modified in order to ensurea uniform axial confinement in this new trap. Using an aspheric collimating lensof focal length f1 = 15mm4 , a cylindrical lens of focal length fx = 400mm5 andkeeping the f2 = 45mm Fourier lens results in an transverse waist of ωx0 = 165µm(Rayleigh range: 16.3cm) and an axial waist of ωy0 = 5.2µm (Rayleigh range:
160µm). This configuration maintains the good matching between the transverse
waist and the axial Rayleigh range, and in addition does not affect our ability
to use 1-inch optics throughout. The preliminary testing, described below, wasperformed using the original design.

F.4.1 Preliminary trials
We initially tested the setup without the cylindrical lens and using a f2 = 1000mm
lens as the final Fourier lens in order to obtain a larger focussed waist for better
visibility. The camera was placed in the lens focal plane and images were recordedwith and without the phaseplate present. Figure F.4 shows cuts through theseimages, together with the theoretical profile predicted for the same peak intensity.
1
2
3
4
5

Thorlabs, SM1Z
Thorlabs, KCB1This beamsplitter is mounted in a gimbal mount (Thorlabs, KC45D)
Thorlabs, AL1815-A
Thorlabs, LJ1363RM-A or LJ1363L1-A

F.4 Experimental setup

225

1.0

a)

0.4

b)

0.8

0.3

I/I0

I/I0

0.6

0.4

0.1

0.2

0.0

0.2

-600

-400

-200

0

200

400

600

0.0

-600

-400

Position (µm)

-200

0

200

400

600

Position (µm)

Figure F.4: (a) Intensity profile at the focus of a 1000mm Fourier lens with(red) and without (blue) the phaseplate in the beam path. (b) The profile withthe phaseplate, together with the theoretical prediction for a beam of equal peakintensity.

We then made use of the custom ‘horizontal’ imaging objective described in section 3.14 to provide a higher spatial resolution. The profile with and without
the phaseplate was recorded both in the presence and in the absence of thefx = 500mm cylindrical lens, using the f2 = 45mm lens as the Fourier lens.
Figure F.5 shows the recorded profiles.
a)

1.0

c)

e)
0.8

I/I0

I/I0

0.6

b)

d)

0.4

0.2

0.0
-40

-30

-20

-10

0

10

20

30

40

Position (µ(µm)m)
Position

Figure F.5: Focal-plane intensity profiles with, ((b) and (d)), and without, ((a)
and (c)), the phaseplate, in the presence, ((c) and (d)), and absence, ((a) and (b)),
of the cylindrical lens. A 45mm lens was used as the Fourier lens. The field ofview of the LHS images is 0.32 × 0.26mm (H×W) and that of the RHS images is0.37 × 0.9mm (H×W). Figure (e) shows a vertical cut through the shifted (red)and unshifted (blue) profiles in the focal plane. The measured 1/e2 gaussian waist
is 7.8µm.

The magnification of the imaging system in this configuration was found to be
M = 2.35 by focussing on a standard resolution chart (1951 USAF Resolution

226

A tightly confining potential for studies of two-dimensional Bose gases

Test Target1 ). A Gaussian fit to the intensity profile yielded a waist in the
focal plane of 7.8µm, compared to the expected waist of 5.1µm. In order tolook into this discrepancy, we measured the profile of the beam after the firstf = 15.3mm collimating lens and found it to consist of only an approximate
Gaussian, superposed with a sharper peak of roughly half the width.
We suspected that this might be due to clipping of the beam by the small diameterof the collimating lens (5mm) and in order to investigate this replaced it by alarge-diameter, f = 20mm, asphere2 . This noticeably improved the gaussianity of
the collimated beam and gave rise to a waist in the focal plane of the 45mm Fourier
lens of ωy0 = 5.8µm, implying a collimated waist of roughly 1.3mm, compared tothe expected 1.97mm. This discrepancy may be due to an overestimate of thefibre’s numerical aperture.
Loading procedure
The simplest protocol for transferring atoms into the 2D potential involves ramping up the 532nm beam in the presence of the CDT. In order to try and avoid
heating, the Paris group ramp up their ∼ 3.6kHz trap over 1.5s3 , while the Oxford
group ramp up their ∼ 2.2kHz trap over roughly 300ms. After this ramp-up, it islikely that some atoms will be trapped in the wings of the potential, within the
CDT but outside the 2D plane. These atoms can be made invisible by transferring them to a state that does not interact with the imaging beam. This can bedone using an RF sweep in the presence of a quadrupole magnetic field or, as in
the Paris group, by selectively illuminating these atoms with |F = 2i → |F 0 = 2i
pumping light to transfer them into the |F = 1i hyperfine state, which is notimaged. In order to transfer more atoms into the 2D potential, a preliminary
stage can be added to the sequence, wherein a red-detuned gaussian sheet is firstturned on in order to confine the atoms to a thin slab. The 2D potential can thenbe ramped up in order to provide the final tight confinement. An elaborationon this method was described in appendix C, whereby an SLM can be used togradually decrease the distance between two initially well-separated blue-detuned
sheets. See [291] and [321] for more details regarding the loading of atoms into
the 2D potential.
1
2
3

Thorlabs, R3L1S4P.
Thorlabs, AL2520-A.
This ramp-up is carried out in the presence of an RF evaporation field and a magnetic
TOP trap.

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