Persistent Currents in Bose-Einstein Condensates
Stuart Moulder
Trinity College
University of Cambridge

This dissertation is submitted for the degree ofDoctor of Philosophy

September 2, 2013

To my parents Kevin andChristine, and my sister Carren.

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

Persistent Currents in Bose-Einstein Condensates
Stuart Moulder
This thesis describes a set of experiments which probe the superfluid nature of both
single and two component Bose-Einstein condensates of 87 Rb in a multiply connected
toroidal geometry.
The first part of this thesis describes the preparation and detection of persistentcurrents in a toroidal Bose-Einstein condensate (BEC). The apparatus and procedure
for cooling of a dilute vapour of 87 Rb atoms to degeneracy is briefly outlined. We thenexplain how the condensate is transferred to a purely optical ring trap, formed at theintersection of a horizontal sheet beam to confine against gravity, and a vertical tube
beam which produces the multiply connected geometry required for persistent flow.
Multiply charged superfluid flow is induced using a two photon transfer, and the final
angular momentum is detected kinematically in time of flight.
We study the metastability and decay of multiply charged superflow in a ringshaped BEC. High-charge superflow persisting for over a minute is observed, and weclearly resolve a cascade of quantised steps in its decay. These stochastic decay events,
associated with vortex-induced 2π phase slips, correspond to collective jumps of atoms
between discrete angular momentum states. By numerical calculation of the excitation
spectrum and superflow velocity, we show the supercurrent decays rapidly if the flow
velocity approaches a critical velocity in agreement with the local sound speed. Forsuperflow below this limit, stochastic phase slips are also observed to occur at a muchslower rate.
This work is extended to study the stability of supercurrents in a toroidal twocomponent spinor gas consisting of 87 Rb atoms in two different spin states. We showthat for a large spin-population imbalance we recover the long-lived metastable behaviour of the single component case, with superflow limited only by atom-number
decay. However we find that the supercurrent is unstable for spin polarisations below
a well defined critical value. The role of phase coherence between the two spin components is investigated, and it is shown that only the magnitude of the spin-polarisationivector, rather than its orientation in spin space, is relevant for supercurrent stability.
By continuously coupling the magnetic levels of the ground state we generate an azimuthal vector potential, corresponding to a non-zero magnetic flux threading the ring.
By varying this effective magnetic field we demonstrate the creation of a limited subset
of dressed states with coupled angular motion and spin composition. Application ofsuch a setup is presented by explaining a proposed superfluid fraction measurement.
With a view to future work, the technical limitations of our current setup are thendiscussed.

ii

Acknowledgements
I owe a huge debt of thanks to all the people who have helped and supported me throughthis PhD. Those that know me well will attest to the fact that I’ve never been particularly forthcoming with gratitude, reserving it for truly landmark occasions. Sincesuch an occasion has come, I sincerely hope that my brandishing the stick of sarcastic
cynicism for the last three and a half years, has made the carrot of humble gratitude all
the sweeter. The biggest thanks must go to my supervisor Zoran Hadizibabic, whose
endless drive and enthusiasm for his work first won me over to the field of degenerate
ultracold gases, and have been a continuing source of inspiration and motivation eversince. In particular, conversations with Zoran always seem to conclude with feverish
nodding of heads and adamant belief in our revised plan to ‘get famous’ and ‘conquer
the world’; and while we seem to have differing ideas of what constitutes famous, I was
always instilled with a renewed sense of excitement and conviction for my work.
A huge thanks must go to Scott Beattie who was the Tango to my Cash, the Gooseto my Maverick, and without whom I would often have been left floundering. For three
years we enjoyed an unnatural amount of time together, and though tempers occasionally frayed, Scott must take a large portion of the credit for making my time in
Cambridge so enjoyable. A great friend both in and out of the lab, and ever willing
partner in too many sports to mention1 , I wish him and Erin all the best back in their
homeland of Canada.
This work would never have been possible without the help of the three original
founding members of this lab, Naaman Tammuz, Robbie Campbell, and Rob Smith,whose tireless efforts built the system which I used in this thesis. I must thank themfor making the lab such a welcoming and friendly place, and for readily passing on somuch wisdom which they themselves learnt through hard graft. One must particularly
applaud the efforts of Naaman to introduce a sense of culture to the lab, and forgive
Robbie his efforts to introduce musical theatre to the lab. Special thanks go to Rob
whose wealth of experience and knowledge have made him an invaluable asset uponwhich the entire fate of the lab hinges. . .
I would also like to thank the newer members of the group for their assistance,
the so called A-team: Toby (explosives), Igor (the muscles), Alex (gadgets), and Rich(effort?). In particular the enthusiasm of Alex and Rich to get involved and bring new
ideas to the table revitalised our studies on persistent currents, and I leave knowing thelab has a promising future. Thanks are also in order to Nigel Cooper, Stefan Baur, and1

football, tennis, squash, table tennis, badminton, Robot Unicorn Attackiii

Jean Dalibard for numerous insightful and helpful discussions along the way. A special
mention has to go to Pam Hadder, who seamlessly dealt with every administrative
problem, half completed claims form, and misguided order I was able to muster.
Outside of the lab there are a few people who offered more support and guidance
than I could ever have deserved. Flying in the face of the dreaded thesis acknowledgement curse I simply have to thank Marion for being an absolute rock and an endlesssource of compassion and understanding. Finally, and surpassing all others, I thankmy family for their endless support and belief.

iv

Contents
1 Introduction

1

1.1

Persistent currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Theory of Bose-Einstein condensation and superfluidity
2.1

2.2

2.3
2.4

2.5

5

Condensation of an ideal Bose gas . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Uniform trapping potential . . . . . . . . . . . . . . . . . . . . .

6

2.1.2

Harmonic trapping potential . . . . . . . . . . . . . . . . . . . .

7

Weakly interacting Bose gas . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Basic scattering theory . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.2

Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . .

10

2.2.3

The Thomas-Fermi regime . . . . . . . . . . . . . . . . . . . . . .

12

Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3.1

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

13

Bose-Einstein condensation and superfluidity . . . . . . . . . . . . . . .

16

2.4.1

The Landau criterion. . . . . . . . . . . . . . . . . . . . . . . .

16

2.4.2

Superfluid component . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4.3

Vorticity and irrotational flow . . . . . . . . . . . . . . . . . . . .

21

Excitations in a uniform Bose gas

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Experimental methods and BEC production

26
27

3.1

Sequence overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2

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

29

3.2.1

Two-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.2.2

Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . .

33

3.2.3

Two photon processes . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2.4

Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . . .

37

3.2.5

Optical forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.3

Laser cooling and trapping. . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.1

87 Rb hyperfine structure . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.2

Magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . . .

42

3.3.3

Sub-Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.4

Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.5

Atoms in an external magnetic field . . . . . . . . . . . . . . . . . . . .

46v

3.5.1
3.6

Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . . .

Evaporative cooling48. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.6.1

Microwave system . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.6.2

RF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.7

Optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.8

|F = 1, mF = 1i state preparation . . . . . . . . . . . . . . . . . . . . .

60

3.9

Absorption imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.9.1

Imaging systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.9.2

Atom number calibration . . . . . . . . . . . . . . . . . . . . . .

65

3.9.3

Imaging the |F = 1i state . . . . . . . . . . . . . . . . . . . . . .

66

3.10 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Preparation and detection of persistent currents
4.1

4.2

4.3

4.4

68
69

Previous ring trap experiments . . . . . . . . . . . . . . . . . . . . . . .

70

4.1.1

Previous ring traps . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.1.2

Applications of ring BECs . . . . . . . . . . . . . . . . . . . . . .

71

Optical ring trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Sheet beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2.2

Laguerre-Gauss beam . . . . . . . . . . . . . . . . . . . . . . . .

77

4.2.3

Spatial light modulator (SLM) . . . . . . . . . . . . . . . . . . .

79

4.2.4

Pseudo-Laguerre-Gauss beam . . . . . . . . . . . . . . . . . . . .

82

4.2.5

Correcting for abberations . . . . . . . . . . . . . . . . . . . . . .

85

4.2.6

Loading procedure . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Imparting angular momentum . . . . . . . . . . . . . . . . . . . . . . . .

92

4.3.1

Rotation procedure . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.3.2

Raman wavelength . . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.3.3

Raman beam setup . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.3.4

Coherent Raman Rabi oscillations . . . . . . . . . . . . . . . . . 101

Detecting angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1

Matter-wave interference . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.2

Kinematic detection in time-of-flight . . . . . . . . . . . . . . . . 105

4.5

Vortex dynamics in a connected geometry . . . . . . . . . . . . . . . . . 108

4.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Quantised decay of persistent currents

113

5.1

Previous studies on supercurrents . . . . . . . . . . . . . . . . . . . . . . 115

5.2

Supercurrent quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3

Long-lived superflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4

Superflow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1

BEC density in a ring trap . . . . . . . . . . . . . . . . . . . . . 122

5.4.2

Effect of azimuthal roughness . . . . . . . . . . . . . . . . . . . . 125vi

5.4.3
5.5

Imaging resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.1

Feynman velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.5.2

Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5.3

Surface modes velocity . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5.4

Excitation spectrum of a ring BEC . . . . . . . . . . . . . . . . . 134

5.5.5

Critical velocity discussion . . . . . . . . . . . . . . . . . . . . . . 138

5.6

Statistics of stochastic phase slips . . . . . . . . . . . . . . . . . . . . . . 141

5.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Persistent currents in spinor condensates
6.1

6.2

145

Multi-component condensates . . . . . . . . . . . . . . . . . . . . . . . . 145
6.1.1

Spinor condensates . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.1.2

Mixtures6.1.3

Dynamic stability

6.1.4

Phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.1.5

Spin domain formation . . . . . . . . . . . . . . . . . . . . . . . . 156. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149. . . . . . . . . . . . . . . . . . . . . . . . . . 151

Two-component supercurrent . . . . . . . . . . . . . . . . . . . . . . . . 158
6.2.1

Single versus two-component supercurrrent . . . . . . . . . . . . 162

6.2.2

Stability phase diagram . . . . . . . . . . . . . . . . . . . . . . . 162

6.3

Spin dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.4

Coherently coupled two-component supercurrent . . . . . . . . . . . . . 166

6.5

Spin symmetric phase diagram . . . . . . . . . . . . . . . . . . . . . . . 167

6.6

Two-state barrier model . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7 Towards an azimuthal gauge potential

173

7.1

Linear gauge potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.2

Azimuthal gauge potential . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2.1

7.3

Experimental progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.3.1

7.4

Superfluid fraction measurement . . . . . . . . . . . . . . . . . . 179
Technical challenges . . . . . . . . . . . . . . . . . . . . . . . . . 184

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8 Summary and outlook
8.1

187

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Appendix A Analytic form of pseudo-LG beam

191

Appendix B Numerical calculation of ring BEC excitation spectrum

193

Bibliography

201

vii

viii

Chapter 1

Introduction
The theory of quantum mechanics has underpinned many of the greatest scientific andtechnological advances in the last century. Nevertheless, for much of the time we prefer
to understand the world through classical mechanics, a high temperature approximation, where the wave-like nature of particles can be neglected. Occasionally however,the quantum nature of the world refuses to be ignored, and in order to progress we areforced out of our comfort zone. Superfluidity and superconductivity offer two particularly striking examples where classical physics falls short. Superfluidity, the phenomenaof mass currents without viscosity, and superconductivity, the phenomena of charge currents without resistance, are fundamentally related and arise as a consequence of theexistence of a macroscopic wavefunction describing the system.
Such a wavefunction was first postulated by Einstein in 1924 [1] following the workof Bose on blackbody radiation [2]. Einstein showed the existence of a phase transition for a gas of non-interacting bosons at low temperatures, arising purely from thequantum statistics of the particles involved. As the temperature is lowered, a criticalthreshold is reached where the interatomic distance is on the order of the coherence
length of the matter waves, and the classical viewpoint breaks down. At this point a
Bose-Einstein condensate (BEC) forms, containing a macroscopic number of particles
all occupying the same quantum state. These particles are thus described by the same
macroscopic wavefunction and the system constitutes a quantum fluid. The properties
of such a quantum fluid were shown to be vastly different from those of a classicalfluid, providing an explanation for the unfamiliar behaviour observed in superfluid andsuperconducting systems.
At this point, the first such quantum fluid had already been observed, with thediscovery of superconductivity in 1911, when Onnes [3] found the resistance of solid
mercury dropped to zero at temperatures below 4 K. The connection between superconductivity and BEC wasn’t fully made however until later, ultimately leading to theformulation of the landmark BCS theory in 1957 [4], which describes superconductivity
as a microscopic effect caused by the condensation of Cooper pairs of electrons in aboson-like state. Instead, Einstein’s prediction of a macroscopically occupied condensed
state remained unproved experimentally until the discovery of superfluidity in liquid11.1 Introduction

helium in 1938 [5, 6], when it was found that the viscosity of liquid helium vanishes
suddenly below 2.17 K. In spite of the fact that liquid helium is strongly interacting,it was soon argued by F. London that this transition was an example of Bose-Einsteincondensation [7].
With this insight, superflow could now be understood as an adiabatic transformation of the condensed state brought about by changes in the macroscopic boundary
conditions. Such motion would not require the presence of intermediate excited states,
and therefore could occur without energy dissipation. This interpretation lead directly
to the two-fluid model for liquid helium [8, 9], which succeeded in explaining and predicting many of the physical phenomena of superfluid helium. Fundamentally however,a full theoretical understanding of superfluid helium is difficult due to the strength of
the atomic interactions present in the liquid phase which exclude the possibility of a
perturbative expansion from the non-interacting Bose gas considered by Einstein.
The realisation of Bose-Einstein condensation in dilute atomic gases in 1995 represented a breakthrough in the history of quantum fluids. This scientific milestone
was first achieved by the groups of E. Cornell and C. Wieman at Boulder [10], and a
few months later by W. Ketterle at MIT [11]. The significance of this achievement,
together with the decades of work on laser-cooling and manipulation of atoms which
preceded it, were recognised by the Physics Nobel prizes of 1997, 2001, and 2005. Suchsystems provide the ideal test beds for simulating and understanding the broad arrayof many-body quantum phenomena observed in condensed matter physics. In part thisis due to the ever expanding arsenal of experimental tools which have now been developed to allow one to effectively construct a desired Hamiltonian. Additionally, due tothe low densities required to prevent the gas transitioning to a solid, the inter-atomic
interactions present in these systems are often sufficiently weak to permit an accurate
perturbative treatment. This is in stark contrast to the superfluid experiments of Helium, where the macroscopic condensed state accounts for less than 10% of the total
density [12].
The combination of a clean and controllable environment together with a tractable
theoretical framework, mean experiments on atomic condensates offer an excellent playground for probing superfluidity, and by extension, superconductivity. From this, it’s
hoped that progress can be made in outstanding problems such as high temperature
superconductors, the connection between superfluidity and dimensionality, and the precise connection between superfluidity and BEC. The specific questions tackled in thisthesis relate to understanding the metastability and decay of persistent currents insuperfluids.

2

1.2 Persistent currents

1.1

Persistent currents

The significance of persistent currents can be seen in an alternate definition of superfluidity and superconductivity, which is given by the response of the system to rotation,or equivalently an external magnetic field1 . Such a definition is explained in [13], where
the author considers liquid helium in a multiply-connected annular geometry. Startingabove the transition temperature to superfluidity, Tλ , the annulus is set into rapid rotation, causing the helium in its normal state to come into rotation with the container.
The liquid is then cooled through Tλ and the container is brought to a stop. The liquid
helium, which is now in its superfluid state, will continue to rotate indefinitely, forming a persistent mass current. For a stationary container, this rotating state cannot
be the thermodynamic equilibrium, and so is an example of an extremely long-lived
metastable state. Such behaviour is characteristic of the superfluid state only, whereangular momentum is conserved.
The equivalent phenomena in superconducting systems is the conservation of magnetic flux. Starting with a cylindrical superconductor above the transition temperature
Tc , an external magnetic field is applied along the axis of symmetry. The sample isthen cooled below Tc and the field removed. The analogous effect in superconductors is
now observed, where a persistent surface charge current forms, generating a magnetic
field such that the flux penetrating the sample remains unchanged. In both cases, thesystems remain in an excited metastable state due to the presence of a free energy barrier to decay, the origin of which is the presence of a macroscopically occupied quantum
state.
The stability and eventual decay of such supercurrents have been studied for decades
in both helium and superconductors, however the decay mechanism is still not fully understood. Persistent currents in dilute atomic BECs therefore offer an ideal platform
with which to make further progress, and in addition, to explore the relationship between condensation and superfluidity.

1.2

Thesis outline

This thesis details our work on the stability and decay of persistent currents in dilute annular BECs confined to a toroidal geometry. The remainder of this thesis is
structured as follows: Chapter 2 provides a theoretical background to the phenomena of Bose-Einsteincondensation and superfluidity.
 Chapter 3 covers the experimental methods necessary for producing condensatesof 87 Rb, as well as the physical principles relevant to preparing and studying
1

One can show that the Hamiltonian for a neutral atom in a rotating frame of reference is equivalent
to that of a charged particle in a magnetic field3

1.2 Thesis outline

persistent currents.
 Chapter 4 describes our experimental setup for creating an all-optical ring trap

and preparing a state of definite rotation using a two-photon transition to imprint
an azimuthal phase winding onto the condensate wavefunction. The method
for kinematic detection of rotation by time-of-flight expansion is also presented.
The significance of a multiply-connected geometry for long-lived supercurrents
is illustrated by studying the rapid loss of angular momentum through vortex
dynamics in a simply-connected rotating condensate.
 Chapter 5 presents our first results on the metastability and decay of persistentcurrents, which are published in [14]. We demonstrate the extreme metastabilityof multiply-charged superflow persisting for over a minute. The persistent current
is found to decay in a cascade of quantised decay steps which unambiguously confirms that 2π phase slips are the supercurrent decay mechanism. The dynamics
of these phase slips is shown to be stochastic, with a probability which increases
as the flow velocity approaches a critical velocity set by the local sound speed.
 Chapter 6 extends our studies to a two-component spinor condensate, the resultsof which are published in [15]. We identify a regime at large population imbalanceswhere the two-component supercurrent is fundamentally stable and exhibits thelong-lived metastability observed in the single-component case. Below a welldefined critical population imbalance however, we show the presence of bothcomponents means the supercurrent is fundamentally unstable.
 Chapter 7 discusses our current progress towards creating an azimuthal vector

potential, analogous to generating a rotating frame of reference. We present
some preliminary experimental results, and describe the technical hurdles which
remain.
 Finally, in chapter 8 we summarise our results and mention possible directions

for future work.

4Chapter 2Theory of Bose-Einstein condensation andsuperfluidity
This chapter presents some of the theory behind Bose-Einstein condensation (BEC) andattempts to illuminate the intrinsic connection between the phenomena of condensationand superfluidity. We start by discussing how the BEC phase transition arises in theideal Bose gas purely as a result of the quantum statistics of the particles. The role ofinteractions is then introduced and the excitation spectrum of the weakly interacting
Bose gas is calculated. From this we show the concept of superfluid flow as adiabatic
motion of the condensate leads directly to the ideas of a critical velocity and quantised
vortex states.

2.1

Condensation of an ideal Bose gas

For non-interacting bosons in thermodynamic equilibrium, the mean occupancy of asingle-particle state of energy  at temperature T is given by the Bose-Einstein distributionf () =

1,
exp [( − µ)/kB T ] − 1(2.1)

where kB is the Boltzmann constant and µ is the chemical potential. The chemical
potential is determined as a function of both the total atom number N and temperature
T , such that particle number is conserved. The total atom number is then given by
the product of the mean occupancy f () with the density of states g() summed over
all available energy levels. Assuming that the confining potential is such that theenergy level spacing is much less than kB T , we can replace the sum with an integral.
Consequently, the system can be described by a continuum of excited states plus thediscrete ground state which must be explicitly retained [16, 17]:Z ∞N = N0 + Nex = N0 +df ()g(),

(2.2)

min

where N0 is the number of particles in the ground state, and Nex the number in the
excited states of the system. As the temperature is lowered the chemical potential rises
52.1 Condensation of an ideal Bose gas

and the mean occupancy of the allowed states changes. For the distribution f () to bepositive for all states, the chemical potential is limited by µ < min , and as a result,while the occupation of the ground state can be arbitrarily large, the occupation numberof any excited state ν is constrained to be less than 1/(exp[(ν − min )/kB T ] − 1). If thetotal number of atoms in the excited states, Nex , is less than the total atom number, N,
the remaining atoms are accommodated in the ground state and form a condensate of
N0 atoms. We therefore quantify the phase transition to BEC as occurring at a criticalatom number Nc for a given T , or equivalently, at a critical temperature Tc for a givenN:

Z ∞Nc = Nex (T, µ = min ) =dg()
min1
exp [( − min )/kB T ] − 1

.

(2.3)

Within the semi-classical approximation we can define a local distribution function,
where fp (r)dpdr/(2π~)3 denotes the mean number of particles in the phase space volume dpdr:
fp (r) =1
exp [(p (r) − µ)/kB T ] − 1(2.4)

The particle energies are those of a free particle at point r
p (r) =p2+ V (r),2m(2.5)

where V (r) is the external potential. With this definition we calculate the density ofparticles occupying the excited states of the system asZ
dp1
nex (r) =3(2π~) exp [(p (r) − µ)/kB T ] − 1
g3/2 (z(r))
=
,
λ3T

(2.6)

where λT is the thermal de Broglie wavelength associated with the finite extent of eachparticle wave packet
s
λT =

2π~2
,
mkB T

(2.7)

z(r) = exp (µ − V (r))/kB T is the fugacity, and g3/2 is the polylogarithm function
defined bygγ (z) =

∞
X
zn
n=1

2.1.1

nγ

.

(2.8)

Uniform trapping potential

In a uniform system the three dimensional free particle density of states is given byg() =

V m3/2 1/2
 ,21/2 π 2 ~3

6

(2.9)

2.1 Condensation of an ideal Bose gas

where V is the volume of the system. Inserting equation (2.9) into (2.2), gives the result(2.6), with an external potential V (r) = 0, as expected. For a uniform ideal gas µ ≤ 0,
and as µ approaches zero the fugacity z(r) approaches unity, and hence g3/2 (z(r)) tends
to g3/2 (1) = 2.612. This gives an upper bound on the density of particles in the excitedstates and gives the criteria for Bose condensation as
nλ3T > max(nex λ3T ) = 2.612,(2.10)

where n is the total particle density. This equation illustrates that condensation occurs
when the inter-particle spacing, n−1/3 , becomes smaller than the wavelength associatedwith each particle, λT .
The critical temperature at which condensation occurs is given by (2.6) with z(r) =1 and nex = n,
2π~2
Tc =
mkB



n
g3/2 (1)2/3.

(2.11)

This gives the temperature at which condensation first occurs; combining (2.6) with
(2.11) gives the functional form of the condensate fraction for T ≤ Tc
N0
=1−
N

2.1.2



T
Tc

3/2
.

(2.12)

Harmonic trapping potential

To first order we can expand any trap in three dimensions around the minimum andapproximate it to a harmonic trap with confining potential
V (r) =m 2 2
(ω x + ωy2 y 2 + ωz2 z 2 ).
2 x

(2.13)

This yields a quantised energy spectrum of single-particle energies that are labeled by
the non-negative quantum numbers nx , ny , nz ,
nx ,ny ,nz = ~(nx ωx + ny ωy + nz ωz ) + 0 ,(2.14)

where 0 is the zero-point energy ~(ωx + ωy + ωz )/2. The onset of condensation ina harmonic trap therefore occurs at µ = 0 . Using the harmonic oscillator quantum
numbers, equation (2.2) can be written asZ ∞dnx dny dnz
N = N0 +,exp [~(ωx nx + ωy ny + ωz nz )/kB T ] − 10(2.15)

where again we approximate the sum to an integral, assuming kB T  ~ω0 , whereω0 = (ωx ωy ωz )1/3 is the geometric mean of the trapping frequencies. This integral can

72.2 Weakly interacting Bose gas

be performed by changing variables ~ωi ni /kB T = ñi to giveN = N0 + g3 (1)

kB T
~ω0

3
.

(2.16)

With the condition N0 = 0, we rearrange this expression to find the critical temperature
Tc =

~ω0
kB



N
g3 (1)

1/3,(2.17)and the functional form of the condensate fraction for T ≤ Tc
N0
=1−
N



T
Tc

3
.

(2.18)For an ideal gas in the absence of interactions, the density distribution of the condensate
will reflect the ground state wavefunction ψ0 (r), which for a harmonic trap is given by"   2  2 #2
 mω 3/2xyz
0,(2.19)exp −−−n0 (r) = N0 |ψ0 (r)|2 = N0
π~
ax
ay
az
where ai =

q

~
mωi

is the oscillator length. Neglecting the zero point energy, 0 , thedensity distribution of the excited states for T < Tc is given by (2.6) with µ = 0g3/2 (e−V (r)/kB T )g3/2 (exp −m(ωx x2 + ωy y 2 + ωz z 2 )/2kB T )nex (r) =
=
.
λ3T
λ3T

(2.20)For typical experimental parameters, the condensate width is much narrower than
the width of the thermal distribution. As a result, in harmonically trapped gases,
condensation occurs both in momentum space, with the macroscopic occupation of theground state, and in real space, with the condensate appearing as a sharp peak in thecentral region of the density distribution.

2.2

Weakly interacting Bose gas

So far we have considered the ideal Bose gas, which even in the absence of interactions
exhibits a purely quantum-statistical phase transition to BEC. In reality particles will
always interact, and even in the weakly interacting limit a real Bose gas behaves qualitatively differently from an ideal Bose gas. Interactions between atoms will modify theequilibrium shape and dynamics of the condensate, and play an essential role in the
superfluid properties studied in this thesis.

2.2.1

Basic scattering theory

Here we outline the basic scattering theory in the context of ultracold gases, which
has been extensively covered in textbooks and lectures [18–20]. We consider a collision82.2 Weakly interacting Bose gas

process between two particles, 1 and 2, of mass m, interacting through the sphericallysymmetric potential V (r1 − r2 ). Defining the relative coordinates r = r1 − r2 andp = (p1 − p2 )/2, the Hamiltonian for the relative motion is given byp̂2+ V (r̂) ψk (r) = Ek ψk (r),2mr

(2.21)

where mr = m/2 is the reduced mass, and Ek = ~2 k 2 /2mr is the incident energy.
Assuming V (r) → 0 at large distances we look for solutions at |r| → ∞ with theasymptotic form:ψk (r) ' eik·r + f (θ)

eikr
,
r

(2.22)

where θ is the angle between the incoming and outgoing waves. The physical meaningof this collision state is the superposition of an incident plane wave with momentum k,
and a scattered wave function. The form of the scattering potential, V (r), determines
the scattering amplitude f (θ). At the very low energies considered in ultracold gases
experiments, only partial waves with zero angular momentum (l = 0) will contributeto the outgoing wavefunction. In this s-wave scattering limit the scattering amplitude
has no angular dependence and approaches a constant, denoted by −a. Therefore in
the low energy, k → 0, limit, the wavefunction becomesaψ(r) = 1 − .
r(2.23)

The constant a is the scattering length, and gives the intercept of the asymptotic
wavefunction with the r axis. To relate the scattering length and the interatomic
potential V (r), we must solve (2.21) and take the same low energy limit. Rewriting(2.21) as(∇2 + k 2 )ψk (r) = U (r)ψk (r),(2.24)

where U (r) = 2mr V (r)/~2 , the general solution can be written asZψk (r) = φk (r) + d3 r0 G0 (r − r0 )U (r0 )ψk (r0 ),(2.25)

where φk (r) = eik.r is the solution to the free particle Hamiltonian, (∇2 + k 2 )φk (r) = 0,
and G0 (r) is a Green’s function of the Laplace operator(∇2 + k 2 )G0 (r) = δ 3 (r)=⇒G0 (r) = −1 eikr.
4π r(2.26)

To compare to (2.22), we take the far field limit where r is much greater than the rangeof the interaction.
r · r0|r − r | ' r −
r
0

0

=⇒

9

eik|r−r |eikr −ik0 ·r0'e,|r − r0 |r(2.27)

2.2 Weakly interacting Bose gas

where the vector k0 = kr̂ is orientated along the direction of the scattered particle.
Within this approximation we obtain the far field form of (2.25),
Z
eikr
0 0
d3 r0 e−ik ·r U (r0 )ψk (r0 )ψk (r) = φk (r) −4πreikr= φk (r) −hφk0 | U (r0 ) |ψk i ,4πrand by direct comparison with (2.22), we identify f (θ) as
Z
1
1
0 0
0
d3 r0 e−ik ·r U (r0 )ψk (r0 ).
f (θ) = − hφk0 | U (r ) |ψk i ≡ −4π
4π
0(2.28)(2.29)

0

In the low energy limit |k0 .r0 |  1, and we can replace eik .r by 1, removing the angular dependence in the scattering amplitude. Equation (2.25) is an iterative equation,
however making the first Born approximation by inserting the incoming wave on theright hand side gives fBorn (θ) = hφ0k | U (r0 ) |φk i /4π. Taking the k → 0 limit we obtainan expression for the s-wave scattering length,Zmd3 rV (r).
a=4π~2(2.30)We therefore see that in the low energy limit the details of the interaction potential areintegrated out and can be replaced by an effective potential,
V (r − r0 ) = gδ(r − r0 ),(2.31)

where the coupling constant, g, and the s-wave scattering length, a, are related byg=

4π~2 a.
m(2.32)

Low energy collisions are therefore characterised by a single parameter, the s-wave
scattering length, independent of the details of the two-body potential.

2.2.2

Gross-Pitaevskii equation

The many-body Hamiltonian describing N interacting bosons confined in an externalpotential Vext (r) is given by [16],~2 2Ĥ = drΨ̂ (r) −∇ + Vext (r) Ψ̂(r)2mZ
1+
drdr0 Ψ̂† (r)Ψ̂† (r0 )V (r − r0 )Ψ̂(r0 )Ψ̂(r),
2
Z

†



(2.33)

where Ψ̂† (r) and Ψ̂(r) are the creation and annihilation operators for a boson at positionr, and V (r − r0 ) is the two-body interatomic potential. The operators Ψ̂(r) and Ψ̂† (r)10

2.2 Weakly interacting Bose gas

obey the usual Bose commutation relations:[Ψ̂(r), Ψ̂† (r0 )] = δ(r − r0 ),[Ψ̂(r), Ψ̂(r0 )] = 0,and[Ψ̂† (r), Ψ̂† (r0 )] = 0.

(2.34)The time evolution of the field operator is given by the Heisenberg equation with the
many-body Hamiltonian:i~∂Ψ̂(r, t) = [Ψ̂, Ĥ]
∂t
 2 2

Z
~ ∇
0 † 0
0
0
= −
+ Vext (r) + dr Ψ̂ (r , t)V (r − r)Ψ̂(r , t) Ψ̂(r, t).
2m(2.35)For T  Tc , when the ground state is macroscopically occupied, we may use the
Bogoliubov mean-field description [21] and replace the field operator byΨ̂(r, t) = Φ(r, t) + δ Ψ̂(r, t),(2.36)

where Φ(r, t) = hΨ̂(r, t)i is a complex function defined as the expectation value of the
field operator, and δ Ψ̂(r, t) is a small perturbation due to atoms not in the condensate.
The function Φ(r, t) is an order parameter and is referred to as the wavefunction of
the condensate. For the mean of the field operator, hΨ̂(r, t)i, to be non-zero, a welldefined phase relation must be enforced, otherwise the average over all phases of the
wavefunction would result in zero. From this, a generalised criterion for Bose condensation can be formed as a constraint on the first order spatial coherence function
G(1) (r, r0 ) = hΨ̂† (r)Ψ̂(r0 )i [22, 23],lim

|r−r0 |→∞hΨ̂† (r)Ψ̂(r0 )i = Φ∗ (r)Φ(r0 ) 6= 0.

(2.37)

Condensation can therefore be viewed as the onset of long-range off-diagonal order.
To zeroth order, the excited state perturbation can be ignored, and the field operator
is simply replaced by the condensate wavefunction. At low energies where s-wave
collisions dominate, the interatomic potential, V (r0 −r), can be replaced by the effective
potential (2.31). Using these two approximations we obtain the time dependent GrossPitaevski equation (GPE)∂
i~ Φ(r, t) =
∂t




~2 ∇2
2
−
+ Vext (r) + g|Φ(r, t)| Φ(r, t).
2m(2.38)

The condensate density and number are then given by
n0 (r, t) = |Φ(r, t)|

2

Z
=⇒

11

N0 =

dr|Φ(r, t)|2 .

(2.39)

2.3 Excitation spectrum

Separating out the time dependence of the condensate wave function [24], Φ(r, t) =Φ(r)e−iµt/~ , where µ is the chemical potential, we obtain the time-independent GPE
 2 2

~ ∇
2
−
+ Vext (r) + g|Φ(r)| Φ(r) = µΦ(r).
2m(2.40)

This has the form of a nonlinear Schrödinger equation, where the effect of interactions
is to add a mean field potential, proportional to the condensate density.

2.2.3

The Thomas-Fermi regime

For most experimental parameters where N0 is sufficiently large, the kinetic energy
term is much smaller than the interaction energy term and can be neglected [19]. In
this regime the GPE simplifies to
Vext (r) + g|Φ(r)|2 Φ(r) = µΦ(r),(2.41)and the condensate density ton0 (r) = |Φ(r)|2 =µ − Vext (r)g=0

for µ > Vext (r)for µ < Vext (r).

For the case of a harmonic trap the condensate density profile is parabolic


µ
x2
y2
z2
n0 (r) =1− 2 − 2 − 2 ,
g
Rx Ry
Rz
where Ri =(2.42)(2.43)q2µ/mωi2 is the Thomas-Fermi radius. The chemical potential, µ, is foundfrom normalising the condensate wavefunction according to (2.39)~ω0µ=2where a0 =



15N0 aa0

2/5,(2.44)

p~/mω0 is the harmonic oscillator length associated with the mean trap-

ping frequency. In reality the condensate wavefunction will deviate from this inverted
parabola at the surface, otherwise the kinetic energy associated with the discontinuity
at the Thomas-Fermi radius diverges.

2.3

Excitation spectrum

In deriving the GPE we made use of the zeroth order Bogoliubov approximation bytreating the wavefunction of the condensate as a classical field. To take into accountfluctuations about the condensed state, we now retain the first order perturbation in(2.36), δ Ψ̂(r, t), which is assumed to be small. To conserve particle number on average,
the appropriate operator to consider is K̂ = Ĥ − µN̂ , where the Hamiltonian, Ĥ, is the122.3 Excitation spectrum

same as (2.33). To second order in the fluctuations
Z
K̂ = Ĥ − µN̂ =E0 − µN0 +~2 2
dr −δ Ψ̂† (r, t)∇ δ Ψ̂(r, t)
2m+ [Vext (r) + 2g|Φ(r, t)|2 − µ]δ Ψ̂† (r, t)δ Ψ̂(r, t)

g
2
†
2
∗
2
2
,
+
Φ(r, t) [δ Ψ̂ (r, t)] + Φ (r, t) [δ Ψ̂(r, t)]2(2.45)

where E0 is the energy of the condensate wavefunction only. The equations of motion
for the operators δ Ψ̂ and δ Ψ̂† are given by
i~

∂δ Ψ̂
= [δ Ψ̂, K]
∂t

and

i~

∂δ Ψ̂†= [δ Ψ̂† , K].
∂t(2.46)

This gives the two coupled equations:


∂δ Ψ̂
~2 2
i~
= −
∇ + Vext (r) + 2n0 (r)g − µ δ Ψ̂ + gΦ(r)2 δ Ψ̂†
∂t
2m


~2 2
∂δ Ψ̂†= −∇ + Vext (r) + 2n0 (r)g − µ δ Ψ̂† + gΦ∗ (r)2 δ Ψ̂
−i~
∂t
2m

(2.47)(2.48)

To solve these equations we look for solutions which are periodic in time, of the formδ Ψ̂(r, t) =X[ui (r)αi e−ii t/~ − vi∗ (r)αi† eii t/~ ],(2.49)iwhere the operators αi† and αi create and destroy bosons in the ith excited state.
The requirement that αi† and αi satisfy the Bose commutation relations leads to theconditionZ

dr[|ui (r)|2 − |vi (r)|2 ] = 1.

(2.50)

Noting that the condensate wavefunction is real, we obtain the Bogoliubov equations:~2 2∇ + Vext (r) + 2n0 (r)g − µ − i ui (r) − n0 (r)gvi (r) = 0(2.51)
−
2m



~2 2
∇ + Vext (r) + 2n0 (r)g − µ + i vi (r) − n0 (r)gui (r) = 0−2m(2.52)

Solving to find the eigenvalues i , the operator K̂ can then be expressed in the form
P
K̂ = i αi† αi + constant, which simply has the form of the excitation energy times thenumber of excitations. This is most easily solved in the case of a uniform system where
the excitations have the form of plane waves.

2.3.1

Excitations in a uniform Bose gas

For a uniform Bose gas, Vext (r) = 0, and from (2.42) the chemical potential µ = n0 g.
From the translational invariance of the system the solutions to (2.51) and (2.52) can

13

2.3 Excitation spectrum

be written in the fromeiq·r
ui (r) = uq √Veiq·r
vi (r) = vq √ ,Vand(2.53)

where V is the volume of the system. Substituting these solutions into the Bogoliubov
equations gives0q + n0 g − q uq − n0 gvq = 00q + n0 g + q vq − n0 guq = 0,(2.54)(2.55)

where 0q = ~2 q 2 /2m is the free particle energy. This equation can be solved by therequirement that the determinant of the coefficients of uq and vq equals zero, leading
to the excitation energyq0q + n0 g

2

− (n0 g)2 ,(2.56)

and using the normalisation condition (2.50), we find that!
1 0q + n0 g
2
+1
uq =
2
q
!
1 0q + n0 g
2
−1 ,
vq =
2
q

(2.57)

q =(2.58)

where the phase freedom of the solution permits us to choose uq and vq to be real andpositive. Having found the excitation spectrum for the uniform system, we use theexpressions for uq and vq in the first order perturbation operatorX  eiq·r
e−iq·r † iq t/~−iq t/~δ Ψ̂(r, t) =uq √ αq e− vq √ αq e
,
V
V
q6=0

(2.59)

to express the number conserving Hamiltonian (2.45) in terms of the operators αq andαq† :K̂ = E0 − µN0 +X

q αq† αq −

q6=0


1X 0
q + gn0 − q .
2

(2.60)

q6=0

It can now be seen that the system behaves as a collection of non-interacting bosons
with an energy given by the Bogoliubov spectrum (2.56). In the long wavelength limit(q 2  2mgn0 /~2 ) the dispersion relation reduces to a linear function of q,q ' c~q,(2.61)and the spectrum is sound like. Excitations are phonons and the speed of sound isgiven by
r
c=

14

µ
.
m

(2.62)

2.3 Excitation spectrum

In the short wavelength limit (q 2  2mgn0 /~2 ) the spectrum reduces to that of a freeparticle in the mean field potential of the other atoms
q ' 0q + gn0 .

(2.63)The transition from phonon to free particle excitation occurs when the wavelength of√the excitation ∼ ~/ 2mn0 g. This length scale is termed the healing length ξ, anddenotes the shortest distance over which the wavefunction tends to its bulk value whensubject to a local perturbation. If we consider the density varying from 0 to n over alength scale ξ, then the competing kinetic and potential terms will balance when
~2
= ng
2mξ 2

ξ2 =

=⇒

1
.
8πna

(2.64)

We can therefore understand that on length scales larger than ξ, the atoms are able
to move collectively as phonons, whereas on shorter scales they will behave as freeparticles.
Thermal component
We now consider the depletion of the condensate due to excitations. The total particle
number is given byZN̂ = N0 +drδΨ† (r)δΨ(r).

(2.65)

Using the form of the perturbation wavefunction in terms of plane waves, this takes
the formN̂ = N0 +

X
q6=0

vq2 +

X

(vq2 + u2q )αq† αq ,(2.66)

q6=0

where we have made use of the Bose commutation relations for αq . This expression
shows that the condensate is depleted due to interactions, even at T = 0 when no real
excitations are present. The two-body interaction mixes the ground state components
with higher energy states, and hence the ground state of the interacting gas does nothave all atoms occupying the zero momentum state. The zero temperature depletionof the condensate is then given by
nex
8 √ 3
= √
na ,
n
3 π

(2.67)

which for most experimental parameters is less than a percent.
When a real excitation is added to the system, keeping the total number fixed, thecondensate is depleted by an amountνq = u2q + vq2 =15

0q + gn0.
q(2.68)

2.4 Bose-Einstein condensation and superfluidity

For large momenta, when the excitations behave as free particles, this value tends
towards unity as expected. For non-zero temperatures we can calculate the thermal
distribution of such excitations using the Bose-Einstein distribution function, fq (2.1).
Since the addition of an excitation does not change the total particle number, the chemical potential is set to zero. From this, we get the thermal depletion of the condensate
density asZ
nex (T ) = nex (T = 0) +dq 0q + gn0
1
.
3
(2π)
q
exp(q /kB T ) − 1(2.69)In deriving this expression we have used the Bogoliubov approximation, which assumes
the system is close to its ground state, and excitations are independent of one another.
As a result this expression is only valid at low temperatures; for temperatures closer
to the transition temperatures we must include higher order terms corresponding tointeractions between excitations. Such extensions are used in the Hartree-Fock andPopov approximations, as explained in [16, 19].

2.4

Bose-Einstein condensation and superfluidity

The phenomena of superfluidity is fundamentally linked with the existence of a condensate; a single macroscopically occupied quantum state. In 1938, London publishedthe idea that superfluidity could be understood intuitively as an experimental manifestation of BEC [7]. Superfluid motion, viewed as the motion of the condensed particles
as a whole, is therefore a collective phenomenon in which particles move together topreserve the macroscopic occupation of a single state. Modification of the entire condensate wavefunction would have a macroscopic energy cost, therefore superfluid flow
can only be altered by depletion of the condensate as a result of the creation of excitations. Below a certain critical velocity the energy to create such excitations is notavailable, and the superfluid is able to flow unperturbed by its environment, and hence
without viscosity.

2.4.1

The Landau criterion

In the previous section we calculated the excitation spectrum for the interacting Bosegas. Landau showed that if the excitation spectrum satisfies certain criteria, the motionof the fluid does not cause energy dissipation [25]. To see this, consider the motion ofa uniform condensate, flowing at velocity vs . In the frame of reference stationary withrespect to the condensate, the excitation spectrum is the same as that found before inequation (2.56). To find the Hamiltonian in the frame of the observer we use a Galilean

162.4 Bose-Einstein condensation and superfluidity

transformation to obtain
H0 =X (pi + mvs )2
2m

i

=

X (pi )2i

where p =

P

2m

+

+

1X
V (ri − rj )
2
i6=j

1X
N mvs2V (ri − rj ) + p · vs +,2
2

(2.70)

i6=j

i pi is the net momentum of the particles in the condensate frame.

For

the fluid in its ground state (E = E0 , the energy of the pure condensate, and p = 0)
the energy of the system in the frame of the observer is
E00 = E0 +N mvs2.
2(2.71)

If we now add a single excitation to the ground state of energy p and momentum p as
measured in the condensate frame, the new system energy in the observer frame is
E10 = E0 +N mvs2+ p + p · vs = E00 + p + p · vs .
2

(2.72)

It therefore follows that an excitation cannot be created provided
p + p · vs > 0.

(2.73)

This is the Landau criterion, and for flow without viscosity it needs to hold across theentire spectrum for all p. If the Landau criterion is violated, the relevant excitation
will have negative energy in the observer frame, and therefore will grow exponentiallywith time. Energy and momentum from the coherent motion of the condensate will betransferred to incoherent quasi-particles, and the motion of the superfluid will become
unstable. This criterion is more commonly presented as a critical velocity pvc = min,p(2.74)

where for vs < vc , there is no mechanism for transfer of superfluid flow to heat, andthe fluid flows without viscosity. The ground state of any system is clearly for thecondensate to be stationary with respect to the laboratory frame. Superfluid flow istherefore a long-lived metastable state, protected from decay to the true ground stateby the lack of low energy excitation states.
Taking the excitation spectrum for an interacting uniform condensate, (2.56), plotted in figure 2.1, we see that the critical velocity is the speed of sound, corresponding
to exciting long wavelength phonons. Therefore a uniform BEC has a non-zero critical velocity, and at low flow velocities will behave as a superfluid. The concepts ofsuperfluidity and Bose-Einstein condensation are therefore fundamentally connected,
with the Landau criterion linking the onset of viscosity in a superfluid to the creationof excitations in a condensate. For the case of a non-interacting gas, the spectrum is17

2.4 Bose-Einstein condensation and superfluidity

Figure 2.1: Excitation
spectrum for the uniform
weakly interacting Bose
gas: This illustrates that
the Landau critical velocity is equal to the speedof sound, and viscosity is therefore brought
about by the excitation oflong wavelength phononmodes.

simply that of a free particle (p) = p2 /2m, and hence the critical velocity disappears.
Interparticle interactions are therefore essential for the existence of superfluidity, andintuitively we would then expect that decreasing the strength of interactions must decrease the critical velocity.
In reality the Landau criterion is actually only a necessary, but not sufficient condition for superfluid flow without viscosity, with many systems exhibiting lower energy
excitations involving vortex like motion of the superfluid leading to superfluid turbulence and unstable flow. In the classic experiments on strongly interacting He II flowing
through narrow channels, it was found that the critical velocities were typically around
two orders of magnitude lower than those expected from the Landau criterion, whichin this case corresponds to roton excitations. As discussed in Section 2.4.3, this is due
to low energy vortex states which involve a macroscopic transformation of the condensate wavefunction, and therefore were not initially predicted by the perturbative
method used by Bogoliubov. As first predicted by Feynman, vortices nucleate inside
the channel and carry energy away from the superfluid via the phase-slip mechanism[26, 27]. The critical velocity for such a process is still set by equation (2.74), only now
the relevant energy and momentum are those of the vortex state.

2.4.2

Superfluid component

In describing the theory of BEC we used a two-component picture of the particle density, where the cloud could be considered to be composed of a condensed component
and a thermal component. Superfluids are also described by a two-component model
where the superfluid is made up of two interpenetrating fluids: the normal fluid associated with excitations, and the superfluid which is associated with the condensate. Since
particles are converted between the two components the two fluids are not physicallydistinct, and one cannot separate the fluid into two such parts. More accurately we cansay that a quantum fluid can execute two motions at once, each with its own effective
mass. One of these motions is normal (ie. has the same properties as an ordinary
viscous fluid), but the other is the motion of a superfluid. This concept is referred toas the two-fluid model of superfluids and is used to successfully explain many observed18

2.4 Bose-Einstein condensation and superfluidity

phenomena in liquid helium experiments, as well as predict the phenomenon of second
sound.
In many ultracold gas experiments it is common to simply equate the condensed
fraction and the superfluid fraction, however in general these two quantities take very
different values. Although superfluidity and condensation are strongly related, the definition of superfluidity as dissipationless flow and the definition of condensation as themacroscopic occupation of the ground state are clearly distinct. For low temperature
4 He experiments the condensed and superfluid fractions are believed to be approxi-

mately 10% and 100% respectively [12], while in two-dimensions the superfluid fraction
can be non-zero even if the condensate vanishes [28]. To formulate a strict expression
for the superfluid and normal component we consider a gas at rest and calculate themomentum density carried by the excitations of the system
Z
dp
jex =
pfp ,
(2π~)3(2.75)

where, as in (2.69), we have used the Bose-Einstein function to find the distributionof thermal states. Transforming to a frame moving with a velocity −vs , in which thecondensate is now moving with velocity vs , the total momentum density in this newframe is given byj = ρvs + jex ,(2.76)

where ρ = nm is the total mass density. Using (2.72), the energy of an excitationin this moving frame is given by p − p.(vn − vs ), where vn is the velocity of theexcitation in the original frame. As a result this transformed excitation energy enters
the Bose-Einstein function and the momentum density of the excitation spectrum isgiven bywhereZ
ρn (v) =jex = ρn (|vn − vs |)(vn − vs ),(2.77)

dp p · v
1.
(2π~)3 v 2 exp [(p − p · v)/kB T ] − 1(2.78)

Substituting this into the expression for the total momentum density (2.76), we obtainj = ρn (vn − vs ) + ρvs .

(2.79)

If we define the superfluid density as the difference between the total and normal
densities, ρs = ρ − ρn , the total momentum density can be written in the formj = ρs vs + ρn vn .

(2.80)

This clearly takes the form of two interpenetrating fluids, and we can now compare
the normal fluid fraction (2.78) to the thermal fraction derived earlier (2.69). We notein general that the normal fluid and thermal component are not equivalent, and we19

2.4 Bose-Einstein condensation and superfluidity

Figure 2.2: Superfluid
fraction and thermal fraction of a weakly interacting uniform Bose gas as afunction of temperature:
The thermal fraction in
red is calculated using
the Hartree-Fock versionof (2.69), whereas the superfluid fraction in blue iscalculated using (2.78).

also find that the superfluid fraction is velocity dependent. The superfluid fraction ofa uniform weakly interacting Bose gas can be calculated using the excitation spectrum
calculated earlier (2.56) and is shown in figure 2.2 as a function of temperature. The
corresponding thermal fraction is also shown. This is calculated using the Hartree-Fockform of (2.69), which is more accurate at higher temperatures. For this situation thecondensed fraction and superfluid fraction closely track one another, and to first ordercan be considered equal.
To confirm this, we consider the case of small relative velocities where we can
expand the distribution function to first order as
Z
ρn '
Z
'

dp
(p · v̂)2(2π~)3dp p2(2π~)3 3

−

∂fp0
−
∂p
!

∂fp0
∂p

!

,

(2.81)(2.82)

where fp0 = [exp(p /kB T ) − 1]−1 is the zero velocity distribution function. Integrating
this by parts we obtainZρn '4πdp ∂(2π~)3 ∂p

 4

p ∂p
fp0 .
3 ∂p

(2.83)For temperatures T  T ∗ = nU0 /kB the dominant contribution comes from highmomentum states, where the energy of an excitation is approximately equal to that ofa free particle p ' p2 /2m. Equation (2.83) then simplifies to
Z
ρn '

dp
mfp0
(2π~)3' mnex ,20(2.84)(2.85)

2.4 Bose-Einstein condensation and superfluidity

where we have used the T  T ∗ form of the thermal particle density in (2.69), neglecting
the zero-temperature quantum depletion. We therefore see that the thermal component
and normal component are only equal at high temperatures, where excitations behave
as free particles and contribute precisely one particle to the thermal component andparticle mass m to the normal component. At lower temperatures excitations arecollective and this relation is no longer valid.

2.4.3

Vorticity and irrotational flow

In this section we show that the connection between superfluid flow and condensate
motion constrains superfluid flow to be irrotational. This leads to quantised circulationof the entire condensate, and the existence of quantised vortices; lines of zero density
about which the phase of the condensate wavefunction must wrap around an integer
multiple of 2π. One can show that the condensate in a rotating frame is analogous to
the physics of a charged particle in a magnetic field, and hence the irrotational natureof the condensate maps onto the Meissner effect in superconductors [29].
Quantised circulation
Starting from the time dependent GPE (2.38) we derive the equation,


~
∂|Φ|2
+∇·
(Φ∗ ∇Φ − Φ∇Φ∗ ) = 0.
∂t
2mi(2.86)

Identifying the condensate density n0 = |Φ|2 , this takes the form of the continuity
equation for particle density,∂n0+ ∇ · (n0 v) = 0,∂t(2.87)

where we distinguish the velocity of the condensate asv=~ (Φ∗ ∇Φ − Φ∇Φ∗ )
.
2mi
|Φ|2

(2.88)

Splitting the condensate wavefunction into its amplitude and phase, Φ = f eiφ , thevelocity is then simply the gradient of the phase
v=

~
∇φ.
m

(2.89)

From this it then follows that the condensate velocity is irrotational,∇ × v = 0,21(2.90)

2.4 Bose-Einstein condensation and superfluidity

and hence the motion of the condensate is more restricted than that of a classical fluid.
Calculating the circulation, Γ, around a closed path we findI
Γ = v · dlI~~=
∇φ · dl = 2π`,mm

(2.91)

where from the single-valuedness of the wavefunction, the phase of the wavefunction
must change by an integer multiple of 2π around a closed loop. This immediately
leads to the quantisation of circulation of the BEC. However, to create such a state
we have created a phase singularity within the closed loop, about which the phasewinds and the wavefunction is many-valued. Such a state can therefore only exist ina multiply-connected geometry, where the condensate density vanishes at this phasesingularity. Such a multiply-connected geometry can be created either by the externaltrapping potential, or by the formation of vortices; lines of zero density about which
the condensate phase winds. Using Stoke’s theorem the general form of (2.90), allowing
for such phase singularities, is∇ × v = ẑ`h 2
δ (r − r0 ),m(2.92)

where ` is referred to as the charge of the singularity, δ 2 is a two-dimensional delta
function in the xy plane, and r0 is the location of the singularity. This tells us that acondensate can rotate, but at the expense of producing such phase singularities. Suchphase singularities have been experimentally observed in BECs in several experiments
including as density singularities in [30–32] and as phase singularities in [33, 34].
Quantised vortices
If we consider purely azimuthal flow of a condensate in a system with rotational symmetry about the z axis, from (2.91) the flow velocity is given byvθ = `~,
mr(2.93)

where r is the radial distance from the trap axis, and the condensate wavefunction mustvary as eilθ , where θ is the azimuthal angle. This velocity profile is that of a vortexline, and for the kinetic energy to remain finite the condensate density must vanishalong the axis of the trap (r = 0). Such quantised vortices were first proposed by
Onsager [35] and Feynman [36] in the context of liquid helium experiments, and werefirst experimentally verified by measuring the attenuation of second sound propagating
perpendicular to the axis of rotation, due to scattering of the normal fluid by vortex
lines [37]. Since the experimental realisation of BEC, several experiments have beenable to generate and study both a single vortex [30], and vortex arrays [31, 32] inrotating condensates.
22

2.4 Bose-Einstein condensation and superfluidity

An important distinction to note is that although the circulation of a condensate
is quantised for all flow configurations, the angular momentum per particle of thecondensate need not be. To demonstrate this we now consider a cylindrically symmetrictrap of radius R rotating about its axis z. The angular momentum about the axis of
the trap is equal toZ
Lz = n2D m

rdrdθvθ r,(2.94)

where n2D is the columnar particle density integrated along the axis of rotation, whichis assumed constant. The angular integral can be written in terms of the circulation
ZIdθvθ r = v · dl,(2.95)
which is equal to `h/m if the contour contains the vortex line, and zero otherwise. Fora system with a vortex along the axis of symmetry this simply evaluates to Lz = N `~,and therefore the angular momentum per particle is also quantised. If the vortex is
now displaced a distance d from the axis, the total angular momentum is given by
Z R
Lz = nhdd2
rdr = N `~ 1 − 2
R



,

(2.96)and the angular momentum per particle need not be quantised.
Energy of a single vortex
The creation of vortex states is clearly of significance as it allows the bulk condensate torotate by the formation of phase singularities where the irrotational condition is allowed
to break down. To find whether such vortices form, we first calculate the energy perunit length of a single vortex of charge ` on axis in a uniform condensate of density
n, and radius R. This can be found approximately by simply integrating the kinetic
energy associated with the flow,
Z
Z
Z
nm 2π R 2
πN 2 2 R drE=
vs rdrdθ =
` ~
2 0
m
r
0
0
πn 2 2 R=` ~ ln .
m
rc(2.97)The integral diverges for r = 0, hence a cutoff rc is introduced, below which theexpression for the velocity breaks down. This cutoff is typically taken to be the healing
length, ξ in (2.64), however more accurate variational calculations give rc = ξ/1.46[19]. In principle, the potential energy due to the density profile of the vortex should
also be included, however this can be shown to be a small contribution and is typically
included in the uncertainty of rc . Because the energy of a vortex scales as `2 , the lowest
energy configuration is for the multiply charged vortex to break up into an array of `,singly-charged vortices. The vortices will attempt to form a regular hexagonal array tomaximise their separation and minimise the total energy.
232.4 Bose-Einstein condensation and superfluidity

We now consider the response of the condensate in a frame rotating with angularvelocity Ω about the axis. The energy of the vortex in the rotating frame is given by
E 0 = E − ΩLz ,(2.98)

where Lz is the angular momentum of the condensate in the laboratory frame. For thecase considered of a singly charged vortex on axis, the angular momentum is simply
Lz = N ~, where the total atom number N = nπR2 . The condition for the vortex
solution to become energetically favourable is then when the rotation velocity exceeds
a critical value given by
~
E
Ωc =
=
ln
Lz
mR2

 
R
.
ξ

(2.99)

Therefore, for Ω < Ωc , the superfluid will remain at rest while the normal component
will be brought into rotation. As Ω is increased above Ωc it will become energeticallyfavourable for the condensate to rotate and form a vortex, and at even higher Ω additional vortices will enter the system and a vortex array will form. Such an experiment
was performed in [38], where the angular momentum of the condensate was measured as
the stirring rate of an auxiliary laser beam was increased. For a large range of stirring
frequencies Lz remains zero, corresponding to stirring below the critical frequency. AtΩ ' Ωc an abrupt jump in Lz was observed corresponding to the nucleation of a vortex
at the condensate surface which quickly moves to the central axis, giving Lz /N = ~.
As Ω is increased further Lz is found to increase continuously, which is facilitated by
nucleation of a secondary vortex at the surface and the motion of the first vortex off axis
due to mutual interactions between the vortices. At even higher Ω additional vortices
enter the system and Lz , determined in a non-trivial manner from their arrangement,
increases continuously. Above a certain Ω the vortex array breaks apart and the formation of a turbulent state is observed where Lz no longer increases monotonically withΩ.
From this we see that the counter situation of a rotating condensate in a stationary
frame is an excited state, and that the ground state is the non-rotating vortex-free state.
While it’s simple to understand the ground state of such systems, the mechanism bywhich a condensate gains or loses vorticity is far from trivial, and still an active area of
research. If the appropriate excitation for the system to acquire or dissipate vorticity
is not energetically accessible, then the metastable state is protected from decay to thetrue ground state.
Feynman critical velocity
We now switch to a different flow geometry, relevant to our studies on persistent currents
in annular condensates and studied extensively in superfluid helium experiments. Herewe consider superfluid flow through a long narrow channel of radius R (figure 2.3). In
such a flow channel it was found that the measured critical velocity was significantly
24

2.4 Bose-Einstein condensation and superfluidity

lower than that predicted by the Landau criterion, and was also dependent on thechannel geometry. It was Feynman who first suggested that these low critical velocities
could be due to the formation of vortices produced by friction with the walls [39]. The
mechanism by which such vortices are created then determines the critical velocity,and the fate of these vortices determines the nature of the supercritical flow. We canenvisage two possibilities for nucleated vortices:
1. For velocities larger than vc , vortices nucleate as small vortex rings attached tothe container wall, which then interact with the normal fluid through mutual
friction. This allows the vortex energy to dissipate into heat and the vortex ring
shrinks steadily until reaching atomic size and decaying into quasiparticles. Sucha scenario converts superfluid kinetic energy into heat and therefore appears asviscosity, identical to that considered by Landau.
2. Alternately, the vortex ring may grow and enter into the fluid, creating a complicated tangle of vortex lines which are characteristic of a turbulent state. Suchsuperfluid turbulence will act to damp the superfluid flow, but via a different
mechanism to that in scenario 1.
Numerical calculations suggest that for a weakly interacting BEC the first scenario
applies [40]. Experimental evidence indicates that the superfluid flow of liquid helium
above vc is a superfluid turbulent state, corresponding to the second scenario. Understanding turbulent flow is a notoriously difficult problem, and as a result there are still
many open questions [41]. The mechanism by which vortices nucleate off the walls, enter into the bulk of the fluid, and how turbulence damps the superfluid flow are all still
not fully understood. Nevertheless we can formulate a crude model outlined in [39, 42]
to obtain an order of magnitude estimate of the critical velocity for both scenarios.
To estimate the critical velocity we simply ask at what flow velocity is it energetically favourable to place a vortex structure into the flow channel. It turns out thatthe lowest energy structure to consider is a vortex ring, illustrated in figure 2.3. For avortex ring of radius r, the energy Er and momentum Pr are given by [43]Er = 2π 2 r

n~2 r
ln
m
ξ

(2.100)Pr = 2π 2 n~r2 ẑ,(2.101)

where ẑ is the unit vector normal to the plane of the vortex ring. Using (2.72) the
energy of a vortex ring in the plane perpendicular to a background flow of velocity vsis given by
Er0 = Er + Pr · vs = 2π 2 r2 n~



~
ln
mr

 

r
± vs ,
ξ(2.102)where the sign characterises the sense of rotation of the vortex ring. This expression isminimised by letting r attain its largest value, which is R, the radius of the container.
25

2.5 Conclusion

Figure 2.3: Formation of a vortex ring in a flow channel: The vortex ring is a single vortex
connected back onto itself, and therefore at distances far from the ring the velocity tends tozero. As a result, unlike a single vortex, a vortex ring has a finite energy dependent on its size
and not the size of the container it is in.

For vs > vc this energy will be negative, and hence we obtain the Feynman estimate of
the critical velocity,
vcF =

R
~
ln .
mR
ξ

(2.103)

We note that this is only an estimate as it neglects interactions of the vortex with thewalls, the energy barrier present to actually reaching such a state, and any density
modification due to the vortex and edge effects. Nevertheless (2.103) gives the correct
order of magnitude of critical velocity for liquid helium experiments, and gives theobserved scaling of vc with pipe radius R. Although here we considered the case of avortex ring, the same expression is obtained for the case of a vortex-antivortex pair,
separated by a distance 2R. For the case of r  ξ the curvature of the vortex can beneglected, and hence a vortex ring of radius r maps onto a vortex-antivortex pair oflength πr.

2.5Conclusion

In conclusion we have introduced some of the theory of BEC and superfluidity relevant
for discussing persistent currents. Of particular importance is the concept of superflow
as the translation of an interacting condensate which permits flow without viscosity below a critical velocity. The value of this critical velocity is set by the allowed excitationsof the system, and in particular by the formation of vortex states. This is discussed
further in Chapter 5 for the case of a ring BEC, where the role of quantised vortices isfound to be essential in explaining the stability and eventual decay of superflow.

26

Chapter 3

Experimental methods and BEC
production
This chapter covers the experimental methods and parts of the experiment used for the
production of 87 Rb BECs. Most of the work in designing and building the system forachieving both 87 Rb and 39 K condensates was carried out by students before me, so I
will attempt to only briefly cover the aspects relevant for cooling to quantum degeneracy. For a more detailed and exhaustive explanation of this part of the experimentrefer to [44–46]. The specific sections relevant to studying persistent currents I will
cover in greater depth. This includes (i) the theory of atom-light interactions (relevant
for the production of an optical ring trap and transfer of angular momentum to thecondensate), (ii) the interaction of atoms with an external magnetic field (relevant for
defining the basis of states used in transfers and multi-component studies), and (iii)
absorption imaging (relevant for the quantitative analysis of persistent current stability).
The experimental setup used in this work was primarily designed for the productionof 39 K condensates via sympathetic cooling with 87 Rb. The broad Feshbach resonancepresent in 39 K allows us to tune the interparticle interactions across a wide range. Thishas been used in several studies performed on this system, to study the thermodynamics of ultracold Bose gases [47–49] as well as condensates out of equilibrium [50, 51].
The work presented here only made use of 87 Rb condensates, however due to the significance of the 39 K experiments, the system remained optimised for 39 K condensate
production for much of my time.
The structure of this chapter will be as follows. The first section will outline theexperimental sequence and techniques used in cooling a room temperature vapour of
87 Rb to degeneracy. The sections that follow explain some of the physics and technicaldetails behind these stages, with additional information provided for those stages relevant to studying persistent currents. The ordering of these sections will try and reflect
the order in which they appear in the experimental sequence.

27

3.1 Sequence overview

Figure 3.1: Diagram of the vacuum system, translation stage and quadrupole coil support.
Components labeled include (1) ion pumps, (2) turbo pump, (3) atom source module, (4) MOT
cell and quadrupole coils, (5) quadrupole coil mounting arm, (6) translation stage, (7) Ti-Sub
pump, and (8) science cell.

3.1

Sequence overview

The experiment is performed in an ultra-high vacuum chamber which is illustrated infigure 3.1 [46]. 87 Rb vapour is released into the Magneto-Optical Trap (MOT) chamber(4) by heating getters fitted in (3). Following laser cooling, the atoms are magneticallytrapped and transported to the science cell (8) where they are cooled to degeneracy.
This dual-chamber setup is beneficial for two reasons. Firstly, the MOT chamber
requires sufficiently high 87 Rb vapor pressure, on the order of 10−9 mbar, for rapid
loading of the MOT, whereas the science cell requires very low background pressure, in
the region of 10−11 mbar, to ensure the long lifetime of the atoms required for efficientevaporative cooling and long BEC lifetime. This is achieved by splitting the system
into two regions, connected by a low-conduction section. The second advantage of thisdesign is to increase optical access to the BEC, which is required for imaging, trapping,and manipulation of the condensate.
The ground state of the atoms is the 52 S1/2 state, which is split into the F = 2
and F = 1 hyperfine levels, with the final BEC produced in the |F = 2, mF = 2i state.
For studies on persistent currents we then transfer the BEC to the |F = 1, mF = 1i
state using a microwave transfer. The sequence outline is as follows:
1. 87 Rb vapour is loaded into the MOT (4) and Doppler cooled (Section 3.3.2)
2. The quadrupole field is removed and the molasses stage cools the atoms further,maximising phase space density (Section 3.3.3)
3. The atoms are then optically pumped into the |F = 2, mF = 2i magnetically trappable state (Section 3.4)
28

3.2 Atom-light interactions

Figure 3.2: The two-level atom:
The two energy levels are separated by a transition with frequency ωeg and the atom is driven
by a monochromatic plane wave of
frequency ω. The variable parameters are the detuning of the lightfield from resonance ∆ = ω − ωeg ,and the strength of the atom-field
coupling Ω.

4. The atoms are transferred to a magnetic quadrupole trap and transported to thelow pressure science cell (8) (Section 3.5.1)
5. The quadrupole trap is transformed to a QUIC trap and the gas is evaporatively
cooled (Section 3.6)
6. The atoms are loaded into an optical dipole trap and the depth is gradually
lowered to cool the gas to BEC (Section 3.7)
7. (Optional) An adiabatic microwave sweep transfers the atoms from|F = 2, mF = 2i → |F = 1, mF = 1i (Section 3.8)
8. All optical and magnetic fields are removed and the cloud is absorption imaged
after a variable time of flight (Section 3.9)
All subsequent experiments on persistent currents follow the same steps (1-7) to prepare
the initial BEC.

3.2

Atom-light interactions

The interaction of light and atoms is of great significance in the field of ultracoldgases. When an atom is placed in a light field the oscillating electric field induces anelectric dipole moment in the atom which then couples to the field. The effect of this is
typically divided into two process which are used extensively in the field of laser coolingand trapping. Firstly the light may induce transitions between atomic states involving
the absorption and emission of photons. Secondly this coupling will modify the atomic
eigenstates, creating new dressed states with different eigenenergies. To understandthese processes we consider a simple two-level model of the atom.

3.2.1

Two-level atom

The two-level atom we consider is shown in figure 3.2, where the atom is simply composed of a ground state |gi and an excited state |ei separated by a transition of frequency
ωeg . Incident on the atom is a time varying electric field E = êE0 cos(ωt). The dipole29

3.2 Atom-light interactions

moment of the atom, d = −er, couples to the electric field producing a perturbation
H 0 = −d · ê

E0 iωt(e + e−iωt ).
2

(3.1)

The atomic wavefunction at any time can be written as a superposition of the unperturbed ground and excited states,
Ψ(r, t) = cg (t) |gi + ce (t) |ei e−iωeg t .

(3.2)

Substituting this into the time-dependent Schrödinger equation with the Hamiltonian
H = H0 + H 0 , where H0 is the unperturbed Hamiltonian, gives the coupled equations
dcg (t)Ω∗  i(ω−ωeg )t= ce ~e+ e−i(ω+ωeg )tdt
2
dce (t)Ω  i(ω+ωeg )ti~= cg ~e+ e−i(ω−ωeg )t ,
dt
2

i~

(3.3)
(3.4)

where we have defined the Rabi frequencyΩ=E0
he| d · ê |gi ,
~

(3.5)

which describes the strength of the coupling between the atom and the electric field. If
we assume the field is close to resonance we can make the rotating wave approximation(RWA) by neglecting terms which oscillate at twice the driving frequency since theirtime dependence will average out to zero compared to the much slower evolution of ce
and cg . In this regime the equations simplify todcg (t)ei∆t= ce ~Ω∗dt
2dce (t)e−i∆ti~= cg ~Ω
,
dt
2

i~

(3.6)
(3.7)

where we have introduced the detuning from resonance ∆ = ω − ωeg .
Rabi oscillations
To obtain the time dependence of ce and cg we solve the coupled equations by differentiating (3.6) and (3.7) in time to obtain the uncoupled second order equations
d2 cg
dcg
Ω2
−
i∆
+
cg = 0
dt2
dt
4
d2 ce
dce Ω2+i∆+ce = 0.
dt2
dt
4

(3.8)
(3.9)

Such equations are simply solved, and assuming only the ground state is populated at
t = 0 when the field is turned on, the population of the excited state at time t is given30

3.2 Atom-light interactions

Figure 3.3: Rabi Oscillations: This figure shows the time dependent oscillations of the excitedstate population for different detunings. As the detuning increases the frequency increases andthe maximum population transferred decreases.

byΩ2|ce (t)| = 2 sin2
W
2



Wt
2


,

(3.10)

where we define the effective Rabi frequency
W =pΩ2 + ∆ 2 .

(3.11)The population of the excited state therefore undergoes Rabi oscillations at the frequency W . The efficiency of transfer to the excited state depends on the detuning ∆,with complete transfer to the excited state only possible on resonance. If we apply thelight field on resonance for duration T = π/Ω the atom undergoes a π pulse whereby
the state is flipped completely from pure |gi to pure |ei. Likewise for a light pulse ofhalf the duration, the atom undergoes a π/2 pulse and is left in an equal superposition√of ground state and excited state (|ei ± |gi)/ 2. Such operations are used extensively
in this work.
AC Stark shift
In addition to affecting populations, the perturbing radiation also changes the energy
levels of the states. We show this by defining the new coefficients c̃g = cg e−i∆t/2 andc̃e = ce ei∆t/2 which removes the time dependence from (3.6) and (3.7). The equationsof motion then take the formdc̃gΩ∆= c̃e + c̃g
dt
2
2
dc̃e
Ω
∆
i
= c̃g − c̃e ,
dt
2
2

i

31

(3.12)(3.13)

3.2 Atom-light interactions

Figure 3.4: Light shift of a twolevel atom interacting with an external electric field: The energy
shift of the ground and excitedstate are shown as a function of increasing Rabi frequency. The redline is for negative detuning, andthe blue line is positive detuning.
In the limit of zero coupling the energy levels revert back to their unperturbed levels. The effect shown
here is exaggerated for clarity.

and the Hamiltonian is written as
"
#
~ ∆ Ω
Ĥ =
2 Ω −∆(3.14)

Diagonalising this Hamiltonian, the eigenvalues are
λ=±

~p 2
Ω + ∆2 ,2(3.15)and the new eigenvectors can be written as
e0 = cos θ |gi − sin θ |ei

g 0 = sin θ |gi + cos θ |ei ,,(3.16)

where tan 2θ = −Ω/∆. These new states are the dressed states of the excited state andthe ground state plus a photon. At ∆ = 0 there is an avoided crossing and the dressedstates are split by ~Ω. The composition of the dressed states varies as a functionof detuning and on resonance the dressed states are an equal superposition of theoriginal ground and excited states. For far detuned light, ∆ → ±∞, the dressed states
revert to the pure unperturbed states, |gi and |ei. Reversing the various coefficienttransformations we have done until this point, the energies of the excited and groundstate in the original basis are then given by~∆ ~ p 2±Ω + ∆22
2~∆ ~ p 2
Eg = −∓Ω + ∆2 ,22
Ee = ~ωeg +(3.17)(3.18)

where the top sign choice is for negative detuning (∆ < 0 corresponding to red detuning)
and the bottom sign choice is for positive detuning (∆ > 0 corresponding to blue
detuning) to give the correct unperturbed energies in the limit Ω → 0. The perturbed
states experience a shift which is dependent on the strength and detuning of the lightfield, and the overall sign of the shift depends on the sign of the detuning. Thisis illustrated in figure 3.4 where the light shift is plotted as a function of increasing
electric field strength. This is known as the AC Stark shift and is used to optically32

3.2 Atom-light interactions

trap atoms by spatially varying the energy of the ground state to create a trapping
potential.

3.2.2

Optical Bloch equations

In the discussion so far, an atom in a light field undergoes oscillations between the
ground and excited states, with conservation of energy requiring that the energy istransferred coherently to and from the external field by absorption and stimulated
emission. To include the process of spontaneous emission which limits the lifetime ofthe excited state, we use the density matrix representation of the atom to account for
the loss of energy to a background field. The density operator is defined for our two
state atom asρgg ρgeρ̂ = |Ψi hΨ| =

ρeg

ρee

!
=

cg c∗g cg c∗e
ce c∗gce c∗e!
.

(3.19)

Diagonal elements give populations, while off diagonal elements are coherences. If a
state has non-zero coherences then there exist observables whose value depends on therelative phase between the states |gi and |ei. The time evolution of the density matrix
is given by von Neumann’s equation
i
dρ
ih
=
ρ̂, Ĥ .
dt~(3.20)

The advantage of this formalism is that we can now include the effect of spontaneousemission into the time evolution of the density operator. If the lifetime of the excitedstate |ei is 1/Γ, it can be shown that the complete equation of motion for the density
matrix is given by [52]
i
dρ
ih
=
ρ̂, Ĥ −
dt
~

−Γρee

Γ
2 ρge

Γ
2 ρeg

Γρee

!
.

(3.21)

Using the Hamiltonian derived in (3.14) we obtain the coupled differential equationsiΩ(ρ̃ge − ρ̃eg ) + Γρee
2iΩρ̇ee = −(ρ̃ge − ρ̃eg ) − Γρee2iΩΓρ̃˙ ge = −(ρee − ρgg ) − i∆ρ̃ge − ρ̃ge22iΩΓρ̃˙ eg = (ρee − ρgg ) + i∆ρ̃ge − ρ̃eg .
22

ρ̇gg =(3.22)

These equations are further constrained by the sum of the populations totalling one(ρee + ρgg = 1), and the fact that the coherences are complex conjugates of each other(ρ̃ge = ρ̃∗eg ), reducing the number of independent variables to three. These are usually33

3.2 Atom-light interactions

Figure 3.5: Evolution of the Bloch vector: (a) Evolution of the Bloch vector for the parametersΩ = 1, ∆ = 0.3, and Γ = 0. In the absence of decay the Bloch vector is of unit length and theatom remains in a fully coherent state. (b) Evolution of the Bloch vector for the parametersΩ = 1, ∆ = 0.3, and Γ = 0.1. The inclusion of spontaneous emission causes the Bloch vector
to shorten and the system to decohere to a statistical mixture.

formulated as the components of the Bloch vector R = (u, v, w):
u = (ρ̃ge + ρ̃eg ),v = i(ρ̃eg − ρ̃ge ),w = (ρgg − ρee ).

(3.23)The equation of motion of the Bloch vector is found from (3.22), to give the standard
form of the optical Bloch equationsu̇ =∆v −

Γ
u
2

Γ
v
2
ẇ = − Ωv − Γ(w − 1).
v̇ =∆u + Ωw −(3.24)In the absence of spontaneous decay (Γ = 0), the evolution of the Bloch vector can be
expressed in the more intuitive formṘ = R × W,

(3.25)

where we define the vector W = (Ω, 0, ∆). From this it follows that Ṙ · R = 0
and Ṙ · W = 0 and hence the Bloch vector R precesses about W at a fixed angle
at a frequency equal to |W|. This time evolution is shown in figure 3.5(a) for nonzero detuning. This offers an intuitive method for visualising Rabi oscillations andthe sequential application of π and π/2 pulses. In figure 3.5(b) we show the effect ofspontaneous emission on the evolution of the Bloch vector. The Bloch vector spirals in34

3.2 Atom-light interactions

towards W and the Bloch sphere collapses down such that the Bloch vector no longer
has unit length. This can be viewed as the decoherence of the initial pure state with
|R| = 1 towards an incoherent mixture with |R| < 1.
Steady state solutions
The inclusion of spontaneous emission means that at long times (t  Γ−1 ), the atoms
will no longer undergo Rabi oscillations but will reach steady state populations. Setting
the time derivatives in (3.24) to zero gives the steady state solutions
 


u
Ω∆
 


1
v  = ΩΓ/2  .
  ∆2 + Ω2 /2 + Γ2 /4 w
∆2 + Γ2 /4(3.26)

From this we find the steady state population of the excited state is given by
ρee =

1
s
1−w
=
,
2
2 1 + s + 4∆2 /Γ2(3.27)

where we have defined the on-resonant saturation parameter s = I/Isat = 2(Ω/Γ)2 .
Here the intensity of the light field is given by I = 12 c0 E02 , where 0 is the permittivityof free space and c is the speed of light. The saturation intensity, Isat , describes thestrength of the transition and is given in terms of the dipole moment as
Isat =

0 cΓ2 ~2.
4|ˆ · d|2(3.28)We therefore see that in the limit of high intensity, the population of the excited state
saturates at 1/2. Using the steady state excited population (3.27), the total photon
scattering rate over all directions and frequencies is then given by Γρee , which equals
Rsc =

Γ
s
.
2 1 + s + 4∆2 /Γ2(3.29)√
We note that the scattering rate has a Lorentz lineshape with width Γ0 = Γ 1 + s,
hence at higher intensities the width increases and the line is said to be power broadened.

3.2.3

Two photon processes

A Raman transition is a two-photon process involving simultaneous absorption andstimulated emission by an atom. Figure 3.6 illustrates the coherent Raman transitionof an atom addressed by two lasers between the levels |gi and |ei. The atom is excitedfrom the ground state |gi to an intermediate state |ii, far detuned from the single photontransition, by the absorption of light of frequency ω1 from one beam. Simultaneously
the atom emits a photon of frequency ω2 into the second beam by stimulated emission

35

3.2 Atom-light interactions

Figure 3.6: A Raman transition between levels |gi and |ei driven by two laser beams via
an intermediate state |ii. Provided the detuning ∆ of the virtual state is large, single photon
excitation is negligible and the atom is coherently transferred from |gi to |ei.

and ends in the excited state, |ei. Raman transitions are extensively used in ultracoldgases with applications including Raman cooling [53] and electromagnetically induced
transparency [54]. This work uses two-photon Raman transitions to coherently transfer
the BEC between two rotational states without imparting linear momentum to thesystem (See Section 4.3).
To calculate the effect of both laser beams on the atom we extend the two-level
discussion and show that for large detuning of the intermediate state |ii, the systembehaves simply as a two-level atom with coherent Rabi oscillations between |gi and |ei
and negligible occupation of the intermediate state, |ii. Unfortunately for the situations
we consider, ∆  Γ, the dynamics of the intermediate population are very fast and canbe difficult to calculate numerically. From arguments presented in [55], one can showthat in the limit ∆  δ, the interaction of the atom with both lasers yields an effective
Rabi frequency, Ωeff , which depends on the individual Rabi frequencies of each beam,Ωgi1 and Ωei2 , as well as the detuning from single-photon resonance, ∆:Ωeff =Ωgi1 Ωei2.
2∆(3.30)The Raman coupling gives rise to coherent Rabi oscillations between |gi and |ei of theform in (3.10), with Ω replaced by Ωeff , and the relevant detuning being the two-photon
detuning δ, illustrated in figure 3.6.
The significance of the Raman transfer comes from the fact that the transfer efficiency depends on the two-photon detuning, δ, while the rate of spontaneously emitted
photons depends on the single photon detuning, ∆. For δ = 0 the duration of a π-pulse
is tπ = π/Ωeff . The rate of incoherent photon scattering is the sum of the scattering36

3.2 Atom-light interactions

rates for each beam, given by (3.29). For large detunings (∆  Γ) the total number ofphotons scattered in a single π-pulse is then found to beΓ
2π∆(Ω2 + Ω2ei2 )4∆2 gi1Ωgi1 Ωei2
Γπ Ω2gi1 + Ω2ei2 1=.
2 Ωgi1 Ωei2 ∆Rsc tπ =(3.31)

Therefore we can almost eliminate spontaneous emission events and maintain coherence for many Rabi oscillations by detuning both beams far from the single-photon
resonance. Unfortunately this comes at the cost of reduced effective Rabi frequency,
which scales as 1/∆2 .

3.2.4

Adiabatic rapid passage

The task of coherently transferring a BEC from an occupied state to an empty one
can be achieved by applying a π-pulse to exactly flip the state vector. This requiresa pulse of radiation whose duration, power and frequency with respect to the atomic
transition need to be closely controlled for efficient transfer. Another method is touse an adiabatic rapid passage (ARP), where the radiation is tuned above or below
the resonance frequency, and the radiation or level itself is swept through resonance.
Provided the process is performed adiabatically the atoms will remain in a single dressed
eigenstate and the dressed state itself, which is a function of the detuning, will transformfrom the initial state to the final state. When the coupling between the states is bya Raman transfer this process is referred to as stimulated Raman adiabatic passage(STIRAP)[56]. ARP transfer schemes tend to be more robust and able to achieve
higher transfer efficiencies.
This process is illustrated in figure 3.7. If one starts in the ground state |gi andapplies a coupling field with frequency far below resonance, this effectively loads thecondensate into the lowest band. Sweeping the frequency across resonance the systemremains in the lowest band, and the composition of the lowest dressed state changes topure |ei for detuning far above resonance. Rapidly removing the coupling field projects
the dressed state into the original bare states. If one sweeps the frequency in the otherdirections the same state transfer is observed, but this is achieved by loading into theupper band.
The possibility of adiabatic crossing of energy levels was first considered by Landau
and Zener. They showed that for a system of two coupled states of the form
Hφ1 = 1 φ1 + 12 φ2(3.32)

Hφ2 = 12 φ1 + 2 φ2 ,(3.33)

37

3.2 Atom-light interactions

Figure 3.7: Dressed states energy bands and decomposition: The energy levels of the dressedbands as a function of detuning are obtained from diagonalising (3.14). The color of the band
indicates the relative decomposition of the dressed states in terms of the bare states, with blue
being pure |gi, and red pure |ei. In the limit of far detuning the dressed states tend toward thecorresponding labeled bare states.

the probability of adiabatically crossing from one state to the other is given by [57]P = 1 − e−2πγ ,

γ=

212
2π
.
d
~ dt
(1 − 2 )(3.34)

Identifying the terms 1 = ~∆/2, 2 = −~∆/2, and 12 = ~Ω/2, the adiabatic criterion
is given as
Ω2 

1 d∆
.
π 2 dt(3.35)

Sweeping faster than this will cause admixture of the other dressed state and imperfectstate transfer.

3.2.5

Optical forces

So far we have derived the AC Stark shift which exerts an optical dipole force used
for optical trapping, and we have derived the scattering rate which exerts a scattering
force used in Doppler cooling. To connect these two forces and include the effect ofspontaneous emission in our discussion of the optical dipole potential, we now derive
the complex polarizability and show that the real and imaginary parts correspond to
the dipole and scattering force respectively. This also elucidates the imaging of atomic
clouds, which behave as an object with complex refractive index.
The electric field we have considered so far has the form E(r, t) = ê(E0 (r)/2)eiωt +c.c. This induces a time-varying dipole moment of the form d(r, t) = ê(d0 (r)/2)eiωt +c.c., where the the dipole moment amplitude, d0 , and the field amplitude E0 , are relatedby the complex polarizability
d0 = α(ω)E0 .

38(3.36)

3.2 Atom-light interactions

Here we have assumed we are dealing with linear optics and that α is simply a scalar.
The dipole potential and scattering rate are then related to α by [58]:11
Udip (r) = − hd · Ei = −Re(α)I(r),
2
20 c
1
hḋEi
=
Im(α)I(r).
Rsc (r) =~ω~0 c(3.37)(3.38)The energy level shift therefore results from the in-phase interaction between the electric field and the induced dipole, while the out of phase interaction is responsible forabsorption. The expectation value of the dipole moment is given by hdi = hΨ| d · ê |Ψi,where the wavefunction Ψ is given by (3.2). Substituting in this expression for the
wavefunction gives
hdi = deg ρeg eiωeg t + c.c.
= deg ρ̃eg eiωt + c.c.
= deg(u − iv) iωt
e + c.c.,2(3.39)

where we define deg = he| d · ê |gi = ~Ω/E0 . By comparison with (3.36) we can thenidentify1
hdi = E0 (αeiωt + α∗ e−iωt )21= (deg (u − iv)eiωt + dge (u + iv)e−iωt )
2
deg
⇒α=
(u − iv).
E0(3.40)

Using the steady state solutions of the Bloch vector (3.26) this directly gives the polarizability as


~c0
1
iΓ
α=−
∆−
,
Isat 1 + I/Isat + (2∆/Γ)2
2(3.41)

and from this we obtain
2∆
I/Isat∆Γ ~Ω−−−→,
2 1 + I/Isat + (2∆/Γ)24∆2
ΓI/Isat∆Γ ΓΩ
Rsc (r) =−−−→,2 1 + I/Isat + (2∆/Γ)2
4∆2

Udip (r) = ~(3.42)(3.43)

where the expressions are also given in the limit of large detuning, ∆  Γ. The
expression (3.43) is the same result as (3.29) derived earlier. We note that these results
were derived using the RWA approximation, and as a result for very large detunings
from resonance one needs to include a small correction due to the co-rotating term oforder (ω − ωeg )/(ω + ωeg ).
The optical dipole force is simply obtained from the gradient of the potential, andfor the common case of large detunings used in traps (|∆|  Γ), the dipole force reduces393.3 Laser cooling and trappingto the expression

Fdip = −∇Udip ' −∇

~Ω2
4∆


.

(3.44)

This approximate expression is also obtained by taking the large detuning limit of the
AC Stark energy shift derived in (3.15). In a field with spatially varying intensity or
detuning, the atom will experience a net force, the direction of which depends on thesign of the detuning. In this work red-detuned traps (∆ < 0) are used, where the atom
is attracted to regions of high intensity. The scattering force is simply obtained fromthe product of the scattering rate, Rsc , with the momentum per photon, ~k,
Fsc = ~k

3.3

s
Γ
.
2 1 + s + (2∆/Γ)2(3.45)

Laser cooling and trapping

Having covered the interaction of light with atoms, we are now in a position to describe
the experimental techniques used to slow and spatially constrain atoms. The first ofthese is the magneto-optical trap which uses the scattering component of the radiative
force to cool atoms down to the Doppler limit. The second technique is the opticalmolasses stage, which employs Sisyphus cooling. To relate these techniques directly tothe experiment we first consider the 87 Rb D2 transition hyperfine structure.

3.3.1

87

Rb hyperfine structure

So far we have considered a simple two-level atom, however real atoms possess a far
more complicated electronic level structure with multiple ground and excited states andseveral decay paths. Applying the principles of laser cooling to real atoms therefore
seems like a daunting task. Thankfully alkali atoms, which have a single valence electron, permit a perturbative and accurate calculation of their electronic level structure.
In 87 Rb suitable transitions are found in the D2 transition between the 52 S1/2 groundstate and 52 P3/2 excited state.
The hyperfine structure of these states is illustrated in figure 3.8, along with thetransitions used in the production of a BEC. The nuclear spin for 87 Rb is I = 3/2,
hence the ground state |L = 0, J = 1/2i is split by the hyperfine interaction into the
F = I + J = 1, 2 levels. Similarly the excited state |L = 1, J = 3/2i is split into theF 0 = 0, 1, 2, 3 levels. For the MOT and Molasses stages described in Section 3.3.2 thecooling transition is the cycling |F = 2i → |F 0 = 3i transition. To prevent pumping
to the |F = 1i dark state, repump light is added to continuously transfer the |F = 1i
population to the |F 0 = 2i state. For the pumping stage described in Section 3.4, the|F = 2i → |F 0 = 2i transition is used to cycle the atoms over to a magnetically trappable state. The microwave transition between the two ground state hyperfine levels|F = 1i → |F = 2i is also indicated by the green arrow. This transition is used for40

3.3 Laser cooling and trapping

Figure 3.8: 87 Rb D2 hyperfine structure: Levels are labeled by their total angular momentum,
F , and the approximate Landé g-factor, gF . The appropriate cooling and repump transitions
used in the MOT and molasses stage are indicated by the red and blue arrows respectively. The
transition for optical pumping before magnetic trapping is indicated by the purple arrow. The
microwave transition used in both evaporative cooling and the final state transfer is shown by
the green arrow.

evaporative cooling described in Section 3.6 and in the final state transfer explained in
Section 3.8. Absorption imaging makes use of the same |F = 2i → |F 0 = 3i transition
used for laser cooling.
The laser setup used to generate the cooling, repump, pumping, and imaging light is
described in detail in [45, 46]. The lasers used are two temperature controlled commercial external cavity diode lasers (ECDL). Both are locked using saturated-absorption
spectroscopy, with the cooling laser locked to the |F = 2i → |F 0 = 2, 3i crossover peak,
and the repump laser locked to the |F = 1i → |F 0 = 1, 2i crossover peak. The precise
frequencies required are then obtained using acousto-optic modulators (AOM) to shift
the laser frequencies. Sufficient cooling power is achieved by using the cooling laser
to seed an integrated master-oscillator-power-amplifier which provides several hundred
mW of cooling light used in the MOT beams.

41

3.3 Laser cooling and trapping

3.3.2

Magneto-optical trap (MOT)A MOT is a combination of light and magnetic fields used to both cool and spatiallyconfine atoms, and was first demonstrated in [59]. The ability of the MOT to capture
and cool a room temperature atom vapour down to a few hundred µK means that it
has become the starting point for most experiments on ultracold gases. A MOT isformed by three pairs of counter-propagating red-detuned laser beams, overlayed witha magnetic quadrupole field, such that the magnetic field strength increases along all
directions from the origin. This is illustrated in figure 3.9 for one dimension for theexample of cooling the F = 0 state using the F = 0 → F 0 = 1 transition. If one
initially neglects the magnetic gradient there is no shift of the Zeeman levels and thelasers are red-detuned of resonance for all z. However, the one-dimensional velocityof the atom along the direction of the beams, v, causes a Doppler shift of ±kv of theobserved frequency of the laser light. As a result the motion of an atom towards one of
the beams causes it to Doppler-shift into resonance and the resultant scattering force
due to absorption slows the atom, creating a viscous damping force. This effect is called
Doppler cooling and is used to cool the atoms in a technique called optical molasses.
To spatially confine as well as cool the atoms a magnetic gradient must be included,
creating a MOT.
As described in Section 3.5, the quadrupole field has a magnetic zero at the origin,and the magnitude of the field increases linearly with distance. The effect of thisgradient is to cause the energy levels of the three sub-levels of the F = 1 level to varylinearly with the atom’s position. In a MOT the polarisation of the pair of beams withrespect to a fixed direction in space is chosen to be opposite: σ + and σ − . The effectof this is that when an atom is displaced from z = 0, the selection rules mean thatabsorption from the beam that will push the atom back towards z = 0 will dominate
over absorption from the opposite beam. This imbalance of radiative force caused by
the Zeeman effect creates a position dependent force, confining atoms at the origin.
Using the scattering force derived in (3.45), we can write the overall scattering
force in the MOT configuration as




µB 0
µB 0
σ−
σ+
FMOT = Fsc ∆ − kv −z − Fsc ∆ + kv +
z
~
~


∂Fsc
µB 0
= −2
kv +
z ,
∂∆
~

(3.46)(3.47)

where µ is the dipole moment of the atom, and B 0 is the magnetic field gradient. By
expanding the expression to first order we see that the motion of an atom in the MOT
is equivalent to damped simple harmonic motion. The MOT configuration has a highcapture velocity and therefore is able to trap and cool room temperature atom vapour.
The limit to which an atom cloud can be cooled in a MOT is given by the Doppler
limit. This limit arises due to the recoil energy associated with random fluctuations42

3.3 Laser cooling and trapping

Figure 3.9: Mechanism for a 1D magneto-optical trap illustrated for the F = 0 to F = 1
transition. Two counter-propagating circularly polarised beams, red-detuned of the zero fieldtransition, are overlayed with a magnetic field gradient. The handedness of the beams is thesame with respect to their direction of propagation, but opposite with respect to the fieldaxis. The magnetic gradient adds spatial confinement, bringing atoms far from the origin into
resonance with the appropriately polarised beam.

about the mean number of photons absorbed and spontaneously emitted [55]. This
gives rise to the Doppler cooling limit
T > TD =~Γ
,
2kB(3.48)

which for the case of 87 Rb gives a temperature of 146 µK.
As mentioned in Section 3.3.1 the cooling transition used is the |F = 2i → |F 0 = 3itransition. Because the laser polarisation with respect to the field direction is σ − , theprecise transition is the |F = 2, mF = −2i → |F 0 = 3, mF 0 = −3i transition, which is a
closed cycling transition since |F 0 = 3, mF 0 = −3i should only decay back to|F = 2, mF = −2i. Due to the continuous absorption and emission of photons, off
resonant excitation to the |F 0 = 2i state can occur, which can then decay to the |F = 1i
ground state which is optically dark to the cooling light. To prevent all the atoms
accumulating in this state, repump light which is resonant with the |F = 1i → |F 0 = 2itransition is also present in the MOT stage and clears out this dark state.
For our MOT we typically use 130 mW of cooling light, red-detuned 3.1Γ below
resonance, where Γ ≈ 6 MHz is the linewidth of the excited state. The 1/e2 beam
diameter of our MOT beams is ≈ 3 cm. Overlapped with the cooling beams is 10 mWof resonant repump light. The magnetic gradient is 6.4 G cm−1 along the verticaldirection and 3.2 G cm−1 in the horizontal plane. Typical MOT loading time is about
15 seconds and the atom number saturates at ≈ 5 × 109 .

43

3.3 Laser cooling and trapping

3.3.3

Sub-Doppler cooling

Sub-Doppler cooling was first achieved using the standard molasses technique describedabove [60]. This unexpected effect arises due to the fact that real atoms have a groundstate with degenerate magnetic sublevels which were neglected in the previous discussion. The first explanation of temperatures observed below the Doppler cooling limit
was provided by the mechanism of Sisyphus cooling [61], which can be used to explain
many of the sub-Doppler cooling techniques which have been developed. To briefly explain the process, we consider a simplified system consisting of a J = 1/2 ground state(with degenerate mJ = ±1/2 magnetic sublevels) and a J 0 = 3/2 excited state (withdegenerate mJ = ±3/2, ±1/2 magnetic sublevels). This atom is dressed in zero magnetic field with two counter-propagating red-detuned laser beams, linearly polarisedorthogonal to one another 1 .
The interference of the two beams creates a polarisation standing wave, where thepolarisation varies from σ + to π to σ − with a period of λ/2. If the ground state has
degenerate magnetic sublevels, the light shift experienced by each sublevel will be polarisation dependent, and the degeneracy will be lifted. For the case of mJ = +1/2,
the electric dipole transition for the σ + transition to mJ 0 = 3/2 is stronger than the πtransition to mJ 0 = 1/2 and stronger still than the σ − transition to mJ 0 = −1/2, withthe ratio 3 : 2 : 1. This relates directly to the Rabi frequency, Ω, hence the light shift
will vary spatially along with the polarisation, being largest where the light is pureσ + and smallest where it’s pure σ − . For the mJ = −1/2 ground state the ordering of
the ratios is flipped and the light shift will also vary spatially, but π out of phase withrespect to that of the mJ = +1/2 state. This effect is illustrated in figure 3.10.
The other effect of the polarisation standing wave is spatially selective pumpingof the ground states. In regions of pure σ + polarisation the mJ = +1/2 state canonly transition to mJ 0 = 3/2, where it spontaneously decays back to its original state.
Conversely the mJ = −1/2 is pumped to the mJ = +1/2 ground state since it can onlytransition to the mJ 0 = +1/2 excited state, where there is a possible decay path tothe mJ = +1/2 ground state. We therefore see that in regions of σ + polarisation theatoms are pumped towards the lower energy mJ = +1/2 ground state, and similarly inregions of σ − polarisation the atoms are pumped towards the lower energy mJ = −1/2
ground state. This is illustrated in figure 3.10, where we see that an atom moving
in this potential will repeatedly climb the potential barrier of one state and then be
pumped to the potential minima of the other state, thus giving up its kinetic energy to
the background field. The limit of such a cooling method is the recoil energy associatedwith spontaneous emission which provides a lower threshold for the temperature called
the recoil limit:
T > Trec =1~2 k 2.
kB m(3.49)

This is the textbook lin ⊥ lin configuration, but the effect is also present for two circularly polarised
beams of opposite handedness44

3.4 Optical pumping

Figure 3.10: Sisyphus cooling mechanism in the lin ⊥ lin configuration: The counter propagating beams create a polarisation standing wave, where the polarisation varies between σ + ,π, and σ − with period λ/2. Due to the different electric dipole transitions of the different
magnetic sublevels this creates a periodic shift in the ground state energies. The polarisation
standing wave preferentially pumps atoms from the upper ground state to the lower, hence as
an atom moves it climbs a potential barrier and is then pumped to a potential minima. This
removes the kinetic energy of the atoms, cooling the cloud.

For the case of 87 Rb this gives a recoil temperature of 180 nK, well below the limits ofDoppler cooling.
The typical experimental sequence used in our optical molasses involves removing
the magnetic field gradient and further detuning the MOT beams to −4.1Γ. The powerof the cooling light is also reduced to about 60 mW. The molasses stage typically only
lasts about 6ms, after which the 87 Rb atoms reach ≈ 50 µK. Our molasses stage is
optimised to maximise phase space density, rather than actually achieve cooling to
Trec .

3.4

Optical pumping

At the end of the MOT and molasses stages the atoms are distributed between thevarious mF states of the F = 2 manifold. As explained in Section 3.5, only stateswith mF = 2, 1 are magnetically trappable, hence to avoid losing over half the atoms
the cloud is optically pumped to the |F = 2, mF = 2i state before the quadrupole field
is ramped up. Transferring the atoms to the stretched state is also beneficial for twoother reasons. Firstly the increased magnetic dipole moment of the mF = 2 stateprovides tighter magnetic confinement. This improves efficiency of transport to thescience cell, and increases the elastic collision rate which is important for efficient453.5 Atoms in an external magnetic fieldevaporative cooling. Secondly, the presence of other spin states allows inelastic spinexchange collisions which can lead to additional heating and losses. While this effectis only modest in the case of 87 Rb, removal of atoms in the |F = 2, mF = 1i statesignificantly improves the sympathetic cooling of 39 K.
The pumping sequence starts after molasses by removing the cooling beams but
retaining the repump light, allowing the atoms to fall under gravity. A guide field of
several Gauss is ramped up over 800 µs to provide a quantisation axis for the atoms,
defining the polarisation of the pumping light. The pumping light is then flashedon for approximately 80 µs with an intensity of about the saturation intensity. The
pumping light is resonant with the |F = 2i → |F 0 = 2i transition, as illustrated in figure
3.8. The pumping light is chosen to be σ + polarised, exciting transitions of the type
|F = 2, mF i → |F 0 = 2, mF + 1i. While the excited F 0 = 2 state can spontaneously
decay back to the F = 2 state via any ∆mF = 0, ±1 transition, the polarisation of thepumping light ensures the atoms migrate towards the |F = 2, mF = 2i state. Once inthe |F = 2, mF = 2i state, the atoms are dark to the light since there is no ∆mF = +1
transition, therefore for perfect σ + polarisation the atoms accumulate in the stretchedstate, where they are then transparent to the pumping light. The repump light ispresent to transfer any atoms which decay to the F = 1 state back into the pumping
cycle.
To test the efficiency of the pumping stage we load the atoms into the magnetic
trap and then back into the MOT. Only those atoms pumped into a magneticallytrappable state will remain, causing the MOT to load from a non-zero population. Withpumping this starting population can reach ≈ 80% of the saturated MOT population,compared to only 40% without pumping. We can test the correct sign of the pumping
polarisation by reversing the direction of the guide field and observing anti-pumping tothe |F = 2, mF = −2i state, and a much lower starting MOT value.

3.5

Atoms in an external magnetic field

The Zeeman Hamiltonian describing the interaction of an atom with a static external
magnetic field B is given by
HB =µB(gS S + gL L + gI I) · B,~

(3.50)

where S is the total electronic spin operator, L is the total electronic orbital angularmomentum operator, I is the internal nuclear spin operator, and gS , gL , and gI are thecorresponding Landé g factors. The interaction of the electron spin with the effective
magnetic field of the nucleus leads to the spin-orbit coupling term in the atomic Hamiltonian which couples S and L into the total angular momentum J = L + S. With J as46

3.5 Atoms in an external magnetic fielda good quantum number, the magnetic Hamiltonian now takes the form
HB =µB
(gJ J + gI I) · B,~

(3.51)

where gJ is given in terms of gS and gL [55]. If we now include the hyperfine termin the atomic Hamiltonian, arising due to the interaction of the nuclear spin with theeffective magnetic field of the electron, the relevant Hamiltonian to diagonalise is
H = AI · J +µB(gJ J + gI I) · B,~

(3.52)

where A is the hyperfine structure constant, and is equal to A = h × 3.417 GHz for
87 Rb. This can be easily solved in the two limiting cases of low and high field. In bothcases we neglect the nuclear Zeeman term since the coefficient gI is smaller than gJ by
about three orders of magnitude.
Low field: When the energy shift due to the field is much smaller than the hyperfine
term, the coupling between I and J dominates over that with the magnetic field
and we may define F = I + J, where F and mF are good quantum numbers. The
Zeeman terms now act as perturbations to the states labeled by |F, mF i and theenergy levels are
1
EAZ = A[F (F + 1) − I(I + 1) − J(J + 1)] + gF mF µB Bz ,2

(3.53)and the corresponding gF is given in terms of gJ [55]. The first term is the usual
hyperfine splitting, while the second term gives the field dependence. Different
mF states split linearly with magnetic field according to∆E|F,mF i = µB gF mF Bz ,(3.54)and hence the energy spacing E|F,mF i − E|F,mF ±1i is equal for all mF in a given
F manifold.
High field: When the energy shift due to the field is much larger than the hyperfine
term, I and J will couple strongly to B and hence the states will be labeled by|mJ , mI i. The energy levels are then given by
EP B = AmJ mI + gJ mJ µB Bz ,(3.55)and now mJ determines the scaling of the energy with magnetic field.
For intermediate field strengths the full Hamiltonian (3.52) must be diagonalised. Thiscan be done by expanding the hyperfine term into raising and lowering operators using
the identity

1
I.J = Iz Jz + (I+ J− + I− J+ ).
2
47(3.56)

3.5 Atoms in an external magnetic field

Figure 3.11: Energy shift in a magnetic field: (a) Hyperfine structure of the 52 S1/2 groundstate of 87 Rb in an external magnetic field. States are labeled with their low field quantum
number |F, mF i. (b) The energy difference ∆E = E(|F = 1, mF = 0i → |F = 1, mF = 1i) −E(|F = 1, mF = −1i → |F = 1, mF = 0i).

For the state we care about, which is the ground state of the D line transition with
I = 3/2 and J = 1/2, there exists an analytical form of the solution called the BreitRabi formula [62]:
∆Ehfs
∆Ehfs
E|J=1/2,mJ ,I,mI i = −+ gI µB mB ±2(2I + 1)
2


1/2
4mx
2
1+
+x
,
2I + 1(3.57)

where ∆Ehfs = A(I +1/2) is the zero field hyperfine splitting, m = mI ±mJ = mI ±1/2,
andx=(gJ − gI )µB B.
∆Ehfs(3.58)

This is plotted in figure 3.11(a) for the ground state of 87 Rb, 52 S1/2 . We see thatthe energy splitting is approximately linear up to about 500 G, and therefore the low
field approximation for the energy splitting (3.53) can be used for most experimental
parameters. We note however in figure 3.11(b) that the energy differences E|F =1,mF =1i −E|F =1,mF =0i and E|F =1,mF =0i − E|F =1,mF =−1i differ by several kHz even at a modest
field of a few Gauss. This has strong implications when coupling the F = 1 manifold.
If the Rabi frequency of the coupling field is less than this difference, the coupling canonly be resonant for one of the energy splittings at any one time, and hence the threelevel system mF = +1, 0, −1 is reduced to a two level system.

3.5.1

Magnetic trapping

In our experiments we can assume the linear form of the Zeeman energy
U (r)|F,mF i = µB gF mF B(r),48(3.59)

3.5 Atoms in an external magnetic field

where we have replaced Bz by B = |B|, since for sufficiently large magnetic fields the
dipole of the atom will adiabatically follow the field direction. If we allow the magnetic
field to vary spatially the atom will experience a force
F(r) = −∇U (r) = −µB gF mF ∇B.

(3.60)

From (3.59) it follows that a local field minimum would act as a trap for low-fieldseeking states with gF mF > 0. For 87 Rb, the Landé factors for the ground state
are gF =1 = −1/2 and gF =2 = 1/2, and hence the magnetically trappable states are|F, mF i = |1, −1i , |2, 2i, and |2, 1i.
Once in a magnetic trap it is possible to cool the gas down to degeneracy using
microwave or RF fields to selectively transfer the most energetic atoms to untrappedstates. The advantage of using magnetic traps is that firstly they can be very deep,
allowing them to efficiently collect the atoms at the end of the laser cooling stage.
Secondly the lifetime in such traps can be extremely long, limited only by collisions
with the residual background gas. This is particularly useful in our setup when we needto sympathetically cool 39 K with 87 Rb, which has a long thermalisation time due to thesmall inter-species scattering length, aK−Rb . Finally evaporative cooling in a magnetic
trap can be state and species selective, which again is utilised in the sympathetic coolingof 39 K to only remove 87 Rb atoms and maximise the phase space density of 39 K.
Quadrupole trap
In our experiment, following laser cooling in the MOT and molasses stages, the atoms
are transferred to a quadrupole trap produced by a pair of opposed Helmholtz coils asshown in figure 3.12(a). Defining the axis of symmetry as the z direction, using theMaxwell relation ∇ · B = 0, the magnetic field near the origin is given by
B = B 0 (x, y, −2z).

(3.61)The strength of the field varies linearly with distance from the origin, and due to thesymmetry of the trap the gradient is twice as strong along the vertical direction thanalong the radial. The full field calculation is shown in figure 3.12(b), and shows thatthe trap is only a local minimum with the trap depth given by the difference between
the minimum field at the origin and the maximum field along the radial direction. The
field lines shown in figure 3.12(a) illustrate that the field direction is the same as that
required for the MOT stage, hence we use the same coils for both roles.
In this experiment, the quadrupole coils are a pair of water-cooled copper coils in
anti-Helmholtz configuration mounted on a track mount, which is used to translatethem between the MOT chamber and the science cell. Following optical pumping, thetrap is loaded by jumping the field to 40 G/cm along the strong axis, and then ramping the field over 50 ms to its maximum value of 160 G/cm. This tightly confines the49

3.5 Atoms in an external magnetic field

Figure 3.12: Quadrupole trap: (a) Direction of the magnetic field in the xz plane created bya pair of coils in Anti-Helmholtz configuration. The vertical spacing between the coils is chosenequal to the diameter of the coils. (b) Potential created in the xz plane by the quadrupole trap.

atoms, allowing the track to move the coils to the science cell, carrying the atom cloudwith it down the narrow channel and into the lower pressure environment.
The main disadvantage of the quadrupole trap is that the magnetic field vanishes
at the origin. In this region of very low field, states with different mF become degenerate and the magnetic field no longer defines the quantisation axis for the atoms to
adiabatically follow. This allows non-adiabatic transfers between different mF states tooccur, and if these new states are not magnetically trapped they are lost from the system, reducing the cloud lifetime. Such spin flips are called Majorana spin flips [63] andcause trap losses and heating. Ultimately this process dictates that cooling to quantum
degeneracy in a quadrupole trap alone is not possible [64]. Several solutions to thisproblem have been found, however the solution used in our experiment is to transform the quadrupole trap into a QUIC (Quadrupole-Ioffe configuration) trap where themagnetic minimum is offset from zero.
QUIC trap
To overcome the problem of Majorana spin flips several solutions were implemented inour system and used to cool to degeneracy. These include:
Optically plugged quadrupole The magnetic zero in the quadrupole trap is overlapped with a blue-detuned repulsive optical potential which acts to “plug” thehole. This method was first used successfully at MIT in 1995 [11].

503.5 Atoms in an external magnetic field

Figure 3.13: QUIC trap: This illustrates the configuration of the quadrupole, Ioffe and antibias coils in the QUIC trap relative to the science cell. The arrows on the coils indicate thedirection of current.

Time-averaged orbiting potential (TOP) trap The quadrupole field is superimposed with a rotating, spatially uniform magnetic field which causes the magnetic
zero to move out from the original origin and orbit about it. If the frequency of
the rotation is chosen to be slow enough for adiabatic following to prevent transitions between mF states, but faster than the motion of the atoms, the atoms
move in the effective time-averaged potential of the instantaneous potential: aharmonic trap with non-zero field everywhere [10, 65].
Optical dimple hybrid trap A red-detuned attractive optical potential is positioned
approximately a beam waist from the quadrupole magnetic zero. At very low temperatures close to degeneracy the optical “dimple” dominates over the magnetic
field minimum and the coldest atoms are held away from the magnetic zero, andtherefore protected from spin flips [66].
Of these three the optical dimple trap was the optimal method for 87 Rb, producing
condensates of 5 × 105 atoms with less than 10 s of evaporation. Unfortunately formuch of the time during this work it was necessary to optimise the experiment for
production of 39 K condensates, which requires very long evaporation sweeps due to therelatively small K-Rb interspecies scattering length of aK−Rb ' 28a0 [67], compared toaRb ' 99a0 [68] for 87 Rb. The solution which worked best for sympathetic cooling of
39 K was found to be to evaporate in a QUIC trap, and as a result the same method is51

3.6 Evaporative coolingused to produce 87 Rb BECs.
The QUIC trap is a relatively simple Ioffe-Pritchard type trap first implemented in[69], where the quadrupole field is modified to offset the trap minimum to a positive
value of magnetic field, and convert the trapping potential from linear to harmonic.
This QUIC trap is formed by adding the field from a single coil, known as the Ioffe coil,orientated perpendicular to the quadrupole coils, which produces the offset harmonic
potential. An additional uniform field known as the anti-bias field opposes the field
generated by the Ioffe coil and provides greater control in the position of the magnetic
field minimum along the direction of the Ioffe coil. The orientation of these coils is
illustrated in figure 3.13. For full details of the setup, potential, and loading procedureof our QUIC trap see [45, 46]. The main disadvantage of the QUIC trap is that theconfinement along the direction of the Ioffe coil is quite weak, and as a result evaporation
sweeps must be slow to allow thermalisation.

3.6

Evaporative cooling

The phase space densities achieved at the end of laser cooling are still typically manyorders of magnitude below unity and hence degeneracy. As a result, evaporative cooling is a necessary stage of nearly all ultracold experiments, and allows one to achieve
arbitrarily low temperatures. The basic principle of evaporative cooling is to selectively
remove only the highest energy atoms in a cloud and then allow the remaining sample
to rethermalise at a lower equilibrium temperature. By repeatedly truncating the thermal distribution in this way the temperature can be lowered indefinitely, limited onlyby the discrete particle number. Provided the evaporation is efficient in removing only
the most energetic atoms and background loses and heating are limited, the atom loss
and temperature reduction result in a net increase in phase space density.
The first evaporative cooling stage is done in a magnetic trap once the atoms
have been transported to the science cell. This itself is split into two steps: an initial
evaporative sweep in the quadrupole trap, followed by a slower sweep in the QUIC
trap. The principles for both steps are the same. The energy levels of the F = 1 andF = 2 magnetic sublevels are shown in figure 3.14 as a function of field strength which,for a quadrupole trap, is proportional to distance from the magnetic zero. We apply amicrowave (MW) field of a frequency ∆M W greater than the hyperfine splitting, ωHF ,indicated by the green arrow. For an atom to move into resonance with the MW field
it must move away from the origin to a region of higher field. If we choose ∆M W to be
sufficiently large, only the most energetic atoms will reach this region, at which point
they will undergo a magnetic dipole transition to the untrapped |1, 1i state and arelost.
We therefore have a method for selectively ejecting the high energy atoms from thetrap, hence by gradually sweeping the MW frequency downwards we can evaporatively

523.6 Evaporative cooling

Figure 3.14: Evaporation in a magnetic trap: The left figure shows the hyperfine energy levels
as a function of magnetic field. The F = 1 and F = 2 manifolds are coupled by a microwave
field (green arrow) detuned ∆M W above the zero-field hyperfine splitting, ωHF . The grey lines
indicate the F = 1 manifold shifted by the microwave photon energy, hence we see the firststates coupled are |2, 2i and |1, 1i plus a photon. The microwave dressed states are shown in
the upper right panel. The F = 2 magnetic sublevels are coupled by an RF field (blue arrow)of frequency ωRF . Here we assume the field is low and the states are all equally spaced suchthat the RF couples all five sublevels at once. The RF dressed version of the F = 2 manifold
is shown in the bottom right.

cool the cloud, with the final temperature set by the final MW frequency. The same
effect can be achieved by applying a radio-frequency (RF) field of frequency ωRF . Thisdrives ∆mF = 1 transitions between the different mF levels of the F = 2 manifold
until ultimately the atoms transfer to the untrapped mF = 0, and the anti-trappedmF = −1, −2 states and are lost from the trap. To create an equivalent trap depth tothe case of MW, the RF frequency is given by ωRF = ∆M W /3. The main advantage
of using microwaves is that it allows selective cooling of 87 Rb only, which is used tosympathetically cool 39 K.
A more informative picture can be obtained using the dressed state formalism developed in Section 3.2.1 [70]. If we label states as |F, mF ; ni, which is the |F, mF i stateplus n photons, the |1, mF ; 1i manifold is indicated by the grey levels in figure 3.14.
We therefore see that the evaporation point is set by where microwaves first couple
the |2, 2i and |1, 1; 1i states. At higher fields it is clear that additional couplings will53

3.6 Evaporative cooling

occur, which can lead to production of atoms in the |2, 1i state. The Hamiltonian of
the coupled |2, 2i and |1, 1; 1i states in a magnetic field is an intuitive modification ofthat in (3.14):"#Ω~ ∆M W − µB B(r)
Ĥ =
,
2
Ω
−∆M W + 2µB B(r)(3.62)

where we have used that gF = 1/2 (−1/2) for the F = 2 (F = 1) states. This isdiagonalised to obtain the dressed state eigenvalues which are plotted in figure 3.14 as afunction of distance from the trap zero. The ground state forms a trap, whose depth canbe reduced by reducing the MW frequency, or increasing the field gradient. Provided
the atoms cross the edge sufficiently slowly they remain in the dressed ground state
and are transferred to the |1, 1i state and are lost. Atoms which cross the boundary
too quickly can be transferred to the upper band, and can then be trapped at thecloud’s edge. To prevent this, the coupling strength between the states, Ω, must be
sufficient for the adiabatic crossing criteria (3.35) to apply. We identify the detuningas ∆ = (3µB B 0 x/2~), where B 0 is the field gradient and x is the position in the trap.
p
Estimating the typical atom velocity as v = kB T /m the adiabatic condition is givenasrΩ  Ω0 =3µB B 02π 2 ~



kB T
m

1/4
.

(3.63)For typical parameters at the start of the quadrupole evaporation, this gives a valueof Ω0 ∼ 7 kHz. If this is satisfied, atoms crossing the resonance point will leave
the trap. If one evaporates with insufficient power, hot atoms can oscillate past theresonance point several times before being ejected, causing heating of the remaining
cloud and inefficient evaporation. This process is extended to calculate the RF dressed
F = 2 manifold which is also shown in figure 3.14. The relevant states coupled are the|2, 2i , |2, 1; 1i , |2, 0; 2i , |2, −1; 3i and |2, −2; 4i. An atom adiabatically following thelowest band transitions from the trapped |2, 2i state to the anti-trapped |2, −2i state.
Because this is a higher order process, the necessary RF power for adiabatic following
is higher than that required for MW, however we note that atoms transferring to thesecond and third band will still be lost from the trap. The requirement for higher power
follows somewhat intuitively from observing the reduced level spacing present in the
RF dressed states for the same Ω.
Following magnetic transport to the low pressure science cell we use a linear 2 s
MW forced evaporation sweep in the quadrupole trap. The trap is then transformed
into the QUIC trap and we then employ a 32 s forced evaporation using an exponential
frequency sweep. Once this concludes we typically have about 4 × 107 atoms at atemperature of about 3 µK [44]. This is sufficiently cold to be loaded into an all optical
dipole trap, where the final evaporation is done to achieve condensation. We now briefly
describe the microwave and RF systems used in the experiment, since these are used
extensively to perform state transfers which are integral to studying persistent currents.

54

3.6 Evaporative cooling

Figure 3.15: Microwave system: (a) The generator used for forced evaporation is the Agilent
N5183A MXG function generator. The secondary generator (Agilent/HP 83622A) is used for|2, 1i clean up and state manipulation. Both can output a maximum of 14 dBm and are
combined using a power combiner. This is then passed through an RF switch and a custom
made 28 dB amplifier before going to the microwave antenna. (b) Custom made helix antenna
designed by I. Gotlibovych [71].

3.6.1

Microwave system

The microwave setup was designed by Igor Gotlibovych for a master’s project andis detailed in [71]. Figure 3.15 (a) illustrates the basic configuration. The system is
optimised to radiate the maximum power at a frequency equal to that of the hyperfine
splitting, 6.8 GHz. The signal for forced evaporative cooling is derived from a function
generator1 which is then passed through a custom made amplifier built by Microwave
Amplifiers Ltd. to give a maximum output power at the antenna of 37 dBm= 5 W.
A secondary synthesizer2 is power coupled3 to this transmission line and provides theoption to perform |2, 1i clean up sweeps during evaporation, and is used to perform
state manipulations of the BEC. The output of both generators is controlled by an
external TTL-controlled RF switch4 which provides timing resolution of 1 µs, limitedby the resolution of our sequence control software.
The antenna is illustrated in figure 3.15 (b) and uses a helix design to try andincrease directionality of the radiated power. The first 1/4 of the helix turn closest
to the ground plate is deformed to impedance match the antenna with the cable andreduce reflections. The antenna radiates circularly polarised light, with a directionality,
D = 4.5. The directionality is defined as the maximum intensity over the average
1

Agilent N5183A MXG
Agilent/HP 83622A
3
Mini-Circuits ZX10-2-71-S+
4
Mini-Circuits ZYSWA-2-50-DR
2

55

3.6 Evaporative coolingintensity,P
G(θ, φ)4πr2
D = maxθ,φ G(θ, φ),I(r, θ, φ) =(3.64)(3.65)

where P is the total power, and r is the distance from the antenna. From this the
maximum possible Rabi frequency can be calculated.
Electric dipole transitions between |F, mF i states are forbidden, so the relevant
matrix elements are magnetic dipole transitions. The coupling is given by~Ωij = hΨi | µ · B |Ψj i(3.66)

withµ = µB (gI Î + gL L̂ + gS Ŝ).

(3.67)

Using L = 0 and gI  gS , we simply need to compute matrix elements of the spin
operator. Using Clebsch-Gordan coefficients we can decompose the hyperfine states
into eigenstates of the spin operator|F, mF i =X

CmI ,mS |I, mI ; S, mS i ,(3.68)

mI ,mS

where CmI ,ms are the Clebsch-Gordan coefficients. We define the spin operator asŜ = (σ̂x , σ̂y , σ̂z )/2, where σx,y,z are the appropriate Pauli spin matrices. The magnetic
dipole moment between two states, (1) and (2), is then given by
D

D
E
E
X
F (1) , mF (1) µ F (2) , mF (2) = gS µB
δm (1) ,m (2) C (1) C (2) S (1) , mS (1) Ŝ S (2) , mS (2)I

I(3.69)
Relevant non-zero magnetic dipole matrix elements are listed below

Element
h2, 2| µ |2, 1i
h2, 1| µ |2, 0i
h2, 2| µ |1, 1i
h1, 1| µ |1, 0i

Value1µB (1, −i, 0)T2√
6µB (1, −i, 0)T√4
3T
2 µB (1, −i, 0)1− 2√µ (1, −i, 0)T
2 B

Using the relation between intensity and magnetic field, I = B 2 c/2µ0 , the maximum
Rabi frequency for the |2, 2i → |1, 1i transition is then given byr√
3µ0 P D~Ω|2,2i→|1,1i =µB≈ 14 kHz,22πr2 c56(3.70)

3.6 Evaporative cooling

Figure 3.16: RF system: The waveform generator used to produce the RF signal can output
frequencies up to 30 MHz with a power of up to 24 dBm. The usable power is restricted to 7
dBm so as to prevent damaging the amplifier. The output state is controlled by a switch which
provides time control of pulse times required for Rabi flopping. The RF signal is boosted by 30
dB by an amplifier before going to a single loop antenna, giving a maximum power of 36 dBm(4 W).

where we assume a distance r = 10 cm. This is an upper limit on the attainable
Rabi frequency since we have neglected reflections at the antenna due to unmatched
impedances which will reduce the useable power. Comparison with (3.65) suggests thisRabi frequency is sufficient for evaporative cooling.

3.6.2

RF system

The RF system is no longer used for evaporation, but is still used to perform statetransfers between magnetic sublevels. The RF setup is illustrated in figure 3.16. The
waveform is produced by a 30 MHz waveform generator1 . To achieve short bursts(∼ 100 µs) required for Rabi flopping, the output of the generator is controlled by an
external TTL-controlled RF switch2 which provides timing resolution of 1 µs, limitedby the resolution of our sequence control software. This signal is amplified by 30 dBm
using a broadband RF amplifier3 and then sent to the antenna. The maximum poweris limited to 36 dBm (4 W) due to the threshold of the amplifier.
The antenna is a single copper wire loop of radius r = 18 mm with wire radiusa = 0.5 mm, mounted horizontally less than z = 20 mm directly above the atoms. To
estimate the magnetic field at the atoms and hence the Rabi frequency we first calculate
the impedance of the coil. Magnetic state transfers are typically done in a field of 10 G,
corresponding to an RF frequency of about 7 MHz. At this frequency the wavelengthof the radiation is large compared to the dimensions of the experiment, justifying theuse of magneto-static methods to calculate the field. The current distribution acrossthe wire is mostly confined to the surface with a skin depth of
r
2ρ
,
δ=
ωµ0
1

Agilent, 33521A
Mini-Circuits ZYSWA-2-50-DR
3
Delta RF Technology, LA2-1-525-30
257(3.71)

3.7 Evaporative coolingwhere ρ is the conductivity, which for copper is ρ = 1.68 × 10−8 Ωm. This gives atypical skin depth of δ = 24 µm. The Ohmic resistance, R, can be easily calculated,
and the inductance of the coil, L, can be found using standard formulae [72]rρaδ   8r−2 .
L = µ0 r lnaR=

(3.72)(3.73)For our RF coil and typical operating frequency we obtain ZR = R = 0.025 Ω and|ZL | = |iωL| = 3.64 Ω, giving a total impedance of Zloop = ZR + ZL . The antenna isconnected to a coaxial cable of impedance Z0 = 50 Ω. If the waveguide carries a signalof peak voltage Vin towards the antenna, the impedance mismatch will cause reflections
and therefore the transmitted voltage is given by
Vloop =

2ZloopVin
Zloop + Z0

(3.74)

Vloop
2Vin
=
.
Zloop
Zloop + Z0

(3.75)

giving a current in the antenna of
Iloop =

The amplitude of the magnetic field at the atoms can be found using the standard
formula for the on axis field from a current loop
r2.
(r2 + z 2 )3/2

(3.76)

2Pin Z0 , the relevant Rabi frequencies at 7 MHz arer1
Pin~Ω|2,2i→|2,1i = µB B ≈ 58 kHz
,
2
1W
r
1
Pin
~Ω|1,1i→|1,0i = √ µB B ≈ 41 kHz
.
1W
2 2

(3.77)

B = µ0 |Iloop |Using the relation Vin =√(3.78)

At such low frequencies the inductive impedance of the coil is sufficiently low thatvery high Rabi frequencies are possible. Typically we run the waveform generator
well below its maximum power, producing a Rabi frequency of approximately 5 kHz,
corresponding to an input power at the antenna of approximately 12 dBm (16 mW).
For higher frequency applications, where the inductance of the loop becomes significant,the output power can be enhanced by insertion of a capacitor in series. By tuning thevalue of this capacitor such that (ωC)−1 = ωLloop , the reactance of the loop can becanceled to increase the power output. Due to its significance in MRI experiments,
extensive literature exists on RF coil design for both transmission and pick-up [73].

583.7 Optical dipole trap

Figure 3.17: Crossed optical dipole trap: 1070 nm laser light from a fibre laser is passed
through an AOM and focused down onto the atoms. The light is recycled in a bow-tie configuration and a second beam is focused onto the atoms, orthogonal to the first beam, producing
a crossed dipole trap which confines the atoms in all three dimensions. The power of the trap
is stabilised by picking off a weak reflection towards a photodiode, providing input for a PID
which controls the AOM efficiency to attain the desired trap power.

3.7

Optical dipole trap

Following evaporative cooling in the QUIC trap, the atoms are transferred to an all optical dipole trap (ODT) illustrated in figure 3.17. The trap is formed using red-detuned
1070 nm laser light from a 10 W linearly polarised ytterbium fibre laser1 . The linewidthof the laser is 1.2 nm, corresponding to a coherence length of only ∆x ≈ λ2 /∆λ ≈ 0.8
mm. This short coherence length ensures no interference effects which could modify
the trapping potential. The crossed trap is constructed using a bow-tie configuration,
where the light is focused onto the atoms and then recycled to form a second, focused,orthogonal beam. This provides additional trap depth and confinement along the direction of propagation of the first beam.
It is important to stabilise the beam power to prevent long-term drifts in trap depth
and high frequency modulations which can cause additional heating. To achieve thisthe beam is passed through an AOM2 and the first diffraction order is used to form the1
2

IPG Photonics YLM-10-LP-SC
Crystal Technology 3110-197

59

3.8 |F = 1, mF = 1i state preparation

trap. Following the bow-tie configuration the laser is sent to a beam dump, but before
this a glass slide picks off a weak reflection towards a monitor photodiode. The voltage
signal obtained from the photodiode is compared with a set voltage generated by thecontrol software using a PID controller (Proportional-Integral-Derivative controller).
The PID outputs an error signal which suitably attenuates the AOM driving signal
to increase or decrease the AOM efficiency to achieve the desired trap power. This
AOM-PID setup is used on several more occasions in this experiment to stabilise beam
powers.
For such far-detuned trapping light the expressions for the dipole potential and scattering rate derived in (3.42) and (3.43) respectively using the rotating-wave-approximation
are no longer accurate. Instead, a weakly perturbative method can be used to derive thefollowing expressions, relevant in the case of large detunings and negligible saturation
[58]:


Γ
Γ
3πc2
+
I(r),
Udip (r) = − 32ωeg ωeg − ω ωeg + ω

2
 
3πc2
ω 3
Γ
Γ
Rsc (r) =
+
I(r).
3
2~ωeg
ωeg
ωeg − ω ωeg + ω(3.79)(3.80)In the limit of large detuning the optical dipole force dominates over the scattering
force. While Udip scales as I/∆, Rsc scales as I/∆2 , hence far detuned optical traps
benefit from reduced scattering rates and heating.
The ODT is loaded by linearly ramping the beam power up to 6.8 W, corresponding
to a trap depth of ∼ 35 µK. The QUIC trap is then overlapped with the ODT andabruptly switched off, achieving transfer efficiency of ∼ 60% with a temperature of≈ 6 µK. The antibias field remains on at a fixed field of approximately 3 G. The final
evaporative cooling to degeneracy is performed in the ODT by gradually reducing thebeam power and trap depth over 6 s. The final temperature and condensate fraction
are controlled by choosing the PID voltage and correspondingly the beam power. With
this protocol pure BECs of 87 Rb can be achieved with up to 8 × 105 atoms, however
when the system is optimised for production of 39 K BECs, the 87 Rb BEC atom number
is typically about half this value.

3.8|F = 1, mF = 1i state preparation

After cooling to BEC in the |F = 2, mF = 2i state we transfer the atoms to the|F = 1, mF = 1i state while holding in the ODT. The original purpose of this was toreduce the number of magnetic sublevels from five to three, and hence simplify thesystem for coupling the different mF states. Another advantage of changing states isan observed increase in the BEC lifetime. Due to a reduced three-body recombination
rate the lifetime of the |1, 1i state is found to be slightly longer than that of the |2, 2i

60

3.9 |F = 1, mF = 1i state preparation

Figure 3.18: Adiabatic state transfer |2, 2i → |1, 1i: Plotted are the dressed state energy
levels as a function of external bias field. The bias field is swept from 3.12 G→ 3.48 G in 100
ms in the presence of a microwave field of Ω ≈ 13 kHz which causes adiabatic following of
the lowest band. This transfers the atoms from the initial |2, 2i state to the |1, 1i state with
95% efficiency. The upper insets show 18 ms time-of-flight absorption images of the BEC in
the presence of a vertical Stern-Gerlach gradient to spatially separate the different magnetic
sublevels.

state. For the three F = 1 sublevels, the lifetimes are found to be approximately equal,
while conversely the lifetime of the |2, 1i state is observed to be considerably shorter
than that of |2, 2i, and the |2, 0i state lives shorter still. While the extended lifetimeof the stretched state can be understood as the absence of spin-changing collisions, thediscrepancy between F = 2 and F = 1 states is less clear. Although not understood,
nevertheless we use this feature to our advantage by operating in the F = 1 manifold
which provides a long-lived basis of states.
This state transfer is achieved using an adiabatic Landau-Zeener sweep illustrated infigure 3.18. With the antibias field providing a uniform field of 3.12 G, an off-resonant
microwave field is applied at fixed frequency 6841.98 MHz and Rabi frequency 13 kHz.
The antibias field is then swept across resonance by 0.36 G in 100 ms before removing
the microwave field. The efficiency of this transfer is detected by spatially splitting thetwo states in time-of-flight by applying a vertical magnetic gradient. Typical transfer
efficiency is about 95%. The remaining atoms in the |2, 2i state are removed by a short(10 µs) resonant pulse of imaging light which scatters the |2, 2i atoms out of the trap,
but is far detuned for the |1, 1i state.

61

3.9 Absorption imaging

3.9

Absorption imaging

Our experiment uses absorption imaging to extract information about the density profile, atom number, and temperature of our atoms. Absorption imaging simply relies on
illuminating the atoms with a laser pulse and measuring the spatial attenuation of thebeam profile, which relates directly to the density profile of the atom cloud. This isdone by imaging the shadow cast by the atoms onto a CCD camera.
To calculate the change in laser light due to the atom cloud, we use the relationshipbetween the refractive index, nref , and the complex polarizability, αnref =pnα
1 + nα/0 ≈ 1 +,20(3.81)

where n is the atomic density of the cloud. Since the polarizability is complex, this leads
to a complex refractive index with real and imaginary parts, nR and nI respectively:
n~cΓ
2∆/Γ
4Isat 1 + I/Isat + (2∆/Γ)2
1
n~cΓ
.
nI =
4Isat 1 + I/Isat + (2∆/Γ)2nR = 1 −(3.82)(3.83)A constant refractive index will modify the propagation of a plane wave as follows:
E = E0 ei(kz−ωt) → E0 ei(nref kz−ωt) = E0 ei(knR z−ωt) e−nI kz . The imaginary part leads toa density-dependent exponential decay of the amplitude which is utilised in absorption
imaging. For non-zero detuning, the real part of the refractive index generates a densitydependent phase shift of the incoming light. This phase shift can be used to imagetransparent objects by phase-contrast imaging, where the information encoded in thephase shift of the light is converted into intensity before detection. In this work, thereal part of nref leads to unwanted diffraction effects when imaging dense clouds intrap. To remove this effect we can image on resonance (∆ = 0), where the phase shift
vanishes, however to reduce the optical density to within the dynamic range of ourimaging system, we only transfer a fraction of the atoms from the |1, 1i state to theimageable |2, 2i state.
Absorption of photons leads to a rate of decay of intensity with propagation distancegiven bydI
= −2nI kI = −nσI.
dz(3.84)

From this we extract the scattering cross-sectionσ=

σ0,
1 + I/Isat + (2∆/Γ)2(3.85)

~Γω
.
2Isat

(3.86)

where
σ0 =

62

3.9 Absorption imaging

The same result can be obtained by equating the rate of decay of intensity, dI/dz, to therate of energy scattered from the cloud, ~ωRsc n, and using the form of the scattering
rate in (3.29). In the limit of low intensity, the effect of saturation in σ can be neglected,
and equation (3.84) can be integrated to give the intensity after propagating a distance
z,



Z z
n(x, y, z 0 )dz 0 ≡ I0 (x, y)e−D ,I(x, y) = I0 (x, y) exp −σ(3.87)

0where the optical density, D, is defined as
Z zI(x, y)00n(x, y, z )dz = − lnD(x, y) = σ.
I0 (x, y)0(3.88)

Therefore by taking an image of the incident light with atoms to obtain I(x, y), and an
image without atoms to obtain I0 (x, y), it is possible to calculate the optical density,
which is proportional to the column density along the line of sight. To account for any
stray light entering the camera, additional dark images are taken with no imaging lightand are subtracted from the atom shadow image and the imaging beam image. The
final optical density is then given byD(x, y) = − ln

3.9.1

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

(3.89)

Imaging systems

Our experiment makes use of a horizontal and vertical imaging setup, both of whichwere designed and tested by Alexandre Dareau using the OSLO optimisation package
and are detailed in his project report [74]. The horizontal direction is used to makequantitative measures of atom number and temperature, whereas the majority of thedata taken on persistent currents is done using the vertical imaging system, along which
vortices are visible. In both instances the imaging bias field is in the horizontal plane
along the direction of the horizontal imaging. This bias field is the same used during
state transfers and hence is quite large at approximately 10 G.
Horizontal imaging Our horizontal objective is based on one proposed by Wolfgang
Alt [75]. The objective uses five commercially available 1” optics, the first four ofwhich are designed such that their own respective aberrations combine to cancel
one another out and the spherical aberrations introduced by the science cell wall.
The fifth lens is a diffraction limited achromatic doublet used to focus the image
onto the camera. This complete objective achieves a resolution of less than 2 µm
and a magnification of Mh = 2.78. The atoms are imaged with a broad beamof σ + polarised light which is resonant with the |2, 2i → |3, 3i cycling transition.
The steady state resonant cross-section (3.86) for σ + polarised light is given by
[76]
+

σ0σ =
63

3λ2
.
2π

(3.90)

3.9 Absorption imaging

Vertical imaging Our vertical objective is similar in spirit to the horizontal objective,
however the additional constraint of the quadrupole coils forces the lenses to be
spaced further apart. A custom meniscus is closely mounted near the sciencecell to maximise the numerical aperture, and the subsequent optics used are 2”
in diameter to further increase light collection. This objective also achieves aresolution of less than 2 µm and a magnification of Mv = 5.2. Due to design
constraints where the vertical direction is used for addressing the atoms with theRaman beams, the vertical imaging light is π polarised with respect to the bias
field. As a result, during the imaging pulse, the atoms become distributed over arange of mF states which makes calculation of the scattering cross-section more
complicated. Calculations in [76] and [77] show that the steady state saturation
intensity is given by Isat = 3.053 mW/cm2 , giving a resonant cross-section of+

σ0 = 0.55 σ0σ . Typically we tend to calibrate the vertical imaging using thehorizontal imaging since the transition is much simpler.
The same type of camera1 is used for both imaging systems. This camera has a relatively
modest quantum efficiency of ∼ 28% and a pixel area of 6.45×6.45 µm2 . For horizontal
imaging this gives an effective pixel size of 2.32 × 2.32 µm2 , just above the resolution
limit, which is calibrated using the motion of the centre of mass of the cloud in free fall.
For vertical imaging in trap this gives an effective pixel size of 1.2 × 1.2 µm2 , just under
the resolution limit, calibrated by comparing cloud sizes in both imaging systems. We
also calibrate the vertical imaging at the plane of 29 ms TOF, which is used extensively
for detecting angular momentum, which gives a larger pixel size of 1.42 × 1.42 µm2 .
Due to the relatively low efficiency of the camera, to achieve a high signal-to-noise
ratio we use quite high intensity and long imaging pulses such that each atom scattersmany photons. Our imaging pulses are typically 80 µs long with an intensity I/Isat ≈0.3, which corresponds to scattering about 350 photons per atom. The disadvantageof this is the high intensity leads to some saturation of the imaging transition, and thescattering force during imaging can lead to a Doppler shift of resonance and an effective
detuning [45].
Stern-Gerlach
To image different magnetic states we apply a Stern-Gerlach magnetic gradient during
time of flight using the quadrupole coils. This exerts a state dependent force whichcauses the different mF states to spatially separate as they fall. For vertical imaging
we apply a gradient of 11 G/cm, which due to the bias field predominantly splits thestates in the horizontal direction. The gradient is ramped off a few ms before imaging,
just leaving the imaging bias field on, so as not to create a spatially varying detuning.
For horizontal imaging we apply a gradient of 16 G/cm and a strong vertical field of 90
1

PCO Pixelfly QE Double Shutter

643.9 Absorption imaging

G. This splits the states in the vertical direction, and once again the fields are ramped
off a few ms before imaging.

3.9.2

Atom number calibration

As discussed in the previous section, various experimental factors will cause the true
scattering cross-section to differ from σ0 , often causing an underestimate of atom number. Typical factors include imperfect polarisation, small detunings from resonance,
and in our case, saturation and acceleration of the atoms during imaging. For quantitative studies of BEC this needs to rectified, and the scattering cross-section is scaled bya ‘fudge-factor’, f = σ0 /σ, to compensate for the discrepancy. The way this is done isto note that while the atom number is altered by f , the temperature, which is extracted
from the width of the thermal distribution, is not. Since the BEC phase transition is afunction of both temperature and atom number, the onset of condensation can be used
to find the fudge factor as follows
f Ntotal = Ncid + f N0 ,(3.91)

since for an ideal gas the thermal atom number saturates at the critical number givenby (2.16)
Ncid = 1.202



kB T
~ω0

3
,

(3.92)

where ω0 is the geometric mean of the trapping frequencies. The uncorrected total
number, Ntotal , and condensate number, N0 , are found by fitting the absorption image,and the critical number is calculated using the fitted temperature, thus allowing us
to find the fudge factor, f . As discussed in [16, 47] the effect of interactions leads tonon-saturation effects which modify (3.91) and need to be considered for an accurate
determination of f . For small condensates this can be suitably captured using HartreeFock theory which gives the total atom number as
Ntotal = N0 + Ncid + 1.37Ncid

µ
,
kB T

(3.93)

where the chemical potential, µ, is a function of the condensate number (2.44). The
fudge factor is then found by solving the equations
f Ntotal = f N0 + Ncid + 1.37Ncid

µ
,
kB T

(3.94)

with the chemical potential given by
~ω0
µ=
2



15f N0 a
a0

65

2/5
,

(3.95)

3.9 Absorption imaging

Figure 3.19: Convergence of saturation intensity and scattering cross-section with scattering
events: The evolution of the saturation intensity, Isat (red), and the on-resonance cross-section,
σ0 (blue), are shown as the expected populations change with absorption of σ + polarised imaging light and spontaneous emission. These are calculated for each of the three starting population distributions obtained by repumping |1, 1i, |1, 0i, and |1, −1i. The vertical axis is thevalue of the parameters over their steady state values.

where a is the s-wave scattering length, and a0 is the harmonic oscillator length associated with the mean trapping frequency. Higher order corrections are considered in[48] by considering modification of the critical temperature due to interactions. For
our horizontal imaging system we observe a typical reduction in the cross-section bya factor of f = 1.5 ± 0.3, consistent with that expected from our range of imaging
parameters.

3.9.3

Imaging the |F = 1i state

As described in Section 3.8, our experiments on persistent currents are performed in
the F = 1 states, whereas our imaging light is resonant for the F = 2 state. To image
the F = 1 state we quickly transfer the atoms to the F = 2 states just before theimaging pulse. This can be done in two different ways.
1. The first is to flash a 10 µs pulse of π polarised repump light just before theimaging pulse, after releasing the atoms from the trap. This transfers the atoms
into the F 0 = 2 state, and those that decay into the F = 2 manifold become dark
to the repump light and remain there. Since each atom should end up in a darkstate we are able to saturate the transition many times over and rapidly move thepopulation into the imaging state. This method will distribute the atoms acrossseveral mF states, however due to the significant number of scattering events
during imaging, the atoms quickly reach their steady state populations and theeffective cross-section is only slightly affected by the starting distribution.
663.9 Absorption imaging

To calculate this effect we use the appropriate branching ratios [78] to find thesteady state populations at the end of repumping
Transition

Final populations mF = (−2, −1, 0, 1, 2)|1, 1i → |2, 1i(0.01, 0.14, 0.36, 0.25, 0.23)|1, 0i → |2, 0i(0.03, 0.42, 0.09, 0.42, 0.03)|1, −1i → |2, −1i(0.23, 0.25, 0.36, 0.14, 0.01)

From these starting populations we now simulate absorption of σ + polarised imaging light and spontaneous emission back to the F = 2 magnetic sublevels. This
gives the expected population distribution as a function of scattering events, fromwhich the saturation intensity and hence the on-resonance cross-section can becalculated. The result of this calculation is shown in figure 3.19, and shows thatfor all starting populations, after about 10 scattering events the population isalmost entirely in the |2, 2i state and the cross-section is equal to that found forthe cycling transition (3.90). Since the imaging pulse involves about 350 scattering events, cycling the atoms into the steady state population distribution only
introduces a small correction to the effective cross-section. For typical imaging
parameters I/Isat = 1/3 and a pulse time of 80 µs, the effective cross-sections
are 0.995 σ0 , 0.993 σ0 , and 0.990 σ0 for imaging the mF = 1, 0, and −1 states
respectively. We can therefore be confident that we image all sublevels equallyand can also directly compare to the |2, 2i state. Similar corrections are foundwhen considering vertical imaging with π polarised imaging light.
2. The second method is to use a microwave π-pulse to flip the population from
|1, mF i → |2, mF + 1i. The disadvantage of this method is that it requires very
stable magnetic fields, and hence cannot be used in conjunction with a SternGerlach gradient used to spatially separate the magnetic states. The pulse is also
only able to transfer one mF state to an imageable state, however this can beused to our advantage to measure the spatial distribution of each mF state insitu. Another advantage is that we need not transfer all the population to theimaging state, providing a simple, non-invasive way to reduce the optical density[79].
The use of absorption imaging for quantitative analysis is typically restricted tosamples with an optical density of less than 3. For optical densities above this,the image is said to be “blacked out” and the level of transmitted light is too
low for reliable inference of optical densities. The cross-section can be reduced bydetuning the imaging light from resonance, however this can lead to significant
refraction of the imaging light beyond the aperture of the imaging system. Thiscan affect the absolute density measurement, and importantly for in situ images
where the density varies on small length scales, it can appear as false absorption
signals in the relative density [80]. Partial transfer absorption imaging simply
673.10 Conclusion

involves only transferring a fraction of the atoms to the |2, 2i imageable state.
This uniformly reduces the optical density across the entire sample, bringing it
within the dynamic range of our setup and providing an accurate representationof the density distribution. Since the imaging light does not affect the remaining
atoms this can also be used to take several pictures of the atoms in a single
sequence.

3.10

Conclusion

In conclusion we have outlined the theory and experimental stages used to produce our
87 Rb BEC held in an all optical dipole trap.

The theory and experimental apparatus

relevant to later chapters has been covered in more detail, in particular that of Rabi
oscillations, optical trapping, and the creation of dressed states by atom-light coupling.
The experimental protocol described in this chapter provides the starting point for all
our experiments on persistent currents, which are described in the chapters to follow.

68Chapter 4

Preparation and detection of persistentcurrents
In the previous chapter we covered our experimental sequence to create a BEC in the|1, 1i state held in a crossed ODT (CDT). We now explain how we load this BEC into
an optical ring trap to create a multiply-connected toroidal BEC, and then impartangular momentum to the BEC to create a superfluid persistent current in analogy tothose created in superconducting [81–85] and superfluid liquid helium [27, 39, 86–88]
experiments. Superfluid flow of a BEC in a multiply-connected ring geometry is thearchetypal metastable many-body state. As discussed in Section 2.4.3, the phase of themacroscopic BEC wavefunction must wind around the ring by an integer multiple of
2π, corresponding to the charge q of a vortex trapped inside the ring. Different q states
are topologically distinct and therefore transitions between them involve a macroscopictransformation of the BEC wavefunction and so are energetically unfavourable. As aresult even in a stationary trap where the ground state is q = 0, a q 6= 0 persistentcurrent can be extremely long-lived since there may be no energetically accessible decay
channel. This chapter details how we prepare and detect such persistent currents, thestability and decay of which are studied in the subsequent chapters.
The outline of this chapter is as follows. Firstly we detail past ring trap experiments,
describing alternate methods for their creation, previous studies on persistent currents,
and applications of BECs in such geometries. The next section explains how we form
our optical ring trap by intersecting a sheet beam with a tube-like Laguerre-Gauss(LG) beam. This includes the methods used to form the LG beam and how we correct
for abberations in the LG beam phase profile. The next two sections describe how wethen impart angular momentum to the BEC, setting up the persistent current, and how
we detect the state of this persistent current following its creation. The final section
illustrates the significance of the multiply-connected geometry by studying the vortex
dynamics when we remove the ring trap, transforming to a simply-connected trap.

69

4.1 Previous ring trap experiments

4.1

Previous ring trap experiments

Atomic BECs trapped in a ring geometry are attractive for creating stable states of nonzero circulation. As discussed in Section 2.4.3, rotation of a BEC requires a multiplyconnected geometry which introduces vorticity into the system. Such vortices are fundamentally unstable and are observed to break apart and migrate towards regions of
low density at the edge of the cloud [31, 32, 89], reducing the angular momentum of
the system. In Section 4.5 we discuss our own observations of dynamically unstable
multiply charged vortices. A ring geometry allows the study of truly stable persistentcurrents by pinning the vorticity to the zero density region in the centre of the ring
[90, 91]. As discussed in chapter 5, the ring geometry introduces low energy states,
corresponding to placing vortices inside the ring, separated by energy barriers which
prevent a continuous deformation of the BEC wavefunction and loss of angular momentum. Ring shaped BECs also have several other promising applications such as
interferometry, atomtronics, and fundamental studies of BECs with periodic boundary
conditions.

4.1.1

Previous ring traps

Several schemes have been implemented to create multiply-connected geometries forultracold atoms. These can be broadly divided into three types: magnetic, RF dressing,and optical. The scale of these traps are generally divided into smaller traps where thequantum coherence extends all around the ring, suitable for persistent current studies
and atomtronics, and larger traps for matter-wave interferometry.
Magnetic
Various current structures have been generated for achieving large area ring
waveguides based on dc magnetic fields to create ring traps suitable for interferometry [92, 93]. The disadvantage of such traps is the existence of field zeros
which lead to losses by Majorana spin-flips. This can be negated by using timeaveraged magnetic traps such as in [94] which reported the first BEC loaded in acircular waveguide. The disadvantage of such traps for our studies is their large
scale, of the order of several mm. The low mean field of a BEC occupying such alarge trap makes it unfeasible to create a continuously connected BEC, unbroken
by residual azimuthal roughness.
RF dressing
Modifications to these magnetic traps have been proposed which use RF fields todress the atoms and create time-averaged ring potentials [95, 96]. Such schemes
involve using RF fields to create a dressed band with a minima isosurface at
a given magnetic field value. The position of this surface can then be moved
to create a time-averaged ring potential, the parameters of which can be modi-70

4.1 Previous ring trap experiments

fied by changing the RF frequency. Such traps are inherently flexible, and can
produce small ring traps suitable for superflow studies. Such a trap was recently
demonstrated for a BEC in [97]. The disadvantage of this solution is the trapping
potential is state dependent.
OpticalSeveral experiments (including this one) make use of optical forces to create ringtraps. The clear advantage of this is the trap is equal for different magnetic
states, and the creation of small ring geometries is easily achieved. The first suchtraps used a repulsive blue-detuned beam aligned along the symmetry axis ofa magnetic trap to create a multiply-connected geometry. This was done withboth a quadrupole magnetic trap [98] and a TOP magnetic trap [99]. This setup
is limited by heating and losses due to drifts in the relative alignment of themagnetic field and the plug beam.
The experiment of Ryu et al. [99] evolved into an all optical trap which madeuse of an intersecting sheet beam and tube-like Laguerre-Gauss beam [100] (seeSection 4.2). This setup is very similar to ours, and both experiments [99] and[100] were also used to study persistent currents. As such, in chapter 5 we will
directly compare our finding to those in [99, 100]. Other all optical ring traps
have used time-averaged painted potentials [101, 102]. These experiments use a
two-directional AOM to deflect a beam and trace out the trapping potential. Thisis done with a kHz refresh rate, allowing one to create an arbitrary time-averaged
potential. Such traps show great promise for creation of dynamic atomtronic
circuits.

4.1.2

Applications of ring BECs

While numerous novel proposals have been developed which make use of multiplyconnected BECs [103, 104], we will limit ourselves to briefly explaining those applications which have been experimentally realised.
Persistent current studies
Persistent flow in a BEC is a striking demonstration of superfluid behaviour andfacilitates studies into critical velocities and the connection between superfluidity
and Bose condensation, which are still not fully understood. The only experiments
to demonstrate persistent flow in a BEC are those of the NIST group [99, 100, 102],
and those presented in this work [14, 15]. The NIST experiments demonstrated
long-lived superflow of up to 40 s, and were able to theorise a possible decay
mechanism. The results of these experiments will be contrasted with our own
findings in chapter 5.
Atom interferometry
Atom interferometry is performed by coherently splitting an atom cloud into two71

4.2 Optical ring trap

parts which then travel spatially different paths before recombining and interfering. If the parameter of interest acts differently on the two paths, a relative phasedifference will accumulate which is read out upon interference as a population difference. Such experiments have allowed precision measurements of various quantities, including rotation [105–107], acceleration [108], magnetic gradients [109],and fundamental constants [110]. Interferometric detection of rotation makes useof the Sagnac effect which states that two interfering propagation paths enclosing
an area A will experience a relative phase shift of ∆φ = 4π
λv Ω · A, where λ isthe particle wavelength, which for a massive particle is the de Broglie wavelength
λdB = h/mv, v is the velocity of the particle, and Ω is the rotation of the external
system. As a result, the inherent sensitivity of a matter-wave gyroscope exceeds
10that of a photon-based system of equal area by a factor of mcλh ∼ 10 , where

here λ is the typical photon wavelength.
Sagnac interferometry in a ring trap is attractive for several reasons: (i) for a complete loop of the trap the two paths are identical (apart from their rotation sense)and have common-mode rejection, and (ii) the area, A, can be trivially extended
by allowing multiple revolutions before readout. Such a proposal is outlined in[111] for a two component BEC. This makes use of a π pulse at half the integration
time termed an echo pulse, which cancels phase accumulation due to atom-atominteractions. Recently a proof of principle rotation sensor was demonstrated in[112] which uses interference of collective excitations in a toroidal-shaped BEC.
Atomtronics
Atomtronics focuses on the creation of atom analogs of electronic materials, devices, and circuits. The analogy between a strongly interacting Bose gas in a
lattice potential and electrons in a solid-state crystal, has led to the proposal of
P-type and N-type material analogs in ultracold systems, and from these building blocks possible diode and transistor configurations are proposed from whichatom circuits could be constructed [113]. Though quite a speculative field at thepresent time, the first realisation of an atomtronic circuit was recently achievedin [102] with a BEC analog of a SQUID circuit. This experiment uses a toroidal
BEC with a rotating weak link which is able to spin up the atoms and introduce
vortices. Such an experiment is then shown to map onto a superconducting loopwith a weak link in the presence of an external magnetic field.

4.2

Optical ring trap

Our ring trap is illustrated in figure 4.1. The trap is formed by the intersection of a reddetuned (λ = 1070 nm) horizontal sheet beam (Section 4.2.1) which provides verticalconfinement and a red-detuned (λ = 804 nm) vertical LG beam (Section 4.2.2) toprovide annular confinement. The use of a separate sheet beam was chosen to decouple
724.2 Optical ring trap

Figure 4.1: Schematic of optical ring trap: Vertical confinement is provided by an oblate
sheet beam in the horizontal direction, red-detuned at λ = 1070 nm. The ring trap is formed
by a tube-like Laguerre-Gauss beam in the vertical direction, red-detuned at λ = 804 nm. The
trap is formed in the green region at the intersection of the two beams. Inset is a top view
absorption image of a BEC loaded in the ring trap. The orientation of the uniform bias fieldused during state transfers is indicated by B.

vertical and horizontal confinement.

4.2.1

Sheet beam

When we evaporatively cool in the CDT we ramp the beam power down from 6.8 W
to approximately 1.6 W, leaving over 5 W of λ = 1070 nm trapping light available.
This power is used to produce our sheet beam, since we only load into the sheet after
reducing the CDT power. The optics used to generate the sheet beam are shown infigure 4.2. The zeroth order beam from the CDT AOM is sent to a separate AOM1 ,which cleans up the beam profile and provides independent control of the sheet power.
A microscope slide picks off a weak reflection which is sent to a photodiode to provide
a signal for a PID controller. This provides stabilisation and control of the sheet beam
power as explained for our CDT in Section 3.7. A 3 : 2 telescope is inserted to provide
independent control of the horizontal dimension of the sheet trap at the atoms. The
beam is translated up to the atoms by a periscope and passes through a polarising
beam splitter cube (PBS) which cleans up the polarisation and also aids in alignment.
A free space prism pair is used to extend the beam profile in the vertical plane by a1

Crystal Technology 3200-115

73

4.2 Optical ring trap

Figure 4.2: Sheet beam optics: The sheet beam uses discarded light from our CDT setup.
The output of the fibre laser passes through a telescope resulting in a beam that is size-matchedto the active aperture of the 110 MHz AOM. First order light is sent to the CDT optics, whilezeroth order light is sent to the sheet beam optics. A 200 MHz AOM cleans up the beam profileand provides power control, with the undeflected light sent to a beam dump. A microscopeslide picks off a weak monitor beam sent to a photodiode used for PID power stabilisation. A
3 : 2 telescope gives added control to collimate the sheet beam at the atoms in the horizontal
direction without adjusting the CDT setup. A periscope brings the light level with the atoms
and then a polarising beam splitter cube (PBS) cleans up the polarisation and allows overlapof guide beams for alignment. The beam is stretched in the vertical direction by a factor 4 : 1
using a free space prism pair and is then finally focused in the vertical direction only onto theatoms using a f = 75 mm cylindrical lens.

factor of 4, creating the profile shown in figure 4.3 (b), with a vertical waist of wv = 2mm and a horizontal waist of wh = 0.5 mm. This is focused in the vertical direction
only by a cylindrical lens of focal length f = 75 mm1 , giving an estimated verticalbeam waist at the focus of w0 = f λ/(4wv ) ≈ 10 µm.
1Thorlabs LY1703RM-C

74

4.2 Optical ring trap

Figure 4.3: Sheet beam: (a) Side view of the cylindrical lens focusing the beam in the verticaldirection only, to create a weak horizontal trap and tight vertical trap. (b) Contour plot of
intensity for the input beam before the cylindrical lens. The beam is extended in the verticaldirection by the prism pair to focus more tightly and strongly confine the atoms in the verticaldirection. Dimensions shown are 1/e2 diameters. (c) Vertical absorption image of atoms loaded
into the sheet beam only. The dimensions of the cloud are larger than our typical ring traps
which provides an error margin should the sheet and ring drift out of alignment.

Trapping parameters
The sheet beam power used is 100 mW, corresponding to a calculated trap depth of 550
nK. This shallow trap depth ensures evaporation of any thermal atoms created duringloading and a high condensate fraction, typically over 85%. For powers below 50 mW
gravity dominates and the trap minimum disappears. The trapping frequencies of the
sheet are measured by translation of the BEC from the trap minimum by applicationof a quadrupole magnetic gradient, the orientation of which is selected by applicationof an appropriate uniform bias field. The fields are then abruptly removed and the
BEC is allowed to freely oscillate in the sheet trap. To amplify the signal we measureoscillations in the BEC position at long time-of-flight (TOF) (18 ms). The measured
trapping frequencies across the sheet, along the sheet, and vertically are ωx = 2π × 6.3
Hz, ωy = 2π × 7.1 Hz, and ωz = 2π × (400 ± 13) Hz respectively. Such weak in-plane
trapping frequencies are important for creating a level ring trap and are also foundto be beneficial for detection of angular momentum. Typically we load about 2 × 105
atoms into the sheet which, using (2.44), gives a chemical potential of µ ≈ 460 Hz.
This gives the corresponding Thomas-Fermi radii as Rx = 52 µm, Ry = 46 µm, andRz = 0.8 µm, in rough agreement with the observed density profile in figure 4.3(c).
Alignment
Due to the short Rayleigh range, zR = πw02 /λ = 0.3 mm, alignment of the sheet beam
proved very difficult. The optimal solution to finding the sheet beam was to overlap
75

4.2 Optical ring trap

Figure 4.4: Repump sheet beam for alignment: A separate repump beam is overlapped withthe sheet beam on the PBS cube before the prism pair. The repump beam produces a strong
signal, pumping atoms into the imageable F = 2 state. (a) The repump beam is flashed on
at various points during TOF. The visibility of the atoms allows us to map out the repump
beam position and profile, and hence find the overlapped sheet trapping beam. (b) A horizontal
absorption image of a cloud which has only partially fallen into the repump beam, hence only
the bottom half is in the visible F = 2 state.

the beam with a repump beam on a PBS cube before the prism pair. Unlike the sheet
beam, which has to be in close proximity to the atoms in trap to significantly distort
their trapping potential and produce an observable signal, any F = 1 atoms which passthrough the repump sheet will be pumped to the imageable F = 2 state. We therefore
flash on the the repump beam during TOF, and vary TOF to determine the height ofthe beam. By flashing the repump on just before imaging, only a slice of the cloud ispumped to the F = 2 state, hence we can find the focus by minimising the thicknessof this slice. Figure 4.4(a) illustrates this technique, and figure 4.4(b) is an absorption
image of a cloud where only half the cloud has passed into the repump sheet. This
technique was able to get the alignment close enough to observe distortion of the BEC
profile due to the trapping sheet. The alignment of the sheet trap was then chosen tominimise heating upon transfer. Any tilt in both directions was removed by rotationof the cylindrical lens, and vertical walking of the beam.
A disadvantage of using an attractive sheet beam for vertical confinement is thepresence of weak fringing near the focus. In our case we observe two local minima
about the focus separated by ≈ 200 µm along the direction of propagation. Private
correspondence with the NIST group [100] revealed that a similar problem was observedin their setup. Despite numerous attempts, the cause of the fringing could not beestablished and the solution was simply to load into one of the fringes. The approximate
spacing of the fringes seems to be weakly dependent on the angle of incidence, hence
we theorise the cause could be due to a reflection, however the fringe spacing does
not fit with interference from a counter-propagating reflection, nor reflection within
any optical coating. Another possible cause could be chromatic aberrations due to thebroad linewidth of the fibre laser, however we would expect the spectral distribution
to only have one maximum.

76

4.2 Optical ring trap

Figure 4.5: Laguerre-Gauss intensity profile and phase winding: Top figures show the intensity
profile and bottom figures show the azimuthal phase winding of an LG beam for ` = 5, 3, 1 fromleft to right. From this we observe the trend of increasing annulus radius with increasing `.

4.2.2

Laguerre-Gauss beam

Our ring trap is an optical dipole trap formed by a red-detuned Laguerre-Gauss (LG)laser mode. LG modes are a complete basis set for paraxial light beams with circularsymmetry about their axis of propagation. Two indices identify a given mode denotedby LG`p , where p is the radial order which we are not concerned with and set to be 0, and` is the angular order. In contrast to the planar wave fronts of the Hermite-Gaussian(HG) modes, for ` 6= 0 the LG beams have an azimuthal phase term ei`θ which resultsin helical wave fronts [114]. This phase winding about the beam’s axis means an LG
beam can be thought of as an optical vortex of topological charge `.
The LG beam is integral to our experiment for two reasons: (i) an LG beam
has a doughnut intensity profile [115] which creates the toroidal optical dipole trap,and (ii) the helical wave front means LG modes possess an orbital angular momentumof `~ per photon [116], quite distinct from the spin angular momentum associatedwith the polarisation state. As discussed in Section 4.3, this is used to impart angularmomentum to the ring BEC and generate a persistent current. The relation between theorbital angular momentum of an LG beam and the circular polarisation of a photon
is analogous to the orbital angular momentum and spin of an electron. The orbital
angular momentum is associated with the spatial mode, while spin is intrinsic to theparticle.
The field amplitude of an LG laser mode is given by [115]

 2

 
−ikr2 z−rzexp
exp −i(2p + ` + 1) arctan
2
2
2
w
zR
2(zR + z )√ !`  2 
r 2
2r
× exp[−i`φ](−1)p
L`p
,
(4.1)
w
w2

E(LG`p ) ∝ exp

where z is the distance from the beam focus, zR is the Rayleigh range, k is the wave
number, w is the Gaussian beam waist, r is the radius, φ is the azimuthal angle, and77

4.2 Optical ring trapL`p is the generalised Laguerre polynomial. From this we find the intensity profile
2
I(LG`p ) =P`! πw2



2r2
w2

` 

L`p



2r2
w2

2


−2r2
,
exp
w2


(4.2)

where P is the total beam power. The intensity profile as a function of ` is shown infigure 4.5. This doughnut profile has clear applications in forming ring traps, as wellas other features of note. The radius of the intensity minimum scales with topologicalpcharge as R` = w `/2, and we also observe that a true intensity zero is created within
the ring due to the phase singularity. The intensity profile inside the ring scales as r2` ,hence a blue-detuned LG beam has applications in creating novel trapping geometries
with high power laws [117–119].
In recent years LG beams have attracted much interest, and found a host of diverse
applications. LG beams are used extensively in optical tweezers, which use the refraction of light about an object with higher refractive index than the surrounding liquid to
optically trap and manipulate samples ranging from simple dielectric spheres to cells,
chromosomes, and protein motors. LG beams offer distinct advantages in being able totrap both high and low-index particles [120], holding samples in low intensity regions
where cell damage is reduced, and rotating samples due to absorption of angular momentum from the trapping LG beam [121]. LG beams were used in a recent experiment
which demonstrated the possibility to use two beams of incoherent radio waves, transmitted on the same frequency but encoded in two different orbital angular momentum
states, to simultaneously transmit two independent radio channels [122]. LG beams
have also found numerous other applications including astronomy [123], phase contrast
microscopy [124], and in secure encoding of information in orbital angular momentum
states [125].
Several methods have been developed in the last two decades for the productionof LG beams, or more often for the production of fields with a single dominant LG
mode. While LG modes can be produced directly from a laser resonator [126], it isoften simpler to obtain these modes by externally converting the HG mode output
from a conventional laser. The creation of pure LG modes is possible using a cylindrical lens telescope with an input beam mode HGm,n , aligned at 45◦ to the principal
axis of the lens, provided either m 6= 0 or n 6= 0 [127, 128]. Approximations to LG
beams can be produced using spiral phase plates [129, 130] and computer-generated
holograms [131, 132]. Such devices introduce a screw-phase dislocation in the centre
of a Gaussian beam, which to first order approximates the spiral wavefront of an LG
beam. On propagation the phase dislocation causes destructive interference leading to
the annular intensity profile and a high percentage of a given LG component, with `set by the phase pitch of the spiral phase plate, or correspondingly the order of thedislocation in the hologram.

78

4.2 Optical ring trap

4.2.3

Spatial light modulator (SLM)

Our technique for creating LG beams uses a spatial light modulator (SLM). An SLM
is a rapidly emerging tool which provides highly customisable control of light fields by
locally modulating light according to a fixed spatial pixel pattern. Modulation options
include amplitude only modulation used in many projectors, but to create a phase winding we require a phase modulating SLM. Simultaneous amplitude and phase modulation
is available but at the cost of some coupling between the two parameters. Several SLM
technologies exist, including optically-addressed SLMs and digital micromirror arrays,
however the most common SLM implementations are based on liquid crystal displays(LCD), which we use here. In an LCD-based SLM the alignment of the nematic liquid
crystal layer at a given pixel dictates the phase shift imparted to the light passing
through that pixel. Higher phase shifts are achieved by retro-reflecting the light off a
dielectric mirror positioned behind the display, to double the total phase shift. This
provides a flexible way to create the phase profiles generated by spiral phase plates
or holograms, without the difficulty of fabricating devices, and with greater control
to achieve smoothly varying phase profiles and hence higher coupling to a single LG
mode. SLMs also provide the tempting possibility to dynamically vary optical fields
and create time-varying trapping potentials.
In principle, to create an LG beam we simply need to display a phase winding on
the SLM where the pixel value azimuthal increases about a central point, and reflect
a Gaussian beam off of it. In addition the linearity of phase imprinting can be used
to add several other phase patterns on top of this and simultaneously achieve several
operations:eiφwinding eiφgrating eiφlens eiφcorrection = ei(φwinding +φgrating +φlens +φcorrection ) ,(4.3)

where our commonly used phase operations are listed below.
Operation

Phase profile

Phase winding

φwinding = mod (`θ, 2π)

Blazed grating
Lensing

φgrating = mod (kx x + ky y, 2π)
2 +y 2
φlens = mod −k x 2f
, 2π

Correction

see Section 4.2.5

The phase winding φwinding imprints the desired angular momentum onto the beam,
the blazed grating φgrating directs the beam, the lensing φlens controls the focus of thebeam, and the correction phase pattern φcorrection cancels imperfections on the SLM
and other parts of the optical setup. This is illustrated in figure 4.6. Since the phaseshift of the SLM is in the range {0, 2π}, the functions are all mod (2π) to make use of
the equivalence between 0 and 2π phase shift. We note that if the functions were allmod (α2π), where α < 1, the effect would be to diffract more of the light into otherorders. This means that the first diffraction order we use only contains LG modes of a79

4.2 Optical ring trap

Figure 4.6: Decomposed SLM pattern and corresponding LG beam intensity profile: Left
images show pixel values displayed on the SLM, where black = 0 and white = 2π. The LG
beam phase winding is created by displaying an azimuthally increasing pixel value, in this casefor ` = 3. Additional operations are achieved by simply adding further phase profiles. A blazed
diffraction grating creates a bright first order beam, spatially separated from the undiffracted
light and other orders. The focus of the first order beam is adjusted by adding a lensing phaseprofile formed from a radially varying pixel value. Finally a correction pattern is added which
compensates for abberations present in the SLM and subsequent imaging optics. In this casethe provided Hammamatsu correction is used, only correcting for unwanted phase shifts on the
SLM. As illustrated on the right this creates a focused LG beam in the first diffraction order,
here presented for ` = 20. Residual imperfections are due to the incomplete correction patternused (see Section 4.2.5).

single ` value, the value of which is dictated by the periodicity of the phase winding,
not the absolute phase value. Due to their design, SLMs inherently reflect a portionof the incoming light without applying any phase shift. The inclusion of a diffraction
grating spatially separates the light field we want from this unmodulated light andother diffraction orders.
SLM Choice
Having decided upon a reflective, phase modulating SLM, our choice was reduced tothree manufacturers: Hamamatsu, Holoeye, and Boulder Nonlinear Systems. The first
criteria is efficiency, i.e. the maximum amount of light possible to diffract into a single
order. This is a strong function of the back mirror, and the fill factor (the fraction of the
SLM area which is usable pixels). Both Hamamatsu and Boulder provide high filling
factors and high efficiency dielectric mirrors, offering efficiencies as high as 95%. Holoeye offers lower efficiency, but much higher screen resolution enabling smoother phaseprofiles. Although not essential to this work, the possibility to dynamically vary the80

4.2 Optical ring trap

light field depends on the refresh rate of the SLM. SLMs also suffer from phase droop,
where the crystal tends to relax between refreshes, creating a time varying phase profile.
Both Hamamatsu and Holoeye run the SLM as a second monitor via a DVI connection
and hence only refresh at 60 Hz, which is slower than the liquid crystal’s free relaxation
rate. In practical applications this translates to a toggle rate for dynamically changing
the phase pattern of about 10 Hz. Boulder SLMs interface via PCI Express, and hence
offer a refresh rate of over 6 kHz and a usable toggle rate of about 1 KHz. Boulder
SLMs therefore offer a clear advantage for dynamic processes, however the PCI Express interface makes the SLM considerably more difficult to communicate with, andthe response time comes at a financial cost. Experimentally, the phase droop of theHamamatsu SLM was not observable and we could not attribute any erroneous heating to the effect. We therefore conclude that for high-efficiency, static light fields the
Hamamatsu SLMs provide a good choice and use the LCOS-X10468-02 with dielectric
mirror for λ = 750 − 850 nm.
The implementation of the SLM is illustrated in figure 4.7. As explained in Section
4.3, the LG beam is one beam of a two-photon Raman transition used to impart angular momentum to the atoms. Both Raman beams exit the same fibre with orthogonal
polarisations and frequency shifted from one another by the relevant Zeeman energy.
The horizontally polarised beam, referred to as the LG beam, is reflected off the SLM
and hence picks up a phase winding and angular momentum. The vertically polarisedbeam, referred to as the G beam, does not, and hence the difference in angular momentum of the two beams is imparted to the atoms during the Raman transition.
Concentrating on the SLM setup for the LG beam, the beam is partially collimatedout of the fibre by an achromatic doublet lens1 . The beam is still diverging when it
reaches the SLM, at which point it completely fills the SLM screen with a waist of
6 mm, before reflecting off at a shallow angle of less than 10◦ to the normal. Before
reaching the SLM the light passes through a neutral optical density filter. Although
not essential, the required power in the LG beam is several orders of magnitude lessthan that in the G beam, hence the filter allows the balance of power between the two
Raman beams to be more comparable, and therefore small drifts in the polarisation
have a smaller relative effect. The phase imprinted on the beam is shown in figure 4.6.
A lensing pattern is added to the SLM which collimates the incoming beam, and adiffraction grating spatially separates the various orders. Both Raman beams are thenrecombined on a cube and focused onto the atoms using an aplanatic meniscus lens2
and companion doublet3 with a combined focal length of f = 86 mm.
1AC254-100-B-ML
CVI Melles Griot 01LAM555
3
CVI Melles Griot 01LAO555
2

81

4.2 Optical ring trap

Figure 4.7: SLM optics: The two beams for the Raman transition exit the fibre with orthogonal polarisations and frequency shifted with respect to one another. The beam with verticalpolarisation, referred to as the Gaussian beam, is split off a PBS cube and telescoped down so
as to have a large waist at the atoms. The beam with horizontal polarisation, referred to as
the LG beam, reflects off the SLM at a shallow angle, imprinting the phase displayed. Bothbeams are recombined on a PBS cube and sent towards the atoms. The inset plane illustrates
the various orders of the LG light field before the final imaging lens. The light is focused down
onto the atoms and the Gaussian beam is overlapped with the 1st order LG beam.

4.2.4

Pseudo-Laguerre-Gauss beam

The SLM allows us to imprint the necessary helical wavefront onto a Gaussian beam,
however this is not a true LG beam since the field intensity at the SLM does not havethe characteristic annular profile. As the beam propagates, destructive interference atthe phase singularity will give rise to a pseudo-LG beam with a high mode overlap withthe corresponding LG` mode. To see this we consider the paraxial Helmholtz equation
given by
∂ 2 fz∂ 2 fz
∂fz
+
+ 2ik
= 0,
2
2
∂x
∂y
∂z

(4.4)

where fz is the field at position z, and k = 2π/λ is the wavevector. We also define thefourier transform of the field given by
Z ∞Z ∞Fz (u, v) =fz (x, y) exp [−2πi(ux + vy)] dxdy.
−∞

−∞

82

(4.5)

4.2 Optical ring trap

From this we can easily show the effect of
1. Propagation in free space a distance z:

2πi 2Fz (u, v) = F0 (u, v) exp −(u + v 2 )z
k


(4.6)

fz (x, y) = f0 (x, y) ∗ h(x, y)

(4.7)



1
ik 2
2
h(x, y) =exp(x + y ) .
iλz
2z

(4.8)

f (x, y) → f (x, y) exp(iφL (r))

(4.9)

where

2. Action of a lens

where φL is the phase shift introduced by the lensφL (r) ≈ −

kr2
2f

for

r  f.

(4.10)

Using these principles we numerically propagate the field at the SLM and find the fieldat the focus. For our experiment the distance from the SLM to the imaging lens isapproximately 1 m, and we predominantly use ` = 3. These calculations are shownin figure 4.8. During free propagation we observe destructive interference along theoptical axis and after 1 m the field appears strongly LG-like. Significant radial ringing
is observed which we can attribute to the contribution of higher p radial modes. Afterthe imaging lens the radius of the ring intensity profile is smallest at the focus, however
the central dark singularity is preserved at all positions. From these calculations weanticipate a Gaussian waist of w ≈ 5.3 µm, and hence a ring radius for ` = 3 of
≈ 6.5 µm. We find contribution from higher p modes can be significantly reduced byradially masking the SLM with a doughnut aperture, the outer radius of which is set
to be the vertical height of the SLM and the inner radius is chosen to minimise ringing
for a given `. We implement this by simply only applying the blazed grating to an
appropriately chosen doughnut area on the SLM, hence light outside this mask is notdeflected towards the atoms. This increases the calculated Gaussian waist to w ≈ 9µm and the ` = 3 radius to ≈ 11 µm. This is close to our observed radius of 12.2 µm.
The properties of such pseudo-LG beams for a fixed field at the SLM are displayedin figure 4.9. In figure 4.9 (a) we compare the intensity profile of the pseudo-LG beamwith a pure LG beam of equal winding, where the waist parameter is chosen to giveequal radius and the profiles are normalised. The two profiles are closely matched, withthe pseudo-LG beam exhibiting small radial oscillations beyond the primary peak. The
intensity profile as a function of ` is shown in figure 4.9 (b). Unlike for a true LG beam,83

4.2 Optical ring trap

Figure 4.8: Numerical calculation of pseudo-LG beam propagation: (a) By propagating thefield produced at the SLM a distance of 1 m we observe destructive interference along theoptical axis and the formation of an annular intensity profile. About the focal point of the lens
the radius of the annular intensity profile reduces, but the intensity zero at the centre nevervanishes. (b) Intensity profile of the field at the lens after free propagation for 1 m. The profile
is LG-like, but the multiple rings indicate a significant contribution of higher radial modes with
p 6= 0. (c) Intensity profile at the focal point. The LG mode at the focus appears cleaner thanthat before the lens.

the ring radius scales linearly with ` as shown in 4.9 (c). This radius scaling is confirmed
by our experimental measurements, and is observed independently in [133]. For a fixed
intensity at the SLM the peak intensity at the focus scales as `−0.8±0.1 , and the radialtrapping frequency scales as `−0.45±0.05 . These powers are consistent with the trapping
minima being independent of ` if the peak intensity is held fixed.
An analytic expression for the field produced at the focus is derived in Appendix A


 2 
 2 
2
r2
rr`−1 √ f w r
π 2
I `−1
− I `+1
Ef (r, θ) = E0 i
exp − 22
2
2
z w0 w0
2w0
2w0
2w02


iπ f − z 2
× exp
r exp(i`θ),(4.11)
fλ fwhere f is the focal length of the imaging lens, z is the free propagation distance between the SLM and the lens, In (z) is a modified Bessel function of the first kind, and
w0 = f λ/πw is the diffraction limit, where w is the waist of the initial Gaussian beam.
Comparison with the field for a pure LG beam in (4.1), our pseudo-LG beam displays84

4.2 Optical ring trap

Figure 4.9: Numerical calculations of pseudo-LG beam properties produced by phase imprinting a Gaussian beam: (a) Comparison of pseudo-LG beam with true LG beam intensity (4.2).
The respective waists have been chosen to give equal ring radii, and both profiles have beennormalised. (b) Scaling of intensity profiles of pseudo-LG beam with `. For all `, the input
field at the SLM is held constant, only the phase winding imprinted is changed. (c) Scaling ofpeak radius with ` for pseudo-LG beam. The red points are experimental data, and the blueline is a numerical calculation which has been scaled to lie on the data.
√ The two show good
agreement and both predict a linear relationship, in comparison to the ` scaling for a true LG
beam. (d) Numerical calculations of peak intensity and radial trapping frequency, ωr , scaling
with `. The points are fitted to a power law function to obtain scalings of `−0.8 and `−0.45 , forpeak intensity and ωr respectively.

the important helical wavefront, ei`θ , associated with orbital angular momentum. Tak1z n
ing the small argument limit of In (z) ∼ Γ(n+1)
2 , retrieves the same small r limit ofEf ∝ r` .

4.2.5

Correcting for abberations

As shown in figure 4.6, the initial LG beams we generated differed from those in our
calculations. We observed azimuthal variations in the intensity which had significant
implications on the smoothness of our trapping potential, and hence the stability ofsuperflow around the ring. These variations are introduced by unwanted phase aberrations present in the SLM and the optics along the LG beam path. Significant timeand effort was therefore invested in minimising such variations to create the ideal trapping environment for observing persistent currents. Helpfully, due to their wide range
of applications, considerable attention has already been paid to the estimation andcorrection of abberations in SLM-based optical systems. A phase pattern is provided
with our SLM to correct for known aberrations measured at production, but further85

4.2 Optical ring trap

Figure 4.10: Schematic of the SLM-based Shack-Hartmann sensor (copied with permission
from [134]): The Shack-Hartmann array produces a grid of spots, where aberrations affectthe position of each spot. The corresponding reference array produces a grid of spots, whereaberrations affect spot shape but not position. By tracking the displacement of the spot centres
between the two arrays we construct a vector map which can be fit to directly recover the
wavefront.

aberrations introduced by our setup needed to be minimised. Below we outline thetwo algorithms we implemented to detect and correct such aberrations, and our final
solution based on manual correction.
Shack-Hartman algorithm
Shack-Hartmann sensors employ an array of lenslets to focus a collimated beam into
an array of spots. The displacement of each spot is proportional to the tilt of the
wavefront at that point, and the resultant tilt information can be integrated to recoverphase information. An equivalent SLM-based Shack-Hartmann sensor is described in[134], and illustrated in figure 4.10. The Shack-Hartmann array is first displayed on
the SLM, which is composed of a grid of circular apertures, each of which contains a
blazed diffraction grating of different pitch, such that each aperture focuses to a different spot in the focal plane. Any aberrations present along a given spot’s optical path,will displace the spot from its true position. To find these true positions the reference
array is then displayed, which creates a grid of spots where each spot comes from thewhole SLM. In this case aberrations will primarily affect the shape of the spots andnot the geometry of the grid. By measuring the displacement of the spots between the
Shack-Hartmann and the reference array, we construct a vector map of wavefront tilts
which can be fitted to recover the aberration present in the optical system. We thencorrect for this aberration by simply displaying the negative of this phase map on the
SLM, which we refer to as the correction pattern.
The vector map of spot displacements are fitted to a linear combination of Zernike
polynomials, listed in table 4.1. Firstly this ensures a smoothly varying correction
pattern, and secondly it also ensures the correction pattern cannot include phase singularities. This is significant as it guarantees the correction pattern cannot change the864.2 Optical ring trap

Figure 4.11: Diagram of Gerchberg-Saxton algorithm for finding phase errors: The initialconditions consist of the modulus of the field amplitude at the SLM plane |ESLM |, which in thiscase is a Gaussian, and an initial phase guess φ0 , which in this case is an ` = 1 phase winding.
The field A1 is propagated to the imaging plane by a fast Fourier transform (FFT) creating
the field A2 . The amplitude of the field is replaced by the modulus of the target intensity|Eimage |, in this case an image of the uncorrected LG beam, to create the field A3 . This field
is inverse propagated back to the SLM plane to create the field A4 , the phase of which is our
improved estimate of the phase required to create the target image. This process is iterated
until convergence.

winding number of the SLM and hence the topological charge of the LG beam produced. The Shack-Hartmann algorithm provides a robust way to cancel aberrations
and significantly improved the quality of our LG beams. For bench-top tests, the algorithm allowed us to obtain highly circular LG beams with a peak-to-peak variation inazimuthal intensity of approximately 30%.
Unfortunately this algorithm has a major drawback for cold-atom experiments
where the focal plane of interest is always within a vacuum system and hence inaccessible. To observe the relevant spot arrays at the focal plane of the atoms we had
to use our vertical imaging systems, hence the algorithm cannot distinguish between
aberrations introduced before the focal plane, and those inherent in the imaging system. To efficiently track spot displacements it is advantageous to have a large grid ofmany spots filling up as much of the field of view as possible, but as we discovered ourvertical imaging system introduces significant distortion when imaging off the opticalaxis. This resulted in the algorithm introducing a lot of lensing aberration at the focal
plane of the atoms to compensate for the distortion of the imaging system, and hence
the algorithm could not be implemented in our system.
Gerchberg-Saxton algorithm
The Gerchberg-Saxton (GS) algorithm is a phase retrieval technique designed to deal
with the problem of finding the phase φSLM (x, y) of a light field by just knowing the87

4.2 Optical ring trap

modulus |Eimage (kx , ky )| of its Fourier transform, and the modulus of the light field|ESLM (x, y)|:|Eimage (kx , ky )| exp[iφimage (kx , ky )] = F F T {|ESLM (x, y)| exp[iφSLM (x, y)]}(4.12)In the paraxial approximation the light field at the imaging plane of a lens and the lightfield at the SLM plane are related by a scaled Fourier transform. From this, one cansee the general problem of finding the SLM phase to apply to a known field modulus inorder to achieve a desired field modulus at the imaging plane, is a member of the classof problems described by (4.12). As such, the GS algorithm (and extensions thereof)
has been implemented by several groups to calculate SLM patterns required to create
complex light fields and trapping potentials [118, 135, 136]. We note that incomplete
knowledge about the phase at the imaging plane means there may exist many phasefunctions with amplitudes close to |Eimage (kx , ky )| which define local minima that may
prevent us from finding the global minimum. GS is a computationally efficient optimisation method for finding phases φSLM (x, y) which minimise the difference between|Eimage (kx , ky )| and |F F T {|ESLM (x, y)| exp[iφSLM (x, y)]}|.
The basic GS algorithm is sketched in figure 4.11. One provides the field modulus
at the SLM, |ESLM (x, y)|, the target field modulus at the imaging plane, |Eimage (x, y)|,and an initial phase guess at the SLM plane, φ(0) . The initial field (A1 ) is propagatedto the image plane by a fast Fourier transform (FFT) and the modulus of this fieldis replaced by the target field modulus, |Eimage (x, y)|. This field (A3 ) is back propagated to the SLM plane by an inverse FFT and the modulus of this field is replaced by
the known SLM field modulus, |ESLM (x, y)|. This process is iteratively repeated untilconvergence and the algorithm provides a best estimate of the phase to display on the
SLM which most closely solves (4.12).
This algorithm places no restrictions on the phase of the image field, which helps by
providing additional freedom to achieve the target modulus. However for the case of our
LG beam the phase at the image plane is vitally important as we need to preserve thephase winding associated with the orbital angular momentum of the beam. As detailedin [137] one can still use the GS algorithm to detect and correct for aberrations by using
an image of an uncorrected LG beam as the target amplitude and the phase winding
φ(0) = `θ as the initial phase guess. The GS algorithm will then converge on the phaseprofile φSLM = `θ + φaberration , providing the correction pattern φcorrection = `θ − φSLM .
Due to the phase freedom inherent in the GS algorithm, convergence to this particular solution is strongly dependent on the amount of aberration present in the system.
Uncorrected LG beams which differ too greatly from a perfect LG beam will converge to phase solutions differing from the addition of a phase winding and smooth,singularity-free aberrations, and cannot be corrected using this method. Experimental
implementation of this method also suffers from several other pitfalls, namely accurate
knowledge of the input field modulus, |ESLM (x, y)|, and precise centring and scaling of88

4.2 Optical ring trap

the target image to match the propagated SLM field intensity. These uncertainties canlead to unnecessary lensing and phase singularities near the centre of the correction
pattern. Though difficult to implement, we were able to achieve peak-to-peak variations in azimuthal intensity of approximately 50%, and could use this method in theactual system.
Manual correction
The two algorithms described above were instrumental steps along the way to achievingsmooth LG beams, but in the end we utilise neither. Our final correction procedure
is based on observations made during our attempts to implement these algorithms.
As shown in table 4.1, we can identify the addition of a given Zernike mode witha characteristic distortion of the LG beam (column 4), and correspondingly in theatom density (column 5). We can therefore approximately identify from the LG beam
shape and intensity profile which of the low order Zernike polynomials is present in
the aberrations and compensate by subtracting them from the correction pattern. The
general procedure is as follows
1. Careful alignment of the SLM optics is essential for getting sufficiently close to
implement any correction procedure. All optics should be kept clean and any
transmissive optics should be positioned as close to normal to the beam as possible. The quality of the initial beam on the SLM is very significant and any
fringing, or diffraction rings due to the fibre output package should be minimised.
2. Apply the desired phase winding, diffraction grating, and lensing to the SLM (Thecorrection is not universal and will not work for different windings or substantiallydifferent gratings and lensings.) Also ensure the supplied Hamamatsu correction
pattern is added which corrects for most of the aberrations.
3. By looking at an image of the light on the camera one can remove nearly all of
the Z(2, −2) and Z(2, 2) Zernike modes by walking the beam in the horizontal
and vertical direction. The target should be to produce an LG beam as close tocircular as possible.
4. Addition and subtraction of the eight Zernike modes listed in table 4.1 can bedone in real time by looking at the camera image and trying to produce as smoothan LG beam as possible. Using this technique we can achieve azimuthal peakto-peak variations in the intensity of . 10%. Use of higher order Zernike modes
is not necessary and identifying their presence is beyond our imaging resolution
and not feasible to do by eye.
5. Aberrations present in our imaging system mean the true ring intensity profile
differs slightly from that imaged on the camera. Therefore, as we are concerned894.2 Optical ring trap

Zernikemode

FunctionZ(2, −2)r2√
sin(2θ)6

Z(2, 2)r2√
cos(2θ)6

Z(3, −1)3 −2r
3r√sin(θ)
8

Z(3, 1)3 −2r
3r√
cos(θ)
8

Z(3, −3)r3√
sin(3θ)8

Z(3, 3)r3√
cos(3θ)8

Z(4, −4)4√r sin(4θ)
10Z(4, 4)4√r cos(4θ)
10Phase profile

Effect

on

LG

beam

(theory)

Observed
effecton

atoms

Table 4.1: Table of Zernike polynomials: The first three columns list the Zernike polynomials
which we use to correct for aberrations, including the functional form and phase profile. The
fourth column is a numerical calculation to illustrate how addition of the corresponding Zernike
polynomial to the SLM phase modifies the LG intensity profile at the focus. The fifth column
is an absorption image of atoms loaded into the corresponding LG beam. This distinctive
mapping between atomic density profile and Zernike polynomial allows us to manually subtract
excess Zernike orders, and hence cancel aberrations and achieve a smooth density profile.

with optimising the true trapping potential, we now modify our correction patternfurther by identifying Zernike modes present in the actual atom density, as shownin column 5 of table 4.1. This stage is the slowest as correction cannot be done in90

4.3 Optical ring trap

real time as a new BEC must be loaded into the ring trap and destructively imaged
in situ to measure the effect of modifying the correction pattern. Typically within
20 iterations we estimate azimuthal peak-to-peak variations in LG intensity are
reduced to < 10%. Due to diffraction effects discussed in Section 3.9.3, to obtaina true measure of the density profile it’s important to image on resonance, andhence we use partial transfer absorption imaging to reduce the optical density[79].
While laborious, this manual correction procedure is necessary for producing smoothring traps needed for sustaining superflow with very low atom density or in deep ringtraps using a high intensity LG beam.

4.2.6

Loading procedure

The normal criteria for loading a trap are maximising atom number transfer and minimising heating. When loading our atoms from the CDT to our ring trap, these criteria
are satisfied for a range of loading procedures, however we must consider a third,
stricter criterion. Due to the multiply-connected geometry of the ring trap, any oscillations induced during transfer which would normally dissipate and contribute toheating, can now lead to unwanted, long-lived superflow. Our initial loading procedure
was done over 2 seconds, as no visible gains were observed for longer loading times.
When we then impart angular momentum, ~`, to the atoms by a two-photon Raman
transition (see Section 4.3) we observe a scatter of angular momentum states with amean magnitude of ~(` + 1). To prove this scatter was induced at loading we reverse
the frequencies of the Raman beams, hence reversing the ordering of absorption andstimulated emission, and thereby imparting angular momentum −~` to the atoms. The
observed angular momentum states retain the same scatter but the mean magnitude is
reduced to ~(` − 1). Hence we conclude the loading procedure quasi-randomly stirs thecondensate with an average angular momentum +~. We therefore increased the loading time to 6 seconds as shown in figure 4.14 and no longer observed any mechanicalstirring on loading.
Unfortunately we observe a slow drift in the position of our LG beam when theexperiment is running. Typically the experiment needs to run for over an hour (approximately 40 cycles) before the LG beam stabilises, by which point the beam has
moved approximately 30 µm at the atoms from its original starting position. The magnitude of this motion is sufficiently large so as to misalign the LG beam with respect tothe sheet potential and create a pooling of atoms to one side of the ring trap. The originof this drift could not be discovered, so instead the system is allowed to “warm-up” fora couple of hours before overlapping the ring and sheet potentials, by which point theresidual drift is much smaller and slower.

91

4.3 Imparting angular momentum

4.3

Imparting angular momentum

Traditionally, orbital angular momentum has been transferred to BECs by some formof mechanical stirring such as by a laser beam or an optical lattice. Such methods
have proven successful for many studies, including the creation of large vortex lattices[31, 32], critical velocity studies [138–140], and the observation of vortex dynamics andnucleation [141]. In general however, mechanical methods impart an unknown amountof orbital angular momentum to the system and cannot be used to deterministically
prepare states of a given circulation. Recently an exception to this was achieved in[102], where a well defined and deterministic ladder of circulation states was observed
as the rotation frequency of a rotating weak link was increased.
Another class of methods exist which involve imprinting a well defined phase winding onto the BEC wavefunction to deterministically prepare a state of known circulation. The first generation of a vortex in a BEC employed such phase engineering byusing a rapidly rotating Gaussian laser beam to couple the external motion to internal
Rabi oscillations, allowing creation of a state with exactly one unit of circulation [30].
Doubly quantised vortices have also been topologically imprinted in condensates by
inversion of the axial bias field in a Ioffe-Pritchard magnetic trap [89, 142]. As the bias
field is inverted, the atomic spins adiabatically follow the field direction, remaining inthe same state with respect to the local magnetic field, but transitioning to a different
magnetic state with respect to the fixed lab frame. The nature of the field rotation
varies azimuthally and hence a topological phase factor is imprinted on the atoms which
can be interpreted as a manifestation of Berry’s phase [143]. This technique has also
been used to generate novel spin structures such as coreless vortices [144] and Skyrmion
states [145].
As discussed previously, LG beams carry well defined quanta of orbital angularmomentum along their direction of propagation, associated with their azimuthal phasewinding. As a result, several experiments have made use of this feature by coherently
transferring atoms with such a beam, and thereby imparting this angular momentum
to the atoms, or equivalently, imprinting the phase winding onto the condensate. To
coherently transfer the atoms to a stable state, these experiments use two-photon transfers, which avoid population of the short-lived excited state and return the BEC to thestable ground state, as explained in Section 3.2.3. The total angular momentum transferred in such a transition is then the difference between the two beams, which for theGaussian (G) plus LG beam configuration is simply `~ per atom, where ` is the angularmode of the LG beam. For efficient transfer and to prevent multi-photon processes, theinitial and final state have to be energetically separated from one another. The energy
associated with rotation is typical of the order of a few Hz and is therefore insufficient
to prevent additional coupling of the final state.
The first experimental observations of such transfers are reported in [99, 146]. These

92

4.3 Imparting angular momentum

Figure 4.12: Scheme for two-photon Raman transfer using G plus LG beams: The two Raman
beams are composed of our trapping LG beam which forms the ring trap, and a second, broad
Gaussian beam, co-propagating in the vertical direction. The basis of states we transfer between
are the mF = 1 and mF = 0 magnetic states of the F = 1 hyperfine level which are energeticallyseparated by ωeg ≈ 7 MHz in an external bias field of ≈ 10 G orientated in the plane of the
sheet trapping beam. The frequency difference between the G and LG beam is equal to the
Zeeman splitting, ωeg , plus the two-photon detuning, δ. The Raman beams are both far red
detuned of the D1 and D2 transitions by an amount ∆D1 and ∆D2 respectively to prevent
population of the upper state. With respect to the field the G beam is π polarised and the LG
beam is equal parts σ + and σ − polarised.

experiments were similar to Bragg diffraction, in that the initial and final states aredifferent momentum states of the same atomic state differing by 2~k, coupled by acounter-propagating Gaussian and LG10 beam. Atoms which undergo the transfer absorb a photon from one beam and stimulatedly emit a photon into the second beamacquiring both linear momentum of 2~k, and orbital angular momentum of ~. The
recoil energy associated with this linear momentum, Er = 4(~k)2 /2m, is sufficiently
large to energetically separate the initial and final states, however the linear momentum
kick causes oscillations of the final state within the trap. This problem is circumvented
in [99] by initially Bragg scattering the atoms into the 2~k state, and then after halfa trap oscillation period, the LG plus G transfer removes this linear momentum andimparts orbital angular momentum.
For studying the decay of persistent currents it’s imperative that we prepare thesame circulation state every time. Therefore we employ the phase-engineering approachdescribed above, utilising the phase profile of an LG beam. Our method involves Raman coupling two different magnetic sublevels of the 52 S1/2 , F = 1 hyperfine state,|F = 1, mF = 1i and |F = 1, mF = 0i, using co-propagating LG and G beams, as illustrated in figure 4.12. This method is similar to that used in [100] and [147]. Unlike[100], we use our LG trapping beam as one of our Raman beams, automatically ensur-

934.3 Imparting angular momentum

Figure 4.13: Protocol for imparting rotation to ring shaped BEC: (a) While still in the CDT
the bias field of 10 G is applied to give time for the field to stabilise (step 3 in figure 4.14). (b)As described in Section 4.2.6 the BEC is loaded into the ring trap, where the LG beam acts
as both trapping beam and one of the Raman beams (steps 4-6 in figure 4.14). (c) The second
Gaussian Raman beam is flashed on for ∼ 200 µs. The atoms undergo a two-photon π pulse to
the |1, 0i state, acquiring `~ units of orbital angular momentum. (d) The small, non-rotating
remnant (≈ 5%) left in the |1, 1i state is immediately microwave π pulsed to the |2, 2i state in≈ 40 µs. (e) A short imaging pulse (≈ 10 µs) scatters the |2, 2i atoms out of the trap leaving
only |1, 0i atoms rotating in the ring trap. Steps (c)-(e) are all performed sequentially in step
7 in figure 4.14.

ing maximal spatial overlap between the BEC and the Raman beams. A second broad
Gaussian beam couples the two mF states, with both beams far red-detuned from boththe D1 and D2 transitions to minimise spontaneous emission and heating. The mF = 1
and mF = 0 states are energetically split by ωeg ≈ 7 MHz by the application of a uniform bias field of about 10 G. The frequency difference between the Raman beams istherefore equal to ωeg plus the two-photon detuning, δ, which we aim to equal zero.
As discussed in Section 3.5, at this field the Zeeman splitting E(|1, 0i → |1, 1i) differsfrom E(|1, −1i → |1, 0i) by 14.4 kHz, hence provided the two-photon Rabi frequency issignificantly smaller than this energy difference, the mF = −1 state can be considered
far detuned and the F = 1 manifold reduces to a two state, or spin-1/2 system. The
polarisation of the Gaussian beam with respect to the field direction is π, and the LG
is equal amounts of σ + and σ − polarisation. Due to the energy difference of the beams
only the σ + polarised light contributes to the Raman transition and hence the twophoton Rabi frequency, however both polarisations contribute to the dipole potential
and inelastic scattering.

4.3.1

Rotation procedure

The rotation protocol used in Chapters 5 and 6 is outlined in figure 4.13.
94

4.3 Imparting angular momentum(a) While still in the CDT the states |1, 1i and |1, 0i are Zeeman split using theantibias1 field to apply a uniform field of 10 G. Early application gives time forthe field to stabilise and eddy currents to dissipate during loading of the ringtrap.
(b) The BEC is then loaded into the ring trap, where the trapping LG beam also
acts as one of the two Raman beams.
(c) The second Raman beam, a broad co-propagating Gaussian beam, is flashed on
two-photon resonance (δ = 0) for a time tπ = π/ΩR ≈ 200 µs, performing a
π pulse and flipping the state vector from pure |1, 1i to pure |1, 0i. The atoms
absorb a photon from the Gaussian beam and stimulatedly emit one into the LG
beam, acquiring orbital angular momentum −`~, but no linear momentum. The
short pulse time and broad Gaussian minimise any oscillations induced by thesudden change in the dipole potential.
(d) Uncertainties in the Raman beam power and field detuning mean a small fraction(≈ 5 − 10%) of the atoms remain in the |1, 1i state. This small component isfound to fundamentally reduce the stability of persistent currents and has to be
removed. This is done by first transfering this remnant to the |2, 2i state using
a microwave π pulse of tπ ≈ 40 µs. The high Rabi frequency of the microwave
coupling dominates any fluctuations in detuning and ensures perfect transfer.
(e) We then apply a short resonant imaging pulse of ≈ 10 µs which scatters the |2, 2i
atoms out of the trap without affecting the rotating |1, 0i state.
Following this rotation protocol the LG beam intensity is then typically ramped to its
final value over 3 seconds. The complete sequence diagram is shown in figure 4.14. We
note that cleaning up the non-rotating remnant is of vital importance to the stability of
the resultant superflow. This is in contrast to the results of Chapter 6 where we study
the stability of co-rotating mixtures and find all mixtures are stable for a few seconds.
Here we find that due to the relative rotation of the remnant in |1, 1i and the bulk in|1, 0i any instability manifests itself within about 100 ms, and so we must remove theremnant immediately following the Raman transfer.
A limitation of this setup is that the transfer procedure prepares a BEC with circulation equal to `, the winding of the LG beam. To change the prepared circulationof the BEC we change the azimuthal mode of the LG beam, and hence change thetrapping potential. Direct comparison of BECs with different initial phase winding inthe same trapping potential is therefore not possible.
1

Power supply: Delta Elektronika SM 18-50

95

4.3 Imparting angular momentum

Figure 4.14: Experimental protocol for preparing a persistent current: Red lines show the
ODT power, purple lines show the antibias field, blue lines show the sheet beam power, andgreen lines show the LG beam power. The stages of the protocol are as follows: (1) 7 s:
evaporative cooling in CDT by ramping power down to 2.5 W (Section 3.7). (2) 200 ms:
adiabatic state transfer |2, 2i → |1, 1i (Section 3.8) and clean up remnant |2, 2i atoms. (3) 200
ms: ramp antibias field to final value of 10 G for state transfers. (4) 1 s: ramp up sheet power
to 100 mW. (5) 1 s: ramp down CDT. (6) 5 s: hold in sheet only for 1 s, then ramp up LG
beam over 3 s and hold for 1 s. (7) ≈ 200 µs: two-photon raman transfer |1, 1i → |1, 0i to
rotating state. (8) 3 s: ramp LG power to final hold value.

4.3.2

Raman wavelength

The absolute wavelength of the Raman beams is important for determining not only
the Rabi frequency, but also the dipole potential, Udip , and the inelastic scattering
rate, Rsc . This is especially important in Chapter 7 where we consider continuous
application of the Raman coupling. Using the far detuned limit for the dipole potential
and scattering rate derived in (3.42) and (3.43) respectively, and the form of the twophoton Rabi frequency, ΩR , given in (3.30), the relevant parameters of interest are
Udip =X ~(Ω2LG,D1 + Ω2G,D1 )
4∆D1

D1

Rsc = Γ4∆2D1

X ΩLG,D1 ΩG,D1
D1

2∆D1

X ~(Ω2LG,D2 + Ω2G,D2 )
D2

X (Ω2LG,D1 + Ω2G,D1 )
D1

ΩR =

+

+

+

4∆D2

X (Ω2LG,D2 + Ω2G,D2 )
D2

4∆2D2

X ~ΩLG,D2 ΩG,D2
D2

,

2∆D2

,

(4.13)!
,(4.14)(4.15)

where Ωi,j is the Rabi frequency for the single beam i = {G, LG} for the transitionPP
j = {D1, D2}, and D1 and D2 are sums over the excited states of the D1 andD2 transitions respectively. Here we have explicitly considered the D1 and D2 lines964.3 Imparting angular momentum

separately as we will use a single-photon detuning comparable to the energy difference
between the two transitions.
To calculate the Rabi frequency we sum over all allowed upper states in the twophoton |1, 1i → |1, 0i transfer, including both D1 and D2 transitions. The appropriate
dipole matrix elements are found in [76] and lead to the expression
"
r
r r
r r !
ILG IG 2| hJ = 1/2| er |J 0 = 1/2i |2
1 1
1
1
ΩR =
−
+
2 ~2 0 c
2∆D1
4 4
12 12
r r
r r !#| hJ = 1/2| er |J 0 = 3/2i |2
5
5
1 1
+
.
−
2∆D2
24 24
8 8
Here we have included a factor of(4.16)p
1/2 as only half the LG polarisation is of thecorrect handedness, and we assume we are sufficiently detuned that we can neglect
energy splittings within the hyperfine levels. We can simplify the expressions for the
scattering rate and dipole potential by using the relation [76]hJ| er J 0 ≡ hLSJ| er L0 S 0 J 0(
J 0 +L+1+S= hL| er L0 (−1)

p(20 + 1)(2L + 1)

L

L0

1

J0

J

S

)
,

(4.17)

where the final term is a Wigner 6 − j symbol. Using this we relate the two dipole
matrix elements
hJ = 1/2| er J 0 = 1/2 = hL = 0| er |L = 1i√hJ = 1/2| er J 0 = 3/2 = 2 hL = 0| er |L = 1i .

(4.18)(4.19)

Making use of the relationship between linewidth and dipole matrix element
Γ=

3
ωD1
|d|2 ,
3π0 ~c3(4.20)

where d = hL = 0| er |L0 = 1i is the matrix dipole element and ωD1 is the D1 transition frequency, we obtain the general expressions for the dipole potential and inelastic
scattering rate [58]πc2 ΓUdip (r) =
3
2ωD1




2 + P gF mF
1 − P gF mF
+
I(r)
∆D2
∆D1


πc2 Γ2 2 + P gF mF
1 − P gF mF
Rsc (r) =
+
I(r),
3
2~ωD1
∆2D2
∆2D1

(4.21)(4.22)

where P characterise the laser polarisation (P = 0, ±1 for linearly and circularly σ ±polarised light respectively). Our Gaussian light is π polarised (P = 0) and the LG
light is equal σ + and σ − polarised, hence the dipole potential (4.13), scattering rate97

4.3 Imparting angular momentum(4.14), and Rabi frequency (4.15) for the Raman setup simplify toπc2 Γ
Udip (r) =
3
2ωD1


1
2
+
(ILG (r) + IG (r))∆D2 ∆D1
πc2 Γ2
2
1
Rsc (r) =(ILG (r) + IG (r))
+
3
2~ωD1
∆2D2 ∆2D1

r
πc2 Γ
1
1
ILG (r)IG (r)ΩR (r) =−.
3
2
2π~ωD1 ∆D2 ∆D1(4.23)(4.24)(4.25)The Rabi frequency, dipole potential, and scattering rate are plotted in figure 4.15
as a function of Raman beam wavelength. Both the scattering rate and Rabi frequency
scale as 1/∆2 , hence to minimise heating and decoherence effects, we wish to detune
far from single-photon resonance, so long as the laser power is still sufficient to drive
the transition. The lower threshold for the Rabi frequency is limited by the field stability. As shown in (3.11), the BEC undergoes Rabi oscillations at an effective frequency
which is dependent on the on-resonance Rabi frequency, ΩR , and the detuning fromresonance, δ. Therefore to accurately perform a π pulse, ΩR must dominate fluctuations in the detuning. We empirically estimate our field fluctuations to be ≈ 300 Hz,
and hence aim for a Rabi frequency of about 3-4 kHz.
The optimal configuration to maximise the Rabi frequency is to have equal intensities in the LG and Gaussian beams, however residual roughness in the LG beam profile
demands that to maximise superflow stability we must use the minimum LG power
required to create a multiply-connected geometry. This will also minimise the heating
rate while the superflow persists in the ring trap, and hence maximise the BEC lifetime.
The LG power required to produce a ring trap is only on the order of tens of µW, hence
to achieve high Rabi frequencies we need the majority of the power in the Gaussianbeam. To minimise the dipole effect produced during the π pulse when the Gaussian
beam is flashed on, a large Gaussian beam waist of ≈ 100 µm is used to minimise theforce from gradients in the dipole potential. Based on these criteria we chose a Raman
wavelength of λ = 804 nm which produces a low scattering rate. The power in the LG
beam is limited to tens of µW by the requirement of a smooth ring potential, and hence
to compensate for the low Rabi frequency at this wavelength we use several hundred
mW of power in the Gaussian beam.
We also note that due to interactions with both D1 and D2 lines, a special point
exists at λ = 790 nm where the dipole potential vanishes, but the Rabi frequency remains large. This has potential applications for achieving strong Rabi coupling without
modifying the trapping potential of the atoms.

98

4.3 Imparting angular momentum

Figure 4.15: Effect of Raman beam wavelength on dipole potential, two-photon Rabi frequency, and inelastic scatting rate: The left plot illustrates the effect of Raman wavelength on
dipole potential, which scales as 1/∆ for large detuning. The middle plot shows the two-photon
Rabi frequency and the right plot shows the inelastic scattering rate, both of which scale as1/∆2 far from resonance.

4.3.3

Raman beam setup

We generate the two Raman laser beams using the setup illustrated in figure 4.16. The
laser light is produced using a Ti:Sapphire laser1 pumped with an 18 W green laser2 .
The Ti:Sapphire is a CW laser with tunable wavelength over the range 700 to 1030
nm, providing complete freedom in the wavelength of our Raman beams. Importantly
the rms linewidth of the laser is < 75 kHz, equivalent to a coherence length of over 4
km. This ensures coherence between the two Raman beams regardless of their relative
path difference. The large output power of the Ti:Sapphire of up to 4 W permits us toachieve high Rabi frequencies and compensates for significant losses within the dualpass AOMs and fibre coupling.
We create our two Raman beams of orthogonal polarisation and relative frequency
difference ωeg , by passing the light through an 80 MHz AOM3 and splitting it off a
PBS cube into two beams of orthogonal polarisation with relative power controlled bya half-waveplate. We then shift the frequency of each of these beams using a dual-pass
AOM4 setup, with the Gaussian beam frequency shifted by 2 × 80 MHz and the LG
beam frequency shifted by 2 × (80 MHz−ωeg /2), thus matching the two-photon resonance condition. Using two dual-pass AOM setups allows us to vary the frequencyof the Raman beams without altering the alignment into the fibre. To ensure phasecoherence between the two beams we drive the two AOMs by function generators5 connected in a phase-lock-loop configuration, where one acts as the master oscillator forthe other. To provide precise control of the AOM switching times needed for π pulses,
we use an RF switch6 to control the signal to each AOM.
Power stabilisation of the two Raman beams is achieved by picking off weak reflec1

Coherent Ti:Sapphire MBR-110
Coherent Verdi V18
3
Crystal Technology Inc. 3080-125
4
Crystal Technology Inc. 3080-125
5Agilent 33250A
6
Mini-Circuits ZYSWA-2-50-DR
2

99

4.3 Imparting angular momentum

Figure 4.16: Overview of Raman laser system: The Raman laser light is generated by an
optically pumped tuneable wavelength Ti:Sapphire laser capable of outputting up to 4 W oflaser light. An AOM provides switching for both Raman beams and power stabilisation for
the Gaussian beam. The relative power distribution between the LG and Gaussian arms iscontrolled by a half-waveplate before the first PBS cube. Both LG and Gaussian light thengo through a dual-pass AOM, shifting the frequency of the Gaussian light by 2 × 80 MHz andthat of the LG light by 2 × (80 MHz−ωeg /2) to match the energy difference between the |1, 1i
and |1, 0i states. Both arms include microscope slides which pick off weak reflections towards
photodiodes used for PID-based power stabilisation. The frequency shifted Raman beams arethen recombined on the PBS cube and are coupled into a polarisation-maintaining fibre. The
output of this fibre is that shown in figure 4.7.

tions towards monitor photodiodes using microscope slides inserted in each beam path.
The photodiodes provides a signal for a PID which alters the AOM efficiency to matchthe monitor signal to a reference signal provided by the control software. The same
setup is used to stabilise the CDT (Section 3.7) and sheet beam (Section 4.2.1) powers. The LG beam power is stabilised by changing the LG frequency AOM efficiency
while monitoring the light level after the dual-pass AOM. This cannot be done for the100

4.3 Imparting angular momentum

Gaussian arm, as the G frequency AOM is only flashed on for a few 100 µs during theRaman π pulse which is too fast for the PID to respond to. We therefore only stabilise
the power incoming to the Gaussian dual-pass AOM by altering the efficiency of thefirst AOM which is common to both Raman beams.
After passage through the dual-pass AOMs the two polarisations are then recombined on the PBS cube and coupled into a single-mode polarisation maintaining fibre1 .
To ensure the polarisations of the different frequency components remain orthogonal
the input polarisation is aligned with the axis of the fibre using a half-waveplate. While
we do observe some contamination of the polarisation for each frequency at the fibre
exit, this is on the order of a couple percent and has no discernible effect on the Raman
transfer.
The two Raman beams exit the fibre shown in figure 4.7. As described in Section
4.2.3 the LG beam with horizontal polarisation and frequency shifted −ωeg with respect to the Gaussian beam is sent towards the SLM. The Gaussian beam with verticalpolarisation and frequency shifted +ωeg with respect to the LG beam is telescoped
down ≈ 10 : 1. This produces a small beam which focuses to a large beam at theatoms. This minimises the optical force exerted during the π pulse and helps ensure
the Rabi frequency is equal at all points around the ring. The vertical imaging light isoverlapped with the Gaussian beam on a 90 : 10 glass sampler inserted in the Gaussianpath, thus discarding 10% of the Gaussian light and 90% of the vertical imaging light.

4.3.4

Coherent Raman Rabi oscillations

To test the suitability of our Raman beams for performing coherent Rabi oscillations wefirst used two Gaussian beams in a modest field of 1.67 G, corresponding to a Zeeman
splitting of 1.17 MHz. At this field the splitting between the |1, 1i, |1, 0i, and |1, −1istates are essentially equal and the Raman beams couple all three states. Our initialattempts at Rabi oscillations are illustrated by the plot in figure 4.17 (a), where weobserved clear decoherence and the system quickly tended towards equal populations
in all three states. The phase coherence between the Raman beams was checked byinterfering the two Raman beams and observing phase stability well below 1 Hz.
The cause of the decoherence was indicated by the absorption image in figure 4.17(b), which is an 18 ms TOF image of the Raman-coupled BEC without a Stern-Gerlach
field applied. We observe two satellite lobes above and below the BEC, the position ofwhich is not due to any magnetic gradient acting on different magnetic sublevels. The
position of these lobes is consistent with atoms receiving a momentum impulse of ±2~kduring the Raman coupling, hence the co-propagating Raman beams were able to Bragg
scatter some fraction of the BEC. This is consistent with the Raman beams interactingwith a weak counter-propagating reflection, enabling coupling to different momentum
states. In addition we observed significant heating and atom loss when applying Raman
1

Thorlabs P3-630PM-FC-10

101

4.4 Imparting angular momentum

Figure 4.17: Plots of both incoherent and coherent Raman Rabi oscillations: (a) Strongly
damped, incoherent, Raman Rabi oscillations are observed due to weak counter-propagatingreflections coupling different momentum states. (b) 18 ms TOF horizontal absorption imageof the BEC following Raman coupling. The position of the small clouds above and below
the BEC are consistent with the momentum states ±2~k. Such momentum states are due to
Raman coupling from the Raman beams and weak counter-propagating reflections from thevertical imaging camera, and are absent if the camera path is blocked. (c) Coherent Raman
Rabi oscillations with the vertical camera reflection blocked. The data is fitted to a three-statesolution of the rate equations, from which we extract the two-photon Rabi frequency, Ω = 20
kHz, and the two-photon detuning, δ = 1.8 kHz.

coupling near resonance. The problem was solved by placing a shutter in front of thevertical imaging camera which was the main source of the reflection. A short-pass filter(λ < 790) nm was also placed in front of the camera to transmit imaging light but
block Raman light which would otherwise saturate the camera.
The resultant Raman Rabi oscillations are shown in figure 4.17 (c). Without
coupling to different momentum states we now observe coherent population transfer
over multiple cycles. Fitting this data with the appropriate three state model, we
find a Rabi frequency of ≈ 20 kHz and a two-photon detuning of ≈ 1.8 kHz. Similar
coherent transfer is observed for our Gaussian and LG setup with lower Rabi frequency,
permitting use of a coherent π pulse to impart angular momentum to the BEC.

102

4.4 Detecting angular momentum

4.4

Detecting angular momentum

Following our rotation procedure we transfer over 90% of the atoms to the |1, 0i state,
which flows around the ring trap with orbital angular momentum `~ per atom. Fundamentally though, the atomic density observed is identical to the stationary case,and therefore measuring the angular momentum present can be challenging. Thisis especially true when considering the energy associated with the rotation is Er =(`~)2 /(2mR2 ), where m is the atomic mass, and R is the radius of the ring. For ourtypical trap parameters (R = 12 µm) this only corresponds to Er /h ≈ 0.4`2 Hz.
Circulation in a BEC is usually detected by a kinematic method, where an absorption image is taken at long TOF and density dips are observed associated withvortices. The characteristic size of a vortex in trap is equal to the healing length,pξ = 1/8πna as derived in (2.64), which is typically less than a µm, and hence cannot
usually be resolved by an in situ image. By allowing the condensate to freely expand
before imaging, the velocity associated with the azimuthal flow around a vortex corecauses the hole to expand to a size above imaging resolution. The presence of a vortex
in a condensate has also been detected by measuring the resultant precession in theaxes of the quadrupole mode in the direction of the circulating vortex flow. The circulating velocity breaks time-reversal symmetry and splits the two otherwise degeneratemodes which contribute to quadrupole oscillations. The magnitude of the splitting isdependent on the angular momentum present in the system and can be directly inferredfrom the rate of precession of the mode [38]. The phase winding present in a rotatingcondensate has also been directly observed using matter-wave interference [30, 33].
Here we detect the circulation present in our ring BEC by both a matter-wave interference method, and by observing the formation of density dips at long TOF. Dueto the necessity of a non-rotating, reference state for the interferometric method, thequantitative studies on persistent currents presented in Chapters 5 and 6 exclusively
use the kinematic detection method.

4.4.1

Matter-wave interference

The procedure for converting the phase winding to a density modulation by matterwave interference is shown in figure 4.18 (a). The initially stationary ring BEC in the|1, 1i state is coherently transferred to an equal superposition of the rotating |1, 0i andstationary |1, 1i states by a Raman π/2 pulse. A subsequent RF π/2 pulse mixes thetwo states, creating an equal superposition of stationary and rotating atoms in eachspin state. The resultant interference creates a density modulation, with the numberof interference peaks equal to `. Imaging just one of the spin states using a microwave
transfer to the F = 2 state, we obtain the absorption images shown in figure 4.18(b), confirming the phase imprinted by the LG plus G Raman transfer. The density
modulations of the two spin states are π out of phase, hence imaging the total density103

4.4 Detecting angular momentumof both states by repumping to the F = 2 state, we only observe the initial smoothdensity profile.
We can quantitatively explain this procedure by describing the BEC in a coherent
superposition of internal states 1 and 2, labeled by the two-component vector!
Ψ1 (θ)
Ψ̄(θ) =,
Ψ2 (θ)
which for our initial state is1Ψ̄0 (θ) = √2π

10(4.26)!
.

(4.27)

We can then define the Raman π/2 and RF π/2 operations by the matrix operators
1
U(Raman),π/2 = √
2

1

e−`θ

e−i`θ

−1

!
,

and

1
U(RF),π/2 = √
2

1

1

!

1 −1

(4.28)

respectively. With this notation the final state of our interference protocol isΨ̄(θ) = U(RF),π/2 U(Raman),π/2 Ψ̄0 (θ)!
1 + e−i`θ1= √.
2 2π 1 − e−i`θ(4.29)(4.30)

The final density distributions of the two spin states are then given by1
sin2 (`θ/2)
2π
1
cos2 (`θ/2),n2 (θ) = |Ψ2 (θ)|2 =
2π

n1 (θ) = |Ψ1 (θ)|2 =(4.31)(4.32)

in agreement with our experimental findings. In principle one can include a delay
between the Raman and RF pulses. The number of density fringes is then a measureof the circulation which persists beyond the delay time, and the observed precessionof the fringes is sensitive to rotation of the frame of reference, acting as a Sagnac
interferometer. In reality we observe a rapid degradation of the interference pattern
for delays longer than 50 ms due to the presence of two effectively counter-rotating
spin states. This is true even with the inclusion of an “echo” RF π pulse at half thedelay time, which swaps the spin states and should negate the accumulated relative
phase between the two states1 . This is in contrast to the calculations of [111] where thefringe pattern is predicted to be stable due to the relatively close inter and intra-state
scattering lengths. The effect of having two spin states in the ring is discussed further
in Chapters 6 and 7.
1

Exact cancelation of the accumulated phase is only expected in the single mode approximation
where only the zero energy mode contributes [148]104

4.4 Detecting angular momentum

Figure 4.18: Interferometric detection of the imprinted phase winding: (a) Interference protocol for interfering a rotating BEC with a stationary BEC. A Raman π/2 pulse coherently
transfers only half the population into the rotating |1, 0i state. A subsequent π/2 RF pulse,which carries no angular momentum, mixes the states so that in each spin state we get an interference of rotating and non-rotating atoms. This matter-wave interference converts the phasewinding into a density modulation in each spin state, with the number of density peaks equalto `. (b) Absorption images of the original ring density profile and the resultant pattern in the|1, 1i state following the interference protocol for ` = 3, 5, 10, and 15. Bottom plots show theatom density of the interference fringes as a function of the azimuthal coordinate, normalised
to the original atom density.

4.4.2

Kinematic detection in time-of-flight

As discussed in Section 2.4.3, the flow velocity about a vortex scales as 1/r and hence
is associated with an angular momentum barrier, ~2 `2 /2mr2 , leading to a density singularity along the vortex core. Due to the mean-field energy of the cloud in trap,
the atoms can climb this 1/r2 potential and acquire kinetic energy as tangential flow
velocity. When the BEC is released in TOF the mean-field energy is given up to ballistic expansion, however the angular momentum barrier due to circulation persists andcauses the density dip due to the vortex core to expand and become resolvable. Numerical simulations of such expansion from a rotating annular BEC are done in [149],105

4.4 Detecting angular momentum

Figure 4.19: Kinematic detection of circulation in TOF expansion: (a) By ramping down the
LG beam and transforming to the simply connected sheet trap, the BEC pushes up against theangular momentum barrier. The size of the vortex hole, ξ, is set by equating the interactionenergy, given by µ, and the kinetic energy associated with rotation, given by ~2 `2 /2mr2 . Removing the trapping potential causes atoms to move away from the angular momentum barrier
and a central hole to form in TOF, the size of which depends on `. (b) Top view absorption
images of non-rotating (left) and ` = 3 rotating (right) BECs after 29 ms TOF expansion fromthe reconnected trap in (a). We use the fitted radius, R, to quantify the rotation of the cloud.
(c) Calculation of central density hole radius versus TOF for the circulation states ` = 3 (blue),` = 2 (green), and ` = 1 (red). This calculation assumes ballistic expansion of the cloud andcalculates the minimal radius at which the density is non-zero. (d) Azimuthally averaged radial
density profiles of the images in (b) for both the rotating (red) and non-rotating (blue) clouds.
This illustrates the sharp density cutoff due to rotation, which is in reasonable agreement withthe cutoff predicted by our simple calculation, R`=3 , indicated by the dashed black line.

where the size of the hole formed in TOF is shown to be dependent on `, and hence
can be used to detect the rotational state of the condensate in the trap.
In our ring trap the atoms are held far from the axis of rotation by the ring trapping
potential and hence have a low azimuthal flow velocity. For our typical parameters of
ring radius rM = 12 µm, and circulation ` = 3, the flow velocity is vs = `~/mrM ≈ 0.18
mm/s. When the atoms are released from the ring trap, the interaction energy is converted to kinetic energy and the atoms expand in all directions, including inwards
towards the axis of rotation. The initial radial velocity dominates over the initial tangential velocity associated with any rotation present, allowing atoms to approach very
close to the vortex core. While the density singularity at the centre never vanishes, this106

4.4 Detecting angular momentum

method requires very long TOF before the density dip due to rotation is observable.
In order to resolve the density hole from rotation we find it necessary to first gradually remove the LG beam and transform the trap into a simply-connected geometry
formed by the sheet trap alone. This has the advantage of bringing atoms close to
the axis of rotation and increasing the azimuthal flow velocity. It also reduces themean field energy, reducing the expansion velocity upon release, and since the verticalconfinement now dominates over the horizontal confinement, the mean field energy is
predominantly released in the vertical direction. The LG beam is ramped off slowly
over about 100 ms, before the sheet beam is turned off a few ms later. As discussed in
Section 4.5, once the ring trap is removed the rotating state is no longer protected andeventually decays. To gain an accurate measure of the angular momentum, the BEC
is only briefly held in this reconnected trap before the vortices have a chance to breakapart to lower-angular momentum states. The BEC is then imaged at 29 ms TOF along
the axis of rotation. Figure 4.19 (b) shows a typical absorption image of a non-rotating
BEC, and a BEC transferred to the rotating state using ` = 3. The non-rotating BEC
has a density maximum at the centre as expected, whereas the rotating BEC exhibitsa central density hole. We fit the high density ring surrounding this hole to an ellipse,
and quantify the rotation of the cloud by the geometric mean of the major and minor
axes, R. The circular symmetry of the density ring in TOF is found to be strongly
dependent on the alignment of the ring trap with the sheet trap. We therefore alignthe LG beam with the sheet trap minimum looking at the density ring in TOF, withthe aim of having circularly symmetric density.
Figure 4.19 (a) illustrates a BEC held in the reconnected sheet trap with a centrifugal barrier at the origin due to rotation with ` units of circulation. We can obtain areasonable estimate of the radius of the density hole due to rotation at long TOF byconsidering the expansion as ballistic, and neglecting the effect of interactions during
TOF. To first order this is correct since the vertical trapping frequency in the sheet
dominates over that in the plane, and therefore nearly all the mean field energy quickly
goes into the vertical expansion of the cloud. Equating the force due to the potential
gradient and the radial acceleration, we obtain the equation of motion for atoms at
position r~2 `2d2 r(t)=.
dt2
m2 r3 (t)(4.33)

Solving this for the initial conditions, r(t = 0) = r0 and dr(t)/dt = 0 gives
r2 (t) = r02 +~2 `2 2t .
m2 r02(4.34)

To illustrate the density profile we might expect to evolve during TOF we consider the
trajectory of the point in the trap, r∗ , which travels the shortest distance from the axis

107

4.5 Vortex dynamics in a connected geometryof rotation during time-of-flight, t. This if found by solving dr(t)/dr0 = 0:
 2 2 2 1/4
~ ` t
r =
.
m2
∗

(4.35)

In our simple picture, atoms located at r∗ in the trap end up closest to the axis ofrotation in TOF, with all other positions in-trap travelling further from the origin.
Using our expression for r∗ as the starting point r0 in equation (4.34), we obtain the√radius, R` (t) = 2r∗ , of the density hole in TOF within which the density is zero due
to rotation. We therefore find that the area of the hole formed in TOF is proportional
to the angular momentum state of the atoms, R(`)2 ∝ `. The radius R(`) is plotted
as a function of t for the circulation states ` = 3, 2 and 1 in figure 4.19 (c). From thiswe anticipate that at 29 ms TOF the zero-density hole will have a radius 6.5 µm, 9.2µm, and 11.2 µm for the circulation states ` = 1, 2, and 3 respectively. Calculating
the ballistic expansion of the initial Thomas-Fermi density for a BEC in the sheet plus
centrifugal barrier potential shown in figure 4.19 (a), we find that the density in TOF
monotonically increases with decreasing radius, until reaching the cutoff value R` (t).
In figure 4.19 (d) the azimuthally averaged radial density profiles of the rotating
and non-rotating clouds at 29 ms TOF are shown. The black dashed line indicates
our calculated density cutoff, R`=3 = 11.2 µm, which is in surprisingly close agreementwith the inflection point of the profile. Due to the effect of interactions during TOF
which we neglected, this cutoff is rounded off and the peak density is actually shiftedto a slightly larger radius than R` . The size of this discrepancy is empirically found to
be a function of atom number, with the value of the peak radius for a given circulation
state decreasing with atom loss, and tending towards values in agreement with R` .

4.5

Vortex dynamics in a connected geometry

In the next chapter we unequivocally show that superflow in a ring shaped BEC isfundamentally stable and protected from decay due to the trapping geometry. We
first conclude this chapter with a brief discussion of vortex dynamics and decay in a
simply-connected BEC. Vortices are excited states of motion and therefore energetically unstable towards relaxation to the ground state where the condensate is at rest.
However, as discussed in Section 2.4.3, quantisation of circulation constrains the decay,
and a vortex in a BEC cannot simply fade away by continuously dissipating its energy. As calculated in equation (2.97) the energy of a multiply charged vortex scales as
the square of the circulation, and therefore it is energetically preferable for a vortex ofcharge q to break up into q singly charged vortices. It has been shown both numerically[150] and experimentally [89] that multiply charged vortices in a harmonic potential
are not energetically stable. Multiply quantised vortices are found to decay into singly
quantised vortices and transfer their kinetic energy to excitation modes. Such insta-

108

4.5 Vortex dynamics in a connected geometry

bility is referred to as dynamic instability and is driven by atomic interactions even atzero temperature, and is different from dissipation due to interactions with the thermal
component [151]. The timescales of such dynamic instabilities is fairly short, quickly
establishing the large vortex lattices observed in [31, 32].
Such multiply charged vortices can become stable under sufficient rotation of thesystem [152] or by the presence of localised pinning potentials [90]. For pinning locationsof sufficient size, the energy reduction of localising several vortices at the density dip
and creating a ‘giant’ vortex state can outweigh the cost of having multiple circulation.
Such multiply quantised vortices have been observed at pinning sites in superconducting systems [153] and BECs [91]. Equivalently, the large density hole we measure in
TOF in figure 4.19 (b) is essentially a multiply quantised vortex which forms due tothe pinning potential of our ring trap, where the vortices are pinned inside the ring
trap where the density vanishes. In [154], giant vortex states are prepared by applying
a blue-detuned laser to a vortex aggregate of many singly charged vortices which pins
many vortices creating a giant vortex core of up to 60 phase singularities. Other proposals for the formation of stable multiply charged vortices involve rapid rotation of a
BEC in a high power-law trap [150, 152].
A singly charged vortex is dynamically stable and therefore the vortex state canonly decay by annihilation with another vortex of opposite circulation or in the presenceof dissipative processes. At finite temperatures the vortex scatters from thermal excitations, transferring energy from the vortex to the thermal cloud causing the vortex tospiral outwards towards the border of the condensate where it then decays to elementary
excitations [155]. There have been several experimental studies on vortex dynamics,
including single vortices [156], small clusters of co-rotating vortices [89, 157, 158], andvortex dipoles [141] which are of particular interest for understanding the BerezinskiiKosterlitz-Thouless (BKT) transition [159].
In figure 4.20 we present our observations of the decay of a multiply charged vortex held in a reconnected trap with initial charge q = 5 and q = 3 for (a) and (b)respectively. We first prepare superflow in the ring trap using the rotation procedure
explained in Section 4.3.1, using LG beams of different azimuthal modes ` = 5 and` = 3 for the two sequences. The LG beam is then ramped down over 3 seconds andthe BEC is held in the simply connected sheet trap only. In the ring trap, motion ofvortices through the BEC has an associated energy cost, hence the circulating stateis metastable and does not decay. Once the ring trap is removed this protection is
also removed and the lowest energy configuration is for the vortices to redistributethemselves within the BEC. For both sequences the initial multiply quantised vortex
of charge q = ` takes about 5 seconds to break up into ` singly charged vortices, after
which these vortices move to the BEC edge and decay over the following 10 seconds or
so. The dynamics timescale for vortices to break apart and the dissipative timescale
we observe for vortices to decay are both quite long compared to those reported in109

4.6 Conclusion

Figure 4.20: Vortex dynamics in reconnected sheet trap: For both sequences we prepare
superflow in the ring trap with ` units of circulation, where ` = 5 and ` = 3 for (a) and (b)respectively. The LG beam power is then immediately ramped down over 3 seconds and therotating BEC is held in just the sheet trap for a variable time before we absorption image at
29 ms TOF. (a) The giant vortex with charge q = 5 persists for a couple seconds but quickly
distorts in shape. Within 7 seconds the giant vortex has completely broken up into 5 singly
charged vortices. Over the next 10 seconds the vortices gradually migrate to the cloud edge
and dissipate due to interactions with the residual thermal atoms. The final single vortex isfound to typically persist for over 20 seconds. (b) The same sequence is repeatedly for a giant
vortex with initial charge q = 3. The dynamics and decay in this case occur in a similar fashionand timescale to that observed for the q = 5 vortex.

[89, 158]. Several of the images exhibit pleasing symmetry which we could interpret as
the formation of stable vortex structures as reported elsewhere [157, 158], however thiswould require further statistics or real-time imaging as done in [156] to confirm.
For both cases we observe images where the giant vortex has broken up into singly
charged vortices, the number of which equals the initial circulation of the BEC, `. Apartfrom being a quasi-random process, this is not a true measure of the angular momentum state, L, of the BEC like the kinematic detection method in Section 4.4.2. As
explained in equation (2.96), L for a cylindrically symmetric BEC containing a vortex
is only equal to ~ per atom if the vortex is on the axis of symmetry. If the vortex moves
towards the BEC edge, although the circulation is still quantised, L decreases and isdependent on the vortex position. Both the number of vortices and their position are
dependent on L, so simply counting phase singularities in TOF cannot unambiguously
tell us the angular momentum state of the BEC in the ring.

4.6

Conclusion

This chapter has outlined the experimental implementation of our ring trap, and theprocedure we use for both preparing and detecting angular momentum states. Particular care has been taken to explain the aspects of the experiment we found integral to110

4.6 Conclusion

preparing a state of well defined and known circulation. This forms the backbone forour studies on persistent currents, in which we always infer any decay events by assuming the initial state of the BEC. We have provided a brief overview of ring traps andtheir uses in ultracold gases, and have illustrated their significance to the preservationof persistent currents by observing vortex dynamics and decay in a simply connected
trap. In the following chapters the ring trap is kept on and the methodology explained
above is used to prepare persistent currents and study their stability.

111

112

Chapter 5

Quantised decay of persistent currents
Having discussed how we prepare a persistent current of well defined initial angularmomentum in Chapter 4, in this chapter we study the stability and decay of suchsuperflow. To clarify our discussion we introduce the parameter q, where the phase of
the BEC wave function, Φ(r), winds around the ring an integer multiple, q, times 2π:Φ(r, θ, z) = Φ(r, z)eiqθ ,(5.1)

where we assume azimuthal symmetry of the condensate density. This is to distinguishq from `, the phase winding of the LG beam which forms our ring trap, and which is
initially imprinted on the condensate wavefunction. Hence at t = 0 the phase windingof the condensate and LG beam are equivalent, q = `, but as the superflow decays q
will decrease, and it is this evolution which we study.
The physical origin of supercurrent metastability is qualitatively illustrated in figure 5.1 (a). For N atoms held in a ring trap, the angular momentum for any atom
is quantised, however the average angular momentum per particle in general need not
be. For a superfluid gas, the presence of the condensate means that such quantisationis energetically preferred, and hence we can map the rotational state of a ring shaped
superfluid onto the parabolic washboard landscape in figure 5.1 (a) which depicts theenergy E of the superfluid system for different fixed values of the total angular momentum L1 . The local minima of E correspond to topologically distinct metastablestates with L/N = q~. A direct ∆q = 1 transition between two such minima involves a
discontinuous 2π phase slip in the condensate wave function, occurring when a singly
charged vortex crosses the annulus. The energy barrier between two adjacent minimacorresponds to the barrier felt from a vortex core when trying to move toward a regionof higher density, arising from the nonlinearity of the GPE.
The origin of such metastable energy minima at integer q can be demonstratedusing a simple toy model presented in [13]. Here we consider a weakly interacting Bosegas at T = 0 trapped in a narrow annulus with thickness much smaller than its radius
R. If we only considers occupation of the non-rotating state |0i, and the state with angular momentum ~ per particle, |1i, the energy of this system has the familiar Noziéres
1

At nonzero temperature the same picture holds but with E replaced by the free energy.

1135.0 Quantised decay of persistent currents

Figure 5.1: Origin of metastability and decay of supercurrents: (a) Energy landscape of aring-shaped superfluid. Local minima correspond to metastable states with quantised angularmomentum per particle, L/N = q~. Energy barriers correspond to placement of a vortex
within the superfluid, reducing the total angular momentum and modifying the condensate
wavefunction. (b) Decay between the discrete q states involves a vortex-mediated phase slip,
illustrated here for q = 5 → 4.

form [12]
E=

g
~2 N1
+
(N02 + N12 + 4N0 N1 ),
2
2mR
2V

(5.2)

where V is the system volume, R is the radius of the ring trap, g is the strength ofcontact interactions (2.32), and N0 and N1 are the number of particles in the |0i and |1i
states respectively. Hence in general, fragmentation of the condensate is energeticallyinhibited due to interactions, leading to energy minima at quantised angular momenta
for N1 = N and N0 = N , similar to that shown in figure 5.1 (a). For suitably weakinteractions the energy minimum at N = N1 disappears and the system cannot support
persistent currents, as was shown in [160, 161]. The criteria for stability of q = 1
supercurrents is then simply given by∂E<0∂N1 N1 =N
gN
~2
<
2
2mRr
V
2gN
vs <
,
mV

(5.3)
(5.4)
(5.5)

where we have used the form of the superflow velocity vs = ~/mR. From the expressionp
for the sound speed in a uniform system (2.62) c = gN/mV , we see, apart from√a factor of 2, this takes the familiar form of the Landau criterion with a criticalvelocity set by the speed of sound, vs < c. Such a simple calculation captures the
behaviour of our system surprisingly well, but we note that even at zero temperature,
due to interactions one should really consider the population of many higher angularmomentum rotation states.
More generally, a superfluid can, in principle, also shed angular momentum in waysthat break the L/N quantisation. Such a proposed decay mechanism is the formationof grey solitons which allow a continuous change in angular momentum [162]. Another114

5.1 Previous studies on supercurrents

way is condensate fragmentation, where the BEC splits into two or more condensateswith differing angular momenta per particle, however such a mechanism is expected tobe suppressed due to the factor of 2 in the interaction energy arising for distinguishable
particles. The dominant superflow decay mechanism depends on the system’s geometry,temperature, and the strength of interactions. Associated with this decay mechanism
is a critical velocity, vc , where for flow speeds vs > vc , the superflow is unstable in
the thermodynamic sense and decay occurs inevitably and on the millisecond scale. In
contrast for vs  vc the decay is strongly suppressed and the metastable superflow
can be almost perfectly stable, as for example observed in bulk superconductors. In
between these two extremes, metastable superflow should persist for much longer thanthe characteristic microscopic time scale of the physical system, but rare stochastic
decay events can still occur through quantum or thermal fluctuations [82–84].
In this chapter we study the supercurrent decay of condensates initially preparedin the q = ` = 3 state. We observe q = 3 superflow persisting for up to a minute ina multiply-connected trap (Section 5.3) and explicitly show that the supercurrent is
indeed quantised (Section 5.2). The persistent current therefore decays in a cascadeof quantised decay steps which unambiguously confirms that 2π phase slips are thesupercurrent decay mechanism. We find that the critical velocities for different q states
quantitatively agree with numerical simulations (Section 5.5). Our ability to resolve
quantised rotational states opens the possibility to study the dynamics of phase slips,
allowing us to observe both rapid q → q − 1 decay events for vs (q) ≥ vc (q), andstochastic phase slips for vs < vc (Section 5.6).

5.1

Previous studies on supercurrents

The persistent current in our condensate is analogous to electrical current flowing without resistance in superconductors, and dissipationless flow of superfluid 4 He in a toroidal
container. The stability and decay of supercurrents have long been studied in 4 He superfluids and thin-wire superconductors, and as yet are still not fully understood. Muchof the difficulty stems from the existence of several breakdown mechanisms, the relative importance of which depends strongly on the system parameters. In superfluid
helium experiments in a toroidal geometry, the critical velocities observed in practise
were often significantly less than the Landau value [163]. It is suspected however, thatin most cases this was due to the presence of a tangle of vortices which formed during
the quench through the BEC transition. For superfluid 4 He flow through small orifices,Anderson predicted that dissipation can occur when a quantised vortex passes acrossthe orifice [26]. If the vortex crosses all the flow lines passing through the orifice, thequantum phase difference across the hole changes by 2π. Such phase slips have since
been observed in several experiments [27, 88, 164]. One therefore anticipates the onsetof such phase slips is given by the Feynman critical velocity, vcF , (2.103) derived in Sec-115

5.1 Previous studies on supercurrents

tion 2.4.3, which gives the velocity above which vortex rings can enter the flow channel.
However, vcF neglects the nucleation mechanism and subsequent trajectory of such vortex rings and cannot give quantitative agreement with experimental observations [27].
In addition, experimental studies of critical velocities for phase slips in narrow channels
for high flow velocity [27, 88, 164] also show a temperature dependence which cannot be
explained by the Feynman criteria. In this regime the normal component is negligible,
and therefore it is proposed that the dependence arises due to the energy barrier for
nucleating vortices. Hence as the temperature decreases the effective critical velocity at
which decay occurs is observed to increase as the rate at which vortices are thermally
nucleated decreases. Below about 200mK the observed critical velocity becomes almost
temperature independent, and quantum tunneling dominates the phase-slip nucleation
process [88]. The fundamental theoretical challenge is then understanding the nucleation process, and as yet no theory connects all these observations [165–167].
Phase slips in the superconducting order-parameter have been shown to cause residual electrical resistance in superconducting nanowires [84, 85]. In analogy to superfluid
4 He, at higher temperatures phase slips between current-carrying states occur though

the process of thermal barrier-crossing by the order-parameter field [81], whereas at
low temperatures quantum phase slips occur by topological quantum fluctuations of
the superconducting order-parameter field [84]. Such superconducting nanowires haveimportant applications for realising qubits [168], and the use of coherent quantum phaseslips to build a current standard has been proposed in [169].
Persistent currents in annular BECs clearly offer a complementary route to further
understanding these experiments on superfluid 4 He and superconducting nanowires. In
contrast to liquid helium, since BECs are dilute and relatively weakly interacting, they
are often described very well by the mean-field GPE which provides a reliable theoretical model for studying the instability mechanisms and dynamics. Prior to this work,
the only realisation of supercurrents in a BEC was achieved in [99, 100], where q = 1
superflow persisting for up to 40 s was observed, and studies of flow through a weak linkcreated by a potential barrier revealed a well defined superflow critical velocity vc . The
observed vc was found to be consistent with the Feynamn estimate vcF , from which it
was suggested that the dominant supercurrent decay mechanism was a vortex-induced
phase slip, analogous to decay in superconducting and liquid 4 He experiments.
Our work extends these studies to multiply charged superflow (q > 1). This allows
us to prove supercurrent quantisation, and from this, we unambiguously confirm phaseslips as the decay mechanism, independent of any calculations. We calculate the flow
velocity at decay and show it to be consistent with the order of magnitude Feynman
estimate. Further to this, by explicit calculation of the excitation spectrum of the ring
BEC we find quantitative agreement with a different critical velocity, which more accurately predicts the decay in our system. These calculations suggest decay by formationof vortex-antivortex pairs due to phonon-like excitations. Our use of q > 1 superflow

1165.2 Supercurrent quantization

also opens the possibility to study dynamics and correlations between phase slip events.

5.2

Supercurrent quantization

To demonstrate the quantised nature of the supercurrent decay we initially prepare
the system in the q = 3 rotational state using the protocol explained in Chapter 4.
Following the transfer of orbital angular momentum to the BEC, we then increase thepower of the LG beam by a factor of 2.4 over 3 seconds to ≈ 200 µW. From this point
we then hold for various times and then release the cloud and absorption-image it after
29 ms of TOF. We detect the angular momentum state of the condensate by fitting
the radius of the density ring, R, due to rotation, as explained in Section 4.4.2. The
purpose of increasing the ring power is to increase the significance of the residual azimuthal roughness due to imperfections in the profile of our LG beam. The potential
barriers due to roughness will scale linearly with LG intensity, ILG , whereas the chemical potential, µ, only scales as ILG 1/4 through the radial trapping frequency, hence
the superflow stability decreases with increasing LG power. This has the advantage ofsignificantly reducing the supercurrent lifetime to below the atom number decay time.
As discussed in Section 4.4.2, the fitted hole radius in TOF, R, has a weak dependence
on atom number, N , so increasing the ring power minimises this effect, providing uswith the cleanest signal in which to observe the quantised nature of the supercurrent.
In figure 5.2 (b) we plot the evolution of the radius R with time after the superfluid
was set into rotation. The quantisation of R, and therefore of the angular momentumof the condensate, is strikingly obvious, and we can assign a q state to each individual
image with high fidelity. To further illustrate the quantised nature of the superflow,in figure 5.2 (c) we plot a histogram of R values, from which we clearly identify three
separate peaks corresponding to the three states of q = 3, q = 2, and q = 1. The
shaded colour backgrounds indicate our q-value assignments, matching the data-point
colours in the scatter plot. Sample absorption images of the four allowed rotation states
are shown in figure 5.2 (a), illustrating that the quantisation of supercurrent states issufficiently evident as to allow the q states of individual images to be called by eye.
We also mention that we find such quantisation in several thousand other images taken
under different experimental conditions.
We consider the quantisation of the supercurrent decay the primary experimental
evidence for vortex-induced phase slips as the decay mechanism. Condensate fragmentation or collective excitations such as solitons would break the quantisation of R [162],
while individual particles which break away from the superflow would gradually fill up
the hole in the centre of the expanding cloud; we see no clear evidence of this occurring.
Observation of quantised decay events is also used as experimental proof of phase slips
in liquid helium experiments [27, 88, 164], and benefits from not relying on any criticalvelocity calculations.

117

5.3 Long-lived superflow

Figure 5.2: Quantised superflow decay: A superfluid prepared in the q = 3 state is held
in a ring trap of depth Vr ≈ 4µ. (a) TOF absorption images of the respective q = 3, 2, 1,
and 0 states. (b) Top panel: radius R as a function of hold time t. The R values fall into 4
distinct bands corresponding to (top to bottom) q = 3 (blue), 2 (green), 1 (red), and 0 (black).
Bottom panel: atom number, N versus t for the same data set. (c) High-contrast histogramof the measured R values confirms that we can assign a q value to each individual image withnear-unity fidelity. The shaded backgrounds indicate our q-value assignments.

The initial q = 3 state is observed to be fundamentally stable for approximately
2 seconds, corresponding to ≈ 5 complete revolutions of the condensate. In Section
5.3, by using a low power LG beam, we show that this metastable state can be extended to over a minute. We therefore conclude that the decay of the supercurrent isbrought about by the gradual decay of N plotted in the bottom panel of figure 5.2 (b).
This is discussed further in Section 5.5 as the gradual loss of atoms bringing about thecondition vs > vc . The broad q = 2 and q = 1 plateaus in figure 5.2 (b) show thatthe intermediate 0 < q < ` states are metastable even after the supercurrent decay is
initiated by the first phase slip. In the analogy with a particle moving in a washboard
potential as in figure 5.1 (a), this corresponds to a strongly damped motion: when thesystem escapes from a local energy minimum, it gets trapped in a new local minimum
rather than rapidly decaying to the q = 0 ground state.

118

5.3 Long-lived superflow

Figure 5.3: Long-lived q = 3 superflow: R is plotted as a function of hold time in a shallow
ring trap, showing persistent current for longer than a minute. The blue dashed lines are guides
to the eye, indicating the bands of R values corresponding to different q states. The inset shows
the decaying BEC atom number, N , for the same data set: the solid line is a double-exponential
fit to the data with characteristic decay times of 2.5 s and 40 s corresponding to the rate ofthree-body decay and background losses respectively.

5.3

Long-lived superflow

Having demonstrated the quantised nature of supercurrents, we now test the limits ofsupercurrent metastability in our setup by performing the same experiment in a very
shallow ring trap. Following the Raman transfer we now reduce the power of the LG
beam by a factor of 2.47 over 3 seconds to ≈ 30 µW, such that the depth of the ring trap,
Vr , is approximately equal to the chemical potential, µ. This ensures the trap is still
multiply-connected, protecting the system from vortex dynamics. Since the roughnessof the trapping potential scales with Vr , reducing the ring depth to ≈ µ results in thesmoothest trap we can achieve. This makes the condensate density almost perfectly
uniform around the ring and minimises the probability of weak links where the local µ
diminishes and phase slips are more likely.
In figure 5.3 we show the evolution of R for a superfluid prepared in the q = 3 state
and rotating in a shallow ring trap. The non-zero superflow (R > 0) now persists formore than a minute and decays only once the condensate number itself has decayed toabout 30% of its starting value. As discussed in Section 4.4.2, the radius R shows aweak dependence on the atom number, N , making the supercurrent quantisation lessstriking than in figure 5.2, where the fractional variation of N over the relevant timescale is smaller. However, we can still see that the R values fall into distinguishable
bands corresponding to the q = 3, 2, and 1 states. This allows us to conclude that theq = 3 state is perfectly stable for ∼ 40 seconds and can persist for up to a minute. We
119

5.4 Superflow velocity

confirm that the slow bending of the q bands with time is just a consequence of the
weak dependence of R on the decaying N (for fixed q) by preparing the initial q = 3
state with deliberately reduced initial atom number.
It’s worth mentioning at this point that in subsequent experiments carried out withgreater care and understanding, the q = 3 state could be made perfectly stable for up toa minute with no observable decay. As moreover will be discussed in Chapter 6, by using
an RF π pulse to flip the BEC back to the |1, 1i state following the Raman transfer,
we observe the q = 3 state to be perfectly stable for up to 90 seconds, equivalent toover 200 revolutions of the condensate. By this point the condensate number decays toapproximately 12% of its initial starting value. These results demonstrate the extreme
metastability of such states, persisting far longer than any relevant timescale of thesystem, and limited only by the decaying atom number. In similar experiments inhigher ` traps, the larger trap volume due to the larger ring radius reduces the local
condensate density, and hence reduces the stability of superflow. Nevertheless, even for` = 10 we can still observe superflow persisting for up to 20 seconds before any decay
events occur.

5.4

Superflow velocity

Having identified the decay mechanism as vortex-induced phase-slips, we now turn toa quantitative study of the dynamics of the supercurrent decay for different superflow
speeds. Generally as the number of atoms in a rotating BEC decays with time, superfluidity becomes less robust. In the spirit of the Landau criterion, we identify this withthe flow velocity approaching the critical velocity, vs → vc . Once vs equals the criticalvelocity for a given decay mechanism, the energy barrier to create such an excitation in
the moving frame of the condensate vanishes, accompanied with the onset of viscosity.
In our setup, the relevant decay mechanism has been shown to be a phase slip which
reduces the angular momentum of the condensate by N ~, reducing the flow velocity vsback below vc . As a result we anticipate a staircase of q values, where atom loss brings
about the condition vs (q) > vc , where vs (q) is the flow velocity of a given q state. This
causes a ∆q = −1 phase slip, reducing the flow velocity to below the critical velocity
for further phase slips, vs (q − 1) < vc , and hence the q − 1 state persists until further
atom loss brings about the condition vs (q − 1) > vc , and the next phase slip. . .
To confirm such a picture we need to calculate the flow velocity of the BEC in agiven q state for our ring trap. In an ideal case of a perfectly smooth ring trap withcylindrical symmetry the flow velocity is azimuthally uniform around the ring, andgiven by the condition for quantised circulation,vs (r) =

~q
θ̂.
mr

120

(5.6)

5.4 Superflow velocity

In our real trap, variations in the local density result in changes in the flow velocity. While the condition for quantised circulation must always hold, the circulationis achieved by a spatially varying velocity. In general, calculating the dynamics of asuperfluid is a difficult problem. This difficulty stems from the fact that the fluid iscomposed of two components with a total momentum density given byj = ρs vs + ρn vn .

(5.7)

As was shown in Section 2.4.2, the density of the two components depends on thetemperature, excitation spectrum, and relative velocity of the two components, vs −vn .
A complete discussion of how to go about solving such a problem is given in [170]. Here
we make the simplifying assumption that the normal component can be ignored, andhence any particle flux is due to the motion of the superfluid only. This assumption
is based on the fact that our condensate fraction is typically over 85%, and secondly,
we are only interested in the long-time steady state of the system at which point thenormal component can be considered at rest. Using these assumptions we obtain thezero temperature hydrodynamic equations [170]∂n+ ∇ · (nvs ) = 0
∂t


∂vs
1
2
m
+ ∇ µ + Vext + mvs = 0.
∂t
2

(5.8)
(5.9)

The first of these equations is simply the conservation of particle flux, where here we
consider n = ns . The decay events we are interested in occur several seconds after the
initial transfer of angular momentum to the system, hence it’s sufficient to only find thesteady state solutions. This assumes that the gradual atom loss is sufficiently slow that
we may consider the properties of the system to be given by those of the equilibrium
state. We also make the simplifying assumption that the energy associated with theflow velocity in (5.9), is negligible compared to the mean field energy. From this we
obtain the two relationsvs (r) ∝1n(r)µlocal (r) = µ0 − Vext (r),(5.10)(5.11)

where µ0 is the global chemical potential for the system. The first of these relations
simply states that to conserve particle flux, the flow velocity in regions of lower density
must be higher than that in regions of higher density. The second relation is equivalent
to the local density approximation, relating the local chemical potential and hence
density, to the external trapping potential and the global chemical potential, set by the
total particle number. In steady state one can define a flow path, l(r), forming a closed
loop around the ring, where r defines the position along the flow path. To constrain121

5.4 Superflow velocity

Figure 5.4: Particle flux in an angular segment: We consider an angular segment, dθ, within
which we can define a chemical potential, µ(θ). Within this segment we consider the flow
velocity to be parallel to the boundaries, varying as 1/r across the radial extent of the segment.

the constant of proportionality in (5.10), we use the condition for quantised circulation
along such a path,Ivs .dl = 2πq~.
m(5.12)

To simplify the problem further we consider an angular segment of the ring, dθ,illustrated in figure 5.4. This allows us to define a chemical potential µ(θ) for thatgiven segment, which can be found from the local density (5.21). For a sufficiently
slowly varying trapping potential we can consider the flow velocity in such a segment
to be purely in the angular direction. The local velocity in such a segment has no z
dependence, but due to the velocity profile about a vortex line, varies as 1/r across thewidth of the annulus,vs (r, θ) =v0 (θ).
r(5.13)The requirement of particle flux conservation now requires the flux through any segment, dθ, to be equal, and hence equation (5.10) takes the form
Z Zn(r, θ, z)vs (r, θ)rdrdz = constant,(5.14)

where the velocity is given by (5.13). To make further progress we need to consider thedensity profile within an annular trap.

5.4.1

BEC density in a ring trap

The density profile of a condensate in an annular trap is covered extensively in thethesis [171] relating to [100]. We model the ring trapping potential as a harmonic1225.4 Superflow velocity

oscillator about the trap minimum in both the radial and vertical directions,
1
1
Vext (r, θ, z) = mωz2 z 2 + mωr2 (r − rM )2 ,2
2

(5.15)

where rM is the radius of the ring potential. As discussed in Section 4.2.1, we measure the vertical trapping frequency from the sheet potential as ωz = 2π × 400 Hz.
The radial trapping frequency due to the ring potential, ωr , is measured by applying
a weak attractive potential at the centre of the ring trap using the CDT, pulling theatoms towards the inner radius. Quickly turning off this potential, the atoms oscillate
freely in the radial direction of the ring trap and we extract the trapping frequency byobserving oscillations in the radius of the density distribution at short TOF. From this
we find ωr = 2π × 180 Hz for the ring trap used to obtain the data in Section 5.2, andpresented in figure 5.2. The radius of the ring trap is equal to rM = 12.3 µm.
The condensate wavefunction is found by solving the GP equation (2.40) with external trapping potential given by (5.15),

 2 2
~ ∇
2
+ Vext (r) + g|Φ(r)| Φ(r) = µΦ(r).
−2m(5.16)

To simplify this problem one typically makes the Thomas-Fermi (TF) assumption (Section 2.2.3) where, for large atom number, the kinetic energy is assumed to be negligible
compared to the interaction energy. In this regime the condensate wavefunction takes
the form of the inverted trapping potential. This assumption is valid provided the extent of the condensate is much larger than the healing length [172]. For an annular trap
this corresponds to the condition µ  ~ωi , for the condensate to be in the TF regime
along the ith direction. In the opposite regime where the kinetic energy dominates overthe interaction energy the many particle ground state is identical to the single particle
ground state, which for a harmonic trap is a Gaussian wavefunction.
For our experiment we find that at the initial atom number, N = 180, 000, for all
points in the ring µ(θ) > ~ωz > ~ωr , and hence the system is fully in the TF regime.
As the number decays the chemical potential decreases at all points in the ring, andwe find that for the relevant parameters over which phase slips occur, the density at
certain points in the ring is sufficiently low that ~ωz > µ(θ) > ~ωr , and the dynamics inthe z direction are frozen out and locally the BEC is two dimensional. The condensate
therefore has a TF profile in the radial direction, but a Gaussian profile in the verticaldirection. To fully solve the flow velocity profile at all points in the ring, we therefore
need to relate the density and chemical potential for both of these regimes:1235.4 Superflow velocity

Fully TF regime
In the fully Thomas-Fermi regime the particle density is given by


µ(θ)
z2
(r − rM )2nT F (r, θ, z) =1− 2 −
for Vext (r, θ, z) < µ(θ)
g
Rz
Rr2
=0
where Rz =otherwise,(5.17)(5.18)pp2µ(θ)/mωz2 is the vertical Thomas-Fermi radius, Rr = 2µ(θ)/mωr2 is

the radial Thomas-Fermi radius, and µ(θ) is the local chemical potential defined asµ(θ) = µ0 − Vext (r = rM , θ, z = 0). The relevant quantities are the 2D column density
integrated along z, and the 1D angular density integrated along both z and r:
Z
n2D (r, θ) = nT F (r, θ, z)dz
3/2
(r − rM )24µ(θ)
Rz 1 −
=
3g
Rr2
Z
dN
n1D (θ) = n2D (r, θ)rdr =dθµ(θ)Rz Rr rM π=2g(5.19)(5.20)

Noting that Rz and Rr also depend on the local chemical potential,
n1D (θ) =µ2 (θ)πrM.
mgωr ωz(5.21)

These expressions allow us to relate an observed local density to a local chemical potential which we will use to calculate how the flow velocity evolves as the atom number
decays. The equation for 1D flux conservation in (5.14) can now be calculated for the
TF regime density profileZ Z
nT F (r, θ, z)v0 (θ)
rdrdz = constant = C,r

(5.22)

where we label the constant as C, and have made use of the 1/r form of the flow velocity(5.13). Using the result of (5.19) we findC=

n1D (θ)v0 (θ),
rM(5.23)and hence the local velocity constant, v0 (θ) is simply inversely proportional to the
local 1D density, n1D (θ). We now use the condition for quantised circulation along a124

5.4 Superflow velocity

connected path, l(r), to determine the constant C,I~vs · dl = 2πqmIIv0 (θ)
rMrdθ = Cdθ,r
n1D (θ)(5.24)(5.25)

therefore
I
2πq~
1
C=
dθ,
rM m
n1D (θ)I
12πq~ 1v0=dθ.
vs (θ) =r
mr n1D (θ)
n1D (θ)(5.26)(5.27)

This equation shows us that due to the geometry of the system, the flow velocity at
a given point depends on the global density profile of the entire trap and not just the
local density at that point.
Radially TF regime
In the radially Thomas-Fermi regime, anticipated when atom loss reduces the local
chemical potential to below the vertical trapping frequency, the particle density isgiven bynRT F (r, θ, z) = e−z 2 /a2z µ(θ)=0where az =g(r − rM )2
1−
Rr2


for

|r − rM | < Rrotherwise,(5.28)(5.29)

p~/mωz is the vertical harmonic oscillator length. Once again we calculate

the 2D and 1D densities in the same manner√µ(θ) πaz(r − rM )2n2D (r, θ) =
1−
g
Rr2
√
4µ(θ) πaz rM Rr
n1D (θ) =3gr
4 2π µ3/2 (θ)=
rM az .
3g m ωr(5.30)(5.31)(5.32)We therefore see that the local 1D density has a different power law dependence on
the local chemical potential than in the full TF regime. The relation between the flow
velocity and 1D density is the same as in the fully TF regime derived in (5.27).

5.4.2

Effect of azimuthal roughness

With the relations above we can now deduce the flow velocity by simply measuring theazimuthal profile of the 1D density. The procedure for this is outlined in figure 5.5. As
shown in figure 5.5 (a), taking an absorption image of the condensate in the ring trap,1255.4 Superflow velocity

Figure 5.5: Calculation of azimuthal variation in flow velocity: (a) In trap absorption imageof the condensate density. The one dimensional density is obtained by summing pixels within
an angular segment, centred at the angle θ. (b) Extracted one dimensional particle density
versus angle around the ring trap. (c) Using (5.27) one can relate the 1D density profile toa flow velocity profile, here plotted for the inner radius of the trap, r = rM − Rr , where thevelocity is highest. The correlation between regions of low density and regions of faster flow isclear. The black dashed line is the inner radius velocity for a perfectly smooth ring trap withthe same µ0 .

we obtain the 1D density by summing pixels within an angular segment, dθ, centred
at angle θ. The angular width of this segment is 2π/48, chosen to span approximately
the imaging resolution at the atoms, however the final result is the same for smaller
segments. To smooth the density profile obtained we use 144 segments which eachhave 2/3 overlap with the adjacent segment. The extracted profile of n1D (θ) is shownin figure 5.5 (b). Using this and equation (5.27), we can immediately calculate thevelocity vs (r, θ) at any point in the ring. We plot the velocity on the inner edge ofthe ring for r = rM − Rr (θ) as a function of θ in figure 5.5 (b), accounting for thevariation in the radial TF radius, Rr , due to changes in the local chemical potential.
In our discussions we always consider the flow velocity on the inner surface since thisis where the condensate flows fastest, and hence will reach critical velocity first. From
this we can see that to conserve particle flux, regions of low density have the highest
flow velocity, and hence the superflow will destabilise here first.
In our experiment, for the initial atom number, we find the superflow is stable
and no phase slips occur, corresponding to vs < vc . To understand how atom loss
destabilises the flow, we need to calculate the time evolution of the flow velocity profile
plotted in figure 5.5 (c). Using the local density approximation (5.11), we relate thelocal chemical potential to a global one byµ(θ) = µ0 − Vext (r = rM , θ, z = 0)(5.33)

In figure 5.6 (a) we plot µ(θ), calculated from n1D (θ), using the relations (5.21) and(5.32). We define the global chemical potential as the average of µ(θ), shown by thedashed black line, and therefore the difference gives us the local trap roughness, Vext (r =126

5.4 Superflow velocity

Figure 5.6: Local chemical potential in a rough trap: (a) To calculate the flow velocity at all
points in the ring as a function of total atom number we extract the local chemical potential,µ(θ), plotted as a function of the angle θ around the ring. (b) River analogy of figure (a). We
now consider the local mean field as the difference between a global chemical potential, µ0 ,(the water level), and a spatially varying trap roughness (the river bed). As the atom number
decays the value of µ0 decreases (water level falls), but the trap roughness (river bed) stays
constant.

rM , θ, z = 0) = µ0 − µ(θ).
To illustrate this further, in figure 5.6 (b) we plot µ0 and Vext (r = rM , θ, z = 0)
as a function of θ. In analogy to fluid flow in a river, µ0 defines the global fluid
level, and Vext (r = rM , θ, z = 0) defines the roughness of the river bed. The local
chemical potential gives us the depth of the fluid at a given point. As the atom number
falls the global fluid level falls, but the channel roughness remains the same, fixed byimperfections in the light which forms our trapping potential. This allows us to calculate
how µ(θ) (the depth of the fluid) evolves with changing atom number, and from thiswe recalculate the 1D density profile, n1D (θ), and the velocity profile, vs (r, θ)1 .
Before applying this calculation to find the flow velocity evolution during decay, wemust first consider the effect of imaging resolution on the calculation. This was foundto have a significant effect on our results, and must be considered to bring the dataclose to our models of the critical velocity [173].

5.4.3

Imaging resolution

The in-trap absorption images from which we have estimated the condensate density
profile are inherently blurred from the true density profile due to the effects of finite
imaging resolution and pixelation of the image. The true columnar density profile isrelated to the imaged columnar density profile by convolution with a 2D point spread1

Due to the non-trivial relationship between total atom number and µ0 , this calculation is actually
done in reverse. Changing µ0 slightly, we then find µ(θ) using equation (5.33) and the fixed trap
roughness. From this we can find the 1D density, and hence the total atom number which corresponds
to the revised µ0 .

127

5.4 Superflow velocity

Figure 5.7: Effect of imaging resolution on flow velocity: (a) To experimentally determine
the width of the point spread function (PSF) due to our imaging resolution and pixelation, we
fit the radial profile of the ring to n2D at various powers of the LG beam and extract a fitted
width, W . Plotting the square of 2W against the square of the calculated value of Rr , we canextract the width of the PSF from the intercept with the axis. Top-left inset: Comparison ofGaussian profile with waist given by equation (5.36) (dashed red) versus blurred image producedby convolution with PSF (solid blue). Bottom-right inset: Example radial fit, from which W is
extracted. (b) The original image is ‘deblurred’ using the measured width of the PSF to obtainan estimate of the true density profile. The validity of this step is confirmed by convolving
the deblurred image with the PSF and restoring the original image. Plotted is the azimuthal
variation in the trap roughness for both the original and deblurred image.

function (PSF) which we assume to be Gaussian,(image)(r, θ) = [n2D ∗ P SF ](r, θ),(5.34)


1
r2
P SF (r, θ) =exp − 2 ,2πσ 22σ(5.35)

n2D

where sigma defines the waist of the point spread function. The true columnar density
in trap has a TF profile, given by n2D in equation (5.19). The radial profile of theimaged columnar density, found by convolving n2D with the PSF, is well approximatedby a Gaussian profile with a waist given by
s 
Rr2 2
W =
+ σ2.
2

(5.36)

This is illustrated in the top-left inset of figure 5.7 (a). The Gaussian profile with awaist given by W is indicated by the dashed red line. The radial profile formed byconvolution of the original TF profile of radius Rr with the PSF is shown in blue. The
good agreement between the two suggests we can find the σ of our imaging system byfitting the radial profiles of images to extract W , and calculating Rr from the measured
atom number and trapping frequencies of our system.

128

5.4 Superflow velocity

This is done by measuring the fitted Gaussian width, W , for 10 different values of
the true Thomas-Fermi radius, Rr . We vary Rr by changing the intensity of the LGtrapping beam to alter the radial trapping frequency, ωr . For each value of ωr we fit
the radial density profiles at 8 different angles θ, and define the average Gaussian width
W , as the root mean square of the fitted widths. This value is compared against the
average of the true Thomas Fermi radius Rr , which we calculate from our total atom
number and measured trapping frequency. In figure 5.7 (a) is plotted (2W )2 versus thesquare of Rr , fitted to a straight line with a fixed gradient of 1. Each data point is anaverage of two images. The close agreement between the data and the fixed linear fit
is confirmation of equation (5.36). The intercept at the y axis gives the width of the
PSF as σ = 2.6 ± 0.3 µm. This result is reasonable when compared to our estimatedimaging resolution of 2 µm and pixel size of 1.2 µm.
To find the true columnar density profile we now need to deconvolve the image witha PSF of width σ. This is clearly a non-trivial problem, since many density profiles canbe convolved with the same PSF to obtain profiles close to our imaged density. This
step is carried out using in-built tools in Matlab designed for such a problem. To prevent the procedure from introducing small features into the deconvolved image whichcannot be inferred from the original image, we provide the function with a measure of
the signal-to-noise ratio in the image, calculated from the ratio of the variance at theatoms compared to that in an empty control shot. An example of this deconvolving
process is shown in figure 5.7 (b). By repeated calculations of trap roughness fromsuccessive images, we estimate our uncertainty in the calculated trap profile at ≈ 5%,
due to atom number and trap fluctuations, and variations in our methodology.
Importantly, this uncertainty does not account for systematic errors. The deconvolution process we have used here simply provides us with an estimate of the atom
density which is consistent with our observations. To confirm this estimate, we checkthat reconvolving with the measured PSF produces a density profile in good agreementwith our images. In addition, we also adjust the parameters of the deconvolution step
such that the Thomas Fermi radius of the deconvolved image is consistent with our calculated value. Nevertheless, this stage in our calculations introduces a large source ofsystematic uncertainty. Unfortunately, accounting for finite imaging resolution proves
to be an important step in accurately calculating the flow velocity and critical velocity,
but one must keep in mind the inherent uncertainties that such a process involves.
The trap roughness calculated from the deconvolved image is anticipated to becloser to the true trap roughness, and can clearly be seen to be a more sharply peaked
version of the profile extracted from the original image. The larger range of this trap
roughness leads to faster maximal flow at the weakest parts of the ring, and slower flowat the deepest parts of the ring. The complete protocol for extracting the flow velocity
as a function of N is then given as
1. Deblur image by deconvolving with PSF of σ = 2.6 µm
129

5.5 Critical velocity

2. Extract n1D (θ) by summing pixels in angular segments such thatR

n1D (θ)dθ = N

3. Extract µ(θ) from n1D (θ) using equations (5.21) and (5.32)
4. Extract global mean field, µ0 , from the average of µ(θ)
5. Extract trap roughness, Vext (r = rM , θ, z = 0) = µ0 − µ(θ)(a) Recalculate µ(θ) = µ0 − Vext (r = rM , θ, z = 0)(b) Recalculate n1D (θ) from µ(θ) using equations (5.21) and (5.32)(c) Recalculate vs (r, θ) from n1D (θ) using equation (5.27)
R(d) Recalculate atom number, N = n1D (θ)dθ(e) Reduce µ0 slightly and repeat steps 5a→5e
The results of this calculation are shown in figure 5.8 (a), where we plot the maximal
flow velocity versus atom number, N , for the narrowest point in the ring, where the local
density and µ(θ) are lowest. We plot this for the q = 3, 2, and 1 states, indicated by
the solid blue, green, and red lines respectively. As expected, the loss of atom number
further constricts the flow at the narrowest parts causing the velocity to increase.
For N approaching 50, 000, the flow velocity diverges, corresponding to the condensate
density fragmenting due to the global chemical potential, µ0 , falling below the trapping
potential at the roughest part of the trap.
For comparison with our data on phase slips presented in figure 5.2, we eliminate
the time variable and plot the observed R (and equivalently q) values versus N (figure
5.8 (b)). As in figure 5.2, the same colour code is used to indicate different q states.
We find each rotational state is observed up to a critical atom number Nc (q), beyond
which decay to a lower angular momentum state is inevitable. We thus empiricallyassociate this critical atom number with the condition vs (q) = vc . Using figure 5.8 (a),we relate this critical atom number to a maximal flow velocity for each q state, vmax (q).
From the data in figure 5.8 we find vmax (3) = 1.29 mm/s, vmax (2) = 0.96 mm/s, andvmax (1) = 0.78 mm/s. The changing maximal flow velocity for the three q states istherefore due to the critical velocity decreasing with decaying atom number. In figure
5.8 (b) we also observe phase slips for vs < vc . This is discussed further in Section
5.6, where we attribute this to rare stochastic decay events, most likely due to thermal
activation over the energy barrier to decay which is still present for vs < vc .
To see whether we can understand these results further, we now consider whatmechanism defines the critical velocity in our ring BEC.

5.5

Critical velocity

Having calculated the evolution of the superfluid velocity over which we observe phaseslips, we are now in a position to contrast our data to a possible model of the critical130

5.5 Critical velocity

Figure 5.8: Evolution of peak flow velocity for decaying atom number: (a) Peak flow velocities
for the q = 3, q = 2, and q = 1 states are illustrated by the solid blue, green, and red lines
respectively. The shaded regions about each line, illustrate the variation introduced by theestimated 5% fluctuations in the trap roughness and atom number. The vertical dashed lines
indicate the lowest N at which we observe each q state, with the same colour coding applied.
(b) We plot the same data as in figure 5.2, now plotted against atom number. We see that eachq plateau extends up to a minimum N , which using figure (a) we can now relate to a maximum
flow velocity vs (q).

velocity. To do this we first consider three different analytic calculations of vc : theFeynman velocity, vcF , the speed of sound, c, and the surface critical velocity, vcs . Of
these three we find our data agrees closest with a critical velocity given by the speed ofsound, a result predicted in several numerical studies solving the GPE for an annular
BEC with a repulsive barrier potential [40, 173–175]. The propagation speed of surface
modes localised at the edge of the cloud, vcs , also gives reasonable agreement with ourdata, since over the parameter space considered, vcs and c are very similar. In Section
5.5.4 we numerically calculate the excitation spectrum of an annular BEC and show
the relevant excitation which sets vc switches from surface modes to phonons below acertain local chemical potential. We therefore conclude that the phase slips we observein figure 5.8 (b) are indeed nucleated by phonon modes, and hence the relevant criticalvelocity is given by c. Finally, we note that the Feynman critical velocity, vcF , only
provides an order of magnitude estimate and does not provide quantitative agreementwith our measured critical velocity, nor predict the reduction in vmax (q) with decreasing
N , which we observe.

5.5.1

Feynman velocity

In Section 2.4.3 we derived the Feynman critical velocity by considering the minimum
energy cost of inserting a vortex ring into a flow channel of radius R:
vcF =

~
R
ln ,
mR
ξ
131

(5.37)

5.5 Critical velocity

where ξ is the vortex core size, given by the healing length. In a toroidal flow sucha vortex ring will form at the walls of the channel before shrinking in on itself andannihilating, completing a full single phase slip [40]. The geometry of this excitationis three dimensional and applies to cylindrical flow channels. Our annular BEC isasymmetric, with a vertical trapping frequency over twice the radial. For the densities
over which we observe phase slips, the condensate at the weakest part of the ringwhere the decay events occur is quasi 2D, with µ(θ) < ~ωz . Indeed for the very lowest
densities at which we observe q = 1 → q = 0 phase slips, the condensate is approaching
1D at the weakest point. The relevant excitation to then consider in a two dimensional
geometry is not vortex rings, but straight vortex lines of vortex-antivortex pairs aligned
along the vertical direction, which either cross the annulus and annihilate one anotherin the bulk, or nucleate in the bulk and move apart towards the edge of the condensate,decaying to elementary excitations at the surface. While a single vortex line can also
cause a 2π phase slip, the energy associated with a vortex-antivortex phase slip is lower.
For a vortex-antivortex pair separated by a distance D in a background flow of velocity
vs , the energy in the frame moving with the flow is given by
E 0 = 2π

~2 n2D D
ln − 2π~n2D Dvs ,mξ(5.38)and hence the critical velocity is given by
vcF =

~
D
ln .
mD
ξ

(5.39)For our annular flow, D = 2Rr , and the healing length ξ, is an effective healing length,
ξ1D , integrated over the vertical and radial dimensions to account for the varying density
across the annulus.
Using the expression for the healing length in a uniform condensate in (2.64), andthe expression for the sound speed in a uniform condensate in (2.62), we can show thatthe two quantities are related by
ξ=√

~
.
2mc

(5.40)

Using this we can rewrite the Feynman velocity (5.39) in the form
√ ξ1D
D
vcF
= 2
ln
,
c1D
D
ξ1D

(5.41)

where c1D is similarly an effective sound speed averaged over the cross section of theannulus. Since the expression for the Feynman velocity is only valid for the conditionD  ξ1D , where the channel width is much larger than the vortex core size, we havethe condition that the Feynman velocity is always much lower than the sound velocity.
The maximal value of the Feynman velocity is therefore found to be vcF = 0.52c1D ,occurring for a channel width D = 2.7ξ.

132

5.5 Critical velocity

5.5.2

Speed of soundThe speed of sound is defined as the flow velocity above which it is energetically beneficial to excite phonon modes in the condensate. In Section 2.3.1 we considered perturbations in the wavefunction of a uniform condensate to derive the sound velocity
as
c=

p
ng/m,

(5.42)

where n is the particle density, and g is the interaction parameter given in (2.32). In
real experiments with inhomogeneous density, the sound velocity is maximal at thetrap centre and falls to zero at the surface. To compare this criteria to our system
we therefore need to define an effective sound velocity c1D , which integrates over thevarying density in the z and r direction [176]. Such an effective sound velocity is givenby [149]r
c1D =n1D ∂µ,m ∂n1D(5.43)

where n1D is the 1D density defined in (5.21) and (5.32). Since our phase slip eventsoccur where the condensate is in the radial TF regime, using equation (5.32), we findan expression for the local effective sound velocity asr2µ(θ)
c1D (θ) =.
3m(5.44)We therefore anticipate a local Landau criterion, where for max(vs (r, θ)) > c1D , phononmodes will be excited and the supercurrent will decay. The precise mechanism by whichsuch phonon modes then go on to nucleate vortices and cause a phase slip is still not
fully understood [173].

5.5.3

Surface modes velocity

In experiments on simply-connected rotating atomic gases [31, 32, 177, 178] it was often
found that the critical velocity for a vortex entering the condensate was higher than thatpredicted purely by global energetic arguments [179]. This higher vc is associated withdynamic instabilities of surface excitations, which provide the necessary microscopic
route for vortex nucleation. The surface mode model aims to find the local criticalvelocity over the TF surface above which such unstable surface excitations form. Thismodel produces a general theory based on the idea that the TF surface of a large
condensate enters a bulk regime in which the physics of vortex formation is local. In
general, an energetically unstable surface mode must first appear for vortices to enter
the cloud, therefore the Landau criterion is universal to vortex formation. For large
enough condensates where the surface length scale is much smaller than their overall
extent, the local Landau critical velocity may be determined by solving one universal
equation.

133

5.5 Critical velocity

For a large enough condensate one can neglect the curvature of the TF surface andapproximate the trapping potential to a linear ramp, V = F r, where F is the potential
gradient at the TF surface given bypdV= 2µmωr2 ,dr r=RrF =(5.45)

for a harmonic trap. One then considers the instability to form in a surface boundaryof depth

δ=

1/3

~2
2mF

.

(5.46)The surface mode model is then valid in the regime where Rr  δ, which simplifies to
Rr
=2
δ



µ
~ωr

2/3
 1.

(5.47)The relevant GPE equation then reads
 2 2

∂Φ
~ ∇
2
i~
= −
+ F r + g|Φ| Φ,∂t2m(5.48)

where the origin of r is at the TF surface. The excitation spectrum of such a system
was solved numerically in [180], where it was found that the critical velocity is given
exactly by
vcs =



2~F
m2

1/3

r
=

2~ωr
m



µ
~ωr

1/6
.

(5.49)

The arguments of the surface model are local and hence the theory should be equallyapplicable to rotating annular condensates as long as both the width of the annulus
and its inner radius are much large than the surface depth δ. Indeed the surface mode
theory was extended to a 2D ring geometry by Dubessy et al. [181]. By explicit
calculation of the excitation spectrum of an annular condensate, the calculated valueof the critical velocity was shown to be equal to vcs . In Section 5.5.4 we repeat thiscalculation, arriving at the same conclusion for sufficiently large condensates. Herehowever, we show that for low densities the nature of the excitation changes. Belowa certain chemical potential the radial extent of the annulus becomes comparable toδ and the instability changes from a surface mode with vc = vcs , to a phonon mode
located in the condensate bulk with vc = c1D .

5.5.4

Excitation spectrum of a ring BEC

We now calculate the excitation spectrum of a two dimensional Bose gas trapped ina smooth ring potential, as was done in [181]. From this we find the critical angularvelocity above which the system must be rotated before vortices are excited. By varying
the total atom number in the trap we calculate the value of this critical velocity as afunction of the chemical potential (here µ(θ) ≡ µ0 , since we consider a cylindrically1345.5 Critical velocity

symmetric trap). In this case of a rotating trap and a stationary condensate, the criticalvelocity is first exceeded at the outer radius of the condensate. In our experiment ofa rotating superfluid and a stationary trap, the same critical velocity applies, but now
the fluid velocity is largest on the inner radius.
We consider the GPE at zero temperature for a 2D ring. This most closely matches
our geometry of interest, where we find phase slips occur in the quasi-2D regime. As
before, the trapping annular potential is written as a harmonic potential of frequency
ωr centred at radius rM . We use the associated scales for energy (~ωr ), time (ωr−1 ),pand length ar = ~/(mωr ). The dimensionless 2D GPE then reads


∆ 1
2
2
i∂t Φ = − + (r − r0 ) + g|Φ| Φ,
2
2(5.50)

where the condensate wavefunction, Φ = Φ(r, θ, t), is normalised to unity, ∆ = ∂r2 +∂r /r + ∂θ2 /r2 is the Laplacian in polar coordinates, r0 = rM /ar is the dimensionless√ring radius, and g = N 8π aaz is the dimensionless 2D interaction constant, where N is
the atom number, a is the s-wave scattering length, and az is the harmonic oscillator
length in the z direction [182].
From the rotational invariance of (5.50), we consider solutions of the form
Φ(r, θ, t) = e−iµt [Φ(r) + δΦm (r, θ, t)],(5.51)where∗δΦm (r, θ, t) = um (r)e−i(ωt−mθ) + vm (r)∗ ei(ω t−mθ) .

(5.52)

Φ denotes the stationary ground state of the system, µ is the global chemical potential,and δΦ is a small perturbation parameterised by the angular wavenumber m. A detailed
explanation of how this problem is solved is given in Appendix B. We first calculate theground state for a given N by imaginary time propagation of a test TF profile, stopped
when the relative variation in µ falls below 10−12 . The Bogoliubov-de Gennes equations
for um (r) and vm (r) are then solved by diagonalisation. We obtain real frequencies witha dispersion relation ω = ω(m), the lowest branch of which allows us to compute thecritical angular velocity Ωc = min(ω(m)/m). The density profile of the excitation isfound by δnm (r) = 2Re[Φ(r)∗ (um (r) + vm (r)∗ )].
The results of this calculation are shown in figure 5.9. Here we plot the lowest
branch of the dispersion relation for three different values of the total particle number,
N . The critical angular velocity is indicated by the dashed grey line, ω = mΩc . At
small m values the lowest branch is linear, and can be associated with rotating soundlike
waves. For larger atom numbers, at higher m values, this branch exhibits a small
negative curvature that makes the critical angular velocity smaller than the angularspeed of sound. The radial density profile of the excitation, shown in the inset, is135

5.5 Critical velocity

Figure 5.9: Numerical calculation of 2D annular BEC excitation spectrum: The three plots are
the eigenfrequencies, ω, and the angular wavenumber, m, for the lowest branch of the dispersion
relation, plotted for three different values of the total particle number. The black data points
are the results of our numerical calculation, obtained by diagonalisation of the Bogoliubov-deGennes equations. The grey dashed line is the critical angular velocity, Ωc = min(ω(m)/m).
The solid red line is the critical angular velocity, as predicted by the surface model. The inset ofeach plot shows the radial density profile of the condensate (black dashed line) and the density
profile of the excitation (red line), normalised to have the same peak value.

strongly localised on the outer radius and we thus expect that the mode corresponding
to the critical angular velocity will be correctly described by the surface mode modelin Section 5.5.3. The critical angular velocity within the surface mode model is simply
given by
Ωsc =

vcs
,
re

(5.53)

where re is the radius of the excitation. For this case of a rotating trap and stationary
BEC, the critical velocity is first reached on the outer surface, and hence re = rM + Rr .
This critical angular velocity is indicated by the red line in figure 5.9. We therefore
see that for the largest atom number, where the assumptions of the surface model aremost valid, Ωc and Ωsc are in excellent agreement as was concluded in [181].
As illustrated in figure 5.9, as the particle number decreases the negative curvature
at large m values disappears. Eventually at low numbers the critical velocity is equal to
the speed of sound, set by the linear part of the spectrum near m = 0. The excitations
are then phonons and are no longer localised at the surface but distributed over theentire condensate profile. This is associated with a failure of the surface model, withthe value of Ωc significantly less than Ωsc . This disagreement with the surface model at
low N is intuitive, since at low densities the extent of the condensate becomes comparable to the surface depth.
The critical velocity as determined by the numerically calculated excitation spectrum is simply found from the critical angular velocity as
vce = Ωc (rM + Rr ),136(5.54)

5.5 Critical velocity

Figure 5.10: Comparison of critical velocity models: Plotted is the critical velocity versus
local chemical potential for all models considered: Feynman velocity, vcF (green), effective
sound speed, c1D (blue), surface modes velocity, vcs (red), and the critical velocity extracted
from numerical calculations of the excitation spectrum of an annular BEC, vce (black triangles).
We observe a transition at ≈ 1kHz, above which the critical velocity is determined by surface
mode instabilities, and below which it is set by excitation of phonon modes. We note that ourdata on phase slips is in the regime where the critical velocity is sound like.

since the maximum velocity at a given rotation Ω is at the outer radius. In figure 5.10,
the value of vce is plotted as a function of chemical potential, as indicated by the black
triangles. To compare this result with our analytic models, the Feynman velocity, vcF ,the effective sound velocity, c1D , and the surface mode velocity, vcs , are also plottedon the same axes. For large µ, the critical velocity agrees with vcs , in agreement withthe results of [181]. For values of µ below around 1kHz, the sound velocity falls below
the surface mode velocity, and the relevant excitation becomes phonon like and is nolonger localised at the surface. This deviation from the surface model occurs when theratio of the condensate extent to the surface depth falls below Rr /δ . 6.3. The valueof vce then closely follows the effective sound velocity, c1D , with the small discrepancy
possibly due to the continuous change in dimensionality, and hence averaging of c. Thisagreement between the speed of sound and vce found by solving the GPE is in agreementwith the results of [40, 173–175]. We also observe that the Feynman critical velocity,
vcF , is considerably lower than the others and only gives reasonable agreement with thenumerical result at very low chemical potentials.

1375.5 Critical velocity

Figure 5.11: Comparison of critical velocity models with flow velocities calculated from datafor different q states: The data points are the calculated flow velocities versus atom number forthe data points in figure 5.8, where the blue, green and red points indicate the q = 3, 2, and 1
states respectively. The solid lines in blue, green, and red plot the functions vs (q) versus N for
the q = 3, 2 and 1 states respectively. The shaded areas around each q data set illustrate theuncertainty in flow velocity due to 5% fluctuations as in figure 5.8 (a). The dashed black line isthe critical velocity versus N for the model considered. The shaded area around this line again
illustrates the error introduced by the fluctuations. The models used are the Feynman criticalvelocity, vcF , in (a), the surface mode critical velocity, vcs , in (b), the averaged speed of sound,
c1D , in (c), and the numerically calculated critical velocity, vce , in (d).

5.5.5

Critical velocity discussion

To test our calculations of vc we now need to compare them to our data in figure 5.8,
where we have the distribution of q states with atom number N , and our calculationsof how to convert a given N and q to a peak flow velocity vs (q, N ). The results of thiscomparison are plotted in figure 5.11 for all four critical velocities: Feynman, vcF , surface mode, vcs , averaged sound, c1D , and the numerical result, vce , in figures (a) → (d)respectively. The data points are our measured vs (q, N ) values versus N , where theblue, green, and red points indicate the q = 3, 2, and 1 rotations states, originally assigned in figure 5.2 using the quantisation of R. The shaded area around each q data
set illustrates our uncertainty in calculating vs (q, N ) due to fluctuations, as in figure1385.5 Critical velocity

5.8(a). The right edge of this area therefore indicates the highest N /lowest vs (q) at
which we observed each q state and the left edge indicates the lowest N /highest vs (q)at which we observe each q state. We therefore identify the left edge with the conditionvs (q) = vc . The respective critical velocity is also plotted as the dashed black line, andthe black shaded area similarly represents the effect of fluctuations in calculating vc .
Good agreement between our data and a given vc model would then be indicated by
the maximum vs (q) for each q state (left edge of the coloured shaded areas) lying close
to the critical velocity (black shaded area).
The Feynman critical velocity (5.39), plotted in figure 5.11 (a), is clearly too low
and predicts immediate decay of all q > 1 states and decay of the q = 1 state at
N ≈ 120 × 103 . We also note that the form of the Feynman velocity has a very weak
dependence on N and therefore cannot explain the decrease in vmax (q) with decaying
N which we observe.
The critical velocity of the surface model (5.49), plotted in figure 5.11 (b), shows
better agreement with the data, though predicts a larger vc than we observe. For ourdata, the ratio of the condensate extent to surface depth Rr /δ is in the range 2 − 4,and therefore the criteria for applying the surface model is marginal. In our original
discussion of these results in [14], the deconvolution method adopted was slightly different and we allowed the width of the PSF to vary to best match our data, finding
good agreement between the data and the surface model for σ ' 2.9 µm, in reasonableagreement with our measured σ. With the additional insight from the numerical calculations in Section 5.5.4, we now see that in the regime of our results, the condition forthe surface model to hold is not strictly met, and we anticipate the critical velocity to
be in better agreement with the local sound velocity. In this discussion, we therefore
fixed σ at its measured value to directly compare our data to both models. This original
agreement with the surface model does however highlight the systematic uncertainty
introduced by the deconvolution step in our calculation.
The averaged speed of sound (5.44), plotted in figure 5.11 (c), shows excellent agreement with our data, closely predicting the minimum N , and therefore maximum vs (q),for all three q states. The critical velocity extracted from our numerical calculations(5.54) is plotted in figure 5.11 (d), and as expected is very close to the result for the 1D
sound velocity. From this we conclude that our data is in agreement with the criticalvelocity predicted by the GPE, which for our regime is very closely related to the local
1D sound speed.
To further confirm our results we perform another experiment in which we exploit the fact that the roughness of our ring potential grows with Vr . We calculate
how much further we would need to increase the trap roughness, by increasing theintensity of the LG beam, before the q = 3 and q = 1 state were inherently unstable and decayed immediately at the starting atom number. Our calculations predictthat the an increase of LG intensity by 30 ± 10% would cause immediate decay of139

5.5 Critical velocity

the q = 3 state (vs (q = 3, N = 180 × 103 ) > c1D (N = 180 × 103 )), and an increase of LG intensity by 70 ± 10% would cause immediate decay of the q = 1 state(vs (q = 1, N = 180 × 103 ) > c1D (N = 180 × 103 )). Experimentally we find agreementwith the values in the upper end of this range, typically needing to increase the LG
intensity by 40% to deterministically kill q = 3 flow, and by 80% to deterministicallykill all superflow. This suggests that the true value of our superflow velocity lies towards the lower end of our error region, and the critical velocity lies towards the upper
end of the error region. Since the two quantities have negative correlation, this would
be consistent with a small systematic underestimate in determining atom number, oroverestimate in determining trap roughness.
While our results do show excellent agreement with a critical velocity given by
vc = c1D , it’s important to note that our calculations of the flow velocity and criticalvelocity are built upon several steps, each of which introduces uncertainties. While wehave tried to account for errors in our analysis arising from fluctuations in atom number
and trapping potential, possible systematic effects such as atom number calibration,
trapping frequency measurements, and accounting for our finite imaging resolution, canalso effect our results. In our original analysis for example, by considering a slightly
larger PSF we found reasonable agreement between our data and the surface model
calculation of the critical velocity. It is therefore important to keep in mind that thedeconvolution step which accounts for our finite imaging resolution, only provides anestimate of the true density profile. Such systematic uncertainties make it difficult toreliably distinguish between vs and c1D which, over the range where we observe phaseslips, offer similar predictions of the critical velocity. Nevertheless, the fact that ourdata is in such close agreement with vc = c1D for a range of atom numbers and flow
velocities, is strongly suggestive that the critical velocity in our system is set by the
local effective sound velocity.
Comparison to previous measurements
We now compare our findings to those in [100], which provides the only other experimental measure of the critical velocity in an annular condensate. In this experiment,
q = 1 superflow was prepared using phase imprinting and a repulsive potential barrier
was raised, reducing the density at a point in the ring. For q = 1 superflow, the detection was binary, and hence they plotted the survival probability of circulation versus
barrier height. For a barrier height above a certain value they observed a dramatic
reduction in survival probability, associated with flow above the critical velocity in thebarrier region. Using similar methods to estimate the superflow velocity at the barrier, they find their measured vc to be consistent with the Feynman velocity vcF , andfrom this infer the decay mechanism to be phase slips. In contrast, for our experiment
we found the Feynman velocity did not provide an accurate prediction of the criticalvelocity we observed, and more importantly could not explain the variation in critical140

5.6 Statistics of stochastic phase slips

velocity we observed with changing atom number. At first glance then these two results
seem incompatible, however closer inspection actually reveals agreement between thetwo measurements, and only a difference in interpretation.
Importantly for the experiment in [100], the local chemical potential in the barrier
region is only on the order 100-200 Hz, and hence is bordering the quasi-1D regime.
For such a low chemical potential, the small radial extent of the annulus means theratio of the Feynman critical velocity to the effective sound velocity is close to its maximal value (5.41). While their measured critical velocity is found to agree with vcF , intheir case this actually corresponds to vc ≈ 60%c1D , and therefore the discrepancy is
not as large as first seemed. The remaining difference in critical velocity between theirexperiment and ours can be explained by the differing definitions of the critical velocity
which were used.
In our experiment we define the critical velocity as the flow velocity above which
decay is ultimately inevitable and occurs on the millisecond scale. Therefore we assign
the condition vs (q) = vc to the largest flow velocity we observe for a given q state. For
vs < vc we still observe phase slips, however we interpret these as rare stochastic decay
events which can occur through thermal or quantum fluctuations, as observed in liquid
helium and superconducting nanowire systems. In contrast, the condition for reaching
the critical velocity in [100] is defined as the velocity at which hqi = 1/2. While they doobserve some width to this transition, they attribute this to technical fluctuations rather
than any stochastic nature of the decay events. To distinguish this definition from ourown, we label such a velocity as an effective critical velocity v¯c . This effective criticalvelocity does not therefore define the maximal allowed flow velocity without decay, butinstead defines the velocity above which supercurrent decay becomes likely. Looking
at our data in figure 5.11, we observe that such a definition decreases the critical flow
velocity we consider by around 20%, and increases the sound velocity we consider byaround 25%. Therefore we find our data is also consistent with v¯c /c1D ≈ 0.6 − 0.8.
This suggests that the two data sets are not necessarily inconsistent, simply that theinterpretations of flow reaching the critical velocity are different. Importantly however,since our experiment uses data from several q states with decay at higher values of
the chemical potential, we are able to conclusively discount the Feynman criteria as an
accurate quantitative measure of the critical velocity. This enables us to interpret ourdata as stochastic phase slips as vs → vc , and a hard cutoff for superflow above thelocal sound velocity, c1D , where decay is inevitable.

5.6

Statistics of stochastic phase slips

In our discussion of the critical velocity, we defined the critical velocity as the minimum
superflow velocity of the system above which the superflow would decay both rapidly

1415.6 Statistics of stochastic phase slips

and inevitably. This is consistent with numerical solutions of the GPE which find thatfor flow above the critical velocity, phase slip events occurring via vortex-antivortex
annihilation take place within about 0.5 ms [173]. At absolute zero temperature in theabsence quantum tunneling of the order parameter, one would then expect a given q
state to remain perfectly stable until gradual atom loss brought about the conditionvs (q) = vc , at which point a phase slip would occur to the q − 1 state. Returning to theparticle in a washboard analogy in figure 5.1, the condition vs = vc then corresponds tothe energy barrier disappearing, and the local energy minima at L/N = q~ vanishing,causing decay to the next local energy minima.
Looking at our data in figure 5.11 (d), we clearly observe a less deterministic process.
We observe stochastic phase slips occurring for vs < vc , corresponding to particles
escaping their local energy minima before the energy barrier reaches zero. Purely
experimentally, this is directly observed in the horizontal overlaps of the different q
plateaus as a function of N in figure 5.8 (b). Similar overlaps are seen in the time
domain in figure 5.2 (b), showing that the observed q is not a deterministic functionof either t or N . From figure 5.11 (d), we see that only a small fraction of this overlap
can be attributed to technical fluctuations in our experiment. Factoring in the atom
number and trap fluctuations indicated by the shaded areas in figure 5.11 together withshot-to-shot variations in atom number detection of ∼ 6%, such technical fluctuations
can only account for ≈ 25% of the observed overlap in q-state plateaus.
We therefore conclude that there exists a significant parameter space where thesuperfluid flow is subcritical but stochastic phase slips still occur on a time scale ofseconds. We can then consider the supercurrent state, q, for vs (q) < vc , as in an energy
minimum where there still exists an energy barrier preventing decay to the q − 1 state.
As vs → vc , the size of this barrier tends towards zero, and hence the probability ofcrossing the barrier and causing a phase slip increases exponentially. As was observed inhelium and superconducting nanowire experiments, the nature of this barrier crossing
could be due to either thermal or quantum fluctuations.
The possibility of thermal activation of phase slips for vs < vc was considered in[175], where a numerical implementation of the truncated Wigner approximation wasused to simulate the experiment of Ramanathan et al. in [100] at finite temperature.
These calculations found the observed reduction in the effective critical velocity v¯c =0.6c1D from the zero temperature GPE value of v¯c = vc = c1D was consistent with atemperature of ≈ 35 nK. For our experiment we find an effective critical velocity for theq = 3 → 2 transition1 of v¯c = 0.73c1D , and we independently measure our temperatureby fitting the thermal component in TOF as T = 29 ± 3 nK. Such a result is certainlyof a similar order to that of the calculation and, provisionally at least, suggests thestochastic decay events we observe may be explained by thermal activation of the1

We cannot easily define similar critical velocities for subsequent phase slips since the initial population in these q states depends on decay from higher q states.

1425.7 Conclusion

Figure 5.12: Counting statistics of phase slips: (a) For the data shown in figure 5.2 we plot
the distribution of the observed q values as a function of rotation time t. Each data point isan average over a 0.8 second time bin. The inset shows the smooth evolution of hqi with t. (b)Histogram of q values for four representative rotation times.

system over an energy barrier. Such a hypothesis could then by tested by repeating
this experiment as a function of temperature. With increasing temperature the system
could be excited over a larger energy barrier at smaller values of vs /vc , and hence one
would expect to observe phase slips at larger values of N .
Our ability to resolve q states allows us to study the evolution of the q distribution
in time. This in essence provides full time-resolved counting statistics of phase slips,
and should provide excellent input for further theoretical modeling and understandingof our observed stochastic decay dynamics. In figure 5.12 we present such an evolution
in time for the same data set as in figure 5.2, which forms the basis of our criticalvelocity analysis. We note that this accelerating decay process is not Markovian since
the phase slip probability grows as vs /vc increases through the gradual decay of N . For
larger data sets this opens the possibility to fit this to solutions of the rate equation,and thereby extract how the phase-slip probability evolves with time. Such results
could provide insights into the form of the energy barrier, and therefore shed light on
the decay process and the role of thermal and quantum fluctuations. We also note thatthe mean rotational state, hqi, decays smoothly with time (inset of figure 5.12), so ourability to experimentally resolve different q states with high fidelity will be essential tofurther understanding phase slip dynamics.

5.7

Conclusion

In conclusion, we have demonstrated and studied long-lived multiply charged superflow
in an annular atomic BEC. We resolve with high fidelity quantised steps in the decay1435.7 Conclusionof the supercurrent, which corresponds to vortex-induced 2π phase slips. This result
allows us to unambiguously identify the decay mechanism as phase slips, independentof any calculations. The supercurrent decays rapidly if the flow velocity approaches acritical velocity that is in agreement with our numerical calculations of the excitation
spectrum. Our numerical calculations identify a transition from a large condensate
regime where the relevant excitations are surface instabilities and vc = vcs , to a small
condensate regime where the relevant excitations are phonons localised in the condensate bulk and vc = c1D . For our data on phase slips we are in the small condensate
regime, and hence observe a critical velocity in agreement with the averaged speed ofsound, as predicted by numerical solutions of the GPE.
While our calculated vc is in good agreement with the maximal flow velocity weobserve, we also find stochastic phase slips occur at a much lower rate, for lower flow
speeds. An important question for future work is whether these rare phase slip eventsoccur via quantum or thermal fluctuations. This obviously points towards further experiments studying the dynamics of phase slips as a function of temperature. Such an
experiment would be analogous to those performed in liquid helium [88] and superconducting nanowire [84] systems, which have been able to identify two distinct regimes
dominated by either thermal or quantum fluctuations. It should also be possible toreach the regime of one dimensional superflow, where it’s possible the supercurrent
decay could be fundamentally different.
Another important direction for future study would be to measure the critical velocity for higher charge superflow. One of the main sources of uncertainty in both
our experiment and that in [100], is that for modest q values the flow velocity only approached the critical velocity for low values of the local chemical potential. In particularfor our experiment, the deconvolution step introduces a systematic uncertainty whichis quite small as a fraction of the trap roughness. However, when compared to the local
chemical potential at which we observe decay, this uncertainty becomes significant. In
contrast, high charged superflow would decay at larger values of the local chemical potential where uncertainties in calculating the flow velocity and critical velocity would
be lessened. For now however, we decided to explore a different direction by studying
the role of the spinor degree of freedom on supercurrent stability, the results of which
are presented in the following chapter.

144

Chapter 6

Persistent currents in spinor condensates
In the previous chapter we studied the stability and decay of persistent currents insingle-component condensates. We showed that decay occurred through stochastic,
vortex-induced phase slips, the probability of which increased as the flow velocity approached the critical velocity. In this chapter we extend our studies to multi-component
systems, in particular those involving two spin states. Such an extension is essential
for the further understanding of superfluids with a vectorial order parameter, such as
those in d-wave and p-wave superconductors, and certain phases of superfluid 3 He.
These experiments are also pertinent to the application of ultracold atom systems,with persistent currents in multi-component systems a necessary aspect of many atominterferometers. Persistent currents in two-component Bose gases have been studied
theoretically [183–187] but many issues remain open. Even the central question ofwhether, and under what conditions, this system supports persistent currents has notbeen settled.
In this chapter we study the stability of supercurrents in a toroidal two-component
gas consisting of 87 Rb atoms in two different spin states. For a large spin-population
imbalance we recover the long-lived metastable behaviour of the single component casein Chapter 5, with superflow persisting for over two minutes and limited only by theatom-number decay. However at a small population imbalance the onset of supercurrent decay occurs within a few seconds. We demonstrate the existence of a well-defined
critical spin polarisation separating the stable and unstable current regimes. We also
study the connection between spin coherence and superflow stability, and show that in
our system only the modulus of the spin-polarisation vector is relevant for the stabilityof the supercurrent.

6.1

Multi-component condensates

Multi-component condensates are defined as systems where two or more internal quantum states are macroscopically populated. Such systems then possess internal degrees
of freedom relating to the relative population and phase of each component, in addition to the usual external degrees of freedom of a single-component condensate. The
145

6.1 Multi-component condensatesinterplay between these external and internal degrees of freedom in multi-component
systems can then lead to a range of phenomena not present in scalar condensates. In
this section we aim to introduce some of the theory of multicomponent systems. We
emphasise that a complete theoretical understanding of our observations regarding thestability and decay of a two-component superflow is still lacking. The topics covered
here are therefore not intended to offer an explanation of the measurements which follow, but simply touch on some of the theory which we feel may be relevant.
The requirements for studying multi-component condensates are two fold. First thepossibility of studying multi-component condensates is dependent on the collision properties of the given mixture. Many configurations suffer from the presence of scatteringresonances and scattering channels with negative s-wave scattering length, leading toheating and condensate collapse respectively. Second, one requires a trapping potential capable of confining both states with non-trivial overlap. For magnetic traps thisequates to studying states which posses equal magnetic moments. In a conventional
magnetic trap, the spins of the atoms are frozen, constrained to follow the field direction. As a result, even though the atoms carry spin, they behave like scalar particles.
Possible multi-component condensates in magnetic traps are then restricted to different
hyperfine states of the same isotope, and mixtures of two different species of bosons.
Conversely, in an optical trap the spins of the atoms are essentially free, providing uswith an additional non-trivial degree of freedom. Possible multi-component condensates in optical traps then also include different magnetic states of a single hyperfine
level. This leads to the natural distinction of multi-component condensates into two
categories: mixtures and spinors.

6.1.1

Spinor condensates

A spinor is defined as a multi-component condensate containing all magnetic sublevelsof a manifold of states with a single value of the hyperfine spin F . For the F = 1 state
which we consider here, this corresponds to a condensate in the three hyperfine spinstates mF = 1, 0, and −1. The internal states of a spinor condensate are then related byrotational symmetry in spin space. Experimentally, long-lived spinor gases have beenrealised in three spin manifolds: the F = 1 hyperfine manifold of both 23 Na [188] and87 Rb [189], and also the upper F = 2 manifold of 87 Rb [190]. The significance of sucha system is that while the total particle number is conserved, due to the possibility ofspin-exchange collisions the particle number for any one mF state is not. For example,
an atom in the mF = 1 state may inelastically scatter with another in the mF = −1
state to give two atoms in the mF = 0 state, or vice versa.
The general formalism of the low energy Hamiltonian for a spinor condensate isgiven in [191, 192]. Collisions in the system are described by a pairwise interactionthat is rotationally invariant in the hyperfine spin space and preserves the hyperfine
spin of the individual atoms. Two identical bosons with F = 1 in an s-state of their146

6.1 Multi-component condensates

relative motion can couple to make states with total angular momentum, F = 0 or
2, since the requirement that the wave function is symmetric under particle exchange
rules out collisions with F = 1. The general form of the interaction is then given by
V (r1 − r2 ) = δ(r1 − r2 )(g0 P0 + g2 P2 ),(6.1)

where gF = 4π~2 aF /m, the operators PF project the wave function of a pair of atoms
onto a state of total angular momentum F, and a0 and a2 are the corresponding scattering lengths for the singlet and triplet channels respectively. This is usually expressedin terms of the operators for the angular momentum of the two atoms, denoted by F1and F2 . The contact interaction may then be expressed asg0 P0 + g2 P2 = c0 + c2 F1 · F2 ,
where
c0 =

4π~2 2a2 + a0
,
m
3

c2 =

4π~2 a2 − a0.
m3

The Hamiltonian in second quantised form is then 2
Z~Ĥ = dr∇Ψ̂†a · ∇Ψ̂a + Vext Ψ̂†a Ψ̂a
2m

c2
c0
+ Ψ̂†a Ψ̂†a0 Ψ̂a0 Ψ̂a + Ψ̂†a Ψ̂†a0 Fab · Fa0 b0 Ψ̂b0 Ψ̂b ,
2
2

(6.2)

(6.3)

(6.4)

where Ψ̂a is the field annihilation operator for an atom in the hyperfine state mF = a(a = 1, 0, −1) at point r, and the angular momentum matrices are written as






0 1 0
0 −1 0
1 0 0




1 
i 
Fx = √ ,
Fy = √ ,
Fz = 
1 0 1
1 0 −1
0 0 0





.
2
2
0 1 0
0 1
0
0 0 −1(6.5)

We note that in the absence of an external magnetic field to break the symmetry, theHamiltonian is rotationally symmetric in spin space.
The time evolution of the field operator can be found by using the Heisenberg

1476.1 Multi-component condensates

equation with the Hamiltonian (6.4):
#
"
X †
∂ Ψ̂1
~2 2
i~
Ψ̂i Ψ̂i Ψ̂1=−∇ Ψ̂1 + [Vext − µ1 ]Ψ̂1 + (c0 + c2 )
∂t
2m
i


†
†
+ c2 Ψ̂−1 Ψ̂0 Ψ̂0 − 2Ψ̂−1 Ψ̂−1 Ψ̂1 ,
"
#
X †
∂ Ψ̂0
~2 2
i~
=−
∇ Ψ̂0 + [Vext − µ0 ]Ψ̂0 + (c0 + c2 )
Ψ̂i Ψ̂i Ψ̂0
∂t
2m
i


+ c2 2Ψ̂†0 Ψ̂1 Ψ̂−1 − Ψ̂†0 Ψ̂0 Ψ̂0 ,
"
#
X †
~2 2
∂ Ψ̂−1=−∇ Ψ̂−1 + [Vext − µ−1 ]Ψ̂−1 + (c0 + c2 )
Ψ̂i Ψ̂i Ψ̂−1
i~
∂t
2m
i


+ c2 Ψ̂†1 Ψ̂0 Ψ̂0 − 2Ψ̂†1 Ψ̂1 Ψ̂−1 ,

(6.6)

(6.7)

(6.8)

where the three chemical potentials µi , are defined such that the total particle number
and magnetisation are conserved. In this formalism one can explicitly identify thespin mixing terms. Importantly, such spin exchanging collisions coherently transfer
populations between different spin states, with coherent oscillations observed in mF
populations due to such collisions [193, 194]. Coherent spin dynamics preserve thecoherence between the different internal state components of a spinor BEC, and manyof the characteristic phenomena of spinors, such as spontaneous magnetisation andsymmetry breaking, are physical consequences of this coherence. In a mixture wheresuch spin-exchange terms are absent, while there can still be coherence between theinternal state populations, such coherence does not affect how the physical systemevolves. Therefore, while a mixture can be considered to evolve as an incoherent overlapof individual condensates, such a picture is not valid for spinor gases.
The significance of spin dependent interactions is determined by the value of c2 ,which is generally much smaller in magnitude than c0 . Nevertheless, the sign andmagnitude of c2 is found to have significant implications on the properties of a spinorgas, including its ground state structure, spin wave modes, and possible topologicalspin structures such as coreless vortex states. So far we have neglected any possible
effect due to long-range magnetic dipole-dipole interactions (MDDI). For most ultracold
atom experiments the MDDI is negligible in comparison to the spin-independent contact
interaction1 . For spinor gases however, since the MDDI is both spin-dependent andlong-ranged, it is expected to play a significant role in determining the low field groundstate [197], and the long range formation of spin structures [198]. Experimental evidence
for the role of MDDI in spinor condensates has even been observed in [199], where thespontaneous dissolution of long-wavelength spin textures in a 87 Rb F = 1 spinor BEC
was attributed to long-range magnetic dipole interactions. As we have alluded to,1

This is not the case for condensates of 52 Cr [195] and 164 Dy [196], which have exceptionally large
magnetic dipole moments, where the energy associated with long-range dipole interactions can becomecomparable to that of the mean-field interaction energy.

1486.1 Multi-component condensates

spinor condensates exhibit a dizzying array of static and dynamic properties. Reviewsof some of the pertinent experiments and directions of research in this expansive field
are available in [200] and [201]. We now consider the often simpler case of a condensate
mixture, where spin exchange collisions are forbidden.

6.1.2

Mixtures

A mixture is defined as a multi-component condensate where the particle number foreach species is strictly conserved. If we consider a mixture of two-components labeled
1 and 2 with equal mass, which experience the same trapping potential, the GP energyfunctional is given by [19]~2~2|∇Ψ1 |2 + Vext (r)|Ψ1 |2 +|∇Ψ2 |2 + Vext (r)|Ψ1 |2
2m
2m

g11
g22
+
|Ψ1 |4 +|Ψ2 |4 + g12 |Ψ1 |2 |Ψ2 |2 .
2
2


Z

E=

dr

(6.9)

The interaction parameters g11 , g22 , and g12 = g21 are related to the respective scattering lengths a11 , a22 , and a12 by gij = 4π~2 aij /m. Minimising this energy for fixed
particle number one obtains the time-independent coupled GP equations~2 2∇ Ψ1 + Vext (r)Ψ1 + g11 |Ψ1 |2 Ψ1 + g12 |Ψ2 |2 Ψ1 = µ1 Ψ1
2m
~2 2
−
∇ Ψ2 + Vext (r)Ψ2 + g22 |Ψ2 |2 Ψ2 + g12 |Ψ1 |2 Ψ2 = µ2 Ψ2 .
2m−

(6.10)(6.11)We therefore see that for a mixture, each component simply experiences the mean fieldpotential arising from both populations. The coupling between the two components
through the intercomponent mean field as determined by the value of g12 , yields structures and dynamics which are not present in a single-component BEC.
The two-component persistent current we study is a mixture of atoms in the|F = 1, mF = 1i and |F = 1, mF = 0i magnetic states, and therefore is a spinor gas.
We note however that our system is in an external magnetic field of 10 G, and therefore
due to the quadratic Zeeman shift, the energy gap between the mF = −1 and mF = 0
states is 14.4 kHz larger than the energy gap between the populated mF = 1 andmF = 0 states. Spin exchange collisions to the mF = −1 are therefore energeticallyprohibited, and for an initial state where the mF = −1 state is not populated, the populations of the mF = 1 and mF = 0 states are conserved. Taking the time dependent
GPE equations for a spinor in (6.6) and (6.7) and removing spin-exchange terms, one
retrieves the equations of motion for a mixture with inter and intra-species scattering
lengths given by
a11 = a2 ,a00 =2a2 + a0,3149

a10 = a2 .

(6.12)

6.1 Multi-component condensates

We therefore anticipate our spinor condensate to be described by a mixture with theappropriate scattering lengths given above in terms of the singlet and triplet scattering lengths. Such a view is confirmed experimentally by the fact that we observe no
population of the mF = −1 state due to coherent scattering from the mF = 0 state.
Calculation of the scattering lengths in terms of singlet and triplet scattering lengths
can also be done by decomposing the states of total angular momentum |F, mF i using
the appropriate Clebsch-Gordan coefficients|F, mF i =

mX
1 =F1

mX
2 =F2

|F1 , mF 1 , F2 , mF 2 i hF1 , F2 ; mF 1 , mF 2 | |F, mF i ,(6.13)

m1 =−F1 m2 =−F2

where F1 = 1 and F2 = 1. One can then express the state of the two incident particles|F, mF 1 , F, mF 2 i in terms of states of their combined total angular momentum F:|mF 1 = 1, mF 2 = 1i = |F = 2, mF = 2irr
21
|F = 2, mF = 0i −|F = 0, mF = 0i
|mF 1 = 0, mF 2 = 0i =3
3
1
√ (|mF 1 = 0, mF 2 = 1i + |mF 1 = 1, mF 2 = 0i) = |F = 2, mF = 1i .
2(6.14)(6.15)(6.16)

from which one confirms the results of (6.12). The values of a0 and a2 for the F = 1
state of 87 Rb are given as [68, 202] a0 = (101.78 ± 0.2)aB and a2 = (100.4 ± 0.1)aB ,where aB is the Bohr radius. From this one calculates the relevant inter and intraspecies scattering lengths for our system as
a11 = (100.4 ± 0.1)aB ,a00 = (100.86 ± 0.13)aB ,a10 = (100.4 ± 0.1)aB .

(6.17)

As we discuss in Section 6.1.4, the fact that these values are so close has importantconsequences for the ground state and dynamics of our system. Strictly speaking, one
may only decompose the scattering lengths in terms of singlet and triplet channels
at low magnetic fields where F is a good quantum number [191]. We assume thisdecomposition is valid in our experiment since the associated Zeeman energies aremuch smaller than the hyperfine splitting.
In this chapter we use the pseudo-spin representation, where our two-component
mixture is represented by a spin-1/2 BEC [203]. Introducing the normalised spinor
χ(r) = [χ1 , χ2 ]T = [|χ1 |eiθ1 , |χ2 |eiθ2 ]T , we can decompose the wave function of eachpcomponent as Ψi = n(r)χi (r), where n(r) is the total density of both components.
Defining the spin density as S = χ(r)† σχ(r), where σ is the Pauli spin matrix, the unit
spin vector is then defined as


Sx





2Re(χ1 χ∗2 )



  

Sy  =  2Im(χ1 χ∗ )  .
2 
  
Sz
|χ1 |2 − |χ2 |2150(6.18)

6.1 Multi-component condensates

We therefore represent our two-component spinor BEC as a unit vector on the Bloch
sphere, where mF = 1 and mF = 0 are the up and down components of the spin 1/2
spinor respectively. This formalism implies a relative phase coherence between the up
and down states. Decoherence then corresponds to a shortening of the state vector
from |S| = 1. The energy functional (6.9) can then be written in the form
√
mn 2
~2
~2 n
(∇ n)2 +(∇S)2 +
v
2m
8m
2 eff

n2
+Vext + (γ0 + γ1 Sz + γ2 Sz2 ) ,
2


Z
E=

dr

(6.19)

where we define the coupling constants
g11 + g22 + 2g12,
4
g11 − g22,
γ1 =2
g11 + g22 − 2g12
γ2 =
,
4

(6.20)(∇S)2 = (∇Sx )2 + (∇Sy )2 + (∇Sz )2 ,(6.21)

γ0 =the spin stiffness

and the effective velocity~(χ∗ ∇χ1 − χ1 ∇χ∗1 + χ∗2 ∇χ2 − χ2 ∇χ∗2 )2im 1Sz (Sy ∇Sx − Sx ∇Sy )~∇(θ1 + θ2 ) +,=2m
Sx2 + Sy2

veff =(6.22)(6.23)

which depends both on the gradient of the total phase and that of the pseudo-spin. This
form illustrates that for a general mixture where g11 6= g22 6= g12 , the system experiencesa fictitious interaction-induced linear and quadratic Zeeman shift which breaks therotational symmetry in our pseudo-spin space. The form of the effective velocity also
has important connotations for the stability of persistent currents in a Bose mixture.
In general we note that there is now an interplay between mass currents, associatedwith the gradient of the phase, and spatial variations in the spin vector. Formationof spin textures is energetically costly due to the presence of the (∇S)2 term, which
introduces a spin stiffness to the system. This equation is analogous to the classicalnonlinear sigma model which describes Heisenberg ferromagnets.

6.1.3

Dynamic stability

In Chapter 5 we found that the onset for superflow decay by phase slips could beunderstood by considering the excitation spectrum of the system. In that same spirit
we now consider the excitation spectrum for a two-component mixture described byequations (6.10) and (6.11). For a uniform system the time-dependent GPE takes the1516.1 Multi-component condensates

form


∂Ψ1
~2 2
2
2
i~
= −
∇ + g11 |Ψ1 | + g12 |Ψ2 | Ψ1
∂t
2m


∂Ψ2
~2 2
2
2
i~
= −
∇ + g22 |Ψ2 | + g12 |Ψ1 | Ψ2 ,∂t
2m

(6.24)(6.25)

where the wavefunctions are normalised as |Ψi |2 = ni , where ni = Ni /V is the particle
density for component i. As before, when we calculated the excitations of a single
species uniform condensate in Section 2.3.1, we consider plane wave perturbations to
the ground state of the form(0)Ψi (t) = e−iµi t/~ (Ψi + δΨi (t)),(6.26)

δΨi (t) = ui e−i(ωt−q·r) + vi∗ ei(ωt−q·r)(6.27)

whereµi = gii ni + gij nj ,(6.28)

for the two components {i, j} = {1, 2}. Inserting this into the coupled, time-dependent
GP equations we obtain the matrix equation


u1





q + g11 n1

g11 n1   v1   −g11 n1−q − g11 n1
 
~ω 
√
u  =  g √n n
g12 n1 n2 2   12 1 2√√−g12 n1 n2 −g12 n1 n2
v2√
g12 n1 n2√−g12 n1 n2
q + g22 n2−g22 n2

√
g12 n1 n2√−g12 n1 n2 ,
g22 n2 −q − g22 n2(6.29)

where q = ~2 q 2 /2m is the free particle energy. The energies of the excitations are thengiven by the determinant of this matrix as
1
1q 2
2 ,
(~ω)2 = (21 + 22 ) ±(1 − 22 )2 + 162q n1 n2 g12
2
2(6.30)

where21 = 2g11 n1 q + 2q(6.31)

22 = 2g22 n2 q + 2q .

(6.32)In the limit that we remove the inter-species coupling (g12 = 0), or the density of
one of the components goes to zero, we retrieve the excitation spectrum for a uniform
single species condensate which we derived in Section 2.3.1. The condition for dynamic
stability is that all the eigenfrequencies are real. From (6.30) this correspond to thecondition
g11 g22 > g12 2 .

152(6.33)

6.1 Multi-component condensates

This is often expressed as the criterion for the two components to be miscible. If thiscondition is not met, the mean field energy each component experiences from the otherspecies is greater than that which it experiences from itself, and hence the lowest energy
configuration is for the two-components to separate spatially. If this is the case, thetime scale of this phase separation is given by the value of the largest imaginary eigenvalue, and the length scale of the spin textures which form is given by the wavenumber
q associated with this eigenfrequency. The dynamics of such phase separation havebeen observed experimentally in [204–206].
For our two components, mF = 1 and mF = 0, the scattering lengths (6.17) aresuch that we anticipate our system to be dynamically stable and therefore miscible.
As the scattering lengths are so close however, the maximum energy barrier towards
phase separation, occurring for equal populations in each component, is only 3 Hz per
particle. We therefore find that our two-component system is extremely sensitive toexternal factors such as magnetic field gradients and the polarisation of our opticaltrap, which can differentiate between the two states and overcome this small energy
barrier towards phase separation. The cancelation of such effects is discussed in Section
6.1.4. Due to fluctuations of both the experiment, and inherent thermal and quantum
fluctuations of the system [148], over very long times we find the system is unstabletowards the formation of spin textures. Our measurements of this effect are presented
in Section 6.1.5.
Our calculations of the excitation spectrum for a two-component condensate ina ring are shown in figure 6.1. This calculation was done by extending our numericalcalculation for the 2D single species BEC in a ring (explained in Appendix B) to thetwo-component case. A similar result is obtained by mapping the excitation spectrum
calculated for a uniform two-component condensate in (6.30), to that of a one dimensional system on a ring. This amounts to simply replacing the interaction parameters,
gij , with effective 1D parameters averaged over the radial and vertical directions. Dueto the very small spin dependent energy, we find that the dispersion relation is very
close to quadratic. The energy barrier to formation of long-wavelength excitations istherefore very small, however the restriction that the wavefunction is single valued leads
to quantised values of the angular wavenumber m, which increases this somewhat. The
form of the excitation spectrum is very similar for all ratios of the two populations. The
inset, which shows the spectrum at small values of m for three different ratios of thetwo populations, actually shows that the case of equal populations in each component
is the most stable. As the population imbalance becomes larger the system becomesless stable.
Unlike the single species case, inferring the stability of persistent currents from calculations of dynamic stability is clearly not correct for the two-component case. All
this calculation tells us is that if one adds a small admixture of a second component
to the system, the energy barrier to creating spin textures is very small. As the ad-1536.1 Multi-component condensates

Figure 6.1: Excitation spectrum for an annular two-component BEC: Plotted is the excitationenergy (in units of ~ωr ) versus angular wavenumber for numerical solutions of the 2D GPE for a
two-component mixture in a ring with equal populations in the mF = 1 and mF = 0 states. This
calculation predicts an excitation energy close to quadratic in m, and therefore a small energy
barrier to the formation of long-wavelength excitations. Inset is the initial part of the slope forthe three different mixture populations N (mF = 1) : N (mF = 0) = 0.5 : 0.5, 0.75 : 0.25, and0.98 : 0.02, as indicated by the black dots, blue triangles, and red triangles respectively. The
spectrum is close to identical under exchange of the relative populations of the two components,
N (mF = 1) and N (mF = 0).

mixture becomes larger, the energy cost of creating such spin textures increases in atrivial manner. The stability of the system against forming such excitations does notdirectly tell us about the stability of the system against dissipating angular momentum.
Indeed such a calculation shows us that if we take the single species system, which wehave shown to possess a large critical velocity, and add a vanishing small admixture of
the second component, this critical velocity all but vanishes. Such a discontinuity inbehaviour is unphysical, and clearly indicates that the excitations we are consideringin the two-component case are distinct from those in the single-component case, anddo not relate directly to the decay of supercurrents.
The primary tool by which we were able to understand the onset of superfluid instability in the single species case does not then seem to apply to the two-component
case. As we will show, while we have been able to accurately measure the conditions
under which a two-component persistent current becomes unstable, as yet we do not
fully understand this condition. This problem is therefore perfectly poised for further
theoretical work to both explain our findings, and point the way to future experiments.

6.1.4

Phase separation

To observe long-lived persistent currents in a two-component condensate, it was firstimportant to remove environmental factors which would differentiate between the two1546.1 Multi-component condensates

Figure 6.2: Phase separation: (a) Absorption images of the mF = 1 (|↑i) and mF = 0 (|↓i)
states phase separating in the presence of a weak magnetic field gradient. The surface plotshows the total density, which is observed to remain constant during the phase separation. The
colour of the surface indicates the relative population of the two components. (b) Absorption
image of the mF = 1 state only, in an ` = 5 LG beam ring trap in the presence of a small
amount of leakage Gaussian light. The density of the mF = 0 state is the inverse of this, suchthat the total density is uniform around the ring.

states. For the mF = 1 and mF = 0 states which have different magnetic moments,
the primary concern is magnetic field gradients. In the presence of in-plane gradientsthe two-component gas is extremely susceptible to phase separation. In figure 6.2 (a)we show in-trap absorption images of the mF = 1 (|↑i) and mF = 0 (|↓i) states phaseseparating due to a small field gradient. The total density is uniform around the ring,and only by simultaneously trapping two components can one detect the presence of themagnetic gradient. The orientation of the phase separation is repeatable, and therefore
we were able to cancel this gradient by the use of two orthogonal compensation coils
positioned close to the atoms. The sensitivity of the two-component gas is remarkable,
and by ensuring the two states showed perfect and repeatable spatial overlap even after
10 seconds hold time, we minimised in-plane magnetic gradients to < 5 × 10−5 G/cm,
roughly equivalent to an energy difference across the ring of 1 Hz per particle.
A second, less obvious, cause of phase separation turned out to be the imperfectpolarisation of our ring trap light. As derived in Section 4.3.2, the dipole potential fora multilevel system depends on the product of the polarisation of the trapping lightand mF state of the atom. Our LG trapping beam has equal components of σ + and σ −polarised light, and hence should trap all mF states equally. In reality, slight misalignment with respect to the field direction or polarisation dependent reflections will cause
a marginal imbalance in the polarisations, and hence one mF state will see a deeper
trap than the other. For most situations this simply manifests as a small global shift
in the potential and has no effect on the experiment. Due to our Raman setup (figure4.7), a small amount of LG light propagates along the Gaussian path and does notpick up a 2π` phase winding. Interference of these two beams at the atoms creates a155

6.1 Multi-component condensates

small azimuthal corrugation in the LG intensity, which for a single species condensate
is undetectable. If we trap two species however, the combination of this interference
and the imperfect polarisation, cause a phase separation of the like shown in figure 6.2(b). The state which experiences the slightly deeper potential moves into the minimaof the corrugation, and the other spin state fills the gaps to preserve the total density.
This effect appears identical to the matter-wave interference of Section 4.4.1, but issimply due to the trapping potential. To remove this effect we place a shutter in thepath of the Gaussian light, which is only briefly opened during the Raman transfer.
Having canceled these two effects, we observe the two states remain mixed after
10 seconds hold time. As shown in the next Section, at even longer times we find thesystem is unstable towards forming variations in the densities of the two individualcomponents. Neither the direction nor degree of such spin textures is repeatable however, and therefore we cannot identify the source of these dynamics as being either
technical, or as inherent fluctuations present in the two-component condensate.

6.1.5

Spin domain formation

There have been many studies on spin mixing dynamics in degenerate spinor Bose gases[200]. Such dynamics are associated with the redistribution of atomic populations
among spin states and the formation of spin domains and spin textures. Of thoseexperiments, [206] and [207] also studied two-component F = 1 spinor gases in thepresence of large quadratic Zeeman shifts, where spin-exchange scattering to the third
mF state was energetically eliminated. Both of these studies used immiscible mixtures,
where the gas was dynamically unstable to phase separation. Due to the small spin
dependent energies, such systems take several seconds to reach equilibrium, and exhibit
interesting features such as the growth and coarsening of large scale spin domains which
are nucleated by initial fluctuations.
In our case of mF = 1 and mF = 0, the two states are miscible. To characterise our
two-component state we define the longitudinal spin polarisation
Pz =N↑ − N↓,N↑ + N↓(6.34)

where N↑ (N↓ ) is the number of atoms in the mF = 1(mF = 0) state. Using an√
RF π/2 pulse we prepare an initial superposition state, |Ψi = (|↑i + |↓i)/ 2 withequal magnetic state populations (Pz = 0), and then hold for various times. We thenmeasure the local variation of Pz by imaging only the mF = 1 state using a microwave
π pulse to transfer the population to the imageable |2, 2i state. In our system, for thestarting atom number, q
the width of the annulus (Rr ≈ 6 µm) is comparable to the spin2
≈ 4 µm)1 [199]. As the atom number decays during
healing length (ξs = 1/ 8πn a0 −a3
1
The spin healing length is similarly determined as the length below which the kinetic energy ofspins wins over the spin-dependent interaction energy.

156

6.1 Multi-component condensates

Figure 6.3: Spin domain formation: (a) Using an RF π/2 pulse at t = 0 we prepare a spinsuperposition state with equal populations in each magnetic state (Pz = 0). Allowing thestate to evolve in time, the superposition decoheres and the transverse spin polarisation decays(Section 6.3). We also observe a spreading in Pz values about the mean which we detect byimaging only the |↑i state using a microwave transfer. (b) Evolution of the variance of Pz withhold time. Each data point is an average of 8 images, and normalised to control shots which
account for variations in the total density due to the imperfect trapping potential. (c) The
distribution of measured local Pz as a function of hold time. Insets show typical absorption
images of the |↑i state only, at short and long hold times. At all times the total density is
uniform.

evolution, the width of the annulus decreases while ξs increases, and hence the spin
degree of freedom becomes increasingly one-dimensional. We therefore approximatethat the spin structure only varies significantly azimuthally around the ring and not
across the width of the annulus. Hence we integrate the density over the radial direction
when measuring Pz .
The results of this measurement are shown in figure 6.3. While the total Pz isconserved, we observe the formation of local spin domains of high and low Pz , thevariance of which is plotted against hold time in figure 6.3 (b). We find the fluctuationsin Pz grow over the first 20 seconds and then saturate. The saturation value is close to
the value of 1/3, which one would obtain if the distribution of Pz values was uniform157

6.2 Two-component supercurrent

between −1 and 1. In figure 6.3 (c) we plot the evolution of the Pz distribution withtime. At short times the Pz values are tightly distributed about the initial Pz = 0
value, corresponding to all state vectors lying on the equator of the Bloch sphere.
With increasing hold time an increasing range of Pz values is observed, corresponding
to atoms moving off the equator of the Bloch sphere. At very long times the distribution
is close to uniform for all Pz values, corresponding to a uniform distribution of statevectors over the Bloch sphere. This particular data is taken for an initial superposition
state which is rotating with q = 3 at t = 0. The same evolution of Pz values is observed
for the non-rotating case. The fact that the distribution of Pz values shows no angulardependence around the ring suggests this is not a technical artifact resulting fromremnant magnetic gradients.
If one only considers the s-wave contact interaction, then our spinor BEC should
be dynamically stable and the observed formation of spin textures at long hold times
is difficult to explain. The fact that the energy barrier to phase separation is so small
though, means additional mechanisms which one usually neglects may have a significant
effect. In the experiment of Vengalatorre et al. [199] for example, it was found for a
87 Rb F

= 1 spinor condensate that the role of magnetic dipole-dipole interactions

could have a significant effect on the stability of spin structures. The influence ofdipolar interactions on the spinor gas was evidenced by the spontaneous dissolution ofdeliberately imposed long-wavelength spin structures, in favour of a finely modulated
pattern of spin domains. This effect was ascribed to the magnetic dipole energy which
disfavours the homogenously magnetised state. One could therefore hypothesise thatthe spin dynamics we observe may similarly be driven by magnetic dipole interactions.

6.2

Two-component supercurrent

To prepare our two-component supercurrent we first prepare a single-component persistent current using the same method explained in Chapter 4. As before, we load
our condensate into the ring trap created by intersecting the sheet laser beam with anLG` beam of ` = 3. Due to changes in the sheet beam optics, the vertical trapping
frequency is reduced to ωz = 2π × 340 Hz. Initially we load about N ≈ 150 × 103
atoms into the ring trap, which we then hold for 6 seconds to reduce the starting atom
number for our measurements to N ≈ 100 × 103 . This reduction in atom number has
no discernible effect on the supercurrent stability, but is found to make the angularmomentum state of the two-component supercurrent easier to measure. Due to spin
dynamics, the presence of both spin states leads to azimuthal density variations in thedensity holes which we observe at long TOF. These density variations can make theradius of the hole, and hence the angular momentum of the state, harder to accurately
quantify. Empirically we find a reduced atom number improves our fitting procedureand enhances the signal of quantisation.

158

6.2 Two-component supercurrent

Figure 6.4: Preparation of a two-component supercurrent: (a) Supercurrent is induced in asingle component condensate by a Raman transfer of atoms between the two spin states |↑iand |↓i, using the LG beam and an auxiliary Gaussian (G) beam. A two-component gas isthen created by coupling the same |↑i and |↓i states with an RF field which imparts no angularmomentum to the system. Here we denote the energy difference between the stationary androtating |↑i state of q 2 Er , with Er /h ≈ 0.4 Hz. (b) Plot of measured spin polarisation, Pz ,versus RF pulse time. By varying the time of the RF pulse between tRF = 0 and tRF = tπ = 140µs we can prepare a state of any given Pz . Also shown is the Bloch sphere representation of therotation of the initial state vector of our spin 1/2 system with RF pulse time. The blue arrowdenotes our two-component state vector, while the green arrow denotes the effective magnetic
field created by the RF coupling, which the state vector then precesses about.

To set the gas into rotation the auxiliary Gaussian beam is flashed on briefly, transferring the atoms via a two-photon Raman transfer between the initial |F = 1, mF = 1i
state and the final |F = 1, mF = 0i state with angular momentum 3~ per atom. Any
atoms which are not transferred are removed from the trap by a microwave transfer andresonant imaging pulse, as described in Section 4.3.1, creating a pure single-component
persistent current. To amplify any difference between the single and two-component
supercurrent, we maximise the single-component superflow lifetime by reducing the intensity of the LG trapping light to create the smoothest trap possible. Our final ring
trap has radius 12 µm, radial trapping frequency ωr = 2π × (53 ± 5) Hz, and the trap
depth is about twice the BEC chemical potential, µ0 /h ≈ 750 Hz. At this point theazimuthal variations in the local chemical potential are less than 10% and the single
component superflow persists for up to two minutes, limited only by the eventual atomloss.
The |F = 1, mF = 1i and |F = 1, mF = 0i states also define the |↑i and |↓i states,
respectively, of the spin space for our two-component supercurrents. Since the mF = 1
and 0 states are split in an external field of 10 G, the mF = −1 state is detuned from
Raman and RF resonances by 14.4 kHz. We therefore observe negligible population of
the mF = −1 state during state transfers, and population of the mF = −1 state byspin exchange collisions is prohibited by energy conservation. To first order, the large

1596.2 Two-component supercurrent

Figure 6.5: Detection of two-component supercurrent: (a) Time-of-flight absorption image ofthe atoms, with spin states separated using a Stern-Gerlach gradient. The rotational state q
is deduced from the radius R characterising the central hole in the density distribution. The
longitudinal polarisation Pz is measured directly from the image using the relative populationsof the two spin states. The image shown was taken after t = 4 s of rotation; Pz = 0.44 andq = 3 for both spin states. (b) Histogram of ≈ 900 measurements of R at various Pz and t.
Once again the background colours indicate the corresponding q state for a given R, with q = 3,
2, 1, and 0 indicated by the blue, green, red, and black regions respectively.

bias field has eliminated the presence of the mF = −1 state, reducing the parameter
space of our system of our F = 1 spinor to a two-component mixture.
The protocol for preparing a two-component supercurrent is illustrated in figure
6.4 (a). After preparing a pure |q = 3, ↓i rotating state, we couple |↑i and |↓i by an
RF field which carries no orbital angular momentum and does not affect the motional
state of the atoms. As shown in figure 6.4 (b), by varying the length of the RF pulsetRF in the range 0 ≤ tRF ≤ tπ , one can prepare a state of arbitrary longitudinal spinpolarisation, Pz . The prepared state is then in a superposition of the mF = 1 andmF = 0 state, both rotating in the same direction with angular momentum 3~ peratom:|Ψ(φ)i = (sin(φ/2) |↑i + cos(φ/2) |↓i)e−iqθ .

(6.35)

Here φ = ΩRF tRF , where ΩRF is the RF Rabi frequency. In this state Pz = cos(φ) andthe gas is still fully spin polarised with polarisation vector P~ = [sin(φ), 0, − cos(φ)] and|P~ | ≡ P = 1.
After preparing a co-rotating (q = 3) two-component supercurrent in a specificspin state, we let it evolve in the ring trap for a time t and then probe it by absorption
imaging after 29 ms time-of-flight expansion as explained in Section 4.4.2. We separate
the two spin components with a Stern-Gerlach gradient and directly detect the longitudinal spin polarisation, Pz , by measuring the populations of the two spin states. As
before, the rotational state, 0 ≤ q ≤ 3, is seen in the size R of the central density hole160

6.2 Two-component supercurrent

Figure 6.6: Single versus two-component supercurrent stability: (a) In a pure |↑i state with
Pz = 1 supercurrent is observed to persist for over 2 minutes, with no phase slips occurring for≈ 90 s. (b) At Pz = 0 the first phase slip occurs within 5 seconds and we observe no rotation
beyond 20 seconds. (c) Total atom number decay for Pz = 1 in open red symbols, and Pz = 0
in solid blue symbols. Dashed lines are double-exponential fits.

in the atomic distribution, arising due to the centrifugal barrier. An example image isshown in figure 6.5 (a). A histogram of fitted R values for the full range of Pz values
and hold times in the range 0 ≤ t ≤ 20 s1 is shown in figure 6.5 (b). The R values
are clearly quantised for all Pz , allowing us to determine q with near perfect fidelity.
For spin polarisations sufficiently close to zero, where we are able to assign a q state
to both spin states, we always observe the two components to be in the same q state.
This explicitly identifies that the superflow decay mechanism is once again a 2π phaseslip which affects both spin states simultaneously. Our observation do not however
preclude the possibility that the supercurrent may first only decay in one of the spincomponents, after which the relative flow velocity between the two states causes a rapid
secondary phase slip in the faster component.

1616.2 Two-component supercurrent

6.2.1

Single versus two-component supercurrrent

In figure 6.6 we illustrate the dramatic difference between superflow stability in a Pz = 1
single-component gas and a Pz = 0 two-component system with equal populations ineach spin state. The two different Pz states are created respectively, by a (140 µs) πand (70 µs) π/2 RF pulse at t = 0. In the pure |↑i state (figure 6.6 (a)) the current
persists for over two minutes, with the BEC always remaining in the q = 3 state for≈ 90 s. In contrast at Pz = 0 (figure 6.6 (b)) the first phase slip occurs within 5 seconds
and the current completely decays within 20 seconds.
As we demonstrated in Chapter 5, supercurrent stability generally depends on thenumber of condensed atoms and at Pz = 0 the atom number per spin state is halved.
However, from the N decay curved in figure 6.6 (c) we see that this alone cannot explain
the difference in superflow stability. At Pz = 1 rotation still persists for N ≈ 104 whileat Pz = 0 it stops already at N > 4 × 104 . Moreover, if we apply a π/2 pulse at t = 0
but then immediately remove all the |↑i atoms from the trap using a combination of
microwave π pulse and resonant light pulse, we observe the pure |↓i state with half theinitial atom number is once again stable, with supercurrent again persisting for over aminute. This unambiguously confirms that in figure 6.6 (b) the superflow is inhibited by
the presence of both spin components. We therefore conclude that the two-component
superflow decay mechanism is facilitated by the internal degrees of freedom introducedby the pseudo-spin, and is different from the single-component mechanism.

6.2.2

Stability phase diagram

We now turn to a quantitative study of the supercurrent stability as a function of thespin-population imbalance (see figure 6.7). We tune Pz by varying the length tRF of
the RF pulse applied at t = 0, and measure the q state of the majority spin component,|↑i, as a function of t. Whenever the radius R is fittable for the minority component
we get the same q state for both spin components in > 99% of cases. However, forN↓ < 104 we cannot determine q for the minority component.
Based on over 1600 measurements of q(Pz , t), in figure 6.7 we reconstruct the complete current stability phase diagram for 0 ≤ Pz ≤ 1. The contour plot of hq(Pz , t)i
is obtained by spline interpolation through a 3D mesh of data points with integer q
values. The blue-shaded region corresponds to rotation times for which no phase slips
occur, whereas the black-shaded region corresponds to times at which all rotation has
decayed. We clearly distinguish two qualitatively different regimes. For large Pz thesuperflow is fundamentally stable and limited only by the atom number decay. In thisregime the additional internal degree of freedom is insufficient to facilitate decay beyond that present in the scalar BEC case. For low Pz the current starts to decay within
1Over this time the decay in particle number is sufficiently small that the weak dependence of
R on N can be neglected. To account for this effect at longer hold times we establish quantisation
independently for a given t before assigning q states.

162

6.3 Spin dephasing

Figure 6.7: Supercurrent stability in a partially spin-polarised gas: The statistically averaged
supercurrent state, hqi, of the majority spin component is shown as a function of Pz andthe evolution time t. The contour plot is based on over 1600 measurements of q(Pz , t). The
transition between stable and unstable current regimes occurs at 0.6 < Pz < 0.7. In the stableregime the current eventually decays due to the atom number decay, equivalent to the eventual
decay of the scalar-condensate superflow studied in Chapter 5.

a few seconds. A sharp transition between the two regimes occurs at 0.6 < Pz < 0.7.
Based on an additional ≈ 500 measurements we find close to symmetric behaviour for
the Pz < 0 portion of the diagram. One small difference is that no phase slips occur for
about 70 seconds for Pz → −1, as apposed to 90 seconds for Pz → 1. The transition
between the two stability regimes is observed to occur at approximately the same value
of |Pz |. This slight asymmetry between the two spin states might be attributed to theexistence of the mF = −1 state, or the presence of small terms in (6.19) which depend
explicitly on Sz , breaking the symmetry between the mF = 1 and mF = 0 states.

6.3

Spin dephasing

To fully understand these observations, we need to distinguish a coherent superpositionof |↑i and |↓i states from an incoherent mixture. The RF pulse at t = 0 corresponds torotation around the y axis on the Bloch sphere and puts the BEC into a superposition
state, |Ψ(φ)i, with a well defined phase relation between the two components. Importantly, this new state is still a pure BEC corresponding to the macroscopic occupationof a single state. Due to the near perfect spin-symmetry of our two state basis arising from the almost equal inter and intra-state scattering lengths, this rotated stateshould have almost identical properties to the original state. The significant difference
however, is that the superposition state is no longer an eigenstate of the system. We
therefore find that while the longitudinal polarisation of the state Pz is a conserved1636.3 Spin dephasing

Figure 6.8: Phase diffusion: (a) Ramsey-style experiment to study the decay of transverse
spin polarisation with time. Starting in the pure rotating (q = 3) |↓i state, we apply an RF π/2
pulse to rotate the state vector onto the purely transverse spin state, P~ = (1, 0, 0). Allowing
this state to evolve for a variable hold time, we then apply a second RF π/2 pulse which maps
the decaying |P~ | into Pz . The Pz signal which we observe varies sinusoidally with the relative
RF phase δRF , and with an amplitude equal to the length of the global state vector, |P~ |. (b) By
measuring the variance of the Pz values observed, we obtain a direct measure of the transverse
polarisation decay. The solid red line is a double-exponential fit which gives the decay function
f (t).

quantity, the relative phase between the two components undergoes a complicated evolution causing the transverse polarisation to decay. This decohered state can now nolonger be considered a single pure BEC, but corresponds to a mixture of two macroscopically occupied |↑i and |↓i states.
To study this decay of transverse polarisation, we perform a Ramsey-type experiment, illustrated in figure 6.8 (a). Starting in the pure |↓i state we apply two π/2
RF pulses separated by time t and then measure Pz . The first pulse rotates the statevector on the Bloch sphere about the y axis, creating a purely transverse spin state
P~ = (1, 0, 0) with a well-defined relative phase between the two components. The
relative phase between the the two states evolves at a rate proportional to the local
difference in chemical potentials between the two states ω21 , which in general is a func164

6.4 Spin dephasingtion of both time and space. During the evolution time, couplings to the environment,
quantum fluctuations [148], and finite temperature effects [208] can induce additional
diffusive precession of the relative phase, leading to an uncertainty in its value. We
illustrate this uncertainty in the relative phase as a probability distribution functionof the state vector on the Bloch sphere which spreads with time and acts to shorten
the global state vector for the whole system. During the same time, the RF coupling
drive accumulates a phase ωRF t. The second RF π/2 pulse with phase δRF relative tothe first pulse then maps the transverse polarisation onto longitudinal polarisation Pz ,which we measure by separating the two states in TOF with a Stern-Gerlach field.
Z t
ω21 dt − ωRF t.

δRF (t) =(6.36)

0Pz then varies sinusoidally as a function of δRFPz (t) = V(t) cos(δRF (t)),(6.37)

where the coherence between the two state is quantified by the visibility of the Ramseypfringes, V(t) = P⊥ , where P⊥ = P 2 − Pz2 is the transverse polarisation.
Due to the very long spin-coherence time we observe, shot-to-shot variations in therelative phase δRF prevent us from measuring V directly by observing Ramsey fringes.
Instead we take 40 shots at a given hold time t and infer V(t) from the standard
deviation of the Pz values we observe. Assuming sufficient fluctuations that we sample
δRF uniformly in the range 0 to 2π, the probability distribution of Pz values is givenbyP(Pz → Pz + dPz ) =dPzp.
πV 1 − (Pz /V)2

(6.38)The variance in Pz is then directly related to the Ramsey visibility, and hence transverse
polarisation, by
σ 2 (Pz ) =

P⊥2
.
2

(6.39)

Using this method, we measure the decay of P⊥ as a function of hold time, as shown infigure 6.8 (b). We see the phase coherence decays very slowly with a characteristic spincoherence time tcoh ∼ 10 s. This means that in the unstable regime in figure 6.7 phaseslips occur already at t . tcoh , when we cannot equate P and Pz . By fitting a double
exponential to P⊥2 (t) we obtain a decay function f (t) which allows us to calculate thetotal spin polarisation as a function of time
P (t) =

p
Pz2 + (1 − Pz2 )f (t).

(6.40)

Here we have assumed to first order that we can neglect the dependence of f (t) on thevalue of Pz , simply using the characteristic decay function we measure at Pz = 0 toinfer the decay of transverse polarisation at all Pz .

165

6.4 Coherently coupled two-component supercurrent

Figure 6.9: Adiabatic dressing of the spin state: (a) By sweeping the RF frequency from far
above resonance to zero detuning, the effective magnetic field rotates from the −z direction
to the y direction. Adiabatic following transforms the initial |↓i with Pz = −1 to the dressed
Pz = 0 state |yi, which is now an eigenstate of the effective Hamiltonian. Retaining the
RF coupling during the evolution time, the dressed state does not undergo phase diffusion andremains a pure single component state. (b) This dressed state now exhibits long-lived persistentcurrents, even though Pz = 0. This confirms that the relevant quantity is the absolute spinpolarisation P , and not Pz .

6.4

Coherently coupled two-component supercurrent

To further understand the role of transverse coherence in preserving supercurrent stability we also perform a complimentary experiment in which we adiabatically dress therotating BEC with the RF field. In the presence of the RF field of frequency ωRF ,~ eff = (2~/µB )(0, ΩR , −δ), where µB is the Bohr magthe effective magnetic field is B
neton and δ = ωRF − µB B/(2~) is the detuning from resonance (derived in Section~ eff ∝ −~z, and on resonance B~ eff ∝ ~y . Starting in the
3.2.2). For large detuning, B
initial rotating |↓i state, we apply an RF field far detuned (100 kHz) above resonancewith a Rabi frequency ΩR ≈ 25 kHz. At such a large detuning the RF dressed stateis effectively equivalent to the |↓i state, hence we have loaded into the lowest band of
the dressed state. At t = 0 we adiabatically sweep δ to zero over 200 ms, rotating
the direction of the effective magnetic field from −~z to ~y . The system remains in thelowest dressed state, adiabatically following the effective field, thus preparing a Pz = 0√
superposition state |yi = (|↑i+i |↓i)/ 2 (figure 6.9 (a)). At this point, |yi is equivalent
to the |Ψ(π/2)i state prepare by an RF pulse, which does not show long-term current

1666.5 Spin symmetric phase diagram

stability, decaying form the q = 3 state within a few seconds. However, if we leave theRF field on during the in-trap evolution, |yi is an eigenstate of the Hamiltonian andthe coherence between |↑i and |↓i does not decay1 . In this case Pz = 0 supercurrent isstable and persists for more than a minute, as shown in figure 6.9 (b).
These experiments clearly show that for analysing current stability in a partiallypolarised gas we must distinguish Pz and |P~ |. The long phase decoherence times wemeasure suggest that the non-zero lifetime of supercurrents we observe at all Pz arises
due to the non-zero time taken by the system to evolve from an initially pure singlecomponent BEC with |P~ | = 1, to an incoherent mixture with |P~ | < 1. Secondly, thelong lived supercurrrents observed in the Pz = 0 RF-dressed state confirms that, due
to the spin-symmetry of our system, the value of Pz alone has no bearing on the decayof superflow. Instead, the supercurrent stability can be attributed to the RF coupling
modifying the Hamiltonian such that the Pz = 0 state is now an eigenstate of thesystem, and therefore in the absence of phase decoherence |P~ | remains equal to unity.

6.5

Spin symmetric phase diagram

With this understanding, we now quantitatively characterise the onset of the supercurrent decay in figure 6.7 by the time τ at which the probability that the first phase slip
(q = 3 → 2) has occurred is 50%; this closely corresponds to the border between theblue and white-shaded regions. This is done by binning the data in Pz and plotting theoccurrence of q = 3 states as a function of evolution time. To quantify the transition
between times when q = 3 superflow is likely, to later times when q = 3 superflow has
decayed, we fit this data to a sigmoid function
1

P(q = 3) =t−τ,

(6.41)

1+e σ

where τ characterises the typical decay time, and σ the width of the transition in time.
The resultant τ are plotted as a function of Pz in figure 6.10 (a). The horizontal errors
are given by the standard deviation of Pz for a given bin, and the vertical errors are
given by the fitted uncertainty in τ . We see that τ rapidly increases for Pz & 0.64,
saturating at 100 seconds due to N decay. However, our observations of phase diffusion
and long-lived RF-dressed supercurrent suggested that the absolute spin polarisationis the significant quantity, and not Pz . We therefore now combine our measurementsof τ , defining the characteristic onset of decay, with the transverse polarisation decayfunction f (t) (figure 6.8 (b)), to calculate the characteristic total spin polarisation at
the onset of supercurrent decay:
P (τ ) =p
Pz2 + (1 − Pz2 )f (τ ).

1(6.42)We checked for t up to 100 seconds that we can always convert |yi into pure |↓i by sweeping δ farfrom resonance.

1676.5 Spin symmetric phase diagram

Figure 6.10: Critical spin polarisation Pc : (a) Characteristic time of the first phase slip, τ ,versus Pz . Vertical blue line marks Pc , accurately determined in (b). (b) Modulus of the spinpolarisation vector at the onset of the supercurrent decay, P (τ ), versus Pz . Fit to the points
with P (τ ) > Pz (horizontal solid blue line) gives Pc = 0.64(1). (c) Stability diagram on the
Bloch sphere. The blue-shaded region in figure 6.7 maps into the outer shell |P~ | > Pc = 0.64.
(A cut through the sphere is drawn for clarity.)

P (τ ) is plotted as a function of Pz in figure 6.10 (b). We can now clearly distinguish
two distinct regimes: one where P (τ ) is constant within errors and one where P (τ ) =Pz . We thus complete our physical picture and accurately determine the critical spinpolarisation Pc . The two regimes thus correspond to:
1. If Pz > Pc , then |P~ | can never drop below Pc . The supercurrent is fundamentallystable and long-lived, τ  tcoh and therefore P (τ ) = Pz .
2. If Pz < Pc , supercurrent decay starts at τ . tcoh , when the decaying P becomesequal to Pc .
From all the data in the second regime we obtain Pc = 0.64(1). For 0 ≤ Pz ≤ Pc thepvalue of τ varies from 4 to 15 seconds and the orientation of P~ (τ ) = ( P 2 − P 2 , 0, Pz )czin spin space varies from purely transverse to purely longitudinal, but the onset of thesupercurrent decay always occurs at the same |P~ |. We thus conclude that the regionof supercurrent stability is in fact the outer shell of the Bloch sphere where |P~ | > Pc ,as shown in figure 6.10 (c).
168

6.6 Two-state barrier model

Figure 6.11: Two component barrier model: (a) Schematic of one component creating adensity barrier which constricts the flow of the second component. The total density at all
points around the ring is constant, however the density of the majority component (indicatedin red) and that of the minority component (indicated in blue) can vary azimuthally. Herewe consider the worst-case scenario where one component forms a density peak of minimum
width W (ξs ). (b) Calculated flow velocity of the majority component as a function of P for
the q = 3, 2, and 1 states at the peak density of the minority component for W = 6.5 µm.

The spin-rotational symmetry of our stability criteria arises intuitively from the nearequivalence of the inter and intra-component scattering lengths. As we saw in equation(6.19), the small difference leads to symmetry breaking terms dependant on the valueof Sz , however these terms are ≈ 0.2% of the spin-independent term. We do observea very slight asymmetry between the stability of Pz > 0 and Pz < 0 supercurrents,
however we cannot reliably attribute this difference to the small spin-dependence ofthe system. We also note from our data the existence of critical polarisations for theq = 2 and q = 1 states at Pc (q = 2) ≈ 0.60 and Pc (q = 1) ≈ 0.55. Due to the factthat we focused most of our data on studying the q = 3 → 2 transition, the errors onthese values are much larger. We also note once again that these states are themselvesformed from decay of the initial q = 3 state, and therefore are constrained to posses
lower values of Pc .

6.6

Two-state barrier model

We now present a simple model which provides a possible explanation for the criticalspin polarisation which we observe. Starting from a stable single-component superflow,
we consider the addition of a minority component. If the density of this minority
component is uniform around the ring, this has no effect on the flow velocity of either
component. Here we consider the worst-case scenario, where the density of the minority
component is strongly localised at one point in the ring, effectively acting as a barrier tothe flow of the majority component. The minimum width of the the minority component

169

6.6 Two-state barrier model

density profile, W (ξs ), is dependent on the spin healing length, ξs . Since ξs gives thelength scale over which the value of the spin changes between its extremal values,
one would expect the minimum minority component width to be in the range ξs <W (ξs ) < 2ξs . For our system and starting atom number the spin healing length isξs ≈ 4 µm, and then increases as the atom number decays. With the constraint thatthe total density is uniform around the ring, the presence of the minority component
creates a local reduction in the majority component density, and by the requirementof particle flux conservation, this leads to an increased flow velocity of the majority
component, as shown in figure 6.11 (a). Our simple model is then to fix the width ofthe minority component density bump at W (ξs ), and calculate how the majority flow
velocity increases as we reduce the population imbalance.
For the purposes of providing a qualitative explanation, we assume a Gaussian
density distribution of the minority component of the form


−x2
N1
√
,
exp
n1 =
2W 2
2πW 2

(6.43)

where N1 is the atom number in component 1, x is the azimuthal direction, and wehave integrated out the vertical and radial directions, only considering variations inspin polarisation in the azimuthal direction. By restraining the total density to beuniform, the majority component density is simply given by
n2 = ntot − n1 .

(6.44)

We now apply the same criteria to calculate the flow velocity as we did in Section 5.4,
namely conservation of particle flux
n2 v2 = constant = C,

(6.45)and quantised circulation
Iv2 .dl = 2πq~.
m(6.46)

The flow velocity is then given byv2 (x) =

C
L

,
N
L
1
N 1 − 1 √ exp −x2
N W

2π(6.47)

2W 2

where the constant is given by
,Z


−1
L/2
~N
N1 L 1−x2√ exp
C = 2πqdx 1 −mL
N W 2π2W 2−L/2(6.48)We therefore see that the flow velocity is independent of the total atom number N , andonly depends on the ratio N1 /N and W/L. In figure 6.11 (b), we plot the calculated flow
velocity of the majority component for W = 6.5 µm, for the rotation states q = 3, 2,
170

6.7 Conclusion

and 1. This value of W is chosen to best match the critical spin polarisation which
we observe, and is consistent with our anticipated range of allowed W values. We
see that as the spin polarisation decreases from 1, the size of the barrier we can form
with the minority component increases, and hence the peak flow velocity increases. At
around P = 0.6, the density in the minority component is sufficient to fragment the
majority component and all flow stops. Comparing this to typical values of the local
sound velocity c ≈ 1mm/s, we see that such a healing length gives the correct value ofPc (q = 3), and predicts the relatively close values of Pc for the q = 2 and q = 1 states.
Although quite a crude calculation, it does illustrate that the formation of outof phase density fluctuations can lead to a non-trivial critical spin polarisation. The
fundamental questions left by this model are two fold. Firstly, we ask what drives
the formation of such spin structures? In spite of our two components being miscible
and eliminating any systematic differences in the potentials of the two-states, as shownin Section 6.1.5, we do still observe the formation of spin structures. Modeling suchspin diffusion could also explain whether both components always decay together, or
whether above a certain population imbalance the minority component fragments andstops rotating, while the majority persists. The second question is then what sets
the critical velocity? While one would assume the local sound velocity sets an upperbound on vc , it may be possible that the coexistence of two components also facilitatesa different, lower critical velocity. This may be especially true since the local flow
velocity increase experienced by one component will correspond to a local flow velocity
decrease in the other component, and hence a large relative velocity between the twocomponents. As shown in [209–211], sufficiently high counterflow velocity between
two components can lead to excitations with imaginary frequencies and the onset ofdynamic instability.

6.7

Conclusion

In summary, we have observed persistent currents in multiply connected spinor condensates, demonstrated the existence of a critical spin polarisation for stable superflow, and elucidated the role of spin coherence in supercurrent stability. Empirically wehave shown that the supercurrents in both spin states are constrained to decay quasisimultaneously, and by establishing two-component superflow quantisation, we confirm
the decay mechanism is still 2π phase slips, as observed in the single condensate case.
By studying the stability of supercurrents as a function of the total spin polarisation P ,we have identified and accurately measured the transition between two distinct regimes
with different decay mechanisms. For P > Pc the superflow is fundamentally stableand decays in a manner identical to that of a pure single-component supercurrent. For
P < Pc the internal spin degree of freedom facilitates an additional decay mechanism
which causes the supercurrent to be fundamentally unstable.

1716.7 Conclusion

The existence of a critical population imbalance for superflow stability was predicted in [185–187], assuming equal intra and inter-component interactions and nointer-component coherence. The supercurrent instability was associated with out-ofphase density fluctuations in the two components. However, an agreement on the valueof Pc has not been reached. In references [185, 186] it was predicted that any q > 1
flow is unstable for essentially any P < 1, but according to [187] such current was
found to be stable above some nontrivial interaction-dependent Pc . The latter conclusion qualitatively agrees with our observations, however none of the existing theories
is quantitatively applicable to our experiments, since they are limited to the simplified
cases of reduced dimensionality and very weak interactions.
The simple two-state barrier model we have presented provides at least a qualitative description of how a non-trivial critical spin polarisation of the form observed canarise. This model also highlights the need for further theoretical work to understandthe dynamics of phase decoherence and spin fluctuations in our system. In general
however, we cannot yet establish how the spin degree of freedom and the supercurrent decay mechanism are microscopically related. In one picture the spin degrees of
freedom evolve independent of the rotation, and once sufficient spin fluctuations haveformed the supercurrent decays in a manner similar to that explained by the two-statebarrier model. A second, more exciting, alternative exists, where the supercurrent decay mechanism intrinsically uses the additional spin degree of freedom to unwind thescalar phase associated with superflow. Such decay mechanisms could relate to darkbright solitons or magnons on a 1D Heisenberg chain. For these mechanisms, the global
value of the spin polarisation could permit sufficient freedom in the local spin vector
for accumulation of a Berry phase resulting in a phase slip.
Much of the interesting behaviour of our system then seems to arise from the nearequality of the scattering lengths involved. Within the single-mode-approximation, thephase decoherence of a two-component system results from a spread in the distributionof populations in the initial state which are converted into phase fluctuations by thenonlinear interactions during the evolution. Therefore the rate of decoherence should√
be proportional to the difference in scattering lengths as = (a11 + a00 − 2a10 )/ 2 [212].
Similarly the spin healing length also depends on the difference in scattering lengths,
and as a result our mixture exhibits long coherence times and large scale spin structures.
These two features respectively explain the lifetime of persistent currents of several seconds for all values of Pz , and also, at least in the two-state barrier model, the significant
deviation of Pc from 1. An important next step would then be to study supercurrents ina two-species system with significantly different intra and inter-component interactions
to confirm these conclusions.

172

Chapter 7

Towards an azimuthal gauge potential
Ultracold atoms offer an appealing system for the study of many-body correlated states
relevant to condensed matter physics. One of the primary challenges however, is toengineer a Hamiltonian for which neutral atoms behave as charged particles in a strong
magnetic field. Such a technique is required to realise exotic topological phases, suchas the fractional quantum hall effect (FQHE) in two-dimensional systems, which only
exists at high magnetic fields where the ratio of flux quanta to particles is of orderunity. The standard way to produce an artificial magnetic field is to rotate an atomic
cloud which produces a non-trivial vector potential in the rotating frame of reference[29]. Such an approach however is limited to only modest fields, and as a result there
have been a wealth of proposals to create effective magnetic fields without rotation.
These can broadly be divided into two categories: asymmetric tunneling in opticallattices [213–215], and geometric gauge potentials. We will be focusing on the second
approach, whereby atoms with two or more ground states are optically dressed, andthe position dependence of the dressed internal states leads to geometric vector andscalar potentials [216].
The field of geometric gauge potentials has been driven forward primarily by theexperiments of Spielman et al [66, 217–220]. By dressing the F = 1 magnetic sublevelsof a BEC with two counterpropagating Raman laser beams they were able to couple theinternal spin states with linear momentum differing by twice the photon momentum.
This gives rise to a spatial gradient of the phase difference between spin componentsof the dressed state, achieving the first light-induced vector gauge potential [66]. By
subsequently varying this gauge potential both spatially, and temporally, they wereable to synthesise both artificial magnetic [218] and electric [219] fields respectively.
By engineering a dressed band with two minima, they have also managed to realise an
ultracold atom analogue of spin-orbit coupling [220].
In this chapter we discuss our progress in realising an azimuthal gauge potential,
where the internal spin components of the dressed state are coupled to the angularmomentum. We start by reviewing the theory behind linear gauge potentials, andthen show how our setup is a simple mapping from linear to azimuthal motion. Our
azimuthal gauge setup exhibits additional exciting possibilities which arise due to the173

7.1 Linear gauge potentials

Figure 7.1: Linear gauge potential: (a) The BEC is dressed by two counterpropagating Raman
beams with frequencies ω and ω + ∆ω, linearly polarised along orthogonal axes. The Raman
beams couple the mF magnetic sublevels of the F = 1 manifold, which is split in an external
magnetic field B. (b) Population of the three mF spin states in the lowest dressed band as afunction of Raman detuning, δ, for ~Ω = 5Er and kx =kmin . (c) Position of the band minimum
kmin in momentum space as a function of Raman detuning for ~Ω = 5Er . The insets show theenergy of the dressed bands (blue) as a function of kx for the cases of positive and negative
detuning. The dashed black lines indicate the original bare states. The red dot indicatesthe minimum of the lowest dressed state, which due to the confining potential, the BEC will
adiabatically follow.

translational symmetry afforded to us by our ring geometry. To illustrate this, we thendescribe a proposed superfluid fraction measurement, where the spin-orbit couplingallows us to relate the rotational state of the system to the spin populations. The
final section presents the experimental progress made so far, as well as the technicalchallenges which remain.

7.1

Linear gauge potentials

In this section we briefly explain the linear gauge setup realised by Spielman et al. anddetailed in [66, 217]. We start by dressing a BEC in the F = 1 ground state with two174

7.1 Linear gauge potentials

Raman beams of frequency difference ∆ω, counterpropagating along x̂, as shown infigure 7.1 (a). These beams couple states |mF , kx i differing in their internal state by∆mF = ±1 and differing in linear momentum ~kx by 2~kr , where ~kr = h/λ is the
single-photon recoil momentum. We also define Er = ~2 kr2 /2m as the associated recoil
energy. The spin and momentum states |mF , kx i coupled by the Raman beams aregrouped into families of states labeled by the momentum ~kx :Ψ(kx ) = {|−1, kx + 2kr i , |0, kx i , |+1, kx − 2kr i}.

(7.1)

In the rotating wave approximation for the frame rotating at ∆ω, the Hamiltonian
expressed in the state basis of the family Ψ(kx ) is given by~(k + 2kr )2 − δΩ/20 2m x~2
H = ~
Ω/2Ω/22m (kx ) − 
0

Ω/2

~
2
2m (kx − 2kr ) + δ



.


(7.2)

Here δ = gF µB B~ is the detuning from Raman resonance, Ω is the resonant Raman
Rabi frequency, and  accounts for the quadratic Zeeman shift. For each kx , diagonalising H gives three eigenvalues Ej (kx )(j = 1, 2, 3). For dressed atoms in state j,
this eigenenergy as a function of kx is the effective dispersion relation, and depends
on the experimental parameters, δ, Ω, and , as shown in the insets of figure 7.1 (c).
Concentrating on the lowest energy dressed band, the number of energy minima andtheir positions kmin are thus experimentally tuneable. Around the band minimum kmin ,the dispersion relation can be expanded asE(kx ) ≈~2(kx − kmin )2 ,
2m∗

(7.3)

where m∗ is the effective mass. Considering the Hamiltonian for a particle of charge q
in a magnetic vector potential A
H=(p − qA)2
,
2m

(7.4)

we can then identify kmin with the light-induced vector gauge potential
qA = ~kmin x̂.

(7.5)

In figure 7.1 (c) we calculate the value of kmin , and therefore the effective magnetic
vector potential qA/~, for the lowest dressed band as a function of the Raman detuningδ.

175

7.2 Azimuthal gauge potential

The mechanical momentum is defined as
∂H
∂pcan
m ∂H
=
,
~ ∂kx

pmech = mẋ = m

(7.6)
(7.7)

where pcan is the canonical momentum. We therefore identify the mechanical motionof the dressed state as the gradient of the energy band with respect to kx . In [66] thecondensate was adiabatically loaded into the minimum of the lowest dressed band forzero Raman detuning, δ = 0, corresponding to the band minimum at kmin = 0. Fromthe populations of the mF states plotted in figure 7.1 (b) as a function of δ, we see
this dressed state corresponds to a superposition of all three mF states. From equation(7.7), for such a state at the band minimum, the mechanical momentum is zero andthe dressed atoms are stationary in the lab frame. The momenta of the individual spincomponents however are nonzero. To see this, one can project the dressed state onto
its original spin and momentum components (shown by the dashed black lines in figure
7.1 (c)) by abruptly removing the Raman coupling. One then decomposes the dressedstate into the mF = 1, mF = 0, and mF = −1 states with mechanical momenta −2~kr ,0, and 2~kr respectively.
If one first adiabatically sweeps the Raman detuning to ~δ = −2Er (Inset of figure
7.1 (c)), the minimum of the energy band is shifted to a non-zero kmin , corresponding toapplication of a vector gauge potential in the x̂ direction. Due to the trapping potential
the dressed state of the atoms are confined to follow the minimum of the band, andhence the mechanical momentum of the dresses state is restricted to be zero in the labframe. From figure 7.1 (b), we see that the spin composition of the dressed state haschanged to predominantly the mF = 1 state. The presence of a non-zero kmin is thenconfirmed by projecting into the bare states, where one then observes the appropriate
decomposition of the mF = 1, mF = 0, and mF = −1 states, but now with respective
mechanical momenta −2~kr + ~kmin , 0 + ~kmin , and 2~kr + ~kmin .
These experiments showed that one can utilise light fields to create vector gauge
potentials for neutral atoms, which can be easily manipulated by varying the detuning of
the Raman coupling. This basic setup was then extended by the addition of a magnetic
field gradient to create a spatially varying A(r), and therefore an effective magnetic
field B = ∇ × A(r) [218]. Varying the value of the detuning and therefore A(t) in timethey also created an ultracold atom analogue of an electric field E = −∂A(t)/∂t [219].

7.2

Azimuthal gauge potential

In this section we detail our experimental setup, which though similar in spirit, instead
couples the internal spin state to angular, rather than linear, momentum. We startby considering a BEC held in a ring trap formed by a red-detuned LG beam, which176

7.2 Azimuthal gauge potential

Figure 7.2: Azimuthal gauge potential: (a) The ring BEC is dressed by two copropagating
Raman beams with frequencies ω and ω + ∆ω, linearly polarised along orthogonal axes. The
Raman beams couple the mF magnetic sublevels of the F = 1 manifold, which is split in an
external magnetic field B. One of the Raman beams is a broad Gaussian beam, while thesecond is an LG beam of phase winding ∆`. (b) Population of the three mF spin states in
the lowest dressed band as a function of Raman detuning, δ, for ~Ω = 5Erot and `=`∗ . (c)Position of the band minimum `∗ in angular momentum space as a function of Raman detuningfor ~Ω = 5Erot . The insets show the energy of the dressed bands (blue) as a function of ` forthe cases of positive and negative detuning. The dashed black lines indicate the original barestates. The red dot indicates the minimum of the lowest dressed state.

also acts as one of the Raman beams. To generate an azimuthal vector potential, weRaman-couple the mF magnetic states of the F = 1 ground state using the LG beamand secondary Gaussian beam, co-propagating in the vertical direction ẑ, perpendicular
to the toroidal trap. In this sense, our setup is identical to the one we introduced in
Chapter 4 to create persistent currents. The only distinction is that now rather thanusing the Raman transition to Rabi flop to a state with different angular momentum,
we will continuously couple the magnetic states to create new dressed states. Our
experimental setup is illustrated in figure 7.2 (a).
Our Raman transition couples states |mF , `i differing in their internal state by∆mF = ±1 and differing in angular momentum ~` by ~∆`, where ~∆` is the angular177

7.2 Azimuthal gauge potential

momentum carried by the LG beam. The relevant energy is now the rotational energy
Erot = ~2 (∆`)2 /2mR2 , where R is the radius of the ring trap. The family of coupledstates is now given by:Ψ(`) = {|−1, ` + ∆`i , |0, `i , |+1, ` − ∆`i}.

(7.8)

In the rotating wave approximation for the frame rotating at ∆ω, the Hamiltonian
expressed in the state basis of the family Ψ(`) is given by~(` + ∆`)2 − δΩ/2 2mR2~

H = ~
Ω/2(`)2 − 
2mR2
0

Ω/2

0



Ω/2


.


~
(` − ∆`)2 + δ
2mR2

(7.9)

This Hamiltonian is identical to that for the linear momentum case in (7.2) with thefollowing mapping
~2
~2
→
;
2m
2mR2

kx → `;

2kr → ∆`.

(7.10)

Once again we diagonalise H to obtain the dressed state energy bands, which we
expand about the minimum asE(`) ≈~2(` − `∗ )2 .
2m∗ R2(7.11)(pθ − qAθ )2,2m(7.12)

This is equivalent to the HamiltonianH=

where we have identified the azimuthal momentum and azimuthal vector potential as
pθ =

~`
θ̂;
R

qAθ =

~`∗
θ̂.
R

(7.13)As shown in figure 7.2 (c), the position of the band minimum can be varied in therange −∆` → ∆` by changing the Raman detuning δ, which also changes the spincomposition of the dressed state as shown in figure 7.2 (b).
We note that unlike for the linear gauge case, the fact that our vector potential isin the azimuthal direction means it automatically corresponds to a non-zero magnetic
flux threading the ring. If we shift the band minimum to `∗ 6= 0, we create an azimuthal
vector potential, equal in magnitude at all points on the ring. Using Stoke’s theorem
we see this corresponds to a flux, Φ, threading the ring
Z Z
I
2π~`∗
.
Φ=
B · dS = A · dl =q

178

(7.14)

7.2 Azimuthal gauge potential

The form of the effective magnetic field is then given by
1 ∂RAθ
ẑ
R ∂R

1 ∂
~`∗
=
ẑ.
R ∂R
q

B=∇×A=(7.15)(7.16)

This is in general a nontrivial function, since `∗ depends on Ω which varies with theintensity of the Raman beams. To first order though ∂`∗ /∂R = 0 at the atoms since
the maximum Ω also defines the trap minimum, and hence the magnetic field canbe considered zero at the atoms. The angular momentum acquired by the dressedstate, ~`∗ , can therefore be viewed as an Aharanov-Bohm type effect. Alternately one
can also explain it as a cold atom analogue of Faraday induction, where the angularmomentum acquired as we vary the effective magnetic field is equivalent to generating
an electrical current in a coil as the magnetic flux enclosed is varied in time. We
now further explain the properties of this system by describing a proposed experiment
to measure the superfluid fraction of an atomic gas which makes explicit use of theazimuthal nature of our created vector gauge potential.

7.2.1

Superfluid fraction measurement

As discussed in Section 2.4.2, while the concepts of superfluidity and BEC are intrinsically linked, the superfluid fraction and condensed fraction of a gas or fluid in generaltake very different values. While the condensed fraction can be measured by mapping
the momentum distribution to real space in time-of-flight expansion, a quantitative
measure of the superfluid fraction is harder to achieve. Such a measurement however
is important for investigating the properties of interacting Bose gases. Specific casesof interest include strong interactions which can lead to condensate depletion without
loss of superfluid fraction, and the Kosterlitz-Thouless phase transition in a quasi-2D
geometry which manifests itself in a universal jump in the superfluid density [221, 222].
Recently the first measure of superfluid fraction in a quantum gas was realised in a degenerate Fermi system by measuring the speed of second sound [223]. Here we present
an alternate approach using our azimuthal gauge potential, which was first proposedin [104].
The basic principle of the experiment is shown in figure 7.3. Starting for example
in the pure mF = −1 state, one can load into the lowest dressed band by adiabaticallyturning on the second Gaussian Raman beam with the detuning δ  0. As shownin the left panel of figure 7.3 (a), this loads both normal and superfluid components
into the band minimum at `∗ = 0. Since the gradient of the dispersion relation is zeroat the minimum, pmech = 0, both components are at rest in the lab frame. By thensweeping the Raman detuning towards zero, we shift the band minimum and impose
our azimuthal vector potential (right panel figure 7.3 (a)). Due to interactions with thetrapping potential, the normal component is constrained to follow the band minimum
179

7.2 Azimuthal gauge potential

Figure 7.3: Superfluid fraction measurement: (a) Starting with the Raman coupling far
detuned, |δ|  Ω, we adiabatically load the pure mF = −1 state into the minimum of thelowest dressed band by ramping on the Gaussian beam. Both normal component (green circle)
and superfluid component (red star) are then at the band minimum ` = `∗ = 0. By sweeping
the detuning towards resonance, the band minimum shifts and the spin composition changes.
The normal component interacts with the walls of the trap and relaxes, following the band
minimum such that pmech = 0 and ` = `∗ . The superfluid component cannot relax and remains
at the original ` = 0, and therefore starts to rotate in the lab frame. (b) As well as a difference
in mechanical momentum between the two components, the spin composition of the dressedstate at ` = `∗ differs slightly from that at ` = 0, which allows one to perform a spectroscopic
superfluid fraction measurement. Plotted is this difference as a function of the final detuning.

where it is stationary with respect to the lab frame. In contrast the superfluid component is unable to relax and follow the band minimum, remaining at the original angularmomentum ` = 0. The gradient of the dispersion relation at this point increases as thedetuning is changed, causing the superfluid component to rotate in the lab frame withpmech 6= 0. This distinction provides the definition of the superfluid fraction [13]. Thisbehaviour in a toroidal trap with an azimuthal vector potential is in marked contrast to
the case of a linear vector potential, where, due to the lack of translational invariancealong the direction of the vector potential, both components must relax and follow theband minimum.
A key element of this proposal is that one can measure this distinction between
normal and superfluid behaviour using spectroscopic methods. The wavefunction of
the lowest band is a linear superposition of the three magnetic levels, with amplitudes

1807.3 Experimental progress

which vary with `. For a given detuning, the spin composition of a normal state at
the band minimum ` = `∗ , differs slightly from the spin composition of a superfluid
state which remains at ` = 0. One can quantify this by the difference in the number ofparticles in the mF = ±1 states∆p(`) =

N−1 − N1.
N

(7.17)

In figure 7.3 (b) we plot the difference in ∆p between the normal component (` = `∗ )and the superfluid component (` = 0). Initially the population imbalance is identicalfor both the purely normal and purely superfluid states, however for final detunings
close to zero, the distinction between ∆p peaks at a few percent, before tending towards zero again at δfinal  0. This difference in population imbalances forms thebasis of the proposed measurement. Sweeping the detuning to δfinal = 0.2Ω where thesignal peaks, if our gas was purely normal we would measure a population imbalance∆p(`∗ ). Alternately if our gas was purely superfluid we would measure a different population imbalance ∆p(0), differing from ∆p(`∗ ) by a few percent. If our gas had bothcomponents, one could therefore calculation the superfluid fraction as∆p(`∗ ) − ∆pρs=,ρ∆p(`∗ ) − ∆p(0)(7.18)

where ∆p is the measured imbalance, while ∆p(0) and ∆p(`∗ ) are calculated using the
known values of Ω, δfinal , and .
This proposed experiment illustrates how one can make use of an azimuthal vector
potential to directly link superfluid motion and spin composition. While in principle
we have the experimental setup required to perform this experiment, a key step is toprove that the Raman coupling is coherent and that the atoms remain in the lowest
dressed state only. Incoherent processes can lead to population of higher bands which
will alter the measured spin composition, destroying the few percent signal we need toobserve. In the next section we detail the progress we have made so far in implementing
such an azimuthal vector potential, and also some of the technical issues which remain.

7.3

Experimental progress

The current status of our experimental progress is illustrated by figure 7.4. By coupling
the F = 1 manifold with our ` = 3 LG and Gaussian Raman setup, we have managed toengineer the azimuthal vector potential proposed above. Through varying the detuningof the Raman coupling we can demonstrate the changing mechanical rotation and spincomposition of the superfluid state as the band minimum is shifted. We can also force
the superfluid state to relax to the band minimum by ramping up a repulsive green
laser beam overlapped with the toroidal trap to create a barrier to the superflow. Using these tools we now demonstrate six distinct dressed states. The appropriate band181

7.3 Experimental progress

Figure 7.4: Far-detuned azimuthal vector potential: (a) Relevant dressed state dispersion
relations E(`). The angular momentum ` is indicated for the superfluid component by the redstar, and for the normal component by the green circle. As we work with a pure condensate
we only ever observe behaviour consistent with a pure superfluid. (b) Absorption images ofthe dressed state projected onto the original basis states at 29 ms TOF in the presence of a
Stern-Gerlach field which splits the different mF states along the vertical direction. The q stateof each image is measured kinematically by fitting the size of the central density hole formed
in TOF. The protocol to prepare each measurement (i)→(vi) is explained in the text.

structure for these states is shown in figure 7.4 (a), and the projection of the dressedstate onto the original basis |mF , `i is shown in figure 7.4 (b).
In this experiment we prepare the dressed state in an external field of 10 G, where
the second order Zeeman shift  ≈7 kHz, and use a modest Rabi frequency of onlyΩ ≈ 500 Hz. As a result our Raman beams do not simultaneously couple all three
mF states, and therefore for a window of detuning near zero, the spin composition of
the dressed state is purely mF = 0. The reasoning behind limiting ourselves to thissimplified case will be explained in the following section.
We now describe the six experimental sequences, relating to the appropriately labeled band diagrams and absorption images in figure 7.4. In all cases the dressed state1827.3 Experimental progressis projected into the |mF , `i basis just before TOF by abruptly jumping the Raman
detuning far from resonance. The spin composition of the dressed state is measured byseparating the mF states during TOF by application of a Stern-Gerlach gradient. The
angular momentum ` of the dressed state is measured by fitting the size of the density
hole formed in TOF, as explained in Section 4.4.2.
( i ) Starting in the pure mF = 1 state, the Raman coupling is ramped up with δ far
above resonance: This loads the condensate into the lowest dressed band with spincomposition Ψ(` = −3) = {|mF = 1, ` + 3i , |mF = 0, `i , |mF = −1, ` − 3i} ={1, 0, 0} and band minimum `∗ = −3. This is confirmed by observing only the
non-rotating |mF = 1, ` = 0i state in TOF.
( ii ) Sweeping the detuning to below the mF = 1 → mF = 0 resonance: This shifts theband minimum to `∗ → 3, however our superfluid remains at ` = −3 and therefore
we prepare the dressed state with spin composition Ψ(` = −3) = {0, 1, 0}. Thisis confirmed by observing only the |mF = 0, ` = −3i state in TOF.
(iii) Sweeping the detuning far below both the mF = 1 → mF = 0 and mF = 0 →mF = −1 resonances: Our superfluid remains at ` = −3 and we prepare thedressed state with spin composition Ψ(` = −3) = {0, 0, 1}. This is confirmed byobserving only the |mF = −1, ` = −6i state in TOF.
(iv) Preparing state (iii), we then kill rotation by ramping up and down the repulsive
barrier beam over 500 ms: The superfluid velocity exceeds critical velocity at thebarrier and relaxes to the band minimum at `∗ = 3. The relaxed dressed statethen has spin composition Ψ(` = 3) = {0, 0, 1}. This is confirmed by observing
only the non-rotating |mF = −1, ` = 0i state in TOF.
( v ) Preparing state (iv), we then then sweep the detuning above the mF = 0 →mF = −1 resonance: This shifts the band minimum back to `∗ → −3, but our
superfluid which has relaxed to ` = 3 by application of the barrier, does not follow.
We therefore prepare the dressed state with spin composition Ψ(` = 3) = {0, 1, 0}.
This is confirmed by observing only the |mF = 0, ` = 3i state in TOF.
(vi) Preparing state (iv), we then sweep the detuning far above both the mF = 0 →mF = −1 and mF = 1 → mF = 0 resonances: Our superfluid remains at ` = 3
and we prepare the dressed state with spin composition Ψ(` = 3) = {1, 0, 0}.
This is confirmed by observing only the |mF = 1, ` = 6i state in TOF.
In principle then, we have demonstrated a tuneable light-induced azimuthal vector
potential. The next important step would then be to prepare a dressed state with nontrivial occupation of more than one mF state. Unfortunately our research so far has
highlighted several technical difficulties which prevent us from further manipulating
and quantifying dressed states composed of multiple spin states.
183

7.4 Conclusion

7.3.1

Technical challenges

In the hope of guiding future work on this project, we now outline some of the technicalproblems observed in our setup which remain unsolved:
1. Dressed states prepared near resonance, which therefore are composed of morethan one spin state, are observed to decohere on timescales of ≈100 ms. This isconfirmed by preparing and holding such a state, and then sweeping the Raman
detuning far below resonance. We observe occupation of more than one mF state,corresponding to population of higher dressed states. The rotational state of themF = −1 component is also found to decrease the longer we hold near resonance.
2. To detect the rotational state we need to accurately measure the size of thedensity hole formed in TOF due to the presence of a centrifugal barrier. This
technique requires us to first ramp down the ring trap over ≈ 100 ms before
release, for the hole to expand sufficiently over the 29 ms TOF (See Section 4.4.2).
We find that for spin mixtures prepared by projecting the Raman dressed state,complex dynamics occur over this timescale which prevent us from measuring therotational state of each spin component upon projection. The reason why this isin such stark contrast to the mixtures prepared by RF coupling in Chapter 6 is
not clear.
3. To deterministically kill rotation and force the dressed state to relax, the repulsive
potential barrier must be raised and lowered very slowly so as not to induce an
unknown quantity of circulation through mechanical stirring. While we have beenable to readily achieve this in the limit of large detuning, due to the decohering
processes we find near resonance, the time to do so is not available for dressedstates with non-trivial spin composition.
4. Finally, we also note that our Raman beams exert a significant dipole potential
on the atoms, and therefore the high intensities required to generate large Rabi
frequencies will alter the ring potential. Due to the requirement that our ring potential remains sufficiently smooth, circularly symmetric, and multiply-connected
for superflow to persist, the maximum Rabi frequency we can achieve is limited.
This problem can be solved by changing the Raman wavelength to nearer 790 nm
where the dipole potential is smaller, but at the cost of increased light scattering.

7.4

Conclusion

In conclusion, by Raman coupling different mF states with different angular momenta,
we have managed to generate an azimuthal vector potential. Through varying thedetuning of the Raman beams we have demonstrated control of this vector potential,
and therefore the magnetic flux which threads the ring. Due to the toroidal geometry
184

7.4 Conclusionof our system, we have shown that in contrast to the case of a linear vector potential,
we can prepare both dressed states which conserve angular momentum by rotatingin the lab frame, and those which relax to the minimum of the dispersion relation
and are stationary. Technical challenges still need to be addressed however, including
maintaining coherence near resonance, deterministically forcing the dressed state torelax, and measuring the rotation of multi-component states.

185

186

Chapter 8

Summary and outlook
It is my hope that the work in this thesis has both demonstrated, and in some partfurthered our understanding of, the intricate physics at play in persistent currents. In
many regards the study of metastable superfluid flows and their decay feels quite traditional, building on a century of experimental and theoretical research. As we discovered
while researching the mechanisms at play in our toroidal condensate superfluid, manyof the questions which we have tried to address in this work have been posed decades
before in the context of either superfluid helium or superconducting nanowire experiments. The hope then is that while the physics may be universal and long-standing,
the realisation of persistent currents in dilute atomic condensates, presented both in
this work and that of NIST, will offer a unique alternate route to further probe these
phenomena.
The key achievements of our work can be summarised as follows: We have created a long-lived multiply-charged persistent current in a toroidal-

shaped BEC. By using an SLM to generate our LG beam, we are able to imprint a well-defined and variable phase winding onto the condensate wavefunction, preparing a state of known angular momentum. The SLM also allows us
to correct for abberations in our system, enabling us to achieve a ring trap withazimuthal variations of less than 10%. With this setup we have demonstrated
supercurrents persisting for two minutes, limited only by the gradual decay of
atom number.
 For the first time we have demonstrated that the angular momentum state of

the persistent current decays in a quantised fashion, corresponding to collective
jumps of the atoms between metastable minima. This unambiguously confirms
the decay mechanism as vortex-induced 2π phase slips.
 Our ability to resolve individual phase slips also opens the possibility to study

the dynamics of phase slips. We find that the supercurrent decays rapidly if
the superflow speed exceeds a critical velocity in good agreement with numericalsimulations, and we also observe rare stochastic phase slips for superflow speeds

1878.1 Outlook

below the critical velocity. From this, we attribute the onset of decay to theexcitation of phonon modes when the superflow velocity exceeds the local soundvelocity, in agreement with numerical solutions of the GPE.
 We study persistent currents in a toroidal two-component Bose gas for the firsttime. In contrast to many theoretical predictions, we find that the supercurrent is stable for spin polarisations above a well defined, and non-trivial, criticalvalue. Below this value we find the supercurrent is unstable, and decays dueto the presence of the second component. We also investigate the role of phasecoherence between the two spin states and show that only the magnitude of thespin-polarisation vector, rather than its orientation in spin space, is relevant forsupercurrent stability.

8.1

Outlook

The outlook for future work on this system falls into two categories. The first of theseare experiments which further the understanding of our observations so far. One of
the main question posed by our observations of stochastic phase slips is the role oftemperature in the stability of persistent currents. By studying supercurrent decay as
a function of temperature it might be possible to understand how the energy barrier
to decay changes as a function of atom number. One could also extrapolate the data
down to T = 0, possibly allowing further confirmation of the zero temperature GPE
result, or perhaps one might find evidence of quantum tunneling, as was observed insuperfluid helium and superconducting nanowire systems. Studying how the criticalvelocity changes, if at all, as the dimensionality of the system changes, could also offer
further insight into the connection between superfluidity and BEC.
Persistent currents in multi-component condensates offer a wealth of complex physics,
and here the challenge is more in trying to restrict oneself to a limited subset of thepossible mechanisms at play. Even in the absence of superflow, the evolution of a
two-component condensate with periodic boundary conditions is an interesting andchallenging problem. One of the main conclusions of our studies was the significanceof the inter and intra-species scattering lengths. For the two components we used, thefact that they were almost equal, lead to a spin-symmetric stability criteria and longtimescale decoherence and spin dynamics. Clearly it would be beneficial to put our
results in context, by studying either a two-component condensate with significantly
different scattering lengths, or even a dynamically unstable mixture. This could perhaps
also further our understanding of the mechanisms which give rise to spin structures,and whether these out of phase density fluctuations do indeed play a significant rolein disrupting the supercurrent stability, as predicted by our simple two-state barriermodel.
The second category of future experiments are those which use our setup to branch
1888.1 Outlook

out into new areas of physics. As we discussed in Chapter 7, one of the most exciting
possibilities is the opportunity to generate an azimuthal vector potential, not least asa means of measuring the superfluid fraction. While we have been able to demonstratesome preliminary results, the truly interesting physics comes about once the dressedstate has a non-trivial spin composition. In such a situation one can demonstrate
the coupling between spin-composition and angular momentum, and even counterintuitively induce rotation in one spin-state by forcing the atoms to relax in the dressedband. Several mappings have been proposed for the response of this system, includinginterpretations as Faraday induction, an Aharanov-Bohm type effect, and the Einsteinde Haas effect. While several technical challenges currently bar our progress, thissystem is certainly interesting enough to merit further persistence.
An alternate direction we have considered, is to use our ring trap to study the
Kibble-Zurek mechanism [103]. The principle of this measurement is that in a secondorder phase transition, the relaxation time of the system diverges near the critical point,and hence every such transition traversed at a finite rate is a non-equilibrium process.
If one considers Bose-Einstein condensation in a ring trap, then the phase of the macroscopic wavefunction is first established locally, with a domain size dependent on therate at which one quenches through the transition. As the condensate then comes into
equilibrium, these regions merge and can potentially create topological defects forming
quantised vortices. Such a mechanism has been observed in [224]. If one performs thisquench in a ring trap, it is possible to test the predicted scaling laws of the Kibble-Zurek
mechanism, by measuring the distribution of supercurrent states formed as a functionof quench rate.
Finally, we also note that our ring trap also forms an ideal system for fundamental
studies in a periodic potential. One such study, which would also further illuminate
our results on multi-component persistent currents, would be to study spin diffusion
on a 1D periodic lattice. By reducing the radial extent of our trap to below the spin
healing length, we can effectively create a 1D chain of spin domains. Such a system
would then be analogous to a Heisenberg spin chain, allowing us to study the system’s
response to both rotations of the global spin, and the propagation of local spin defects.
Such a setup could then also be used to study both single and two-component solitons
in a periodic potential. Many of these proposed experiments can also be done with
39 K, where the additional freedom to precisely tune the scattering length over a wide

range opens up a wealth of further possibilities.

189

190

Appendix A

Analytic form of pseudo-LG beam
To compare our pseudo-LG beam to the true LG field we now derive an analytic
expression for the field at the focus. As illustrated in figure A.1, we define the field at
the SLM as f0 (x0 , y0 ), the field directly before the lens as fz (x1 , y1 ) after propagatinga distance z, and the field at the focus as gf (x2 , y2 ). Using the rules defined in Section
4.2.4, we identify these fields as
 2
r
(A.1)
f0 (x0 , y0 ) = E0 exp − 02 exp (−`θ0 )
w


Z Z
1
ik
fz (x1 , y1 ) =f0 (x0 , y0 ) exp((x1 − x0 )2 + (y1 − y0 )2 ) dx0 dy0 (A.2)
iλz
2z


Z Z
1
ik 2
2
gf (x2 , y2 ) =fz (x1 , y1 ) exp − (x1 + y1 )
iλf
2f


ik
2
2
× exp
((x2 − x1 ) + (y2 − y1 ) ) dx1 dy1
2f

 

ik 2
x 2 y21
2
exp(x + y2 ) Fz
,
=
iλf
2f 2
fλ fλ

 



1
ik 2
x2 y2
2πi (x22 + y22 )z2=
exp(x + y2 ) F0
,
exp −
iλf
2f 2
fλ fλ
k
f 2 λ2 
1 f
iπ f − z 2x 2 y2=exp(x2 + y22 ) F0
,
,
(A.3)
2
iλ z
fλ f
fλ fλ
where

F0

x2 y2
,
fλ fλ





Z Z
=


2πi
(x0 x2 + y0 y2 ) dx0 dy0fλf0 (x0 , y0 ) exp
 2
Z Z
r
= E0
exp − 02 exp(i`θ0 )w

2iπr0 r2× expcos(θ0 − θ2 ) r0 dr0 dθ0fλUsing the identity
Jn (z) =

1
2πin

Z 2π
exp(iz cos θ) exp(inθ)dθ,
0

191

(A.4)

A Analytic form of pseudo-LG beam

Figure A.1: Schematic of field propagation within LG setup: The field at the SLM, f0 (x0 , y0 ),is composed of the incoming Gaussian intensity profile with the phase of the SLM imprinted.
For this calculation we only consider the effect of the azimuthal phase winding. The field after
freely propagating a distance z is defined as fz (x1 , y1 ). The action of the lens is to impart a
spatially varying phase delay, φL , defined in (4.10). The field of interest is that at the focal
point, gf (x2 , y2 ).

where Jn (z) is the Bessel function of the first kind. One then finds
2πi`−1 f
iπ f − z 2
gf (x2 , y2 ) = E0exp
r2 exp(i`θ2 )
λ z2
fλ f




Z
r02
2πr0 r2
× r0 exp − 2 J`
dr0 .
w
fλ

(A.5)

We have now extracted the azimuthal phase winding, showing that the field at thefocus carries angular momentum. The radial function can be slightly simplified using
the identities [225]
Z ∞
0

 2 
 2 
√
π
b
b
I
,
exp
Jn (bz) exp(−p z )dz =2p
8p2 n/2 8p2
2 2

(A.6)

where In (z) is a modified Bessel function of the first kind, andJn−1 (z) − Jn+1 (z)dJn (z)
=
.
dz
2

(A.7)

Integrating the radial function by parts we obtain the final expression


 2 
 2 
2
r22
r2
r2`−1 √ f w r2
gf (x2 , y2 ) = E0 i
π 2
exp − 2I `−1
− I `+1
2
2
z w0 w0
2w0
2w02
2w02


iπ f − z 2× exp
r2 exp(i`θ2 ),
(A.8)
fλ f
where we define the diffraction limit w0 = f λ/πw. We note that unlike the numerical calculations presented in Section 4.2.4, this result does not account for the finite
apertures of the SLM and the imaging lens which are found to reduce radial oscillations.

192

Appendix B

Numerical calculation of ring BEC
excitation spectrum
To consider the relevance of the various analytic models of critical velocity consideredin Section 5.5, we independently calculate the Bogoliubov excitation spectrum for anonrotating, two dimensional Bose gas confined in a ring potential. This calculation ispresented in [181], but here we present further details of our implementation. Repeating
this calculation was necessary to extend the results down to low atom numbers, relevant
to our experimental conditions where we observe phase slips. In doing so, we find theconclusion of [181], that the spectrum is well approximated by the surface mode modelof [180], is only valid above a critical chemical potential, µc . Below µc we find that thecritical velocity for sound excitations becomes the lowest, and hence relevant, criticalvelocity. This is accompanied by a change in the nature of the excitations from surface
instabilities, to phonon modes.
The condensate is described by the 2D GPE


∆ 1
2
2
i∂t Φ = − + (r − r0 ) + g|Φ| Φ,
2
2

(B.1)

where we use the associated scale for energy (~ωr ), time (ωr−1 ), and length ar =p~/(mωr ). The condensate wavefunction, Φ = Φ(r, θ, t), is normalised to unity,∆ = ∂r2 + ∂r /r + ∂θ2 /r2 is the Laplacian in polar coordinates, r0 = rM /ar is the√dimensionless ring radius, and g = N 8π aaz is the dimensionless 2D interaction constant, where N is the atom number, a is the s-wave scattering length, and az is the
harmonic oscillator length in the z direction [182].
The initial ground state of the system is found by propagation in imaginary time.
If we consider the wavefunction as a superposition of eigenstates φm (r) with timedependent amplitudes am (t) and eigenenergies Em (t), by making the substitution ∆t →−i∆t, the time evolution operator leads to an exponential decay of the wavefunction,
and a corresponding decay of the eignestates via
Φ(r, t + ∆t) =Xam (t)φm (r) exp(−Em ∆t).

m

193

(B.2)

B Numerical calculation of ring BEC excitation spectrum

Figure B.1: Convergence of ground statecalculation: Plotted is the fractional change
in the calculated chemical potential and afunction of iteration number. The relativechange in µ exponentially decays with iteration number, and the calculation stops when
the desired accuracy of 10−12 is obtained.
Inset is an example plot of the calculated
ground state density.

Crucially, the eigenenergy governs the decay rate, and so the eigenstate with the lowest
energy decays slowest. From some trial wavefunction and by renormalisation of the
wavefunction during the imaginary time propagation, the wavefunction will tend towards the ground state of the system.
We perform this calculation on a mesh grid in both real space and Fourier space.
We split the evolution into two steps, the first acts on the real space wavefunction
with the real space energy terms associated with the external trapping potential andmean field interactions. The second step Fourier transforms the wavefunction to themomentum space and acts upon it with the momentum term. Fourier transformingback to real space and renormalising, we obtain the wavefunction propagated in imaginary time. For sufficiently small time steps the wavefunction will then converge tothe ground state. The corresponding Matlab code for this calculation is presented in
CalculateGroundState.m. An example of the obtained convergence to the ground stateis presented in figure B.1.
To find the excitation spectrum we use the rotational invariance of B.1, and consider perturbations of the form
Φ(r, θ, t) = e−iµt [Φ(r) + δΦm (r, θ, t)],

(B.3)

where
∗

δΦm (r, θ, t) = um (r)e−i(ωt−mθ) + vm (r)∗ ei(ω t−mθ) .

(B.4)

Φ denotes the stationary ground state of the system, µ is the global chemical potential,and δΦ is a small perturbation parameterised by the angular wavenumber m. Insertionof this trial solution into the GPE we obtain the Bogoliubov-de Gennes equations tofirst order in δΦ:


m2 1
1
2
2
2
ωu(r) = −∂r − ∂r + 2 + (r − r0 ) + 2g|Φ| − µ u(r) + gΦ2 v(r),
r
r
2


2
1
m
1
2
2
2
ωv(r) = ∂r + ∂r − 2 − (r − r0 ) − 2g|Φ| + µ u(r) − gΦ∗ 2 u(r).
r
r
2
194

(B.5)
(B.6)

B Numerical calculation of ring BEC excitation spectrum

Figure B.2: Typical calculation of 2D annular BEC excitation spectrum: Plotted isthe eigenfrequency, ω, versus the angularwavenumber, m, for the lowest branch of
the dispersion relation. The black data
points are the results of our numerical calculation, obtained by diagonalisation of the
Bogoliubov-de Gennes equations. The reddashed line is the critical angular velocity,Ωc = min(ω(m)/m). The inset shows the radial density profile of the condensate (black
dashed line) and the density profile of theexcitation (red line), normalised to have thesame peak value.

To solve this problem we make use of the cylindrical symmetry of the problem andreduce it to a one dimensional problem in the positive radial direction. For our meshof points in the radial direction r1 ...ri ...rN , we define the vector of length 2N , Xi =[u(r1 ), ...u(ri ), ...u(rN ), v(r1 ), ...v(ri )..., v(rN )]. The Bogoliubov-de Gennes equations
can then be expressed as the matrix problem
Mi,j Xi = ωXj ,(B.7)

where Mij is a two dimensional matrix obtained from equations (B.5) and (B.6). To
discretise the gradient operators we use the standard form∂r u(ri ) ≈u(ri+1 ) − u(ri )
∆r

(B.8)

∂r2 u(ri ) ≈u(ri+1 ) − 2u(ri ) + u(ri−1 )
∆r2

(B.9)

and

Calculating the eigenenergies of the excitations, ω, then amounts to finding the eigenvalues of a 2N × 2N matrix, the lowest branch of which gives the critical angularvelocity, Ωc = min(ω(m)/m). One can also obtain further insight from the density
distribution of the excitation, given by δnm (r) = 2Re[Φ(r)∗ (um (r) + vm (r)∗ )].
An example excitation spectrum is shown in figure B.2, along with the density
profile of the excitation in the inset. We note that to extend our calculations to very
low atom numbers, large mesh sizes are required to minimise the introduction of errors
as the width of the annulus becomes comparable to the grid spacing. A more efficient
formalism could make use of variable mesh spacing, implementing a coarse mesh in theempty regions of the problem, and a finer mesh at the atoms.
Matlab code for calculating the ground state:1function Phi = CalculateGroundState()

195

B Numerical calculation of ring BEC excitation spectrum

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

%Set up mesh grid for calculation%Size of grid in radial oscillator lengths
L = 50;%Number of points per dimension in grid
Ngrid = 512;
%Iteration step size in imaginary time
tau = 0.04;%Distance between points in grid∆x = L/Ngrid;%Real space grid[x,y] = meshgrid(−L/2:∆x:L/2);%Momentum space grid
[px,py] = meshgrid(−pi*Ngrid/L:2*pi/L:pi*Ngrid/L);%Real space grid in cylindrical polar coordinates[theta, r] = cart2pol(x,y);%Convergence required for ground state calculation
lim = 10ˆ−12;

20
21
22
23

%=================Find the Ground State================%%Guess initial trial wavefunction based on 2D TF profile
Phi0 = real(sqrt(real((1−((r−r0)/RTF).ˆ2).ˆ(3/2))));

24
25
26
27

Phi0 = Normalise(Phi0);
Phi = Phi0;∆u = 1;28
29
30%Iterate to find ground state
while (abs(∆u)>lim)

31
32
33
34
35
36
37
38
39
40

ustart = chemPot(Phi);
Phi = exp(−tau*(0.5*(r−r0).ˆ2+g*Phi.*conj(Phi))).*Phi;
Phi = fftshift(fft2(Phi));
Phi = exp(−tau*(0.5*px.ˆ2+0.5*py.ˆ2)).*Phi;
Phi = ifft2(ifftshift(Phi));
Phi = Normalise(Phi);%Calculate chemical potential of new wavefunction
ustop = chemPot(Phi);∆u = abs((ustop−ustart)/ustart);

41
42
43

end
%=======================================================%

44
45
46

%==================Normalise Wavefunction===============%
function p = Normalise(Phi)

47
48
49
50
51
52

if (sum(sum(Phi.*conj(Phi))) 6=0)
p = Phi*sqrt(1/sum(sum(Phi.*conj(Phi)*(∆xˆ2))));else
p = Phi;end

196

B Numerical calculation of ring BEC excitation spectrum

53

end%=======================================================%

54
55
56

%==============Calculate chemical potential=============%
function u = chemPot(Phi)

57
58
59

if (sum(sum(Phi.*conj(Phi))) 6=0)
%Real space energy
V = (∆xˆ2)*sum(sum(conj(Phi).*((0.5*(r−r0).ˆ2).*Phi...
+g*(Phi.*conj(Phi)).*Phi)));%Kinetic energy
K = (∆xˆ2)*sum(sum(conj(Phi).*ifft2(ifftshift((0.5*px.ˆ2+...
0.5*py.ˆ2).*fftshift(fft2(Phi))))));
u = (V+K)/((∆xˆ2)*sum(sum(conj(Phi).*Phi)));else
u = 0;
end

60
61
62
63
64
65
66
67
68
69
70

end

71

%=======================================================%

72
73
74
75

end

Matlab code for calculating the excitation spectrum:1function Phi = CalculateExcitationSpectrum(Phi)234%Angular wavenumber of excitations to consider
m values = 0:60;

5
6
7
8
9
10
11
12
13

%Get Phi as a function of radial variable
PhiR = Phi(Ngrid/2+1,:);
R = x(Ngrid/2+1,:);%Crop R and PhiR to exclude zero
PhiR = PhiR(logical(R>0));R = R(logical(R>0));∆R = L/Ngrid;n = size(R,2);

14
15%=================Find the excitation spectrum================%

16
17
18

for i = 1:size(m values,2)m = m values(i);

19
20
21
22
23
24

%Generate sparse matrix for diagonalising%Create on diagonal terms relating to angular kinetic energy,% trapping potential, mean field interactions, and chemical% potenial
OnDiag = [(((m)ˆ2)./(2*R.ˆ2)+0.5*(R−r0).ˆ2+...

197

B Numerical calculation of ring BEC excitation spectrum

2*g*(PhiR.*conj(PhiR))−u)...
(−((m)ˆ2)./(2*R.ˆ2)−0.5*(R−r0).ˆ2−...
2*g*(PhiR.*conj(PhiR))+u)];

25
26
27
28

%Create terms corresponding to double derivative term in the%kinetic energy, dˆ2/drˆ2.
OnDiagKineticSecond = −0.5*[−1/(∆Rˆ2) −2*ones(1,n−2)/(∆Rˆ2)...
−1/(∆Rˆ2) 1/(∆Rˆ2) 2*ones(1,n−2)/(∆Rˆ2) 1/(∆Rˆ2)];
OneUpDiagSecond = −0.5*[ones(1,n−1)/(∆Rˆ2) 0 ...
−ones(1,n−1)/(∆Rˆ2)];
OneDownDiagSecond = −0.5*[ones(1,n−1)/(∆Rˆ2) 0 ...
−ones(1,n−1)/(∆Rˆ2)];

29
30
31
32
33

34

35

%Create terms corresponding to single derivative term in the%kinetic energy d/dr.
OnDiagKineticFirst = −0.5*[−1./(R(1:n−1)*∆R) 1/(R(n)*∆R)...
1./(R(1:n−1)*∆R) −1/(R(n)*∆R)];
OneUpDiagKineticFirst = −0.5*[1./(R(1:n−1)*∆R) 0 ...
−1./(R(1:n−1)*∆R)];
OneDownDiagKineticFirst = −0.5*[zeros(1,n−2) −1/(R(n−1)*∆R)...
zeros(1,n−1) 1/(R(n−1)*∆R)];

36
37
38
39
40

41
42
43

OnDiagKinetic = OnDiagKineticSecond+OnDiagKineticFirst;
OneUpDiag = OneUpDiagSecond+OneUpDiagKineticFirst;
OneDownDiag = OneDownDiagSecond+OneDownDiagKineticFirst;

44
45
46
47

%Create the cross terms between the u(r) and v(r) equations
NOffDiag = g*PhiR.ˆ2;

48
49
50

%Build matrix from individual vectors created so far. All ...
elements%not specified are set to zeroM = diag(OnDiag)+diag(OnDiagKinetic)+diag(OneUpDiag,1)+...
diag(OneDownDiag,−1)+diag(NOffDiag,size(R,2))+...
diag(−conj(NOffDiag),−(size(R,2)));

51

52
53
54
55
56
57

%Calculate eigenvectors and eignevalues of matirx M[V,D] = eig(M,'nobalance');
W = diag(D);%Extract lowest energy eigenvalue and the corresponding ...
eigenvector[W,index] = sort(W);Omega(i) = W(1);Vector = V(:,index(1));
Vectors{i} = Vector;

58
59
60
61

62
63
64
65
66
67

end

68
69
70

%Anglin surface mode vc
w surface = abs(m values)*(sqrt(2)*(uˆ(1/6))/(r0+sqrt(2*u)));198

B Numerical calculation of ring BEC excitation spectrum%Local speed of sound vc
w sound = abs(m values)*(2*sqrt(u*hbar*wr/(3*mass))/(r0))/(ar*wr);%vc from excitation spectrum[CritW,mcrit] = ...
min(abs(Omega(abs(m values)>0)./m values(abs(m values)>0)));w spectrum = abs(m values)*CritW;

71
72
73
74

75
76
77

end

199

200

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