Thermodynamic properties of a Bose gas withtuneable interactions
Robert Lorne Dugald Campbell
Jesus College
University of Cambridge

This dissertation is submitted for the degree ofDoctor of Philosophy

November 2011

To my parents Lorne and Lucy,and my brother Neill.

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 already been oris being concurrently submitted for any such degree, diploma or
any other qualification. This thesis does not exceed the wordlimit of sixty thousand words, including tables, footnotes, bibliography and appendices, set out by the Faculty of Physics andChemistry.

Abstract

Thermodynamic properties of a Bose gas withtuneable interactions
Robert Lorne Dugald Campbell
This thesis describes the design, construction and optimisation of a new set of apparatus
for the creation of 39 K Bose-Einstein condensates. This system is used to investigate
the effects of repulsive inter-atomic interactions on the saturation of excited states andthe shift in the critical temperature of harmonically trapped Bose gases.
Bose-Einstein condensates of over 4×105 atoms have been produced in a crossed optical dipole trap after sympathetic cooling with 87 Rb in a quadrupole-Ioffe-configuration
magnetic trap. Condensation is made possible by tuning the scattering length of 39 K topositive values by means of a broad magnetically-tuneable Feshbach resonance centred
at 402.5 G.
The ability to tune the scattering length in this system is used to scrutinise theconcept of saturation of the thermal component of a Bose gas. We observe strong deviations from the ideal theory over a wide range of interaction strengths. Extrapolation
to the non-interacting limit shows that the saturation picture is recovered for an idealgas.
High-precision measurements of the shift in critical temperature of a harmonicallytrapped Bose gas are performed, using a novel differential measurement method. In the
weakly-interacting limit the data show excellent agreement with mean-field theory. For
stronger interactions an additional positive shift is observed, characteristic of beyondmean-field effects due to critical correlations. Non-equilibrium behaviour is observedin the limits of very-strong and very-weak interactions.

Acknowledgements
Firstly, I would like to thank my supervisor, Zoran Hadzibabic, for taking me on as afounding member of our now-flourishing group. The task of building up the lab fromscratch has at times been a daunting one, but has been greatly eased by the neverending stream of ideas and optimism from above. I would also like to thank Jean
Dalibard for his inspiring presence during his brief sabbatical. I couldn’t have asked
for more illustrious or pleasant company in the office.
The presented work could never have happened without the joint efforts of all of thepast and current members of the group. I would especially like to thank Naaman andRob. I am most fortunate to have spent my entire time here with Naaman, who has
been the perfect companion through all the highs and lows of experimental research.
I’m glad that we shared so many interests (our mutual appreciation of Bob Dylan saw
us through many a long day’s work), and to have gained so many new ones from him.
Rob has been around for almost as long, and has been an invaluable asset to the group.
I cannot thank him enough for his encyclopaedic knowledge and willingness to answer
questions (however daft) - a winning combination. Life in the lab wouldn’t have been
half as fun without either of them.
The arrival of Scott and Stuart two years ago came as a welcome addition to theteam, not least because of the formation of a full doubles pairing for tennis. Eachhas been a great help to me, and I now can’t imagine the place without them. I must
particularly thank Scott for his expert assistance when in the painful throws of rewriting
our first paper for the umpteenth time. Stuart, who clearly has an alternative career
looming with his surprising blu-tack sculpting skills, must also be praised/censured forrecklessly introducing us all to the Empress pub quiz. Long live the lunchtime practice‘quiz of the day’.
The newest members of the group, Tobias, Igor, Alex and Richard, are alreadymaking their mark, and I thank them for the assistance they have already provided.
Essential contributions to the experiment have also been made by the array of coworkers
to have passed through the group. Jun-Young, Phil, Nils, Alexandre, Mehdi, Diana,
and Mingjie have all helped to add plenty of colour to life in the lab. I am grateful to
Michael Köhl and all of his students, who have provided invaluable support, whether
by lending equipment or through helpful discussions.
I would like to thank all of the staff in the Cavendish, especially those working in themain workshop and stores, for all the support they have provided. A special mention
is due to Pam Hadder for her impeccable efficiency and organisation. I am muchix

indebted to Nigel Palfrey in the student workshop for all his patient (and sometimes
not-so-patient) assistance. Though it seems a while since I had cause to give him any
real bother, I am sure he has had enough of me to last him until his retirement.
Outside the lab, I wish to thank all the G&S/musical theatre crew and my housemates past and present (Lucy, James, Danielle and Sybil) for making life in Cambridge
such a pleasant experience. Also the old Univ crowd, for giving me the opportunity to
let off some steam in our all-too-infrequent London gatherings (especially Laura andAna for regular soft surfaces to spend the night). Claire is to blame for the contentious
appearance of P. G. Wodehouse throughout this thesis (and many thanks to the manhimself for keeping my morale high while writing up!) Finally I express my sincere
gratitude to Siân for her magnificent attempts to keep me sane, and even more for her
partial success.
Above all else I would like to thank my family for their unfaltering support. My
brother Neill has acted as sibling, housemate, fellow cast-member, bank (I owe you£10. . . ) and friend throughout the course of my PhD. My parents Lucy and Lorne
have always been there for me with kind words and wise heads. Thank you all for yourlove and encouragement.

xContents
1 Introduction1

1.1

Why 39 K?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

The semi-classical approximation . . . . . . . . . . . . . . . . . .

5

1.2.2

Ideal Bose gas in a harmonic trap . . . . . . . . . . . . . . . . .

7

The Bose-Einstein condensate . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3.1

Ideal condensate . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3.2

Condensate with repulsive interactions . . . . . . . . . . . . . . .

9Properties of alkali atoms . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4.1

Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4.2

Interactions with magnetic fields . . . . . . . . . . . . . . . . . .

12

1.4.3

Interactions with light . . . . . . . . . . . . . . . . . . . . . . . .

15

Collisions between ultra-cold atoms . . . . . . . . . . . . . . . . . . . . .

20

1.5.1

Scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.5.2

Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.5.3

Inelastic processes . . . . . . . . . . . . . . . . . . . . . . . . . .

26

1.3

1.4

1.5

2 Experimental setup
2.1

2.2

2.3

2.4

27

Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.1

Differential pumping design . . . . . . . . . . . . . . . . . . . . .

28

2.1.2

Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.1.3

Assembly and bake-out . . . . . . . . . . . . . . . . . . . . . . .

37

Magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.2.1

Coils around the MOT cell . . . . . . . . . . . . . . . . . . . . .

43

2.2.2

Coils around the science cell . . . . . . . . . . . . . . . . . . . . .

44

Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.3.1

Optical Molasses . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.3.2

Doppler limit and sub-Doppler cooling . . . . . . . . . . . . . . .

47

2.3.3

Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.3.4

Laser cooling for real atoms . . . . . . . . . . . . . . . . . . . . .

49

Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.4.1

Saturated absorption spectroscopy . . . . . . . . . . . . . . . . .

52

2.4.2

Frequency locking of lasers . . . . . . . . . . . . . . . . . . . . .

54

2.4.3

Required frequencies . . . . . . . . . . . . . . . . . . . . . . . . .

56

xi

2.4.4Laser table summary . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.4.5

Fibre-port cluster . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.4.6

Optics around the MOT cell . . . . . . . . . . . . . . . . . . . .

60

Magnetic transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.5.1

Coil holder design . . . . . . . . . . . . . . . . . . . . . . . . . .

62

2.5.2

Translation stage . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

2.5.3

Atom transportation . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.5.4

Majorana spin-flips . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.6

Quadrupole Ioffe-configuration trap . . . . . . . . . . . . . . . . . . . . .

65

2.7

Evaporative cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

2.7.1

Sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . .

70

2.7.2

Thermalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

2.7.3

Microwave antenna . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Crossed optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.8.1

CDT laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

2.8.2

Forming the trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

2.8.3

Trap potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

State transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

2.9.1

RF antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

2.10 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

2.10.1 Optical density . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

2.10.2 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

2.10.3 Imaging light . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

2.10.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

2.10.5 Imaging sequence . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

2.10.6 ‘Fudge-factor’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

2.11 Image Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

2.11.1 Thermal distribution . . . . . . . . . . . . . . . . . . . . . . . . .

88

2.11.2 Thomas-Fermi distribution . . . . . . . . . . . . . . . . . . . . .

89

2.11.3 Fitting routines . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

2.12 Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

2.12.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

2.12.2 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

2.13 Field and resonance calibration . . . . . . . . . . . . . . . . . . . . . . .

92

2.13.1 Field calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

2.13.2 Resonance measurement . . . . . . . . . . . . . . . . . . . . . . .

93

2.13.3 Zero crossing measurement . . . . . . . . . . . . . . . . . . . . .

95

2.5

2.8

2.9

xii

3 Experimental sequence
3.1

97

Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

3.1.1

Monitoring MOT loading . . . . . . . . . . . . . . . . . . . . . .

97

3.1.2

MOT optimisation . . . . . . . . . . . . . . . . . . . . . . . . . .

98

3.1.3

Dual-species MOT . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.1.4

Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2

Magnetic trapping and transport . . . . . . . . . . . . . . . . . . . . . . 102

3.3

Sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.4

3.3.1

Choice of trapping potential . . . . . . . . . . . . . . . . . . . . . 102

3.3.2

Transfer to the QUIC trap

3.3.3

Cooling in the QUIC trap . . . . . . . . . . . . . . . . . . . . . . 103

3.3.4

Sympathetic cooling results . . . . . . . . . . . . . . . . . . . . . 105. . . . . . . . . . . . . . . . . . . . . 103

Quantum degeneracy in the CDT . . . . . . . . . . . . . . . . . . . . . . 107
3.4.1

Cooling methods in the dipole trap . . . . . . . . . . . . . . . . . 107

3.4.2

Transfer to the CDT . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.4.3

Evaporation in the CDT . . . . . . . . . . . . . . . . . . . . . . . 112

3.4.4

Number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.4.5

Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.4.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4 Lack of saturation in a Bose gas

121

4.1

Saturation: another approach . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2

Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3

Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3.1

Image fitting for saturation data . . . . . . . . . . . . . . . . . . 126

4.3.2

Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.4

Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.6

4.5.1

Popov approximation . . . . . . . . . . . . . . . . . . . . . . . . 133

4.5.2

Small condensate limit . . . . . . . . . . . . . . . . . . . . . . . . 134

4.5.3

Large condensate limit . . . . . . . . . . . . . . . . . . . . . . . . 135

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Effect of interactions on the critical temperature
5.1

5.2

139

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1.1

Uniform system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.1.2

Harmonic confinement . . . . . . . . . . . . . . . . . . . . . . . . 142

5.1.3

Past experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.1.4

Aims of the investigation . . . . . . . . . . . . . . . . . . . . . . 143

Mean-field shift in a harmonic trap . . . . . . . . . . . . . . . . . . . . . 144
xiii

5.3

5.2.1

Analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2.2

Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2.3

Introducing critical effects . . . . . . . . . . . . . . . . . . . . . . 148

Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.1

Image fitting routine . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.3.2

Differential measurement process . . . . . . . . . . . . . . . . . . 154

5.4

Critical temperature shift results . . . . . . . . . . . . . . . . . . . . . . 154

5.5

Assessment of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.5.1

In-trap momentum distribution . . . . . . . . . . . . . . . . . . . 156

5.5.2

Interaction energy . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.6

Non-equilibrium effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6 Conclusion

165

6.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2.1

Condensed fraction induced by critical correlations . . . . . . . . 166

6.2.2

Out-of-equilibrium studies . . . . . . . . . . . . . . . . . . . . . . 167

6.2.3

Two-dimensional Bose gases . . . . . . . . . . . . . . . . . . . . . 167

6.2.4

Annular Bose-Einstein condensates . . . . . . . . . . . . . . . . . 168

6.2.5

Power-law traps

6.2.6

Superconducting magnetic trap . . . . . . . . . . . . . . . . . . . 168

6.2.7

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169. . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A Useful Physical Quantities

171

B Small system

175

C Popov Approximation

181

D Assessment of absolute errors

185

D.1 Finite imaging time

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

D.2 Feshbach field curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

xivChapter 1

Introduction“ I don’t know if you have had the same experience, but the snag I
always come up against when I’m telling a story is this dashed difficult
problem of where to begin it.”– P. G. Wodehouse (Right Ho, Jeeves)
The quantum regime in which a Bose-Einstein condensate (BEC) can exist seems a far
cry from day-to-day experiences of the world around us. Classical mechanics provides an
excellent description for our interactions with macroscopic objects, which we naturallyfind to be intuitive. However, the quantum mechanical description takes into accountthe true wave nature of matter. Yet for a gas at room temperature, the extent ofthese wave-packets is much smaller than the inter-particle separation. The atoms andmolecules can be thought of as hard billiard balls colliding with one another, and the
behaviour of the gas is also well-described by a classical statistical treatment.
If, however, the spacing is decreased or the wave-packets spread out such thatthey start to overlap with one another, a classical description is not sufficient, andquantum mechanical behaviour takes hold. In the case that the particles are identicalbosons (which have integer quantum mechanical spin), the remarkable result is that,
below a critical finite temperature, they start to macroscopically occupy the singleparticle quantum ground state of the system. At this point the particles are completely
indistinguishable, forming a single matter-wave; in other words a BEC.
This phase transition was predicted by Einstein [1], following from the work of Bose[2], in 1925. However, it wasn’t until 70 years later that it was first realised in a gas of
atoms in the group of Wieman and Cornell [3] and soon after in the group of Ketterle[4]. It took so long to achieve because of the extreme low temperatures required. To
avoid the formation of a solid the atoms must be kept extremely dilute, several million
times lower than the density of air. The thermal wavelength of the particles must
therefore be increased to an extent that can only be achieved by cooling the atoms
to near the photon recoil limit (the temperature associated with the momentum ‘kick’of a single photon). This is typically around 1/10,000,000 of a degree above absolutezero, where atoms move at speeds of millimetres per second rather than at hundreds of
1

10−3

1
103
Temperature (K)Sun’s core

Sun’s surface

Room temperature

Cosmic microwave
background radiation

Doppler limit

Recoil limit
10−6

106

Figure 1.1: Logarithmic temperature scale. Bose-Einstein condensation typically occurs near
the recoil limit in a dilute atomic gas.

metres per second (as they do at room temperature). This is put into context in Fig.
1.1 where it may be seen that on a logarithmic scale we live at a temperature closer tothat at the centre of the sun than at which a BEC is formed.
The bulk of the cooling of atoms from room temperature to quantum degeneracy isachieved with a combination of the scattering force of laser light (the same force that
directs a comet’s tail away from the sun) and evaporation of the most energetic atoms
in the latter stages (the process that our bodies employ to keep cool via perspiration).
The development of laser cooling and trapping techniques for neutral atoms was recognised with the 1997 Nobel prize in physics, while the 2001 prize was awarded for theattainment of the BEC itself.
The motivation to produce these condensates in the lab has not simply been driven
by the desire to verify Einstein’s prediction. Though quantum effects naturally originate at the atomic scale, they also lead to some more familiar macroscopic behaviour,such as superconductivity, superfluidity and semiconductor properties. Such manybody quantum phenomena can be studied using ultracold gases to simulate relevant
theoretical models, where the experimentalist has a high degree of control over nearly
all of the system parameters. Often the inter-atomic interactions are weak in thesesystems, and the corresponding theoretical problems are easily solved. The system parameters can then be tuned to a regime in which these theories break down, providing
new insight into previously impenetrable processes.
Recent additions to the experimental arsenal include the introduction of opticalconfinement to reduce the dimensionality of the system and to create strong periodic
potentials, and the ability to directly tune the strength of interactions using a Feshbachresonance. These methods allow the study of strongly correlated systems, more typically associated with quantum liquids (such as superfluid helium). The overall pictureis that of a system in which we have great control over most parameters, enabling thestudy of a wide range of phenomena in different regimes. An extensive review of recent
experimental and theoretical progress concerning many-body physics with ultracold2

1.1 Why 39 K?

gases is provided in [5].
Before we are able to experimentally study these many and various fascinating topics, we first have to build a system that allows us to reach the ultracold regime. Thisrequires the combination of many technological disciplines, including those of ultrahigh vacuum, laser light, electromagnetic coils, radio frequency (RF) and microwave
radiation, while carefully synchronising the operation of the components by means ofcomputer control. After giving the arguments for the choice of atomic species usedin our system the remainder of this chapter is given over to basic theory of the Bose
condensed state and interactions of bosons with electromagnetic fields and one another.
Chapters 2 and 3 describe the experimental setup and process respectively, while Chapters 4 and 5 report on the first new science to have been studied with the system. Finallya summary of our findings and future directions for research is provided in Chapter 6.

1.1

Why 39 K?

As a brand new group setting out to build a complete system for the study of quantum
gases it would be understandable to stick to a ‘safe’ atomic species. Though our field
is still young there are now well over a hundred research groups worldwide producing
BECs every day. Most of these groups use isotopes of lithium, sodium, rubidium andcaesium in their experiments, providing a wealth of information to aid the prospective
atomic physicist. In contrast the successful condensation of 39 K was first reported
only a matter of months before the formation of our group, after a series of pioneeringexperiments at LENS in Florence [6, 7]. This achievement itself was founded on previous
work with the other isotopes of potassium [8, 9]. Potassium was a latecomer to thequantum degenerate regime due to unfavourable atomic properties which hamper its
cooling and in the case of 39 K condensation itself.
Initially we concentrated on the relatively straight-forward task of condensing 87 Rb,
whilst preparing for the possibility of cooling 40 K to work with fermions in the future.
However, it became apparent that the advantages of working with 39 K might well outweigh the inherent difficulties. Our overall goal is to study universal phenomena in
many-body physics; this should not depend on the particular properties of the atom weare using, only its more generic properties. The beauty of working with ultracold gases
to this end is that we have the tools to control the temperature, spin, confinement,
motion, rotation and other such properties of the system. The final ‘knob’ we would
like control of is the inter-atomic interaction strength. Remarkably we may play withthis too, via the phenomenon of Feshbach resonances. These allow us to control thescattering length of our atoms by simply applying a uniform magnetic field of appropriate strength. However, their accessibility and usefulness (for our purposes) vary widely
between species.
The behaviour of the scattering length with magnetic field strength near a Fesh3

1.2 Ideal Bose gas

B∞ (G)∆ (G)abg /a0

abg /(a0 ∆) (G−1 )

7 Li

736.8

-192.3-25

0.13

Pollack et al., 2009 [10]

23 Na

1195

-1.4

62

-44.3

Stenger et al., 1999 [11]

39 K

402.5

-52

-29

0.56

Zaccanti et al., 2009 [12]

41 K

51.4

-0.3

60

-200

Kishimoto et al., 2009 [13]85 Rb155.04

10.7

-443

-41.4

Claussen et al., 2003 [14]87 Rb1007.4

0.21

100

476

Dürr et al., 2004 [15]133 Cs

-11.7

28.7

1720

59.9

Lange et al., 2009 [16]

Atom

Ref.

Table 1.1: Characteristic parameters of selected Feshbach resonances in stable bosonic alkali
atoms. These are experimentally verified resonances with the most promising properties forcontrol of interaction strength for each respective atomic species. The broad resonances in 7 Li
and 39 K stand out as prime candidates. References are to the most recent known experimental
measurements. The data was collated in Ref. [17], some values being the result of unpublishedwork.

bach resonance may usually be characterised by just three parameters1 (see §1.5.2): a
resonance centre B∞ , the corresponding resonance width ∆ and background scattering
length abg . Near the resonance the scattering length diverges, while there is what is
known as a zero-crossing (where the scattering length goes to zero) for a field value
B0 = B∞ + ∆. Near this zero-crossing the variation in scattering length with field
is roughly linear, with gradient abg /∆. Thus prime resonances for good experimental
control of the scattering length have a small magnitude for this figure of merit. As canbe seen in Table 1.1, the quoted resonances for 7 Li and 39 K have gradients at the zerocrossing about two orders of magnitude smaller than any of the others. By this criterion
alone 7 Li would appear to have the best experimental control, and indeed is being usedwith great success [10]. All things considered, there isn’t much to choose between these
two species, and with the more convenient lower-field resonance of 39 K (and with ourexperiment already shaping up to cater for cooling rubidium and potassium) it was theobvious choice to concentrate our efforts on.

1.2

Ideal Bose gas

Bose-Einstein condensation is fundamentally different to all other phase transitions between different states of matter in that it occurs even in the absence of interactions
between particles, relying on purely statistical processes2 . In a normal classical gas thethermal energy kB T is far greater than the level spacings of the system, and the behaviour of the gas can be described with standard Maxwell-Boltzmann statistics, where
the bosonic or fermionic nature of the particles is insignificant. However, Einstein ob1
2

This is valid as long as there are no other resonances nearby.
Although thermal equilibrium is assumed, generally requiring interactions between particles.

4

1.2 Ideal Bose gas

served in 1925 [1] that this description is inadequate for thermal energies for which
the spreading of the atomic wavefunction is comparable to the inter-particle separation, where the indistinguishable nature of the identical particles becomes important.
Einstein generalised Bose’s work on the statistics of (massless) photons [2] to consider
massive particles, describing what is now referred to as Bose-Einstein statistics.
The essential feature is that the mean occupation of single-particle states i for noninteracting bosons in thermal equilibrium is given by the Bose-Einstein distributionfunctionf (i ) =1e(i −µ)/kB T − 1

,

(1.1)

where i is the energy of the single-particle state i and µ is the chemical potential,
which is set by particle number conservation. From this the total number of particles
is easily evaluated as
N=X

f (i ) .

(1.2)

i

Equation (1.1) also provides the important constraint µ < 0 for the ideal Bose gas,
where 0 is the energy of the lowest single-particle state. As µ → 0 the occupation of
the ground stateN0 ≡ f (0 ) =1e(0 −µ)/kB T − 1(1.3)

increases and becomes comparable to N . In fact, this is the mechanism behind BoseEinstein condensation, occurring under specific conditions as discussed below.

1.2.1

The semi-classical approximation

We can treat (1.3) as a special case such that the total number of atoms is given by
N = N0 + NT

(1.4)

where NT is the number of atoms in excited states (i.e. the number of thermal atoms)
given byNT =X

f (i ) .

(1.5)

i6=0

In general it is not straightforward to evaluate such summations. However, as long as
the thermal energy kB T is much larger than the energy level spacing of the system we
may convert the sum into an integral by replacing the discrete i with the continuous
variable (r, p) dependent on position r and momentum p, which corresponds to theclassical energy associated with the single-particle Hamiltonian for the system. This
does not take into account the contribution from i = 0. In three dimensions there ison average one quantum state per volume of phase space (2π~)3 . Integrating over allphase space and dividing by this factor thus yields the total number of occupied excited51.2 Ideal Bose gas

states,1NT =(2π~)31ZZe((r,p)−µ)/kB T − 1

dpdr .

(1.6)

This transformation to an integral is known as the semi-classical approximation. For aR
gas in a box, (r, p) is simply p2 /2m and we may separate the integral dr = V , where
V is the volume of the containing box. We may then use the substitution x = p2 /2mkB T
to calculate the number density as a function of µ
Z ∞
1
1
1
NT
=
nT ≡
4πp2 dp = 3 g3/2 (eµ/kB T ) ,2 /2m−µ)/k T
3(pBV(2π~) 0 e
λT
−1
where

s
λT =

2π~2
mkB T

(1.7)

(1.8)

is the thermal de Broglie wavelength. g3/2 (z) is a special case of the set of polylogarithmic functions
1
gs (z) =
Γ(s)

Z ∞
0

∞

X zk
xs−1
dx
=
z −1 ex − 1
ks

(1.9)

k=1

where the Gamma function is defined asZ ∞Γ(s) =
xs−1 e−x dx ,(1.10)

0giving Γ(3/2) =√π/2. The value of nT obtained from (1.7) increases with increasingµ. However, there is a fundamental limit from our constraint µ < 0 = 0 for a freeparticle in a box. This means that as N increases the available excited states become
saturated at a critical densitync ≡Ncζ(3/2)
2.612
=
≈
,
3
V
λT
λ3T

(1.11)

where ζ(s) = gs (1) is the Reimann zeta function, with all further atoms forced to occupy
the ground state, forming a Bose-Einstein condensate. In this ideal case the numberof thermal particles is fixed at NT = Nc for N > Nc . Note that this phenomenon does
indeed occur at a point where interparticle separation is comparable to the spread of−1/3

the atomic wavefunction with nc
nλ3T∼ λT . The critical value of the phase space densityof 2.612 is a key prediction of Einstein’s theory and provided a useful target for

the experimental pioneers aiming to realise these extreme conditions in the laboratory.
We may reformulate (1.11) using (1.8) to give the corresponding result for the criticaltemperature,
2π~2kB Tc =mn
ζ(3/2)

2/3

6≈ 3.313~2 n2/3.
m(1.12)

1.2 Ideal Bose gas3

nc Λ3T

ΖI 3 M
2
2

1

0
-3

-2

0
rRT

-1

1

2

3

Figure 1.2: Density distribution (solid line) of a harmonically trapped Bose gas at the criticalpoint in terms of phase space density. The peak density is equal to the critical phase spacedensity for a uniform Bose gas. The dashed line shows the form of the harmonic trapping
potential.

1.2.2

Ideal Bose gas in a harmonic trap

Most experimental setups produce ultracold atomic samples that are trapped in aharmonic potential. We may describe such a system by now setting
(r, p) =wherep2+ V (r) ,2m(1.13)

1
V (r) = m ωx2 x2 + ωy2 y 2 + ωz2 z 2
2(1.14)

describes a generic harmonic potential with trapping frequencies ωk . Using (1.13)
with (1.6) and performing only the integral over momentum space gives us the spatial
distribution of thermal atoms in the trap asnT (r) =hi
1(µ−V (r))/kB T
g
e
.
3/2
λ3T

(1.15)

Once more the upper bound for our thermal atom number is acquired in the limit ofµ → 0 ≈ 01 . This gives a critical density distribution (as displayed in Fig. 1.2)
(
"
!#)
1
x2
y2
z2
1
nc (r) = 3 g3/2 exp −++ 2,(1.16)
2 RT2 x RT2 y
λT
RT z
where the characteristic widths of the distribution RT k =q
kB T /mωk2 are known as

the thermal radii. We immediately recover the critical density given in (1.11) at thecentre of the trap by setting x = y = z = 0. Performing instead the r integral of(1.6) would give a similar expression for nT (p). To obtain the critical number we must
integrate (1.16) over r, finding
1

This ignores the zero point energy of the potential.

71.3 The Bose-Einstein condensateNc = ζ(3)

kB T
~ω̄

3


≈ 1.202

kB T
~ω̄

3
,

(1.17)

where the geometric mean trapping frequency ω̄ = (ωx ωy ωz )1/3 is employed. Onceagain above this critical atom number additional atoms must occupy the ground state
as a Bose-Einstein condensate. Alternatively we can describe the critical point in termsof a critical temperature Tc for a fixed atom number given bykB Tc = ~ω̄



N
ζ(3)

1/3

≈ 0.940 ~ω̄N 1/3 .

(1.18)

The above expression may be used to more closely scrutinise the applicability of thesemi-classical approximation. The level spacings of a quantum harmonic oscillator being~ωk , the condition kB T  ~ωk must hold. It is apparent then from (1.18) that thesemi-classical approximation is only compatible with the quantum degenerate regime
if N 1/3  1.

1.3The Bose-Einstein condensate

The preceding section gives us an adequate description of the distribution of the thermal
atoms of an ideal gas below the critical temperature, but we would also like to know
about the distribution of the condensate. In order to calculate this we may invoke
the Bogoliubov approximation [18] to describe the wavefunction of the condensate as√Ψ0 = N0 ϕ0 , where ϕ0 is the single-particle ground state wavefunction of the system,obeyingZ
N0 =|Ψ0 (r)|2 dr .

(1.19)

Ψ0 is also known as the order parameter, being zero above the critical temperature andfinite below it1 .

1.3.1

Ideal condensate

For an ideal Bose gas, Ψ0 solves the familiar Schrödinger equation


~2 2
−
∇ + V (r) Ψ0 (r) = µΨ0 (r) ,2m(1.20)

where the chemical potential µ replaces the usual energy term as we keep the numberof particles fixed. It is easy to check that for the harmonic potential given in (1.14)
1
The Bogoliubov treatment is justified for a large number of particles occupying the same state,
where the non-commutativity of field operators is not important and a classical function may be usedinstead.

8

1.3 The Bose-Einstein condensate

6

5

n Λ3T

4

3

2

1

0
-2

-1

0
rRT

1

2

Figure 1.3: Density distribution in terms of phase space density for a non-interacting Bosegas in a harmonic trap, where Nc = 106 and N0 = 104 . The solid line gives the full distribution
while the dashed line shows that of the thermal gas only. The x-axis is in units of thermal
radii while the y-axis is given in terms of phase space density. The sharp peak in density is asignature of the onset of condensation in a harmonically trapped Bose gas. It is clearly visible
despite the condensed fraction N0 /N of only ∼ 1%.

this is solved by the Gaussian wavefunction
!#
"
√
y2
z2
N0
x2
1
Ψ0 (r) =
+
+ 2
,
exp −
3/2
2 a2ho x a2ho y
aho z
π 3/4 ā(1.21)

hopwhere aho k = ~/mωk is the oscillator length along the k axis and āho is the geometric√mean (aho x aho y aho z )1/3 . This is not surprising, being simply N0 times the lowest
single-particle state wavefunction for a harmonic potential. This directly gives theideal condensate density distribution"N0n0 (r) = |Ψ0 (r)|2 = 3/2 3 exp −π āho

x2
y2
z2
+
+
a2ho x a2ho ya2ho z!#.

(1.22)

Note that the characteristic length scales aho and RT , for the ideal condensate andthe thermal distribution respectively, differ by a factor [ζ(3)/Nc ]1/6 ≈ 0.1 for Nc =106 . This means that in a harmonic trap a sharp peak in the spacial distribution is asignature of the onset of Bose-Einstein condensation, as shown in Fig. 1.3 for Nc = 106
and N0 = 104 .

1.3.2

Condensate with repulsive interactions

The introduction of interactions complicates matters, but we are fortunate to be considering dilute systems in which the inter-particle separation is much larger than therange of the interatomic forces. We may generally then just consider the effects of pairs91.3 The Bose-Einstein condensateof particles interacting, and we may also use the asymptotic expression for the wavefunction for the relative motion of the atoms which is fixed by the scattering amplitude.
In the ultracold regime the particles have very low momenta such that the scattering
amplitude is fully determined by the s-wave scattering length a (this is discussed in alittle more detail in §1.5). We may also assume that the variation of Ψ0 (r) is small over
the range of the inter-atomic force (i.e. for a harmonic trap aho /a  1), such that theeffects of interactions are fully captured by introducing a self-interaction term g|Ψ0 (r)|2
to the Schrödinger equation, where the interaction coupling constant is defined asg=

4π~2 a.
m(1.23)

The addition of this non-linear term results in the time-independent Gross-Pitaevskii
equation,



~2 2
2
−
∇ + V (r) + g|Ψ0 (r)| Ψ0 (r) = µΨ0 (r) .
2m(1.24)

This important equation is a key tool for investigating the behaviour of nonuniform
ultracold Bose gases. A thorough derivation can be found in [19]. This provides astarting point to find the condensate density in a self-consistent manner.

A remarkable further simplification may yet be made. If we take as a starting
point the wave function (1.21) for g → 0 we can compare the expectation values of
the interaction and kinetic energy terms of (1.24). Assuming no extreme anisotropy of
the trapping potential, the ratio of the two terms is ' N0 a/āho . This means that theinteraction term swamps the kinetic energy provided the condition
N0

a
1
āho

(1.25)

holds. Once more under standard experimental conditions this criterion is met, and wemay use the so-called Thomas-Fermi approximation to drop the kinetic energy term of(1.24). This may then be simply arranged in the form
n0 (r) = |Ψ0 (r)|2 =µ − V (r)g(1.26)

up to the point in the trap where µ = V (r), while the trivial solution |Ψ0 (r)|2 = 0
holds otherwise. For a uniform system this gives the result µ = gn0 (0), being positive
for a > 0 below the critical temperature, in contradistinction to the ideal case. We canalso see that the condensate density is drastically altered from the ideal case, reflecting
the shape of the confining potential. Going back to our harmonic potential, the density
may be written in the form
n0 (r) =

µ
g


1−

x2
y2
z2
−
−
Rx2
Ry2
Rz2
10


,

(1.27)

1.3 The Bose-Einstein condensate

1.0

0.8

Ideal

nid
0 H0L

n0 HrL

0.6

0.4

a = 135a0 H50´L

0.2

0.0

0
raho

-5

5

Figure 1.4: In-trap density distributions for pure interacting (a = 135 a0 , solid line) and ideal(dashed line) 39 K condensates of 400 000 atoms with ω̄ = 2π × 70 Hz. Densities are normalised
to the central density of the ideal condensate, approximately equivalent to 1016 cm−3 . Notethat the central density in the interacting case is only ∼0.8% that of the ideal case, so thedisplayed distribution has been multiplied by a factor of 50.

where Rk =q
2µ/mωk2 is the Thomas-Fermi radius at which the condensate density

drops to zero along the k axis. The relation of N0 and µ may be found by integrating(1.27) over the ellipsoid with semi-axes Rk . Doing so gives
N0 =8π µ 3R̄ ,15 g(1.28)R̄ representing the geometric mean (Rx Ry Rz )1/3 . This may be rearranged to give theThomas-Fermi chemical potential in the form1
µ = ~ω̄2



15N0 a
āho

2/5
,

(1.29)

which we shall make good use of in Chapter 4. We can calculate n0 (0) by dividing(1.29) by g, while the Thomas-Fermi radius is given in terms of N0 as

R = aho

15N0 aaho1/5,(1.30)

showing how significantly the condensate spreads out in comparison to the ideal case,where the size is independent of N0 . Fig. 1.4 compares the in-trap distributions for purecondensates of 39 K in the ideal case (a = 0) and when a = 135 a0 . Interactions suppress
the central density by a factor of ≈ 125 with some typical experimental parameters.
11

1.4 Properties of alkali atoms

1.4

Properties of alkali atoms

The production of BECs is only possible due to the interactions of atoms with electromagnetic fields, so it is best to have an idea of the properties of the atoms themselves
before getting into the practical applications of such interactions.

1.4.1

Hyperfine structure

The electronic ground state configurations of the alkali atoms (group IA of the periodic
table) are characterised by a single electron in their outer shell, for example:
K

1s2 2s2 2p6 3s2 3p6 4s,Rb

1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s

for potassium and rubidium. It is enough simply to refer to the outer electron todescribe electronic states for our purposes as it is the only one that will be affectedon the energy scales used (the inner electrons are much more tightly bound). Thus
the potassium and rubidium ground states are referred to as 42 S1/2 and 52 S1/2 respectively. Here the superscript 2 refers to the number of spin components of the angularmomentum S (equal to 2S + 1), the ‘S’ refers to the orbital angular momentum L (S,P, D. . . for 0, 1, 2. . . ) and the subscript 1/2 refers to total electron angular momentumJ.
We concern ourselves with this ground state and the lowest excited nP states. Theinteraction between the spin and orbital components of the angular momentum (theorigins of which are discussed in [20] as well as many other texts) create a fine splittingof the excited state into a doublet with J = 1/2 and 3/2. The two frequencies associatedwith the energy gap between the ground state and these two excited states are referred
to respectively as the D1 and D2 lines.
Each of these states is split further by the interaction between the nuclear spin I
and J, described by the HamiltonianĤhf = Ahf I · J ,(1.31)

forming the hyperfine structure. Assuming J ≤ I, this produces 2J +1 levels labelled by
the total atomic angular momentum F , taking the values F = I −J, I −J +1, . . ., I +J.
For our experiments we shall mainly be concerned with the D2 line and the subsequent
hyperfine splitting, which is shown in Fig. 1.5 for 39 K and 87 Rb. Conveniently, both
species have I = 3/2, allowing us to consider just one configuration of S and I.

1.4.2

Interactions with magnetic fields

The degeneracy of the hyperfine levels may be lifted by applying a magnetic field.
Each is split into 2F + 1 sublevels, giving a total of (2J + 1)(2I + 1) for each fine12

1.4 Properties of alkali atoms

2

2

39

87

K

6

6

′

F =3

14.4

F′ = 1F′ = 0?

F′ = 2F′ = 1??

F′ = 06

F =2780.2412 nm
384.2305 THz

766.7019 nm
391.0157 THz

6
6

F =2

173.1

2563.0

6

42 S1/2F′ = 3

302.1

??

?

F′ = 2

229.9

19.4

16.1

?

6

495.8

?

52 P3/2

72.9

33.8

6

6.7

42 P3/2

6

193.7

6

Rb

?

52 S1/2?
6834.7461.74271.7

288.6

?

?

F =1
?

Figure 1: Caption

Figure 1: Caption

?

F =1

Figure 1.5: Hyperfine structure of the 39 K and 87 Rb D2 lines. Both species have I = 3/2
resulting in similar structures, with the frequency splitting between hyperfine levels given inMHz. It should be noted however that the scale of the entire 87 Rb structure is scaled down
by a factor of ten with respect to that for 39 K, while the scale of the lower hyperfine splitting
for both species is scaled down by a factor of ten with respect to that of their upper scales.
Potassium data is taken from [21, 22], while the rubidium data is collated in [23].

131.4 Properties of alkali atoms

structure configuration. For low magnetic fields (where the interaction with the external
field is weaker than the hyperfine interaction) each sublevel may be labelled by thegood quantum numbers F and its projection along the quantisation axis mF . In thestrong field regime however we must use instead the quantum numbers J, I and theirprojections mJ and mI . Using m = mF for low fields and m = mJ + mI for highfields provides a consistent parameter to label states over the full range of magnetic
field strength.
It is possible to find expressions to describe the behaviour of the hyperfine sublevelsin the weak- and strong-field regimes by considering the magnetic interaction as aperturbation of the hyperfine interaction eigenstates and vice versa, but to examine thebehaviour in the intermediate region one must diagonalise the full hyperfine-magnetic
HamiltonianĤ = Ĥhf + ĤB = Ahf I · J − µJ · B − µI · B ,(1.32)

where µJ = −gJ µB J /~ and µI = −gI µB I/~ are the angular momentum and nuclear
magnetic moments respectively, where gJ and gI are Landé g-factors and µB is theBohr magneton. For J = 1/2 , this results in the Breit-Rabi formula [24],∆Ehf− gI µB mB
EBR (F± = I ± 1/2, m) = −2(2I + 1)
r1
2 + 4m (g − g )µ B∆E + (g − g )2 µ2 ,
±
∆Ehf
J
I B
J
I
hf
B
2
2I + 1(1.33)

where ∆Ehf is the hyperfine splitting for B = 0. Turning to dimensionless variables
B̃ =B(gJ − gI )µB
EBR (F, m)=B
and (F, m) =
B0
∆Ehf
∆Ehf

(1.34)

this may be written in the dimensionless formm
1
1
(F± , m) = −−B̃ ±2(2I + 1) gJ /gI − 1
2

r
1+

4m
B̃ + B̃ 2 .
2I + 1(1.35)

This is plotted in Fig. 1.6 for I = 3/2, representing the 39 K and 87 Rb electronic groundstates. In the weak-field (B̃  1) limit, and neglecting gI ∼ gJ × 10−3 , we recover thefamiliar Zeeman effect,
EZ = EBR (F, mF , B) − EBR (F, mF , B = 0) = µF B ,(1.36)

where µF = gF µB mF . The factor gF arises from the projection of J onto F , which
for J = 1/2 is simply given by
gF± = ±1gJ .
2I + 1(1.37)

Magnetic trapping of atoms therefore depends on creating an inhomogeneous magnetic field such that a potential minimum is created. Given that Maxwell’s equations
14

1.4 Properties of alkali atoms

2

1

1
mJ =+
2

m=+2

E BR

m=-2

0

Ε=

D E hfs

F=2

F=1 m=-1
1
mJ =-

m=+1

2

-1

0

0.5

1

1.5

B
B=
B0

2

2.5

3

Figure 1.6: Energies of the magnetic sublevels of the n2 S1/2 fine structure ground state ofan alkali atom with nuclear spin I = 3/2. In the low-field regime the states are grouped by F
while in the high-field regime they are grouped by mJ . Note that B0 is 2438.8 G for 87 Rb while
it is 164.7 G for 39 K.

prohibit the formation of a magnetic maximum in space, only the atoms in states where
magnetic potential increases with field strength are magnetically trappable, i.e. thosefor which gF mF is positive. Note that magnetic potentials are proportional to |B|, as
the direction of the field defines the precession axis of the system, such that there will
always be a potential minimum where B = 0 .

1.4.3

Interactions with light

The development of techniques to cool and trap atoms with laser light was the key toopening up the world of ultracold atomic gases, being recognised with the bestowal of
the 1997 Nobel Prize for Physics on Steven Chu, Claude Cohen-Tannoudji and William
Phillips. That light has an effect on an atom stems from the fact that a dipole d may
be induced in an atom in the presence of an external electromagnetic radiation field E,
in general given by the expectation valuehΨ| d |Ψi = αE ,(1.38)

where α is the complex polarisability of the atom. The effects of light on atoms can be
split into two categories: reactive effects and absorptive effects, the former being more15

1.4 Properties of alkali atoms

important when the frequency of the external field is far from resonance, the latter
when the frequency is near resonance. These two effects are directly related to the realand imaginary parts of α. Much information may therefore be garnered from finding α,
for which we need to know the electronic wavefunction Ψ. This may be approximatedby considering the interaction as a perturbation of the isolated atom system, with thetotal HamiltonianĤ = Ĥ0 + Ĥdip ,(1.39)

where Ĥ0 is the Hamiltonian describing the atom in the absence of the field, and the
Hamiltonian representing the perturbing dipole interaction is given byĤdip = −d · E ,(1.40)

noting that d = −er, where we are considering the coordinate system where the atomic
nucleus is located at r = 0. This shows that the dipole moment has odd parity which
is used in the following discussion.

As is standard in perturbation theory we set about finding the solution to thetime-dependent Schrödinger equationĤΨ(r, t) = i~dΨ(r, t)dt(1.41)

by considering the wavefunction to be a superposition of the eigenstates of the unperturbed system:
Ψ(r, t) = ρ1 (t)Ψ1 (r, t) + ρ2 (t)Ψ2 (r, t) ,(1.42)withi

Ψi (r, t) = e− ~ Ei t ϕi (r) .

(1.43)

Here we have simplified the full description to a two-level system (lower state 1, upper
state 2) where the time dependent coefficients ρi (t) are normalised such that the sumof the state populations |ρ1 |2 + |ρ2 |2 = 1. The quantities Ei are the energy eigenvaluesof the unperturbed Hamiltonian:Ĥ0 Ψi = Ei Ψi .

(1.44)

Using (1.41), (1.42), (1.43) and (1.44) we obtainĤdip (ρ1 Ψ1 + ρ2 Ψ2 ) = i~(Ψ1 ρ̇1 + Ψ2 ρ̇2 ) ,(1.45)

where ρ̇i ≡ dρi /dt. If we assume that the applied radiation field oscillates harmonicallyin time with frequency ω/2π we can write
E(t) = E 0 cos(ωt) ,16(1.46)

1.4 Properties of alkali atoms

where it is also assumed that any spatial variation of the field amplitude is insignificant
over the length scale of the atom. To eliminate the spatial dependence of (1.45) we canmultiply through by the complex conjugate wavefunction Ψ∗i and integrate over space.
Doing this for both i = 1 and i = 2 yieldsii
ρ2 (t)d12 · E 0 cos(ωt)e− ~ (E2 −E1 )t~iiρ̇2 (t) = ρ1 (t)d21 · E 0 cos(ωt)e ~ (E2 −E1 )t .
~

ρ̇1 (t) =Here we have introduced the dipole matrix elementsZ
dij = hi| d |ji = ϕ∗i (r)d(r)ϕj (r)dV(1.47)(1.48)(1.49)

coupling the two states. For convenience we may choose the phases of the wavefunctions
such that the dipole matrix elements are real. We can then define the Rabi frequencyΩ (a measure of the strength of the coupling between the light and the applied field)asΩ=|d12 · E 0 ||d21 · E 0 |=.
~~(1.50)

This is not quite the full story as we have not included the effect of spontaneousemission. This is easily rectified by introducing a damping term (Γω /2)ρ2 (t). Also
defining the resonant transition frequency ω0 = (E2 − E1 )/~ we may rewrite (1.47) and(1.48) asΓωρ2 (t) = iρ2 (t)Ω cos(ωt)e−iω0 t2Γωρ̇2 (t) +ρ2 (t) = iρ1 (t)Ω cos(ωt)eiω0 t .
2ρ̇1 (t) −(1.51)(1.52)

Γω = (ω 2 /ω02 )Γ represents the off-resonant damping coefficient [25] which reduces toΓ close to resonance (or indeed in the absence of an applied field). Note that setting
E 0 = 0 (i.e. Ω = 0) in (1.52) gives the result that, in the absence of external radiation,
the upper state population decays at a rate Γ, and|ρ2 (t)2 | = |ρ2 (0)2 |e−Γt ,(1.53)

as expected. However, we are more interested in the steady-state solutions in thepresence of an external field, i.e. the case where the state populations are constant:d|ρi (t)2 | = 0 .
dt(1.54)For near resonant light we may make the rotating-wave approximation, δ = ω−ω0 
ω + ω0 , such that in the steady-state situation the time-dependence of ρ1 ρ∗2 is simply
171.4 Properties of alkali atoms

given by the factor exp(iδt). With these conditions, and usingd(ρi ρ∗j ) = ρi ρ̇∗j + ρ̇i ρ∗j ,dt(1.55)

we may use (1.51) and (1.52) to obtain the equationsΩe+iδt ρ2 ρ∗1 − Ωe−iδt ρ1 ρ∗2 − 2iΓρ2 ρ∗2 = 0(1.56)Ω(ρ1 ρ∗1 − ρ2 ρ∗2 ) + (2δ − iΓ)e−iδt ρ1 ρ∗2 = 0 .

(1.57)

We may solve these directly to obtain, for example,ρ2 ρ∗2 = |ρ2 |2 =

Ω2
Γ2 + 4δ 2 + 2Ω2(1.58)

for the upper state population, which reaches a maximum of 1/2 as Ω → ∞. We may
now find a value for α by considering the expectation value of the dipole moment,ZhΨ| d |Ψi = Ψ∗ d Ψ dV = ρ∗1 ρ2 d12 e−iω0 t + ρ1 ρ∗2 d21 eiω0 t = αE 0 cos(ωt) .
(1.59)In the near-resonance case that we have been discussing we may use the steady-statesolutions obtained from the above equations to findαnear =

−δ + iΓ/2
d12 d21.
~ Γ2 /4 + δ 2 + Ω2 /2(1.60)In the far-detuned case we can no longer make the rotating-wave approximation, butinstead we may assume Γ2  δ 2 and that the coupling Ω is weak. Similar arguments
eventually lead to an expression for the polarisability in this regime,
d12 d21 2ω0 ω02 − ω 2 − i(ω 3 /ω02 )Γ
.
αfar =~(ω02 − ω 2 )2

(1.61)The reactive dipole force may be found from

where

Fdip = −∇Udip(1.62)

1
1
Udip = − hd · Eit = −Re(α)I(r) .
220 c(1.63)

h. . .it represents the time average and the factor of 1/2 appears as the dipole is induced
rather than permanent. We have also used I(r) = (1/2)0 c|E 0 (r)|2 and the fact thatthe time-averaged value of cos2 (ωt) is 1/2. The r dependence is now on the macroscopic
scale, not the atomic scale, and we see a force that is proportional to the gradient of
the intensity of the external field. As we are taking the real part of the polarisability
we can see that this force is vanishing in the near-resonant limit.
18

1.4 Properties of alkali atoms

The absorptive scattering force is found from
Fsc = ~kΓsc ,(1.64)

where k is the wave-vector of a photon of the laser field (so its direction is that of
the beam propagation), and Γsc is the scattering rate of the atom. The force is thenconsidered to arise from the atom receiving a kick of momentum ~k each time it absorbs
a photon, while we consider the momentum kicks from re-emission to average to zero.
This picture will be considered later in §2.3 when discussing the limits of laser coolingmethods. Γsc can be found by considering the power absorbed by an atom and dividing
by the energy attributed to an incident photon, ~ω:
Γsc =

1
1
Pabs
=
hḋ · Eit =Im(α)I(r) .
~ω
~ω
~0 c

(1.65)

Near-resonant light
Taking all of the above into account we can introduce the saturation intensity Isat where2Ω2
Γ2(1.66)

I/IsatΓ.
2 1 + (2δ/Γ)2 + I/Isat(1.67)

IIsat=

to finally obtain the scattering force as
Fsc = ~kThis force is applied in the early stages of the experiment to trap and cool atoms as is
discussed in §2.3. Note that Γsc could also have been found by multiplying the value of|ρ2 |2 given in (1.58) by Γ, being simply the steady-state population of the upper state
multiplied by the decay rate from this state.

Far-detuned light
By using the quantum mechanical result for the spontaneous decay rate of the upper
state (see e.g. [26, 27])Γ=ω03|h1| d |2i|23π0 ~c3(1.68)

with (1.61) and (1.63), we may obtain an expression for the dipole potential
3πc2
Udip (r) = − 3
2ω0



Γ
Γ
+
ω0 − ω ω0 + ωI(r)(1.69)

and using (1.65) the corresponding scattering rate
3πc2
Γsc (r) =
2~ω03



ω
ω0

3 

Γ
Γ
+
ω0 − ω ω0 + ω

192I(r) .

(1.70)

1.5 Collisions between ultra-cold atoms

The dipole force is stronger than the scattering force in the far-detuned regime, but due
to the typical temperatures of atoms in traps formed by dipole potentials it is useful
to know the scattering rate of the atoms, as only a few scattering events can cause
losses from the trap. Assuming for a moment that the detuning is small enough that1/δ  1/ω0 , we can see that Udip is roughly proportional to I/δ while the scattering
rate is roughly proportional to I/δ 2 . It is therefore generally preferable to detune
the laser beam far from resonance while compensating for the potential strength byincreasing the power in the laser as far as possible, for greater lifetimes and lower
heating rates in a dipole potential. Also note from (1.69) that if ω < ω0 (red-detuning)a potential minimum can be formed by an intensity maximum, while for ω < ω0 (bluedetuning) a potential minimum is formed by an intensity minimum.

1.5

Collisions between ultra-cold atoms

Collisions play a central role in both the formation and behaviour of ultra-cold atomic
gases. Elastic collisions enable thermalisation of a cloud of atoms, allowing cooling via
evaporation in the final step towards quantum degeneracy in many experiments (§2.7),
while inelastic collisions limit the effectiveness of such a technique. The modificationof the shape of the BEC due to collisional interactions has already been discussed in§1.3.2, while inelastic processes ultimately limit the maximum phase-space density it ispossible to achieve in a cooled gas.

1.5.1

Scattering length

In general, collision theory is a complex topic, but the fact that we are dealing withthe ultra-cold regime greatly simplifies the description. For now we shall consider themost pertinent case of elastic collisions between just two particles. The easiest way todescribe such a system is to ignore the centre-of-mass energy and to consider just therelative position and momentum operators r̂ and p̂:r̂ = r̂ 1 − r̂ 2 , p̂ = (p̂1 − p̂2 )/2 ,(1.71)

where the subscripts 1 and 2 refer to the individual atoms, using the notation of [28].
The Hamiltonian for the relative motion of the two particles then naturally follows as 2p̂+ V (r) ψ(k, r) = Ek ψ(k, r) ,(1.72)
2mrwhere V (r) represents the inter-atomic potential, k is the relative wavevector and mris the reduced mass, equal to m/2 in the case of atoms of the same species. We assume
that V (r) → 0 as |r| = r → ∞, such that the eigenstates of the system have energy
Ek =~2 k 2.
2mr

20(1.73)

1.5 Collisions between ultra-cold atoms

If we assume that the potential is spherically symmetric, the solution to (1.72) in ther → ∞ limit takes the asymptotic form of a superposition of an incident wave and ascattered wave, given byψ = eikz + f (θ, k)

eikr
,
r

(1.74)

where the z direction has been chosen for the propagation of the incident wave. Heref (θ, k) is the scattering amplitude which depends on the angle θ between the relative
incident and scattered wave-vectors. Due to the axial symmetry with respect to z, thegeneral form of the wavefunction can be expanded as ‘partial waves’ in terms of the
Legendre Polynomials Pl (cos θ), taking the formψ=∞X

Al Pl (cos θ)Rkl (r) .

(1.75)

l=0

Using the usual quantum mechanical result for motion in a centrally symmetric field(see e.g. [26]), the radial wavefunction must satisfy the equationl(l + 1) 2mr V (r)d2 (rRkl (r))
2
+ k −
−
rRkl (r) = 0 .
dr2
r2
~2

(1.76)For k → 0 , the relative particle energy will be far lower than the barrier formed by
the addition of the centrifugal potential ~2 l(l + 1)/2mr r2 , such that only the partial
wave for l = 0 will ‘see’ the potential V (r), and all other waves will not contribute to
the scattering behaviour1 . The scattering amplitude for l = 0 is independent of θ, andso is referred to as the s-wave scattering length, which is written as a. We therefore
expect in the limit r → ∞ and k → 0 to find the solution
ψ0 = 1 +a.
r(1.77)

This is simply the result for a hard-core potential of radius a, giving a means of interpreting the scattering length. It can be shown [28, 29] that in the low k limit a isindependent of k, so low energy scattering is completely determined by a. This applies
to the total scattering cross-section σ, which for identical bosons is given by
σ = 8πa2 .

(1.78)The extra factor of two over the usual expression results from the symmetry of the
wavefunction under the exchange of the two sets of particle coordinates for identicalbosons. The scattering length also gives the condition for a gas to be considered dilute,
being the regime where an1/3  1. The quantity an1/3 is therefore referred to as thediluteness parameter.
As the behaviour of the atoms depends only on the scattering length, we can choose
1Technically this depends on the form of r as r → ∞. When this is given by the Van de Waals
potential ∝ r−6 it is a good approximation. See [26] for details.

21

1.5 Collisions between ultra-cold atoms

an effective pseudopotential to model the system as long as the scattering length a isretained. The simplest model describing interactions between two particles uses thecontact potential
V (r) = gδ(r) ,(1.79)

where δ(r) is the Dirac delta function centred at r = 0 . The proper way to treat thecontact interaction is discussed in [30], and it leads to a total s-wave elastic scatteringcross-section for polarised Bosons of [28]σ(k) =

8πa2,
1 + k 2 a2

(1.80)

where the contact energy g is found to be the interaction coupling constant given in(1.23). This gives two very different results in the small and large ak limits:(
σ(k) '

8πa2

for ka  1

8π/k 2 for ka  1 .

(1.81)

The first case is the result (1.78) for identical bosons at ultra-low energy, while thesecond case reveals the upper limit to the cross-section for s-wave scattering. This is
known as the unitarity limit, the regime in which the inter-particle spacing is considerably greater than the range of the two-body interaction, but considerably smaller thanthe magnitude of the scattering length. In this scenario both the form of the interatomic potential and the precise scattering length cease to matter when considering
few-body scattering. This leads to intriguing universal properties, such as the shallow three-body bound ‘Efimov states’ in the vicinity of a Feshbach resonance [12, 31].
This limit can be reached in ultra-cold atoms only when the scattering length can takedivergent values. For the most-part the former cross-section is appropriate.
Real potentials (unlike hard-core potentials) can result in either positive or negative
a . Calculation of the scattering length requires a precise knowledge of the shape of
the interaction potential, being very sensitive to small alterations, in particular whenthe energy of a quasi-bound state is closely matched to the potential depth U0 . This is
illustrated in Fig. 1.7, where the quantity rR0 (r) has been plotted for three different
scenarios in the style of [32]. The interatomic interaction has been modelled using a
Lennard-Jones potential, which takes the form

V (r) = −C6 r−6 + C12 r−12

(1.82)

where C6 and C12 are positive. Fig. 1.7 a shows a standard scenario where there isno near-threshold bound state, producing a moderate positive scattering length. If thetrap depth is increased, a quasi-bound state with energy slightly greater than U0 causes
the wavefunction to start to ‘turn over’, and the extrapolation from the large r limit
designates an increasingly large negative scattering length, as shown in Fig. 1.7 b. As
22

1.5 Collisions between ultra-cold atoms

a

0

a

b

0

V (r), rR0 (r)c0rFigure 1.7: Three different cases showing the effect that small changes in the interaction
potential can have on the resultant scattering length. The shape of the potential V (r) is inblack, while the radial wavefunction rR0 (r) is in red. The well depth U0 is increased froma to c. a In a simple case where there is no bound state near the dissociation energy limit
the scattering length takes a moderate positive value a as indicated. b As a bound state
approaches resonance the extrapolation of the wavefunction from the large r limit (dotted redline) indicates an increasingly negative scattering length, diverging completely on resonance. cPast the resonance position an extra node appears in the wavefunction and a positive scattering
length is once again found. These plots are in the style of those found in [32].

the bound state comes into resonance with U0 the scattering length diverges, until a
new bound state exists and an extra node appears in the wavefunction. Just past theresonance the scattering length is therefore large and positive as, depicted in Fig. 1.7 c.
In practice, the shape of the interatomic potential is too complicated to directly
calculate, and experimental methods such as photoassociation spectroscopy [33], as
well as the observation of collisional dynamics in cold atom clouds (as studied in [6]for 39 K), are used to determine the scattering length. The scattering properties also
depend on the internal states of the atoms. The total electronic spin S in a binary
system can be either 0 (singlet) or 1 (triplet). Each case gives rise to a different
interatomic potential, resulting in different scattering lengths as and at respectively.
For collisions between maximally polarised atoms (i.e. for mF = F ), the single-channel
scattering is fully determined by the triplet potential, but the scattering length in other23

1.5 Collisions between ultra-cold atoms

states is a mixture of as and at due to the effects of the hyperfine interaction. However,circumstances can conspire such that a second channel plays a large role in determining
the scattering properties, as now described.

1.5.2

Feshbach resonances

The case in which there are two scattering channels to consider is illustrated in Fig.
1.8. In black is the usual open channel, which for large separation corresponds to two
free atoms in a gas, while in red is the closed channel. In the ultra-cold regime theatoms have little kinetic energy, and the total atomic energy tends to that of the open
channel potential at large separation. Atoms must therefore enter and exit in the open
channel as they do not have enough energy to separate in the closed channel. Thismeans there is no coupling at first order between the channels. However, if the closed
channel supports a bound state with an energy close to that of the atoms then an
extra scattering pathway is available, and the atoms may briefly scatter into the closed
channel before emerging in the open channel. In this case even weak coupling can leadto strong mixing between the channels and a large change in the scattering length fromthe pure single-channel scenario. This second-order coupling scenario is a familiar one,and from perturbation theory we would expect a sum of contributions to the scattering
length of the form [29]a∼C,
E0 − Eres(1.83)

where E0 is the energy of the particles in the open channel and Eres is the energy inthe closed channel. For channels with different magnetic moments, as is the case forthe singlet and triplet potentials, the application of a magnetic field allows the relative
energy of the bound state to be tuned and brought to resonance. This is known asa magnetically-tuneable Feshbach resonance, the benefits of which have already been
discussed in §1.1. To reflect the ability to tune with a magnetic field the total scattering
length near a particular Feshbach resonance is written in the form
∆a(B) = abg 1 −,B − B∞(1.84)

as introduced in [34], relating the scattering length in the presence of a magnetic fieldof strength B to the off-resonant value abg via a resonance position B∞ and width ∆.
Note that this form holds as long as the resonance is well isolated from other resonances.
In this case the sum of the rest of the terms with the form of (1.83) will effectively beconstant, and their effect is absorbed into the value abg . The above equation is derived
in a considerably more rigorous fashion in [35] and [17], where it is shown that thewidth parameter ∆ may be written∆=m(2π~)3|hψres |Ĥex |ψ0 i|2 ,
4π~2 abg µres
24(1.85)

1.5 Collisions between ultra-cold atoms

V (r)Closed Channel
0Open ChannelrFigure 1.8: Basic two-channel model for a Feshbach resonance. The phenomenon occurs whenatoms incident upon one another in the open channel (black line) have an energy approachingthat of a bound state (red dashed line) in the closed channel (red line). Mixing between statesleads to a divergent scattering length when the bound state is resonant with the colliding atoms.
200

a Ha0 L

100

0
abg

ÈDÈ
-100

-200
250

300

B0

B¥

450

500

550

B HGL

Figure 1.9: Characteristic shape of a magnetically tuneable Feshbach resonance. Here thecase for the 39 K resonance centred at 402.5 G is shown.

very suggestive of the perturbative origins of (1.84). Here Ĥex is the spin exchange
coupling Hamiltonian, ψ0 and ψres are the open and closed channel energy states respectively, and µres is the difference between the magnetic moments of the molecularbound state and the asymptotically separated non-interacting atoms.
The resonance profile (1.84) is shown in Fig. 1.9, using the parameters for the
39 K resonance in the |F, m

F i = |1, 1i state listed in Table 1.1.

Note that both thebackground scattering length and the resonance width are negative in this case. The
zero-crossing point (where the scattering length goes to zero) is given by B0 = B∞ + ∆.
The existence of such resonances was experimentally confirmed in a BEC of sodium
atoms in 1998 [36]. Since then, access to such resonances has become an extremely
desirable tool for many experimentalists, while for the studies described in this thesis
it is essential.
251.5 Collisions between ultra-cold atoms

1.5.3

Inelastic processes

We conclude this chapter with a brief discussion of inelastic collisions which are, forthe most part, undesirable but inevitable in ultra-cold atoms experiments. Most obviously, the imperfect vacuum of any experimental system leads to collisions between thetrapped gas and the background vapour. This vapour is at room temperature such thatthe energies of the constituent particles are far greater than that of the trapped atoms.
This causes exponential decay of the number of trapped atoms, at a rate determinedby the background pressure in the vacuum chamber. This is usually the dominant loss
mechanism for clouds at low density.
At higher densities collisions between three particles become more frequent, increasing the probability of three-body recombination, in which two particles form a molecule
while the third carries away the excess energy. This is often particularly undesirable,
as not only are all three atoms lost, but the coldest atoms are preferentially affected.
This is often the dominant loss mechanism for condensates due to their high densities.
The process has also been shown to scale as a4 [37], so can be greatly enhanced near a
Feshbach resonance.
Finally, dipole-dipole interactions between polarised atoms in a magnetic field canlead to two-body spin relaxation [38], where the spin state of one or both of the colliding
atoms may be flipped. Due to the Zeeman splitting of spin states, this is energeticallyfavourable. When a cloud is magnetically confined this process can lead to atom lossby the scattering of atoms into untrappable states. These three processes all contributeto the overall heating and atom loss which act to oppose the desired increase in phasespace density required for the formation of BECs, so the cooling mechanisms used must
work at a higher rate to overcome them.

26

Chapter 2

Experimental setup“ The only way out that I can think of is to ask the old gang to let theirattention wander for a bit - there are heaps of things they can be doing;
washing the car, solving the crossword puzzle, taking the dog for a run while I place the facts before the newcomers.”– P. G. Wodehouse (Much Obliged, Jeeves)
The construction of a working experimental setup to reliably create and detect quantum
degenerate gases is no mean feat in itself, and typically still takes a new group a periodon the order of years to achieve. With the increasing global knowledge of these systems
start-up times are coming down, but the practical details are still inevitably numerous.
This chapter provides the particulars of the experimental methods and setup in somedetail, with the hope that it will prove useful to future students setting out on a similarroad to ours. For those better acquainted with cold atoms research it may be convenient
to use as a reference, with the more concise details of our experimental sequence toproduce 39 K BECs being given in Chapter 3.
The system in which we have condensed and experimented with 39 K is, in fact, thesecond BEC machine in our lab. A trial single-chamber system was first created, themain purpose of which was to familiarise ourselves with the basic experimental methods
required for the laser-cooling of atoms by creating a magneto-optical trap (see §2.3).
The setup exceeded our expectations though, and was pushed further to produce 87 Rb
BECs of ∼ 105 atoms in an optically-plugged quadrupole trap [4]. Working with asingle chamber creates inherent limits on what is experimentally achievable (e.g. therelatively short lifetimes of trapped atoms due to the pressure required for loading amagneto-optical trap), but it is still hoped that this system will prove fruitful in the
future with the aid of planned improvements. To go into detail here would be to straya little too far from the main story, but a brief description of the trial system may befound in Appendix B. The experimental details given in the remainder of this chapter
apply only to the second system.
27

Atom Sources

Pumps

Pumps

2.1 Vacuum system

Dipole Trap BeamTrans

MOT Cell

M

OT

Be

am
s

Science Cells

rt Coilpo

Figure 2.1: Simplified representation of our experimental setup showing a horizontal crosssection through the vacuum system. On the right is the MOT cell, at a relatively high pressure(∼ 10−9 mbar) for efficient collection of atoms, while on the left is the ‘science cell’ where
evaporative cooling occurs, and the pressure is ∼ 10−11 mbar. Once the requisite number ofatoms has been loaded into the MOT, they are transported to the science cell by the transportcoils.

2.1

Vacuum system

In contrast to our trial setup the main system has a two-chamber arrangement asshown in Fig. 2.1. The essential stages in trapping and cooling atoms are a lasercooling stage in a magneto-optical trap (MOT), and an evaporative cooling stage. The
former requires a relatively high pressure to collect a large number of atoms in the firstplace, while the latter requires as low a pressure as possible to avoid undesired atomloss. The first chamber is therefore referred to from here on in as the MOT cell, whereatoms are trapped and laser-cooled from a background vapour. The second chamberis referred to as the science cell, which is held at a lower pressure than the MOT cell,providing the extended lifetimes required for our experiments1 . The pressures required
throughout the system are in what is known as the ultra-high vacuum (UHV) regime,
typically defined as ∼ 10−12 to 10−9 mbar. The difference in pressure between thetwo regions is maintained by the continual operation of two ion pumps separated by
low-conductance tubes. The background vapour of alkali atoms in the MOT cell issustained by regular deposition on the chamber walls from atom sources located in an
adjoining vacuum tube.

2.1.1

Differential pumping design

The basic design of a two-chamber vacuum system depends on the desired ratio ofpressures between the two regions. Applying the old adage that it is better to be safe
than sorry, we decided to aim for a drop of around three orders of magnitude across1
The system is displayed in figures such that the MOT cell is on the right while the science cell ison the left. This is due to its orientation in the laboratory - the familiarity of several years’ work is
hard to shake.

28

2.1 Vacuum system

s2
p2

c2

s1
p1

c1

p0

Figure 2.2: Representation of our differential pumping system (a simplified representationof Fig. 2.1). The pressure p0 is assumed to be constant due to the vapour pressure from therubidium and potassium on the walls of the MOT chamber which is regularly replenished by
the atom sources. The low conductances c1 and c2 allow a large stable pressure drop to thescience cell at p2 , maintained by the relatively high pumping speeds s1 and s2 to the ion pumps.

the system. This could be achieved with a single low-conductance tube and a single
pump, but the tube would either have to be excessively long or narrow, which would
be detrimental to the transportation of atoms along the system. Instead we chose a
two-stage differential-pumping setup, a simplified depiction of which is shown in Fig.
2.2. Pressures are denoted p, conductances c and pumping speeds as s. p0 is thepressure in the MOT cell, p2 in the science cell. In reality the system is made up of
many components in series, but we may simply calculate total conductances cT using
X1
1
=
.
cT
ci

(2.1)

i

As we use identical pumps with similar conductances at their inlets we make the simplification s1 = s2 = s. By considering the system to have reached a steady state it ispossible to equate total pumping speeds to and from each vacuum region, to give thepressure ratio from MOT cell to science cell as
p2
c1 c2
=
.
p0
c1 c2 + c1 s + 2c2 s + s2

(2.2)

Generally c  s , such that (2.2) may be approximated as p2 /p0 ≈ c1 c2 /s2 . This may
be compared with the fraction c/s for a single pumping stage. The total pressure drop
is not the only consideration for the design of the differential pumping system however.
Once we have collected the desired number of atoms in the MOT chamber, we want
to transport as many of them as possible to the science cell at as low a temperature
as possible, so any collisions between trapped atoms and the background gas in thevacuum chamber are undesirable. The main initial concern therefore is dropping thepressure quickly. We could of course do this by means of a very small aperture (andhence low conductance) between the MOT cell and the first pump, but there is a limit
to how small an aperture we can safely transport the atoms through. In the molecularflow regime1 the conductance of a long circular tube is given (in litres per second) by
1
The molecular flow regime is roughly defined to be where the mean free path of particles in a gas
is greater than the size of the confining system, such that generally the particles travel from chamber
wall to chamber wall without encountering other particles. The system is well within this regime.

292.1 Vacuum system

100.0

100

50.0

80
60
40
20

10.0
H%L

1

2

3

4

p0

p

0

5.0

1.0
0.5

0

1

2

x=

3

4

x
l0

Figure 2.3: Pressure drop along a theoretical vacuum system as described in the text, where
the final pressure reached is fixed to be 0.1% of p0 . l1 varies from 0.25 l0 to 2.0 l0 in steps of
0.25 l0 . The black line represents the case where l1 = l2 = l0 , while red and blue lines represent
l1 < l0 and l1 > l0 respectively. There is a clear trade-off between rate of pressure drop-off andthe pressure reached after the first stage of differential pumping. This general scaling informed
the choice of vacuum components in our system. Inset is the same information on a linearlinear scale, illustrating the linear scaling of the pressure with distance along a tube of uniform
cross-section.

[39]
−1

c (l s) ≈ 12.4

d
1 cm

3 

1 cm
l



T
m N2
×
295 K
m

1/2
,

(2.3)

where d is the diameter and l the length of the tube, m is the particle mass and T thetemperature. Fixing d to be the minimum diameter for atom transportation resultsin conductances inversely proportional to l. The pressure drop for the first stage ofpumping is given byp1
c1 (c2 + s)=,
p0
c1 c2 + c1 s + 2c2 s + s2

(2.4)

while the pressure drop along a tube is simply linear. The result of all this is that if
we fix the total pressure drop p2 /p0 then the pressure gradient is greater initially forshorter length l1 of the first tube (corresponding to conductance c1 ∝ 1/l1 ). This is
illustrated in Fig. 2.3 on a logarithmic pressure scale for p2 /p0 = 1/1000, where theblack line represents the case where l1 = l2 = l0 , while red and blue lines illustrate cases
where l1 < l0 and l1 > l0 respectively. The plot shows the pressure drop along the firsttube immediately followed by the pressure drop along the second ignoring any interim
space, x̃ = x/l0 representing distance along the system. Here it is clear that the fullpressure drop is achieved in the shortest distance for l1 ≈ l2 , but faster initial drops areachieved for l1 < l2 . Of course if the first tube is too short then the pressure p1 won’t
30

2.1 Vacuum system

be much lower than p0 , defeating the purpose of using a small l1 . A happy medium isachieved when p1 is acceptably low. Our final design has a theoretical p1 /p0 ≈ 1/27
and p2 /p0 ≈ 1/1200. It can be seen on the cross-section shown in Fig. 2.1 that we haveadhered to the spirit of the above arguments as far as possible, with the customised‘reducing cross’ piece on the right having a wider flange towards the MOT cell so thatthe pressure is reduced more quickly (corresponding to an effective low l1 ), with thenarrower flange making up part of the longer second conductance tube.
The simple model we have been considering does not hold so well once we areworking at extremely low pressures (in the 10−11 mbar regime at the science cell) due
to outgassing from the walls of the vacuum chamber becoming a limiting factor. This is
reduced as far as possible as discussed in §2.1.3, but nonetheless in reality the pressure
at the science cell is closer to only two orders of magnitude lower than at the MOT
cell. The effect is mainly on p2 , and as stated earlier it is worth leaving some leeway inthe design. The lifetimes of several minutes achieved for trapped gases in the sciencecell are extremely satisfactory.

2.1.2

Components

The complete vacuum system is shown in Fig. 2.4 with a breakdown of the key component parts. Most of the pieces are standard items bought from various vacuum companies, while others, such as the ‘4-way reducer crosses’, were customised on request
for non-standard lengths and tube diameters. All of the metal near the atom-transport
axis is 316LN stainless steel as opposed to standard 304L, the main advantage being
the lower magnetic susceptibility of the material (0.001 instead of 0.03). This is tominimise any possible distortion of the magnetic fields we create in the system withexternal coils. All components in the system are UHV compatible, connected together
with standard UHV conflat (CF) flanges and may be baked to well over 200 ◦ C (see§2.1.3). Below the important features of some of the vacuum components are discussed.

Pumps
As has already been alluded to, UHV is maintained by the workhorse pumps of thesystem - a pair of Gamma Vacuum TiTan 75S ion pumps, which have a pumping speedof ∼ 60 l s−1 each. The pumps operate by ionising gases and using an electric field
to accelerate these ions into solid electrodes, trapping the particles by chemisorption.
As such, no material physically leaves the system, so the pumps can only be operated
below ∼ 10−5 mbar to have a reasonable lifetime. The effective pumping speed to thevacuum chamber is only around 20 l s−1 , which is limited by the conductance between
the pump and the main chamber, so an increase in pump size would have little effect.
In order to reach a pressure at which the ion pumps can operate we use an Oerlikon
Leybold Turbovac TW70 H turbomolecular (turbo) pump. The turbo pump itself only31

2.1 Vacuum system
TCUDORP  LANOITACUDE  KSEDOTUA  NA  YB  DECUDORP

c

g
a

f

b

d

i

h
e

Figure 2.4: An isometric projection of the vacuum system. a Ion pumps. b Titanium sublimation pump. c Turbomolecular pump. d MOT cell. e Science cell. f Atom sources. g All-metalright-angle valve. h Bayard-Alpert pressure gauge. i ‘Cube’ with windows.
TCUDORP  LANOITACUDE  KSEDOTUA  NA  YB  DECUDORP

operates with a maximum forevacuum of ∼ 20 mbar, so it is backed by an Oerlikon
Leybold SC 5 D scroll pump. The main advantage of using a scroll pump as a roughingpump is that it is oil free (unlike most alternatives), so there is no chance of oil feedingback into the system which would prevent the attainment of UHV. The turbo pump
can itself be used to reach ∼ 10−9 mbar, well below the pressure required for ion pump
operation, and is operated during baking to reduce the load on the ion pumps whilepressures in the system are still relatively high. Once the turbo pump has completed
its job we use the VAT all-metal angle valve to close off the vacuum chamber and we
may detach the turbo pump and scroll pump. The ion pumps are supplemented byoccasional use of a Varian 916-0050 series titanium sublimation (Ti sub) pump. This
operates by forming a reactive coating on the walls of the vacuum chamber, effectively
removing molecules such as H2 which are less readily dealt with by the ion pumps.
We apply a current of 47 A for ninety seconds roughly twice a year to replenish thedeposited titanium.
32

TCUDORP  LANOITACUDE  KSEDOTUA  NA  YB  DECUDORPTCUDORP  LANOITACUDE  KSEDOTUA  NA  YB  DECUDORPa

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

2.1 Vacuum system

92
56
44

PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

  110

  38

Figure 2.5: Annotated diagram of the MOT cell with viewports attached. Some importantdimensions are shown (in mm), such as the 92 mm total clearance required for the transportcoils.

MOT cell
PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCT

Vacuum chambers for laser cooling require optical access from several angles, and therefore typically fall into one of two broad categories: an all-glass chamber (as used in thetrial setup - see Appendix B), or a steel body with viewports bolted on. We took thelatter path for the sake of robustness and the fact that viewports can easily be antireflection (AR) coated on both surfaces, while typically all-glass chambers can only becoated on the outer surface (once the manufacturing process is complete). We chose touse a standard vacuum flange size of 70 mm diameter for the kodial viewports, whichhave 38 mm clear window diameter to allow for large MOT trapping beams.
At the point of designing the MOT cell we were yet to settle on the exact configuration of the complete system, most importantly the method of transportation of theatoms to the science cell. We had decided to implement magnetic transportation, but
this may be done in two ways: Either with a series of coil pairs along the transportation
axis, ramping up each pair in turn such that the magnetic minimum travels smoothly
along the system [40], or by using a single coil pair mounted on a translation stage
such that the magnetic minimum travels with the trapping coils as they are physically
moved [41]. The design therefore had to cater to either eventuality. In either case a low
vertical profile is advantageous, so the cell was designed to have a main body heightof 44 mm as shown in Fig. 2.5, just slightly larger than the viewable diameter of thewindows. This allowed enough margin for tubes to be welded to the main body, extending out to CF flanges for attaching the horizontal viewports. Added to this 44 mm
33

2.1 Vacuum system

Figure 2.6: Constructed MOT cell with the atom sources in the background. The cell isdisplayed with some of the supports used to mount the chamber during construction which were
subsequently replaced by a single, less obstructive support to allow room for the translation
stage and transport coils.

was the minimum flange height to accommodate threaded holes to securely attach thevertical windows, resulting in a total chamber height of ∼92 mm once the height of the
securing bolt heads are taken into account. This limits the trapping coil separation if
the translation method is used, while the possibility of a series of coils is catered for byallowing the first pair (with inner diameter greater than 70 mm) to nestle against the
MOT cell body with only 44 mm separation.
In the end we opted for the translation method for atom transportation as describedin §2.5. The finished constructed MOT cell is shown in Fig. 2.6. It allows access forthe three orthogonal pairs of beams in a standard MOT configuration with a coupleof spare viewports for monitoring fluorescence from the cell, a connection to the atom
sources and a narrow connection along the transportation axis to reduce conductanceto the rest of the system.
Science cellAt the science cell the requirements are slightly different as we want to be able totake high resolution images of the atoms requiring close-proximity lenses, while also
allowing for access for further lasers (e.g. the optical dipole trap) and surrounding
magnetic coils. Due to the low profile and optical access that this requires, we decided
to buy a high-quality glass cell from Starna Scientific as shown in Fig. 2.7. The cell has
a 30 × 30 mm square outer cross-section, which allows a reasonable numerical apertureto the lens systems for both horizontal and vertical imaging, and is 110 mm long with
5 mm thick quartz walls which are AR coated on the outside (as mentioned above, the34

2.1 Vacuum system

Figure 2.7: The science cell shown with its connection to the rest of the vacuum system.

manufacturing technique precludes AR coating of the inner surface). A glass-metal
connection extends to a small (34 mm) CF flange for connection to the rest of thevacuum system.

Atom sources
While we are trying to create such low-pressure conditions in the vacuum system, we
also need a supply of our intended atomic species to study. As we have to bake thesystem to a temperature well above the boiling points of both rubidium and potassium
to achieve UHV we cannot just leave a lump of material in the system. One solution
is to intern a sample in a glass ampule, so that it is contained during baking, then to
break the ampule once we are back at room temperature. The release of atoms into
the system can then be controlled to some degree by heating or cooling the sample. A
more controllable method is to use commercially available vapour sources which rely ona reaction which is activated at high temperatures (> 300◦ C), above the required bakeout temperature. We chose to use sources from Alvatec, a company now specificallycatering to cold atom experimentalists. These particular sources are based on an alloyof the alkali metal and have an indium seal which allows exposure to the atmosphere
during construction of the vacuum system without detrimental effects. The indium
seal melts at ∼ 160◦ C, after which the sources may be operated by passing a current
through the material causing only the alkali metal to sublimate. In order to be able
to pass the required current of around 4-5 A through the sources we spot weld themto contacts on a UHV electrical feedthrough, allowing connection to an external power
supply. When a current is applied the temperature rises in the sources, eventually
releasing atoms to coat the walls of the vacuum chamber.
35

2.1 Vacuum system

IsotopeNatural abundance (%)

39 K

93.26

40 K

0.012

41 K

6.73

85 Rb

72.17

87 Rb

27.83

Table 2.1: Natural abundances of the isotopes of potassium and rubidium.

A major advantage of using rubidium and potassium for cold atoms experiments
is that we may rely on the background pressure from the resultant coating to directly
load a MOT. This is due to their relatively high vapour pressures at room temperatureof around 4 × 10−7 mbar for rubidium, 2 × 10−8 mbar for potassium, compared with,for example, sodium for which it is about 4 × 10−11 mbar [42]. To reach an acceptablepressure to load a MOT, sodium needs substantial heating to temperatures where theBoltzmann distribution of velocities requires that only a small fraction of atoms willbe travelling slowly enough to be captured. This then demands an extra atom coolingstage1 , increasing the complexity of the experiment.
It is usually enough for day-to-day operation in the lab simply to activate an atom
source once every twenty-four hours to keep the pressure up, but for more criticalexperiments we can use shorter pulses each experimental cycle to keep the atom supply
as steady as possible over the course of a day. We actually have two sets2 of four different
sources in the experiment - natural abundance rubidium and potassium sources andenriched 87 Rb and 40 K versions. The natural abundances are shown in Table 2.1,
illustrating another advantage of studying 39 K (93.26% abundant). It is clear that it isessential to use enriched sources for studying 40 K, those in our system having a > 10%
abundance in preparation for the possible future work with fermions. It is not essential
to enrich rubidium, but it is relatively cheap in terms of extra expenditure to potential
gain in system performance given that the enriched sources have a 98% abundance of
87 Rb, and that MOT performance scales with the ratio of partial pressures of the atom

we are trapping to residual gases.
With these four different sources we can in principle have reasonable control over
the relative pressures of each species in the MOT chamber. It is not quite straightforward as the sources coat each other, meaning that an increase in pressure is seen
1
Typically a Zeeman slower is used, which relies upon the combination of the Doppler and Zeeman
effects to keep a beam of decelerating atoms in resonance with a laser acting at a constant frequency.
This generally adds a meter of vacuum chamber and coils to the system, along with an oven to heat
the atoms.
2
A word of caution - we mounted one set such that the sources are pointing downwards without
noting the recommendation that this not be done in the Alvatec information manual. This is the likely
cause of faulty operation of some of our sources.

362.1 Vacuum system

for all species whenever a single source is fired, but repeated firing of a single source
does result in the expected alteration in composition of gas in the MOT cell. Eachsource contains around 50 mg of its respective alkali, enough to coat the MOT cell ina monolayer several thousand times over, but the lifetime of a source depends on thepumping rate from the MOT cell. We are yet to exhaust a source after two years’operation.

Pressure Gauge
To monitor the pressure while pumping down the system we have included a BayardAlpert style ion gauge from Vacom. This works very similarly to a very low speed
ion pump, monitoring the ion current to gauge pressure. The current is highly species
dependent however, and so the gauge is only useful as a rough guide, though it is quoted
to work down to 3 × 10−11 mbar. Due to our chosen positioning for the gauge (see Fig.
2.4) we can only operate it while the system is not in use due to the obstruction causedby connecting the controller cable. The rest of the time we can use the ion pumpcontrollers to give us a rough estimate of the pressure in the system.

Cube
The main curiosity-piece of our design is the central ‘cube’ with connected windows(see Fig. 2.4 i). This was included mainly to enable the imaging of atoms part-way
down the system for diagnostic purposes. It was also considered that we might initiate
evaporative cooling here with additional magnetic coils to increase the trap gradient
before continuing to the science cell. In the end we have not really taken advantage ofit, though this owes something to the fortunate lack of major problems with the vacuum
system. If we were to make a new system based on this design it would be tempting
to leave the cube out to simplify the alignment of the system along the transportation
axis.

2.1.3

Assembly and bake-out

Our prior knowledge of UHV systems was thin on the ground at the outset of the
project and so much of the procedure for the cleaning, assembly and baking of thechamber was acquired from discussion with members of the extended AMOP group,
PhD theses and in fact anyone who would care to give advice in the department at large.
A particularly thorough source worth mentioning is the PhD thesis of Kevin Birnbaum[43] and additional associated information available online from the Quantum Optics
Group at Caltech. We also experienced a steep learning curve with the construction of
the trial system before setting to work on the main event.
37

2.1 Vacuum system

Cleaning
Briefly then, most vacuum parts were cleaned prior to construction of the vacuum
chamber, either by hand with acetone soaked lens paper or by immersion in acetone inan ultrasonic bath where possible, handling components with clean nitrile gloves at all
times. These items were then rinsed with isopropanol to remove any acetone residue
and wrapped in clean aluminium foil once dry to await incorporation into the vacuum
system. Exceptions were the right-angled valve and ion pumps which were left as we
received them, and the viewports and science cell which were cleaned with methanol
and isopropanol to avoid damage to the AR coatings.
Assembly
Smooth assembly of the system was aided by laying out all necessary parts prior toconstruction, i.e. nuts, washers, bolts, spanners, copper gaskets, vacuum components,
and additional lens paper and methanol for last-minute cleaning of the CF flanges.
The sealing mechanism involves a sharp knife edge on each of the pair of flanges thatare being joined together. A copper gasket is placed in between, and as the bolts aretightened the knife edges bite into the copper in a complete ring to form a leak-tightseal. A final clean and close inspection of the knife-edges is essential given that this iswhere the seal is made.
In order to allow deconstruction of the system in the future it is important tolubricate the bolts to avoid seizure after baking. This can be a messy business if using
a liquid lubricant which could accidentally end up inside the system, so we have usedself-lubricating silver-coated bolts throughout to avoid the hassle. When forming a
joint each bolt was made finger-tight before using spanners to gradually tighten thebolts in a star-shaped pattern to ensure an even seal. Once complete the flange faces
should not quite be flush, allowing possible adjustment of the seal if a leak becomesapparent. The assembled vacuum system is shown in Fig. 2.8.
Bake-out
Even with the most stringent solvent-based cleaning of vacuum parts it is impossible
to reach UHV without baking. This is because in the UHV regime the majority ofparticles in the system are adsorbed to the walls of the vacuum chamber rather thantravelling between walls. The contaminants on the walls desorb at a high enough rate
to limit the pressure in the chamber but too slowly to be effectively pumped away
at room temperature. A dynamic model of desorption rate [44] based on adsorbed
molecules requiring an activation energy to leave the surface of the vacuum chamber(analogous to vapour pressure as described by the Clausius-Calpeyron equation) predicts an adsorption lifetime proportional to exp(HA /RT ), where HA is the enthalpy ofadsorption and R is the gas constant. This is inversely proportional to pressure in the38

2.1 Vacuum system

Figure 2.8: Assembled system before bake-out. The turbo pump is removed once the bake-out
is over.

molecular-flow regime, allowing us to relate temperature and pressure via the equation

 
∆HA
1
ln(P ) = −
+c .
(2.5)
R
T
If there is a fixed amount of material in the system we may increase the temperature toincrease the pressure, and with each order of magnitude increase in pressure the pumping rate effectively increases by an order of magnitude (most pumps have a relatively
constant pumping speed in terms of volume).
Prior to baking the scroll and turbo pumps are turned on for the initial evacuationof the system. To prepare for the baking (Fig. 2.9) we first cover the science cell with aprotective aluminium tube and the viewports with aluminium foil ensuring there is no
direct contact with the metal. This is to avoid direct local heating of glass which could
cause damage to the AR coatings or possibly even a leak. We then attach thermocouples
at key points around the system to monitor temperatures as it is important to avoid
large temperature gradients (thirteen thermocouples were used for our bake-out). To
allow independent control of heating rates for different sections of the system we wrap
several heating tapes (PTFE coated resistance wires) around the system and attacheach to a variable AC autotransformer (variac) to control the voltage applied to thewires. The heater tapes used were mainly Isopad TeMS 6 tapes, supplying a maximum
heating power of 150 W m−1 . The ion pumps came with their own heating elements
which can be bolted directly to their casings. Insulating the system well is vital toreach the desired baking temperature, so we first wrap the system in glass fibre tapes
and blankets before adding several layers of aluminium foil.
Once the system has been fully prepared we can start to turn up the voltages on39

2.1 Vacuum system

a

b

c

d

Figure 2.9: Preparing the system for baking. a The science cell is sheathed by an aluminium
tube and the viewports are wrapped carefully in aluminium foil to avoid direct heating ofglass components. Thermocouples are attached at strategic points to monitor the temperature
throughout the system. b Several heating tapes are wrapped around the system enabling
independent control of heating rates in different regions. c Insulation tape and blankets arewrapped around the system. d Final additional insulation is provided by layered aluminium
foil.

the variacs while monitoring temperatures throughout the system, aiming to keeping
the rate of increase in temperature below the recommended limit of 1 ◦ C per minute.
It should be noted that the turbo pump should not exceed ∼ 80 ◦ C, so we try to keep
as gentle a gradient between it and the rest of the system as possible while adheringto this criterion. We do not turn the ion pumps on until we have passed the point at
which the indium seals of the atom sources have melted at ∼ 160 ◦ C due to the sudden
increase in pressure in the system at this point. Once we are sure we are past this point
and the pressure has reached a reasonable level once more below 1 × 10−6 mbar we turnthe ion pumps on to increase the overall pumping speed.
The first two bake-out attempts of the system were prematurely halted by broken
heater tapes, most likely due to excessive local heating breaking a connection somewhere along the tape. As this would lead to a local cold spot we were forced to graduallybring the temperature in the rest of the system back down to room temperature before
stripping away the insulation to replace the faulty part. Our third attempt went far
more smoothly, and we brought the temperature of the system up to a steady temperature of 220 ◦ C. All the components in the system were quoted as being bakeable to250 ◦ C, so we went as far as we were prepared to risk whilst allowing a safety margin.
The baking period was determined by observing a plateau of the system pressure whenheld at the constant temperature. During the bake-out period we occasionally fired theTi sub pump to see if it would help matters, and ran a low current through the atom
sources to help degas them and make sure they were at a higher temperature than40

2.1 Vacuum system

-13
-14
-15
-16

Ln(P/mbar)

-17
-18
-19
-20
-21
-22
-23
-24
-25
0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034

1/T (1/K)
Figure 2.10: Plot of how the pressure recorded by the Bayard-Alpert gauge varied during
the final bake-out of the system. The black points represent the ramping up of temperature in
the system (here measured by the thermocouple closest to the pressure gauge) and holding thetemperature, while the red points represent the cooling period after baking. Spikes in pressure
coincide with application of the Ti sub pump which causes temporary local outgassing and aconsequent increase in pressure near the gauge. There are clear steps down in pressure where
the temperature was held constant for a long period signifying successful baking. The gradientof the red points gives a ∆HA of around 40 kJ mol−1 .

the surrounding system so as not to localise dirt on the sources. Once the recordedpressure had not changed for a couple of days we began to ease the system back toroom temperature. The resultant plot of this final bake-out is shown in Fig. 2.10. It is
not quite as dramatic as perhaps it might be given that during our two aborted baking
attempts we had already pumped a great deal of material out of the system, so we
appear to return to a similar pressure as we started, but this is around the limit of thepressure gauge so cannot fully be trusted. The suggestion is though that at the gauge
we have reached a pressure in the 10−11 mbar range. The relation of equation (2.5) isobserved to hold reasonably well while decreasing the temperature as expected, withthe measured gradient giving a ∆HA of roughly 40 kJ mol−1 which is fairly typical ofsuch systems [45]. Once a fully operational UHV chamber with atom sources has beencreated, we need means of adding light and magnetic fields in order to manipulate theatoms, as will be discussed in the following sections.
41

2.2 Magnetic coils

2.2

Magnetic coils

It is probably already clear that magnetic fields play a vital role in the cooling andtrapping of atoms. They are required for laser cooling in the MOT, transport to thescience cell and for trapping the atoms during the first stages of evaporative cooling, as
well as for accessing Feshbach resonances, providing guide fields for imaging and opticalpumping of atoms and simply to cancel out the effects of the background magnetic fieldin the lab. The simplest way to produce magnetic fields in a controllable manner is
to pass a current through a wire, and generally the easiest shape to work with (bothfor designing and winding magnetic coils) is a circle. The vast majority of coils in thesystem consist of sets of circular loops, the key aspects of which are discussed below.
Magnetic field from a circular loop
The magnetic field B produced at a position r by passing a steady current I throughan element of wire dl may be calculated from the Biot-Savart law
dB =

µ0 I
dl × r ,
4πr3

(2.6)

where µ0 is the permeability of free space. For a circular loop of radius a we may write
dl = (−a sin θ, a cos θ, 0) dθ(2.7)

r = (a cos θ − x, a sin θ − y, z)(2.8)

where we have defined the origin r = 0 to be at the centre of the loop and the axisof the circle is in the z direction. B is then found by integrating (2.6) around the full
loop. This may be found analytically in the on-axis case where, by symmetry, the x
and y components of the magnetic field Bx and By are zero, while the z component isfound to be
Bz =

µ0 I
a2
.
2 (a2 + z 2 )3/2

(2.9)

We can also infer some information about By and Bz close to the axis from Gauss’ law
for magnetism, ∇ · B = 0, and the fact that by symmetry dBx /dx must be equal to
dBy /dy on axis, such that
2

dBy
dBz
dBx
=2
=−
= B0 .
dxdydz(2.10)

Bias fields
The guide, compensation and Feshbach fields mentioned above should ideally be uniform over the experimental region. A single coil cannot easily provide this, but thefield gradients (and in fact all odd terms in a Taylor expansion for the field) at thecentre-point of two identical co-axial coils with the same current circulation will cancel.
422.2 Magnetic coils

This creates a field that is uniform below second order, which is also true along x andy via (2.10). The bias field produced at the centre of a coil pair separated by a distanced is simply twice (2.9) with z = d/2. It is convenient to work with B in gauss, I in
amps and distances in centimetres, giving
Bbias =a2
4πIG.
10 (a2 + d2 /4)3/2(2.11)

The curvature of the field can be found from differentiating (2.9) twice with respect toz and setting z = d/2. It may be shown that it disappears for d = a such that thefield is uniform to below fourth order expansion terms. This is known as the Helmholtz
configuration, which we have tried to use for our bias coils where possible.
Quadrupole fields
Reversing the direction of current flow in one of the pair of coils will instead produce
a quadrupole field as required by the MOT and for simple magnetic trapping, whereB = 0 at the centre (as now all the even expansion terms cancel). B 0 from (2.10) and(2.9) evaluated at z = d/2 is then the total gradient along the ‘weak’ x and y axes
(twice the x and y gradient of a single coil), while −2B 0 gives the total gradient alongthe ‘strong’ z axis. When again converted to gauss with distances in centimetres this
gives
B0 =

3πI
a2 d
G cm−1 .
10 (a2 + d2 /4)5/2

Analogously to the Helmholtz configuration above, for a separation d =(2.12)√

3a the thirdorder term vanishes at the coil-pair centre, giving the most linear field gradient. This
arrangement is known as the anti-Helmholtz configuration. Also useful for coil design√is the fact that B 0 is greatest for d = a for fixed a and for a = d/ 6 for fixed d.
In practice we use coils made up of several turns each, so one may sum the above
expressions over various a and d to estimate the resultant fields. Further considerationsof coil design include the coil inductance, power dissipation (which manifests itself as
thermal energy) and the consequent water-cooling required.

2.2.1

Coils around the MOT cell

The properties of the coils located around the MOT cell are summarised in Table 2.2.
The requirements are fairly minimal as only laser cooling is carried out at this endof the system. Three pairs of compensation coils (providing tuneable bias fields along
three axes) are used to cancel out the background magnetic field of the order of 1 G,
while the transport coils provide the small field gradient required for magneto-opticaltrapping. The compensation coils are essential to minimise the magnetic field at theatoms during the optical molasses stage (see §2.3.1). One of the horizontal pairs is also
used to provide a guide field for optical pumping (§2.4.3).
43

2.3 Laser cooling

CoilTransport
Comp. (Vert.)
Comp. (Horiz.)

Wire
(mm)
4.0
1.0
1.0

Turns
10×4
3×3
5×8

Dimensions
(cm)
5.1
9.0
7.2

Separation
(cm)
9.8
4.4
20.0

B0 /I
(G/A)

B 0 /I(G/(cm A))
0.40

1.73
0.55

Table 2.2: Parameters of the coils used around the MOT cell. The number of turns are givenin the form Nradial × Naxial , while the given dimensions and separations refer to clear innerdistances. The compensation fields have not been verified by direct measurement.

A current of up to 200 A can be supplied to the transport coils by a Delta Elektronika
power supply for magnetic trapping, producing a maximum field gradient along the
weak radial direction of B 0 = 80 G cm−1 . Such high currents naturally generate a lotof heat, so the coils are made from hollow tubing to allow water cooling. Fail-safes areincluded to stop the coils over-heating; a flow-meter detects whether water is flowing
through the water-cooling tubes and thermistors detect if the coils are over-heating.
The flow-meter is connected to the power-supply itself, setting the output to zero if thewater flow drops below a certain level. Voltage comparators are used in conjunction
with the thermistors to provide a signal to break the electrical connection of the coils
via an insulated gate bipolar transistor (IGBT) if the coils reach above ∼ 60◦ C. The
IGBT is also used for switching of the coils during the experimental sequence. Similarsafety precautions are taken with all the other coils in which high currents can flow.

2.2.2

Coils around the science cell

The properties of the coils located around the science cell are summarised in Table 2.3.
The transport coils also operate in this region, while another three pairs of compensation coils are used to cancel the local background fields at this end of the system.
In addition, two pairs of bias coils are wound on top of the (rectangular) horizontal
compensation coils. The coil pair for providing large bias fields to utilise the Feshbachresonance at 402.5 G is wound from the same hollow copper tubing as the transportcoils to allow water cooling. The coils produce a bias field of 528 G when a current of
200 A is flowing through them. Finally, a small Ioffe coil is used in the formation of
the QUIC trap described in §2.6. It is mounted on a water-cooled copper block. The
layout of the coils around the science cell is shown in Fig. 2.11.

2.3

Laser cooling

The cooling of atoms by lasers relies on atoms absorbing and re-emitting photons. A
simple illustration of this concept is given in Fig. 2.12 for an atom travelling towards
a laser source. Each time the atom absorbs a photon its momentum is reduced. The
subsequent emission of photons occurs in a regular dipole pattern such that these44

2.3 Laser cooling

CoilTransport
Feshbach
Bias (Axial)
Bias (Trans.)
Comp. (Vert.)
Comp. (Axial)
Comp. (Trans.)
Ioffe

Wire
(mm)
4.0
4.0
1.0
1.0
1.0
0.7
0.7
1.0

Turns
10×4
7×2
10
10
7×3
8
8
6×3

Dimensions
(cm)
5.1
7.0
7.4×14.9
6.2×15.2
19.0
7.4×14.9
6.2×15.2
1.0

Separation
(cm)
9.8
3.6
5.6
5.6
7.0
5.6
5.6
1.7∗

B0 /I
(G/A)

B 0 /I(G/(cm A))
0.40

2.64
1.43
1.46
2.28
1.14
1.14

Table 2.3: Parameters of the coils used around the science cell. The number of turns are either
given as a total or in the form Nradial × Naxial . Dimensions (as a diameter for circular coils andas side lengths for rectangular coils) and separations refer to clear inner distances. ∗ ‘Separation’here refers to the distance from the face of the Ioffe coil to the centre of the quadrupole field.

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

Figure 2.11: Layout of the magnetic coils around the science cell. In grey are the transportcoils, black the Feshbach coils and in white the Ioffe coil. The copper coloured coils are the bias
PRODUCED  BY  AN  AUTODESK  EDUCATIONAL  PRODUCTand compensation coils (the rectangular coils each contain two sets of coils, wound around thesame groove of the coil mount).

45

2.3 Laser cooling

(a)

a

p

ħk
-ħk

Atom
Photon(b)p - ħkb(c)p - Nħk

cFigure 2.12: Simplistic view of laser cooling. a A photon with momentum −~k is incident
upon an atom travelling with momentum p. b If the photon is absorbed the momentum of theatom is reduced. c After absorbing N such photons the momentum of the atom is reduced top − N ~k as the momentum kicks from photon re-emission cancel.

a
(a)

b
(b)
ω
Atom

ω

ω-- = ω - kv

ω++ = ω + kv

Atom

v

v
Figure 2.13: The Doppler effect in the case of counter-propagating beams. a The frequencyof the light in the lab frame is ω. b In the rest frame of the atom the frequencies are shifted by±kv.

momentum kicks tend to cancel one another out. Over many cycles the atom is slowed
down, which may be used to reduce the spread of atom velocities for a group of atoms,
cooling them down. This is the effect of the scattering force from near-resonant light
introduced in §1.4.3.
The key point is that to cool an atomic cloud each atom must mainly absorb photons
with momenta in opposition to its direction of motion, but the velocities of atoms in a
gas are randomly distributed. A method of coping with this problem was proposed byHänsch and Schawlow in 1975 [46], now known as optical molasses.

2.3.1

Optical Molasses

Initially considering the one-dimensional case, a counter-propagating beam must be
introduced to allow the possibility of atoms receiving momentum kicks in the opposite
direction. Some mechanism is then required by which atoms will preferentially absorb
photons from the beam that opposes its motion. This is achieved by the neat trickof detuning the beam frequency slightly to the red of resonance. The Doppler shiftthen acts to bring the deceleration beam back towards resonance while shifting theacceleration beam further off resonance, providing a net damping force. Atoms close
to zero velocity are equally off-resonant with both beams and therefore feel no further
force. Here we are once more considering the ideal two-level picture. We may describe46

2.3 Laser cooling

this in terms of the scattering force Fsc of equation (1.67) by replacing δ with δ ± kv =
ω − ω0 ± kv to obtain
Fmol = Fsc (ω − ω0 − kv) − Fsc (ω − ω0 + kv)
∂Fsc
kv
∂ω
' −αv

' −2

(2.13)(2.14)(2.15)

for kv . Γ, where the damping coefficient α is given by
α = −4~k 2(I/Is )(2δ/Γ)[1 + 2I/Is + (2δ/Γ)2 ]2.

(2.16)The extra factor of 2 in intensity in the denominator of α arises because saturationmust be considered in terms of both beams. This is valid for v . 4 m s−1 for potassium
and rubidium. α is positive for red-detuned beams so the resultant force does indeed
always oppose the motion of the atom. This is equivalent to motion in a viscous fluid(such as molasses).

2.3.2

Doppler limit and sub-Doppler coolingThis picture may be extended by the addition of two further sets of counter-propagating
beams along orthogonal axes to provide strong damping in all three dimensions. Thismight be expected to lead directly to temperatures approaching absolute zero. However,
the idea that the scattering via spontaneous emission in Fig. 2.12 c cancels to zero is
not true for individual atoms, and instead they are heated by a random walk in velocity
space where each step is a single recoil velocity vr = ~k/m. The atoms also undergo
a random walk via absorption (along the paths of the beams) once they are at low
velocity. The balance of cooling and heating leads to a limit on the temperature that
it is possible to reach in this picture of optical molasses, which is minimised for afrequency detuning δ = Γ/2. This is known as the Doppler limit [47] and is given by
kB TD =

~Γ.
2

(2.17)

TD is 144 µ K for our atomic species.
This technique was first successfully implemented in 1985 by Chu et al. [48], but curiously it was observed not long afterwards that for larger frequency detunings the temperatures reached were well below the Doppler limit [49]. This remarkable phenomenon
was explained by Dalibard and Cohen-Tannoudji [50] and relies on the multi-level nature of the atoms. The polarisation of the electric field produced by counter-propagating
beams with opposite circular polarisations varies with period λ/2. The Zeeman sublevels of the ground state are perturbed by the electric field by differing amounts that
depend on the polarisation of the field, resulting in energy levels that oscillate in space47

2.3 Laser cooling
2.3 Laser cooling

✻

0

λ/4

λ/2

3λ/4

|e�

λ

a

E

b

c

|e�

✻

|g1 �
|g2 �

E
|g1 �

✻
Ea

|g2 �

d

✻
Ea

Figure 2.13: Caption.

Figure2.13: Caption.
Figure 2.14: Simplified depictionof the mechanism
behind sub-Doppler Sisyphus cooling. a
Length scale in terms of the wavelength λ of the optical molasses beams. b The variation ofpolarisation through space produced by two counter-propagating beams. c Relevant energy
levels, including the splitting of the degenerate ground state which is polarisation dependent.
d The energy Ea of an atom as it travels from left to right. The arrows in c show how an atom
in one of the ground states loses kinetic energy by travelling to a position of higher potential
energy, at which point it is pumped to a ground state at lower energy.

E 2.14). As an atom moves in space it loses
withma phase offset between them (see Fig.
m�

kinetic energy as it travels up the potential hill. Near the peak the circular polarisation−1+1of the light near this point pumps the atom into a sublevel of lower energy. Due tothe short excited-state lifetime the position of the atom is essentially constant during00

this process. The energy loss is accounted
δ for by the difference in energy between theabsorbed−1+1 and emitted photon. Now in a different substate, the process is repeatedwith sublevels and circular polarisation reversed at a position λ/4 further along. This

process hastoω become
ω0 known by the term ‘Sisyphus cooling’ due to the resemblance
ω0+−

the Greek myth of the king σpunished to forever push a boulderσuphill only to watch itroll back down again.
The procedure is repeated until the energy of each atom is comparable to the lightshift amplitude of the energy levels1 , resulting in temperatures of tens of µK, around1
A more rigorous treatment given in [51] suggests the limit is around 40 Er , when the amplitudeof the lower-state light shift is ≈ 100 Er , where Er = mvr2 /2 = ~2 k2 /2m is the recoil energy (it is notpossible to reach arbitrarily low temperatures by simply
reducing the laser inensity).
0
B = B�r48
Figure2.14:PLACEHOLDER
PLACEHOLDER
FIGURE.
Figure
2.14:
FIGURE.

2.3 Laser coolingan order of magnitude lower than the Doppler temperature. The success of this process
was a major step towards cooling atoms to quantum degeneracy. The only downsideis that this technique alone does not actually trap atoms due to the lack of a restoring
force. This was rectified by the development of the magneto-optical trap - a concept
suggested by Dalibard [52] and first implemented at Bell Laboratories, USA [53].

2.3.3

Magneto-optical trap

A weak magnetic field may be used to split the atomic energy states via the Zeeman
effect of equation (1.36) to create an imbalance in the radiation force from counterpropagating beams that is now dependent on position. The beams must have opposite
circular polarisations so that they interact with different electronic transitions such thatan increase in magnetic field strength takes one beam closer to resonance, the otherfurther away. If a quadrupole field is used the resultant linear energy level splitting
allows us to use a very similar treatment to that of Doppler cooling described above.
This principle is illustrated in Fig. 2.15. Replacing δ in equation (1.67) now by δ ±(kv + βz) = ω − ω0 ± (kv + βz), where β~z = (g 0 m0 − gm)µB B 0 z is the energy shift ofa transition resonance, directly leads to the one-dimensional scenario where
FMOT ' −αv −

αβ
z,
k

(2.18)

valid for βz . Γ. This can once again be extended to three dimensions by careful choiceof the beam polarisations of the three beam pairs as shown in Fig. 2.16. The MOT isa very robust trap and can capture atoms with significantly higher velocity than are
slowed by the optical molasses technique, allowing us to load the trap directly from aroom temperature vapour for potassium and rubidium.

2.3.4

Laser cooling for real atoms

This outline of the mechanisms involved in laser cooling condenses many years of dedicated research into a few pages, and it is worth noting the huge significance of thesemajor advances. This was recognised by Steven Chu, Bill Phillips and Claude CohenTannoudji being awarded the 1997 Nobel prize “for development of methods to cooland trap atoms with laser light”. A MOT in some form is used in nearly every cold
atoms experiment in the world.
The above arguments are most accurate for a two-level system, but in reality we haveall of the hyperfine levels to deal with. However, atoms excited to |F 0 =3i decay only
to |F =2i, so using light near the |F =2i → |F 0 =3i transition causes the atoms to cycle
between these two states. If an atom happens to fall out of this cycling (or ‘cooling’)
transition we also require light near the |F =1i → |F 0 =2i ‘repumping’ transition toeject the atom from |F =1i.
87 Rb nicely adheres to this picture with its large hyperfine splittings, where the49Figure 2.13: Caption.

2.3 Laser cooling

E
�

m

m

−1

+1

0

0
δ
−1

+1
ω ω0

σ−

σ+

0

ω0

B = B�r

0

0

Figure 2.15: Simple
scheme fora magneto-opticaltrap for aFIGURE.
singlet ground state and triplet
Figure2.14:
PLACEHOLDER
excited state. As the atoms move away from the centre the Zeeman shift tunes them towards
resonance with the restoring beam and out of resonance with the opposite beam producing a
net restoring force. σ ± polarised light works on the ∆m = ±1 transition.

42
σ+
I
σ−

σ+
B
σ+

σ−

σ−
I

Figure 2.16: Extension of the MOT principle to three dimensions. The reversal of the field
direction from the radial direction to the axial direction (with respect to the quadrupole coils)
requires that the beam polarisations also be reversed. The beams are red-detuned from thecycling transition in order to both cool and trap the atoms. In a real system some repumping
light must also be present to avoid pumping the atoms into a dark state.

502.4 Laser systemforce on the atoms is provided by the cooling light, while just a little repumping lightis needed to compensate for the low decay rate to the lower ground state. As such therepumping light can just be sent in as a single separate low-intensity beam while thecooling light must have the signature six-beam geometry.
Conversely, the small splitting of the 39 K F states means that if we detune thecooling beam as required for laser cooling we actually bring it closer to resonance with
the non-cycling transitions. Pumping into the |F = 1i state occurs at a rapid rate. The
repumping light must therefore be about as intense as the cooling light, be detuned
from the repumping transition and must be counter-propagating as it now provides aconsiderable amount of the force. ‘Cooling’ and ‘repumping’ in this scenario start tolose meaning, but will continue to be used for descriptive convenience. In fact it has
been neatly demonstrated that a 39 K MOT may be formed by using only cooling lighton the horizontal axes and only repumping light on the vertical axis [54].
A more detailed discussion of the properties of the MOT setup as used in our
experiments will be given in Chapter 3. Now we know what laser frequencies are
required for laser cooling we just need a way to reliably produce them.

2.4

Laser system

It is advantageous that today the laser frequencies required for the laser cooling of
rubidium and potassium are accessible by diode lasers which are simple to maintainand operate. The basic principle relies on a high-band-width gain medium (laser diode)in combination with an external diffraction grating. The combination of the medium
gain, the diode internal cavity, the external cavity between the grating and the back of
the diode and the grating profile itself produces high gain at a single frequency witha line-width smaller than the natural line-widths of alkali atoms (Fig. 2.17). The
particular lasing mode is selected by tuning the feedback from the grating via its angle,
while we may scan the frequency continuously by modifying the length of the external
cavity via a piezoelectric transducer (piezo). There is a limit to the tuning range as
the laser will eventually ‘mode-hop’ to the adjacent external cavity mode. This modehop-free tuning range is typically up to around 10 GHz (roughly restricted by the mode
spacing of the external cavity). The temperature and laser current are used to tune
where the mode hops occur, affecting the internal cavity modes. Once the laser is tuned
near its designated frequency the piezo voltage is modulated to scan the laser frequency(used for locking the laser frequency as discussed below).
Standard diode lasers at our required frequencies produce up to ∼100 mW of power.
Given the losses inherent in AOMs, fibre-coupling, reflection off mirrors etc. this is below what we require to have enough cooling light at the MOT cell, so tapered amplifiers(TAs) are used to boost the power. These contain an amplifying chip with a tapered
gain region. The internal modes are controlled by the coupling of a low-power ‘seed’51

2.4 Laser system

Medium gain
Laser emission
Internal cavity mode
External cavity
mode

Grating profile∼ 100 GHz-

Frequency
Figure 2.17: Guide to the frequency scales of the gain profiles of the features that make up a
diode laser. The grating profile can be shifted for coarse tuning of the emitted laser frequency,
while the external and internal modes can be modified for fine tuning as discussed in the text.

laser (i.e. ∼10 mW beam from a diode laser) so the fine line-width is maintained whileallowing amplification to over 1 W of laser power.
The lasers in our system were bought with all the necessary current and temperature
regulators and piezo controls, plus the electronics for frequency locking in a feedback
loop (§2.4.2). Optical isolators are fitted in every laser which, via their main components of a Faraday rotator and a polariser, vastly suppress unwanted feedback into theoptical cavity.

2.4.1

Saturated absorption spectroscopy

The laser frequencies required for laser cooling, optical pumping and imaging of atoms
must be accurate to the MHz level, and are generally in the vicinity of the hyperfine
transitions of Fig. 1.5. A natural course of action therefore is to somehow lock thelaser frequencies directly to the hyperfine features of the relevant atomic spectra. We
can observe the absorption of light by an atomic vapour, but in order to discern the hyperfine features the thermal broadening due to the Doppler effect1 must be suppressed.
Typically this is achieved via saturated absorption spectroscopy.
A typical saturated absorption setup is shown in Fig. 2.18. The transmission ofa weak ‘probe’ beam is recorded by a photodiode and produces the expected broad
absorption signal as the frequency of the laser is scanned. When a strong counterpropagating ‘pump’ beam is introduced at the same frequency there is generally noeffect, as atoms travelling at a general velocity v (along the beam axis) will not be on
1

The thermal broadening produces a feature ∼ 1 GHz in width, making it impossible to observe theunderlying peaks of ∼ 6 MHz line-width.

52

2.4 Laser system+23!"#$%&'()**

+&$,)
+%-#

+.$/$01$0)

ȜFigure 2.18: Example of a saturated absorption spectroscopy setup. In this case a half-wave(λ/2) plate is used to adjust the relative strength of the transmitted and reflected components
from a polarising beam-splitter (PBS) cube, forming the probe and pump beams respectively.
The transmission of the probe beam through the vapour cell is recorded by a photodiode.

resonance with both beams simultaneously. However, atoms travelling at v = 0 will
be resonant with both beams when the laser frequency matches an atomic transition.
In this case the stronger pump beam promotes many of these atoms to the excitedstate and indeed may cause atoms to be pumped into a ‘dark’ ground state. The
consequent reduction in the absorption of the probe beam is observed as a sharp peakin transmission to the photodiode.
In the case of 87 Rb both the ground and excited hyperfine states are split well enoughto be easily resolved. Considering just one of the ground states, this mechanism would
lead to the observation of three peaks for, say, the |F = 2i → |F 0 = 1, 2, 3i transitions.
However, when the laser frequency is exactly half way between two transitions ω1 andω2 , atoms travelling at v = ±(ω1 − ω2 )/2k will again be resonant with both beams
simultaneously, so three strong ‘cross-over’ peaks are seen bisecting the three directtransition peaks. This is shown in Fig 2.19 a, where the black line shows the saturated
absorption spectrum while the red line shows the Doppler broadened spectrum observed
when the pump beam is blocked. The linewidth of the six observed peaks is around
16 MHz, not far off the natural lifetime broadening of ∼ 6 MHz. The mechanisms
producing the spectrum are shown schematically in Fig. 2.20.
The small hyperfine splitting in 39 K leads to a different picture. The lower state
splitting is small enough to be spanned by the Doppler broadening, while the upper
state splitting is so small that even with saturation spectroscopy it is difficult to resolve
the structure. The six peaks of the |F = 2i → |F 0 = 1, 2, 3i transition that are so clear
in 87 Rb are condensed into a single peak in 39 K, peak i of Fig. 2.19 b. Another merged
peak (iv) is seen for the |F = 1i → |F 0 = 0, 1, 2i transitions, while we also see a
deep dip in transmission between the two. This results from a similar argument tothe cross-over peaks, where now we consider the case where we have two ground states
and a single upper state. In this scenario, when the laser frequency bisects the twotransition frequencies there is again a velocity class of atoms for which both the probe
and the pump beams are resonant. However, this time the pump beam acts to pumpatoms back into the complementary ground state (much like the repumping beam in the53

2.4 Laser system

a

b

v
iv
ii

iii

i

vi

i

iv

87 Rb

39 K
ii

-300 -200 -100

0

100

200

300

-400

-200

0

iii

200

400

Frequency (MHz)Frequency (MHz)Figure 2.19: Saturated absorption spectra for the D2 transitions of 87 Rb and 39 K. The
saturated spectra are shown in black, Doppler broadened versions in red and the difference
between the two in blue. a Transitions from the F = 2 ground state of 87 Rb. Peaks i, iii andvi are direct transitions to the F 0 = 1, 2 and 3 excited states respectively, the rest being thecorresponding cross-over peaks. The x-scale shows the frequency difference in MHz from theF = 2 to unperturbed upper state transition. b Full spectrum for the D2 line of 39 K. Peaks i
and iv are the conglomerated transitions from the F = 2 and F = 1 ground states respectively.
The crossover dips ii and iii are to the F 0 = 1 and F 0 = 2 upper states respectively. To
accentuate the features the spectrum was taken using a vapour cell that was heated to 30 ◦ C,
but we have not found it necessary to heat the cell when locking our laser. The small residual
narrow features in the Doppler broadened profile are likely due to reflection of the probe beamin the system. The x-scale shows the frequency difference in MHz from the unperturbed D2
transition.

MOT discussed later), increasing the absorption of the probe beam. We can actuallyjust make out two inverted peaks in Fig. 2.19 b, which correspond to the F 0 = 1 and
F 0 = 2 upper states, being the only two states accessible to both lower states by thestandard dipole transition selection rules.

2.4.2

Frequency locking of lasers

The photodiode in the saturation absorption spectroscopy setup provides us with asignal that is either at a local maximum or minimum when we are resonant with oneof the hyperfine transitions or one of the cross-over transitions. It is then a relatively
straight-forward task to implement a feedback mechanism to lock the laser to one ofthese frequencies. We use cross-over resonances as they provide the strongest signal.
The principle relies on applying some low amplitude jitter to the laser frequency
and comparing the photodiode signal either side of the central frequency. For example,
if we are locking the laser to a peak the signal just either side of resonance will belower than the central signal. When multiplied by the wave-form of the jitter this will54

2.4 Laser system

b

c

e2

e2

e2
ω – kv

ω – kvgg
general v

ω + kve1

ω + kve1

ω + kve1

ω – kvag
v=0ω = ω2v = ±(ω1 – ω2)/2k
ω = (ω1 + ω2)/2

Figure 2.20: Mechanisms producing the features of saturated absorption spectroscopy in thecase of a single ground state and two excited states, as seen for 87 Rb. a For general atom
velocity v the pump has no effect on the probe beam and the Doppler broadened absorption isseen. b For v = 0 both beams are resonant with the same transition when the laser frequency
ω is equal to one of the transition frequencies ω1 and ω2 . Atoms absorb light from the pumpbeam that would otherwise absorb light from the probe beam and the transmission of the probe
beam is consequently increased. c For v = ±(ω1 −ω2 )/2k each beam is resonant with a different
transition when ω = (ω1 + ω2 )/2. As they share the same ground state a peak in probe beamtransmission is again observed.

a

g2

g2

g2

g1

g1

g1

general v

v=0
ω = ω1

ω – kv

ω + kv

ω – kv

ω – kvce

ω + kve
ω + kveb

v = ±(ω1 – ω2)/2k
ω = (ω1 + ω2)/2

Figure 2.21: Mechanisms producing the features of saturated absorption spectroscopy in thecase of a single excited state and two ground states, as seen for 39 K. For the cases a and b
the situation is much the same as in Fig. 2.20 a and b. c For v = ±(ω1 − ω2 )/2k each beamis resonant with a different transition when ω = (ω1 + ω2 )/2. In this scenario the pump beam
replenishes the ground state of the probe beam transition, causing an increase in absorption
such that a dip in the transmission is observed.

55

2.4 Laser systemaverage to zero over a period. If, however, we are slightly above resonance the signal
will be lower for positive jitter than for negative jitter, providing a negative signal fromthe periodic average once multiplied again by the jitter wave-form. This gives a signal
proportional to the gradient of the spectroscopy signal. This signal is fed back to thelaser and the external cavity is modified to return the frequency to resonance.
In our system we apply the high frequency jitter to the laser itself by oscillating thecavity length. As long as the amplitude is below the required line-width this doesn’t
cause a problem. A lock-in regulator extracts the slope of the signal and we can adjust
the proportional/integral/derivative (PID) components to optimise the stability of the
lock.

2.4.3

Required frequencies

The need for cooling and repumping light for the MOT in the vicinity of the |F =2i →|F 0 =3i and |F =1i → |F 0 =2i transitions, respectively, has already been discussed in§2.3.4. Elsewhere in the system we need resonant cycling light for imaging the atoms,
and pumping light resonant with the |F =2i → |F 0 =2i transition for when we want totransfer all the atoms into the |F =2,mF =2i state for magnetic trapping. ‘Resonant’
here includes a small Zeeman shift due to the presence of a uniform magnetic guidefield operated to provide a quantisation axis during imaging and pumping. The pumping is achieved by using σ + polarised light so that once the atoms have entered the|F =2,mF =2i state they will remain there.
Acousto-optical modulators
Having locked our lasers to the cross-over transitions of saturated absorption spectroscopy, we must shift the emitted light by tens and hundreds of MHz to reach thedesired final frequencies, which may be done using acousto-optical modulators (AOMs).
These consist of a crystal in which a sound wave is generated by a piezoelectric transducer. Incident laser light is Bragg diffracted by an angle θ given by sin θ = mλl /2λs ,where λl and λs are the wavelengths of light and sound respectively and m is thediffraction order. The frequency of the sound νs also Doppler shifts the diffracted
beam, altering the frequency of the light to νl0 = νl + mνs . The diffraction efficiency
into the first order can be varied from almost 0 to upwards of 90% by varying thepower of the acoustic waves. This makes AOMs additionally useful as fast switches;the diffracted light can be turned off in the time it takes a sound wave to travel thewidth of the laser beam, typically under 100 ns.
As the angle of beam deflection depends on the frequency shift of the AOM, we need
to use a double-pass configuration in order to dynamically change the laser frequency
without losing beam alignment. This involves placing an AOM at the focal point
between two collimating lenses and reflecting the diffracted beam back through the
AOM as shown in Fig. 2.22. The beam is diffracted a second time, acquiring a total
56

2.4 Laser system

νlPBSνl  +  mνsAOM

νl  +  mνs

λ/4

νl

νl  +  2mνs
f

f

Figure 2.22: AOM in double-pass configuration. Identical lenses a focal length either sideof the AOM ensure that as νs varies the return beam retains its alignment. Retro-reflection
through a quarter-wave plate rotates the polarisation so that the frequency-shifted beam may
be separated from the incident beam path by a PBS cube. Irises are used to block unwanted
diffraction orders

frequency shift of 2mνs and returning along the original beam path. This beam can beseparated from the original beam at a PBS cube by inserting a quarter-wave plate before
the retro-reflection mirror, so that the polarisation of the two beams are orthogonal.
In general the diffraction efficiency is peaked at a particular frequency that depends on
the AOM in question, such that shifting the frequency alters the power of the resultant
beam. However, using the double-pass configuration as part of the laser-lock setup
allows frequency control with no power loss from the main beam, so this method isused for most lasers in our system.
The AOMs in our system are from Crystal Technology and AA Opto-electronic
and are controlled by drivers designed and built in-house. The AOM drivers centre
on a voltage-controlled oscillator (VCO) which outputs a voltage-dependent frequency,
the power being adjusted separately by means of a voltage-variable attenuator. The
design allows frequency and power control both externally from a supplied voltage
and internally via control of chassis-mounted potentiometers. Fast-switching with a
TTL signal is also available. The radio-frequency components in the drivers are from
Mini-circuits.
Table 2.4 summarises the laser lock points (the saturated absorption features used
for frequency stabilisation) and AOM frequency shifts used to get to the final frequencies.

2.4.4

Laser table summary

The entire experiment is carried out on two separate optical tables, designated the laser
and vacuum tables. All of the lasers and AOMs are laid out on the laser table to beisolated from the vibrations caused by the translation stage. Only three actual master
lasers are used, one for potassium (where the ground-state hyperfine splitting may be
bridged using AOMs) and two for rubidium. Specifically, we use a Toptica photonics
DL Pro laser for potassium, while a DL 100 is used for the rubidium repumping lightand a TA 100 (actually a combination diode laser and TA in one box) for the rest. The
master lasers are based on the Littrow configuration.
Both the cooling and repumping light for potassium need to end up at high in57

2.4 Laser system

39

87

K

Rb

Function

Master

Locking line

Cooling

DL Pro|F =1-2i→|F 0 =1i

Repumping

DL Pro|F =1-2i→|F 0 =1i
0AOM shift (MHz)Resultant light2 × (70.5 → 95.5)|F =2i→|F 0 = 3i

−2 × (185 → 215)

−89 → +21 MHz

2 × (70.5 → 95.5)|F =1i→|F 0 = 2i

+40

−50 → 0 MHz

Imaging

DL Pro|F =1-2i→|F =1i

2 × 95 − 2 × 195|F =2i→|F 0 = 3i

Pumping

DL Pro|F =1-2i→|F 0 =1i

2 × 95 − 2 × 206|F =2i→|F 0 = 2i

Cooling

TA 100|F =2i→|F 0 =2-3i

2 × (83 → 108)

|F =2i→|F 0 = 3i

−83

−50 → 0 MHz

RepumpingDL 100|F =1i→|F 0 =1-2i+78|F =1i→|F 0 = 2i

Imaging

TA 100|F =2i→|F 0 =2-3i

2 × 100 − 67

|F =2i→|F 0 = 3i

Pumping

TA 100|F =2i→|F 0 =2-3i

2 × 75 − 83 − 200|F =2i→|F 0 = 2i

Table 2.4: Summary of the laser-lock transitions and AOM shifts used to provide the laser
frequencies used for laser cooling, imaging and optical pumping. The form “|F =1-2i→|F 0 =1i”
denotes locking to a ground-state crossover feature. Use of an AOM in double-pass configuration
results in an overall frequency shift of 2 × νs (all AOM shifts are first order). Note that where
an AOM is used in the spectroscopy setup the light that exits the laser is shifted in the opposite
direction (i.e. a negative frequency shift in the locking setup is counted as a positive frequency
shift).

tensity with the same polarisations, so after the master laser light has been split andfrequency shifted it is recombined on a PBS cube before being rotated and split in two
at another PBS cube such that we end up with two polarised beams each containing
both frequencies. One beam is used for imaging and pumping (requiring relatively little
light) while the other (∼10 mW) beam is used to seed a Sacher Lasertechnik TEC-400
tapered amplifier. Dual seeding means we can’t directly measure the relative powers at
the two frequencies, so a Fabry-Pérot interferometer is used to provide a rough gauge.
This is important as the amplification depends on how well coupled each beam is, andsmall imbalances in the coupling leads to larger imbalances in relative output power.
A total power of up to ∼1.5 W leaves the TA.
The rubidium repumping light is produced at a fixed frequency by the DL 100 witha single-pass AOM in the lock setup, the 80 mW it produces proving to be plenty. The
TA 100 has two output apertures, one for a pre-amplification 10 mW beam, the otherfor the amplified beam. The weak beam is used for locking and imaging, while theamplified beam of around 600 mW is split and shifted for cooling and pumping light.
The light is passed to the vacuum table by means of polarisation-maintaining opticalfibres. In addition to the saturated-absorption spectroscopy components and AOMs
there are several other key components which are briefly discussed below.
582.4 Laser system

Anamorphic Prism pairs
Typically diode lasers produce elliptical beams (∼3 mm × 1 mm for the Toptica master
lasers) which are unsuitable for efficient fibre coupling. Anamorphic prism pairs areused at the outputs of the diode lasers which are used to magnify or shrink the beamin one dimension resulting in a near-circular profile. The exact magnification can be
adjusted by the prism angles so can conveniently be adjusted to suit each particularlaser.

Mechanical shutters
Although fast switching of light is available using AOMs there is always a small fractionof light diffracted away from the zeroth order. Sometimes it is necessary that absolutely
no near-resonant light reaches the atoms and mechanical shutters must be used. Theseact more slowly than AOMs (depending on the shutter speed and once again on the
beam width) and must be triggered in advance so that the shutter reaches the beam at
the desired time if timing is critical. However, they have the advantage of blocking alllight. The efficiency of an AOM is unstable when turned on from ‘cold’, so it is best
to keep them on for the majority of the time. A mechanical shutter can be used toblock the light while the AOM is ‘kept warm’, allowing for higher power stability whenlight is needed once more. The shutters are controlled via TTL signals and mounted
on rubber bases to suppress mechanical vibrations that can affect the laser frequency
lock.

Dichroic wave plate
The two-species MOT requires beams containing light for both species with the samepolarisation. To achieve this we combine orthogonally polarised beams at a PBS cube
before using a dichroic wave-plate to rotate the polarisation of just the potassium light.
This wave plate, manufactured by LENS-Optics, is precisely machined such that it acts
as a 27 + 1/2 wave plate for light at 766.7 nm and a 27 wave plate for 780.2 nm light.
Subsequently only zero-order wave plates are used to keep the rubidium and potassium
polarisation matched.
The entire laser table optical configuration is displayed in Fig. 2.23, tying in all theabove component pieces discussed above. All of the optics are anti-reflection coated tominimise power loss through the system. For clarity, the central AOM frequencies arequoted because the operating frequencies change throughout the experimental sequence(whether the shift is positive or negative is indicated).
59

2.5 Magnetic transport

2.4.5

Fibre-port cluster

In order to split the cooling and repumping light into the six beams required for the
MOT we use a two- to six-way fibre-port cluster from Schäfter+Kirchhoff. This is an
enclosed cage-mounted system, providing excellent alignment stability, as required forthe coupling between the input and output optical fibres. The combined rubidium andpotassium light and the rubidium repumping light are collimated at two input couplers.
PBS cubes and rotatable zero-order half-wave plates are used to first combine andthen divide up the two beams as shown in Fig. 2.24. PBS cubes produce near-purepolarised light in transmission, but the reflected light contains a small but significant
fraction of the incorrect polarisation. Linear polarisers are therefore used to clean up
the polarisation of the three beams which undergo reflection at the last cube theyencounter.
This imperfection of the PBS cubes means it is also impossible to balance the output
beam powers with respect to both inputs, but it may be done for just one or the other.
The force-exerting light can therefore be split into three balanced pairs, while various
amounts of rubidium repumping light ends up in each beam. The light is coupled into
six fibres which transport the light to the required positions on the vacuum table.

2.4.6

Optics around the MOT cell

Each MOT beam (now containing all four frequencies) is allowed to freely expand fromthe end of an optical fibre before being collimated by a lens, producing beams 3 cm indiameter. To produce pure circular polarisation each beam then passes through linear
polarising sheet and a zero-order quarter-wave plate. All of these components are cage
mounted for stability and to be sure that the expanding beam and lens share a common
axis. Two 200 steering mirrors are used to allow adjustment of the angle and positionof each beam at the MOT cell, finally reproducing the picture of Fig. 2.16.

2.5

Magnetic transport

In order to transport the atoms to the science cell we use the tight magnetic confinementof the quadrupole field produced by the pair of transport coils that were used at low
current in the MOT. As mentioned before the coils are mounted on a Parker translation
stage to allow computer-controlled motion along the vacuum system axis. The coils areclamped tightly in place and held on a set of coil mounts that were manufactured onsite to our design.

60

61
λ/4

λ/2

PhotodiodeShutter

Lens

400

+80

λ/2

Rb repumping(science cell)

λ/2

TEC 400

Fibre coupler

AOM

Rb repumping-200

λ/2 Wave plate

λ/2

Iris

λ/2

LP Linear Polariser

75

25

TA 100

λ/4

150

125

λ/2

λ/2

750

λ/2

λ/4

150

150

λ/4

400

λ/2

λ/2@767nm
λ/2

Cooling & K
repumping-95

Pumping

Rb cell-80

DL 100Rb cell

K cellDL Pro-80

λ/2Horizontal
imaging

100

100

-200

λ/2

λ/2

λ/2

λ/2

Verticalimaging

Figure 2.23: Overview of the laser system. The DL Pro is the master laser for potassium, seeding the TEC 400 tapered amplifier. The TA 100 contains
both a master laser and tapered amplifier as described in the text, providing all but the repumping light for rubidium. This is catered for by the DL 100
diode laser. The central frequencies of the AOMs are given, and the destinations of the optical fibres are indicated.

-80

λ/2

LP λ/2

0

λ/2

75

50

+4

150

125

-200

125

λ/4

Prism pair

PBS cube

Mirror

2.5 Magnetic transport2.5 Magnetic transport

Input

Fibre collimator
PBS cube

1/99 beam splitter

OutputPhotodiodePolariser

λ/2

Photodiode
Input
λ/2

Outputλ/2
λ/2

Outputλ/2
λ/2
OutputOutputOutputFigure 2.24: Representation of the Schäfter+Kirchhoff two- to six-way fibre-port cluster. 1%of each input beam is split off to monitor input power at a photodiode. Though the system is
enclosed it is possible to rotate the wave plates in order to obtain the required power balanceat the outputs.

2.5.1

Coil holder design

The coil holders are formed from thirteen separately machined aluminium pieces which
are bolted together with M6 screws as displayed in Fig. 2.25. The base is screwed
directly to the translation stage which comes with an array of threaded holes for thispurpose. The arms are the largest pieces and are 25 mm thick for high rigidity, as
the coils are held far away from the translation stage base. This is to allow plenty ofroom for mounting optical elements around the science cell. The vertical struts are far
enough away that when the coils are at the MOT cell the 45◦ horizontal beam may
pass by.
The coils themselves are held in clamps comprising four pieces each. The two plates
on the inner side are fixed solidly to the arms, fixing the coil separation. The two outerplates clamp onto the inner plates to hold the coils in place. There is a gap in each clamp
to reduce eddy currents in the mounts. Grooves in the support arms provide channels
for the copper tubing to travel down to the water cooling and power connections.
As the tubes for water-cooling and the high-current power cables need to travel up
and down the track with the mount an ‘energy chain’ (resembling caterpillar track)from Igus is used to keep everything tidy; serious damage could be caused by thecables snagging on anything as they are pulled along by the powerful translation stage.
The power cables and tubes for water-cooling must be suitably flexible to cope with62

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCTPRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

2.5 Magnetic transport

Figure 2.25: Transport coil holders. The holders are made from thirteen separate piecesof aluminium, held together by screws and bolted directly to the translation stage. The coilseparation is fixed to just clear the vacuum system during transport, with the outer plates
clamping down to securely hold the coils. The water and electrical connections are mounted off
the side of the structure. The arrow shows the direction of motion when the translation stage
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCTis operated.

the small bending radius and repeated flexing during every experimental run. The
assembled transportation system is shown in Fig. 2.26.

2.5.2

Translation stage

The Parker 406XR series translation stage consists of a carriage (to which the coil
mount is attached) mounted on a ballscrew drive which is driven by an in-line synchronous servo motor. The lead of the ballscrew thread is 25 mm while the motor
rotates at a maximum rate of 6000 rpm, giving a maximum speed of 2.5 m s−1 . The
track position is monitored by a Renishaw RGH24 series digital linear non-contact optical encoder with 0.5 µm resolution. Manual limit sensors are included to stop the
track in the event of a control error. The stage allows for 900 mm of travel and is very
robust, taking a load of up to ∼ 6000 N and a torque of up to ∼ 270 N m, far more thanwe actually subject it to. The repeatability of the track end point is given as ±3.0 µm.
The track is controlled by a Parker Compax3 servo drive, computer-operated via
RS 232 connectors. We found that it was important to position the servo drive far fromthe other electrical components of the system, due to the emission of radio-frequency
noise that affected the production of condensates. This noise has been noted elsewhere[41], though in this case no detrimental effects were seen. Similarly we have foundit essential to de-energise the motor while the atoms are in the science cell. These63

2.5 Magnetic transport

Figure 2.26: The transport coils in position around the MOT cell. The coil holders aremounted to the translation stage and travel towards the observer to reach the science cell. The
Igus ‘energy chain’ used to shepherd the cables can be seen in the bottom right-hand corner ofthe picture.

precautions vastly increased the reproducibility of condensates in the system.

2.5.3

Atom transportation

The process of moving the track back and forth during each experimental run is accomplished by fixing the track motion parameters (i.e. maximum speed, accelerationand jerk) and setting the two target end points. The track can then be operated byusing the ‘±jog’ commands by applying a voltage to the correct pin of the servo drive,
prompting the track to move in the designated direction. The coils travel a distance of
740 mm with a maximum speed of 0.5 m s−1 , acceleration of 5 m s−2 and jerk of 50 m s−3 .
This takes a little under two seconds end-to-end.
The maximum available current of 200 A is passed through the transport coils during translation to confine the atoms as tightly as possible. The ‘trap acceleration’gF mF µB B 0 /m is ∼ 50 m s−2 for rubidium (∼ 110 m s−2 for potassium) at this current,much larger than the maximum acceleration of the track of 5 m s−2 , so heating due
to the macroscopic trap motion is not an issue. It is possible for the more energetic
atoms to reach the sides of the vacuum chamber as they are transported through theminimum aperture of 10 mm, causing cooling and atom loss, which will be quantified
in the following chapter.

2.5.4

Majorana spin-flips

Another possible source of heating comes from what are known as Majorana spin-flips.
The problem with the linear quadrupole trap created by the transport coils is that there642.6 Quadrupole Ioffe-configuration trap

is a magnetic zero at the centre. As described in §1.4.2, only atoms with gF mF > 0
can be trapped at low magnetic fields, and will remain trapped as long as they move
adiabatically. However, at the centre of the trap the magnetic field direction rapidlychanges and the spin of an atom moving through the centre will have a good chanceof becoming mis-aligned with the field direction (undergoing a ‘spin-flip’). Though theregion this occurs in is relatively small, the colder atoms spend more time near thecentre and so are preferentially lost, causing heating as well as atom loss. This loss rateis estimated in [55] as~ΓM ≈ 7.4m

µF B 0kB T

2,(2.19)

where m is the atomic mass. ΓM ≈ 0.15 s−1 for rubidium in our trap at 100 µK, so forthe short time spent in the pure quadrupole field this process is not a major issue. As
the atoms get colder this becomes an overwhelming problem though, preventing thecontinued cooling to the quantum degenerate regime in such a trap.

2.6

Quadrupole Ioffe-configuration trap

The quadrupole Ioffe-configuration (QUIC) trap [56] gets around the problem of Majorana spin-flips by eliminating the magnetic zero altogether, having a magnetic minimum at some finite field strength. This comes at the expense of the sharp trapping
confinement and simplicity of a quadrupole trap, but is still relatively straightforward
to implement. In fact it brings together the three basic concepts discussed in §2.2; thefield from a single coil, the quadrupole field and a bias field. The QUIC trap geometry
with the directions of current flow is illustrated in Fig. 2.27, where the origin of thecoordinate system is located at the centre of the quadrupole coils. The quadrupole
coils ideally produce a purely linear field


x






B quad = B 0  y (2.20)−2z

as previously discussed, where B 0 is now the total field gradient along the weak axes.
The so-called Ioffe coil is located on the x-axis and produces an axial field that opposes
the quadrupole field. For a non-zero trap minimum to exist somewhere along the x-axis
there must be a point xmin at which

and hencedBxIoffe= −B 0dx(2.21)

dByIoffe
dBzIoffe
1== B0 .
dydz
2(2.22)

65

2.6 Quadrupole Ioffe-configuration trap

z

y

x

Figure 2.27: Formation of a QUIC trap. The vertical coils with opposing senses of current
flow (indicated by the arrows) provide the quadrupole field while the single horizontal Ioffe coilprovides the requisite field curvature. The rectangular bias coils can be used to independently
tune the field offset. The resultant trap minimum lies somewhere along the x-axis, about which
the trap displays rotational symmetry.

It immediately follows that the total field gradients in the y and z directions at thispoint are (3/2)B 0 and −(3/2)B 0 respectively. Defining the field offset at this minimum
to be −B0 and the field curvature produced by the Ioffe coil to be −(1/2)B 00 (whereB0 and B 00 are positive to comply with the chosen coordinate system), the total fieldnear the minimum may be written as−B0 − 21 B 00 (x − xmin )2


3 0
 .
B=


2B y

(2.23)− 32 B 0 z

The bias field can be applied to decrease B0 , the relevance of which will shortly become
apparent. The field amplitudes along x, y and z are then simply
1|B|x = B0 + B 00 (x − xmin )22"
 #1/2
3 B0 2
1|B|y = B0 1 +y≈ B0 +
2 B0
2
"
#

 1/2
3 B0 2
1z≈ B0 +|B|z = B0 1 +2 B02(2.24)
3 0 2
2
2B



y

B0

3 0 2
2
2B

(2.25)

B0z .

(2.26)The identical forms of (2.25) and (2.26) gives the trap its fundamental circular symmetry about the axis of the Ioffe coil as illustrated in Fig. 2.27. Using the expression
for the Zeeman shift (1.36) we can obtain expressions for the axial and perpendicular66

2.7 Evaporative coolingtrapping frequencies
r
ωax =
s
ω⊥ =

µF B 00m(2.27)

3 0 2
2B



µF

mB0

.

(2.28)The trapping potential is actually modified slightly by the Earth’s gravitational field,
but this description provides a good estimate of the true values. The perpendicular
trapping frequencies diverge as B0 → 0 which is why it is useful to be able to directly
tune this value with the bias coils. B 00 scales as IIoffe /a3Ioffe [56], hence the tight windingof the Ioffe coil.
We are able to measure B0 by finding the microwave frequency that removes all
atoms from the trap (see section §2.7.1), while B 0 for the transport coils has been calibrated using a gauss meter. During normal operation of the QUIC trap they are 5.3 G
and 64 G cm−1 respectively. B 00 is estimated to be 120 G cm−2 from a suitable modelof the Ioffe coil1 (which is constrained by our knowledge of B0 and B 0 ). This predicts
trapping frequencies ωax = 2π × 21(14) Hz and ω⊥ = 2π × 79(53) Hz for potassium(rubidium). A rudimentary check of the vertical trapping frequencies may be provided
by measuring the difference in ‘sag’ due to gravity ∆z (the distance between the trap
minima) for the two species. Assuming perfect harmonic potentials, the trap minimum
is displaced by a distance g/ωz2 under gravity, where in our case ωz = ω⊥ . This gives
the result
ωz2 K =



mRb
mK



ωz2 Rb =

g
∆z




 
mRb
g
48
−1 =
,
mK
∆z 39

(2.29)

which agrees reasonably well with the predicted values. (2.29) typically slightly underestimates the actual trapping frequencies due to the real anharmonicity of the magnetic
field.

2.7

Evaporative coolingThough the temperatures achieved by laser cooling are very impressive it has not yet
been possible to cool far enough at sufficient atomic density to reach quantum degeneracy by these processes alone. The suggestion of evaporative cooling as a pathway
to higher phase-space densities was first suggested by Hess in 1985 [57], focussing on
atomic hydrogen. This was applied to alkali atoms by groups at MIT [58] and JILA[59] in 1994 and indeed subsequently led to the first realisation of alkali atomic BECs
in 1995 [3, 4].
The concept is strikingly simple. Removing atoms with an energy greater than
the atomic average leaves the remaining atoms with a lower average energy. Upon
1

The coil may be modelled by assuming all eighteen windings have radius 8.5 mm and are situateda distance 18 mm from the centre of the quadrupole field.

67

2.7 Evaporative cooling

a

b

c

d

Figure 2.28: Illustration of the process of evaporative cooling in a harmonic trap. The trap
depth is gradually lowered from a to d such that the most energetic atoms escape the trap andthe atomic cloud rethermalises at a lower temperature.

rethermalisation the remaining atoms are at a lower temperature. This process can be
repeated to obtain temperatures far below the recoil limit (Fig. 2.28). The obvious
drawback is that atom loss is an integral part of the process. However, evaporation canbe extremely efficient, with phase-space-density growth by three orders of magnitudecommonplace for a drop in number by only a single order of magnitude.
In a simple model as described in [60] the process can be viewed as repeated removalof the high-energy tail of the Maxwell-Boltzmann energy distribution1 and rethermalisation. The removal of atoms is achieved by restricting the depth of the trapping
potential to some finite value. This is characterised by the truncation parameter η,giving trap depth ηkB T . As long as η is large the treatment that the system is close toequilibrium is valid. The atoms that are lost from the trap take away an average energy(η + κ)kB T , where κ is a small number usually somewhere between 0 and 1. Thus forlarge η a rough approximation can be made that κ is negligible. For the temperature to
decrease then we must have ηkB T greater than the mean atomic energy (3/2 + δ)kB T ,where the mean kinetic energy is (3/2)kB T in three dimensions and the mean potential
energy is δkB T , and it is easily shown that δ = 3/2 for a harmonic trap. The rate ofenergy loss is therefore simply given bydE= ηkB T ,dN

(2.30)

while the total energy of the harmonically trapped sample is given by
E = 3N kB T .

(2.31)

Equating (2.30) with the derivative of (2.31) with respect to N leads us to the temperature variation with N ,T ∝ N (η−3)/3 ,(2.32)

showing that for temperature reduction we indeed require η > 3. The volume of aharmonically trapped sample V is proportional to T 3/2 , which then directly gives us
1

For the most part evaporation is carried out in the regime where this is still a reasonable approximation.

682.7 Evaporative cooling

Quantity (∝ N x )Exponent, xAtom number, N

1Temperature, T(η − 3)/3

Volume, V(η − 3)/2

Density, n(5 − η)/2

Phase-space density, D = nλ3T4−ηElastic scattering rate, Γel = nσv(6 − η)/3

Table 2.5: Scaling laws for physical quantities (in the spirit of [60]) in terms of truncation
parameter η for a harmonic trap in the simple model described in the text. Within this model
it is clear that for evaporative cooling to be successful (i.e. for phase-space density to increase)η must be greater than 4, while to be in the ‘run-away’ regime (i.e. for the elastic scattering
rate to increase during evaporation) η must be greater than 6.

the number dependence of all other relevant physical quantities in the system as given
in Table 2.5.
More important than the temperature scaling is that of the phase-space density,and also that of the elastic scattering rate to maintain the evaporation process. The
above model suggests that for η > 6 the elastic scattering rate increases as N drops, inwhat is known as ‘run-away’ evaporation. Of course these values of η are not really verylarge, so the model is only used as a guide to the key concepts. A more sophisticatedmodel [61] gives η > 4.6 for the run-away regime (in the absence of any additional loss
processes). Most cooling procedures operate by keeping η relatively constant, forcing
the evaporation as the temperature drops by reducing the trap depth correspondingly.
This can be achieved in a magnetic trap by applying time-varying RF or microwave
frequency radiation (discussed later), and in optical traps by reducing the optical power.
The above model suggests that the efficiency of evaporation has no upper bound,
as in the commonly cited example of waiting for a single particle to have the entire
energy of the system. But there is a trade-off between cooling efficiency and evaporation
timescale, while inherent loss mechanisms (such as collisions with the background gas)
limit the time we may take over evaporation. The key requirement in a real system thenis that the elastic collision rate Γel be much higher than the loss rate Γloss of the sample.
The value of η available experimentally is constrained by the ratio α = Γel /Γloss . If α
is small we can no longer go to high η as the trap losses become overwhelming. The
quantity α may be added into evaporation models to include trap losses as described in[60], where it is seen that the minimum value of α for run-away evaporation1 is ∼ 100
for η = 6, rising to ∼ 300 for η = 8. Note that the value of Γloss will change during
evaporation; for example, the three-body inelastic collision rate scales strongly with√This minimum value is a factor 2 2 lower than given in [60] as there the elastic collision rate wasdefined for the trap centre rather than the cloud as a whole.
169

2.7 Evaporative coolingdensity.
Considering the direct evaporation of rubidium alone in the QUIC trap in our
experimental setup, α is large (> 1000) owing to the long trap lifetime and the large
scattering cross-section, allowing highly efficient evaporation at high η. To quantify theefficiency of the evaporation process we introduce the local quantityγ=−d(ln D)d(ln N )(2.33)

where D = nλ3T is the phase-space density. So in the simple lossless model γ =η − 4. Alternatively, a global value γtot is given by taking the difference between final
and initial conditions rather than the derivative. Typical experimental values of γtot
for direct evaporation are in the range 1 to 3 [13, 60, 62, 63]. This is fairly species
dependent; for example, evaporation of 87 Rb usually gives a value closer to 3 due toits favourable scattering length. This is a good benchmark for comparison with our
results.
It is also shown in [60] that the rate of atom loss due to evaporation in the limit oflarge η isηe−η
Γev = √ Γel .
2(2.34)

If we keep η and Γel constant (for forced evaporation on the cusp of the run-away
regime) then, in the absence of losses, the atom number decays exponentially during
evaporation with time constant 1/Γev . As T ∝ N x the sweep of the trap depth should
also be exponential to maintain constant η with a time constant ∼ 1/(xΓev ). Again
this may be used only as a rough guide, though most experiments (including ours) use
an exponential ramp of the trap depth with an experimentally determined optimum
time constant.

2.7.1

Sympathetic cooling

When direct cooling of a particular species is difficult (as for 39 K) we may make use ofa second to act as a coolant [64]. Both species are loaded into a trap but only one isselectively evaporated, while interspecies collisions allow the other to be sympatheticallycooled (Fig. 2.29). This process breaks down once the atom numbers of the two species
are comparable, so efficient sympathetic cooling requires a large initial ratio of coolant
to load. Ideally, no atoms are lost from the load species while the phase-space density
is greatly increased, so it is possible to create large condensates despite the relatively
low initial numbers in the trap. Naturally, losses do occur in real systems, and thesympathetic cooling efficiency can be characterised by γtot for the load species. The
larger the initial load the lower the cooling efficiency will be [63].
Considering for a moment the situation at the start of sympathetic cooling in the
QUIC trap, we can estimate the elastic scattering rate for potassium atoms colliding70

2.7 Evaporative cooling

a

b

c

d

Figure 2.29: Sympathetic cooling in a harmonic trap. The trap depth is lowered as in Fig. 2.28
for the coolant species (black), while the trap depth remains far greater than the temperatureof the load species (red). As the temperature of the coolant is reduced by evaporation the load
is cooled by inter-species elastic collisions, while no load atoms are intentionally removed. The
load is thus cooled without significant reduction in atom number.

with rubidium atoms to be given by
ΓK-Rb = n̄Rb σK-Rb v̄K-Rb .

(2.35)

This quantity is important, as it gives the limiting rate of thermalisation of potassium√
during sympathetic cooling. Here n̄ = n(0)/(2 2) is the average density in a harmonic
trap, σK-Rb = 8πa2K-Rb is the inter-species scattering cross-section andrv̄K-Rb =

8kB
π

r

TRb
TK
+
mK mRb

(2.36)

is the average relative velocity between atoms of different species. At the start ofour evaporation TK > TRb and in fact TK always lags a little throughout the evaporation process as the elastic scattering rate for collisions between rubidium atoms
ΓRb ≈ 5 ΓK-Rb when TK ≈ TRb . For our experimental parameters we find that initially ΓK-Rb ∼ 4 s−1 owing to the weak trapping in the QUIC trap and the low interspecies scattering length aK-Rb = 36 a0 [65]. However, our vacuum is good enough thatΓloss ∼ 1/300 s−1 , allowing suitably long evaporation sweeps for TK to stay close toTRb (important for maintaining the spatial overlap between the two species for crossthermalisation). A more detailed discussion of thermalisation in mixtures of ultracoldgases may be found in [66].

2.7.2

Thermalisation

The constant unavoidable loss of atoms from the trap means that the system is neverfully in equilibrium. However, as long as α  1 the system is in what is termed
quasi-equilibrium, where the thermal distribution is a very good approximation. Thermalisation of the cloud is usually taken to require ∼ 3 elastic scattering events per
particle [67]. Extreme scattering cross-sections can lead to conditions where thermalisation is no longer able to keep up with atom loss, and the system is deemed to beout of equilibrium. Small cross-sections reduce thermalisation below the collision rate71

2.8 Crossed optical dipole trapwith the background vapour, while large cross-sections drive the three-body loss rate
above the elastic scattering rate. Both of these regimes are accessible with a Feshbachresonance, so it is well to be aware of the contingency.

2.7.3

Microwave antenna

When evaporating a single species it is common to use resonant RF radiation to transfer
high energy atoms into untrappable states (the process is similar to the state transfer
described in §2.9). Signal generators and amplifiers spanning the kHz to tens of MHz
range required for RF evaporation from standard magnetic traps are readily available. However, the Zeeman substate splitting for potassium and rubidium are almost
identical, so it is impossible to use RF evaporation for sympathetic cooling as forced
evaporation would also occur for potassium. The large hyperfine splitting of 6.8 GHz
for the rubidium ground states, though, makes the use of microwaves a viable option, as
frequencies around this point are far off resonance for potassium. In our experiments,
rubidium atoms are transferred from the |2, 2i state to the untrapped |1, 1i state during
microwave evaporation.
The design of the microwave antenna was undertaken by Igor Gotlibovych [68]. The
relatively short wavelength required (∼ 4 cm) is comparable to the antenna distancefrom the atoms, allowing consideration of directional radiation intensity. A free-spacehelix antenna was created to maximise the field strength at the atoms. The circumference of the helix matches the wavelength at 6.8 GHz and consists of ten loops with apitch angle of 13◦ . The signal is provided by either of two microwave signal generators:
an HP HP83622A and an Agilent N5183A MXG. A suitable amplifier for 6.7–7.0 GHz
was built to specification by Microwave Amplifiers Ltd., having a saturated output
power of 39.5 dBm (∼ 9 W).

2.8

Crossed optical dipole trap

In order to make use of Feshbach resonances the atoms must be trapped in a purely
optical potential. This may be done by making use of the dipole potential (1.69) as
discussed in §1.4.3. Trapping is achieved by focussing a gaussian shaped red-detuned
beam to a 1/e2 waist size of w0 to create an intensity maximum in space, forming an
optical dipole trap. The Rayleigh range given by [69]
zR =

πw02
λ

(2.37)

is generally far greater than the beam waist, so for tight confinement in three dimensions
a second beam is added, crossing the first to form a crossed dipole trap (CDT) as
illustrated in Fig. 2.30.
72

2.8 Crossed optical dipole trap

Figure 2.30: Basic two-beam trapping scheme, forming the CDT in our science cell. Crossing
two beams produces tight confinement in three dimensions, as opposed to the single-beam casewhere trapping along the axis of the beam is usually relatively weak.

2.8.1

CDT laser

As suggested in §1.4.3 we must try to fulfil the requirements of a low scattering rate anda large trap depth by balancing the laser detuning with the laser power required. These
parameters also depend on the beam waist of course, but at the time of purchasing thelaser we based our decision on the dipole trap parameters of other research groups. We
settled on a 10 W linearly polarised ytterbium fibre laser from IPG Photonics (YLM10-LP-SC) at 1070 nm. The laser is simple to operate given it produces light at a fixed
frequency with a large linewidth of 1.2 nm. The light is collimated at the end of the
10 m long output fibre with a beam diameter of ∼ 5 mm.

Power stability
The beam power stability is important as large long-term drifts alter the trap depth overtime and high frequency modulations cause extra heating in the trap. We therefore
implement a negative feedback loop in order to stabilise the power. We control thepower by passing the light through an AOM and adjusting the acoustic amplitude in
the crystal to vary the power in the first-order diffracted beam. The power in this beam
is monitored by picking off a fraction of the trapping light once it has passed throughthe system, deflecting it into a fast photodiode. The voltage signal obtained from thephotodiode is compared with a desired voltage (set by a computer-controlled analogue
channel) by a PID controller [70]. The PID increases or decreases, as necessary, its
output voltage, which is sent to the voltage variable attenuator (VVA) in the AOM
driver. This directly effects the RF power sent to the AOM, completing the loop. The
trapping beam power may therefore be controlled by adjusting the voltage signal sent
to the PID.
The disadvantage of such a scheme is that a fraction of beam power is lost in the73

2.8 Crossed optical dipole trap

zeroth order of the AOM which is sent straight to a high-power beam dump. This
fraction is minimised by several measures. Firstly, the beam diameter at the AOM isclosely matched with the active aperture of the AOM (1 mm and 3 mm × 1.25 mm
respectively), as the rise time may be traded off for beam diffraction efficiency byincreasing the beam diameter. This still gives a rise time of only ∼ 300 ns. Secondly,
the voltage controlled oscillator (VCO) in the AOM driver is sent a fixed voltage thathas been tuned (via adjustment of a trimmer resistor) such that it outputs a frequency ofexactly 110 MHz, the central frequency of the chosen AOM where the highest diffraction
efficiency occurs. Thirdly, the output of the laser is oriented such that polarisation is90◦ to the AOM mounting plate as required for optimal diffraction efficiency. Finally,
the AOM angle was carefully tuned to be Bragg matched for first order diffraction, theAOM being mounted on a modified mirror mount to allow simple angular adjustments.
The AOM used is product number 3110-197 from Crystal Technology and was chosen
specifically for the possibility of high diffraction efficiency. The photodiode is a Thorlabs
DET36A/M model which has a large (13 mm2 ) detection area and 14 ns rise time. The
key components of the AOM driver are shown in Table 2.6, and the PID device wasdesigned and built by Philip Richman in the lab. The losses in the system mean thatthe trapping beam has a maximum power of 7.5 W at the atoms, though we usuallylimit this to 7 W to ensure the PID can function properly to provide a stable beam
power.

2.8.2

Forming the trap

An overview of the CDT setup is shown in Fig. 2.31. The laser output coupler isheld securely in a cage mount along with two lenses to form a telescope to obtain thedesired beam diameter for efficient diffraction by the AOM. The zeroth order light isdirected straight into a high-power beam dump, while the first order beam is reflected
off two mirrors onto the optical axis of a second cage system. This contains a PBS
cube to make sure the polarisation is pure, deflecting the orthogonal component of the
Product number

Description

Supply voltage (V)Control Voltage (V)ZX95-200-S+

VCO+12/Gnd.

∼ 4 (for 110 MHz)

ZX73-2500-S+

VVA+5/Gnd.

0→12

ZXSWA-2-50DR

Switch±5/Gnd.

0/5

Table 2.6: Key components of the AOM driver used to control the CDT power. All components
are from Mini-circuits. The VCO has a fixed control voltage as the AOM is only used to control
power. The VVA is controlled from an external voltage input during normal operation, but
may be controlled by a chassis-mounted potentiometer during testing. The switch is included
so that the beam may be turned off rapidly via a TTL trigger. An external RF amplifier (Delta
RF LA2-1-525-30) is used to allow all the components to be run off a ±5/12 V power supply
in the driver box.

74

2.8 Crossed optical dipole trap

light downwards into another beam dump. The two lenses in this cage mount first
expand then focus the beam as is discussed below. A periscope with adjustable mirrors
elevates the beam to the height of the science cell (185 mm above the surface of the
optical table) and allows the beam to be directed into the science cell at 45◦ .
To increase the effectiveness of our available optical power we form the CDT in theso-called ‘bow-tie’ configuration by re-collimating the trapping beam after its first passthrough the science cell, reflecting it off a couple of mirrors and then re-focussing thebeam such that the second focal point coincides with the first. In this way almost thefull beam power is recycled. In our system we cross the beams at 90◦ . The transmission
through the science cell is maximised by having the beam p-polarised with respect tothe cell surface for both passes. With this polarisation the transmission is ∼ 96% per
window (recalling that only the outer surface has been anti-reflection coated). There is
no problem with beam interference as the linewidth is sufficiently large that the path
length of the bow-tie section is far greater than the coherence length.
Specialised optics are not required to form the optical dipole trap, owing to the large
beam waists desired (the system is effectively diffraction limited by design). All of thelenses and mirrors used are broadband anti-reflection coated 1” pieces from Thorlabs.
Tuning the beam waist
When setting up the CDT we wanted the ability to alter the beam waist with minimal
realignment time. We also chose to have the focussing lenses mounted in a cage system
for mechanical stability and to allow simple translation along the optical axis. We werealso constrained by the coil mount around the science cell. The result is that the backfocal length, dBFL , of the lens system must be greater than ∼ 450 mm. This meansthat the beam must first be expanded from the 0.5 mm waist required at the AOM,
otherwise the waist at the atoms would end up being > 270 µm.
In order to calculate the required lenses and separations in the system we can makethe paraxial approximation and use simple ray transfer matrices to find the angle andoff-axis distance of a ray at the beam waist (ignoring the diffraction limit). The position
and angle of this ray are recorded in the form
!
x
x=
,
θ

(2.38)

and we sequentially apply to the initial conditions xi the transfer matrices D for traversing a certain distance d along the optical path and L for the advent of a lens of focal
length f , which take the form
D=

!
1 d
0 1

and L =

1

!
0

−1/f

1

.

For the two-lens system illustrated in Fig. 2.32 the final conditions are given by75(2.39)

2.8 Crossed optical dipole trap

LensesPhoto-detector

MirrorBeam dump

Fibre laser

Periscopic mirror pair
PBS cube
Neutral density filter
Glass plate-25+110 MHz

AOM

12
5

10
+1
z

H
M
300

-5
0

300

25
0

10

0

Science cell

Figure 2.31: CDT setup overview. The output of the fibre laser passes through a telescope
resulting in a collimated beam that is size-matched to the active aperture of the 110 MHz AOM.
Zeroth order light is sent directly to a beam dump. The polarisation of the first order light ispurified using a PBS cube, with light of the incorrect polarisation sent to a beam dump below
the cube. This beam is expanded and then focussed by the next lens pair before being raised tothe height of the science cell by a periscope. The focal point lies at the centre of the science cell.
After the first pass the beam is collimated and refocussed for the second pass, crossing itself
perpendicularly. A small fraction of the beam is then split off and sent to two photo-detectors.
One is used in the feedback loop for stabilising the beam power, the other for independent
monitoring of the beam power. The rest of the light is sent to another beam dump.

762.8 Crossed optical dipole trap

f1

xi

f2

θi = 0

w0
θf

d

BFL

Figure 2.32: Two-lens system for producing the desired focal beam-waist w0 for opticaltrapping.

xf =

xf
θf

!
= DBFL L2 DL1 xi .

(2.40)

The angle of convergence allows us to find the true diffraction limited waist from
w0 =

λ
πθf

,

(2.41)

while setting xf = 0 lets us find the back focal length. This results in expressions forw0 and dBFL dependent on the initial beam waist xi , the lens focal lengths f1 and f2
and their separation d:
f2 (d − f1 ),d − f1 − f2

(2.42)

f1 f2
λ.
π xi (d − f1 − f2 )(2.43)dBFL =
w0 =This enables us to find combinations of lenses that produce the desired beam waist
within the constraints of our setup. In our finalised system we use f1 = −50 mm,
f2 = 100 mm and d = 74 mm, giving w0 = 140 µm and dBFL = 511 mm. We have foundthat using the cage system it is a simple task to modify the beam waist by translationof the lenses and almost as easy to simply switch out a lens to change the accessible
range of beam waists. The above lenses allow tuning of w0 between ∼ 120 and 150 µm.
The focal beam waist was verified by diverting the beam directly into a charge-coupleddevice (CCD) camera and extracting the waist at several points along the optical axis.

2.8.3

Trap potential

The Rayleigh range given by (2.37) for the above trap parameters is ∼ 6 cm and gives
a beam waist at the science cell walls of only ∼ 1.03 w0 . A close approximation to thetrue trapping potential near the focal point is that given by collimated gaussian beams
of waist w0 . With x and y the beam axes and z the vertical axis the total potential77

2.8 Crossed optical dipole trap

may be written as
U (x, y, z) = mgz −A −2(x2 +z 2 )/w02 A −2(y2 +z 2 )/w02
e
− e
,
2
2

(2.44)

where the gravitational potential is included. Here g is the acceleration due to gravity,
m is the atomic mass of a trapped atom and −A is the peak optical potential shift at
the centre of the two overlapped beams. The value of the latter may be found from(1.69);
3πc2
A = −Udip (r = 0) =
2ω03



Γ
Γ
+
ω0 − ω ω0 + ω



2P
,
πw02

(2.45)

where P is the total power of the trapping beams.
A trap minimum is created at x = y = 0 , z = z0 , where gravity displaces theminimum vertically with respect to the centre of the dipole trap beams. The trapping
frequencies can be calculated from (2.44) and are given by
2A −2z02 /w02
e
mw02


z02
2
ωz = 2 1 − 4 2 ωx2 ,w0

ωx2 = ωy2 =(2.46)(2.47)

giving a cylindrically symmetric trap around z . In the high-optical-power limit wherepA  mgw0 , the quantity z0 /w0 → 0 such that ωx tends to 2A/mw02 and ωz tends to√2 ωx . From (2.47) it is clear that in the low-optical-power limit the trap is destroyed
if the trap minimum displacement reaches half a beam waist. The ratio between thevertical and horizontal trapping frequencies is shown for the possible values of z0 in√
Fig. 2.33. When z0 = −w0 /2 2 the trap is spherically symmetric.
1.4
1.2Ωz Ωx

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.1

0.2

0.3
-z0 w00.4

0.5

Figure 2.33: Dependence of the ratio of the vertical to horizontal trapping frequencies of the
CDT on the sag z0 of the trap minimum with respect to the beam waist w0 . Displacement z0
is always negative using the convention that gravitational potential increases in the positive z
direction.

782.9 State transfer

2.9State transfer

To be able to work with the broad Feshbach resonance in the 39 K |F, mF i = |1, 1i state
we need to transfer the atoms from the |2, 2i state. This is possible to achieve directly
via a magnetic dipole transition, using radio frequency (RF) radiation of frequency ωin a uniform magnetic field B. The frequency splitting of the |1, 1i and |2, 2i states at
low magnetic fields is given by ωres ≈ ωhf + 3µB B/2~, where ωhf = 2π × 461.7 MHz is
the zero-field lower-state hyperfine splitting in 39 K and we have taken gF ≈ ±1/2 forthe two hyperfine states. In what is known as the dressed-atom picture we view thelower state as being shifted up by an energy ~ω. However, coupling ~Ω0 /2 (where Ω0is the resonant Rabi frequency) between the two states means that as ω or B is tuned
they undergo an avoided crossing, and the eigenstates of the system are split by anenergy ~Ω0 where they would otherwise cross (as shown in Fig. 2.34). If the magnetic
field is increased adiabatically then all of the atoms initially in the |2, 2i state remainin the adiabatic lower energy state |φl i. Removing the RF field once past the crossing
point leaves the atoms in the |1, 1i state and state transfer has been successful.
For a finite rate of magnetic field change there is a chance that atoms will undergo
Landau-Zener transitions, ending up in the diabatic |2, 2i state. The probability of thisoccurring is given by the Landau-Zener equation
P = e−2πΛ(2.48)

È2,2\

È1,1\

ÈΦu \

EÑ

W0

ÈΦl \

È1,1\

È2,2\

2 Ñ HΩ-Ωhf L
3 ΜB

B

Figure 2.34: The avoided crossing scheme of the |2, 2i and |1, 1i Zeeman levels in the dressedatom picture in the presence of an RF field. Where the diabatic energy levels (black) cross, theadiabatic energy levels |φu i and |φl i (coloured) are split by Ω0 ~. Sweeping the magnetic fieldstrength adiabatically from low to high transfers atoms from one state to the other. The atoms
remain in this state once the RF field is removed.

792.10 Absorption Imaging

where(Ω0 /2)2.
dt (ω − ωres )Λ= d(2.49)

In our experiment we chose to perform the state transfer by sweeping the magnetic
field in the presence of a steady RF frequency of 469.3 MHz. We sweep the field at a rateof 15.4 G s−1 about the 3.6 G resonance position, equivalent to sweeping the detuningω − ωres at a rate of 2π × 32 MHz s−1 . In the Landau-Zener model a transfer efficiency
above 99% is achieved for a Rabi frequency Ω0 & 2π × 2 kHz.

2.9.1

RF antenna

The antenna used to provide the RF is a single 18 mm radius loop of 1 mm diametercopper wire, situated a few centimetres from the atoms. A single loop was chosen tominimise the inductance of the loop which is the main contributor to the impedanceof the antenna. In order to maximise transfer efficiency, Ω0 must be maximised which
is proportional to the amplitude of the RF field. This in turn depends on the power
output from the antenna. To maximise this a tuneable capacitor is used to impedance
match the imaginary part of the impedance of the antenna due to its inductance. The
tuning of the capacitor was achieved by maximising transfer efficiency at low power.
Once the full available power is applied we do not observe any atoms left in the |2, 2i
state.

2.10

Absorption Imaging

Absorption imaging is the standard method of recording the atomic distribution aftersome time-of-flight in cold atoms experiments. The atoms are released from the trapping potential and allowed to freely expand until a resonant or near-resonant pulse oflight is applied. The transmission along the direction of propagation of the light pulse
depends on the number of atoms along the path. The shadow produced is then imaged onto a camera for processing as discussed below. Analysis of the resultant images
provides us with details of the imaged cloud such as temperature and atom number as
discussed in §2.11. The process is unfortunately destructive, so the full experimental
sequence must be run through each time a new image is produced.

2.10.1

Optical density

As a beam of light travels through a cloud of atoms, a fraction of the photons areabsorbed dependent on the total atom number per unit area integrated along the beam
path (referred to as the column density, n2D ) and the atomic absorption cross-section
σ . It is simple to show that the intensity is attenuated according to Beer’s law,
I = I0 e−σn2D ,80(2.50)

2.10 Absorption Imaging

where I0 and I are the respective incident and emergent intensities of the imaging beam.
The quantity OD = σn2D is known as the optical density. The standard method ofdetermining OD requires a set of three images: one where the atoms were present,
Iatoms , one with the unattenuated imaging light (where no atoms were present), Ilight ,and one with no imaging light for normalisation purposes, Idark . The images here arerectangular arrays of electron counts at each pixel of the imaging camera’s CCD. The
counts of Idark are produced by any stray non-imaging light that makes its way to thecamera, so ideally the optical density is given by
Iatoms − Idark.
OD = − ln
Ilight − Idark

2.10.2(2.51)

CameraFor cold atom experiments the choice of CCD camera often hinges on a trade-off between price and quantum efficiency (QE), which is the conversion factor of incident
photons to stored electrons. High quantum efficiency cameras have several advantages, e.g. allowing shorter/lower-intensity imaging pulses and offering a better overallsignal-to-noise ratio (SNR). This is most important in studies of such areas as density
fluctuations in ultracold gases [71, 72] where real fluctuations in atom number must be
deduced by removing systematic noise in the image. A high QE is always advantageous,
but there are other features in cheaper cameras with relatively low QE which can makethem more desirable when QE is not critical.
One such feature is the ‘double-shutter’ mode available in some interlining CCD
cameras. In these cameras the CCD is subdivided into alternating columns of sensor
and storage wells. The storage region is masked, so at the end of the exposure theelectrons that have accumulated in the sensor wells are transferred to the storage wells
in a matter of microseconds. The main advantage for our imaging is that a second image
can be taken almost immediately after the first. A common problem in absorption
imaging is that there are fringes in the light intensity (due to the coherence of thelaser) which shift in time due to mechanical vibrations in the system. If the time
between the two illuminated images is small compared to the period of the vibrations
the fringes will overlap and a clear fringeless optical density picture is acquired when
dividing one by the other. If, however, the fringes shift between pictures they will
also appear in the final optical-density image. This is problematic for image analysis,
especially when trying to detect small condensates. It is possible to remove fringes inpost-processing by Fourier filtering methods [73] or by matching the Iatoms image witha linear superposition of real Ilight images [74], but it is simplest to avoid the issue inthe first place.
Our chosen camera, the PCO Pixelfly QE Double Shutter camera, can take two
images less than 10 µs apart, far shorter than standard full-frame CCD cameras (where
the inter-frame time is roughly equal to the digital read-out time, 88 ms for our camera).
812.10 Absorption Imaging

This is because the second image can be taken as soon as the first has been shiftedto the masked region of the interline CCD. The second exposure must, however, be
at least as long as the read-out time because the first image must be registered before
the second can be shifted to the masked region. This leaves additional time after theimaging light pulse for stray light to be acquired in the image. This means that in ourcase two dark images must be taken for normalisation, one short to match Iatoms , one
long to match Ilight , such that optical density is given by
OD = − lnss
Iatoms− IdarkllIlight− Idark!
.

(2.52)

Occasionally images do nonetheless exhibit fringes. Just as with normal shuttering
where it is possible to have a smooth image, there are times when the oscillationsin the system are unfortunately caught at peak velocity and even double-shuttering
produces a fringed image. The extent and regularity of the problem are, however,much reduced, making it a worthwhile technique.
Despite the relatively low quantum efficiency of the camera of ∼ 28 % for λ =767 nm , this is higher than the QE of standard interline cameras as the CCD is overlayed with an array of microlenses. These act to direct the light towards the sensor
regions so that the light that would otherwise strike the masked storage regions doesn’t
go to waste, hence the ‘QE’ in the camera name.
Another neat trick when double-shuttering is available is the ability to image two
species near-simultaneously during time-of-flight. This is done by imaging one setof atoms with the first double-shutter exposure and the other set of atoms with thesecond. The images with no atoms are taken after a pause during read-out of the initial
exposures, and finally the background images. This brings back the issue of fringes in
the optical density images, but the technique was very useful for trouble-shooting.
For example, observing whether there was a correlation between anomalously hot 39 K
clouds and low 87 Rb atom numbers during sympathetic cooling.

2.10.3

Imaging light

All of our samples are imaged with a beam of σ + polarised light that is resonant withthe |2, 2i → |3, 3i cycling transition. A quantisation axis is provided by a weak (fewgauss) guide field provided by the transverse bias coils around the science cell. It is
important for the imaging beam to be near-uniform as the absorption cross-section
varies with intensity, being given by
1,
σ = σ01 + (2δ/Γ)2 + I/Isat82(2.53)

2.10 Absorption Imaging

as may be deduced from (1.67). The resonant cross-section σ0 for σ + polarised lighton the |2, 2i → |3, 3i transition is [23]
σ0 =

3λ2
.
2π

(2.54)

Some intensity variation across the beam is inevitable, so it is best to use low intensities(I  Isat ) when possible1 . It is also sometimes useful (for dense clouds) to detune thebeam to deliberately reduce σ by a determined amount. However, for our critical measurements we tune the imaging beam to resonance as small variations in the detuning
here have little effect on σ, while its dependence on δ is stronger on the sloped sides of
the Lorentzian lineshape.
The light pulse we use for imaging has a relatively high intensity of around Isat /3
and lasts for 80 µs. This evolved due to the balancing of several factors. The effective
pixel size (discussed below) of our CCD, the quantum efficiency, the conversion factorof electrons to counts and the level of read-out noise from the camera require this muchlight so that the accessible range of OD is not too severely truncated. The opticaldensity can also be limited by any off-resonant light in the imaging beam, causing
saturation at a relatively low value. From observations of OD saturation in our images
we choose to trust only optical densities below 3.

2.10.4

Objective

To resolve features on the small length-scales of the atomic cloud an objective wasdesigned in our group by Alexandre Dareau. The aim was to create an objective outof cheap standard optical elements, but designed specifically to take into account thespherical aberrations introduced by the wall of our science cell. The natural aberrationsof the optical elements are calculated to balance one another (and those of the sciencecell wall). The design used for horizontal imaging (from which the vast majority ofthe data in this thesis is taken) was based on one by Wolfgang Alt [75], designed forsimilar purposes. A full description of the finalised objective may be found in [76], a
representation of which is shown (not to scale) in Fig. 2.35. The design has been shownto have a resolution better than 2 µm. The full set of lenses provides a magnification
factor of 2.78, determined by observing the acceleration of the atoms due to gravity
in a sequence of absorption images. This magnification factor gives an effective pixel
size for our camera of 2.32 × 2.32 µm2 , close to the fundamental resolution limit of theobjective.

1

When imaging on resonance (δ = 0), a 20 % change in I alters σ by ∼ 10 % around I = Isat , but
only by ∼ 2 % around I = Isat /10 .

83Camera

Science cell2.10 Absorption Imaging

Figure 2.35: Structure of the objective lens system used for absorption imaging. The four
blue optics form the objective which collimates the light from the atoms while compensating
for spherical aberrations introduced at the science cell wall. The red lens is a diffraction limited
achromatic doublet used for focussing the light to form an image at the camera. The 1” optics
are held in a lens tube with fixed spacers to accurately set the required inter-lens separations.
The lenses used to acquire the data in this thesis resulted in a magnification factor of 2.78 .

2.10.5

Imaging sequence

Usually the atoms leave the trap in the |1, 1i state, so just before imaging a 100 µs pulseof repumping light transfers the atoms to the |2, 2i state with near-perfect efficiency.
The atoms are then imaged (in the presence of a guide field of a few gauss) with an
80 µs pulse of light to produce Iatoms . With the double shutter technique the next
exposure starts only 10 µs later. However, in this short time the atoms are still in thepath of the image beam. To get around this issue a strong magnetic field is ramped up
to take the atoms far off resonance with the imaging light. This is done rather thanchanging the frequency of the light with an AOM to avoid possible changes in the lightintensity. The field-ramping time means we must leave a 60 µs gap before taking Ilightwith a second 80 µs imaging pulse. The two Idark images are taken over a second later
during which the sequence of events just before the first set of images is mimicked tomake sure the same amount of stray light appears in these images as in the first pair.
The acquired images are then immediately processed into a single OD image by our
MATLAB ‘Pixel Machine’ software (designed by Naaman Tammuz).

2.10.6‘Fudge-factor’

Assuming a perfect absorption cross-section σ0 typically causes underestimation of theatom number [72, 77]. The multiplication factor used to compensate for the difference
between the atom number as calculated using σ0 and the true atom number has colloquially become known as the ‘fudge-factor’, roughly equivalent to σ0 /σ. This factor
for references [77] and [72] is 4.0 and 1.4 respectively. The reduction in cross-section isusually mainly attributed to imperfect polarisation of the imaging light. However, in
our case a significant shift is produced by using a relatively high intensity beam and along imaging pulse. This results in the cross-section dropping due to both saturationof the cycling transition and the Doppler shifting of the atoms off resonance due to theacceleration of the atoms by photon recoil.
The combined effect can be estimated from the scattering force (1.67) and thedependence of the cross-section on intensity and detuning given in (2.53). Assuming84

2.10 Absorption Imaging

Figure 2.36: Time-averaged 39 K absorption cross-section hσit for imaging light that is resonant
for a stationary atom. The variation is shown with light intensity I and imaging pulse duration
τ . The limit τ → 0 shows the standard dependence on intensity, while for I → 0 the ideal crosssection is recovered for all τ . Imaging with this frequency with our I and τ gives hσit ≈ 0.5 ,but this frequency does not give the maximum value of hσit , as discussed in the text.

that initially the cloud of atoms is at rest and the imaging light is resonant leads tothe dependence on intensity and imaging pulse duration τ shown in Fig. 2.36 for 39 K
atoms. This takes into account the change in cross-section and scattering force overtime, resulting in a time-averaged cross-section hσit . In this simplified model the atoms
are assumed not to drop out of the cycling transition. In this picture the reduction in
effective cross-section for long imaging pulses is significant; for our imaging parametershσit /σ0 is predicted to be about 0.5 .
The effect is exaggerated compared to what is seen experimentally as we tune theimaging frequency to give the largest OD. This results in the imaging frequency being
tuned slightly above resonance (for stationary atoms) such that as the atoms are accelerated during the imaging pulse they are first Doppler shifted onto resonance before
being shifted away again. This is illustrated in Fig. 2.37 where the dependence of σwith time is shown for a beam that starts on resonance and for a beam that is initially tuned above resonance, becoming resonant half-way through the imaging pulse.
The latter case produces the optimal effective absorption cross-section. The resultant
effective line-shape for this ideal model is shown in Fig. 2.38 for a range of imaging
parameters. This illustrates the fact that not only does the effective cross-section decrease for a finite pulse duration and intensity, but the peak frequency of the profile isshifted, the shape is significantly skewed and the line-width increased.
It is not possible of course to directly compare the line-shape of our chosen setup85

2.10 Absorption Imaging

0.8

0.7

ΣΣ0

0.6

0.5

0.4

0.3
0

20

40
Time HΜsL

60

80

Figure 2.37: Variation of 39 K absorption cross-section with time when the imaging light isinitially resonant (black line) and when it becomes resonant half-way through the pulse duration(red-line). The total pulse duration is 80 µs and the light intensity is Isat /3 . The time-averaged
value gives the effective cross-section in the OD → 0 limit.

1.0

XΣ\t Σ0

0.8

0.6

0.4

0.2

0.0
-1.5

-1.0

-0.5

0.0
∆i G

0.5

1.0

1.5

Figure 2.38: Imaging beam frequency resonances for 39 K with σ + polarisation on the cyclingtransition for the Doppler model discussed in the main text. The grey dashed line shows theideal lineshape. The black, blue and red lines show the lineshapes where imaging period τ → 0,
40 and 80 µs respectively for I/Isat = 0.1 → 0.505 in steps of 0.015. For τ → 0 only thesaturation of the transition affects the cross-section. The blue and red series of lines show how
increasing imaging pulse duration and intensity distorts the perfect Lorentzian lineshape andreduces the effective cross-section.

862.10 Absorption Imaging

with the ideal form, but we can observe the difference between profiles taken withdifferent sets of parameters. To confirm that the above effects manifest themselves ina real system, two series of images were taken of atom clouds created with identicalexperimental sequences. Within each series the imaging frequency was varied over arange of 8 MHz in 0.5 MHz steps, one set being taken with 40 µs imaging pulses, theother with 80 µs pulses. The series were interleaved during the experiment and thewhole process repeated for a total of six sets to reduce possible errors.
The data, shown in Fig. 2.39, confirms the qualitative trend expected for an increasein imaging-pulse duration, and we can fit the data using a model that assumes thisDoppler-based effect is the only one affecting the cross-section. The frequency spacing
between datapoints was controlled in the experiment (and the pulse duration set). This
left I/Isat , absolute frequency and an absolute cross-section scaling factor as free fitting
parameters. The two datasets were fitted simultaneously (producing a single value foreach of the stated fitting parameters), keeping relative frequency and effective crosssection fixed. The fitted resonance curves are shown as blue and red solid lines in Fig.
2.39, while the predicted resonance curve for a stationary atom (where only the finite
I/Isat affects the ideal lineshape) is given by the black solid line. The ideal Lorentziancase is shown as a grey dashed line.
The value I/Isat ≈ 0.23 extracted from the fit is inconsistent with the recorded
value I/Isat ≈ 1/3 (determined by pixel counts on the imaging camera’s CCD). The
fudge-factor predicted from the fit of ∼ 1.4 is also inconsistent with the value 1.6
calculated by observing the critical point of an atomic cloud with minimal interactions.
This is not surprising, as imperfect imaging beam polarisation must also be taken into
account. This causes the line-shape to be less skewed than for perfect polarisation
for a fixed intensity or effective cross-section. A model that includes this effect isunlikely to produce a more meaningful fit to the data due to the over-abundance of
free parameters. However, it does conveniently explain the underestimated I/Isat andfudge-factor values.
A time-varying σ actually leads to non-linearities in the conversion of OD to n2D ,as what we directly measure in the image is actually the transmission, proportional tohe−n2D σ it , rather than the mean optical density, proportional to hσit . The effect isgenerally small though, with variations of < 1 % for OD < 1 in our scenario. However,
it is important when measuring number fluctuations as mentioned in [72], which is why
they avoid the problem by using a high QE camera. This allows a short low-intensity
light pulse during which the atoms on average absorb only 6 photons as opposed to∼ 300 in our procedure. Future experiments in the group may require a high QE
camera, but for the experiments covered in this thesis the smooth imaging producedby double-shuttering with suitable cross-section calibration is more than adequate. It
is well to be aware, though, of the warping of the line-shape and shift in resonanceproduced when altering imaging timings and intensities.
87

2.11 Image Fitting

1.0

XΣ\t Σ0

0.8

0.6

0.4

0.2

0.0
-1.5

-1.0

-0.5

0.0
∆i G

0.5

1.0

1.5

Figure 2.39: Imaging beam frequency resonance data for 39 K. The grey dashed line again
shows the ideal Lorentzian lineshape. The blue and red points are experimental data series
taken with 40 µs and 80 µs exposures respectively, with statistical error bars shown based on
six readings per point. The solid lines show fits using the model discussed in the text, with theblack line representing the 0 µs exposure lineshape based on the fitted value for I/Isat ≈ 0.23 .
The data clearly shows the expected skewing of the lineshape to higher frequencies and reductionin effective cross-section.

2.11

Image Fitting

All of our experimental information is garnered from the images produced by absorption
imaging after time-of-flight. Most of the time this means extracting the atom number(both condensed and thermal parts) and the temperature. With an appropriate function the number can be calculated from the integral under the fitted distribution, while
the temperature can be found from the widths of the distribution.

2.11.1

Thermal distribution

We have already found the in-trap distribution of an ideal Bose gas, and the expanded
distribution can be found in a similar way, as described in [78]. Assuming free expansion, a particle can only end up at final position r if it has momentum p = m(r − r 0 )/t,where its initial position was r 0 . The expanded distribution can therefore be foundby multiplying the integrand on the right-hand side of (1.6) by δ(r − r 0 − pt/m) andintegrating over both p and r 0 (rather than r). Doing this for a harmonically trappedgas givesnT (r, t) =
(
"


#)
3 
3
ωk2
1 Y
1
µ
m X 2
−
xk
g3/2 exp
.
kB T
2kB T
λ3T
1 + ωk2 t2
1 + ωk2 t2
k=1

k=1

88

(2.55)

2.11 Image Fitting

For the critical density distribution (µ → 0) this can be rewritten as
(
"
!#)
x2
nc (0, t)1y2
z2
nc (r, t) =
g
exp −
+
+
,
ζ(3/2) 3/2
2 RT2 x (t) RT2 y (t) RT2 z (t)(2.56)

where the thermal radii RT k (t) have been redefined asRT k (t) =1 + ωk2 t2
kB T
mωk2

1/2
,

(2.57)

such that they now also apply during time-of-flight. For t → 0 the in-trap criticaldensity distribution (1.16) is recovered. Imaging the atoms gives us a two-dimensional
optical density map, equivalent to integration of the three-dimensional distribution
along y:Z ∞nc (r, t) dy .

nc 2D (r xz , t) =(2.58)−∞

Here r xz is the radial vector in circular polar coordinates in the x-z plane. By using
the sum representation of gs (z) given in (1.9) it is straightforward to show that




z2
1
x2
nc 2D (0, t)+
g2 exp −nc 2D (r xz , t) =ζ(2)2 RT2 x (t) RT2 z (t)where
nc 2D (0, t) =√2πRT y (t)ζ(2)
nc (0, t) .
ζ(3/2)(2.59)(2.60)

By fitting an image with this g2 function we can obtain RT x (tTOF ) and RT z (tTOF )which, along with our trapping frequencies, allow us to calculate the temperature of
the cloud using (2.57). It is worth noting that fitting full thermal distributions ofatoms near the critical point with a gaussian function produces significant errors intemperature measurement as mentioned in [41]. As long as we know the conversion of
optical density to number density the integral under the fitted g2 function also gives
the total thermal atom number as
NT =2.11.2ζ(3)2πRT x (t)RT z (t)nc 2D (0, t) .
ζ(2)(2.61)

Thomas-Fermi distribution

We have seen that, even for very weak interactions, if the number of condensed atoms
N0 is large that the ideal gaussian distribution for the condensate is warped into aparabolic shape. If we are in this Thomas-Fermi regime it can be shown (as discussed
in e.g. [19]) that this form also holds during time-of-flight, such that the distribution
is given byn0 (r, t) = n0 (0, t) 1 −

x2
y2
z2
−
−
Rx2 (t) Ry2 (t) Rz2 (t)89(2.62)

2.12 Computer control

inside the ellipsoid with semi-axes Rk (t), and it is 0 outside this region. Now theThomas-Fermi radii Rk (t) are time dependent, where at t = 0 they take the in-trap
valuess
Rk (0) =

2µ
.
mωk2

(2.63)

Here µ is the Thomas-Fermi chemical potential given by (1.29). In our experiments
the time dependence of these radii is not important as we are only trying to extract
the atom number from the fits to the condensate. Of general interest though is the
aspect ratio in time-of-flight. Considering for simplicity an axially symmetric trap
ωx = ωy = ω⊥ , we can denote the in-trap aspect ratio of the condensate λ = R⊥ /Rz .
It can be shown analytically [79] that for λ  1 (i.e. a ‘cigar-shaped’ trap symmetricaround z) the aspect ratio for long time-of-flight goes to 2/πλ  1, so there is aninversion of aspect ratio during time-of-flight. In general an aspect ratio inversion will
be observed for any asymmetric trap. This of course differs from a thermal cloud where
the distribution tends towards spherical symmetry with increasing time, and is anothersignature of Bose-Einstein condensation.
We must integrate (2.62) over y to yield the two-dimensional distribution
n0 2D (r xz , t) = n0 2D (0, t) 1 −
where

z2
x2
−
Rx2 (t) Rz2 (t)

3/2
,

4
n0 2D (0, t) = Ry (t)n0 (0, t) .
3

(2.64)(2.65)

Integrating over the remaining dimensions yields the total atom number
1
N0 = 2πRx (t)Rz (t)n0 2D (0, t) .
5

2.11.3(2.66)

Fitting routines

The image fitting is carried out by our ‘Analysis GUI’ program, created in M AT LAB
by Naaman Tammuz. Essentially, it applies a set of non-linear least squares fitting
routines to the data, with the specifics of each iteration dependent on the parameter weare trying to extract from the image. These are discussed in more detail in the relevant
experimental sections. The essential parameters for processing the images (such as thetrapping frequencies, apparent pixel size, effective imaging cross-section etc.) are set
by the user. The program returns a database of the fit results and calculated quantities
for all the images that were selected for processing.

2.12

Computer control

Typically our experimental sequences last around a hundred seconds, while some steps
in the sequence need to be timed with microsecond precision. A computer system must90

2.12 Computer control

be used to deterministically trigger and control our experimental equipment, with asuitable user interface for simple construction of complex sequences. With the large
number of experimental components to control during each run this is a somewhat
daunting task. The implementation of such a system in our lab was made much morestraightforward thanks to a set of applications created and maintained by Aviv Keshet
in Wolfgang Ketterle’s group at MIT, which he has made freely available to the scientificcommunity. This software is designed to be compatible with National Instruments
analogue and digital output hardware, and allows communication to devices via RS232 and GPIB protocols.
The system allows digital TTL voltage signals to be sent to components such asmechanical shutters which just require an on/off signal, while analogue voltage signals
are used to set power-supply outputs, AOM amplitudes and so forth. Items such as thetranslation stage and RF generators are controlled via RS-232, while the microwave
generator uses the GPIB protocol for communication.

2.12.1

Software

The control software is comprised of Cicero, the client user interface, and Atticus, theserver. Sequences are constructed in the intuitive Cicero interface, which are thencompiled by Atticus into a command sequence that can be passed on to the output
hardware. Cicero allows batches of sequences to be run where parameter values arecycled according to user-defined lists, simplifying the acquisition of experimental data
sets. The software is naturally attuned to the operation of cold atoms experiments,
with nearly all of the functionality we desire being available. The only major feature
missing is the ability to allow feedback from system inputs, which would require majoralterations and additions to the software and is generally unnecessary.
A significant feature added by Aviv while we were setting up our experiments was
the ability to use a variable time-base. Normally a sequence allowing for microsecond
resolution with a hundred output channels would essentially require a set of a hundred
million commands to be sent for every second of the duration of the experimental
sequence. The amount of data that can be stored in the output buffer generated
by Atticus limits the duration of a sequence created in this way to a rather modest
length, and the large amount of data to be compiled creates substantial delays between
sequences. However, the variable time-base feature vastly reduces the buffer size by
updating outputs only when something is changing. Trigger signals to change state
are generated separately, so when all outputs are constant in the system instead ofsending the same set of commands every microsecond, just one command is sent for
the entire period. This allows us to keep the microsecond resolution where needed while
permitting the sequence length required for our experiments.
As we have made no modifications ourselves to the software there is no need to
discuss its use and structure here. The full functionality is covered in the user manuals
91

2.13 Field and resonance calibration

Part description

Company

Part number

Quantity

ChassisNational Instruments

NI PXIe-1062Q

1

Digital card

National Instruments

NI PXI-6533

1

Digital card

National Instruments

NI PXI-6534

1

Analogue card

National Instruments

NI PXI-6713

2

RS-232 card

IntaShield

IS-400

1

GPIB card

National Instruments

NI GPIB-USB-HS

1

Analogue connector block

National Instruments

NI BNC-2110

2

Table 2.7: Summary of the hardware used for computer control. All items were chosen to becompatible with the Cicero sequence generator software.

provided by Aviv Keshet [80].

2.12.2

HardwareHaving opted to use the Cicero software the choice of hardware was greatly simplified.
A desktop computer was chosen to run the software with 4 GB of RAM to cope with thelarge output buffers. Most of the rest of the hardware is made by National Instruments
to be compatible with the software. The parts used in our system are listed in Table
2.7.
The basic hardware setup is shown in Fig. 2.40. The software is run on a desktop
PC, with the GPIB card and the 4-port RS-232 card directly connected. The analogue
and digital cards are connected to the National Instruments chassis. When a sequence
has been compiled by the Atticus software the sequence data is sent to the chassis
while timing data is sent to a separate FPGA board. The FPGA board synthesises avariable frequency clock to produce the timing triggers which are sent directly to oneof the analogue cards and shared between the analogue and digital cards. During a run
successive groups of signals are sent from the output cards on receiving each timing
signal. Break-out boxes to connect the 32 digital channels of each digital card to BNC
connectors were made, with opto-couplers installed for isolation purposes. Commercial
connector boxes were used for the analogue channels. The output signals are sent on
from here to the experiment. As the RS-232 and GPIB cards do not share the timing
signal of the digital and analogue cards, there is a lack of complete synchronisation,
but precise timing of the RS-232 and GPIB commands are not critical in our setup.

2.13

Field and resonance calibration

The purpose of our experiments is not to investigate the Feshbach resonance itself,
rather simply to use it for tuning interactions. However, it was still necessary to92

2.13 Field and resonance calibration

Analogue Signals

ExperimentNI
BNC-2110Timing Signal

Digital Signals

Break-out
Box

Timing Data

FPGA
Board

NI PXIe-1062Q
ChassisRS-232 Signal

RS-232

Running Cicero clientand Atticus server software
Sequence Data

GPIB Signal

GPIB

PC

Analogue
DigitalFigure 2.40: Simple schematic of the hardware setup for computer control. In our system
there are two analogue and two digital cards connected to the chassis. The analogue cards haveeight channels each while the digital cards have thirty-two.

calibrate the field produced by the Feshbach coils for a given electric current, and tocheck that our own resonance position and width calibrations agreed within errors withthe values found during the more stringent calibrations carried out at LENS [6, 12, 81].

2.13.1

Field calibration

In order to calibrate the field produced by our Feshbach coils we looked at state transfer
of rubidium in the presence of an RF field. The most precise determination of the
39 K |1, 1i resonance position B∞ to date was carried out in [12], giving a value of

402.50(3) G. We chose this as a natural point to calibrate our current-to-field conversion.
Returning to the Breit-Rabi equation (1.33), the splitting between the |2, 2i and |2, 1i
states in 87 Rb at this field value is 249.444 MHz. An RF field at this frequency wasapplied to a thermal cloud of rubidium held in the optical trap while the current in
the Feshbach coils was swept over a 160 mA range in 200 ms. This was repeated while
incrementing the central current in steps of 80 mA . The expectation is that atoms
are transferred to the |2, 1i state if 402.50 G lies within the range of the magnetic fieldsweep. To distinguish between atoms in the |2, 2i and |2, 1i states we spatially separated
them by appling a Stern-Gerlach field gradient during time-of-flight. This produced aclear signal, allowing us to determine the current to produce a field of 402.50 G at theatoms to be 152.46(8) A . This gives the conversion factor given in Table 2.3.

2.13.2

Resonance measurement

Our initial calibration of the resonance position relied on the enhancement of three-bodylosses in the vicinity of a Feshbach resonance. 39 K atoms were prepared in the opticaltrap using a standard sequence to provide a fixed starting number and temperature.
93

2.13 Field and resonance calibration

For the data-set shown in Fig. 2.41 a the field was first swept to 403.4 G (near to theresonance), before sweeping from there to the final value in 20 ms and holding at thisfield for 100 ms1 .
As expected, there is a dip in final atom number near the expected resonanceposition, but it is asymmetric and looks to be centred to the attractive side of theresonance position. This was also found in similar experiments carried out by Massimo
Inguscio’s group at LENS when the first studies of the resonance were made [81]. In
their case, fitting a single gaussian profile to the data gave a resonance position of
403.4(7) G, similar to the apparent shift in our data. The asymmetry in the profile
can be attributed to the enhancement of three-body losses on the attractive side of theresonance, due to the mechanism described in [82]. The losses are greatest on resonancebut decrease more sharply on the repulsive side of the resonance, as can readily be seenin Fig. 1 of [12]. This is compounded by the increase in average density for a cloud withattractive interactions which also enhances three-body recombination. The minimum
remaining atom number actually closely coincides with the accepted resonance position,marked in Fig. 2.41 by the dashed line.
We decided to attempt a more accurate determination of the resonance position
following a similar method to that found in [12]. The idea is to prepare identical
thermal 39 K clouds with the field at a fixed initial value. At some point during timeof-flight the field should be switched to its measurement value. If the scattering lengthis tuned to a large positive value the increase in interaction energy causes a more rapid
expansion of the cloud, while if it is tuned to large negative values three-body processes
heat the cloud. Both of these processes produce an enhancement of the cloud size after
further time-of-flight2 . This produces a sharper feature than our initial calibration as
the density of the cloud after some TOF is greatly reduced, so an effect is only seen forvery strong interactions.
Our first attempts at calibration in this manner failed due to the inability to ramp
the Feshbach coils quickly and precisely, probably due to the large self-inductance of thecoils. We therefore decided to use the vertical bias coils around the science cell to produce a small field offset which could be quickly switched off. A small current producing
a field of about 3.5 G was turned on once a cold thermal gas had been produced in the
CDT. The current in the Feshbach coils was then ramped to its measurement value,with the opposing bias field keeping the atoms away from resonance. After 4 ms TOF
the bias field was switched off quickly, while the current in the Feshbach coils remained
constant until being turned off just before imaging. Fig. 2.41 b shows the cloud-width
measured with varying final field value. The sharp peak is close to the expected value.
This measurement gives a resonance position of 402.3(3), in agreement with the pre1

We carried out several different calibrations with different initial fields and final hold times, andall produced similarly-shaped loss profiles.
2
This effect is useful for calibrating the resonance position, but is detrimental when attempting toaccurately determine temperature after TOF expansion, as discussed in §5.5.2.

94

2.13 Field and resonance calibration

a

1 .0
0 .9

0 .7

N

(a .u .)

0 .8

0 .6
0 .5

b

W id t h ( a . u . )1 .0 00 .9 50 .9 00 .8 5
0 .8 0
3 9 8

4 0 0

4 0 2

4 0 4

4 0 6

4 0 8

4 1 0

F ie ld ( G )Figure 2.41: Feshbach resonance calibration. a Surviving atom number after 100 ms holdtime at the final in-trap field value. The centre of the broad asymmetric resonance feature isshifted to the attractive side of the expected resonance position of 402.50 G (dashed line) due tothe enhancement of three-body processes at negative scattering lengths. b Cloud width at theend of 18 ms TOF after the field is switched to its final value 4 ms after release from the optical
trap. This produces a sharp feature near the expected resonance position. The background risein width with increasing field is due to the pre-TOF field increasing, being offset by 3.5 G fromthe measurement value, causing heating before release from the trap. Error bars are statistical.

cisely determined value of 402.50(3). Having satisfactorily reassured ourselves in thismanner we chose to use the latter for calibration of the scattering length.

2.13.3

Zero crossing measurement

To independently calibrate for the width of the resonance we looked for the zero crossing
position. For attractive interactions the mean-field interaction energy decreases withincreasing density, so BECs are unstable against collapse [83, 84]. We therefore createdcold atomic clouds of 39 K with a small condensed fraction at our standard evaporation
scattering length of 135 a0 , before tuning the field close to the zero-crossing position.
Fig. 2.42 shows a typical series of five images where the final field strength is decremented through the crossing from 351.0 down to 349.3 G. The variation in scattering
length across the images is less than a single a0 (see Table 1.1), so the sharp drop insize observed from image b to image c is attributed to having just passed into the attractive regime. The location of the zero crossing is therefore somewhere in this region,95

2.13 Field and resonance calibrationab

351.0 Gc

350.6 Gd350.2 G
0

1

OD

2

e

349.7 G

349.3 G

3

Figure 2.42: Sample images of Bose-Einstein condensates after the Feshbach field has been
swept close to the zero-crossing point. Images a and b are representative of all images seen
close to but on the repulsive side of the zero crossing. A sharp drop in size is seen between
images b and c, suggesting the zero crossing lies between the field magnitudes for these two
images. The collapse of the condensate is more obvious in images d and e. The colour bar
shows the conversion between optical density and colour, though in these images the opticaldensity saturates at the centre of the BEC due to high atomic column densities.

Figure 2.43: The complete setup on the vacuum chamber table used to reach quantum degeneracy of 39 K.

i.e. B0 = 350.4(2) G, consistent with ∆ = −52 G. Images d and e more clearly show
the collapse of the condensate.
The system is shown in its completed state in Fig. 2.43.

96Chapter 3

Experimental sequence“ You know my methods, Jeeves. Apply them.”– P. G. Wodehouse (Right Ho, Jeeves)Having outlined the techniques involved and some of our particular experimental parameters in the previous chapter, our process for creating Bose-Einstein condensates
and the optimisation of experimental stages are discussed, as summarised in [85]. Muchof the process is similar to that used to produce the first 39 K BEC in the Inguscio groupat the University of Florence [7], although our original intention was to carry out sympathetic cooling in a quadrupole trap, as discussed later in the chapter. As the QUIC
trap wasn’t in the original plan it is not perhaps as well integrated as if we had intended
to use this trap from the outset. However, the long trap lifetime it affords leads to very
efficient sympathetic cooling, resulting in large 39 K condensates in the crossed dipole
trap. At the end of the chapter a summary of the experimental stages is provided in
the form of Fig. 3.13.

3.1

Laser cooling

The first stage of the experiment is to load both 39 K and 87 Rb into the MOT fromthe background vapour provided by the getters. Often the atom sources are only fired
once or twice a day to keep the pressure at a usable level. When taking data though
we switch to applying a low current 30 s burst during every run to keep experimental
conditions as stable as possible. The current is applied only while the experiment isbeing carried out in the science cell so as not to interfere with the loading of the MOT.

3.1.1

Monitoring MOT loading

In order to monitor the loading of the MOT during experimental sequences we use thestandard fluorescence detection method. A lens is positioned close to a spare window of
the MOT chamber to collect fluorescent light and direct it onto a photodiode, producing
a current which is converted to a voltage for display on an oscilloscope. If the MOT
97

3.1 Laser coolingparameters (i.e. magnetic field gradient, light frequency and intensity) remain constant
the number is roughly proportional to the photodiode current. The absolute number
can be gauged from the equation
 
 
Ipd /Rpd
1
4π
,
N=
Ω
T
Γ̄scatt hν(3.1)

where Ω is the solid angle subtended by the aperture of the light-collecting lens, Ipd isthe photodiode current, Rpd is the responsivity of the photodiode (i.e. the conversion
factor of incident light power to photodiode current), Γ̄scatt hν is the mean fluorescentpower of an atom in the MOT, and T is the transmittance of the optics between theatomic cloud and the photodiode. This relationship holds well for low atomic densities,
but breaks down when there is a significant amount of photon reabsorption within thecloud. At this point the fluorescence to number conversion becomes sub-linear. Earlycomparisons with absorption imaging in the MOT suggest that the fluorescence acts
well as a rough guide in our system.
The conversion is fairly straightforward for 87 Rb as Γ̄scatt is relatively simple tocalculate, but the small level splitting in 39 K complicates matters [86]. The number
can be calibrated by absorption imaging, but most of the time relative day-to-day
comparisons are all that we are interested in. It should be noted that at high densities
the linear conversion of fluorescence to number is lost due to photon reabsorption in
the cloud, which also limits MOT densities and temperatures [87]. Additionally a Sonydigital video camera (using ‘nightshot’ mode where the infra-red filter is removed)is used to provide constant live footage of the MOT on a monitor. This facilitatesobservation of the size and position of the trapped cloud for balancing the MOT beams
and optimising the optical molasses cooling stage.

3.1.2

MOT optimisation

NumberThe capacity of the MOT is essentially limited by the proportion of particles in the
background vapour that are capturable to those that are not. The number saturates as
the capture and loss rates become equal. Collisions with the background vapour are themain cause for trap losses at low densities, while light-assisted collisions become more
important at high densities. Re-absorption of spontaneously emitted photons produces
a repulsive force between atoms which limits the cloud density. Full descriptions of
MOT dynamics are fairly complex [88], but generally speaking the capacity can be
maximised by using high intensity cooling beams with large detunings such that thecapture velocity is increased. For bosonic potassium this is slightly complicated by thesmall hyperfine splitting, and both laser frequencies (as both contribute to the trapping
force) must be red-detuned from the entire upper-state manifold [13, 54].
To optimise the number we look at the fluorescence detected at the monitoring983.1 Laser cooling

1.0

NNmax

0.8

0.6

0.4

0.2
-60

-50
-40
-30
Cooling light detuning HMHzL-20

Figure 3.1: Example of 39 K MOT number optimisation as discussed in the text, where thecooling light frequency detuning is being varied. These data were taken with the repumping
light detuned by 31 MHz.

photodiode. Changing MOT parameters generally alters the fluorescence to number conversion factor, so is not a reliable direct gauge for optimising the number oftrapped atoms. However, if an alteration is made quickly with respect to the MOT
loading time, one may observe whether the MOT fluorescence consequently grows or
shrinks, indicating whether the situation has improved or worsened. This is the standard method for rough optimisation of beam alignment, field gradient and so on. Fora more precise comparison we optimised the frequency detunings by letting the MOT
saturate for a given frequency detuning before quickly switching to resonance, recording the transient peak fluorescence. An example set of data for a given detuning of
the repumping light is shown in Fig. 3.1. We found detunings of −7.6 and −4.6 Γ(Γ = ΓK ≈ ΓRb ≈ 2π × 6.0 MHz) for the cooling and repumping light respectively1
and B 0 = 4 G cm−1 (the field gradient along the weak axis) maximised the numberof trapped 39 K atoms. The total power in the six MOT beams is 310 mW where eachbeam has a diameter of around 3 cm, resulting in intensities of several times Isat . When
operated on its own the potassium MOT saturates at around 1 × 109 atoms.
Rubidium is more simple to work with having a large splitting between the |F 0 = 2i
and |F 0 = 3i levels. The 87 Rb MOT is very robust in terms of number. We use
a detuning of −3.1 Γ for the cooling light, with the weak repumping light close toresonance. There is a total of 130 mW of cooling light in the cooling beams, and a totalof 10 mW of repumping light. The number of atoms in the single-species rubidium
MOT saturates at around 5 × 109 .
1

Detunings are measured from the specific transition frequency in question, e.g. for repumping lightthe |F = 1i → |F 0 = 2i transition.

99

3.1 Laser cooling

Figure 3.2: Photo of the fluorescence produced by the dual-species MOT. The horizontal
compensation coils can also be seen, wound around the adjacent MOT chamber windows.

Temperature
The large detunings and low field-gradient used to maximise the amount of potassium in
the MOT unfortunately result in high temperatures due to the small damping coefficientand low densities. This is the opposite of what is desired for evaporative/sympathetic
cooling, so 40 ms from the end of MOT loading B 0 is suddenly increased to 6 G cm−1 ,while the detunings of the cooling and repumping light are jumped to −1.5 and −4.1 Γrespectively. The duration of this stage is short enough to avoid any noticeable loss of
atoms while the temperature is greatly reduced [54]. During this stage the intensity of
the repumping light is also reduced slightly. The cooling frequency ends up within the
excited state manifold in a more analogous situation to standard MOT setups. The
effect is to increase the frictional Doppler force on the trapped atoms, cooling them
down.

3.1.3

Dual-species MOT

When both species are simultaneously loaded the peak number of potassium atoms
is reduced by a factor of ∼ 3 by interspecies light-assisted collisions [89], while theamount of rubidium is only reduced by around 15 %. This is not a great problem as
we desire a large ratio of rubidium to potassium for sympathetic cooling, though it
should be possible to reduce the problem (at the expense of ideal alignment with themagnetic field) by displacing the two MOTs from one another using a ‘push beam’[90] if required. In order to control the amount of trapped potassium, we initially load
rubidium on its own for around 15 s before turning on the potassium light for between
0.5 and 3 s. A picture of the loaded MOT is shown in Fig. 3.2.
100

3.1 Laser cooling

3.1.4

Optical molasses

To cool the atoms further before loading the magnetic trap a brief stage of opticalmolasses is required. At this point, it is particularly important for the six MOT beams
to be well balanced so that when the magnetic field is turned off (and the atoms are
no longer trapped) the expansion of the cloud is slow and symmetric. The standard
technique of matching the MOT position for strong and weak gradients is used toachieve this, while a more quantifiable signal is provided by observing the decay of thefluorescence on the photodiode when suddenly switching off the magnetic field.
During the molasses stage the rubidium cooling light detuning is increased to −4.9 Γ
and is reduced in intensity to allow low temperatures to be reached by sub-doppler
cooling processes. The potassium level-splitting once again complicates matters though.
It seems logical that when the level-splitting, ∆ , between the upper-state cooling level
and its adjacent state is much larger than the natural linewidth Γ that the adjacentlevel will not interfere with sub-doppler processes, which is indeed the case. Moresurprisingly, it has been shown that when ∆ is smaller than Γ , sub-doppler processes
continue to work well, as demonstrated in [91]. However, when ∆ ∼ Γ, as is the casefor 39 K, the proximity of the neighbouring levels can disrupt the sub-doppler processes,
and indeed at the time our work was carried out sub-doppler laser cooling was yet tobe observed for 39 K.
We optimised the final temperature of 39 K within our cooling scheme by dropping
the relative repumping light intensity to roughly a tenth of the cooling light intensityfor 6 ms of molasses, observing a minimum final temperature in the absence of 87 Rb of∼ 200 µK, not far off the Doppler limit TD = 145 µK. When rubidium is also present thetemperature is limited to around 600 µK, probably due to a higher starting temperature
at the end of the MOT phase. Rubidium is cooled well below the Doppler limit to∼ 50 µK in spite of the presence of potassium.
Recent findings [92, 93] have shown that it is in fact possible to reach the subDoppler region for bosonic potassium isotopes with laser cooling. This is achieved bycontrolling the natural pumping of atoms into a dark state, minimising heating due
to the reabsorption of spontaneously emitted photons [94]. Our current method is
not very dissimilar to the successful sub-Doppler schemes, the main development of[93] being a clever ramping of frequency and intensity to address the majority of theatoms1 . This should be relatively easy to implement, and is a major step on the way
to directly cooling 39 K. This progress may induce a boom in 39 K experiments due to
its more favourable properties.
1

It is possible that sub-Doppler cooling is achieved in our system for some fraction of the atoms,
producing a bimodal velocity distribution like that shown in Fig. 3 (b) of [93]. In this picture, lowvelocity atoms are cooled further (‘the rich get richer’) while energetic atoms are unaffected or possibly
even heated.

101

3.2 Magnetic trapping and transport

3.2

Magnetic trapping and transport

At the end of the optical molasses stage the repumping light stays on for both species
for a millisecond after the cooling light is switched off to ensure all the atoms are in the|F = 2i state. At this point the pumping guide field is switched to a few gauss, before
a 200 µs burst of intense pumping light is applied. The circularly polarised pumping
light is resonant with the |F = 2i → |F 0 = 2i transition for rubidium, while it is tuned
slightly above (∼ 2 Γ) for potassium due to the close proximity of the |F 0 = 1i level.
Pumping to the |F, mF i = |2, 2i state is about 80 % efficient for rubidium, 65 % forpotassium.
To capture the atoms, the quadrupole field produced by the transport coils is
switched on at B 0 = 20 G cm−1 (weak axis) and then ramped linearly in 50 ms to
80 G cm−1 to adiabatically compress the cloud. During this ramp up the track is energised to prepare for the transport. The coils then travel the 74 cm to the science cellin under two seconds as described in §2.5.3.
During transport the atomic cloud has to pass through a minimum aperture 10 mm
in diameter. Any atoms that happen to touch the walls as the coils pass along thesystem adhere to the vacuum chamber and are lost. As only the most energetic atoms
can reach the walls, the effect is analagous to an inefficient stage of evaporative cooling.
Atoms with an energy greater than ∼ 2.7 mK are lost, so it is mainly potassium thatis affected due to its higher temperature. Around 40 % of the potassium atoms are
removed via this process. Losses due to Majorana spin-flips are minimal during thisstage as the temperature of the atoms is still relatively high.
Once the atoms have reached the science cell there is a one second wait to ensure
the track settles to the correct location within its precision bounds before the track is
de-energised. During this pause one of the microwave generators is turned on ready tostart evaporating rubidium. Due to some thermalisation between the two species (andthe loss of high energy potassium atoms during transport) the rubidium temperature
has increased slightly and the potassium temperature decreased by this time.

3.3

Sympathetic cooling

3.3.1

Choice of trapping potential

Because of our earlier experiences of direct cooling in a quadrupole trap (see Appendix
B) we initially planned to use this potential for sympathetic cooling. The fact thatpotassium has to end up in a purely optical potential meant that, instead of using
an optical plug to suppress Majorana spin-flips, we intended to offset the CDT fromthe magnetic zero, as done in [55]. In fact this method allowed us to apply a 10 s
evaporation sweep when cooling rubidium alone, before transfer to the CDT for further
forced evaporation, resulting in BECs of 5 × 105 atoms.
102

3.3 Sympathetic cooling

Unfortunately, we were unsuccessful in using this method for sympathetic cooling.
This was due in part to the relatively small K-Rb interspecies background scattering
length1 , which precludes the use of such short evaporation sweeps. Longer sweeps arealso prohibited by inelastic collisions with 87 Rb atoms in the |2, 1i state. These are
created by Majorana spin-flips, in spite of the presence of the ODT. If we were able
to use a stronger quadrupole field the evaporation rate could be increased, perhaps
making this method more viable.
While attempting to find the best solution for sympathetic cooling we were alsoable to condense 87 Rb in an optically-plugged quadrupole trap [4], a time-averaged
orbiting potential trap [59], and directly in the QUIC trap. With our system we foundthe QUIC trap to be by far the most effective for sympathetic cooling because of theelimination of |2, 1i 87 Rb atoms, as discussed below. Before turning on the QUIC
trap a short microwave evaporation sweep is applied in the quadrupole trap to aid theloading efficiency of both species. We use a two-second linear sweep from 6909.68 to
6879.68 MHz, corresponding to removing rubidium atoms with energies above 2.4 to
1.4 mK.

3.3.2

Transfer to the QUIC trap

Transferring the atoms to the QUIC trap is not quite so simple as just ramping up
the Ioffe and bias coils to their final values. Initially, an increase in just the Ioffe coilcurrent draws the atoms towards the coil due to the translation of the zero-field point,
which can bring the cloud into contact with the wall of the vacuum chamber. Once themagnetic zero disappears, any further increase in Ioffe current pushes the atoms backaway from the coil, as the point at which B 0 is cancelled by the Ioffe field gradient gets
further from the Ioffe coil. Additionally, it is important to ensure that the trap does
not become too shallow at any stage.
Taking these factors into consideration resulted in the ramping scheme shown in
Fig. 3.3. The bias coils are used to counteract the initial tendency of the trap to movetowards the Ioffe coil (i-iii) before the current in the quadrupole coils is reduced slightly
to increase the height of the trap near the Ioffe coil. Once the magnetic zero is lost(iv) the coil currents are slowly ramped to their final values (v). The minimum total
trap depth during this process is ' 1 mK, occurring at stage iii. Applying this method
results in 4 × 108 rubidium atoms and up to 4 × 107 potassium atoms in the QUIC
trap.

3.3.3

Cooling in the QUIC trap

With sympathetic cooling there is a natural tradeoff between the size of the sympathetically cooled load and the lowest achievable temperature. Our particular aim is to1

aK-Rb = 36 a0 [65], to be compared with the rubidium intraspecies scattering length aRb = 99 a0
[95].

103

3.3 Sympathetic cooling

a

i

b i

ii

ii

iii

iii

iv

iv

v

v

Figure 3.3: Transfer of atoms from the quadrupole trap to the QUIC trap. a Trapping
potential along the Ioffe coil axis at five stages during transfer. In each plot the solid blackline gives the potential ∝ |B| in terms of thermal energy, while the faint red line illustrates thevariation of the magnetic field B. The dashed lines represent the sides of the science cell. b
The current through the transport (black), Ioffe (blue) and bias (red) coils during transfer tothe QUIC. i Initially current is running through just the transport coils forming a quadrupole
trap. ii The currents in the Ioffe and bias coils are ramped up, reducing the total trap depth.
iii The trap depth reaches its minimum value of ∼ 1 mK. iv The point at which a QUIC trap isformed. v The currents reach their final values, ready for sympathetic cooling.

cool the largest possible number of 39 K atoms to a temperature of about 5 µK. Thisis mainly motivated by the 73 µm difference in gravitational sags (discussed in §2.6)
for the two species in our QUIC trap. This corresponds to the vertical thermal radiifor the two species RTRb ≈ RTK at a temperature of about 4 µK. Even for very small
potassium loads sympathetic cooling will become inefficient at around this temperature
due to the reduced spatial overlap of the clouds [96]. It is also motivated by the loading
efficiency of the CDT from the QUIC, though this is a slightly cyclic argument as the
CDT parameters themselves were also adjusted for this purpose.
The substantial field offset at the trap minimum of the QUIC trap eliminates the
detrimental spin-flips of the quadrupole trap, but during evaporation a finite |2, 1i
state population for rubidium is still observed. One possible mechanism that would
cause this is illustrated in Fig. 3.4. Atoms that are successfully transferred to the |1, 1i
state accelerate away from the trapped atoms, but the evaporating microwave frequency
becomes resonant with the |1, 1i → |2, 1i transition at a particular field strength. Thereis some finite chance that atoms will be transferred to the |2, 1i state and once again
become trapped. This mechanism is suggested in [97].
The field offset of the QUIC trap is once again beneficial however, as a second microwave frequency can be used to transfer any |2, 1i atoms to the |1, 0i state (illustrated
104

3.3 Sympathetic cooling|2, 2i|2, 1i|2, 0i|2, −1i|2, −2i|1, −1i|1, 0i|1, 1iFigure 3.4: Possible mechanism by which rubidium atoms are transferred to the |F, mF i =|2, 1i state, and how they are removed. The variation of the energy levels with distance fromthe trap centre (dashed line) is shown (the gap between manifolds is not to scale). Microwave
radiation used for evaporation to the |1, 1i state is also resonant with the |1, 1i → |2, 1i transition
further from the trap centre (black arrows). The finite field offset at the trap centre allows asecond microwave frequency to be applied to permanently remove atoms from the trap (redarrow).

by the red arrow in Fig. 3.4) which will then leave the trap permanently. We transmit
this second microwave frequency with the same antenna, combining the signals before
amplification. A saw-tooth sweep is used for the clean-up microwave signal, sweeping
between 6838 and 6844 MHz with a 5 s period. This has no effect on the atoms in the|2, 2i state.

3.3.4

Sympathetic cooling results

The results of our sympathetic cooling experiments are summarised in Fig. 3.5. Firstly,
as shown in Fig. 3.5 a, we characterised the rubidium cooling reservoir. Allowing forthe fact that, for the same trap depth, the temperature of rubidium will inevitably behigher in the presence of potassium, we set the target temperature for rubidium to be∼ 3 µK. At the start of evaporation the remaining rubidium atoms have a temperature' 100 µK while the potassium atoms are at ' 200 µK. We therefore use a starting
microwave frequency of 6873.68 MHz, corresponding to a thermal energy of ∼ 900 µK,
providing a high starting η. For efficient sympathetic cooling we evaporate the rubidiumwith a long exponential sweep lasting 56 s1 with a decay constant of 14 s, to a final
1

Shorter sweeps are found to be just as effective for cooling rubidium on its own due to its highrethermalisation rate, but are not sufficient when potassium is also present.

105

3.3 Sympathetic coolingabFigure 3.5: Evaporative and sympathetic cooling in the QUIC trap. a Cooling trajectories
in single- (squares) and two-species (circles) experiments. Open and solid markers represent
87
Rb and 39 K respectively. The two fits show the difference between the steep trajectory for
sympathetic cooling (solid line, γsym = 20) and that for efficient evaporative cooling (dashedline, γ = 3.6). Vertical error bars lie within the markers. b Final temperature reached bysympathetic cooling for different sizes of the 39 K load. The dashed line shows the temperaturereached by 87 Rb for the same evaporative cooling sweep in the absence of 39 K.

frequency of 6847.24 MHz. This is followed by a 6 s period of plane evaporation where
the frequency is held at its final value.
Such a long evaporation sweep results in a very high cooling efficiency with γ = 3.6,
where γ was defined in equation (2.33). Also shown in Fig. 3.5 a is the phase-spacedensity evolution of a typical 39 K cloud used in our experiments. The near-verticalevaporation trajectory is the hallmark of effective sympathetic cooling. In this instancethe cloud reaches a final temperature of about 5 µK while the atom number is reducedby less than a factor of two, from approximately 25 × 106 to 14 × 106 . This gives
an effective cooling efficiency γsym ≈ 20. Around a 20 % reduction in NK is expected
over the evaporation period due to the trap lifetime, so the observed atom loss shows
evidence for the presence of other inelastic processes, but the effect is not too large.
106

3.4 Quantum degeneracy in the CDT

3
87 Rb

2

1
39 K

0
Figure 3.6: Sympathetic cooling in the QUIC trap. The field of view is 3.2 × 2.4 mm , the OD
scale is given on the right. Images of 87 Rb (top) and 39 K (bottom) are taken at three points
through the sympathetic cooling process after a fixed TOF. The potassium atom number does
not decrease much, but the cloud is clearly cooled. The saturation of OD in the rubidium
images is caused by imaging such large atomic samples on resonance.

Also shown in the same panel is the effect that the potassium load has on thecooling trajectory of 87 Rb. The path starts to deviate at the point at which the load
becomes a significant fraction of the total atom number (NRb /NK . 10), bending toalmost coincide with the potassium cooling path towards the end of evaporation. This
convergence also suggests that with this particular potassium load we are reaching thelimits of the effectiveness of sympathetic cooling. Further evaporation past the point at
which the heat capacities of the two species becomes comparable will have little benefit
for the further cooling of potassium.
Fig. 3.5 b shows the temperature achieved at the end of sympathetic cooling forvarious final amounts of potassium in the QUIC trap. For low NK the temperaturetends to the 2.9 µK attained by rubidium when cooled alone. The quantity of potassiumthat we choose to load into the MOT is determined by the amount that can be loaded
into the CDT. This in turn depends on the temperature of the atoms being transferredfrom the QUIC trap, as is discussed below. A sequence of images taken at different
points through the sympathetic cooling process is shown in Fig. 3.6, highlighting theincrease in phase-space density of 39 K with minimal atom loss.

3.4

Quantum degeneracy in the CDT

3.4.1

Cooling methods in the dipole trap

Once the atoms have been transferred to an optical trap they are no longer in a statedependent potential. This gives us the opportunity to bring Feshbach resonances intoplay by transferring the atoms to a suitable state and applying a uniform magnetic field.
We are thus provided with two clear options: continue with sympathetic cooling, or
remove the remaining rubidium and directly evaporate potassium. Sympathetic cooling1073.4 Quantum degeneracy in the CDT

a

b

c

d

Figure 3.7: Illustration of equipotentials for a cut through the bisection point of a crossed
optical dipole trap. The beams are horizontal and are perpendicular to one another, with the
cut made vertically such that one beam is entering the page, the other is parallel to it. The
lighter the colour the higher the total potential. The field of view is 2 w0 square. a High-opticalintensity regime, where energetic atoms are lost along the beams. b Decreasing the intensity
leads to a scenario where atoms may be lost along the beams or from the bottom of the trap.
c Low-optical-intensity regime, where energetic atoms are lost from the bottom of the trap. d
The trap eventually vanishes with further reduction of the intensity.

has the added enticement here of an interspecies Feshbach resonance that may be used
to increase the cross-thermalisation rate [98]. Up until this point our experimental
process has, in essence, closely followed that of the Inguscio group [7], where the former
method is successfully applied. Given theirs is the only system previous to ours to havesuccessfully condensed 39 K, it might seem sensible to bow to experience and continue
to follow their course. This might also seem logical given the high efficiency observed in
the sympathetic cooling stage in the QUIC trap. However, as might be deduced fromthe fact that we aimed to use up all of the rubidium in the previous step, we choose todirectly cool potassium. The considerations that informed this decision are discussed
below.
The main factor is the competition between the optical and gravitational potentials,
shown in Fig. 3.7. In the high-optical-intensity regime the effect of gravity is negligible
and energetic atoms are lost along the beams1 , while in the low-optical-intensity regime
gravity distorts the Gaussian optical potential such that atoms fall from the bottomof the trap. The red detuning of the beam produces a stronger optical potential for
rubidium than for potassium. The effect of gravity is also stronger for the more massive
rubidium atoms. This means that above a certain laser power the trap depth will begreater for rubidium, while below this crossover point it will be greater for potassium.
Sympathetic cooling is only possible in the latter regime, i.e. where mainly rubidium
atoms are evaporated.
In the limit that the distance from the centre of the trap to the wall of the sciencecell is small compared to the Rayleigh length of our focussed trapping beams we may
approximate the combined optical and gravitational potential for our trap geometry as
α −2(x2 +z 2 )/w02 α −2(y2 +z 2 )/w02
z
− e
− e
,
(3.2)
U (x, y, z) = mgw0
w0
2
2
1

The Rayleigh length, zR = πw02 /λ , for typical focal beam waists, w0 , and laser wavelengths, λ , ison the order of centimetres, so trapping along the beam axis is negligible.

1083.4 Quantum degeneracy in the CDT

where α is the ratio A/mgw0 , A being the peak optical potential shift defined in §2.8.3.
Here α has simply been substituted into equation (2.44). By switching to dimensionlessvariables z 0 = z/w0 and U 0 = U/mgw0 this reduces toU 0 (0, 0, z 0 ) = z 0 − αe−2z
and
U 0 (0, ∞, z 0 ) = z 0 −02

α −2z 02
e
2

(3.3)

(3.4)

for the relevant cases we will consider here. We can now find some useful systemparameters. For example, by differentiating (3.3) twice with respect to z 0 we find thatthe minimum gradient of the total potential is at z 0 = −1/2 . By setting the gradientat this point to be zero we obtain the value of α at which the atoms are no longer√
trapped, α0 = e/2 ≈ 0.82 .
To find the crossover point where the trap depth is equal for our two atomic species
we first note that αRb = (ARb mK /AK mRb )αK . The trap depth at this point forpotassium is the difference in potential between the local minima z10 and z20 of (3.4) and(3.3) respectively, while for rubidium it is the difference between the local maximum
z30 and local minimum z40 of (3.3). Casting the equations in terms of αK , the condition
that we are at extrema gives us
ARb mK 0 −2z302
1 0 −2z102
ARb mK 0 −2z402
102= z20 e−2z2 =
=
=−
z e
z e
z e
,
2 1
AK mRb 3
AK mRb 4
4αK
while equating the trap depths at this point means we must also satisfy
αK −2z102   0
02− z2 − αK e−2z2
z10 −
e
2
 


mRb
ARb mK
ARb mK0−2z3020−2z402=αK e− z4 −
αK e
.
z3 −
mK
AK mRb
AK mRb

(3.5)

(3.6)

From (2.44) we may use
ARbΓRb λ4Rb=AKΓK λ4K

λ2K − λ2
λ2Rb − λ2≈ 1.172(3.7)

for λ = 1070 nm. Solving (3.5) and (3.6) simultaneously, we find the value of αK at
which the trap depth is equal for the two species to be ≈ 4.01 (with the corresponding
zi0 values).
Combining this result with the values found for z10 and z20 we find that the trap
depth at the crossover point is given by
Uc ≈ 1.97 mK gw0 .

(3.8)

Considering this in terms of thermal energy, we find Uc /w0 ≈ 0.091 µK µm−1 . To havea reasonable starting η (& 5) in the trap with an initial temperature of 6 µK we need109

3.4 Quantum degeneracy in the CDT

0.0

-0.1

-0.2
z¢

Potassium
Rubidium

-0.3

-0.4

0.8

1.0

1.2

1.4

1.6
ΑK

1.8

2.0

2.2

2.4

Figure 3.8: Displacement of the trap minima for 39 K (solid blue line) and 87 Rb (solid redline) with respect to the centre of the optical potential for varying αK . Displacement is given
in terms of dimensionless distance z 0 = z/w0 . The atoms become untrapped at z 0 = −0.5 ,
i.e. when the minimum has been displaced by half the beam waist. The dashed lines represent
a single vertical thermal radius either side of each trap minimum for η = 8, showing that theclouds no longer significantly overlap for αK . 1.86 .

a trap depth of at least 30 µK. To also have a trap in which sympathetic cooling may
be used straight from loading we must therefore use a beam waist of at least 30/0.091= 330 µm. However, that would require 40 W trapping beams to produce the required
depth, far greater than the power available to us with our laser.
Also worth noting is the fact that, as in the QUIC trap, the difference in the gravitational sag between the two species prevents sympathetic cooling past a certain point.
Fig. 3.8 shows how the trap minima for the two species vary with αK , with the corresponding extent of each cloud to a thermal radius RT , assuming η = 8. Sympathetic
cooling ceases to work when the thermal radii no longer overlap, below αK ≈ 1.86.
Thus sympathetic cooling is only possible in the range 1.86 . αK . 4.01. Fig. 3.9
shows a summary of these results for our chosen waist of 140 µm and laser wavelengthof 1070 nm. The 140 µm waist is the largest we could use to have a 30 µK trap depth
when using the full available optical power.
It is relatively simple to extend the above to calculate equivalent parameters forother species in crossed dipole traps, and it is hoped that the above discussion isuseful for anyone considering sympathetic cooling between any two species in such aconfiguration. Note that the calculations are valid for horizontal beams only, but thatthe bisection angle of the two beams does not matter. The upshot is that we must
abandon any hopes of sympathetically cooling 39 K in the CDT, and instead use the
Feshbach resonance in the |1, 1i state to allow efficient forced evaporative cooling.
1103.4 Quantum degeneracy in the CDT

30

87

Rb

Direct Cooling Only

Untrapped

CDT Trap Depth (µK)

39

20

10

0

Sympathetic
Cooling
Possiblef

K

Inefficient Sympathetic
Cooling of 39K

e
d

0

a

bc

2

4

a
b
c
d
e
f

αK
0.82
1.55
1.86
2.52
4.01
4.79

P (W)
0.63
1.18
1.42
1.92
3.06
3.66

6

Power per Beam (W)Figure 3.9: CDT depth versus laser power for both 39 K (solid line) and 87 Rb (dashed line) fora beam waist of 140 µm and wavelength of 1070 nm. The ‘kinks’ in each plot (at d and f) occur
at the boundary between two regimes where different mechanisms limit the trap depth. At
high power the atoms are least confined along the beams, whereas at low power they are least
confined vertically due to the distortion of the trap due to gravity (see Fig. 3.7). The tablegives the values of αK and corresponding power per beam P at six points of interest. a andb are the points at which 39 K and 87 Rb fall from the trap respectively, d and f the respective
kinks already mentioned, while c is the lower limit of sympathetic cooling of 39 K (see Fig. 3.8)
and e is the upper limit, i.e. the crossover point discussed in the text.

3.4.2

Transfer to the CDT

The centre of the QUIC trap during standard operation is offset from the centre of
the science cell towards the Ioffe coil by several millimetres. Unfortunately, the coil
mounts around the science cell limit the CDT position to a small region near the centreof the science cell, so the QUIC trap must be moved before transfer to the CDT canproceed. The current is reduced in the quadrupole coils and increased in the Ioffe andbias coils over the course of one second, reducing the tightness of the trap slightly, but
repositioning it at the science cell centre. Sympathetic cooling is not carried out in thisposition as the shallowness of the trap would require the use of a longer evaporation
sweep and the Ioffe coil would soon overheat with the increased steady current.
Once the QUIC trap is in position the CDT power is increased linearly to its maximum value over the course of 1 s. The currents in the three QUIC coils are ramped
down by a factor of two in the subsequent second before being turned off abruptly,
leaving the atoms trapped in the CDT, which at this point has a depth of ∼ 30 µK for
potassium and ∼ 35 µK for rubidium (see Fig. 3.9).
The number of 39 K atoms that are transferred to the CDT as a function of thenumber in the QUIC is shown in Fig. 3.10. A constant transfer efficiency of ∼ 60 %is observed for samples with a temperature of up to 6 µK (equivalent to η ∼ 5), above
which the efficiency rapidly drops off. This constant factor is due to the imperfect111

3.4 Quantum degeneracy in the CDT

Figure 3.10: Transfer efficiency from QUIC to CDT. The numbers at each point are thetemperatures in the CDT (QUIC).

overlap between the CDT and QUIC potentials. Given the cigar-like (prolate spheroid)
shape of the QUIC potential and the squashed-sphere (oblate spheroid) shape of the
CDT potential this mismatch is difficult to overcome, although using a large beam
waist helps. The beam waist should be compared with twice the thermal radius of thecloud at the end of sympathetic cooling in the QUIC trap, roughly 160 (500) µm inthe radial (axial) directions. For comparison, the loading efficiency for a 100 µm beam
waist was found to be limited to ∼ 20 %, saturating at around a third of the available
CDT power.
In practice we load 8 × 106 potassium atoms into the CDT from the 13 × 106 thatare left in the QUIC after sympathetic cooling. Any remaining rubidium atoms are
removed from the trap to prevent the occurrence of spin-exchange collisions during thestate transfer of the 39 K atoms from the |2, 2i to the |1, 1i state. This is accomplishedwith a strong 2 ms pulse of resonant imaging light, which does not noticeably affect thetrapped potassium atoms.

3.4.3

Evaporation in the CDT

All of the atoms are transferred to the |1, 1i state by a 35 ms Landau-Zener sweep of
the bias magnetic field while applying RF radiation at 469.3 MHz, as described in §2.9.
To increase the thermalisation rate of the potassium we ramp up the current in the
Feshbach coils in 5 ms to produce a uniform magnetic field of 393.3 G, resulting in ascattering length of 135 a0 . The ramping time is kept short to avoid lingering near the
narrow (∆ = −0.47 G) Feshbach resonance at 25.9 G [81].
Forced evaporation is then carried out with a six-second exponential sweep of the
CDT beam power to its final value with a decay constant of 1.4 s , typically followedby another second of plane evaporation. The beam power is only required to span a112

3.4 Quantum degeneracy in the CDT

single order of magnitude (see Fig. 3.9), so is easily stabilised with our PID feed-backloop. The condensation point is reached for a beam power of ∼ 1.5 W, while further
reduction increases the condensed fraction to create quasi-pure BECs.

3.4.4

Number calibration

In order to accurately determine the number of atoms in the condensate we need tocalibrate the imaging absorption cross section as mentioned in §2.10.6. This requires atheoretical knowledge of the critical number Nc , which for an ideal gas in a harmonic
trap is given by (1.17). We can tune the scattering length to a small value so that theshift of Nc due to interactions (discussed in Chapter 5) is well quantified within meanfield theory. The finite-size shift [99] (of around 2 %) can also be taken into account.
However, the combined optical and gravitational potential of the CDT is by no means
perfectly parabolic. Firstly, the shape of the trap is different (or ‘anharmonic’), andsecondly the trap depth is not infinite. To account for this we must use both the true
expression for the trapping potential in our calculations and a truncated momentum
distribution.
Going back to the semi-classical approximation of §1.2.1 we again set µ = 0 in
equation (1.6), but now use
(r, p) =p2+ V (r)2m(3.9)

where V (r) is given by adding an offset to the potential given in (2.44) such that at
the trap minimum (situated at x = y = 0 , z = z0 ) the potential is zero. This results inV (r) = mg(z − z0 ) −i
A h −2(x2 +z 2 )/w02
2
2
2
2
2
+ e−2(y +z )/w0 − 2e−2z0 /w0 .
e2(3.10)

Making the substitutionsq=p2
Vand γ =
2mkB TkB T(3.11)in (1.6) and integrating over both q and space gives the critical number as
Nc =

2
√
3/2
π
λ
1

T

Z Z qmax

q 1/2

0

eq eγ(r) − 1

dqdr(3.12)

within the bounding surface γ(r) = Vmax /kB T = η . Vmax is the trap depth, such that
η is defined as for evaporative cooling. Assuming the cloud is collisionally transparent,
the maximum momentum for a trapped atom at point r is given by
p2max= Vmax − V (r) ,2m(3.13)

qmax = η − γ(r) .

(3.14)

or equivalently

113

3.4 Quantum degeneracy in the CDT

1.14
1.12

Nc Ncharm

1.10
1.08
1.06
1.04
1.02
1.00
0

0.05

0.10
kB TVmax º1Η

0.15

Figure 3.11: The anharmonic shift in critical number for a set of typical experimental parameters. Black points are calculated with the momentum cutoff, red points represent the upperbound where qmax → ∞ . The true potential has a larger volume than the extrapolated harmonic potential so this tends to increase the critical number. The momentum cutoff becomes
important below η ≈ 10 where the two lines diverge. As T tends to zero the critical number
tends to that of the harmonic case as expected.

Setting this as the upper q limit we can now numerically integrate (3.12) and compare
the critical number with that given by the ideal harmonic trap in (1.17).
In this comparison, the harmonic trap extrapolated from the quadratic terms of
the Taylor expansion around the trap minimum is considered. The effect of the alteredshape of the trap is to increase the trap volume and therefore increase the criticalnumber, while the momentum truncation reduces the available states and therefore
also Nc . The truncation only becomes important when the thermal energy per atom
becomes a significant fraction of the trap depth, at an η of . 10. In reality, theexpression for qmax given above is not exact as collisions do occur within the cloud.
The calculated value of the ‘anharmonic shift’ is therefore a lower bound, and an
upper bound can be found by setting qmax = ∞. However, under typical experimental
conditions the values are similar and we are closer to the collisionless regime.
The resultant anharmonic shift of Nc for typical experimental conditions is positiveand in the range 5-10 %. The value depends strongly on the temperature of the cloud,while the shape of the trap itself also varies depending on the power in the dipole beams.
The shift is therefore calculated for specific sets of experimental parameters as required.
It is shown in Fig. 3.11 for a typical trap geometry over a range of temperatures. As
expected, as T → 0 the critical number tends to the value for the ideal harmonic trap.
The freshly determined Nc can be compared with the total optical density of anabsorption image of a weakly interacting gas at the critical point, allowing calibration of
the imaging absorption cross-section. The total OD can be found either by summing
the OD values of the individual pixels or by fitting the image with the g2 function(2.59). The latter method also allows the temperature of the cloud to be determined,114

3.4 Quantum degeneracy in the CDT

3
2
1

Line density(103 μm-1)

0
4
3
2
1
0

-200

0
200
Position (μm)

-200

0
200
Position (μm)

-200

0
200
Position (μm)

Figure 3.12: Condensation of 39 K atoms. From left to right the number of atoms in the
BEC varies from 1.5 × 105 to 4.2 × 105 , while the condensate fraction grows from 20 % to 80 %.
The bottom panels show the line densities of the clouds, obtained by integrating the images
along the vertical direction, emphasising the signature bimodal distribution of a condensate
and thermal gas. Images are taken after 20 ms time-of-flight for a fixed exposure duration. Thechanging aspect ratio of the condensate reflects the increasing importance of the gravitational
potential at low CDT powers.

which is required to calculate Nc in the first place. Our initial investigations showedthat the cross-section was reduced by factors of 1.9 ± 0.3 and 1.5 ± 0.3 for 39 K and 87 Rb
respectively. These values are liable to change over time, with alterations in the imaging
polarisation for example. A more careful calibration, carried out when investigating
very weakly interacting 39 K clouds during the saturation investigations of Chapter 4,
provided an updated value of 1.6 which has remained stable ever since.
Condensate and thermal atom numbers are extracted from absorption images where
both are present using a bimodal fit, combining the g2 function with the two-dimensional
Thomas-Fermi distribution function (2.64). Typically all fit parameters (e.g. widths,
amplitudes, centres) are allowed to vary within user-defined upper and lower bounds.
These fits are used to assess the results of the final stage of evaporative cooling. Samples are imaged as described in §2.10, usually after around 20 ms TOF. This is near
the limit of the TOF available to us for horizontal imaging as the imaging light passes
through the centre of the Ioffe coil, such that further TOF would cause the atoms tofall out of the path of the imaging beam. Once the raw images have been acquired they
are immediately processed into OD images, ready for fitting.

3.4.5

Condensation

The condensation point is typically reached with ∼ 106 atoms at around 400 nK, atwhich point ω̄ ∼ 2π × 85 Hz . Further reduction of the beam power results in quasi-pure
39 K condensates of over 4 × 105 atoms.

Sample shots for various condensed fraction

are shown in Fig. 3.12. These are the largest 39 K condensates to have been reported
115

3.4 Quantum degeneracy in the CDT

to date, representing a significant improvement on past work; the first 39 K condensates
reported contained 3 × 104 atoms [7], later being increased to around 105 in the same
system [12, 100]. Though condensate size isn’t the be all and end all for a useful cold
atoms system it is useful to be able to experiment on large samples, a basic example
being the small fractional error on Nc which is useful for the experiments discussed
in Chapters 4 and 5. The final atom numbers owe much to the high efficiency of
sympathetic cooling in the QUIC trap.
We have created condensates by evaporation with scattering lengths of up to ∼350 a0 , but the largest and most stable condensates are produced for scattering lengthsof around 100 a0 , similar to that of 87 Rb. In the same system we can directly coolrubidium to produce condensates of up to 8 × 105 atoms when the alignment of the
CDT is optimised for the transfer of rubidium from the QUIC trap.

3.4.6

Summary

The experimental sequence for producing 39 K condensates is summarised in Fig. 3.13.
Typical atom numbers and temperatures are specified at several stages of the process.

116

3.4 Quantum degeneracy in the CDT

117

3.4 Quantum degeneracy in the CDT

a Magneto-opticaltrap

loadedfromvapour of K and Rb. Followed by 40 ms
near-resonant MOT to reduce K temperature.
NK ∼ 2×108 , TK ∼ 1−2 mK
NRb ∼ 4×109 , TRb ∼ 300 µKt ≈ 15 sb Magnetic field removed for short ‘optical molasses’ stage to increase phasespace density.
TK ∼ 600 µK
TRb ∼ 50 µK

t ≈ 6 msc Optically pump atoms into |F, mF i =|2, 2i hyperfine ground state.

K pumping efficiency∼ 65%
Rb pumping efficiency∼ 80%t ≈ 200 µs

d Ramp up quadrupole coil for purely
magnetic trapping,then transport

atoms to the science cell using thetranslation stage.
t ≈ 2se Load atoms into QUIC magnetic trapby ramping up Ioffe and anti-bias coils.

NK ∼ 2.5×107 , TK ∼ 200 µK
NRb ∼ 4×108 , TRb ∼ 100 µK

118

t ≈ 5s3.4 Quantum degeneracy in the CDT

f Evaporate Rb by applying a microwave
radiation sweep to drive the |2, 2i →|1, 1i transition.

K sympatheticallycooled by elastic collisions.
NK ∼ 1.4×107 , TK ∼ 5 µK

6.8 GHzt ≈ 60 s

g Load atoms into dipole trap by ramping up power and turning off QUIC
trap. Remove remaining Rb with resonant light pulse.
NK ∼ 8×106 , TK ∼ 6 µKt ≈ 2sh Transfer atoms to the |1, 1i state bysweeping the bias magnetic field while
applying 469 MHz radiation.

t ≈ 35 ms

469 MHzi Tune scattering length to ∼ 135 a0 with
Feshbach coils. Apply forced evaporation sweep by decreasing optical power.

NK ∼ 106 , n(0)λ3T ≈ 2.6

t ≈ 6sj Release atoms from trap and image
them with a resonant light pulse after∼ 20 ms time-of-flight.

t ≈ 20 ms
Figure 3.13: Summary of the main experimental stages to produce quantum degenerate 39 K.
A brief description of each stage accompanies each illustration, along with typical parametersat the end of the stage. The approximate step duration t is framed in each caption.

119

Chapter 4

Lack of saturation in a Bose gas“ The Right Hon. was a tubby little chap who looked as if he had beenpoured into his clothes and had forgotten to say “when”.”– P. G. Wodehouse (Very Good, Jeeves)

4.1

Saturation: another approach

The phenomenon of Bose-Einstein condensation is inextricably linked to the concept ofsaturation; that is, saturation of the available excited states that the atoms may occupy.
If the number of identical bosons in the system exceeds the number of available excitedstates, the only place for surplus bosons to go is the ground state. At this critical point
the excited states are said to be saturated, heralding the onset of macroscopic groundstate occupation - the BEC.
How saturation comes about for an ideal gas in a three-dimensional harmonic trap
was shown in Chapter 1 by considering the semi-classical distribution. This description has many uses as has already been made apparent (hence its early introduction).
However, a simpler semi-classical concept may be used to clarify under what conditions
a BEC may occur. We concern ourselves with the single-particle energy states of thesystem, but instead of considering the individual quantised states we ‘smooth out’ thefine details by considering a continuous density of states, allowing us to perform an
integral rather than the more difficult sum. Of course, this approximation does nottake account of the ground state, which is again treated as a special case.
To illustrate the concept we may consider once more the three-dimensional harmonic
oscillator, precisely as treated in §2.1.1 of [29]. In this case the potential is given by1
V (r) = m ωx2 x2 + ωy2 y 2 + ωz2 z 2 ,
2

(4.1)

having the familiar quantised energy levels111
(nx , ny , nz ) = (nx + )~ωx + (ny + )~ωy + (nz + )~ωz .
2
2
2
121

(4.2)

4.1 Saturation: another approach

Here the nk are positive integers. At this point we make the semi-classical approximation (recalling as previously stated that it is only good for kB T  ~ω) by introducing
the continuous variables k = ~ωk nk and neglecting the zero-point motion (the inaccuracy introduced by this simplification is addressed in §5.2.1). The total number of
excited states G() up to an energy  is then given by
1
G() = 3~ ωx ωy ωz

Z −x −yZ −x

Z 

0

0

0

dz =

dy

dx

3
,
6~3 ωx ωy ωz(4.3)

from which the density of states g() naturally follows asg() =

dG()
2
.
= 3
d
2~ ωx ωy ωz(4.4)

In order to calculate the total number of available excited states NT , we must multiply the density of states by the Bose occupation factor (or Bose-Einstein probability
distribution function) f () , reproduced here for convenience:f () =1,e(−µ)/kB T − 1(4.5)

before integrating the product over all energy space to giveZ ∞NT =g()f ()d .

(4.6)

0

By inserting (4.4) and (4.5) into (4.6) and taking the limit µ → 0 we retrieve the criticalatom number Nc0 for a given temperature T as
Nc0 = ζ(3)



kB T
~ω̄

3
,

(4.7)

exactly replicating (1.17), the result obtained via the semi-classical distribution.
To extend this description to other trap geometries we consider a generalised versionof (4.1) describing a d-dimensional power-law potential,dX~ωk rk nk
V (r) =
, where ak =
2 ak
k=1

s

~
mωk

(4.8)

and the potential exponent along each axis k is given by nk . It is clear that (4.1) isrecovered simply by setting d = 3 and each nk = 2 . It is shown in [101] and [102] thatin general the density of states for such a system follows a simple power law,g() =

Cs s−1
,
Γ(s)

where

(4.9)

d

d X 1
s= +
2
nk
k=1

122

(4.10)

4.1 Saturation: another approach

and
Cs =

d
2s Y Γ(1 + 1/nk ).
π d/2 k=1 (~ωk )1/2+1/nk(4.11)

So, for example, the three-dimensional harmonic oscillator with d = 3 and nk = 2 gives
an s of 3 and hence the density of states is proportional to 2 , as confirmed by (4.4).
The corresponding value of Cs may be read from (4.4) (recalling that Γ(3) = 2) or
calculated from (4.11). So for the commonly investigated three-dimensional scenariosof a harmonic trap and a hard-walled box we find, respectively,(
1 3for d = 3, nk = 2, s = 3
~ω̄
Cs =

3/2
m
V
for d = 3, nk = ∞, s = 3/2 .
2π~3(4.12)In the latter equation V is the volume of the box, simply given by
V = (2 ā)3 where ā = (a1 a2 a3 )1/3 .

(4.13)

So where does this leave us with the conditions for saturation? The simple conclusion is that if the total number of available excited states is a finite value (however large
or small) then it must be at least theoretically possible for a BEC to exist. We must
therefore evaluate (4.6) using the density of states for the general power-law potential(4.9). Employing the substitution x = /kB T with µ → 0, this evaluates to
Cs (kB T )s
NT =
Γ(s)
= Cs (kB T )s

Z ∞
0
∞
X
p=1

xs−1
dx
ex − 1(4.14)

1,ps(4.15)

where (1.9) has been used. The sum in (4.15) is the well-known hyperharmonic series(a generalisation of the divergent harmonic series) which converges to ζ(s) only fors > 1 , and is divergent otherwise1 . This is illustrated by Fig. 4.1 , where the integrandof equation (4.14) is shown for various values of s. Therefore, macroscopic occupationof the ground state only occurs for s ≤ 1 at T = 0 , while the critical number for s > 1
is given by
Nc0 = Cs (kB T )s ζ(s) .
This leads directly to a simple expression for condensed fraction,
 s
N0
T
=1−
,
N
Tc0

(4.16)(4.17)

a relationship that is often used for temperature calibration. The conclusions of theabove discussion are simple but far-reaching; BECs can occur in any three-dimensional
trap, but not in a uniform two-dimensional system, while in one dimension the trap
1

This is easily shown using the integral test for convergence.

1234.2 Context

4

s=0
3
x s-1 Hãx -1L

9/2
2

3/2
1

0
0

1

2

3

4

5

x

Figure 4.1: Integrand of equation (4.14) for s = 0 → 9/2 in steps of 1/2. Lines representing
values of s under which the area is infinite (s = 0, 1/2, 1) are coloured red, the others black.
The area under these curves is Γ(s)ζ(s). Though the value of the function tends to infinity fors = 3/2 , the bounded area is finite.

must be stronger than a harmonic trap (i.e. nx < 2).
This standard view of Bose-Einstein condensation, as presented in statistical physics
textbooks (see e.g. [19, 29, 30, 103]) and indeed in §1.2 of this thesis, is naturally introduced along the lines of the original predictions of Einstein. In this picture the numberof ideal bosons in excited states, NT , reaches a limiting value, Nc , after which any additional particles must occupy the ground state, forming a Bose-Einstein condensate.
This comes about because the available excited states become saturated. The numberof particles occupying the ground state N0 is microscopic until this saturation point isreached, resulting in the simple occupation dependence on total number N shown in
Fig. 4.2.

4.2

Context

In reality we can never produce the exact conditions prescribed by Einstein. This ideal
scenario requires non-interacting atoms in thermal equilibrium, but interactions arenecessary for thermal equilibrium to exist. In fact, because of this the ideal statisticalphase transition proposed by Einstein has never been experimentally confirmed. However, weakly interacting Bose gases in harmonic traps are generally believed to closely
portray this picture. It is accepted that the presence of weak repulsive interactions
produces slight modifications, such as the small shift in Tc discussed in Chapter 5, butthat the essential description holds true.
As such, a saturated thermal component is often heralded as a signature of BoseEinstein condensation, as it was for the groundbreaking first observations of condensa124

4.2 Context

saturated

NT , N0

Nc

NT

N0

0
0

Nc
N

Figure 4.2: State populations of an ideal Bose gas. At fixed temperature, all the atoms
accumulate in the thermal component until all available excited states are filled, after which all
further atoms populate the ground state.

tion in a solid-state system [104] and of photons in a microcavity [105]1 . An attempt toclosely follow Einstein’s idea of a phase transition brought about at constant temperature while increasing the atom number was also believed to directly show saturation
behaviour [106]2 .
However, strong deviations from the ideal case have already been shown to exist.
The T 3 dependence of Nc in a harmonic trap leads to the expression
N0
=1−
N



T
Tc0

3
(4.18)

for the condensed fraction in an ideal system - equation (4.17) with s = 3 . Investigations carried out shortly after the first demonstrations of condensation in diluteatomic gases seemed to show consistency with (4.18) [107, 108]. More recently though
it has been shown that negative shifts in the detected condensed fraction below Tc of∼ 20 % (larger in relative terms) can be seen in a standard setup [109]. This is far
from a small correction, and directly corresponds to non-saturation of the gas. Also,
while studying the critical point of a harmonically trapped two-dimensional Bose gas
at ENS [110], the authors compared their findings with a three-dimensional gas [111].
They were surprised to see a value of dN0 /dN substantially lower than unity past thecondensation point.
1

However, Fig. 3 of [104] and Fig. 2 of [105] show growing thermal components as a condensate isformed.
2
In this reference, a spinor gas was studied in which condensation was brought about in one spin state
by transferring atoms from two others in which condensates were already present. The thermal atoms
therefore experienced a potential already modified by the existing condensates, probably obscuring any
saturation effects due to the emerging condensate.

1254.3 Experimental procedure

The purpose of the investigation described in this chapter is twofold: firstly to show
just how large an effect repulsive interactions have on the saturation picture of BoseEinstein condensation, and secondly by studying a range of interaction strengths (atwhich thermal equilibrium is ensured) to extrapolate to the non-interacting limit to see
if the saturation picture is recovered. This is made possible by our ability to decouple
the scattering length from other parameters, a feature which is of course not available
in many systems.
Much of the confusion surrounding saturation stems from a frequent lack of differentiation between saturation of atom number and saturation of available excited states.
In an ideal gas these terms are equivalent, but when repulsive interactions are included
they are not. Considering a mean-field description, the presence of the dense condensate leads to a flattening of the effective potential seen by the thermal atoms suchthat the trapping volume is increased, i.e. more excited states become available. In
this picture a condensate does indeed form because of saturation of the excited states,
but as atoms are added at constant temperature the thermal atom number must carryon increasing along with condensate number. We take ‘saturation’ to refer to atom
number, and therefore ‘non-saturation’ to refer to deviations from the ideal theory.

4.3

Experimental procedure

The actual experimental concept is a simple one. Partially condensed clouds of 39 K are
created in the CDT as described in Chapter 3, using the Feshbach resonance to tune
the scattering length to 135 a0 for the final stage of evaporative cooling. At this point
the optical power is ramped to a final fixed value, after which the scattering length istuned to its measurement value (Fig. 4.3). The sample is then held in the trap fora variable amount of time, up to several tens of seconds. During this period atoms
are gradually lost from the trap due to collisions with the background vapour, photon
scattering from the optical trap and three-body recombination. The trap depth foreach series is chosen such that the heating associated with these processes is cancelled
by residual cooling due to plane evaporation, producing a near-constant temperature(Fig. 4.4). Consequently the trapping frequencies vary between datasets, lying in therange ω̄ = 2π × 60-80 Hz. Elastic scattering allows the remaining atoms to rethermalise
at a far greater rate than particles are lost, and an equilibrium distribution between thethermal and condensed components is maintained. The cloud is then released from thetrap and imaged after ∼ 20 ms time-of-flight to allow number and temperature data to
be extracted.

4.3.1

Image fitting for saturation data

Condensate population N0 and thermal population NT are extracted using a bimodal fit
to the OD images, while the total number N is found separately by a direct summation
1264.3 Experimental procedure

Hold time400

2

CDT power

380

1

Feshbach field (G)CDT power (W)Feshbach field

360
0

5

10

15

Time (s)Figure 4.3: Scheme of the relevant final stages of the experimental sequence. The solid lines
show the time dependence of the CDT beam power and Feshbach field for a sample sequence,
while the dashed lines show the range of values covered by the complete set of experimental
series. At the end of the exponential evaporative cooling ramp in the CDT the power is held
for one second before it is ramped to its final value in half a second. The Feshbach field isthen ramped to its measurement value, after which the atom number is allowed to decay overa varying period before time-of-flight imaging.

400
200
0

b

200

T (nK)N (thousands)

a

100
0

Nc
0

20

40

Hold time (s)

60

Nc

0

20

40

Hold time (s)60

Figure 4.4: Variation of system parameters with trap hold time for the sample data series
shown in Fig. 4.5 and considered throughout this chapter. a Total atom number in the trap
N versus hold time. The data is fitted with a double exponential to guide the eye. b The
corresponding temperature T . The temperature shows no drift for N > Nc and has a standard
deviation of only 3 nK.

127

4.3 Experimental procedure

over the density distribution. The value of N0 + NT was found to be within 0.5 %of N for every usable image. A separate g2 fit is used to find the temperature, asinteractions affect the ideal thermal distribution function. A region around the centreof the distribution is excluded from the fit as interaction effects are strongest in the highdensity regions and around the condensate [78, 108]. We choose to exclude the atoms
within one thermal radius of the centre having found this to give stable temperature
measurements after similar studies to those found in §II.B of [73]. A variety of data
was fitted using a range of exclusion radii, and a plateau region was observed where
temperature measurements remained constant. Excluding too large a region produces
noisy data due to the loss of information, so the chosen radius was the minimum thatsafely lay within the plateau. The BEC lies within the exclusion region for all ourimages. The bimodal fit nevertheless gives accurate number information because of thequantity of free parameters.

4.3.2

Data processing

A total of 18 data series were taken with 39 K at temperatures in the range 115-284 nK
and scattering lengths in the range 40-356 a0 . Each series consists of up to ∼ 100 data
points, so to speed up processing images were automatically removed according to aset of criteria. First any blank images and those where the fit is clearly poorly matchedto the density distribution were removed manually. The remaining images were thenmasked if the total peak optical density exceeded 3, the detected N0 was lower than anominal value, and if the condensate or thermal cloud widths lay further than several
standard deviations from their averages over the entire series.
Though the temperature is fairly stable over each series as previously discussed, weapply small corrections to account for the remaining drifts. Instead of assigning each
image its individually measured temperature for this correction, we use a two-stage
fitting process based on image index and N0 . The latter method is used because of thelikelihood that any residual temperature fluctuations recorded are due to the fitting
process rather than real effects. The image index designates the ordinal position ofeach image in the series in terms of the order in which they were acquired. This isneeded as the temperatures of clouds produced by the system tend to decay slightly
over time1 . N0 and image index do not scale together as the hold times for each series
were taken in a random order in an attempt to reduce systematic errors.
The temperature assignment works as follows. The dependence on index tends tobe strongest, so firstly the measured temperature Tmeas is plotted against index i. An
exponential decay is used to fit the data, resulting in Ti (i). We then plot Tmeas /Ti (i)against N0 to reveal to what degree the temperature shifts from the index correction
with condensate size. A linear fit is used to give the factor f (N0 ). Each image is thenassigned a temperature T 0 (N0 , i) = f (N0 ) × Ti (i). The component atom numbers are
1

This is most likely due to the gradual heating of optical components in the system.

128

4.4 Hartree-Fock Approximation

NT

300

NT , N0 (thousands)Ncsaturated

200

100

N0
0
0

100

200

300

400

N (thousands)
Figure 4.5: Lack of saturation of a Bose gas for a sample series taken with a = 135 a0 and
T = 177 nK. NT (red points) and N0 (blue points) against total number N . The solid lines
show the predictions for a saturated gas (as displayed in Fig. 4.2). The critical number Nc ismarked by the vertical dashed line.

scaled with T 0 to the series temperature T , where T is calculated by plotting the raw
temperature against N0 and using a linear fit to give the N0 → 0 limit. The form of
the scaling is based on a theoretical model as discussed later.

4.3.3

Results

A sample dataset taken with a = 135 a0 and T = 177 nK is compared with the results
for an ideal gas in Fig. 4.5. The difference between the two is conspicuous; as the total
atom number is increased from the critical number of ∼ 200,000 to 450,000 only about
half of the additional atoms form a condensate.
The full set of 18 data series is shown in Fig. 4.6, where N0 is plotted against the
number of excess atoms above the critical point, N − Nc . The solid line shows thesituation for a saturated gas, N0 = N − Nc . Deviation from the ideal case is clear
for every set of data, with the effect becoming more pronounced as both T and a are
increased. The dependence of N0 with N is also clearly not linear. To start interpretingthese findings more clearly we look to mean-field theory.

4.4

Hartree-Fock Approximation

The simplest way to include interaction effects within a theoretical construct is by use of
the Hartree-Fock approximation [112]. Thermal atoms are treated as a non-interacting
gas with density nT (r) confined by the combination of the trapping potential V (r)and the self-consistent interaction potential 2g(nT (r) + n0 (r)) . The situation may
be simplified by taking advantage of the small density of the thermal component. If
129

4.4 Hartree-Fock Approximation

122nK, 40a0
126nK, 62a0
115nK, 135a0
121nK, 221a0
157nK, 93a 0152nK, 135a0
184nK, 62a0
181nK, 93a0
177nK, 135a0
171nK, 221a0
188nK, 135a0
186nK, 184a0
181nK, 274a0
233nK, 135a0
260nK, 135a0251nK, 221a0
267nK, 274a0
284nK, 356a0

N0 (thousands)

100

50

0

0

100

200

300

400

N - Nc (thousands)
Figure 4.6: Deviation from the saturation picture for a variety of interaction strengths andtemperatures. We plot N0 versus N − Nc where Nc is experimentally determined for each of
the 18 series. The temperature and scattering length for each series is encoded in the colour ofthe markers and are given in the accompanying legend. The solid line marks the prediction fora saturated gas, N0 = N − Nc .

the effect of thermal atoms on the condensate is neglected then n0 (r) is given by theThomas-Fermi distribution of equation (1.26). If the mean-field energy of the thermal
atoms 2gnT (r) is also neglected1 the resulting effective potential is simply given by
Veff (r) = V (r) + 2gn0 (r) = |V (r) − µ0 | + µ0 ,(4.19)

where the mean-field chemical potential µ0 is given by (1.29) and the T → 0 limit isindicated. µ0 is thus the relevant energy scale parameterising the effects of interactions.
This potential has a three-dimensional ‘Mexican-hat’ shape as shown in Fig. 4.7 .
The number of thermal atoms in the presence of N0 condensed atoms can then be
estimated using the semi-classical approximation. Equation (1.6) is reproduced here
for convenience:1NT =(2π~)31ZZe((r,p)−µ)/kB T − 1

By setting
(p, r) =dpdr .

p2+ Veff (r) ,2m(4.20)(4.21)and noting that the chemical potential is µ0 we can perform the momentum integrationof (4.20) to obtainnT (r) =hi
1(µ0 −Veff (r))/kB T
g
e
,
3/2
λ3T

(4.22)

just as (1.15) was found in §1.2.2.
1Making this approximation means we lose the small mean-field shift in Tc and we find Nc = Nc0 .

1304.4 Hartree-Fock Approximation

0.6

Veff kB T

0.4

2Α0
Α0
0
-1

-

0

2 Α0

1

2 Α0

R = rRT

Figure 4.7: Effective mean-field potential Veff for α0 = µ0 /kB T = 0.1 (solid line) and α0 = 0(dashed line), equivalent to the bare parabolic trapping potential. In practice most of thethermal atoms lie outside the condensate, where V > µ0 and Veff = V .

When the temperature of the system is relatively high (compared with µ0 /kB ) the
majority of thermal atoms lie outside the condensate in the region where Veff (r) > µ0
and Veff (r) = V (r). It is therefore reasonable to approximate the full effective potential
as the bare trapping potential and consider only the region outside the condensate. This
does not mean that the effect of interactions may be neglected, as the chemical potential
has a value that differs substantially from the ideal value. Substituting the harmonic
form of V (r) into (4.22) and switching to dimensionless variablesrµ0
and R =
, where RT =
α0 =
kB T
RT
we can writeNT ≈ 4π

RT
λT

3 Z ∞
√

2α0

r

kB T
mω 2

(4.23)hi
2
R2 g3/2 e−(R /2−α0 ) dR .

(4.24)

Here it is sensible to introduce the variable Q, where
Q2
R2
=
− α0
2
2

(4.25)

to rewrite (4.24) ashi
2α0 1/2
2
NT ≈ 4π
Q 1+ 2
g3/2 e−Q /2 dQ
Q03 Z ∞
h
i

RT
2
≈ 4π
Q2 + α0 g3/2 e−Q /2 dQ ,
λT
0


RT
λT

3 Z ∞

2

(4.26)(4.27)

where the binomial expansion has been evaluated to first order in α0 . The summation
form of the polylogarithmic function given by (1.9) can then be used to evaluate the1314.5 Interpretation of results

integral of (4.27):
 ∞
Z ∞

RT 3 X 1
2
NT ≈ 4π
Q2 + α0 e−kQ /2 dQ
3/2
λT
k
0
k=1
!

3 X
∞
∞
X
1
1
3/2 RT= (2π)
+ α0
λT
k3
k2
k=1
k=1
3

3/2 RT[ζ(3) + α0 ζ(2)] .
= (2π)
λT(4.28)(4.29)(4.30)

The first term on the right-hand side is simply the non-interacting critical number Nc0 .
Dividing through by Nc0 gives the important result
NT
ζ(2)
≈1+
α0 ≈ 1 + 1.37 α0 .
0
Nc
ζ(3)

(4.31)

The approximations used here are the same as those used to derive equations (120) and(122) in [112], from which (4.31) can directly be deduced. In this reference the process
is geared towards a revised expression for the condensed fraction below Tc , a conceptthat is far more prevalent in the field, though strongly linked with non-saturation.

4.5

Interpretation of results

The basic mean-field result (4.31) gives a guide to how parameters in the system should
scale, as well as providing a quantitative result to which we can directly compare ourmeasurements. Firstly, within each series there is a predicted scaling of NT above Nc
2/5

2/5

with µ0 ∝ N0 , indicating that plots of the relationship between NT and N0

shouldbe more informative than the linear-linear format used thus far. Secondly, it gives thetemperature correction alluded to at the end of §4.3.2 from the T dependence of Nc0
2/5

and α0 . This amounts to plotting NT (T /T 0 )3 versus N0 (T /T 0 ) . Finally, to directlycompare our results with the prediction (4.31) we find the expected slope for such aplot within this Hartree-Fock approximation:
SHF =

dNT
2/5

d(N0 )≈ 1.37 X ,

where
2 2/5

X = ξT a

ζ(3)
, ξ=
2



kB
~ω̄

2 

15
aho

(4.32)

2/5.

(4.33)

Fig. 4.8 shows the result of plotting the data of Fig. 4.5 with this new approach.
The blue line shows the slope SHF = 699 with a y-intercept given by the experimental
2/5

value of Nc . The initial linear slope of NT with N0

is in impressive agreement withthe findings of the Hartree-Fock description. As the condensate grows past ∼ 10, 000
atoms, corresponding to µ0 /kB T ∼ 0.1, this agreement starts to break down, and thelack of saturation becomes even more extreme.
1324.5 Interpretation of results

NT (thousands)

320

280

240

200
0

20

40

60

80

100

2/5

N0

Figure 4.8: Quantifying the lack of saturation. The data of Fig. 4.5 is now displayed in the
2/5
form NT versus N0 as guided by Hartree-Fock theory. The dotted line shows the experimentally determined Nc . The blue line shows the predicted slope SHF = 1.37 ξT 2 a2/5 = 699 . This
is in close agreement with the data up to N0 ∼ 10,000 . The red line shows a linear fit to thedata in the range 0.1 < µ0 /kB T < 0.3 between the dashed lines. This gives a non-saturation
slope S = 1283 ± 84 . The black line provides a guide to the eye based on a second orderpolynomial fit, the slope of which is indistinguishable from the Hartree-Fock slope at low N0 .

4.5.1

Popov approximation

The first port of call to account for the divergence from first-order mean-field theory forlarge N0 is to use a more sophisticated theoretical description. Higher order terms canbe retained and the effects of the finite thermal density included in the self-consistent
Hartree-Fock approach, but to go beyond this it is necessary to include the effectsof low-energy excitations in the system. This is done by using the so-called Popovapproximation [113], in which the Bogoliubov terms for creation and destruction ofparticle pairs are included (see e.g. [19]).
The first thing to check with this model is the low-N0 behaviour. A shift in Nc
from the ideal value is of course now detected, but aside from this the initial slope of≈ 1.37 X is indeed recovered. The results of using the Popov approximation for themost strongly and weakly interacting conditions used in our experiments are shown on
top of the data for our sample series and the first-order Hartree-Fock prediction in Fig.
4.9. The computational method used to implement the theory is described in Appendix
C. As condensate size increases it is clear that the Popov theory predicts a departurefrom the basic mean-field picture, but the effect is not great enough to fully explain
the behaviour seen in our experiments.
To deal with these larger (and experimentally more typical) condensates we therefore take a heuristic approach, quantifying the degree of non-saturation by defining the133

4.5 Interpretation of results

1.6
1.5

NT Nc

1.4
1.3
1.2
1.1
1.0
0.00

0.05

0.10

0.15

0.20

0.25

0.30

Α0

Figure 4.9: The predictions of Popov theory compared with first-order Hartree-Fock theory(black line) and the sample data series (grey points) of Fig. 4.5. Points are now normalised to
the relevant Nc values and plotted against α0 = µ0 /kB T . The blue points show the prediction
for the most weakly interacting series, T = 122 nK a = 40 a0 , the red points the prediction
for the most strongly interacting series, T = 284 nK a = 356 a0 . The Popov predictions were
calculated as described in Appendix C.
2/5

slope S = ∆(NT )/∆(N0 ) for the range 0.1 ≤ α0 ≤ 0.3 . This slope is shown for thedata series in Fig. 4.8 by the solid red line. Though the data isn’t perfectly linear,
it is well described by S over this broad experimental range. We therefore have toinvestigate the non-saturation behaviour in both this regime and that of low N0 .

4.5.2

Small condensate limit

In order to test the findings of (4.31) for small condensates, it was necessary to takefurther series of data concentrating on the low N0 limit, using a slightly altered experimental method. Interactions are ‘turned off’ during time-of-flight, limiting the
expansion of the condensate to make it clearly stand out against the background ofthermal atoms in absorption imaging. This technique is described in more detail in
Chapter 5. Unfortunately, this method can not be applied when detecting larger condensates, as the peak optical density becomes extremely large. It is for these same
reasons that we use the slope S to quantify non-saturation for larger condensates. The
alternative method of a second-order polynomial fit to the data requires information at
both very-low and high N0 .
The low N0 data series were taken for a = 56-274 a0 and T = 177-317 nK , combining
to give X values in the range ∼ 500-1300. The experimental non-saturation slope
2/5

S0 = dNT /d(N0 ) was found in the N0 → 0 limit for each set of data for directcomparison with SHF . The results are shown in Fig. 4.10 a , where the blue line shows
SHF = 1.37 X . The agreement between the experiment and theory to within a few
134

4.5 Interpretation of results

S0

a

b

2000
1000
0
4000

S

3000
2000
1000
0
0

400

800

1200

1600

X
Figure 4.10: Recovery of saturation in the non-interacting limit. The non-saturation slopes
S0 and S are plotted against the dimensionless parameter X = ξT 2 a2/5 described in the text.
The S0 data are directly compared with Hartree-Fock theory, SHF = 1.37 X (blue line), withno free parameters. A linear fit to the S data gives dS/dX = 2.6 ± 0.3 and an intercept
S(0) = −20 ± 100 , consistent with complete saturation in the ideal gas limit. The S values are
extracted from the 18 39 K data series shown in Fig. 4.6 and use the same colour code. Two87
Rb data points are also shown as black squares. All vertical error bars are statistical. The
systematic uncertainty in N0 and NT of < 10 % corresponds to < 6 % uncertainty in S0 and
S values. The horizontal error bars include the 3 Hz uncertainty in the trapping frequenciesω̄/2π and (for potassium) an allowance of 0.1 G uncertainty in the position of the Feshbachresonance.

percent shows that, in the non-interacting limit (where µ0 = 0 for all N0 ), S0 would
indeed vanish. This satisfactory agreement with theory doesn’t complete the attempted
proof of saturation in the non-interacting limit, as we must now look at the large N0
regime.

4.5.3

Large condensate limit

The values of S extracted from the 18 data series of Fig. 4.6 are shown in Fig. 4.10 b .
The data falls in a straight line as plotted against X, supporting the assumption that
µ0 /kB T may still be used as the relevant interaction parameter. The red line showsan unconstrained linear fit to this data. In addition, data series were also taken with
87 Rb at two temperatures: 175 and 203 nK. The scattering length for this data is fixed

to 99 a0 by the background triplet scattering length of 87 Rb. The good agreement ofthese two points with the 39 K data serves to validate our experimental approach.
The extrapolated y-intercept of the red line in Fig. 4.10 gives the prediction thatin the X ∝ T 2 a2/5 → 0 limit, S = −20 ± 100, consistent with zero. The bounds on thevalue are given by the confidence margin of the linear fit. This limit can be reached in135

4.6 Conclusions

two very different ways. In the rather mundane T → 0 limit, NT trivially vanishes forany value of N0 , automatically giving S = 0. However, the dependence of S only on X
means that in the more pertinent a → 0 limit (at finite T ) we expect the slope to tend
to zero, with a finite thermal fraction.

4.6

Conclusions

Combining the result of the heuristic approach for larger condensates with the successof Hartree-Fock theory for small condensates confirms the picture of a non-interacting
saturated gas for a broad range of experimentally relevant parameters. We expect
this behaviour to be universal for (relatively) weakly-interacting Bose gases in threedimensional harmonic potentials, characterised by the short-range s-wave scattering
length a. This strongly suggests that condensation is indeed the result of a purely
quantum-statistical phase transition.
The Hartree-Fock prediction varies significantly with the form of the trapping potential. We can repeat the derivation of §4.4 for power law traps (in a three-dimensional
system) of the form V (r) ∝ |rn | for general n with interesting results. Once we makethe approximation that the thermal atoms reside only outside the condensate, the remaining process is exact for a linear trap (n = 1), and we findζ(7/2)
ζ(5/2) 2
NT
1/4
≈1+
α0 +
α , where α0 ∝ N0
Nc0
ζ(9/2)
2ζ(9/2) 0(4.34)and Nc0 is defined for a linear trap. This demonstrates a stronger dependence on N0
than the findings for a harmonic trap. For a cubic trap (n = 3) the assumption thatthe thermal atoms reside mainly outside the condensate is no longer a good one1 , andthe central region must be included explicitly. This gives the prediction√NTζ(3/2)
4 π 3/2
1/2
≈1+
α0 −
α
, where α0 ∝ N0 ,Nc0
ζ(5/2)
3ζ(5/2) 0
3/2

where the dependence is now weaker on N0 . The α0(4.35)

term is included due to its

relatively large effect. From purely geometric considerations we would also expect
stronger non-saturation effects for lower dimensional systems. It would therefore beinteresting to study the low N0 limit for different trap dimensions and potentials, andin particular disordered potentials [114] where the effective dimensionality can take on
non-integer values. It is, therefore, possible that non-saturation scaling could act as amethod for classifying the geometries and interactions in many-body systems.
The high N0 behaviour we see is not, however, due to altered mean-field effects
because of the slight anharmonicity of our system. Numerical simulations are observed
to be consistent with those for a harmonic trap. We hesitate to draw any further
1

In fact using this assumption would lead to the conclusion that the gas is actually saturated.

136

4.6 Conclusions

conclusions from the perceived enhanced non-saturation, aside from noting that beyondmean-field effects due to critical correlations might be expected to scale with the sizeof the critical region. For a harmonically trapped gas in the Thomas-Fermi limit, this√region is the surface of an ellipsoid with semi-axes ∝ α0 , such that the size of thecritical region ∝ α0 . The magnitude of these effects might therefore be expected toalso scale with this parameter.
The hope is that this work has dispelled any expectation that the atom number
should saturate during condensation at constant temperature, and has provided a solid
yard-stick for the scale of non-saturation in a harmonic trap. Indeed, in recent work thelack of saturation of the thermal component has been acknowledged and attributed torepulsive interactions [115]. Our results also highlight the need to be careful when using
ideal laws for calibration of experimental systems, for example using the condensed
fraction to measure the temperature of a trapped gas. Perhaps most significantly
though, we have leant the first direct experimental weight to Einstein’s textbook pictureof a saturated gas. The results of this chapter are summarised in [116].

137

Chapter 5

Effect of interactions on the criticaltemperature“ I laughed heartily to myself, but I was the only one. At the point where
the aunts should have rolled out of their seats as one aunt there occurred
merely a rather ghastly silence as of mourners at a death-bed, which was
broken by Aunt Charlotte asking what I had said.”– P. G. Wodehouse (The Mating Season)

5.1

History

Having investigated the effects of repulsive interactions in a Bose gas beyond the pointof condensation, we now turn to the behaviour of the critical point itself. The questionof how interactions affect this standard textbook result is, in contrast to the saturationquestion, one that has been hotly debated for more than half a century, since thepioneering work of Lee and Yang [117, 118]. This debate was reinvigorated with theadvent of the first experimental realisation of a BEC, creating a surge in theoretical
output on the topic since 1995. Though these gases are harmonically confined in thelab, most theoretical work has historically focussed on uniform systems, where thecritical temperature for a non-interacting system Tc0 is given by (1.12). The simplicityof the derivation given in §1.2.1 belies the complexity of the lowest-order correction
due to interactions. For several decades there was no general agreement on the form,
magnitude or even the sign of the shift, as illustrated by the summary of past work in
Table 5.1. The Tc shift is defined as ∆Tc = Tc − Tc0 .
The difficulty stems from the fact that a purely mean-field picture is not sufficientwhen considering the critical point. Mean-field theories have numerous useful applications for the description of interacting Bose systems, such as the successful prediction of
the broadening of the condensate profile, the dispersion law for elementary excitations
and other implementations reviewed in [112], as well of course as the initial lack ofsaturation of a harmonically trapped gas as discussed in Chapter 4. However, near the139

5.1 History

Year

Prediction

1957∆Tc /Tc0 = C(an1/3 )1/2
∆Tc /Tc0 = can1/3∆Tc /Tc0 = C(an1/3 )1/2
∆Tc /Tc0 = D(an1/3 )3/2

1958
1960
1964
1971

Ref.
C = 2.49

Lee and Yang [117]Lee and Yang [118]C>0

Glassgold et al. [119]D>0Huang [120]∆Tc < 0Fetter and Walecka [121]1982∆Tc /Tc0 = C(an1/3 )1/2

C = −3.5

Toyoda [122]1992∆Tc /Tc0 = can1/3

c = 4.66

Stoof [123]∆Tc > 0Bijlsma and Stoof [124]

1996
1997
1999
1999
1999
1999
2000
2000
2000
2000
2001
2001
2003
2003
2004
2004
2004
2010

∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = C(an1/3 )1/2
∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3∆Tc /Tc0 = can1/3

c = 0.34 ± 0.06

Grüter et al. [125]c = 0.7

Holzmann et al. [126]C = 3.5

Huang [127]c = 2.3 ± 0.25

Holzmann and Krauth [128]c = 2.9

Baym [129]c = 2.33

Baym et al. [130]c = 5.1 ± 0.9

Reppy et al. [131]c = −0.93

Wilkens et al. [132]c = 1.71

Arnold and Tomášik [133]c = 1.32 ± 0.02

Arnold and Moore [134]c = 1.29 ± 0.05

Kashurnikov et al. [135]c = 1.14 ± 0.11

Kleinert [136]c = 1.3 ± 0.4

Davis and Morgan [137]c = 1.27 ± 0.11

Kastening [138]c = 1.23 ± 0.12

Ledowski et al. [139]c = 1.32 ± 0.14

Nho and Landau [140]c = −2.33

Betz and Ueltschi [141]Table 5.1: Various predictions for the Tc shift since midway through the twentieth century.
Results are given in terms of the diluteness parameter an1/3 as used in the majority of thereferences. All results are theoretical, apart from the experimental work of Reppy et al. It
was later shown that the estimation of the scattering length was incorrect in this work [135].
For the most part this data was collated by Seiringer and Ueltschi [142] and in the review byAndersen [143].

140

5.1 History

critical point (in what is known as the ‘critical region’) long-range correlations develop
which must be taken into consideration. Correlations play an essential role in phenomena such as superfluidity [144] and associated vortex properties, as well as explaining
shifts in the frequencies of collective oscillations [145] and quantum depletion of thecondensate [146], and are as such a topic of great theoretical interest.

5.1.1

Uniform system

In a uniform system there is no shift within mean-field theory, and the lowest-ordershift in Tc is solely due to correlations. The lack of a mean-field correction in a uniform
system can be understood by observing that, in this description, interactions lead toa universal shift by 0 = 2gn of the effective potential Veff . Note that 0 is now also
the lowest energy state of the system. In this chapter we shall always consider the casewhere there is no condensate, such that n ≡ nT . By setting (r, p) = p2 /2m + 0 in(1.6) we obtain a modified version of (1.7),
n=

i
h
1
(µ−0 )/kB T
.
g
e
3/2
λ3T

(5.1)

At µ = 0 there is clearly a shift in n from the ideal critical value , but the upper boundon the chemical potential, according to (1.1), is the lowest single-particle energy state
0 . Macroscopic occupation of the ground state now occurs when µ approaches thisvalue. The fact that interactions shift both the critical chemical potential µc and theeffective potential by the same amount results in the ideal nc (and equivalently Tc )being recovered.
Calculation of the shift induced by correlations requires non-perturbative methods
which are difficult to compute [143, 147], hence the long and varied theoretical background. Effects are considered in terms of the small diluteness parameter, defined as
the ratio between the scattering length and the typical inter-particle separation
an1/3 ≡ ζ(3/2)1/3

a
,
λ0

(5.2)

where in the equivalent form λ0 is the thermal wavelength at Tc0 ,

λ0 =

2π~2
mkB Tc0

1/2
.

(5.3)

A general consensus for ∆Tc has only been reached in the last ten years [143, 147–
149], thanks to the Monte Carlo simulations carried out by Arnold and Moore [134]and Kashurnikov et al. [135] in 2001. It turns out that long range correlations act toaid the establishment of coherence between particles, so that condensation sets in at ahigher temperature. The result isa∆Tc≈ 1.3 an1/3 ≈ 1.8
,
0
Tc
λ0
141

(5.4)

5.1 History

where the functional dependence is that proposed by Lee and Yang in 1958 [118].
Though the matter is generally considered settled, it is still not universally accepted,
as may be seen from the latest entry in Table 5.1. It may require direct experimental
evidence to finally put the affair to bed.

5.1.2

Harmonic confinement

In the harmonically trapped case, interactions affect µc and Veff differently due to theinhomogeneity of the system, and the critical temperature is shifted within mean-field
theory. Note though that µc = Veff (0), and the critical phase-space density of ζ(3/2) ispreserved at the centre of the trap. The mean-field shift for a harmonic potential was
first calculated by Giorgini et al. [150] and is discussed in §5.2. The first-order result
is given bya∆Tc,≈ −3.426
0
Tc
λ0

(5.5)

predicting a negative shift in this case. For a harmonically trapped gas, the mean-field
result dominates over the effect due to correlations in a manner that goes beyond thedifference in pre-factors between (5.4) and (5.5). This is because for a confined gas at
the condensation point, the critical region at the centre of the trap is vanishingly small,so correlation effects are expected to effect Tc at higher order in a/λ0 . The mean-field
result of (5.5) is therefore expected to be exact to first order.
It is pointed out in [150] that critical correlations are expected to affect the result at
the order of a2 and above. This is as a result of the Ginzburg-like criterion [103, 151],
predicting that correlations become significant when the correlation length of the Bosegas becomes comparable to the length-scale of nonperturbative physics in the system.
This condition is calculated in §II.C of [134] to beδµ = |µ − µc | .

~2 a2
,
mλ40

(5.6)

where µc is the value of µ at the critical point. This condition is equivalent to the
one that appears in the original mean-field Tc shift paper [150]. The conclusion thatcritical correlations come in only at second order and above is most simply attainedby dimensional analysis, as is carried out in [134], along with other more thoroughderivations. The investigation further shows that a logarithmic term is also expected
to appear, resulting in a shift of the form  2
a
a
a
∆Tc
0
00
= c1
+ c2 ln
+ c2
+ ...
Tc0
λ0
λ0
λ0

(5.7)

where c1 ≈ −3.426 is the mean-field result. Attempts have been made to theoreticallyquantify the beyond-MF corrections to the MF result [134, 149, 152–154], but no consensus has been reached beyond the expectation that the additional beyond-MF shift
should be positive.
142

5.1 History0 .0 0E n s h e r e t a l.( 1 9 9 6 )G e r b ie r e t a l. ( 2 0 0 4 )M e p p e lin k e t a l. ( 2 0 1 0 )-0 .0 2-0 .0 40

∆T c / T c-0 .0 6-0 .0 8M F T h e o ry

-0 .1 0-0 .1 2
-0 .1 40 .0 00 .0 10 .0 20 .0 30 .0 4a / λ0Figure 5.1: Summary of the past experimental determinations of the Tc shift by Ensher et al.
[108], Gerbier et al. [77] and Meppelink et al. [155]. The dashed lines, representing the data of
Gerbier et al., show the single standard deviation bounds as found in Fig. 3 of [154]. The data
covers the range a/λ0 = 0.007-0.024 and is consistent with the MF prediction (blue line).

5.1.3

Past experiments

Since the MF prediction of [150] in 1996, experiments in harmonic traps have ruled out
the ideal prediction Tc0 with ever increasing certainty, as summarised in Fig. 5.1. The
first measurement performed by Ensher et al. [108] found ∆Tc /Tc0 = −0.06 ± 0.05 for
fixed a/λ0 ≈ 0.015 , just excluding the ideal case. Gerbier et al. [77] managed to probea range of a/λ0 from 0.016 to 0.024 by varying the density of the cloud, excluding
Tc0 by more than two standard deviations. Most recently in 2010 Meppelink et al.
[155] measured the Tc shift with deft precision at a/λ0 ≈ 0.007 , finding ∆Tc /Tc0 =−0.026 ± 0.001. All of these investigations found results consistent with the mean-field
theory, but none could discern any corrections beyond this description.

5.1.4

Aims of the investigation

Though in the experiments carried out by Gerbier et al. they were able to tune theinteraction parameter a/λ0 by varying the sample density (and hence critical temperature), it is difficult to span a wide range with this method due to the weak dependencea/λ0 ∝ n1/3 . These experiments were carried out with 87 Rb (and in a magnetic trap),
entailing a fixed scattering length. Their concluding remarks from this work point theway to future progress: “We note to conclude that measuring corrections to Tc beyond
the mean field for our typical experimental parameters would require thermometry withan accuracy of 1 % or better. A more direct route to investigate such effects might be
to measure the critical density near the center of the trap, directly sensitive to the presence of critical fluctuations. Alternatively, these many-body effects could be enhancedin the vicinity of a Feshbach resonance.”
Using the above as a manifesto, we use an improved thermometry technique to143

5.2 Mean-field shift in a harmonic trap

increase measurement accuracy, while using the available Feshbach resonance in 39 K
to investigate from the very-weakly interacting limit up to a regime where we hope
to see beyond-MF effects. With the versatility of the Feshbach resonance we can also
probe more extreme regimes, where thermodynamic equilibrium is lost. Direct probingof the in situ density is beyond our current means, but remains an exciting area ofinvestigation. This technique has recently been used to show strong deviations fromMF theory in excellent agreement with quantum Monte Carlo simulations of the effectsof correlations [156]. We proceed with a discussion of the mean-field effects that usually
dominate the Tc shift in typical BEC setups, before moving on to the experimental
methods geared towards revealing the beyond-MF behaviour.

5.2

Mean-field shift in a harmonic trap

The negative mean-field shift of equation (5.5) can be qualitatively understood byconsidering the effect of repulsive interactions near the trap centre. As the mean-fieldpotential is proportional to the density of the cloud the effect is strongest here. The
net effect then is that interactions spread the atoms out, reducing the central density
in comparison to an ideal Bose gas. In the non-interacting case the critical point isreached when Nc0 atoms are in the trap and, as found in §1.2.2, a critical density equal
to that of a uniform system is reached at the trap centre. For the interacting Bose gas
this critical density is no longer reached with Nc0 atoms due to the flattening of the
distribution at the trap centre, and further atoms must be added before the criticalphase-space density is attained. There is therefore a positive shift in the critical number
at constant temperature (∆Nc > 0), or correspondingly a negative shift in the criticaltemperature for constant atom number (∆Tc < 0). The first-order shift is derived
analytically below, before a self-consistent numerical method is presented to help both
illustrate the effect and for use later in the chapter.

5.2.1

Analytic solution

To calculate the first order effect of interactions on the critical temperature in a harmonic trap we look again at the Hartree-Fock description in the semi-classical limit. In
this case the single-particle energy levels are given by(r, p) =p2+ V (r) + int ,
2m

(5.8)

where int = 2gn(r) , and V (r) takes the parabolic form given by (1.14). We want tofind the shift in critical temperature for a particular value of N , so we consider the144

5.2 Mean-field shift in a harmonic trap

differential dN , given by
Z
∂N
∂N
∂n(r)
dint
dT +
dµ +
∂T
∂µ
∂int
Z
∂N
∂N
∂n(r)
=
2gn(r)dr .
dT +
dµ +
∂T
∂µ
∂int

dN =

(5.9)
(5.10)The last term must be kept in this form because of the spatial dependence of int . We
are interested in the small variations from the ideal case where T = Tc0 , µ = 0 andint = 0 , so the partial derivatives are evaluated at this point. Considering the firstorder shift from the ideal case in terms of the small quantities ∆Tc =Tc − Tc0 , 0 and
2gn(r) allows us to write
∂N
∂N
0=
∆Tc +
0 + 2g
∂T
∂µ

Z

∂n(r)
n(r)dr .
∂int(5.11)The quantity 0 is the true zero-point energy of the system, which is shifted by anamount ≈ 2gn(0) by interactions1 . Up until now we have also neglected the zero-point
motion of atoms in the harmonic oscillator ground state. Its inclusion results in a shiftof the critical temperature even for a non-interacting gas, though it is independentof N (as we shall see) and so in the limit N → ∞ the relative shift tends to zero.
Condensation actually sets in when the chemical potential µ is equal to30 = ~ωm + 2gn(0) ,2(5.12)

where the first term is the zero-point energy of the harmonic oscillator, and ωm =(ωx + ωy + ωz )/3 is the algebraic mean of the trap frequencies.
The first term on the right-hand side of (5.11) can be found from the ideal equation
for N at Tc0 given by (1.18). We immediately find∂NN
=3 0 .
∂T
Tc(5.13)

To calculate the second term we need to use the familiar thermal distribution functionn(r) =hi
1−(V (r)+int −µ)/kB T
g
e
.
3/2
λ3T

(5.14)

It is helpful to use the sum form of the g3/2 function, so that by taking the partial
derivative with respect to µ and integrating over space we obtain
∞

1 X 1
∂N
= 3
∂µ
λT
k 3/2
k=1

Z

∂ h −k(V (r)+int −µ)/kB T i
e
dr .
∂µ

1

(5.15)

This approximation is good as long as aho  RT , i.e. the extent of the lowest single particle
wavefunction is much smaller than the size of the thermal cloud.

145

5.2 Mean-field shift in a harmonic trap

At the ideal critical point this can be written as∞∂N
1 1 X 1=∂µkB Tc0 λ30
k 1/2

Z

0

e−kV (r)/kB Tc dr ,(5.16)

k=1

where

λ0 =

2π~2
mkB Tc0

1/2(5.17)

is the thermal wavelength at Tc0 . Evaluating the above expression and substituting in
N from (1.18) gives the result
ζ(2) N∂N=,∂µζ(3) kB Tc0(5.18)

giving the first-order saturation result of (4.31). The final term of (5.11) can be foundby following a similar process. Using the above expression for n(r) and using the sum
form of the g3/2 function allows us to writeZ∂n(r)n(r)dr =∂int∞ ∞1 XX 1
λ6T j=1

k=1

1

Z

j 3/2 k 3/2∂ h −j(V (r)+int −µ)/kB T i −k(V (r)+int −µ)/kB T
e
dr ,
e
∂int
(5.19)

which, at the critical point, can be evaluated as
Z

∞

∞

∂n(r)
1 1 XX 1 1
n(r)dr = −∂int
kB Tc0 λ60
j 1/2 k 3/2Z0

e−(j+k)V (r)/kB Tc dr(5.20)

j=1 k=1

NS=−,
3
ζ(3) λ0 kB T
where
S=

∞ X
∞
X
j=1 k=1(5.21)

1
j 3/2 k 1/2 (j + k)3/2≈ 1.208 .

(5.22)

Inserting the expressions (5.13), (5.18) and (5.21) into (5.11) and collating similar terms
gives the first order result

~ωm
a
4
−
[ζ(2)ζ(3/2) − S]
0
kB Tc
3ζ(3)
λ0


~ωm
a
≈ −0.684.
− 3.426
kB Tc0
λ0

∆Tc
ζ(2)
=−
0
Tc
3ζ(3)



(5.23)(5.24)

The first term on the right-hand side results solely from the zero-point motion of theharmonic oscillator, and is known as the finite-size shift. From this it is clear that theabsolute shift is indeed independent of the atom number. With the number of atoms inour system the relative shift of Tc is typically ∼ 1 % . The second term is the shift due
146

5.2 Mean-field shift in a harmonic trap

to interactions of (5.5), characterised by the interaction parameter a/λ0 . In typicalcold atoms systems this takes a value on the order of 0.01 , shifting Tc by about 3 % .
Equation (5.24) is often written in the equivalent form
ωm −1/3
a 1/6∆Tc≈ −0.728
N− 1.326
N
,
0
Tc
ω̄
āho
where the conversion factor N 1/6 /aho =√(5.25)

2πζ(3)1/6 /λ0 has been employed for theinteraction term. We choose to work in terms of the original form of (5.24). We are
only interested in the shift due to interactions, so must cater for the finite-size shift insome manner.
It is worth noting that the value 2gn(0) used for the zero-point interaction energy
is the mean-field value. The effect of correlations is to suppress density fluctuations
near the critical point, decreasing int (0). Allowing this to propagate through the above
calculation results in the expected positive shift with respect to the mean-field result(5.24).

5.2.2

Numerical solution

To visualise the effect on the atomic density distribution we return to the semi-classicalHartree-Fock approximation and equation (1.6). The interaction term 2gn(r) must be
added to the expression for the single-particle energy spectrum,(r, p) =p2+ V (r) + 2gn(r) ,2m(5.26)

where for simplicity we consider a spherical harmonic potential such that1
V (r) = mω 2 r2 .
2

(5.27)

By conversion to the dimensionless variables
A=

r
µ
4a
, D = nλ3T , R =and α =λTRT
kB T

(5.28)and integrating (1.6) over momentum space we obtain the implicit equation

2
D(R) = g3/2 eα−R /2−AD(R) .

(5.29)In mean-field theory we know that at the critical point D(0) = g3/2 (1) = ζ(3/2). This
allows us to write down the value of α at the critical point,αc = Aζ(3/2) .
147(5.30)

5.2 Mean-field shift in a harmonic trap3
ΖI 3 M
2

a>0
D

2

a=0

1

0
-3

-2

0
R

-1

1

2

3

Figure 5.2: Mean-field theory predicted critical density distribution for a harmonically trapped
Bose gas with repulsive interactions. The black line shows the non-interacting case (identical to
Fig. 1.2). The blue lines show the distribution for increasing interaction strength, in successive
steps of 0.01 a/λT up to 0.05 a/λT . Here it is clear that Nc increases with increasing a/λ0 .
Note in each case the central phase-space density is ζ(3/2) .

Combining this with (5.29) we obtain an expression for the phase-space density distribution at the critical point2
Dc (R) = g3/2 eAζ(3/2)−R /2−ADc (R) .

(5.31)

Equation (5.31) must be solved self-consistently for a given value of A . This may bedone numerically, and the results are shown for various values of a/λ0 = A/4 in Fig.
5.2. This clearly shows how the central density distribution is increasingly spread out
for higher a/λ0 . The mean-field Nc shift can be determined by integrating (5.31) overR-space. In Fig. 5.3 this result is compared with the first-order result found from (5.5):
∆Tc
a
∆Nc
≈ −3 0 ≈ 10.278
.
0
Nc
Tc
λ0

5.2.3

(5.32)

Introducing critical effects

When critical correlations are also taken into account there are now two competing
effects; mean-field repulsions act to hinder condensation, while correlations act to aid
it. Qualitatively the result can be visualised as in Fig. 5.4. Thinking for now in termsof the atom number at a fixed temperature, the MF effect reduces the central density,
such that more atoms are required to reach a particular peak density than for an ideal
gas. However, correlations act to reduce the critical density, so fewer atoms are needed
to reach the onset of condensation. We do not simply expect the critical density to
be described by the uniform result (5.4), as long-range correlations are not a local
phenomenon. We therefore proceed to determine the effect experimentally, as related148

5.3 Experimental procedure

0.4

DNc Nc0

0.3

0.2

0.1

0
0

0.01

0.02
aΛ0

0.03

0.04

Figure 5.3: Comparison between the first order analytic result ∆Nc /Nc0 = 10.278a/λ0 (blueline) and the numerical method (blue points) used to find the mean-field Nc shift described in
the text.
3
ΖI 3 M

MF

2

2
D

bMF
Ideal

1

0
-3

-2

0
R

-1

1

2

3

Figure 5.4: Illustration of the competing effects induced by repulsive interactions on the critical point of a harmonically trapped Bose gas. Compared with an ideal gas (black line) with thesame Tc , repulsive interactions reduce the critical density, but also broaden the density distribution (red line). Mean-field theory (blue line) captures only the latter effect, thus predicting
an increase in the critical atom number Nc at fixed T , corresponding to a reduction in Tc atfixed N .

below.

5.3

Experimental procedure

To measure the critical point we create clouds with various condensed fractions inorder to extract the value of NT as N0 tends towards zero. This involves a basic
method closely following the one used in Chapter 4, with each data series made up of∼ 100 images (see Fig. 5.5). Condensates are prepared once more using a = 135 a0
before the trap depth and Feshbach field are ramped to their final values, the fixed
trap depth ensuring good temperature stability over each experimental series. In this149

5.3 Experimental procedure

Figure 5.5: Sample measurement series taken for a scattering length of 274 a0 . The cloudis held in the trap for between 4 and 22 s at the measurement scattering length. Hold timeincreases from left to right and from top to bottom. The data is shown with our standard
OD colour map. The red box indicates the image shown in Fig. 5.6, where the range of thecolour map has had to be reduced to be able to distinguish the tiny condensed fraction fromthe thermal cloud. All clouds have a condensed fraction of under 2 %.

150

5.3 Experimental procedureab

Figure 5.6: a A sample optical density imageof a cloud with 450,000 atoms and a condensed
fraction of 0.14 % , taken after 19 ms TOF withinteractions turned off. The range of the colour
map has been restricted to allow the condensate
to be discerned. b Azimuthally averaged column density of the same data, displayed for illustrative purposes. The full 2D optical-density
profile is used for fitting.

set of experiments we use trapping frequencies in the range ω̄ = 2π × 75-85 Hz. The
clouds are held while the atom number decays and elastic collisions redistribute atoms
between the two components. This is interrupted at a-dependent intervals (loss rates
being much greater for high a) by switching off the trap for imaging after 19 ms timeof-flight. In contrast to the saturation measurements we are interested only in thevery-low N0 regime, studying condensed fractions no higher than ∼ 2 % .
Other groups, as discussed in §5.1.3, have measured the Tc shift for values of a/λ0
between 0.007 and 0.024. The ability to tune the scattering length with a Feshbachresonance allows us to expand this range to explore the region 0.001 < a/λ0 < 0.06 .
In addition to this, the resonance provides two major experimental advantages thatare essential for the precision and accuracy of our measurements. Firstly, for eachmeasurement series, taken at a particular a and λ0 , we concurrently take a series ofreference data with a different scattering length aref , using the same ω̄ and very similar
N and hence λ0 . We use aref /λ0 ≈ 0.005 , which is small enough that beyond-MF effects
are expected to be negligible, but also large enough that the system remains close tothermal equilibrium. This allows us to directly measure the small Tc shift due tothe difference in a/λ0 between the two series, essentially eliminating all a-independent
systematic errors that would otherwise affect the absolute measurement of Tc (N, ω̄, a) .
These include the finite-size shift of equation (5.24), absolute number calibration andthe correction due to the trap anharmonicity (see §3.4.4). This method is described in
more detail later.
The second experimental advantage is the ability to turn interactions off during
time-of-flight. This minimises the expansion of the condensate, allowing it to stand out
against the thermal background. A sample image is shown in Fig. 5.6. We achieve thisby ramping the Feshbach field to 350 G (the zero crossing value) in under 2 ms once
the atoms have been released from the optical potential. Along with our image fitting
technique, this allows the reliable detection of condensed fractions as small as ∼ 10−3 .
This allows us to home in on the critical point with high precision.
1515.3 Experimental procedure

5.3.1

Image fitting routine

The details of the Feshbach-field ramp during TOF are discussed in the context oftemperature errors in §5.5.2. The narrow condensates this method produces show
up much more clearly than without the ramp, but the images must still be carefully
processed to accurately detect the condensate numbers. This is important to allow
accurate extrapolation to the critical point. The smallest condensates can easily be
missed with standard fitting techniques, so a method tailored to the purpose is used.
This must be combined with the careful measurements of the cloud temperature andthermal atom number used in our saturation measurements. The result (summarisedin Fig. 5.7) is a seven-step process:1. Data are excluded within a 60 µm radius of the cloud centre (an estimate isentered manually) to be sure to remove the BEC from the fit. A simple g2 fit isthen applied to the thermal cloud.
2. The g2 profile from the first step is fixed, and a Thomas-Fermi profile matchedto the condensate.
3. The condensate width found from the previous step is used to provide a tighter
exclusion region to mask the condensate. The exclusion radius is set to 1.1 Rk(where Rk is the previously fitted Thomas-Fermi radius), with the cloud centre
now defined by the value found in step 1. A new g2 fit is used to extract thethermal atom number NT .
4. Step 2 is repeated using the new g2 profile.
5. The central region out to one thermal radius RT is now excluded and the thermal
atoms fitted with a g2 to find T . This is the most reliable method to find thetemperature as previously discussed.
6. Now just the region between a radius of 35 µm and one RT is used for a g2 fit.
This provides the best fit to the thermal atoms near the cloud centre.
7. N0 is found by summing the difference between the data and the thermal profileof step 6 within 0.5 RT of the centre.
The value of N0 found in the last step is consistent with the value obtained from step 4
for N0 & 2,000 , but for smaller condensates the direct summation is more reliable. The
shape of the condensate is not preserved when a = 0 during TOF, so a Thomas-Fermi
fit is no longer applicable in any case.

1525.3 Experimental procedure

a

b

c

d

NT

e

T

f

g

N0

Figure 5.7: Illustration of our fitting routine. All fitting is carried out with full 2D columnn
densities. Red curves indicate fits to the data, blue curves are fixed profiles. Greyed regions are
excluded from the fitting step. a A g2 fit to the thermal atoms is carried out, excluding data
inside a 60 µm radius (with a manually provided centre) to be sure to exclude the BEC. b A
Thomas-Fermi profile is used to fit the BEC (with fixed thermal profile). c The Thomas-Fermi
radius Rk provided by the previous step allows us to more stringently exclude data within 1.1Rkof the centre, with a new g2 fit giving NT . d As in b, but with the new fixed g2 profile. e
A g2 fit to the wings of the cloud (outside one thermal radius RT ) gives us T . f We focus on
the critical region, fitting a g2 profile to the thermal atoms around the BEC. g N0 is found bysumming the difference between the data and the thermal profile around the critical region.

153

5.4 Critical temperature shift results

5.3.2

Differential measurement process

The first stage of data processing is to compensate for any residual temperature drifts
through each series, using the Hartree-Fock scaling of Chapter 4. This time the processof assigning a temperature to each image is simplified, as images were taken sequentially with decreasing atom number. Temperature is assigned by a fit to the variation
in measured image temperature with N0 . This method is used because we are onlyinterested in the critical point, which can be done by finding both T and NT in the
N0 → 0 limit.
The next stage draws upon the NT scaling with N0 found in the Hartree-Fockregime from the saturation investigation. In order to accurately determine Nc we plot
2/5

NT versus N0

and fit the data with
2/5

NT = Nc + SHF N0,(5.33)

using the definition of SHF (T, ω̄, a) given by (4.32). This method is illustrated in Fig.
5.8. Here the importance of the Hartree-Fock scaling and data at low N0 is clear; linear
extrapolation in Fig. 5.8 a would produce vastly different results.
To correct for the small (few per cent) difference in Nc between the measurement
and reference series, we apply the ideal gas scaling Tc ∝ Nc to the reference series. The
second order correction in ∆Tc due to the small (< 2 %) Tc shift at a/λ0 ≈ 0.005 ismuch smaller than our statistical error bars. The result is a measurement for Tc at
two values of a/λ0 and equal N . The measurement at aref /λ0 is expected to be welldescribed by mean-field theory. This provides an accurate measure of ∆Tc /Tc0 , even if
the absolute system parameters are not as accurately known.

5.4

Critical temperature shift results

The 21 series for which thermal equilibrium was deemed to hold are shown in Fig. 5.9,
where data were taken for N ≈ (2-8)×105 (corresponding to Tc0 ≈ 180-330 nK) in orderto verify that the results depend only on a/λ0 . The results show very good agreementwith the mean-field result of equation (5.5) for a/λ0 . 0.01. For large a/λ0 we see
a clear positive deviation from this prediction. All data points are fitted well with asecond order polynomial,
a
∆Tc
= c1
+ c2
0
Tc
λ0



a
λ0

2
,

(5.34)

with c1 = −3.5 ± 0.3 (consistent with MF theory) and c2 = 46 ± 5. Note that thepolynomial fit was used to calibrate the y-axis of Fig. 5.9 by setting ∆Tc /Tc0 = 0 where
the fit cuts the axis. However, our results are unchanged within errors if we assume
the MF prediction for our reference data at a/λ0 ≈ 0.005 is exact. By fixing the slope
c1 to the mean-field result we can improve the estimate for the quadratic result to154

5.4 Critical temperature shift resultsb

N 0 (thousands)

8

56 a0

6

274 a0NT (thousands)

a

4
2
0
350

400

450

500

400
350

500

274 a0

450

56 a0
0

NT (thousands)

10

20

2/5

30

N0

Figure 5.8: a N0 versus NT for two concurrently taken series. The blue points were taken ataref = 56 a0 , the red points at a = 274 a0 (where λ0 ≈ 104 a0 for both series). The data has
been scaled to the same temperature (T = 240 nK) for illustrative purposes, directly showing
the shift of the critical number ∆Nc (T ). All points shown correspond to a condensed fractionof under 2 % . The solid lines show the extrapolation to N0 = 0 used to extract Nc . b The same
2/5
data is shown plotted in the form NT versus N0 , showing the extrapolation more clearly. In
both plots the filled red square is the data point calculated from the image shown in Fig. 5.6.
00

ΔTc /Tc (%)

-2

-4

-6
MF theory
-8
0.00

0.01

0.02

a/λ0

0.03

0.04

Figure 5.9: Shift of Tc with repulsive interactions. Data points were taken with N ≈ 2 × 105(blue circles), 4 × 105 (black squares) and 8 × 105 (red triangles) atoms. The dashed line is
the mean-field result ∆Tc /Tc0 = −3.426a/λ0 . The solid line shows the result of a secondorder polynomial fit to the data, discussed in the text. Vertical error bars are statistical, whilehorizontal error bars reflect the uncertainty in the position of the Feshbach resonance.

c2 = 42 ± 2.
The value of c2 strongly excludes zero, and is positive as expected for beyondMF correlations. As previously mentioned, this correction is not expected to be purely
quadratic, with the pre-factor expected to take the form c2 = [c02 ln(a/λ0 )+c002 ] according
to (5.7). However, our experimental precision is not high enough to discern the exact
form of the correction. Fitting with the logarithmic term included would only produce
a better fit to the data due to the increased number of fitting parameters.
These results rely heavily on our differential measurement approach to remove manysources of error. However, any errors that scale with a (or a/λ0 ) will systematically155

5.5 Assessment of errors

3

a>0

D

2

a=0

1

0
-3

-2

-1

0
R

1

2

3

Figure 5.10: Modification of the effective trapping potential caused by repulsive interactions.
The black dashed line shows the harmonic trapping potential, while the red line shows theeffective potential produced for Nc atoms at a/λ0 = 0.05. The solid lines show the phase-spacedensity distribution of the ideal (black) and interacting (red) clouds.

distort our findings. The following section describes the sources of such errors and how
they have been quantified. The results have been incorporated in the error bars of Fig.
5.9.

5.5

Assessment of errors

As we are looking for variations on the per cent scale, it is crucial to ensure systematic errors are handled correctly. The results depend heavily on the precise relative
measurement of temperature, as most other sources of error are eliminated by using the differential measurement method, while the dependence of our result on therecorded atom number is less strong and less susceptible to systematic variation. Two
a-dependent sources of temperature error have been identified, and require careful attention. These are discussed below. Two other effects (due to the finite imaging timeand residual Feshbach field curvature) that provide only absolute errors are still ofgeneral interest, and are assessed in Appendix D.

5.5.1

In-trap momentum distribution

The mean-field modification of the ideal in-trap density distribution caused by interactions has been discussed already, and this has a knock-on effect for the momentum
distribution. The effective trapping potential for the atoms is a flattened version of theharmonic potential due to the addition of the interaction potential. This is illustrated
in Fig. 5.10. The flatter trap means more low momentum states are occupied, so fitting
the resultant time-of-flight spatial distribution with the ideal g2 function will naturally
produce errors. Significantly, the error here is dependent on a/λ0 , so the magnitude ofthe effect will differ between our reference data and our measurement data.
156

5.5 Assessment of errors

The modified momentum distribution can be found by using the self-consistent
Dc (R) of equation (5.31) and plugging this back into (1.6), integrating only over space
to give n(p) . Using dimensionless coordinates once more this may be written
Z ∞

R2 dR

n(P ) ∝
0

exp

h

i

P2
R2
2 + 2 + ADc (R) − Aζ(3/2),(5.35)−1

where the T -dependent prefactors have been dropped for clarity as we are only interested in the shape of the distribution. Here we have introduced another dimensionlessvariable P , given by
P =where PT =

√

p
,
PT

(5.36)

mkB T . To see what effect the modified distribution has on our image

fitting we need to integrate along one axis (cf. the imaging axis) to give
Z ∞Z ∞
n2D (P ) ∝0

0

R2 dR dPy

exp

h

Py2

P2
2 + 2

i
,
2
+ R2 + ADc (R) − Aζ(3/2) − 1(5.37)

such that P is now in the x-z plane. Taking A → 0 it is possible to recover the ideal g2
distribution. n2D is fitted with a g2 function, and the extracted radius in terms of PT
gives an estimate of the expected temperature measurement error. PT is used to avoid
confusion between the thermal radius in-trap and in time-of-flight. For large TOF thethermal radius RT (t) → PT × (t/m). As shown in Fig. 5.11, it is clearly important to
exclude data around the cloud centre from the fit, as it is here where the distribution
is most altered. The temperature error expected for the experimental range of a/λ0 isshown in Fig. 5.12, where the fits to the high-energy wings of the distributions exclude
data within PT of the centre. If the fits are carried out on the entire distribution theerrors are about three times as large. We can see that the error is roughly linear in
a/λ0 , and at a/λ0 = 0.04 is ≈ −2.5%. This is too large to ignore, and must be factored
in to our error calculations for the data series.

5.5.2

Interaction energy

When the trapping potential is suddenly turned off to allow the expansion of the cloud,the stored interaction energy can be converted into kinetic energy. The cloud effectively
forces itself apart such that on average the atoms travel further than their initial kinetic
energies would have taken them, giving an apparent temperature higher than the true
value. This produces another a-dependent error, which will not cancel out in our
differential measurement.
To get an idea of the magnitude of the effect, the interaction energy per particle
at the critical point εint can be calculated from the average self-consistent mean-field157

5.5 Assessment of errors

43
n2 D Harb. unitL

a>0
a=0

2

1

0
0.0

0.5

1.0

1.5

2.0

pPT

Figure 5.11: Two-dimensional momentum distribution for an ideal gas and an interacting gas
in a harmonic trap. The black line shows the ideal g2 function, while the red line shows theenhancement of low-momentum states caused by interactions (a/λ0 = 0.05). The distributions
are normalised to the same atom number. This illustrates the need to exclude data from thecentral region when fitting the images. The dashed line shows the experimental exclusion region
for temperature fitting, equivalent to a single thermal radius in time-of-flight.

0% shift

-1

-2

-3
0.00

0.01

0.02

0.03

0.04

0.05

aΛ0

Figure 5.12: ∆T error associated with the modified momentum distribution. The error isroughly linear in a/λ0 . The shift is negative and ∼ 1-2 % for most of our experimental data.

158

5.5 Assessment of errors

0.14
0.12

Εint kB T

0.10
0.08
0.06
0.04
0.02

0

0.01

0.02

0.03

0.04

0.05

aΛ0

Figure 5.13: Variation of εint per kB T with interaction strength. The black points show theresults of equation 5.40, where the modification of the density distribution is taken into account.
The red dashed line shows the result assuming a fixed ideal gas distribution.

value of 2gnc (r) . This is given by
Z ∞
Z ∞
Eint
2
2
εint == 2g
4πr nc (r) dr
4πr2 nc (r) dr
N
0
0
Z ∞
Z
2g ∞ 2
R Dc (R)2 dR
R2 Dc (R) dR
= 3
λ0 0
0

Z ∞
Z ∞
2
2
= AkB T
R Dc (R) dR
R2 Dc (R) dR .
0(5.38)(5.39)(5.40)

0

As the integrals of equation (5.40) depend only on A, the quantity εint /kB T also depends
only on A, so we expect the error to scale with a/λ0 once more. The dependence issuperlinear as the average density increases with a/λ0 due to the central spreading of the
distribution. The dependence is shown in Fig. 5.13, along with the linear scaling thatwould be obtained if the distribution remained ideal. At a/λ0 = 0.04 the interactionenergy per particle is over 10 % of kB T , a considerable fraction. The effect is limited
however by using the zero-crossing of the Feshbach resonance to turn off interactions
during time-of-flight. Unfortunately we cannot completely eliminate the effect becauseof the finite ramping time of the Feshbach field. We also want to avoiding changing theinteraction strength before the trap is turned off which could invalidate our results.
To estimate the residual effect another mean-field model was used. Randomly
distributed starting positions and momenta for 100,000 atoms were generated according
to the Bose distribution (for our particular trapping potential). The final positions werethen numerically calculated given the time-varying interaction potential experienced
during time-of-flight. These positions were then projected along one-axis to produce afake image for fitting. This process was repeated to produce many such images. To keep
it reasonably simple the interaction potential during time-of-flight was modelled on the159

5.5 Assessment of errors

4 0 0

4 0 0
3 8 0

3 0 0
2 0 0

3 6 0

F ie ld ( G )S c a tte r in g L e n g th ( a 0 )

5 0 0

1 0 0
3 4 0
0
-1

T ra p O ff (0 )

1

2

3

4

T im e ( m s )Figure 5.14: Example of our Feshbach field ramp during TOF, used to minimise interaction
broadening of the cloud. Black points show the field strength and red points the corresponding
39
K scattering length. In this example the field is ramped down from close to resonance towards
the zero crossing value of 350 G immediately after the trap is turned off (dashed line). The
apparent quantisation in the scattering length is due to the resolution of the current meter used
to monitor the ramp, and the strong scaling of scattering length with field near the resonance.

density distribution of an ideal gas released from a harmonic trap, given in equation(2.56). The interaction strength could also be varied in time to compare what happens
when interactions persist during the whole TOF, when they are suddenly switched off
at some point, or when they are gradually ramped off.
Under typical experimental conditions the model predicts (when fitted using an
exclusion radius of RT ) that if interactions are sharply switched off after 2 ms TOF
there is no discernible temperature shift for a = 56 a0 (i.e. our reference data), while
for a = 357 a0 there is a residual shift of ∼ 2 % . This latter value corresponds toa/λT ≈ 0.034 for the temperature used in the model of 250 nK. This 2 % shift is arough upper limit, as can be seen by looking at the actual field variation during timeof-flight given in Fig. 5.14.
It was also possible to investigate this effect experimentally by comparing the fitted
temperatures for interactions on during TOF with those attained when ramping off
interactions for various a. The results, shown in Fig. 5.15, confirm the general scalingof the effect with a/λT , and also helped to validate the above model. Note that here
∆T = Ton − Toff , the difference between the observed temperature with interactions
left on during time-of-flight and when they are ramped off, so the residual systematic
error (which we are trying to quantify) is not included.
Fortuitously, this positive shift tends to counteract the negative shift caused by thechange to the momentum distribution discussed in the previous section, resulting in a
net error of at most ±1 % at a/λ0 = 0.04 . This is large enough to take account of in
our error assessments, but not so large as to greatly modify the results themselves.
160

5.6 Non-equilibrium effects

65DTToff H%L

4
3
2
1
0
0

100

200

300

400

500

a Ha0 L

Figure 5.15: Experimental observation of the shift in measured temperature due to interactions during time-of-flight. ∆T is the difference between the temperature measured wheninteractions are left on until just before imaging and the temperature measured when interactions are tuned to zero directly after the trap is turned off. For the right-most data point thisturn-off follows the ramp of Fig. 5.14. All of the data was taken at a temperature of ∼ 260 nK,
so multiplying the x-scale by 10−4 gives a rough conversion to a/λT . The error bars show thestandard error, while the red line is an unweighted linear fit to the data points.

5.6

Non-equilibrium effects

As previously stated, data was taken in the range 0.001 < a/λ0 < 0.06 , but only thatfor which 0.005 < a/λ0 < 0.04 was shown for the Tc shift results. This is because
outside this region the atom-loss rate becomes comparable to the re-thermalisation
rate, and the condition α  1 discussed in §2.7.2 is violated. As thermal equilibrium isnever fully established in a system with continuous dissipation (we can only be ‘close’
to equilibrium) we need a better way to quantify this condition, while also confirmingthat global (rather than just local) equilibrium can be achieved.
The relevant quantities are the elastic collision rate Γel , the hold time1 in the trap
thold , the trapping frequency ω̄ and a characteristic atom-number decay time τ . The
definition of τ is guided by our measurement precision. We measure Nc within ∼ 1 %,so we require that the gas continuously re-equilibrates on a time-scale τ corresponding
to 1 % atom loss. As long as this occurs the distribution of atoms between the thermaland condensed components should reflect the conditions of equilibrium to within themeasurement precision.
This condition is therefore parameterised by the quantity Γel τ . Equilibrium isusually deemed to require about 3 collisions per particle [67]. For all the data used for1

The hold time thold is taken to be the time from which the scattering length a reaches its measurement value to the moment at which the atoms are released from the trap.

161

5.6 Non-equilibrium effects

the Tc shift investigation we made sure that Γel τ > 5 , where Γel is given byΓel = n(0)vth σ0 /2 ,where
n(0) = Nmω̄ 2
2πkB T3/2r, vth =

8kB T
and σ0 = 8πa2 ,
πm(5.41)(5.42)and τ was found from our experiments near Nc . The expression for the elastic scattering
rate is the classical result in a harmonic trap used in [157].
For global equilibrium to hold, the condition thold > τ > 1/ω̄ must be satisfied. This
ensures there is enough time for the measurement conditions to be reflected across thefull extent of the cloud. Global equilibrium will also only hold outside the hydrodynamic
regime (in which the cloud becomes collisionally opaque), governed by the condition
Γel /ω̄ < 1 .
The results of our Tc measurement for N ≈ 4 × 105 (λ0 ≈ 104 a0 ) over the fullrange of a/λ0 values studied is displayed in Fig. 5.16 a. All data shown still satisfy theconditions thold > τ > 1/ω̄ and Γel thold > 5, sufficient for energy to be redistributed
through the system once the measurement conditions have been established. However,if Γel τ is too small the elastic collisions are unable to ‘keep up’ with the continuous
dissipation. This cannot be rectified by simply increasing thold , and the system isinherently out of equilibrium.
In Fig. 5.16 b we plot Γel τ for each Tc measurement, suggesting that quasi-equilibrium
appears to be lost for a universal value of Γel τ , despite Γel ≈ 0.7-1000 s−1 and τ ≈ 21000 ms individually varying by around three orders of magnitude. Also shown is thehydrodynamic parameter Γel /ω̄ .
The recorded values of ∆Tc /Tc0 where Γel τ > 5 is violated have no quantitative
meaning, as they depend heavily on our initial conditions, as is intrinsically true toall non-equilibrium phenomena. We can qualitatively interpret the results based on
our experimental procedure. In the small-a limit there is a smooth apparent rise inTc above Tc0 for a → 0. The loss mechanism in this regime is predominantly singlebody, affecting N0 and NT equally. Elastic collisions act to redistribute the atoms
between the two components to maintain an equilibrium population distribution. In
this case this corresponds to a net transfer of atoms from the condensate to the thermal
cloud. However, as a decreases so does Γel τ , and a condensate is observed for total
N lower than the expected equilibrium Nc (corresponding to a positive measurementof ∆Tc /Tc0 ). Strictly speaking of course T is not defined out of equilibrium, but theobserved effect is sufficiently small that the fitted equilibrium distribution still provides
a good measure of the energy content of the cloud.
In the large-a limit, ‘super-heated’ condensates surviving at a temperature above
the equilibrium Tc are also observed, but the explanation here is less trivial. Three-bodylosses affect the condensed and thermal fractions differently, the thermal component is162

5.7 Conclusionsa0

Equilibrium region0

ΔTc /Tc (%)

4

-4
-8

3

2

10

1

Γel / ω

Γelτ

b

1
0.00

0.01

0.02

0.03

0.04

0.05

0.06

0

a/λ0
Figure 5.16: a ∆Tc /Tc0 for N ≈ 4 × 105 atoms over the full experimental range, where alldata is taken using the procedure which assumes thermal equilibrium. The vertical dashedlines show the bounds of thermal equilibrium. At both low and high a there is sharp deviationfrom the smooth equilibrium curve (red line). b Equilibrium criteria, showing the values of Γel τ(solid squares) and Γel /ω̄ (open circles) that correspond to each data point in a. The horizontal
dashed line shows Γel τ = 5. The onset of the hydrodynamic regime occurs at Γel /ω̄ = 1.

far from saturation and the cloud enters the hydrodynamic regime. Without further
investigation it is hard to give even a proper qualitative explanation for the outcome
observed in our system.

5.7

Conclusions

Using the beneficial properties of a Feshbach resonance in 39 K and a novel differential
thermometry technique, we have taken high precision measurements of the Tc shift
over a far broader range of a/λ0 than has previously been achieved. In doing so wehave shown good agreement with mean-field theory below a/λ0 ≈ 0.01 , and above thisvalue observed positive deviation from mean-field theory, showing clear evidence forthe condensation-enhancing effects of critical correlations for the first time. In addition
we have probed beyond the regime in which equilibrium measurements can be taken.
In both very-weakly and very-strongly interacting limits non-equilibrium effects havebeen observed. The results of this chapter are summarised in [158].
It is hoped that the new information provided by our investigation will act toguide and encourage further theoretical studies of the difficult problem of beyondmean-field effects in a harmonically trapped gas. It would once again be of interest to163

5.7 Conclusions

study the behaviour in a variety of trapping geometries, tending towards a hard box
potential. Combined with direct investigation of local behaviour of a trapped gas using
in situ measurements, this could provide a clearer pathway to investigate the build-up
of coherence across a Bose gas.
It should be possible to extend the range of a/λ0 over which equilibrium measurements can be taken by altering other system parameters in combination with a (withwhich the three-body collision rate scales strongly). However, in our picture more precise measurements necessarily restrict the range that equilibrium measurements may
be taken over, so it is not clear whether it will be possible to reveal logarithmic terms
in the form of the Tc shift with this method.
The non-equilibrium behaviour seen will also vastly benefit from dedicated investigation in order to disentangle the competing effects in the strongly-interacting
regime. This should prove fruitful for comprehending condensation in intrinsically outof-equilibrium systems, such as polariton gases [104, 159].

164

Chapter 6

Conclusion“ Well, it was worth it, Jeeves.”– P. G. Wodehouse (The Code of the Woosters)
In this concluding chapter a brief summary of the accomplishments from each aspect of
the work covered in this thesis is given, before the direction of more recent and future
work is discussed.

6.1

Summary

A timeline of key events in the history of the group since its foundation is shown in
Fig. 6.1. From this it is clear that the majority of experimental time over the courseof this PhD was spent creating the two working sets of apparatus to produce BoseEinstein condensates. Once a stable system for producing 39 K BECs was in place,the scientific investigations progressed rapidly and promise to continue to do so. The
primary achievements of the work described in this thesis are as follows.
• A single-chamber system has been created, allowing the rapid production (∼ 15 s
cycle) of 87 Rb BECs of over 105 atoms in an optically plugged quadrupole trap.

2008

2009

2010

Saturation paper
Tc shift paper

K BEC paper
39

K BEC II39

Rb BEC II

87
Rb MOT II
39
K MOT II
87

87

87

Rb BEC I

Rb MOT I

Group foundedThis system is briefly described in Appendix B.

2011

Figure 6.1: Timeline of key events over the course of this thesis. The suffixes I and II refer tothe single-chamber and two-chamber experimental systems respectively. Dates are marked forthe first occurrence of magneto-optical trapping and Bose-Einstein condensation in each systemand for each atomic species. The publication dates of the papers [85, 116, 158] are also marked.

1656.2 Outlook• A two-chamber system has been created, and is only the second system worldwide
to have produced 39 K BECs. These are held in a crossed-dipole trap, allowing
tuning of the strength of interactions in the system via a Feshbach resonance.
The maximum condensate number of over 4 × 105 is the highest ever reported forthis atomic species. In the same system it is possible to produce 87 Rb BECs ofover 8 × 105 atoms. The setup and operating procedure are described in Chapters
2 and 3 respectively.
• The non-saturation of an interacting Bose gas in a harmonic trap has been quantified, both near the critical point and for larger condensed fractions. Extreme
deviations from the behaviour of an ideal gas are seen, even for what are generally considered weak interactions. This goes against preconceptions that arecommonly held in the field. Excellent quantitative agreement with mean-field
theory is seen in the low-N0 limit, while extrapolation to the non-interacting limit
provides the most compelling experimental evidence to date of ideal saturation
as predicted by Einstein. This work is described in Chapter 4.
• The interaction induced Tc shift for the BEC transition has been experimentally recorded at high precision over a greater range of interaction strengths thanever before. Good agreement with mean-field theory is observed in the weaklyinteracting limit, while for stronger interactions beyond-mean-field effects havebeen observed for the first time. The positive direction of this shift is in agreement with widely accepted theoretical prediction. In addition to this, in both thevery-weakly and very-strongly interacting limits non-equilibrium behaviour has
been detected. This work is described in Chapter 5.

6.2

Outlook

It is hoped that the work carried out in the group thus far is just the tip of theexperimental iceberg. Now that the essential machinery is in place there is a wealthof possible topics to explore, both as direct extensions to this work and other morediverse projects. Several such areas for exploration at various stages of conception are
discussed below.

6.2.1

Condensed fraction induced by critical correlations

An investigation directly related to the work on the Tc shift has already taken place[160]. The downside to the earlier study is that no straightforward link can be madeto the uniform system due to the intrinsic dependence of Tc on the global propertiesof a non-uniform system. In the more recent work this problem is negotiated by observing the condensed fraction f0 = N0 /N in a harmonic trap at the mean-field criticalnumber N = NcMF > Nc . The condensed fraction observed at this point is then solely
1666.2 Outlook

due to the effects of critical-correlations. Moreover, the effect is limited to the quasiuniform central region of the harmonic trap with volume ∝ (a/λ0 )3 . Combined withthe expected scaling n0 /n ∝ ∆nc ∝ a/λ0 in a uniform system leads to a predictedscaling f0 ∝ (a/λ0 )4 . This scaling has been experimentally observed. Furthermore, themeasured effect has been quantitively compared (via the local density approximation)with the Monte Carlo simulated prediction for a uniform system, and shows excellent
agreement.
The ability to tune the strength of interactions with the 39 K Feshbach resonancewas crucial to this investigation. Not only was it necessary to cover a wide range ofa/λ0 values and to use the differential measurement scheme, but also the largest valueof f0 recorded at N = Ncmf was only ∼ 1 %. It was therefore essential to also use themethods for precise measurement of small condensed fractions developed during thesaturation and Tc shift investigations.

6.2.2

Out-of-equilibrium studies

The non-equilibrium behaviour observed during the Tc shift measurement has sparked
further interest in this area. Studies are well underway into the dynamics of a Bosegas equilibrating from an out-of-equilibrium starting point. The method involves the
preparation of an ultra-cold cloud at a relatively high scattering length (where there
is only a small condensed fraction) and then instantaneously reducing the scattering
length. The system that was in thermodynamic equilibrium suddenly is no longer,
as the value of Nc suddenly drops. At this point elastic collisions act to return thesystem to equilibrium by the redistribution of energy between the atoms, resulting in a
rapid rise in the condensed fraction. The energy released by condensation results in theheating of the remaining thermal atoms, which increases Nc , so that we do not simply
acquire the condensed fraction expected for the new scattering length at the original
temperature. The details of this process have recently been submitted for review [161].
It is also planned to start the process from a position where there is no initial BEC, toobserve the phase transition as driven by the change of interaction strength.

6.2.3

Two-dimensional Bose gases

The planned extension of the system to incorporate two-dimensional trapping has been
prepared, and will hopefully be implemented in the near future. This will allow direct
extension of the saturation experiment to reduced dimensionality, and allow detailed
studies of the Berezinskii-Kosterlitz-Thouless (BKT) transition [162, 163] with the ability to tune interactions. Two-dimensional Bose gases come with a rich variety of new
phenomena, where in a uniform system superfluidity exists in the absence of a BEC.
The addition of 2D trapping to the system will therefore open up another branch of
future research topics [164].
167

6.2 Outlook

6.2.4

Annular Bose-Einstein condensates

In a brand new direction, work has been carried out on the two-chamber system to allow
transfer of a BEC from the CDT to a ring-shaped trap, created by the intersection of a
red-detuned hollow Laguerre-Gauss beam with a thin light sheet. A two-photon Ramanprocess [165] is used to impart angular momentum to the 87 Rb condensate. In a toroidal
geometry the BEC total angular momentum is quantised. The superfluid flow of the
BEC has been shown to perpetuate for long periods, showing the high metastabilityof the state. There is currently much interest in the decay mechanism of the multiplycharged quantised angular momentum, with possible parallels in systems ranging insize from superconducting nano-wires to rotating (pulsar) neutron stars. The resultsof these investigations have recently been submitted for review [166].

6.2.5

Power-law traps

The different scaling of physical parameters (such as non-saturation) with the trapping
potential make the study of gases in controllable power-law traps a topic of interest [167]. This should be possible using overlapped blue-detuned beams with various
Laguerre-Gauss modes, enabling trapping from a typical harmonic potential throughto a near-homogeneous hard-walled potential [168]. This would increasingly enhancethe effects of critical correlations in a three-dimensional Bose gas, and may even leadto direct global measurements of a close-to-uniform system. The tools necessary toachieve such a scenario are well under way to being developed, with Laguerre-Gauss
beams already in use as described above. The direct measurements will involve newinteresting challenges, such as accurate temperature determination from the greatly
modified time-of-flight distributions expected from these traps. New techniques developed in this direction should prove interesting to the general community as well ofcourse as the results of the investigations themselves.

6.2.6

Superconducting magnetic trap

The single-chamber system is being redeveloped with the planned addition of a dipole
trap. It is also being used to test magnetic coils wound from high-temperature superconducting tape, which is now commercially available through AMSC. When cooledwith liquid nitrogen, this tape can support current densities of close to 10,000 A cm−2 .
The coils in their present prototype form do not have any advantages over normal
copper-wound coils, but it is hoped that the proof-of-principal will allow advances in
the future to the advantage of cold atoms physicists. Possible benefits might include
the ability to position the coils further from the system than equivalent copper coils,
allowing greater optical access to experiments, or producing large magnitude magnetic
fields to, for example, access high-field Feshbach resonances. Another intriguing possibility would be to use persistent flow in the coils to produce highly-stable magnetic168

6.2 Outlook

fields for experiments.

6.2.7

Final remarks

The outlook is promising for the group and the field as a whole, with many diverting
questions still open. While being proud of the scientific achievements of the last four
years, the real lasting legacy will hopefully be the two functional BEC systems created during this period. The use of 39 K in our system has been key to most of ourexperiments thus far, and fully vindicates our choice to use this difficult species. The
experimental advantages of an accessible Feshbach resonance open up myriad possibilities for the future of many-body research with quantum degenerate gases.

169

Appendix A

Useful Physical Quantities
In this appendix some useful physical quantities are tabulated that are relevant to therest of the thesis. The physical constants in Table A.1 are the recommended CODATA
values [169]. The majority of the properties of 39 K and 87 Rb were collated in [23, 170](see references therein), while the scattering length data is taken from [65, 95, 171].
Quantity

SymbolValue

Speed of light in free spacec2.99792458 × 108 m s−1

Permeability of free spaceµ0

4π × 10−7 H m−1

Permittivity of free space0 = 1/(µ0 c2 )8.854187817... × 10−12 F m−1

Planck’s constanth = 2π~6.62606957(29) × 10−34 J s

Electronic chargee1.602176565(35) × 10−19 C

Electron rest massme

9.10938291(40) × 10−31 kg

Atomic mass unit

u1.660538921(73) × 10−27 kg

Bohr radius

a0 = 4π0 ~2 /(me e2 )5.2917720859(36) × 10−10 mBoltzmann’s constantkB

1.3806488(13) × 10−23 J K−1

Bohr magnetonµB = e~/(2me )9.27400968(20) × 10−24 J T−1

Table A.1: Values for some fundamental physical constants, as recommended by the National
Institute of Standards and Technology [169].

µB /kB≈ 2/3 K T−1= 200/3 µK G−1µB /h≈ 1.4 × 1010 s−1 T−1= 1.4 MHz G−1

h/kB≈ 4.8 × 10−11 K s= 48 µK MHz−1

Table A.2: Useful quantities when considering RF evaporation in a magnetic trap.

171

Property

Symbol

39 K

87 Rb

Atomic number

Z

19

37

Mass number

A

39

87

Relative natural abundanceη

93.2581(44) %27.83(2) %Atomic massm

38.96370668(20) u

86.909180520(15) u

Vapour pressure at 25 ◦ C

Pv

2.42 × 10−8 mbar

5.23 × 10−7 mbar

Nuclear spin

I

3/2

3/2

42 S1/2

52 S1/2

Electronic ground state

Table A.3: General properties of 39 K and 87 Rb.

Property

Symbol

39 K

87 Rb

Frequency

ω0 /2π389.286058716(62) THz

377.107463380(11) THz

Wavelengthλ770.10838505(12) nm

794.978851156(23) nm

Lifetimeτ

26.37(5) ns

27.679(27) ns

Natural linewidthΓ/2π6.03(1) MHz

5.7500(56) MHz

Table A.4: Optical properties of the 39 K and 87 Rb D1 lines.

Property

Symbol

39 K

87 Rb

Frequency

ω0 /2π391.01617003(12) THz

384.2304844685(62) THz

Wavelength

λ766.700921822(24) nm

780.241209686(13) nm

Lifetimeτ

26.37(5) ns

26.2348(77) ns

Natural linewidth

Γ/2π

6.035(11) MHz

6.0666(18) MHz

Recoil velocity

vrec

13.357 mm s−1

5.8845 mm s−1

Recoil temperatureTrec

418.06 nK

361.96 nK

Doppler temperatureTD

144.82 µK

145.57 µK|2, 2i → |3, 3i cycling transition, σ + polarisation:
Saturation intensity

Isat1.750 mW cm−2

1.669 mW cm−2

Cross-section

σ02.807 × 10−9 cm−2

2.907 × 10−9 cm−2

Table A.5: Optical properties of the 39 K and 87 Rb D2 lines.

172

39 K

87 Rb

Property

SymbolElectron spin g-factor

gS

2.00231930436153(53)Electron orbital g-factor

gL

0.99999369

Landé g-factor

gJ

2.00229421(24)2.00233113(20)Nuclear g-factor

gI-0.00014193489(12)-0.000 9951414(10)Hyperfine splitting∆Ehf /h

461.7197202(6) MHz

6834.68261090429(9) MHz

Table A.6: Ground state properties for 39 K (42 S1/2 ) and 87 Rb (52 S1/2 ), which may be usedin the Breit-Rabi equation (1.33).

Property

Symbol

39 K

87 Rb

Singlet scattering lengthas138.49(12) a0

90.0(2) a0

Triplet scattering length

at-33.48(18) a0

98.99(2) a0

Interspecies singlet scattering length

aKRb
s
aKRb
t

Interspecies triplet scattering length824+90−70 a035.9(7) a0

Table A.7: Singlet and triplet scattering lengths for 39 K and 87 Rb.

173

Appendix B

Small system
Much of the first year after our group was founded was spent getting to grips withthe experimental techniques required for atom trapping and cooling, taking the formof the construction of a small single-chambered system. The original intention was tocreate a magneto-optical trap for 87 Rb, which would entail setting up a laser system,a vacuum chamber, magnetic coils and computer control. The lessons learned whiledeveloping this small system would then aid in the construction of the large system. It
soon became apparent though that we might push the limits of this small system, andit was decided that we should attempt to use it to create a BEC. This was achieved
about a year after experimental work commenced in the lab.
The vacuum system is shown in Fig. B.1. The quartz optical chamber was made by
Triad Technology, and features seven radial 1” windows and two vertical 2” windows
for optical access, with a ∼ 1 cm diameter connection to the rest of the vacuum system.
The atom sources are of the same style as the large system, and vacuum is maintainedwith a 40 l s−1 ion pump. The original laser system consisted of a stripped-down versionof that shown in Fig. 2.23. When the large system was first built, light was shared
between the two systems. The small system has since been moved to a different labwith a new dedicated laser setup1 .
The scheme for magneto-optical trapping is shown in Fig. B.2. The splittingof the cooling and repumping light into six beams was initially achieved using the
Schäfter+Kirchhoff fibre-port cluster used in the main setup. This has now been
replaced with a home-built version. The field for the MOT is provided by a pairof water-cooled magnetic coils situated just next to the 2” windows of the chamber.
These are capable of producing a quadrupole field gradient of up to 200 G cm−1 along
the weak axis, and are also used for capturing and holding the atoms magnetically afteran optical molasses stage.
The principal drawback to the single-chamber design is the short trapping lifetime
resulting from the necessity to have a reasonable 87 Rb pressure for MOT loading. Oncethe magnetic trap has been loaded the cooling process must therefore proceed rapidly.
1

The compact nature of the small system meant that we could fit it on a small ‘bread-board’, so it
is easily transportable while still under vacuum.

175

b
a

d

c

e

f

Figure B.1: Small system vacuum setup. a Turbomolecular pump. b Ion pump. c All-metalright-angle valve. d Pressure gauge. e Optical chamber. f Atom sources. There is a window
on the opposite side of the vacuum system to the optical chamber to allow additional opticalaccess.

We thus choose to evaporatively cool the atoms using RF radiation directly in thequadrupole trap, which provides tight confinement, aiding the thermalisation rate of
the cloud. This requires some means of suppressing Majorana spin flips at the magnetic
zero of the quadrupole trap. This we achieve with an ‘optical plug’, this being a tightly
PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

focussed (∼ 30 µm waist) blue-detuned gaussian beam directed at the trap centre. Thisis the method that was used by the Ketterle group to create the first ever sodium BEC[4]. We use plug light with wavelength in the range 700-740 nm produced by a Coherent
MR110 titanium sapphire laser pumped with a Verdi 18 W laser at 532 nm. Despite
the relatively close detuning of the plug beam there is very little heating and atom loss
through light scattering, as the tight focus of the laser produces a very steep barriernear the trap centre.
The potential produced by the combination of the quadrupole field, optical plugand evaporating RF field is shown in Fig. B.3. The potential is displayed as a crosssection perpendicular to the optical axis of the plug and around the centre of thequadrupole field. The plug beam is fired horizontally at the magnetic trap such thatthe asymmetrical magnetic gradients in the vertical and radial directions result in two
distinct trap minima (the red regions in Fig. B.3). If the azimuthally symmetricgaussian beam were instead directed along the vertical axis a ring-shaped minimum
would be formed. Small imperfections in the laser profile would then lead to one or176Photodiode

Input

MOT
verticalaxis

Key
OutputPolarizing beam splitter cube

λ/2

Photodiode
Input

Polarizer
1/99 beam splitter
Wave plate

Output
λ/2

λ/2
Lens

λ/2

2" mirror

Outputλ/2

OutputFree-space optical fibre output

Outputλ/22" wave plate
Fibre collimator

OutputPolarization maintaining optical fibre

λ/4

λ/4
MOT
λ/4

Vacuumsystemλ/4Figure B.2: Schematic of the splitting of the MOT beams and their delivery to the opticalchamber.

177

Figure B.3: Potential produced by the combination of a quadrupole field, optical plug and
RF radiation. The plug beam is directed along one of the weak quadrupole axes. This produces
two distinct trap minima (shown by the red regions). The RF radiation used for evaporation
causes the quadrupole potential to ‘turn over’ as atoms are transferred into untrapped states.

more minima at random locations [172], so it is preferable to have the control provided
by our trap geometry. The resultant anisotropic harmonic trap frequencies for this typeof potential are given in [4].
The standard experimental sequence is as follows. We load rubidium atoms into the
MOT for around 5 s before turning off the magnetic field for 2.5 ms of optical molasses.
A 20 µs burst of pumping light drives the atoms into the |2, 2i state for magnetic
trapping. The atoms are captured by suddenly turning the quadrupole field on witha weak-axis field B 0 = 32 G cm−1 before the cloud is compressed by increasing B 0 to
100 G cm−1 in 0.5 s. An RF field is then turned on and held at 10 MHz for 0.3 s of plain
evaporation. At this point there are typically 108 atoms at 100 µK. The RF frequency
is swept linearly to 2 MHz in 3 s, reducing both the atom number and temperature by afactor of 10. To slow the loss rate due to three-body collisions we then decompress thetrap to B 0 = 40 G cm−1 in 200 ms. Finally, we apply a final exponential evaporation
sweep from 1.4 MHz to a chosen final value in 0.5 s. This results in quasi-pure BECsof over 105 atoms1 . A sample sequence with increasing time-of-flight for a condensed
fraction of ∼ 30 % at ∼ 500 nK is shown in Fig. B.4. The bimodality of the cloud isrevealed as TOF increases and the thermal atoms expand more rapidly than the BEC.
The fact that we see only one BEC is due to the fact that the plug beam is aligned
1
When BECs were being created in this manifestation of the small system we did not calibrate theimaging fudge-factor described in §2.10.6. This atom number is therefore likely to be an underestimationof the true value.

178

Figure B.4: Optical density images of BECs with ∼ 30 % condensed fraction, released fromthe optically-plugged quadrupole trap and imaged after 3-12 ms time-of-flight. As time-of-flightincreases, the bimodal distribution becomes apparent.

Figure B.5: Photograph of the trial system as it is now. The optical chamber can be seenamongst the magnetic coils to the right-hand side of the picture. The original plug beam wasaligned with the axis of the connection between the optical chamber and the rest of the vacuum
system.

slightly off centre such that one trap minima is higher than the other. During the final
evaporation sweep all of the atoms in the higher minima are removed.
After a period of neglect the small system is now under new management and isnearing a usable state once more. The disadvantage of the short trap lifetime is partiallycompensated for by the extremely fast condensation process and overall simplicity of thesystem. It is planned to add an optical dipole trap to the system for a comparison withthe plug, which has proved problematic at times due to the high alignment accuracy
required. The current incarnation of the system is shown in Fig. B.5.

179

Appendix C

Popov Approximation
The Popov approximation extends Hartree-Fock theory to include small deviations
from standard Hartree-Fock theory in the form of Bogoliubov-like excitations. The
background of this theory may be found in several text-books, but the notation usedbelow follows that found in [29] most closely. Here it is used to predict deviations fromthe Hartree-Fock description of non-saturation in a harmonically-trapped interacting
Bose gas. The semi-classical limit is taken once again, so the excited state density inthis picture (following equation (1.6)) is given as the familiar starting point:Z
1
1
nT (r) =dp .
3
(r,p)/kBT − 1
(2π~)
e

(C.1)

The chemical potential has been absorbed into (r, p) for the purposes of describing
the excitations. The process below was laid out by Jean Dalibard.
In the absence of a BEC the excitation spectrum takes the standard form
(r, p) =p2+ 2gnT (r) + V (r) − µ .
2m

(C.2)

When a BEC is present we must first generalise the Thomas-Fermi relationship of (1.26)
to take account of the effects of the thermal atoms on the condensate distribution.
This gives the chemical potential as also required for the self-consistent Hartree-Fockapproach,µ = V (r) + gn0 (r) + 2gnT (r) .

(C.3)

The factor of 2 before the excited state density appears due to the inclusion of theexchange term for atoms in different states, which is thus absent for the condensate.
The form of (r, p) now reflects the revised excitation spectrum, found by diagonalisinga modified form of the Hartree-Fock Hamiltonian1 . This is given by (from equation(8.119) of [29]): 2 1/2hi1/2  p2p
(r, p) = 2HF (r, p) − (gn0 (r))2+ 2gn0 (r)
,
=
2m
2m
1

(C.4)

The added term allows for creation and annihilation of momentum-conserving pairs of particles.

181

whereHF =p2
p2+ 2g(n0 (r) + nT (r)) + V (r) − µ =+ 2gn0 (r) .
2m
2m

(C.5)

In this case the relation (C.1) is modified by a factor to reflect the number of noncondensed particles associated with an excitation:
Z
1
HF (r, p)1
dp .
nT (r) =3
(r,p)/kBT − 1(2π~)
(r, p) e(C.6)

The above equations are local, so we can simplify the situation by omitting the spatial
dependence for now and switching to dimensionless variables
P =√

µ−V
4a
p
, D = nλ3T , and κ =
, A=
.
λ
kB T
mkB T
T

(C.7)

Working in terms of the phase-space density D, we find in the purely thermal case (cf.
§5.2)
DT = g3/2 (eκ−ADT ) .
When a condensate is present this is replaced byr Z ∞2P (P 2 + AD0 )
1
√
√
dP ,
DT =
π 0
P 2 + 2AD0 eP P 2 +2AD0 /2 − 1

(C.8)

(C.9)

where
AD0 = 2(κ − ADT )(C.10)

via equation (C.3). For computational purposes it is advantageous to use the substitution x2 = P 2 (P 2 + 2AD0 )/4 to rewrite (C.9) as
2
DT = I(κ − ADT ) , where I(z) = √
π

Z ∞
0

[(z 2 + x2 )1/2 − z]1/2
dx ,ex − 1(C.11)and to use the integral form of g3/2 (ez ) given in (1.9) for when there is no condensate.
Using z = κ − ADT , we must find the value z = zs that solves the equation
κ−z= F (z) ,A(C.12)

where F (z) is defined for all z by(
F (z) =

g3/2 (ez ) for z ≤ 0 ,I(z) otherwise.

(C.13)

If zs is negative there is no BEC, and DT = (κ−zs )/A . If zs is positive, DT = (κ−zs )/Aand D0 = 2zs /A . Fig. C.1 gives a graphical representation of equation (C.12).
When κ ≤ 0 the solution is straightforward and zs ≤ 0 , corresponding to D0 = 0.
When κ > 0 there are three possible cases, dependent on A, labelled i-iii:• i: If A is larger than some non-trivial value, there is a single solution where no
182

3

iii

ΖI 3 M

FHzL, HΚ - zLA

2

ii
i

2

1

F (z)

κ≤0
0
-4

-2

κ>0

0
z = Κ - ADT

2

4

Figure C.1: Graphical solutions for the self-consistent Popov approximation. Note that thefunction F (z) is continuous at z = 0 . When κ ≤ 0 there is only one solution (as we consider
only positive A). When κ > 0 there are three possible cases, discussed in the text. For case ii,
where there are multiple solutions, we choose the negative z result.

condensate is present.

• ii: For some intermediate value of A smaller than this threshold but larger thanκ/ζ(3/2) there are three possible solutions, two with a condensate and one without.

• iii: For A < κ/ζ(3/2) there again exists only one solution, this time giving a finite
condensate density.

We choose to omit the solutions of case ii for which a condensate is present, based only
on the assumption that the local phase-space density reaches the ideal ζ(3/2) before acondensate appears. We therefore obtain solutions where D0 = 0 for κ ≤ ζ(3/2)A, andD0 > 0 otherwise.
To find bulk solutions for a harmonically-trapped gas we consider the simple isotropic
case where the confining potential is given by V (r) = mω 2 r2 /2 . To carry on working
with dimensionless variables we setrR=, where RT =
RT

r

kB T
,
mω 2

(C.14)

and reinstate the position dependence of the other system parameters. κ is therefore
given byκ(R) = α −R2µ, where α =
,
2
kB T
183

(C.15)

and the number of atoms for each component is related to the phase-space density by
1
N0,T =(2π)3/2



kB T
~ω

3 Z ∞

4πR2 D0,T [κ(R), A]dR .

(C.16)

0

A final simplification is provided by setting S = R2 /2 and giving the results in termsof the ideal critical number Nc0 , resulting in
Z ∞√
N0,T
2
√
=
SD0,T [α − S, A]dS .
Nc0
ζ(3) π 0(C.17)

To calculate the data of Fig. 4.9 this procedure was carried out for the two mostextreme values of A used in experimental series. The data was then normalised forcomparison with both Hartree-Fock theory and the data by dividing by Nc , the valueof NT when N0 becomes finite in the Popov approximation, incorporating the inherent
shift of Nc for finite A.

184

Appendix D

Assessment of absolute errors
D.1

Finite imaging time

When taking a photo of a moving object with a stationary camera (such as a tennis
ball in flight) the finite shutter time on the camera introduces motion blur, and theobject appears to be spread along its axis of motion. The same is true when imaging
the cloud of potassium atoms after tTOF = 20 ms time-of-flight, where it has a finite
velocity gtTOF , while the image is taken with a τimage = 80 µs pulse of light (Fig.
D.1). The effect can be characterised by the ratio of the distance moved in this time,d ≈ τimage × gtTOF , to the thermal width of the cloud RT ,
d0 =

d
.
RT

(D.1)

In our experiment d ≈ 16 µm, while RT ≈ 130 µm for the lowest temperature data
series, giving d0 = 0.12. d here is a reasonable fraction of the thermal radius, andinstinctively one might think this would produce a large distortion in the measured
temperature.
To estimate the magnitude of the effect, the imaged cloud was modelled as an ideal
g2 exponential function. This was convolved with a unit box function of width d in thevertical z direction, and the resulting distribution was fitted with a g2 function once
more where the x and z widths were permitted to vary independently. The data within
a single thermal radius RT was excluded from the fit, which results in a perceived
temperature closer to the true value. The temperature shift is extracted by squaring
the widths given by the fit. The results of this investigation are shown in Fig. D.2.
Temperature shifts ∆Tx and ∆Tz are found for the horizontal and vertical directions,and the average ∆Tav is also shown. Of course the effect is predominantly along thedirection of motion z, but there is also a small shift in the horizontal direction.
Surprisingly the effect is negligible for even our coldest clouds, only exceeding the
per cent level for d0 & 0.5 . The scaling of d0 shows that the effect is enhanced forincreasing τimage or for particularly cold clouds and massive atoms. It is interesting tonote that for long time-of-flight (when the cloud width is much greater than its in-trappwidth) RT ≈ kB T /m × tTOF , meaning d0 is essentially independent of time-of-flight.
185

a

b

c

d

Figure D.1: Illustration of the effect on the perceived cloud width for an atom cloud thatis falling during imaging. The effect is analogous to motion blur in photography. a A tennis
ball falling at ∼ 2.5 m s−1 captured with a shutter time of 1/800 s is sharp, and the size iseasily measured. b When the shutter time is increased by a factor of 8 the image of the ballis noticeably blurred vertically. c A cloud imaged with a short imaging pulse (i.e. d0 ≈ 0)shows the characteristic circular symmetry of free thermal expansion. d If the imaging time isincreased the image is blurred along the z direction, leading to inaccuracies in the temperature
measurement. Here d0 = 0.75 to emphasize the effect.

2.5% shift

2.0
1.5
1.0

DT z

0.5

DTav

0.0

DTx

0.0

0.1

0.2

0.3
d

0.4

0.5

¢

Figure D.2: Temperature measurement error ∆T caused by motion blur with finite imaging
time. The error is dominated by the spreading of the distribution along the z-axis. The greyed
region represents the range of d0 covered by our measurements. The errors of  1% may besafely ignored.

186

The key point for us though is that the shift is  1% for all our measurements, so it
can safely be ignored.

D.2

Feshbach field curvature

The Feshbach coils were designed to be as close to Helmholtz configuration as possible, but they nonetheless produce a finite radial field curvature, calculated to be∼ 0.00325 G cm−2 A−1 . The zero crossing of the Feshbach resonance is reached witha current of 129 A, so during time-of-flight the atoms experience a field curvature−B 00 /2 ≈ 0.42 G cm−2 . This causes radial deceleration of atoms in the expanding
cloud (acceleration for negative curvature) as they fall along the z-axis. Assuming thissimple quadratic form for the field magnitude holds over the region in which the atoms
move we may again use a simple model to calculate the magnitude of the effect of
this curvature. For positive curvature we simply have a very weak harmonic trap withtrapping frequency ωFB and it is straightforward to calculate the radial motion of theatoms. Assuming all the particles start at the centre of this weak potential the final
position is given (in a one-dimensional description) byx=visin(ωFB tTOF ) ,
ωFB

(D.2)

where vi is the initial velocity of the atom. The position the particle would have reachedwithout the field curvature is x0 = vi tTOF . The perceived temperature T divided by
the real temperature T0 is then simply given by
T
=
T0



x
x0

2
=

sin2 (ωFB tTOF ).
(ωFB tTOF )2(D.3)

For negative field curvature a similar expression involving sinh rather than sin is found.
The value for ωFB with the above field curvature for our potassium atoms in the |1, 1i
state is ∼ 2π ×1.2 Hz, so for our 20 ms TOF the field curvature should cause a perceived
shift in Tx of ∼ −0.8%. The cloud temperature is calculated from Tz as well, so thisshift can be halved. This can’t accurately be accounted for as the field won’t be purely
quadratic and all of the atoms aren’t released from the centre of the Feshbach field.
However, the rough magnitude of the effect is still small enough to be ignored, though
it is large enough to be wary of in other experimental setups using large Feshbach bias
fields.

187

Bibliography[1] A. Einstein. Quantentheorie des einatomigen idealen gases. Sitzungsber. Preuss. Akad.
Wiss. 1925, 3 (1925).
[2] S. N. Bose. Plancks gesetz und lichtquantenhypothese. Z. Phys. 26, 178 (1924).
[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman & E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198 (1995).
[4] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn
& W. Ketterle. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett.
75, 3969 (1995).
[5] I. Bloch, J. Dalibard & W. Zwerger. Many-body physics with ultracold gases. Rev. Mod.
Phys. 80, 885 (2008).
[6] L. De Sarlo, P. Maioli, G. Barontini, J. Catani, F. Minardi & M. Inguscio. Collisional
properties of sympathetically cooled 39 K. Phys. Rev. A 75, 022715 (2007).
[7] G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio &G. Modugno. 39 K Bose-Einstein condensate with tunable interactions. Phys. Rev. Lett.
99, 010403 (2007).
[8] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni & M. Inguscio. Bose-Einsteincondensation of potassium atoms by sympathetic cooling. Science 294, 1320 (2001).
[9] G. Roati, F. Riboli, G. Modugno & M. Inguscio. Fermi-Bose quantum degenerate 40 K87
Rb mixture with attractive interaction. Phys. Rev. Lett. 89, 150403 (2002).
[10] S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos & R. G. Hulet. Extreme
tunability of interactions in a 7 Li Bose-Einstein condensate. Phys. Rev. Lett. 102, 090402(2009).
[11] J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M. Stamper-Kurn & W. Ketterle. Strongly enhanced inelastic collisions in a Bose-Einstein condensate near Feshbachresonances. Phys. Rev. Lett. 82, 2422 (1999).
[12] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. Muller, G. Roati,
M. Inguscio & G. Modugno. Observation of an Efimov spectrum in an atomic system.
Nature Physics 5, 586 (2009).
[13] T. Kishimoto, J. Kobayashi, K. Noda, K. Aikawa, M. Ueda & S. Inouye. Direct evaporative cooling of 41 K into a Bose-Einstein condensate. Phys. Rev. A 79, 031602 (2009).
[14] N. R. Claussen, S. J. J. M. F. Kokkelmans, S. T. Thompson, E. A. Donley, E. Hodby& C. E. Wieman. Very-high-precision bound-state spectroscopy near a 85 Rb Feshbachresonance. Phys. Rev. A 67, 060701 (2003).
189

[15] S. Dürr, T. Volz & G. Rempe. Dissociation of ultracold molecules with Feshbach resonances. Phys. Rev. A 70, 031601 (2004).
[16] A. D. Lange, K. Pilch, A. Prantner, F. Ferlaino, B. Engeser, H.-C. Nägerl, R. Grimm& C. Chin. Determination of atomic scattering lengths from measurements of molecularbinding energies near Feshbach resonances. Phys. Rev. A 79, 013622 (2009).
[17] C. Chin, R. Grimm, P. Julienne & E. Tiesinga. Feshbach resonances in ultracold gases.
Rev. Mod. Phys. 82, 1225 (2010).
[18] N. N. Bogoliubov. On the theory of superfluidity. J. Phys. (USSR) 11, 23 (1947).
[19] L. Pitaevskii & S. Stringari. Bose-Einstein Condensation (Oxford University Press, 2003).
[20] C. J. Foot. Atomic Physics (Oxford University Press, 2005).
[21] E. Arimondo, M. Inguscio & P. Violino. Experimental determinations of the hyperfine
structure in the alkali atoms. Rev. Mod. Phys. 49, 31 (1977).
[22] N. Bendali, H. T. Duong & J. L. Vialle. High-resolution laser spectroscopy on the D1
and D2 lines of 39,40,41 K using RF modulated laser light. J. Phys. B: At. Mol. Phys. 14,4231 (1981).
[23] D. A. Steck. Rubidium 87 D line data. http://steck.us/alkalidata/ (23rd December2010).
[24] G. Breit & I. I. Rabi. Measurement of nuclear spin. Phys. Rev. 38, 2082 (1931).
[25] R. Grimm, Weidemüller & Y. B. Ovchinnikov. Optical dipole traps for neutral atoms.
Adv. At. Mol. Opt. Phys. 42, 95 (2000).
[26] L. D. Landau & E. M. Lifshitz. Quantum Mechanics (Elsevier, 1981), third edn.
[27] B. H. Bransden & C. J. Joachain. Physics of Atoms and Molecules (Prentice Hall, 2003),
second edn.
[28] J. Dalibard. Collisional dynamics of ultra-cold atomic gases. In M. Inguscio, S. Stringari& C. Wieman (eds.) Proceedings of the International School of Physics Enrico Fermi,
Course CXL: Bose-Einstein condensation in gases, Varena 1998 (Addison-Wesley, 1998).
[29] C. J. Pethick & H. Smith. Bose-Einstein Condensation in Dilute Gases (Cambridge
University Press, 2008), second edn.
[30] K. Huang. Statistical Mechanics (Wiley, 1987), second edn.
[31] E. Braaten & H. W. Hammer. Universality in few-body systems with large scattering
length. Phys. Rep. 428, 259 (2006).
[32] L. S. Butcher, D. Stacey, C. J. Foot & K. Burnett. Ultracold collisions for Bose-Einsteincondensation. Phil. Trans. R. Soc. Lond., Ser. A 357, 1421 (1999).
[33] H. R. Thorsheim, J. Weiner & P. S. Julienne. Laser-induced photoassociation of ultracold
sodium atoms. Phys. Rev. Lett. 58, 2420 (1987).
190

[34] A. J. Moerdijk, B. J. Verhaar & A. Axelsson. Resonances in ultracold collisions of 6 Li,
7Li, and 23 Na. Phys. Rev. A 51, 4852 (1995).
[35] T. Köhler, K. Góral & P. S. Julienne. Production of cold molecules via magneticallytunable Feshbach resonances. Rev. Mod. Phys. 78, 1311 (2006).
[36] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn & W. Ketterle.
Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392, 151(1998).
[37] P. O. Fedichev, M. W. Reynolds & G. V. Shlyapnikov. Three-body recombination ofultracold atoms to a weakly bound s level. Phys. Rev. Lett. 77, 2921 (1996).
[38] H. M. J. M. Boesten, A. J. Moerdijk & B. J. Verhaar. Dipolar decay in two recent
Bose-Einstein condensation experiments. Phys. Rev. A 54, R29 (1996).
[39] J. F. O’Hanlon. A User’s Guide to Vacuum Technology (John Wiley and Sons Ltd, 2003).
[40] M. Greiner, I. Bloch, T. W. Hänsch & T. Esslinger. Magnetic transport of trapped cold
atoms over a large distance. Phys. Rev. A 63, 031401 (2001).
[41] H. J. Lewandowski, D. M. Harber, D. L. Whitaker & E. A. Cornell. Simplified systemfor creating a Bose-Einstein condensate. J. Low Temp. Phys. 132, 309 (2003).
[42] A. N. Nesmeyanov. Vapor Pressure of the Chemical Elements (Elsevier, 1963).
[43] K. M. Birnbaum. Cavity QED with multilevel atoms. Ph.D. thesis, California Instituteof Technology (2005).
[44] J. H. De Boer. The Dynamic Character of Absorption (Clarendon Press, 1953).
[45] A. Chambers, R. K. Fitch & B. S. Halliday. Basic Vacuum Technology, 2nd Edition(Institute of Physics Publishing, 1998).
[46] T. W. Hänsch & A. L. Schawlow. Cooling of gases by laser radiation. Optics Communications 13, 68 (1975).
[47] V. S. Letokhov, V. G. Minogin & B. D. Pavlik. Cooling and capture of atoms andmolecules by a resonant light field. Sov. Phys. JETP 45, 698 (1977).
[48] S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable & A. Ashkin. Three-dimensional viscousconfinement and cooling of atoms by resonance radiation pressure. Phys. Rev. Lett. 55,
48 (1985).
[49] P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould & H. J. Metcalf.
Observation of atoms laser cooled below the Doppler limit. Phys. Rev. Lett. 61, 169(1988).
[50] J. Dalibard & C. Cohen-Tannoudji. Laser cooling below the Doppler limit by polarization
gradients: simple theoretical models. J. Opt. Soc. Am. B 6, 2023 (1989).
[51] Y. Castin, J. Dalibard & C. Cohen-Tannoudji. The limits of Sisyphus cooling. In L. Moi,
S. Gozzini, C. Gabbanini, E. Arimondo & F. Strumia (eds.) Proceedings of the workshop“Light Induced Kinetic Effects on Atoms, Ions and Molecules”, held in Elba Island, Italy(May 2-5 1990), 5 (ETS Editrice Pisa, 1991).
191

[52] S. Chu. Nobel lecture: The manipulation of neutral particles. (1997).
[53] E. L. Raab, M. Prentiss, A. Cable, S. Chu & D. E. Pritchard. Trapping of neutral sodium
atoms with radiation pressure. Phys. Rev. Lett. 59, 2631 (1987).
[54] C. Fort, A. Bambini, L. Cacciapuoti, F. Cataliotti, M. Prevedellia, G. Tino & M. Inguscio.
Cooling mechanisms in potassium magneto-optical traps. Euro. Phys. J. D 3, 113 (1998).
[55] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman & J. V. Porto. Rapid productionof 87 Rb Bose-Einstein condensates in a combined magnetic and optical potential. Phys.
Rev. A 79, 063631 (2009).
[56] T. Esslinger, I. Bloch & T. W. Hänsch. Bose-Einstein condensation in a quadrupole-Ioffeconfiguration trap. Phys. Rev. A 58, R2664 (1998).
[57] H. F. Hess. Evaporative cooling of magnetically trapped and compressed spin-polarizedhydrogen. Phys. Rev. B 34, 3476 (1986).
[58] K. B. Davis, M.-O. Mewes, M. A. Joffe, M. R. Andrews & W. Ketterle. Evaporative
cooling of sodium atoms. Phys. Rev. Lett. 74, 5202 (1995).
[59] W. Petrich, M. H. Anderson, J. R. Ensher & E. A. Cornell. Stable, tightly confining
magnetic trap for evaporative cooling of neutral atoms. Phys. Rev. Lett. 74, 3352 (1995).
[60] W. Ketterle & N. J. van Druten. Evaporative cooling of trapped atoms. In B. Bederson& H. Walther (eds.) Advances In Atomic, Molecular, and Optical Physics, vol. 37, 181(Academic Press, 1996).
[61] O. J. Luiten, M. W. Reynolds & J. T. M. Walraven. Kinetic theory of the evaporative
cooling of a trapped gas. Phys. Rev. A 53, 381 (1996).
[62] J.-F. Clément, J.-P. Brantut, M. Robert-de Saint-Vincent, R. A. Nyman, A. Aspect,
T. Bourdel & P. Bouyer. All-optical runaway evaporation to Bose-Einstein condensation.
Phys. Rev. A 79, 061406 (2009).
[63] D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Köppinger & S. L. Cornish. Dual-species
Bose-Einstein condensate of 87 Rb and 133 Cs. Phys. Rev. A 84, 011603 (2011).
[64] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell & C. E. Wieman. Production of
two overlapping Bose-Einstein condensates by sympathetic cooling. Phys. Rev. Lett. 78,
586 (1997).
[65] F. Ferlaino, C. D’Errico, G. Roati, M. Zaccanti, M. Inguscio, G. Modugno & A. Simoni.
Feshbach spectroscopy of a K-Rb atomic mixture. Phys. Rev. A 73, 040702 (2006).
[66] M. Anderlini & D. Guéry-Odelin. Thermalization in mixtures of ultracold gases. Phys.
Rev. A 73, 032706 (2006).
[67] C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt & C. E. Wieman. Measurementof Cs-Cs elastic scattering at T = 30 µK. Phys. Rev. Lett. 70, 414 (1993).
[68] I. Gotlibovych. Microwave manipulation of ultra-cold atoms. Master’s thesis, Universityof Cambridge (2010).
192

[69] E. Hecht. Optics (Addison Wesley, 2002), fourth edn.
[70] P. Richman. Development of laboratory equipment for the trapping and cooling of atomic
gases. CPGS report, University of Cambridge (2010).
[71] T. Müller, B. Zimmermann, J. Meineke, J.-P. Brantut, T. Esslinger & H. Moritz. Local
observation of antibunching in a trapped Fermi gas. Phys. Rev. Lett. 105, 040401 (2010).
[72] C. Sanner, E. J. Su, A. Keshet, R. Gommers, Y.-i. Shin, W. Huang & W. Ketterle.
Suppression of density fluctuations in a quantum degenerate Fermi gas. Phys. Rev. Lett.
105, 040402 (2010).
[73] J. Szczepkowski, R. Gartman, M. Witkowski, L. Tracewski, M. Zawada & W. Gawlik.
Analysis and calibration of absorptive images of Bose-Einstein condensate at nonzero
temperatures. Rev. Sci. Instrum. 80, 053103 (2009).
[74] C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw & S. Whitlock. Detection of small
atom numbers through image processing. Phys. Rev. A 82, 061606 (2010).
[75] W. Alt. An objective lens for efficient fluorescence detection of single atoms. Optik 113,142 (2002).
[76] A. Dareau. Optical devices for imaging, trapping and manipulating cold atoms. Master’s
thesis, University of Cambridge (2010).
[77] F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer & A. Aspect. Criticaltemperature of a trapped, weakly interacting Bose gas. Phys. Rev. Lett. 92, 030405(2004).
[78] W. Ketterle, D. Durfee & D. Stamper-Kurn. Making, probing and understanding BoseEinstein condensates. In Proceedings of the International School of Physics - Enrico Fermi(IOS Press, 1999).
[79] Y. Castin & R. Dum. Bose-Einstein condensates in time dependent traps. Phys. Rev.
Lett. 77, 5315 (1996).
[80] A. Keshet.
2011).

http://akeshet.github.com/Cicero-Word-Generator/ (1st September[81] C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Inguscio, G. Modugno & A. Simoni.
Feshbach resonances in ultracold 39 K. New Journal of Physics 9, 223 (2007).
[82] B. D. Esry, C. H. Greene & J. P. Burke. Recombination of three atoms in the ultracold
limit. Phys. Rev. Lett. 83, 1751 (1999).
[83] C. C. Bradley, C. A. Sackett & R. G. Hulet. Bose-Einstein condensation of lithium:
Observation of limited condensate number. Phys. Rev. Lett. 78, 985 (1997).
[84] J. M. Gerton, D. Strekalov, I. Prodan & R. G. Hulet. Direct observation of growthand collapse of a Bose-Einstein condensate with attractive interactions. Nature 408, 692(2000).
193

[85] R. L. D. Campbell, R. P. Smith, N. Tammuz, S. Beattie, S. Moulder & Z. Hadzibabic.
Efficient production of large 39 K Bose-Einstein condensates. Phys. Rev. A 82, 063611(2010).
[86] R. S. Williamson III. Magneto-optical trapping of potassium isotopes. Ph.D. thesis,
University of Winsconsin (1997).
[87] C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane,
P. Szriftgiser, H. Perrin & J. Dalibard. Phase-space density in the magneto-optical trap.
Phys. Rev. A 52, 1423 (1995).
[88] J. Weiner, V. S. Bagnato, S. Zilio & P. S. Julienne. Experiments and theory in cold andultracold collisions. Rev. Mod. Phys. 71, 1 (1999).
[89] W. Süptitz, G. Wokurka, F. Strauch, P. Kohns & W. Ertmer. Simultaneous cooling andtrapping of 85 Rb and 87 Rb in a magneto-optical trap. Opt. Lett. 19, 1571 (1994).
[90] M. L. Harris, P. Tierney & S. L. Cornish. Magnetic trapping of a cold Rb-Cs atomic
mixture. J. Phys. B 41, 035303 (2008).
[91] X. Xu, T. H. Loftus, J. W. Dunn, C. H. Greene, J. L. Hall, A. Gallagher & J. Ye.
Single-stage sub-Doppler cooling of alkaline earth atoms. Phys. Rev. Lett. 90, 193002(2003).
[92] V. Gokhroo, G. Rajalakshmi, R. K. Easwaran & C. S. Unnikrishnan. Sub-Doppler deepcooled bosonic and fermionic isotopes of potassium in a compact 2d+ -3d MOT set-up.
Journal of Physics B: Atomic, Molecular and Optical Physics 44, 115307 (2011).
[93] M. Landini, S. Roy, L. Carcagnı́, D. Trypogeorgos, M. Fattori, M. Inguscio & G. Modugno.
Sub-Doppler laser cooling of potassium atoms. Phys. Rev. A 84, 043432 (2011).
[94] C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie & C. J. Foot. The
temperature of atoms in a magneto-optical trap. Europhys. Lett. 28, 397 (1994).
[95] E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen & B. J. Verhaar.
Interisotope determination of ultracold rubidium interactions from three high-precisionexperiments. Phys. Rev. Lett. 88, 093201 (2002).
[96] G. Delannoy, S. G. Murdoch, V. Boyer, V. Josse, P. Bouyer & A. Aspect. Understanding
the production of dual Bose-Einstein condensation with sympathetic cooling. Phys. Rev.
A 63, 051602 (2001).
[97] D. Xiong, P. Wang, H. Chen & J. Zhang. Evaporative cooling rubidium atoms withmicrowave radiation. Chin. Opt. Lett. 8, 351 (2010).
[98] A. Simoni, M. Zaccanti, C. D’Errico, M. Fattori, G. Roati, M. Inguscio & G. Modugno.
Near-threshold model for ultracold K-Rb dimers from interisotope Feshbach spectroscopy.
Phys. Rev. A 77, 052705 (2008).
[99] W. Ketterle & N. J. van Druten. Bose-Einstein condensation of a finite number of particles
trapped in one or three dimensions. Phys. Rev. A 54, 656 (1996).
194[100] M. Zaccanti. Tuning of the interactions in ultracold K-Rb quantum gases. Ph.D. thesis,
LENS and Università degli Studi di Firenze (2007).
[101] Z. Yan, M. Li, L. Chen, C. Chen & J. Chen. Density of states and thermodynamic
properties of an ideal system trapped in any dimension. J Phys. A: Math. Gen. 32, 4069
(1999).
[102] V. I. Yukalov. Modified semiclassical approximation for trapped Bose gases. Phys. Rev.
A 72, 033608 (2005).
[103] L. D. Landau & E. M. Lifshitz. Statistical Physics, Part 1 (Elsevier, 1980), third edn.
[104] J. Kasprzak et al. Bose-Einstein condensation of exciton polaritons. Nature 443, 409(2006).
[105] J. Klaers, J. Scmitt, F. Vewinger & M. Weitz. Bose-Einstein condensation of photons inan optical microcavity. Nature 468, 545 (2010).
[106] M. Erhard, H. Schmaljohann, J. Kronjäger, K. Bongs & K. Sengstock. Bose-Einsteincondensation at constant temperature. Phys. Rev. A 70, 031602 (2004).
[107] M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee & W. Ketterle.
Bose-einstein condensation in a tightly confining dc magnetic trap. Phys. Rev. Lett. 77,
416 (1996).
[108] J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman & E. A. Cornell. Bose-Einsteincondensation in a dilute gas: Measurement of energy and ground-state occupation. Phys.
Rev. Lett. 77, 4984 (1996).
[109] F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer & A. Aspect. Experimental study of the thermodynamics of an interacting trapped Bose-Einstein condensed
gas. Phys. Rev. A 70, 013607 (2004).
[110] P. Krüger, Z. Hadzibabic & J. Dalibard. Critical point of an interacting two-dimensional
atomic Bose gas. Phys. Rev. Lett. 99, 040402 (2007).
[111] J. L. Miller. Ultracold Bose gases deviate from the textbook picture. Physics Today 64,
16 (2011).
[112] F. Dalfovo, S. Giorgini, L. P. Pitaevskii & S. Stringari. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463 (1999).
[113] V. N. Popov. Functional Integrals and Collective Excitations (Cambridge University
Press, 1987).
[114] L. Sabchez-Palencia & M. Lewenstein. Disordered quantum gases under control. Nature
Physics 6, 87 (2010).
[115] J. Klaers, J. Schmitt, T. Damm, F. Vewinger & M. Weitz. Bose-Einstein condensationof paraxial light. Appl. Phys. B 105, 17 (2011).
[116] N. Tammuz, R. P. Smith, R. L. D. Campbell, S. Beattie, S. Moulder, J. Dalibard &Z. Hadzibabic. Can a Bose gas be saturated? Phys. Rev. Lett. 106, 230401 (2011).
195

[117] T. D. Lee & C. N. Yang. Many-body problem in quantum mechanics and quantum
statistical mechanics. Phys. Rev. 105, 1119 (1957).
[118] T. D. Lee & C. N. Yang. Low-temperature behavior of a dilute Bose system of hard
spheres. I. Equilibrium properties. Phys. Rev. 112, 1419 (1958).
[119] A. E. Glassgold, A. N. Kaufman & K. M. Watson. Statistical mechanics for the nonideal
Bose gas. Phys. Rev. 120, 660 (1960).
[120] K. Huang. Studies in Statistical Mechanics, Vol. II (North-Holland, 1964).
[121] A. Fetter & J. D. Walecka. Quantum Theory of Many-Particle Systems (McGraw-Hill,1971).
[122] T. Toyoda. A microscopic theory of the lambda transition. Annals of Physics 141, 154(1982).
[123] H. T. C. Stoof. Nucleation of Bose-Einstein condensation. Phys. Rev. A 45, 8398 (1992).
[124] M. Bijlsma & H. T. C. Stoof. Renormalization group theory of the three-dimensional
dilute Bose gas. Phys. Rev. A 54, 5085 (1996).
[125] P. Grüter, D. Ceperley & F. Laloë. Critical temperature of Bose-Einstein condensationof hard-sphere gases. Phys. Rev. Lett. 79, 3549 (1997).
[126] M. Holzmann, P. Grüter & F. Laloë. Bose-Einstein condensation in interacting gases.
Eur. Phys. J. B 10, 739 (1999).
[127] K. Huang. Transition temperature of a uniform imperfect Bose gas. Phys. Rev. Lett. 83,
3770 (1999).
[128] M. Holzmann & W. Krauth. Transition temperature of the homogeneous, weakly interacting Bose gas. Phys. Rev. Lett. 83, 2687 (1999).
[129] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë & D. Vautherin. The transition temperature of the dilute interacting Bose gas. Phys. Rev. Lett. 83, 1703 (1999).
[130] G. Baym, J.-P. Blaizot & J. Zinn-Justin. The transition temperature of the dilute interacting Bose gas for N internal states. Europhys. Lett. 49, 150 (2000).
[131] J. D. Reppy, B. C. Crooker, B. Hebral, A. D. Corwin, J. He & G. M. Zassenhaus. Density
dependence of the transition temperature in a homogeneous Bose-Einstein condensate.
Phys. Rev. Lett. 84, 2060 (2000).
[132] M. Wilkens, F. Illuminati & M. Krämer. Transition temperature of the weakly interacting
Bose gas: perturbative solution of the crossover equations in the canonical ensemble. J.
Phys. B 33, L779 (2000).
[133] P. Arnold & B. Tomášik. Tc for dilute Bose gases: Beyond leading order in 1/N . Phys.
Rev. A 62, 063604 (2000).
[134] P. Arnold & G. Moore. BEC transition temperature of a dilute homogeneous imperfect
Bose gas. Phys. Rev. Lett. 87, 120401 (2001).
196

[135] V. A. Kashurnikov, N. V. Prokof’ev & B. V. Svistunov. Critical temperature shift in
weakly interacting Bose gas. Phys. Rev. Lett. 87, 120402 (2001).
[136] H. Kleinert. Five-loop critical temperature shift in weakly interacting homogeneous BoseEinstein condensate. Mod. Phys. Lett. B 17, 1011 (2003).
[137] M. J. Davis & S. A. Morgan. Microcanonical temperature for a classical field: Application
to Bose-Einstein condensation. Phys. Rev. A 68, 053615 (2003).
[138] B. Kastening. Bose-Einstein condensation temperature of a homogenous weakly interacting Bose gas in variational perturbation theory through seven loops. Phys. Rev. A 69,
043613 (2004).
[139] S. Ledowski, N. Hasselmann & P. Kopietz. Self-energy and critical temperature of weaklyinteracting bosons. Phys. Rev. A 69, 061601 (2004).
[140] K. Nho & D. P. Landau. Bose-Einstein condensation temperature of a homogeneous
weakly interacting Bose gas: Path integral Monte Carlo study. Phys. Rev. A 70, 053614(2004).
[141] V. Betz & D. Ueltschi. Critical temperature of dilute Bose gases. Phys. Rev. A 81, 023611(2010).
[142] R. Seiringer & D. Ueltschi. Rigorous upper bound on the critical temperature of dilute
Bose gases. Phys. Rev. B 80, 014502 (2009).
[143] J. O. Andersen. Theory of the weakly interacting Bose gas. Rev. Mod. Phys. 76, 599(2004).
[144] R. Onofrio, C. Raman, J. M. Vogels, J. R. Abo-Shaeer, A. P. Chikkatur & W. Ketterle.
Observation of superfluid flow in a Bose-Einstein condensed gas. Phys. Rev. Lett. 85,
2228 (2000).
[145] E. Braaten & J. Pearson. Semiclassical corrections to the oscillation frequencies of atrapped Bose-Einstein condensate. Phys. Rev. Lett. 82, 255 (1999).
[146] L. Pitaevskii & S. Stringari. Elementary excitations in trapped Bose-Einstein condensed
gases beyond the mean-field approximation. Phys. Rev. Lett. 81, 4541 (1998).
[147] M. Holzmann, J.-N. Fuchs, G. A. Baym, J.-P. Blaizot & F. Laloë. Bose-Einstein transition
temperature in a dilute repulsive gas. C. R. Physique 5, 21 (2004).
[148] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë & D. Vautherin. Bose-Einstein transition
in a dilute interacting gas. Eur. Phys. J. B 24, 107 (2001).
[149] O. Zobay. Phase transition of trapped interacting Bose gases. Laser Phys. 19, 700 (2009).
[150] S. Giorgini, L. P. Pitaevskii & S. Stringari. Condensate fraction and critical temperatureof a trapped interacting Bose gas. Phys. Rev. A 54, R4633 (1996).
[151] V. L. Ginzburg. Some remarks on phase transitions of the second kind and the microscopic
theory of ferroelectric materials. Sov. Phys. Solid State 2, 1824 (1961).
197

[152] M. Houbiers, H. T. C. Stoof & E. A. Cornell. Critical temperature of a trapped Bose gas:
Mean-field theory and fluctuations. Phys. Rev. A 56, 2041 (1997).
[153] M. Holzmann, W. Krauth & M. Naraschewski. Precision Monte Carlo test of the HartreeFock approximation for a trapped Bose gas. Phys. Rev. A 59, 2956 (1999).
[154] M. J. Davis & P. B. Blakie. Critical temperature of a trapped Bose gas: Comparison oftheory and experiment. Phys. Rev. Lett. 96, 060404 (2006).
[155] R. Meppelink, R. A. Rozendaal, S. B. Koller, J. M. Vogels & P. van der Straten. Thermodynamics of Bose-Einstein-condensed clouds using phase-contrast imaging. Phys. Rev. A
81, 053632 (2010).
[156] N. Navon, S. Piatecki, K. Günter, B. Rem, T. C. Nguyen, F. Chevy, W. Krauth &C. Salomon. Dynamics and thermodynamics of the low-temperature strongly interacting
Bose gas. Phys. Rev. Lett. 107, 135301 (2011).
[157] H. Wu, E. Arimondo & C. J. Foot. Dynamics of evaporative cooling for Bose-Einsteincondensation. Phys. Rev. A 56, 560 (1997).
[158] R. P. Smith, R. L. D. Campbell, N. Tammuz & Z. Hadzibabic. Effects of interactions on
the critical temperature of a trapped Bose gas. Phys. Rev. Lett. 106, 250403 (2011).
[159] A. Amo et al. Collective fluid dynamics of a polariton condensate in a semiconductor
microcavity. Nature 457, 291 (2009).
[160] R. P. Smith, N. Tammuz, R. L. D. Campbell, M. Holzmann & Z. Hadzibabic. Condensed
fraction of an atomic Bose gas induced by critical correlations. Phys. Rev. Lett. 107,
190403 (2011).
[161] R. P. Smith, S. Beattie, S. Moulder, R. L. D. Campbell & Z. Hadzibabic. Condensate
growth dynamics in a quantum-quenched super-saturated Bose gas (2011). arXiv:1112.
4457v1[cond-mat.quant-gas].
[162] V. L. Berezinskii. Violation of long range order in one-dimensional and two-dimensional
systems with a continuous symmetry group. I. Classical systems. Sov. Phys. JETP 34,
610 (1971).
[163] J. M. Kosterlitz & D. J. Thouless. Ordering, metastability and phase transitions in
two-dimensional systems. J. Phys. C 6, 1181 (1973).
[164] Z. Hadzibabic & J. Dalibard. Two-dimensional Bose fluids: An atomic physics perspective. In R. Kaiser, D. Wiersma & L. Fallani (eds.) Proceedings of the International Schoolof Physics Enrico Fermi, Course CLXXIII: Nano optics and atomics: transport of lightand matter waves, Varena 2009 (IOS Press, Amsterdam and SIF, Bologna, 2011).
[165] M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson & W. D.
Phillips. Quantized rotation of atoms from photons with orbital angular momentum.
Phys. Rev. Lett. 97, 170406 (2006).
[166] S. Moulder, S. Beattie, R. P. Smith, N. Tammuz & Z. Hadzibabic. Quantised superflow
glitches in an annular Bose-Einstein condensate (2011). arXiv:1112.0334v1[cond-mat.
quant-gas].
198

[167] V. Bagnato, D. E. Pritchard & D. Kleppner. Bose-Einstein condensation in an externalpotential. Phys. Rev. A 35, 4354 (1987).
[168] A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost & E. Charron.
Bose-Einstein condensation in dark power-law laser traps. Phys. Rev. A 82, 023613(2010).
[169] P. J. Mohr, B. N. Taylor & D. B. Newell. The 2010 CODATA internationally recommended values of the fundamental physical constants. http://physics.nist.gov/cuu/
Constants/ (31st December 2010).
[170] T. G. Tiecke. Properties of potassium. http://staff.science.uva.nl/~tgtiecke/
PotassiumProperties.pdf (February 2010).
[171] S. Falke, H. Knöckel, J. Friebe, M. Riedmann, E. Tiemann & C. Lisdat. Potassium
ground-state scattering parameters and Born-Oppenheimer potentials from molecular
spectroscopy. Phys. Rev. A 78, 012503 (2008).
[172] D. S. Naik & C. Raman. Optically plugged quadrupole trap for Bose-Einstein condensates.
Phys. Rev. A 71, 033617 (2005).

199