The superfluid properties of a Bose-Einstein Condensed Gas (2002)

The Superfluid Properties of a

Bose-Einstein Condensed Gas

Eleanor Hodby

A thesis submitted in partial fulfilment ofthe requirements for the degree of

Doctor of Philosophy at the University of Oxford

Christ Church CollegeUniversity of Oxford

Trinity Term 2002

Abstract

The Superfluid Propertiesof a Bose-Einstein Condensed Gas

Eleanor Hodby, Christ Church College, OxfordD.Phil thesis, Trinity 2002

This thesis describes experiments carried out on magnetically trapped Bose-Einstein condensates of 87Rb atoms and the theoretical interpretation of the re-sults. We investigate the superfluid nature of the condensate by observing itsresponse to a variety of torques applied by the trapping potential. Using this di-lute, weakly-interacting system the fundamental relation between Bose-Einsteincondensation and superfluidity is explored directly, without the complications ofstrong interatomic interactions.

The apparatus and procedure used to achieve quantum degeneracy in our dilute87Rb vapour is described briefly, with particular emphasis on the modifications thathave been necessary for the experiments described in this thesis. Condensates areproduced with up to 5 × 104 atoms and at temperatures as low as 150 nK, usinglaser cooling followed by magnetic trapping and evaporative cooling.

Superfluidity imposes the constraint of irrotational flow on a condensate in a ro-tating potential and leads to the formation of quantized vortices at higher rotationrates. Observation of the scissors mode oscillation and the expansion behaviourof a condensate after release from a slowly rotating potential both confirmed thepurely irrotational nature of the condensate flow pattern under conditions where anormal fluid would flow in a rotational manner. The scissors mode oscillation fre-quency was also measured at higher temperatures and was observed to decrease.This result indicates a reduction in the superfluidity of the condensate fractionclose to the critical temperature.

A systematic study of the critical trap conditions for vortex nucleation wascarried out in a purely magnetic rotating potential. This work provided importantdata against which the numerous theories of vortex nucleation can be tested. Theareas in which our results both agree and disagree with current theories will bediscussed. In the final experiment, a superfluid gyroscope was created from a singlevortex line and the scissors mode of the condensate. It was used to measure theangular momentum of the vortex line and the results are in good agreement withquantum mechanics.

i

Acknowledgements

My first thanks to CJ, once Dr, now Prof,There aren’t many supervisors so highly thought of,I hope he’s not stressed that I’m cutting it fine,But I might have submitted in about 8 days time.

Big thanks go to Hoppo, Onof and Jan,For building an experiment that has proved that it canTake on the world at the BEC game....As PRL have acknowledged again and again

With Uncle Gerald we made a great team,With laughter and fun and occasional steamWith different approaches but the same aim at heartGroup discussions could sometimes be heard from the Parks

Now Nathan’s installed I have nothing to fear,He’s salvaged the BBQ already this year,Where once there was silence and no-one dared sneeze,Now the condensates boogie to his MP3s.

As for my office, apologies galoreif you’ve stumbled on underwear, left on the floor.Dona will be knighted soon after I’m gone,For feeding me chocs and putting up with the pong.

Without Auntie Rachel - what will I do?Doling priceless advice in the Clarendon loosNo dilema or panic has defeated her yet,So I’ll seek her advice via the internet.

After altitude training, I’ll be back Mr Mike,And the triathlon sequel won’t be no look-a-like,Rowing songs, Angharad, are at last history,And don’t stress, when you’re famous, I’ll hide the CD.

To the rest of the basement - I’ll miss you allAnd I’ve even grown rather fond of that wall...And about Uncle Graham, what can I say-Who’ll dream up my demos in the US of A?

ii

Thanks to the theorists for staying straight-faced,When my notions of theory were way off the pace,And when your stuck, the best theories I hear,Are inspired in the pub with pork scratchings and beer.

Right there you have it, 4 years condensed,I’d better stop now, while I still make some sense,Thank you for making my D.Phil so funAnd if you’re out in the States then you know where to come.

To Lesley and Cecily - thank you so much for all the fun that we have hadin Oxford, but most of all for your friendship, especially over the last few months(and don’t think that the crazy plans stop just because I’m in the US!). I’d alsolike to thank my housemates Carol, Suzanne, Cecily and Onofrio for all the goodtimes that we have had at 12 Oswestry Road.

Finally thank you to my family, Mum, Dad, Richard and Katharine, for allyour love and encouragement over the last 26 years.

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Contents

1 Introduction 2

2 The BEC Apparatus 4

2.1 Overview of the apparatus . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Maintaining the vacuum system . . . . . . . . . . . . . . . . 7

2.2.2 Improvements to the vacuum system . . . . . . . . . . . . . 8

2.3 Lasers and optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 The master and repumping lasers . . . . . . . . . . . . . . . 10

2.3.2 The magneto-optical traps . . . . . . . . . . . . . . . . . . . 10

2.3.3 Commercial ECDLs . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Frequency control . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.5 The master oscillator power amplifier . . . . . . . . . . . . . 17

2.3.6 The injection-locked slave laser . . . . . . . . . . . . . . . . 18

2.3.7 A new MOPAless laser system . . . . . . . . . . . . . . . . . 20

2.4 The magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 The theory of the TOP trap . . . . . . . . . . . . . . . . . . 22

2.4.2 The TOP trap apparatus . . . . . . . . . . . . . . . . . . . . 25

2.4.3 Calibration of the TOP trap . . . . . . . . . . . . . . . . . . 27

2.4.4 Radio-frequency coils . . . . . . . . . . . . . . . . . . . . . . 30

2.5 The imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.1 The horizontal imaging system . . . . . . . . . . . . . . . . 30

2.5.2 The camera . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.4 Calibrating the imaging system . . . . . . . . . . . . . . . . 34

2.5.5 Further image analysis . . . . . . . . . . . . . . . . . . . . . 35

2.5.6 Non-destructive imaging . . . . . . . . . . . . . . . . . . . . 36

2.5.7 The vertical imaging system . . . . . . . . . . . . . . . . . . 38

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CONTENTS v

3 BEC Production 453.1 Loading the second MOT . . . . . . . . . . . . . . . . . . . . . . . 463.2 Loading the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Compression of the cloud in the MOT . . . . . . . . . . . . 463.2.2 Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . 473.2.3 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . 473.2.4 Loading the TOP trap . . . . . . . . . . . . . . . . . . . . . 47

3.3 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.1 Adiabatic compression . . . . . . . . . . . . . . . . . . . . . 483.3.2 Evaporation using the magnetic field zero . . . . . . . . . . 503.3.3 Radio-frequency evaporation . . . . . . . . . . . . . . . . . . 50

3.4 Detecting the transition . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Optimizing Condensate Production 534.1 Loading the second MOT . . . . . . . . . . . . . . . . . . . . . . . 534.2 Alignment of the second MOT and stray

magnetic field nulling . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Loading the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.1 Compression and molasses parameters . . . . . . . . . . . . 564.3.2 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . 564.3.3 Initial parameters for the magnetic trap . . . . . . . . . . . 59

4.4 Evaporative cooling ramps . . . . . . . . . . . . . . . . . . . . . . . 604.4.1 Adiabatic compression . . . . . . . . . . . . . . . . . . . . . 604.4.2 Evaporation using the magnetic field zero . . . . . . . . . . 624.4.3 Radio-frequency evaporation . . . . . . . . . . . . . . . . . . 64

4.5 Optimization summary . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Condensate Theory 705.1 BEC in a non-interacting gas . . . . . . . . . . . . . . . . . . . . . 705.2 The trapped non-interacting Bose gas . . . . . . . . . . . . . . . . . 725.3 Bose-Einstein condensation with interacting particles . . . . . . . . 735.4 The ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5 The hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . 765.6 Low-lying collective states . . . . . . . . . . . . . . . . . . . . . . . 78

5.6.1 Mode frequencies . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Bose-Einstein Condensation and Superfluidity 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Dissipationless flow and critical velocity . . . . . . . . . . . . . . . . 876.3 The superfluid response to a torque . . . . . . . . . . . . . . . . . . 886.4 Irrotational flow and the reduced moment of inertia . . . . . . . . . 896.5 Vortex theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.5.1 Core size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi CONTENTS

6.5.2 Vortex energetics and metastability . . . . . . . . . . . . . . 94

6.5.3 Quantization of angular momentum . . . . . . . . . . . . . . 95

6.5.4 Kelvin waves . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 The Scissors Mode Experiment 101

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2.1 The scissors mode oscillation of the condensate . . . . . . . 102

7.2.2 Oscillation frequencies of the thermal cloud . . . . . . . . . 104

7.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4 Thermal cloud results . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.5 Scissors mode results for the condensate . . . . . . . . . . . . . . . 109

7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.7 The scissors mode at finite temperature . . . . . . . . . . . . . . . . 111

7.7.1 Experimental procedure and results . . . . . . . . . . . . . . 112

7.7.2 Moment of inertia at finite temperature . . . . . . . . . . . . 114

8 Superfluidity and the Expansion of a Rotating BEC 117

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.3 The rotating anisotropic trap . . . . . . . . . . . . . . . . . . . . . 121

8.3.1 The elliptical TOP trap . . . . . . . . . . . . . . . . . . . . 121

8.3.2 Rotating the trap . . . . . . . . . . . . . . . . . . . . . . . . 123

8.4 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . 125

8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9 Vortex Nucleation 132

9.1 Vortices in He II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9.2 Vortices in a dilute gas Bose-Einstein condensate . . . . . . . . . . 133

9.2.1 Nucleation of vortices . . . . . . . . . . . . . . . . . . . . . . 133

9.2.2 Detection of vortices . . . . . . . . . . . . . . . . . . . . . . 135

9.3 Vortex nucleation in a rotating potential . . . . . . . . . . . . . . . 136

9.4 Experimental method . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.5 Nucleation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.6 Vortex nucleation mechanisms in ourapparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.6.1 Evidence for vortex decay mechanisms . . . . . . . . . . . . 146

9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS vii

10 The Superfluid Gyroscope 14910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14910.2 The theory of the superfluid gyroscope . . . . . . . . . . . . . . . . 15110.3 Exciting and observing the gyroscope . . . . . . . . . . . . . . . . . 15210.4 Gyroscope results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.5 How does the vortex core move? . . . . . . . . . . . . . . . . . . . . 15810.6 Kelvin waves and the gyroscope experiment . . . . . . . . . . . . . 162

11 Conclusion and Future Plans 16411.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 16511.2 Future experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11.2.1 Nucleation of multiply charged vortices . . . . . . . . . . . . 16611.2.2 Exciting and observing Kelvin waves . . . . . . . . . . . . . 16911.2.3 Damping of the m = 2 modes in the presence of a vortex . . 17111.2.4 Critical conditions for nucleating a second vortex . . . . . . 17111.2.5 Anti-vortex production . . . . . . . . . . . . . . . . . . . . . 17211.2.6 Precession with an off-centred vortex line . . . . . . . . . . . 172

A The Properties of a 87Rb Atom. 173

B Clebsch-Gordan Coefficients 175

C Thermal Cloud Formulae 177

List of Figures

2.1 A scale diagram of the BEC experiment . . . . . . . . . . . . . . . 5

2.2 The double MOT vacuum-system . . . . . . . . . . . . . . . . . . . 6

2.3 The improved vacuum system . . . . . . . . . . . . . . . . . . . . . 8

2.4 The hyperfine structure of the 5S1/2 to 5P3/2 D2 transition in 87Rb 11

2.5 A y-z cross-section of the pyramid MOT. . . . . . . . . . . . . . . . 12

2.6 The design of the external cavity diode lasers. . . . . . . . . . . . . 14

2.7 The saturated-absorption signal from one of our Rb vapour cells . . 16

2.8 The Doppler-free spectrum from the upper hyperfine ground statein 87Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 The optical isolation in front of an injection-locked slave laser . . . 19

2.10 The current laser system and a replacement system that uses onlyinjection-locked slave lasers . . . . . . . . . . . . . . . . . . . . . . . 21

2.11 The instantaneous magnitude of the TOP trap magnetic field in thex-y plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.12 A cross-section diagram and photograph of the TOP trap . . . . . . 26

2.13 The circuitry driving the TOP coils for a standard TOP trap . . . . 27

2.14 A plot of the relative optical density of a cloud versus probe detuning 28

2.15 The atom cloud position after excitation of a dipole mode in the xdirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.16 The horizontal imaging system . . . . . . . . . . . . . . . . . . . . . 31

2.17 The power passing a knife edge as a function of its position in ourprobe beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.18 The position of the cloud as it falls under gravity, used to calibratethe magnification of the horizontal imaging system . . . . . . . . . 36

2.19 1D density distributions of expanded condensates at different tem-peratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.20 A diagram of the vertical imaging system . . . . . . . . . . . . . . 39

2.21 A photograph of the experimental MOT and vertical imaging system 40

2.22 A diagram showing how the depth of focus of an imaging systemmay be calculated . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

viii

LIST OF FIGURES ix

3.1 The different magnetic substates involved in our optical pumpingprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 The trap parameters during evaporative cooling . . . . . . . . . . . 493.3 Images of expanded clouds with different condensate fractions . . . 52

4.1 The temperature of the cloud as a function of the vertical nullingcoil current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 The phase-space density, number and temperature of atoms in themagnetic trap as a function of optical pump intensity . . . . . . . . 58

4.3 The number and temperature of the cloud after loading into trapsof different radii but the same stiffness . . . . . . . . . . . . . . . . 59

4.4 Phase-space density versus atom number close to condensation fordifferent minimum values of BT . . . . . . . . . . . . . . . . . . . . 63

4.5 Phase-space density versus B = 0 evaporation time . . . . . . . . . 644.6 Dressed atom energy levels in a strong rf field . . . . . . . . . . . . 654.7 Typical condensate images taken during the gyroscope experiment,

both before and after optimization . . . . . . . . . . . . . . . . . . 674.8 Phase-space density versus atom number during the optimized evap-

oration ramps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.9 The atom number, temperature and phase-space density as a func-

tion of time during the optimized evaporation ramps . . . . . . . . 69

5.1 The density of states as a function of energy in a homogeneous andtrapped gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 The six independent quadrupole mode geometries in an axially sym-metric trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 The Cartesian operators and geometries of the m = ± 2 and m =± 1 modes in an axially symmetric trap . . . . . . . . . . . . . . . . 82

5.4 The spectrum of low-lying, collective modes in an axially symmetricTOP trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.1 Rotational and irrotational velocity fields . . . . . . . . . . . . . . . 896.2 Vortex core radius as a function of expansion time . . . . . . . . . . 936.3 The energy of the first vortex state in the rotating frame as a func-

tion of the vortex position . . . . . . . . . . . . . . . . . . . . . . . 966.4 The forces on a vortex line that result in helical Kelvin wave motion 98

7.1 Direct Monte-Carlo simulations of the ‘scissors’ type oscillation ina non-condensed cloud . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2 The scissors mode excitation procedure . . . . . . . . . . . . . . . . 1087.3 Typical images of the thermal cloud and condensate used for the

scissors mode experiment . . . . . . . . . . . . . . . . . . . . . . . . 1097.4 The scissors oscillation data for the thermal cloud and the condensate110

x LIST OF FIGURES

7.5 The damping rate and oscillation frequencies of the condensate andthermal cloud as a function of temperature . . . . . . . . . . . . . . 113

7.6 The normalized moment of inertia of the condensate fraction, thethermal cloud and the combined system as a function of temperature116

8.1 The trap conditions (εt, Ω) under which each of the three quadrupolemodes of a condensate in a rotating potential exist . . . . . . . . . 120

8.2 The normalized trapping frequencies in an elliptical TOP trap, as afunction of E = Bx/By . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.3 A schematic diagram of the electronics for the elliptical rotatingTOP trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.4 Absorption images of an initially rotating condensate and a staticcondensate at different expansion times . . . . . . . . . . . . . . . . 127

8.5 The angle and aspect ratio of the condensate as a function of ex-pansion time, released from a rotating trap . . . . . . . . . . . . . . 128

8.6 The asymptotic rotation angle of the condensate after long expan-sion times, as a function of the initial trap rotation rate . . . . . . . 129

8.7 The aspect ratio and angle of a thermal cloud as a function of timeafter release from a rotating trap . . . . . . . . . . . . . . . . . . . 130

9.1 Images of the condensate during nucleation and of lattices of 1-4vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

9.2 The mean number of vortices as a function of the normalized traprotation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.3 The mean number of vortices as a function of trap deformation . . . 1419.4 The critical conditions for vortex nucleation . . . . . . . . . . . . . 1439.5 The number of vortices and number of atoms as a function of hold

time is a stationary circular trap . . . . . . . . . . . . . . . . . . . . 147

10.1 The gyroscope motion . . . . . . . . . . . . . . . . . . . . . . . . . 15010.2 Images of single vortices, illustrating our criterion for a centred vortex15310.3 Gyroscope data - excitation in xz plane . . . . . . . . . . . . . . . . 15510.4 Gyroscope data - excitation in yz plane . . . . . . . . . . . . . . . . 15710.5 Sideview images of the gyroscope motion in which the vortex line is

clearly visible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.6 A mechanical model of the trapped condensate . . . . . . . . . . . . 161

11.1 Radial magnetic fields in a quadrupole and I-P trap . . . . . . . . . 16711.2 The scheme for exciting and trapping a vortex with 4 units of cir-

culation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16811.3 Resonance conditions for exciting the lowest Kelvin wave . . . . . . 170

B.1 The Clebsch-Gordan coefficients for the F=2 to F’=3 hyperfine tran-sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

List of Tables

2.1 Details of the injection-locking characteristics of two diode lasers. . 20

3.1 The stages in the TOP trap loading procedure. . . . . . . . . . . . 473.2 The evaporative cooling ramps used to obtain BEC . . . . . . . . . 49

4.1 The stages in the optimized TOP trap loading procedure. . . . . . . 564.2 Transition rates for optical pumping with the original and improved

schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 The optimized evaporative cooling ramps. . . . . . . . . . . . . . . 61

5.1 Formulae for useful condensate parameters . . . . . . . . . . . . . . 77

9.1 Critical rotation rates for vortex nucleation . . . . . . . . . . . . . . 137

A.1 The properties of 87Rb relevant to this experiment . . . . . . . . . . 173A.2 Rb vapour pressure constants . . . . . . . . . . . . . . . . . . . . . 174

C.1 Useful thermal cloud formulae . . . . . . . . . . . . . . . . . . . . . 178

1

Chapter 1

Introduction

This thesis describes the experimental investigation of superfluidity in a dilute gasBose-Einstein condensate of 87Rb atoms. Bose-Einstein condensation occurs whenmany identical particles all occupy the same quantum mechanical state. Thisbehaviour results from the quantum statistics of identical bosonic particles andwas first predicted by Einstein in 1925 [?]. It lies at the heart of phenomena suchas superconductivity and superfluidity of liquid helium [?]. However these systemshave strongly interacting particles that make them much more complex than theideal gas described by Einstein. Realization of a Bose-Einstein condensate (BEC)in an ultra-cold dilute atomic vapour, with properties close to those predicted byEinstein was first achieved by Cornell, Wieman and coworkers in 1995 [?]. This firstBose-Einstein condensate experiment used 87Rb atoms and within a few monthscondensation of 23Na atoms had also been achieved by Ketterle and his group atMIT [?]. Over the last seven years the field has grown exponentially. There are nowover thirty Bose condensate experiments around the world, producing condensatesof six different elements and seven different isotopes (87Rb [?], 23Na [?], H [?], 85Rb[?], 7Li [?], He* [?, ?], 41K [?]).

Although it was always expected that a dilute gas BEC would behave as asuperfluid, it was several years before superfluid effects were observed [?, ?, ?, ?].The most striking signatures of superfluidity involve the response of the systemto rotation, since the presence of a single macroscopic wavefunction constrains theflow patterns allowed within its bulk. Thus to probe the superfluid nature of adilute gas Bose-condensate we have developed a flexible trapping potential thatis able to rotate the condensate about any axis. Each experiment described inthis thesis shows the superfluid response of the condensate to a different appliedtorque.

The structure of this thesis may be summarized as follows: In chapters 2 to 4the apparatus and procedures used to create, excite, image and analyse condensates

2

3

of 87Rb atoms are described. In particular in chapter 4 we describe the detailedoptimization process that was recently carried out prior to the final gyroscopeexperiment, which succeeded in tripling the number of atoms in the condensate.Chapters 5 and 6 summarize the BEC theory that underlies the experiments de-scribed later in the thesis. Chapter 5 concentrates on the collective excitationspectrum of a weakly interacting Bose condensate in a harmonic trap, whilst chap-ter 6 discusses the definition of superfluidity and predicts the superfluid responseof the condensate to a rotating trapping potential. Both here and in chapter 9the behaviour of the dilute gas condensate is compared and contrasted to liquidhelium II, which to date is the most extensively studied superfluid system.

Chapters 7 to 10 present 4 different experiments which each demonstrate thesuperfluid response of the condensate to a different applied torque. The first twoexperiments demonstrate the pure irrotational flow pattern characteristic of a sim-ply connected superfluid in different ways. First we investigated the frequency ofthe small angle oscillations of the condensate relative to the trapping potentialknown as the scissors mode, whilst chapter 8 describes the distinctive behaviour ofan expanding condensate, after it is released from a slowly rotating potential. Theoriginal scissors mode experiment provided some of the first evidence that the con-densate behaves as a superfluid together with the observation of quantized vortices[?, ?] and a critical velocity for superfluid flow [?]. The scissors mode experimentwas later repeated at higher temperatures to investigate the interaction betweenthe condensate and the thermal cloud and how it affects the superfluidity of thecondensate fraction.

The observation of quantized vortices in a purely magnetic rotating potentialis described in chapter 9. The chapter also presents a detailed study of the criticaltrap conditions (eccentricity and rotation rate) for nucleation, which was used todetermine the mechanism by which vortices are nucleated. Finally in chapter 10 wepresent the results of the superfluid gyroscope experiment, in which simultaneousexcitation of the scissors mode and the first vortex state enabled us to measurethe quantized circulation associated with the vortex line. In the concluding chap-ter, these results are summarized and plans for future superfluid experiments aredescribed in detail.

Chapter 2

The BEC Apparatus

This chapter briefly describes the apparatus that was used for creating Bose-Einstein condensates of Rubidium-87 atoms in Oxford. A more detailed descrip-tion of the original set-up is contained in [?], whilst this account focuses on theimprovements made to the apparatus during the course of my D.Phil.

2.1 Overview of the apparatus

The experimental procedure is similar for the majority of BEC experiments. Atomsare collected and laser cooled in a magneto-optical trap (MOT) inside an ultra-high vacuum (UHV) system. The pre-cooled cloud is then transferred to a purelymagnetic trap for evaporative cooling to the BEC transition temperature. Finallythe atoms are imaged on a CCD (charge-coupled device) camera. Figure 2.1 showsthese 4 basic elements - the vacuum system, laser system, magnetic trap and imag-ing system. Precise timing and synchronization of the different stages is requiredto make and image a condensate and so all the different components operate undera single control computer.

2.2 Vacuum system

The evaporative cooling process used to make a condensate relies on elastic colli-sions between atoms in the magnetic trap for rethermalization. The initial collisionrate of ∼ 10 s−1 sets the minimum timescale for the evaporative cooling process toaround 40 s. Since we desire both a low temperature and high number of atomsto achieve BEC, the lifetime of the magnetic trap must be greater than 40 s. Thelifetime of the trap is set by loss from collisions with atoms in the backgroundvapour, and so the magnetic trapping region must be held at a very low pressure

4

2.2. Vacuum system 5

Figure 2.1: A scale diagram of the BEC experiment showing the optical layout in detail.

6 Chapter 2. The BEC Apparatus

Figure 2.2: The double MOT vacuum system

of < 10−11 mbar. Such a low pressure would not be easy to achieve with the rubid-ium source nearby and so we use a double MOT system to separate the rubidiumsource from the magnetic trap (fig. 2.2). Other experiments have used differentsolutions to this problem. The first BEC experiment used a single dark MOT witha very low background pressure of rubidium ∼ 10−11 Torr [?]. Alternatively theuse of a pre-cooled beam of atoms from a Zeeman slower to load the experimentalMOT was demonstrated by the experiments to condense sodium [?] and lithium[?].

In our experiment the first MOT is of a pyramid design (section 2.3.2). It ishoused inside a 10-way cross with a large (100 mm diameter) front window, anda 25 ls−1 ion pump (Varian VacIon Plus 25) maintains a pressure of ∼ 10−9 mbar.The source of rubidium atoms is a dispenser (Saes Getters Rb/NF/7/25 FT10+10)situated just above the pyramid mirrors. When a current passes through the wirea chemical reaction releases rubidium vapour into the cell. The threshold currentfor this reaction is 2.6 A, and we currently run it continuously at 3.7 A. Sincethese getters were designed for rapid deposition of large quantities of rubidium,its lifetime for slow emission is unknown. When it finally runs out it should bereplaced by several dispenser wires so that a new one can be activated withoutneeding to open the vacuum system.

2.2. Vacuum system 7

Atoms are continuously pushed from the first MOT (by one of the MOT beams),through a hole at the back of the pyramid and along a narrow tube to the second,experimental MOT. This MOT is contained in a quartz cell of square cross-section(2× 2× 6 cm), connected to the tube and additional pumps by a custom made T-piece. (see fig. 2.2). The connecting tube is 30cm long, with an internal diameterof 16 mm. Its long thin shape results in a low conductance of 16 ls−1 and enablesus to maintain a pressure differential between the ends of the tube. Thus theexperimental MOT is held at < 10−11 mbar by a 40 ls−1 ion pump (Varian VacIonPlus 40). In addition to the ion pump, we also occasionally (roughly once a month)use the titanium sublimation pump (TSP) (Vacuum Generators ST22) which issituated in the pipework behind the quartz cell. A current of 48 A is passed througha wire in the pump for one minute, causing titanium to sublimate and stick to thesurrounding walls of the vacuum system. This produces a very thin porous layerwhich adsorbs background gas molecules. However the efficiency with which theTSP pumps rubidium is unknown and we have no clear evidence that the secondMOT vacuum improves after it is fired.

2.2.1 Maintaining the vacuum system

The original assembly and bakeout is described in [?]. After about a year of oper-ation, it became difficult to produce condensates reliably, although large numbersof atoms could be collected in the second MOT. We suspected that Rb had builtup on the walls of the cell, which was liberated when the cell was heated by thesurrounding trap coils, producing a high background pressure and inefficient evap-orative cooling. A comparison between the original and current magnetic traplifetimes would have confirmed this idea, however for technical reasons (destruc-tive imaging, heating of cell by trap coils etc) it has never been possible to get areliable measurement of the lifetime. So we wrapped the cell in an aluminium foiljacket and heated it with a hot air gun overnight, gently raising the surroundingtemperature to 80C, to investigate whether production could be temporarily im-proved as some rubidium sublimated from the cell walls and recondensed a littlefurther away on the cold metal pipework. Production did apparently improve forseveral hours of operation adding weight to our suspicions about the Rb back-ground pressure. We decided to rebake the system, which involved removing theMOT optics, building an oven around the system, attaching the turbo pump andbaking at 250C for several days. With hindsight I have the following doubts aboutwhether the bakeout was necessary, which could only be resolved with accuratemagnetic trap lifetime measurements:

• The experiment has since run for 3 years without requiring another bakeout.

• We have since discovered that problems with the laser modes can result ingood loading but erratic condensate production. This observation is probably


Figure 2.3: The vacuum system of the new BEC experiment at Oxford, which incorpo-rates many of the improvements ideas listed in this chapter - namely larger ion pumps,a larger pyramid MOT, a shorter connecting tube, reduced pipework and a new getter

pump.

due to a reduction in laser power at the correct frequency during the molassescooling stage, if the laser is not perfectly single mode (see section 2.3.3).

2.2.2 Improvements to the vacuum system

Improvements could be made to the vacuum system, to increase both the size andrate at which condensates are produced. Several of these have been implementedon the second generation BEC apparatus, shown in fig. 2.3. (This apparatus wasnot used for the experiments described in this thesis).

• A larger pyramidal MOT (base size 6 cm) gives a larger collection region andhence produces an atomic beam of higher flux

• The background gas conductance of the tube connecting the 2 MOTs is givenby (D3/L)×12.1 ls−1 where D is the diameter of the tube and L is the length,both in cm. This is the formula for free molecular flow down a tube at verylow pressures. It assumes that the tube size is much smaller than the meanfree path and that after collision with a wall, the path of an atom followsa cosine law (no specular reflection) thus providing a resistance mechanism.The conductance depends on the number of atoms entering the tube (∝ D2)and is inversely related to the average number of collisions that a particlehas with the walls, (∝ L/D).

2.3. Lasers and optics 9

Given the divergence of the atomic beam leaving the pyramid, the flux of pre-cooled atoms reaching the second MOT is roughly proportional to the solidangle of the tube D2/L. Comparing this to the expression for background gasconductance, one can increase the flux whilst maintaining the same pressuredifferential by using a shorter, narrower tube. In the new experiment, thedistance between MOTs is reduced to 30 cm, and the pressure differential ismaintained by a 5 mm diameter hole in a metal flange of thickness 17 mm.

• Given our doubts about the effectiveness of the TSP, it could be replaced orsupplemented by a non-evaporable getter (NEG) pump, which removes gasmolecules that arrive at the surface of the porous getter cartridge by chemicalreaction. The model used in the new experiment is (SAES CapaciTorr-CF35,Cartridge C-400-DSK-St172).

• Larger ion pumps are available. The new experiment uses 40 ls−1 and 55 ls−1

models (VacIon Plus 40 and Varian 55 Starcell).

• The length of pipework around the second MOT has been reduced to increaseconductance. In particular the vacuum system is mounted on runners on itsown breadboard and so it can be wheeled to the edge of the table to attachthe turbo pump for baking out. The original version requires a long pipereaching to the edge of the table because the vacuum system cannot bemoved.

2.3 Lasers and optics

Laser light has three roles in the production of a BEC:

• Magneto-optical trapping and cooling

• Optical pumping prior to loading the magnetic trap

• Imaging the condensate

The laser light is generated by 2 external cavity diode lasers (ECDLs) andamplified by further semiconductor devices (a MOPA and a slave laser). Theprecise control that we require over the intensity and frequency of each beam isachieved using acousto-optic modulators (AOMs), whilst the absolute frequency isset by a saturated absorption locking scheme. In this section I will briefly describethe role and operation of each element in the present laser system and then outlinechanges that should improve the day-to-day reliability of the system.


2.3.1 The master and repumping lasers

The first role of the laser system is to collect a cloud of pre-cooled atoms in theexperimental MOT. In summary, a MOT consists of three counter-propagating,red-detuned pairs of laser beams, that intersect at the centre of a quadrupolemagnetic field. As atoms move away from the centre of the trap they are Zeemanshifted into resonance with a laser beam that pushes then back again and sotrapping is achieved. The polarizations of the beams must be carefully chosen sothat the correct Zeeman shifted transition is excited. Cooling occurs because atomsare Doppler shifted into resonance with photons traveling in the opposite direction -preferential absorption of this low energy photon followed by spontaneous emissionof a higher energy resonant photon reduces the kinetic energy of the atom. Thereare also other more subtle cooling mechanisms at work (e.g. Sisyphus cooling,polarization gradient cooling) which are described in detail in [?] and lead tocooling below the Doppler limit.

The cooling transition that we use is the 5S1/2 to 5P3/2 transition at 780 nm (theD2-line). The hyperfine structure of this transition is given in fig. 2.4. The masterlaser generates the more powerful beam for laser cooling, which is red detuned fromthe closed F=2 to F’=3 transition by 15 MHz. A few atoms (approximately 1 inevery 250) are off resonantly excited to the F’=2 state, from where they may decayto the lower hyperfine ground state, F=1, and be lost from the cooling process. Toprevent this, a separate laser produces a repumping beam resonant with the F=1to F’=2 transition, that is mixed with the trapping beam and excites atoms backinto the cooling cycle.

2.3.2 The magneto-optical traps

We use 2 different designs of magneto-optical trap, both of which operate using theprinciples described above. The second MOT is a standard design, with 6 circularlypolarized beams, each of 0.8 cm waist and containing 3 mW of trapping light and150 µW of repumping light, which intersect at the centre of the quadrupole field.All six beams originate from the output of the same polarization preserving opticalfibre (OZ optics LPC-02-780-5/125-P-2.2-11AS-40-3A-3-4) - this ensures that therelative powers of the beams remains constant, even though the total power mayfluctuate. The fibre also acts as a spatial filter, ensuring that the beams have asmooth Gaussian profile.

The first MOT is of a pyramidal design. It is housed inside the 10-way crossdescribed in section 2.2. The quadrupole field of 5 G cm−1 (radial) is produced bya pair of anti-Helmholtz coils, wound around the cross and coaxial with the inputbeam. They carry 6 A and require water cooling. Four mirrors form the inside ofa square based pyramid (of base 3.8 cm), which enclose the cloud. The pyramid isilluminated by a single, circularly polarized, wide diameter input beam (of waist1.8 cm), which creates the six MOT beams with correct polarizations by reflection


Figure 2.4: The hyperfine structure of the 5S1/2 to 5P3/2 D2 transition in 87Rb. Coolingand trapping is done on the closed F=2 to F’=3 transition. The repumping laser islocked to the F=1 to F’=2 transition to recycle atoms that are off-resonantly pumped

into the lower hyperfine ground state.


Figure 2.5: A y-z cross-section of the pyramid MOT. By convention, the polarizationsare defined relative to the positive x,y and z axes, rather than the direction of travel ofthe beam or the local B field. In this convention, pairs of MOT beams have oppositepolarization handedness, but the polarization is always the same relative to the local

magnetic field. Helmholtz bias coils exist in all 3 directions, but only one is shown.

from the mirrors as shown in fig. 2.5. (Remember that after reflection from an idealmirror the handedness of polarization relative to the direction of beam travel isreversed). The mirror blanks were custom made by Halbo Optics using BK7 glassand a multilayer dielectric coating was applied by the physics department opticalcoating facility (courtesy of Chris Goodwin). It is important that the coatingproduces equal reflectivities and phase shifts for both s and p polarized light sothat the reflected beams are still circularly polarized. After 1 reflection, less than2 % of the light was found to have the wrong handedness, which is sufficient forMOT operation. A small hole of 1 × 2 mm exists at the apex of the pyramid,through which atoms are continuously pushed by the trapping light towards thesecond MOT. External bias coils enable us to move the zero of the quadrupole fieldand hence the atom cloud over the hole to optimize the loading of the experimentalMOT.

The pyramidal MOT has many advantages over a conventional 6-beam MOTas a large, primary source of cold atoms. With only one input beam, the set


up is much simpler, smaller and cheaper (high quality, wide diameter optics areexpensive). A much larger input beam can used, and so many more atoms canbe captured (the steady state number depends on d4, where d is the dimension ofthe capture region). The laser power is used more efficiently because it is recycledthrough several MOT beams.

Laser design

For most of the work described in this thesis, both the master and repumpinglasers used SDL diodes (SDL-5402-H1). These produce up to 50 mW of power infree running operation at a wavelength of 782 nm. The free running wavelengthis lowered towards 780 nm by using the built in peltier and thermistor to holdthe diode head at ∼ 14C, using a commercial temperature control box (NewportMod.325). The diodes are mounted in an external cavity which provides opticalfeedback to improve the spectral properties of the laser, reducing the linewidthto < 1 MHz and enabling us to select the absolute frequency of the output beam.The design of the ECDL is shown in fig. 2.6 and is similar to that described in[?]. The diffraction grating (Optometrics, 1200 lines per mm, blazed at 750 nm) ismounted in Littrow configuration and reflects the first order back into the diode,whilst the zeroth order forms the output beam. A frequency selective cavity formsbetween the grating and the back face of the diode. The range of possible lasingfrequencies is set by the angle of the grating, whilst the narrow individual modefrequencies are set by the external cavity length. These individual modes may bescanned continuously over ∼ 5 GHz by changing the length of the piezo-electriccrystal on which the grating is mounted. Thus large, slow adjustments (up to1 kHz) may be made to the laser frequency by driving the piezo, whilst small, fastadjustments may be made by varying the laser current. Since the cavity lengthultimately determines the laser frequency, we found it necessary to temperaturestabilize the whole mount to avoid thermal drifts in the cavity length. This wasachieved by placing a peltier element underneath the mount and maintaining itjust below room temperature.

The blazed gratings that are used reflect over 50% of the incident light backinto the grating, providing a strong feedback beam but reducing the maximumoutput power to ∼ 12 mW. Holographic gratings which reflect only 25% into thefirst-order feedback were also investigated, in the hope that higher output powerscould be achieved. However the smaller feedback reduced the stability of the lasermode and so these were abandoned.

2.3.3 Commercial ECDLs

Whilst most of the work in this thesis was done with the ECDLs described above,they were one of the most time consuming aspects of the experiment. Since theexact power and frequency of each beam is critical, each laser had to be operating


Figure 2.6: The design of the external cavity diode lasers. The mount was made in houseand is made of brass. The vertical adjustment screw is used to direct the first order beamback into the diode - feedback is optimized when the lasing threshold is minimized. Thehorizontal adjustment screw and piezo are used for coarse and fine adjustments of the

lasing frequency respectively.

in a clean single mode, centered on the locking frequency and able to scan at least1 GHz either way without mode hopping. Unfortunately the external cavity lengthwas not stable over periods of hours, either due to temperature drifts or relaxationof the mount. This would cause the locking frequency to drift to the edge of thestable mode and the lock to fail. Thus the cavity would have to be realigned severaltimes a day, and the laser temperature and current adjusted to produce a cleansingle mode again. The horizontal and vertical feedback adjustments worked byflexing a thin piece of brass against a very finely threaded screw (see fig. 2.6). Asthe mount aged, the elasticity of the brass reduced and no longer flexed smoothly.Finally the current drivers and locking electronics were not well shielded. Theypicked up electrical noise from other labs and in particular from our own radio-frequency evaporative cooling ramps, which could cause the laser to lose lock, justseconds before it was required to probe the condensate.

There are of course many advantages to laser systems that are built in house -they are cheap, can be repaired and modified quickly and we were able to incorpo-rate our own choice of high quality SDL laser diodes. However with a more reliablelaser lock, we would be able to automate the experiment to run without constantsupervision, which would dramatically increase the data rate and make more effi-cient use of our time. Thus we investigated two commercial ECDL systems, theTUI (now Toptica) DL100 and the Laser 2000 TEC500, both of which cost around£6000 for the complete package, including electronics. The Laser 2000 system wasnot suitable, whilst the Littman mounting was useful because the beam angle does


not change with frequency, the output power was low (20 mW), the adjustmentscrews were awkward to operate and we were unable to produce a pure single modeoutput at the desired frequency, when viewed on a Fabry-Perot spectrum analyser.The TUI system (which we eventually bought and now operates as the repumpinglaser) was more promising. It is mechanically very robust and the cavity onlyrequires adjustment every few weeks. It produces up to 50 mW of output powerand can be made to go single mode at the desired frequencies. However it alsohas the confusing ability to lase cleanly in several different modes, producing agood saturated absorption signal from the mode at the correct frequency but withmost of the power unable to interact with the atoms. This probably occurs dueto the lack of AR coating on the Sanyo diodes that are used by TUI. To be surethat the TUI laser is operating single mode, it is necessary to view the output ona Fabry-Perot spectrum analyser. Finally whilst the system does pick up someelectronic noise, it is more likely to retain lock than our original lasers.

2.3.4 Frequency control

The output frequencies of the master and repumping lasers are locked to specific87Rb hyperfine transitions using a Doppler-free saturated-absorption set up [?](fig. 2.1). Figure 2.7 shows the saturated-absorption signal obtained from oneof our vapour cells, which contains both 87Rb and 85Rb. Figure 2.8 is a close-up of the 87Rb master lines (transitions from the upper ground hyperfine state)indicating the cross-over peak to which the master laser is locked. The derivativesof these peaks provides the error signal for locking. It is generated by dithering thelaser current at 100 kHz and feeding the saturated absorption signal into a phasesensitive detector. After integration and phase correction, the error signal is fed toboth the laser current and the grating piezo. The former provides a fast response(up to 10 kHz), whilst the latter corrects for large slow drifts in frequency. Furtherinformation about the locking circuit may be found in [?, ?, ?].

It can be seen from fig. 2.7 that the repumping saturated absorption lines areweaker than the master lines, resulting in a lower signal/noise ratio in the lockingcircuit and a less stable lock. We have improved the lock by heating the vapourcell to a steady 40C, which increases the rubidium vapour pressure and hencethe saturated absorption signal by a factor of five, compared to a cell at roomtemperature. To provide a stable temperature, the cell is completely enclosed ina metal case. A temperature control circuit (Cebek thermostat module HK00023)with a thermistor and a peltier element attached maintains the temperature at40 ± 1 C.

Figure 2.8 shows that the master laser is locked to the cross-over peak betweenthe F=2 to F’=1 and the F=2 to F’=3 transitions, 214 MHz below the coolingtransition. The beam is split between the first and second MOT, amplified andthen sent to two double-pass acousto-optic modulators (Crystal-Technology 3110-


Figure 2.7: The saturated-absorption signal from one of our Rb vapour cells whichcontains both 87Rb and 85Rb (77% abundant), showing the hyperfine transitions of eachisotope within the D2-line. The Doppler broadened absorption curves containing the

87Rb master (F=2 to F’) and repumping lines (F=1 to F’) are indicated.

Figure 2.8: A close up of the Doppler-free 87Rb master lines with the background Dopplercurve removed by subtraction. All transitions start from the upper ground hyperfinestate (F=2) and go to the F’ state indicated. The master laser is locked to the F’= 1-3cross-over peak. Some power broadening is present in this scan, since the linewidth is

apparently greater than the natural width of 5.76MHz.


140), which control the frequency and intensity of light in each MOT. On eachpass the frequency changes by 110± 24 MHz and so after the double pass we shiftthe laser light back on resonance with a tuning range of about ± 40 MHz. Aswell as doubling the tuning range, a well aligned double-pass AOM produces anoutput beam angle that is independent of frequency. Since the beams are thencoupled down optical fibres, this is very important - we do not want large changesin the coupling efficiency as we ramp the laser frequency during the experiment.In reality, the double-pass alignment is not perfect and so the fibre output poweris independent of frequency over ±10 MHz range and falls off fairly rapidly outsidethis - this had to be taken into account when a precise power level was necessary,for example in the probe beam. The AOMs also act as very fast shutters, witha switching time of ∼ 1 µs, enabling us to generate 5 µs pulses for pumping andprobing. Since they can leak light even when off, they are backed up by slowermechanical shutters. The repumping laser is locked directly to the repumpingtransition and operates at constant frequency and power.

2.3.5 The master oscillator power amplifier

The master laser outputs a high quality beam, which is frequency locked, has anarrow linewidth and good transverse mode structure, However it has only a fewmW of power and must be amplified to run the first and second MOTs, whichrequire 50 and 20 mW of trapping light respectively. Originally the experimentused just one amplification device, the master oscillator power amplifier (MOPA),with the output split between the two MOTs. However this arrangement did notproduce sufficient power reliably and so the second MOT is now run from a slavelaser as shown in fig. 2.1, with all the MOPA power available for the first MOT.

The MOPA is a tapered semiconductor gain medium with AR coated facets(SDL-8630-E). Six mW of master light is focused in and amplified to 500 mW,retaining the narrow linewidth of the master beam (confirmed with a Fabry-Perotspectrum analyser) over a 5 GHz scan. At the time that the experiment was built,there were no sufficiently powerful slave diodes available at 780 nm and so theMOPA chip seemed a cheap and simple alternative to a titanium sapphire laser.The only drawback was that it had only just become commercially available andhence its operational lifetime was unknown [?, ?].

A detailed technical description of the MOPA setup is given in [?]. To optimizethe output power, the injection beam is carefully matched to the shape of thespontaneous emission beam from the input facet of the MOPA by a row of lensesand prisms (see fig. 2.1). Whilst the MOPA output retains the narrow bandwidthof the master beam, it is highly astigmatic and elliptical. The poor transverse modequality means that much of the power cannot be coupled through an AOM crystalor down a fibre and so is wasted. In addition, the transverse mode structure andhence the coupling efficiency of the beam drifts with temperature over a period of


hours. Whilst this problem was temporarily improved by temperature stabilizingthe laser mount, the output mode structure deteriorated again after about 1 yearof use (probably due to damage to the output facet) and the chip was replaced.The MOPA is currently used to run the first MOT only, (where the absolute powerlevels are not critical) after the beam has been spatially filtered by a polarization-preserving, single-mode optical fibre (OZ optics LPC-02-780-5/125-P-2.2-11AS-40-3A-3-4).

2.3.6 The injection-locked slave laser

A couple of years after the experiment was built, high power diodes (80 mW)became available at 780 nm, which could be used as injection-locked slave lasers torun the second MOT. This scheme had many advantages:

• The MOPA beam could be spatially filtered and still provide sufficient powerto run the first MOT alone. The filtering prevented drifts in the transversemode structure affecting the loading rate.

• The slave beam has the same high quality transverse mode structure (mainly 0,0)as the master and so couples efficiently and reliably down optical fibres.

• A high quality slave diode costs ∼ £ 400 and has a lifetime of several years,compared to £ 5000 for a MOPA chip with a lifetime of ∼ 1 year.

• The slave requires a lower power injection beam ∼ 2 mW.

A detailed description of the injection-locking technique is given in [?]. Insummary, a master beam is injected into a high power laser diode (analogousto the grating feedback in the ECDL), seeding the lasing process. Whilst muchof the master power is used up in the process of controlling its frequency andbandwidth, the slave inherits the master beam qualities and so all of its outputpower is available for experiments. The quality of the lock is determined by therange of frequencies over which the slave output follows that of the master (whichcan be tested either using saturated-absorption signals or a Fabry-Perot spectrumanalyser) and is affected by several factors:

• The injection beam must be exactly aligned with the slave output.

• Ideally the injection beam should have the same transverse shape as theoutput beam, but we found that this was not critical.

• The locking range can be increased by raising the injection power above thethreshold value of ∼ 0.5 mW.


Figure 2.9: A schematic diagram of the optical isolation in front of an injection-lockedslave laser. Polarizations are used to ensure that the injected beam can reach the laserdiode, but any back reflections from the slave beam are stopped at cube 2. Note thatthe polarization of the injection beam and the slave beam must be matched at the laser

input.

• The master wavelength must match the internal slave cavity. The opticallength of the cavity can be affected by the slave current and temperature, sofor a given current and hence output power the temperature must be scanneduntil a lock is achieved.

Good temperature stabilization of the slave is essential and was provided by apeltier under the brass mount, a feedback thermistor and temperature control boxEW 1251. The optical arrangement for injecting the slave is shown schematically infig. 2.9. The 45 Faraday rotator with polarization beam splitting cubes mountedon either end transmits the injection and output beams, whilst protecting the slavediode from unwanted back reflections. The diode was conveniently mounted insidea Thorlabs (LT230P5) collimation tube, which locks the diode in position andallows adjustment of the position of the collimation lens on an internal thread,until the spot size on a distant wall is minimized.

Despite the good transverse mode structure of the slave beam, a large frac-tion of its power was lost passing through the isolation optics, double-pass AOMand fibre and so around 100mW of bare output power was required. We tested 2different laser diodes, a Sanyo (DL-7140-201) and a more expensive SLI-CW-9MM-C1-782-0.08S-PD from Laser Graphics which were specified for 75 and 80 mW re-


spectively, to see whether they could operate and lock reliably above the specifiedoutput power. To find the individual maximum power of each laser we added asaw-tooth oscillation of ±10 mA at 100 Hz to the current and looked for a changein the current/output power relationship as the mean current was gradually in-creased. This change marked the damage threshold of that particular diode, butsince the diode spent little time per cycle above this threshold value then dam-age was unlikely to occur (in theory!) [?]. The first Sanyo diode died when theoutput power was around 75 mW. It was replaced and a maximum output powerof 110 mW was successfully recorded at 130 mA. The laser graphics diode showedno sign of a damage threshold at 100 mW and so it was not tested further sincethis was sufficient power for the experiment. Both lasers produced similar lockingcharacteristics, shown in table 2.1. The laser graphics diode was eventually chosenand has run reliably for several years under the conditions shown below.

Diode Max. power Current Temp. Injection Locking(mW) (mA) (C) power (mW) range (GHz)

Sanyo 80 100 17 ∼ 2 4Laser Graphics 100 120 15 ∼ 2 4

Table 2.1: Details of the injection-locking characteristics of two diode lasers.

2.3.7 A new MOPAless laser system

Given the problems with the MOPA described in section 2.3.5, its cost and thefact that cajoling it to an acceptable power level is one of the most time consumingaspects of the experiment, we do not intend to replace the present chip when itdies. Instead, both MOTs will be run using slave lasers and a suitable scheme(assuming 100 mW slaves) is outlined in fig. 2.10. The most significant changeis in the pyramid MOT beam. The AOM (at which a 40% power loss occurs) ispositioned before, rather than after, the amplification stage to obtain sufficientpower at the pyramid and probe fibres. This poses a problem for the probe beam,which currently uses the AOM after the MOPA to generate 5 µs pulses. Hence anew single pass AOM must be added to the probe beam, purely to act as a shutter.A mechanical shutter is sufficiently fast to shut off the pyramid MOT beam oncethe experimental MOT has loaded.

With the improved transverse beam quality of the slave laser, it might bepossible to run the first MOT without a fibre acting as a spatial filter and thushave extra power to increase the loading rate. Whilst this is worth investigating, itis extremely convenient to have the fibres decoupling the alignment of the MOTsfrom the alignment of beams on the laser table.


Figure 2.10: A schematic diagram of the current laser system, and a new laser systemwhich would replace the MOPA with a second injection-locked slave laser. Only those

components at which significant power loss occurs are shown.


2.4 The magnetic trap

Until recently, magnetic trapping was essential for the evaporative cooling stagesof BEC production. Condensation cannot be achieved in a magneto-optical trapsince reabsorption of spontaneously emitted photons limits the maximum densityand photon recoil ultimately limits the minimum temperature that can be achievedin a MOT. (The first all optically trapped BEC was reported last year, in whichatoms are trapped for evaporative cooling using the optical dipole force of a tightlyfocused, far-detuned CO2 laser beam [?]). Magnetic trapping of neutral atoms isbased on the interaction between the atomic magnetic dipole moment µ and aweak inhomogeneous magnetic field B. The interaction energy U may be writtenas:

U = −µ.B = gF mF µB B (2.1)

Atoms that are spin polarized in low field seeking states (those with gF mF > 0)may be trapped at minima in the magnetic field. (It is not possible to trap atomsin high-field seeking states since Maxwell’s equations do not allow a local maximumin the magnetic field.)

2.4.1 The theory of the TOP trap

The magnetic traps currently used in BEC experiments fall into 2 general cat-egories, Ioffe-Pritchard type traps and the TOP trap. The Ioffe-Pritchard trap[?] consists of 4 parallel wires at the corners of a square, which provide a tightradial confinement, and a pair of Helmholtz pinch coils producing axial confine-ment. Variations on this magneto-static scheme include the baseball trap [?, ?],the cloverleaf trap [?], the QUIC [?] trap.

The Time Orbiting Potential (TOP) trap was invented by Eric Cornell [?] andwas used in the first BEC experiment [?]. It is based on a quadrupole field Bq,generated by a pair of anti Helmholtz coils:

Bq = B′q (x x + y y − 2z z) (2.2)

Note that the field gradient is twice as strong in the z direction as in the radialdirections, to satisfy ∇.B = 0.

At first sight, the quadrupole field alone appears to create a suitable confiningpotential, however it contains a field zero at the centre of the trap. At the zero,atoms can no longer stay aligned to the field and undergo transitions into untrappedmF states known as ‘Majorana spin flips’ and are expelled from the trap. Theaddition of a static bias field does not solve the spin flip problem, as it would simplydisplace the centre of the trap to a new position. However if the bias field (or TOPfield), BT rotates sufficiently fast, the atoms cannot follow the instantaneous trapcentre and their translational motion is controlled instead by the time-average of

2.4. The magnetic trap 23

the magnetic potential. This creates a harmonic potential with a minimum fieldof BT at the centre.

In our experiment, the bias field rotates in the xy plane and may be representedas:

BT = BT (cos ω0t x + sin ω0t y) (2.3)

So the total magnetic field is

B = (B′q x + BT cos ω0t) x + (B′

q y + BT sin ω0t) y − 2B′q z z (2.4)

The confining potential, UTOP is the interaction energy of µ and B, averaged overone rotation of the bias field:

UTOP =ω0

2π

∫ 2π/ω0

0mF gF µB |B| dt (2.5)

= µBT +µB′2

q

4BT

(x2 + y2 + 8z2

)+ O(4) (2.6)

where µ = gF mF µB. UTOP is harmonic, with trapping frequencies ωx = ωy = ω⊥and ωz =

√8 ω⊥, where

ω⊥ =

√µ

2m

B′q√

BT

(2.7)

The rotation rate of the bias field ω0 must be much faster than the trap os-cillation frequencies so the translational motion of the atoms is controlled by thetime averaged potential. It must also be much slower than the Larmor frequencyωL = µB/h, so that the atomic dipole moments can adiabatically follow the direc-tion of the instantaneous B field and not make transitions into untrapped states.In our experiment ω0/2π = 7 kHz, trap frequencies are typically ∼ 100 Hz and theLarmor frequency is 1.4 MHz in a minimum field of 1 G and so both of the aboveconditions are satisfied. If we consider the instantaneous picture rather than thetime-averaged one, then the zero of the quadrupole field is moving in a circle ofradius:

r0 =BT

B′q

(2.8)

This locus of B = 0 (colloquially known as the ‘circle of death’) defines the bound-ary of the trap in the radial direction. Atoms that reach this radius are flippedinto untrapped states and lost from the trap (see fig. 2.11).

The TOP trap that we use is the simplest and most symmetric realization of thistrap - the bias field rotates around the axis of the quadrupole field and gravity actsalong the same axis so that nothing breaks the radial symmetry. The trap is stifferin the z direction than the radial one, creating pancake shaped condensates withan aspect ratio (axial size : radial size) of 1 :

√8. (In contrast, most Ioffe Pritchard

traps have tighter radial confinement and produce cigar shaped condensates witha typical aspect ratio of 20:1). However, one of the great advantages of the TOP


Figure 2.11: A diagram showing the instantaneous magnitude of the TOP trap magneticfield in the x-y plane. The total field is an axially symmetric quadrupole field with fieldgradient B′

q, displaced in the x-y plane by a distance r0 = BT /B′q. As the bias field

rotates, the zero of the total magnetic field follows a circle of radius r0.


trap is that it is possible to make physically significant changes to the shape andaspect ratio of the trap simply by changing the symmetry or direction of the TOPbias field. Most of the results in this thesis rely on such modifications, which willbe described in detail later. It is worth noting that a triaxial TOP trap has alsobeen built with a fixed aspect ratio of 1 :

√2 : 2, by allowing the TOP field to

rotate in a plane containing the quadrupole axis [?].

2.4.2 The TOP trap apparatus

When designing a magnetic trap, the factors to take into account are:

• Maximize trap stiffness for efficient evaporative cooling

• Maximize initial trap size to collect a large initial number of atoms

• Ensure that the trap size and stiffness can be matched independently to theatom cloud for minimum heating during loading.

• Minimize switching time for coils for loading and for a clean release prior toimaging.

• Maximize cooling of coils

• Maximize optical access

Figure 2.12 gives a schematic diagram and photograph of our TOP trap. Fur-ther details may be found in [?, ?]. The quadrupole field (which is also used forthe second MOT) is generated by a pair of anti-Helmholtz coils of 450 turns eachwound onto an aluminium former. A slit along one radius of the former minimizeseddy currents. The large number of turns enables us to create a relatively high fieldgradient (200 G/cm radially) with a modest current (9.5 A) supplied by a standardbench power supply (Farnell PSU 3510A). Since the total power dissipated is lessthan 300 W, this low current arrangement can also be cooled easily by pumpingwater from a beer cooler at 13C through the former of the coils. An alternativedesign with high current and low turns would have 2 advantages. Firstly the coilswould be smaller and physically closer to the cell thus increasing the gradient.Secondly the inductance would be lower and so faster switching times could beachieved. We currently switch off the coils using a solid state relay (RS 200-2058)in ≤ 1 ms. However neither of these two factors have been a limitation in theexperiments carried out so far.

The radial TOP bias field is generated by 2 orthogonal pairs of coils, each of 6turns, sandwiched between the cell and the quadrupole former. Figure 2.13 showsthe circuitry driving each coil to create a rotating bias field. The magnetic fieldgenerated by each pair of coils is monitored on single turn pick-up coils woundwithin the TOP coils. These pick-up coils have negligible impedance and so there


Figure 2.12: A cross-section diagram and photograph of the TOP trap


Figure 2.13: The circuitry driving the TOP coils for a standard TOP trap. Impedancematching the output of the audio amplifier is necessary. It generates a maximum of 600 Winto a load of 4Ω, whilst at 7 kHz the TOP coils only have a reactance of 0.35Ω. Anyunequal phase shifts in the x and y channels may be compensated with the adjustable

phase shifter.

is exactly 90 phase lag between the maximum field and maximum voltage on thepick up-coil.

So far I have described the original, axially symmetric TOP trap apparatus. Toperform the experiments described in this thesis, several modifications have beenmade. The radial TOP circuitry has been modified so that the trap can be madeelliptical in the x-y plane and then be rotated. Also a third set of TOP coils (andpick up coil) has been wound around the quadrupole coils (30 turns each), creatingan oscillating bias field in the z direction. These enable us both to tilt the trapand change its aspect ratio.

2.4.3 Calibration of the TOP trap

It is important to know the exact trap frequencies during experiments, whichfrom eqn. 2.7 requires an accurate knowledge of the quadrupole and x,y and zTOP fields at the trap centre. These can only be approximately calculated fromthe known currents and geometries and so must be measured. Given that thereare 4 unknown quantities, then 4 independent measurements must be made. Anabsolute calibration of the y (and z) TOP fields may be made using the Zeemanshift. A cloud of cold atoms is released from the trap and imaged with a probe


Figure 2.14: A plot of the relative optical density of a cloud, imaged along the y direction,versus probe detuning (from the resonant frequency at zero B field) used to calibrateBy. The data was fitted with a Lorentzian curve, centred on ∆max = 26.0 ± 0.2MHz,

indicating that the By field present had a value of 26.0 / 1.4 = 18.6± 0.1G

beam along the y (or z) direction, after all fields have been turned off, except theone under investigation. The detuning of the probe beam is varied and a plot ofoptical density versus detuning is built up (fig. 2.14). The data has a Lorentziandistribution centred on ∆max. Since the mF = 2 to mF = 3 transition is Zeemanshifted by 1.4 MHz/G, then we have:

BTOP =∆max

1.4G (2.9)

Secondly, the trap oscillation frequencies in the x and y directions can be mea-sured by ‘kicking’ a small cold cloud of atoms in the harmonic trap to excite thedipole mode. The kick is achieved by suddenly switching off the nulling fields inthese 2 directions, effectively jumping the position of the centre of the trap by∼ 150 µm. The position of the cloud is recorded in the x and y directions (us-ing a probe beam along the z axis) as a function of time and fitted with a sinewave, from which the oscillation frequency is extracted. Typical data is shown infig. 2.15. To minimize the error on the fitted frequency, the oscillation is sampledfor a few cycles over a wide range of evolution times, from 0 to 3100 ms. Fromthe ratio of oscillation frequencies ωx/ωy, the ratio of the radial TOP fields Bx/By

may be deduced and hence Bx calculated. For many experiments it is importantthat the trap is exactly circular in the x-y plane - these dipole measurements de-termine ωx/ωy to an accuracy of at least 0.5 %. Finally, with Bx and By known,the quadrupole field gradient may be calculated from the measured value of ωx.

Previously, the quadrupole gradient was calibrated by using the effect of gravityon the TOP trap. If we include the effect of gravity in the TOP trap potential we


Figure 2.15: The x position of the centre of an atom cloud (in pixels) after the x dipolemode has been excited. Each pixel is 24µm wide and the magnification is × 1. Only asmall section of the data has been shown. The solid line is a sine wave fitted to the full

data set (up to t = 3100 ms) and gives ωx/2π = 11.83± 0.01Hz

have:

UTOP =ω0

2π

∫ 2π/ω0

0(µ |B|+ mgz) dt (2.10)

≈ µBT (1− ρ2)1/2 +µB′2

q

4BT

[(1− ρ2)1/2(1 + ρ2)r2 + 8(1− ρ2)3/2z2

]

where ρ = mg/2µB′q. This shows that the ratio of the axial to radial oscillation

frequencies is given by:

ωz

ω⊥=√

8×√

1− ρ2

1 + ρ2(2.11)

This ratio does not depend on BT and so measurements of it may be used todetermine B′

q independently. However this method only provides an accurate cal-ibration for small B′

q (≤ 20 G/cm), below which gravity has a significant effect onthe trap. Non-linearities in the relationship between the computer output and thequadrupole field produce errors if the calibration is extrapolated to the values ofB′

q used in the experiment. It is worth noting that gravity provides a simple wayof altering the aspect ratio of the TOP trap and even producing a spherical trap(ωz = ω⊥), which holds some very interesting physics [?, ?]. The disadvantage isthat these aspect ratios can only be achieved in very weak traps.


2.4.4 Radio-frequency coils

A pair of 5 turn coils (radius r = 1.8 cm, wire diameter 0.5 mm, separation z= 12 mm) is positioned above and below the cell (fig. 2.12). These provide therf radiation that is used for evaporative cooling. They are driven by a frequencygenerator (Stanford Research Systems DS-345) with a range of 0 - 30 MHz. Orig-inally a 50 Ω resistor was placed in series with the coils to smooth out resonancesin the power output that occur at certain driving frequencies, however this waslater removed to increase the overall rf power.

2.5 The imaging system

All the information about the size, shape, and temperature of our condensate orthermal cloud is obtained by absorption imaging, essentially looking at the shadowthat the atoms cast in the probe beam. The experiment was originally set up witha single ‘horizontal’ imaging system, probing along the y direction, through theside of our pancake-shaped condensate. However following the development ofan elliptical trap that rotates in the x-y plane (section 8.3), it became essentialto observe the condensate in the plane of rotation. For experiments describedin chapter 8 and beyond, a second ‘vertical’ imaging system was available, thatprobes along the symmetry axis of the cloud (z axis). The optical principles behindboth systems are identical although because of the layout of the coils and vacuumsystem, the vertical system appears more complicated.

Images may either be obtained in the trap, or after a ‘time of flight’ (TOF), i.e.after the trapping potential has been suddenly turned off and the cloud has beenallowed to expand freely for a certain time. Thermal clouds are usually imaged inthe trap, since the cloud is many pixels in size and the optical density is relativelylow. Condensates must be imaged after expansion - in a typical trap they are∼ 3 µm in size and so below the resolution limit of our imaging system.

2.5.1 The horizontal imaging system

The horizontal imaging system is shown in fig. 2.16. The figure shows the beampath both for the unscattered probe light and also for the image of the shadowcreated by the condensate.

A 5 µs probe pulse is delivered down an optical fibre, using the AOM in thepyramid MOT beam as a fast shutter (see fig. 2.1). The timing of this pulse issynchronized with the rotation of the TOP bias field, so that the pulse is centred onthe instant when the field points along the imaging direction (y). The σ+ optimizedprobe light drives the closed |2, 2〉 to |3, 3〉 transition; this has the largest Clebsch-Gordon coefficient, thus gives the strongest absorption (appendix B).

2.5. The imaging system 31

Figure 2.16: The horizontal imaging system. The beam paths are indicated both for theunscattered probe light and also for the image of the shadow created by the condensate.


Following the cell, two 10 cm focal length doublet lenses (Comar 100DQ25)form a 1 to 1 imaging system, that moves the image away from the congestedarea around the cell. The numerical aperture of the first lens sets the resolutionof the system. It has a value of D/f = 0.25, where D is the diameter and fis the focal length of the lens. This is the maximum numerical aperture thatcan be obtained with a standard commercially-available diffraction limited lens.Defining the resolution limit, d as the FWHM of the image of a point like objectformed by our system,we find from simple diffraction theory that d = 1.02λf/D =1.02× 0.78× 10/2.5 = 3.2 µm. To obtain the optimal resolution it was importantthat the two doublet lenses were the correct way round, so that the light wasrefracted equally at each surface, thus minimizing aberrations on the image.

The image is magnified using either a ×4 or ×10 microscope objective, (Comar04OS10, 10OS25). Microscope objectives are designed with the object to imagedistance is constant, so that objectives may be exchanged to change the magni-fication without major refocusing being required - a property which is ideal forour system. The highest magnification that we use is × 10, at which the resolu-tion limit of our system, 3.2 µm is magnified just beyond the size of a camera pixel(24 µm). Further magnification would only restrict the field of view and reduce thesignal to noise ratio of our images, without providing any additional information.Our × 10 objective has a numerical aperture of 0.25 and so does not change theresolution limit. The objective is mounted on a vernier translation stage, so thatthe system may be focused by making fine adjustments to its position along theoptical axis.

2.5.2 The camera

The camera used for these experiments is a Princeton Instruments TE/CCD-512SBwith an ST-138 controller. It has a CCD array of 512 by 512 pixels, each 24 µmsquare. The array is 80% efficient around 780 nm and is peltier cooled, to minimizedark noise. Whilst it is an excellent camera it was originally bought with caesiumexperiments in mind and so some of its specifications are not perfectly optimizedfor our experiment.

The CCD chip is protected by a mechanical shutter. To prevent vibrationsfrom this shutter throwing the lasers off lock, the camera is suspended in a metalcradle from the frame around the experiment. The shutter is opened at the startof each experimental run, but since it is operated in ‘continuous cleans’ mode, anystray MOT light that it detects is wiped from the array well before it is triggeredto take an image. After receiving the trigger, the clean CCD array is exposed for10 ms before being read-out to the controller. The read-out process is slow andlimits the time between shots to 1 s.


2.5.3 Data acquisition

Each condensate picture is constructed from 3 camera images, which are digitizedinto 16 bit integers, and passed to our data acquisition package WINVIEW forMicrosoft Windows. The first shot contains the shadow of the atoms in the probebeam. The second, taken ∼ 1 second later, long after the atoms have dispersedcontains just the probe beam. The third shot gives the dark noise, i.e. the chargethat is thermally excited onto each pixel between cleans. From these three images,the WINVIEW program constructs a false colour image representing the densityof atoms integrated along the imaging direction (y) in the following way:

After passing through the atom cloud, the intensity of the probe beam, I i,arriving at the ith pixel in image 1 is

I i = I i0 e−

∫ni(y)σdy (2.12)

where ni(y) is the number density of atoms along the column of real space imagedonto the ith pixel. The number of counts recorded on this pixel is Ci

1 = αiI i whereαi is the efficiency of the pixel. The intensity at the pixel in the absence of atoms,I i0 is recorded in image 2 so Ci

2 = αiI i0. Thus we account for variations in intensity

across the probe beam and in pixel efficiency. We have assumed that the cloud isoptically thin and so the intensity does not change significantly across it. σ is theabsorption cross-section given by:

σ =Γ

2× 2I0/Isat

1 + 2I0/Isat + 4δ2/Γ2× hωL

I0

(2.13)

= σ01

1 + 2I0/Isat + 4δ2/Γ2(2.14)

with Isat = 3.14 mW/cm2 and σ0, the unsaturated resonant cross-section (6πλ2) =2.9 ×10−13 m2. δ is the detuning from resonance, and Γ is the natural linewidth,both in MHz.

Rearranging eqn. 2.12, we have the total number of atoms recorded on the ith

pixel

N i = A∫

ni(y)dy = −A

σln

Ci1 − Ci

3

Ci2 − Ci

3

(2.15)

where A is the area of real space imaged onto each pixel. If M is the magnificationfactor, then A is given by:

A =24× 24 µm2

M2(2.16)

The dark count of the ith pixel (Ci3) is found from image 3 and subtracted from Ci

1

and Ci2 to minimize the effect of dark noise. All further analysis uses this 2D grid

of the number of atoms per pixel N i.


The total number of atoms in the sample is

Ntotal =∑

i

N i =A

σ

∑

i

lnCi

1 − Ci3

Ci2 − Ci

3

(2.17)

Further image analysis is described in section 2.5.5

2.5.4 Calibrating the imaging system

Equation 2.17 shows that both the absorption cross-section σ and the area imagedonto each pixel A must be accurately known to calculate the total number of atoms.The probe beam intensity I0 and detuning δ must both be accurately known todetermine σ (eqn. 2.14), since I0 is comparable to Isat in our apparatus. Themagnification of the imaging system M must be measured to determine A.

Intensity calibration

We assume that the atom cloud is small and at the center of a Gaussian beamprofile. The intensity of the probe beam at the cloud I0 can be calculated from ameasurement of the total power (Ptot) and the beam waist (w).

Ptot =π

2I0w

2 (2.18)

To measure the beam waist one places a razor blade mounted on a micrometerin the beam just before the cell and measures the power that gets past (P ) as afunction of knife position (x) (positions for the knife and power-meter are indicatedin fig. 2.16).

P = P0 + P1

(1− erf

((x− x0)

√2

w

))(2.19)

Figure 2.17 shows our data for the power as a function of knife position, witheqn. 2.19 fitted, from which the value for the beam waist, w = 4.5 ± 0.1 mm, isobtained. Plugging this value into eqn. 2.18 we find that the probe intensity (inmW/cm2) is related to the measured total power (in mW) by:

I0 = 3.14 Ptot cm−2 (2.20)

The power (and hence intensity) of the probe beam must be checked daily, (re-membering that the coupling efficiency down the probe fibre will be a function ofprobe detuning) if accurate number measurements are required.

For most experiments a probe intensity of around Isat was used, since thisoptimizes the signal to noise of the image formed on the CCD. If a higher power isused then the percentage of light absorbed by the atoms falls and so imperfectionsin the probe beam become significant. If a lower power is used then few photonsare available to be absorbed and so dark noise becomes significant.


Figure 2.17: The power passing a knife edge at position x across our probe beam. Thesolid line fits our data to eqn. 2.19, assuming that the beam has a Gaussian profile. The

extracted beam waist size was w = 4.5 ±0.1mm

Magnification calibration

The magnification of the system, M was determined by releasing a cloud from thetrap, allowing it to fall under gravity and recording its position on the CCD (s) asa function of time. These data were fitted with the function:

s = M(−1

2gt2 + V0t + S0

)(2.21)

where M is the magnification. The constant V0 accounts for the velocity impartedby any kick when the trap turns off and S0 is the initial position. A typical fit tothe data is shown in fig. 2.18, which gives M = 9.7± 0.1.

2.5.5 Further image analysis

To obtain the x and z size of the cloud, the rows and columns of the 2D imagearray were added to produce 2 1D density distributions, which were fitted withGaussian curves.

n(xj) = n0je−x2

j/σ2j (2.22)

For the special case of a Gaussian distribution, the same values for σx and σz areobtained as from a 2D Gaussian fit. The summation increases the signal to noiseand enables us to use a very fast 1D fitting routine for online analysis. Once thecloud size and number of atoms is extracted, the WINVIEW program is able to usethe known trap stiffness and expansion time to calculate the temperature, phase-space density, collision rate and density of a thermal cloud in the trap. This isparticularly useful when optimizing the different stages of the evaporation ramps.


Figure 2.18: The position of the centre of a cloud as it falls under gravity. Comparingthe known acceleration, to the acceleration observed on the CCD array enables us to

calculate the magnification of the imaging system with a horizontal probe beam.

Whilst the number calculation is accurate for both thermal clouds and conden-sates, all other quantities assume the Gaussian distribution of a thermal cloud, bothin the trap and after expansion and so are not accurate for analyzing a condensatein the Thomas-Fermi regime. Condensate images are transferred to a MATLABroutine, developed by Gerald Hechenblaikner, which fits both a 1D and 2D doubledistribution - a Gaussian for the thermal cloud, with a parabolic condensate dis-tribution superimposed. This double distribution is clearly visible in fig. 2.19(a).Knowing the trap stiffness and the expansion time, the program is able to calculatethe temperature both from the condensate fraction and, more accurately, from thewings of the thermal distribution. Most of the experiments in this thesis tested thepredictions of the GP equation, which assumes a pure condensate at T = 0. Thuswe used the coldest condensates possible, at T ≤ 0.5 Tc, where the thermal cloud isno longer visible and the temperature cannot be measured. Figure 2.19(b) showsthe parabolic density distribution of a pure condensate, which lacks the thermal‘wings’ of fig. 2.19(a).

2.5.6 Non-destructive imaging

Our current absorption imaging system destroys the condensate in the processof taking a single image, for two reasons. Firstly it uses a pulse of resonant ornear resonant light, which heats the condensate 2 orders of magnitude above thecritical temperature Tc ∼ 250 nK . Consider a 5 µs pulse of resonant light at Isat.The heating per atom, ∆T is given (approximately) by

∆T = No. of photons absorbed× 2 Trecoil


Figure 2.19: 1D density distribution of an expanded condensate (TOF = 10ms) in theradial (x) direction, at 2 different temperatures. The number of atoms per pixel has beensummed along the z direction. In (a) T = 0.9Tc and the thermal cloud is clearly visible.The condensate sits on a pedestal produced by the wider Gaussian distribution of thethermal cloud. In (b), T ≤ 0.5Tc, only the narrow parabolic condensate distribution is

visible.

∼ I0 t σ0

hω× 2

h2k2

2kBm= 177× 0.18 µK = 32 µK

Secondly, in a typical trap, the condensate has dimensions ≤ 5 µm, close to theresolution limit of the imaging system. Therefore we must release it from thetrap and allow it to expand before imaging. Many of our experiments involveobserving the evolution of a condensate excitation, for example plotting a shapeoscillation and fitting the oscillation frequency. At present this requires makingmany condensates with identical starting conditions and varying the evolutiontime of each one before imaging. This is very time consuming, and inevitablysmall variations in the starting conditions produce noise on the data. It wouldbe desirable to track the evolution of a single condensate, by taking many non-destructive images. This is possible using phase-contrast or dark ground imagingtechniques; these both use the change in phase rather than the change in amplitudeof the probe beam, produced by absorption in the condensate [?]. The phaseshift falls off more slowly with detuning than the absorption rate, so that far-detuned light (∼ 500 MHz), which has a much lower heating rate than resonantlight, may be used. Typically 10s of images of the same condensate may be takenbefore significant heating has occurred. Whilst our imaging system could easilybe adapted for non-destructive imaging, until we create much larger condensates,with an in-trap size significantly greater than the resolution limit, we will alwayshave to release and hence destroy the condensate prior to imaging.


2.5.7 The vertical imaging system

The vertical imaging system has the same basic optical design and components asthe horizontal system. However the beam path is more complicated, partly becauseit must share the route up through the cell with the lower MOT beam and partlybecause the apparatus was not originally designed to allow imaging along a seconddirection.

The scheme is best described pictorially and is shown in the diagram of fig. 2.20and the photograph of fig. 2.21. After exiting the optical fibre the direction of theprobe pulse (horizontal or vertical) is determined by a half-waveplate mounted ona shutter that can swing in and out of the beam, followed by a polarization beamsplitting cube. Since the horizontal path must always be used for optical pumpingat the start of the evaporation ramps, the shutter operates under computer con-trol. The vertical probe beam is then mixed with the lower MOT path, travels upthrough the cell and is separated again at a polarization beam splitting cube atthe top of the cell. After passing through a hanging 1 to 1 imaging system (Comarlenses 160DQ32), the vertical path rejoins the horizontal one through the micro-scope objective and onto the camera. The vertical system is focused independentlyof the horizontal one by moving the second 1 to 1 lens on a vernier mount. Twomain considerations guided the design of this imaging system - firstly, is it possiblefor the MOT and probe beams to coexist with the correct polarizations on thesame path and secondly can sufficient resolution be obtained to resolve structures(vortices) within the condensate?

Polarizations

Before considering the probe beam it is necessary to determine the polarizationsof the upper and lower MOT beams. Many standard texts describe the MOTpolarizations relative to a fixed axis, typically the positive x, y and z axes andunder this convention, beam pairs have circular polarization of opposite handedness(which handedness depends on the direction of current flow in the quadrupolecoils). Two quarter-wave plates, one above and one below the cell produce thecircularly polarized upper and lower MOT beams. In the correct configuration,the fast axis of one plate is aligned with the slow axis of the other, so after passingthrough both, the linear polarization of a beam is unchanged.

Given that the polarizations of the MOT beams are fixed, we must now considerif it is possible to use polarization to mix the vertical probe beam with the lowerMOT beam and separate it out again after the cell onto another path. The schemeis in fact possible and is best described with reference to the diagram in fig. 2.20.The two beams are initially mixed at a polarization beam splitting cube and sohave opposite linear polarizations. These polarizations are maintained after passingthrough the cell and both quarter-waveplates and so the imaging beam is separatedfrom the lower MOT beam by reflection at the polarization beam splitting cube


Figure 2.20: A diagram of the vertical imaging system, showing the polarizations requiredto mix and separate the vertical probe beam and the lower MOT beam


Figure 2.21: A photograph of the experimental MOT with the new vertical imagingsystem in the background.


above the cell. Beyond the quarter-waveplate, both the upper and lower MOTbeams have the same linear polarization and so both may be transmitted throughthis cube in opposite directions.

Resolution

The vertical imaging system was built primarily to study vortices, which appear assmall holes within the condensate, and so optimizing the resolution of the systemwas an important consideration. From Fourier diffraction theory, small objects ofdimension d scatter light within a solid angle ∼ λ/d, and all this light must begathered onto the camera if the object is to be resolved. Light scattered by thecondensate must travel through the hole in the quadrupole coil former, through apolarization beam splitting cube and then into the first imaging lens. The resolu-tion is set by the element with the smallest numerical aperture. The first imaginglens, with a numerical aperture of 0.19, limits the resolution to 4.2 µm. This limitis slightly higher than that for the horizontal system (3.2 µm), since the quadrupolecoil former forces the first imaging lens to be further from the condensate but it isstill adequate for observing vortices.

Focusing and calibration

As mentioned earlier, our condensates can only be resolved after several millisec-onds of free expansion, during which time they (and hence the object plane of thevertical imaging system) are accelerating under gravity. We can estimate whetherthe movement of the cloud (∆u) during 10 ms of free expansion is sufficient to causedefocusing by considering the depth of focus (∆vf ) of our system. The depth offocus is the distance from the focal plane at which the first diffraction minimumappears at the centre of the image of a point-like object. Alternatively it is thedistance at which the image is twice its diffraction limited size. It provides a crudeestimate of the range of image plane positions over which the optical system maybe considered to be focused.

Consider the lens system in fig. 2.22 (focal length f , diameter D) creating animage of a point-like object. In the focal plane, the diffraction limited radius ofthe image Wd is:

Wd ≈ λ

D× v (2.23)

The size of the image as a function of distance ∆v away from the focal plane isapproximately:

W (∆v) ≈ Wd +D/2

v×∆v (2.24)

The depth of focus is the value of ∆v at which the image has doubled in size:

∆vf =2λv2

D2(2.25)


Figure 2.22: A diagram showing how the depth of focus of an imaging system may becalculated. The imaging system is assumed to have a large magnification factor, so thatv À u and u ≈ f . The diffraction pattern in the focal plane and a distance ∆vf away

from it is shown.


Having found an expression for the depth of focus, it will now be used to findthe range of positions ∆uf through which the object may move whilst producinga reasonably focused image on a stationary image plane. From the lens equationwe have

∆u

u2= −∆v

v2(2.26)

Inserting this into eqn. 2.25 gives:

∆uf =2λu2

D2(2.27)

≈ 2λf 2

D2=

2λ

NA2(2.28)

where we have used u ≈ f for a ×10 objective lens. The numerical aperture (NA)of our imaging system with a ×10 objective is 0.19. Putting this into eqn. 2.28gives ∆uf = 43 µm. This is in good agreement with our experimental observationthat changing the expansion time by 1 ms required refocusing, for all expansiontimes longer than about 4 ms. (At 4 ms the condensate is falling at 40 µm ms−1

assuming zero initial velocity). The focal depth of the system could be increasedwithout compromising resolution by using a smaller magnification and a camerawith smaller pixels (Smaller pixels would hold less charge and hence have a smallersignal to noise, but we are far from this limit under current conditions).

Crude focusing was done by minimizing the size of a condensate image and thediffraction rings around it, at the chosen expansion time. Then vortices were madewithin the condensate, that have an expanded diameter of ∼ 6 µm, and detailedfocusing was done by optimizing the depth and clarity of these small structures. Forhorizontal focusing we simply minimize the size of a trapped condensate (∼ 3 µm),but this method is not appropriate in the vertical direction because the trappedand expanded condensates are not in the same object planes.

Calibrating the vertical magnification can be achieved by measuring the x sizeof a cloud with both the vertical system and a calibrated horizontal system. This isaccurate if the average of many identically prepared clouds is used. The intensity ofthe vertical probe beam was only approximately known because we did not use thevertical system for accurate number measurements. It would have been difficultto measure the intensity at the cloud because we placed a very small pinhole inthe vertical beam, of ∼ 2mm diameter about 1 m from the cell. The size of thehole was adjusted until only the central bright diffraction ring could be observedon the camera. This central ring produced a much smoother beam profile at thecondensate than the bare beam and hence reduced the noise on the image.

It is possible to take both a vertical and horizontal image of the same con-densate, simply by sending light down both paths at once and forming 2 imagesat different positions on the CCD. However the quality of each image is compro-mised because the absorption of the atoms must be shared between both images.


If probing occurs in a magnetic field, then the image that has the magnetic fieldcorrectly aligned for mF = 2 to 3 transition dominates the absorption. One wayto improve the system would be to fire two separate probe shots, onto the sameCCD screen, allowing just enough time inbetween for the magnetic field to rotateto be optimally aligned for each. The minimum time would be one quarter ofthe TOP period i.e. 35 µs. After the first shot, the atoms would be heated to atemperature of ∼ 30 µK or a speed of 8× 104 µm/s. Thus in 35 µs they will haveonly moved 3 µm, which is small compared to the size of the condensate. Thus thesecond image would be adequate for observing the shape of the entire condensatealthough not for resolving structures within it. This system is currently beingimplemented, using a new liquid crystal polarization rotator (Displaytech LV2500-OEM) to rapidly change the imaging path. Using a specialized driver (DisplaytechDR95), this waveplate can change the polarization of a beam in ∼ 30 µs and sothe horizontal and vertical probe pulses can be fired separately, when each has themagnetic field optimally aligned.

Chapter 3

BEC Production

This chapter briefly describes the original procedure for condensate production,which was used for most of the experiments in this thesis. Further details may befound in [?, ?]. It should be read in conjunction with chapter 4, which describesin detail how this procedure was recently optimized to triple the number of atomsin the condensate.

The stages in BEC production may be summarized as follows:

• Load second (experimental) MOT in U.H.V. region

• Load magnetic trap

– Compression of the cloud in the experimental MOT

– Optical molasses

– Optical pumping

– Transfer to magnetic trap

• Evaporative cooling

– Adiabatic compression

– Evaporation using zero of magnetic field

– Radio-frequency evaporation

• Imaging after time-of-flight expansion

45

46 Chapter 3. BEC Production

3.1 Loading the second MOT

Laser-cooled atoms are continuously pushed out of the hole in the pyramid MOTby the trapping light. These cold atoms travel to the second MOT with a velocityof ∼ 10 m/s, which is 100 times greater than their transverse speed. The pushbeam tends to optically pump the atoms into a low-field seeking state as theyleave the pyramidal MOT. Hence magnetic guiding, provided by magnetic stripsalong the transfer tube, is used increase the transfer efficiency to the experimentalMOT by a factor of 1.4. Further investigation of the transfer process may be foundin [?, ?].

Conditions in the second MOT are chosen to optimize the number of atomscaptured, balancing the need for a large capture velocity vc, with a low centraldensity [?]. The beams have a clean Gaussian profile and are limited to a waistof 0.8 cm by the cell dimensions. Each beam contains 3.3 mW of trapping light,corresponding to a central intensity of 3.3 mW/cm2 (eqn. 2.18) and has a detuningof −15 MHz or −2.6 Γ. In addition, each beam contains 150 µW of repumpinglight. The radial quadrupole field is 6.5 G/cm. The number of atoms in theMOT is limited by light-assisted, density dependent collisional losses. Thus thenumber may be doubled by the addition of a 2 G rotating TOP bias field, thatlowers the central density [?]. The relative number of atoms in the second MOT ismonitored by the fluorescence collected on a photodiode (fig. 2.1), to ensure thateach experimental run starts with the same number of atoms (∼ 6 × 108). Oncethe required number of atoms is loaded (loading typically takes ∼ 1 minute), thecontrol computer is triggered by the operator and all further steps in the productionof a condensate occur under computer control.

3.2 Loading the magnetic trap

We aim to transfer the maximum number of atoms from the MOT to the magnetictrap at the lowest possible temperature. The important parameters for the transferof atoms from the MOT to the magnetic trap are summarized in table 3.1.

3.2.1 Compression of the cloud in the MOT

The first stage in the transfer of atoms from the MOT to the magnetic trap is tocompress the cloud in the MOT, thus increasing the number of atoms within thespatial boundaries of the magnetic trap. To achieve the compression the bias fieldis turned off suddenly and the quadrupole field gradient is doubled over a periodof 25 ms. Meanwhile the trapping beam detuning is increased to −3.5 Γ and thepower reduced slightly to 3 mW/cm2, to reduce the outward radiation pressurefrom scattered photons at the trap centre. All these changes are made in linearramps for convenience.

3.2. Loading the magnetic trap 47

Stage BT (G) B′q (G/cm) δ/Γ I/Isat Duration (ms)

Second MOT 2 6.5 −2.6 1.1 -Compression in MOT 0 13 −3.5 0.96 25

Opt. molasses 0 0 −5.2 0.80 7Opt. pump.(3 pulses) 1, 2, 3 0 −0.69 1.0 5× 10−3

Mode-matched TOP 25 65 - - -

Table 3.1: The stages in the TOP trap loading procedure. The intensity for the MOT,compression and molasses is the intensity per beam. B′

q is the radial quadrupole gradient,as in eqn. 2.2

3.2.2 Optical molasses

The temperature of the cloud in the MOT will be around or just below the Dopplercooling limit of 138 µK (kB TD = hΓ/2). In zero magnetic field, sub-Dopplercooling mechanisms [?] operate efficiently producing a lower temperature Tmol ∝I/|δ|. Thus the magnetic trap is turned off, δ is increased and I reduced to thevalues in table 3.1 over 2 ms and then held there for a further 5 ms. Temperaturesas low at 60 µK have been measured at the end of this stage.

3.2.3 Optical pumping

The atoms are still evenly distributed over the 5 magnetic substates within theF=2 manifold and must be optically pumped into the F=2, mF =2 state desiredfor magnetic trapping. This is achieved by firing three σ+ polarized pulses of lightfrom the probe fibre, each of 5 µs in length, and synchronized with a TOP biasfield of 1,2 and then 3 G (fig. 3.1). The pumping light has a fixed detuning of−0.69 Γ from resonance in zero field. The increasing B field is designed to reducethe transition rate and hence heating for atoms that have reached the desiredmF = 2 state (although this can be achieved much more effectively as described insection 4.3.2). During optical pumping the trapping light is off, but the repumpinglight is left on.

3.2.4 Loading the TOP trap

Finally all the laser light is turned off and the TOP trap fields are snapped on withBT = 25 G and B′

q = 65 G/cm. These values were chosen to create a large locus ofB=0 which encloses the entire cloud, and a trap curvature that matches the shapeof the cloud to minimize heating. The calculation for mode-matching the trap tothe shape of the cloud is given in [?].


Figure 3.1: The different magnetic substates involved in our optical pumping process.Arrows mark the transitions that we excite using σ+ polarized pumping pulses. Thedetuning of each transition from resonance in zero field is shown. µBB/h has a value of

1.4MHz in a magnetic field of 1 G

3.3 Evaporative Cooling

Immediately after loading into the trap, the phase-space density (n0 λ3dB) is ∼

7 × 10−7. By a process of forced evaporative cooling and trap compression weincrease the phase-space density by 7 orders of magnitude to produce a condensate.During forced evaporative cooling, we selectively remove the hottest atoms fromthe cloud and allow those that remain to rethermalize via elastic collisions at alower mean temperature. Important parameters during evaporative cooling are thephase-space density, temperature, density and collision rate of the cloud. We aimto achieve a regime of runaway evaporation, where the density and collision raterises steadily as a result of the decreasing temperature, despite the necessary lossof atoms. The dependencies of these quantities on number, temperature and trapfrequency are given in appendix C. The evaporation scheme is shown in table 3.2and plotted in fig. 3.2.

3.3.1 Adiabatic compression

The speed of our evaporation ramps is limited by the elastic collision rate becausethe cloud must remain thermalized during the evaporation ramps. The initialcollision rate during evaporation is 10 s−1, resulting in ramp times of the order oftens of seconds. Since atoms are constantly being lost due to background collisions,it is desirable to evaporate in a stiff trap with a high elastic collision rate and sothe first stage is a compression of the trapped cloud. This may be done quickly

3.3. Evaporative Cooling 49

Ramp Stage Dur. B′q BT ω⊥/2π r0 rrf/r0 νrf

(s) (G/cm) (G) (Hz) (mm) (MHz)

TOP load - 75 28 12.8 3.7 - -1 Adiab. comp. 2 120 45 16.1 3.7 - -2 Comp. + evap. 2 194 45 26.0 2.3 - -3 B = 0 evap. 15 194 20 39.0 1.0 1.1 29.44 B = 0 evap. 16 194 2 123.4 0.1 1.0 2.85 Rf evap. 12 194 2 123.4 - 0.24 1.736

Table 3.2: The evaporative cooling ramps used to obtain BEC. The trap conditionsquoted are those at the end of each ramp. The final rf frequency yields the formation of

a pure condensate of about 1.5× 104 atoms.

Figure 3.2: The trap parameters as a function of time during the evaporative coolingramps; the quadrupole field (B′

q), TOP bias field (BT ), radius of death (r0), radial trapfrequency (ω⊥) and rf evaporation frequency (νrf )

.


(2 s) because we keep r0, the locus of the magnetic field zero, constant and sono evaporation occurs. The change is adiabatic (phase-space density constant)provided that the potential energy changes at a lower rate than the kinetic energyof the trapped atoms; this condition can be expressed as:

dω

dt¿ ω2 (3.1)

and is well satisfied for this ramp. The pure compression stage is limited by themaximum value of BT (45 G) that we can achieve without the temperature of thecoils rising too far (i.e. above 50C). The next ramp increases B′

q to its maximumvalue, to achieve the stiffest trap possible. Since r0 decreases during this ramp,some cutting of the cloud occurs.

3.3.2 Evaporation using the magnetic field zero

The hottest atoms are found around the outside edge of the cloud, (just inside thecircular path of the magnetic field zero) since the turning points of their orbits arein the regions of highest potential energy. The initial evaporative cooling methodis to gradually reduce the radius of the magnetic field zero by decreasing BT . Themagnetic field zero induces Majorana spin flips into untrapped magnetic states,removing atoms at this radius (fig. 2.11). This evaporation method is unique tothe TOP trap and is used in ramps 3 and 4. (The process is divided into twoseparate ramps for a purely technical reason; to enable the rf evaporation field tobe switched on partway through). Since the trap frequency depends on B′

q/√

BT

it increases throughout both ramps, helping to maintain a high collision rate. Atvalues of BT < 2 G, the trap lifetime is reduced. This occurs because at smallTOP fields the transition frequencies between different mF states are reduced andnoise on the bias field can drive transitions to untrapped states. Thus this firstmethod of evaporation stops in a 2 G trap, with a phase-space density of ∼ 0.05.

3.3.3 Radio-frequency evaporation

The final evaporative cooling stage uses the spatially dependent Zeeman shiftof atoms in a magnetic trap to selectively remove the hottest atoms. A radio-frequency (rf) field of amplitude Brf is applied, tuned to the ∆mF = −1 transitionat the outside edge of the cloud. The resonance condition is:

νrf =µB gF B

h(3.2)

As the bias field rotates, the rf cutting region moves in a circle about the centreof the trap on the opposite side to B = 0. The exact size of the region over whichatoms are removed will depend on the rf power. To ensure that there is a smooth

3.4. Detecting the transition 51

transition from B = 0 evaporation to rf evaporation, the rf field is turned on at thestart of ramp 4, cutting outside the locus of B = 0. It moves inwards slightly fasterthan the magnetic field zero, so that by the end of this ramp both are cutting atthe same radius rrf/r0 = 1. During the final rf evaporation ramp, the rf cuttingsurface moves inside the circle of death to a position of 0.24 r0 at which point apure BEC is formed. A more detailed discussion of the coupling between trappedand untrapped states in the presence of an rf field is given in section 4.4.3.

3.4 Detecting the transition

The phase-space density is plotted through the evaporation ramps as a function ofatom number in fig. 4.8. Bose-Einstein condensation occurs when the phase-spacedensity is equal to 2.61. Using this evaporation procedure, the transition point isreached with ∼ 5×104 atoms and a critical temperature Tc of 300 nK. After furtherevaporative cooling to T < 0.5 Tc (the lowest temperature that we can measure),we produce a ‘pure’ condensate of ∼ 1.5× 104 atoms.

The onset of condensation is marked by the appearance of a sharp spike atthe centre of the density distribution, as the occupation of the ground state of themagnetic trap becomes macroscopically large (fig. 3.3(b)). Another clear signatureof condensation may be gained from the shape of the cloud after expansion. Theinternal energy of the condensate is dominated by the repulsive interaction betweenatoms, whilst in the thermal cloud it is purely kinetic energy. The thermal cloudexpands isotropically (from equipartition) becoming spherical at large expansiontimes (fig. 3.3(a)). The condensate has more energy in the direction that is mosttightly confined in the trap (axial) and hence expands faster in this direction. Thusa trapped oblate condensate (pancake-shaped) becomes prolate (sausage-shaped)at long expansion times (fig. 3.3(c)).


Figure 3.3: The atomic cloud with different condensate fractions (N0/N) after 15msof expansion from a pancake-shaped trap. The top line of pictures are false colourimages of the density distribution, whilst the lower line of pictures plots the same densitydistribution on the z axis. (a) shows the thermal cloud just prior to condensation,(b) contains a small condensate fraction and (c) is the coldest condensate that we can

produce, with no visible thermal cloud.

Chapter 4

Optimizing CondensateProduction

The gyroscope experiment described in chapter 10 required a long excitation pro-cedure (∼ 2 s) after the condensate had been made, first to produce a vortex andthen excite the scissors mode. For our first attempt at the gyroscope experiment,using the procedure described in chapter 3, we started with 14,000 atoms imme-diately after the condensate was made, which decayed during excitation to lessthan 10,000 atoms for imaging. With such a low atom number, fitting the tiltedparabolic density distribution to extract an angle became very unreliable. Theeffect of the vortex line (with a core size ∝ 1/

√n) on the parabolic density dis-

tribution could not be ignored and imperfections in the imaging system becamesignificant against the weak condensate image. We decided to repeat the experi-ment with larger condensates to ensure that well over 10,000 atoms were alwaysavailable for imaging.

To produce larger condensates it was necessary optimize every step of the pro-duction process, from loading the second MOT, through the transfer to the mag-netic trap and finally the evaporation ramps. This chapter gives an account of themethods used to optimize each stage of the process and the improvements thatwere made.

4.1 Loading the second MOT

Poor second MOT loading may be due to a number of factors - misalignment of thepyramid pushing beam, poorly balanced second MOT beams and poor master orrepumping laser modes are amongst the most common. Alignment of the secondMOT beams will be discussed in section 4.2. The difficulties that we have hadwith external cavity diode lasers (ECDLs), both homemade and commercial, are

53

54 Chapter 4. Optimizing Condensate Production

discussed in detail in section 2.3.3. In summary, the homemade ECDLs sufferproblems with the mechanical mount, which cause the external cavity to driftwhilst the commercial TUI ECDLs have the ability for stable lasing in severalmodes, which, if undetected, results in only a small fraction of the light interactingwith the atoms.

The current arrangement uses a homemade master laser and a commercial TUIrepumping laser. The weak repumping saturated absorption lines are strengthenedby heating the cell (section 2.3.4). The repumping laser mode structure is checkedon a weekly basis using a Fabry-Perot spectrum analyser, rather than just relyingon the saturated absorption signal. We have checked the output spectrum of ourlaser amplifiers, the MOPA and slave, and these appear to reproduce the masterspectrum reliably.

4.2 Alignment of the second MOT and stray

magnetic field nulling

The second MOT must not only load quickly but must also transfer atoms withmaximum phase-space density into the magnetic trap. This requires the MOTto have a regular shape, have the same centre as the magnetic trap and expandevenly and slowly when the quadrupole field is turned off e.g. during the molassesphase. These requirements are affected by several factors including the alignmentand power balance of the six MOT beams and the current in the three nulling coils.Thus we had to develop a scheme to optimize each of these factors independently.

First the second MOT beams were approximately aligned and the power ineach beam was checked. We use equal power in all of the horizontal beams andapproximately 20% extra in the vertical beams. Then we aligned the centre ofeach beam with the centre of the quadrupole field. This was achieved by placingan iris in the centre of each beam in turn, just before the final alignment mirrorand adjusting the mirror to maintain the MOT whilst the iris size was reduced.

With the second MOT roughly aligned, we can then use the molasses stage tooptimize the current in each of the 3 pairs of nulling coils. The molasses stage usessub-Doppler cooling mechanisms to reduce the temperature of the cloud belowthat of the MOT. Sub-Doppler cooling mechanisms require the ground state tohave degenerate magnetic sublevels and so only work effectively in zero magneticfield. We measure the temperature of the cloud after it has been loaded into themagnetic trap as a function of one of the nulling coil currents. The data in fig. 4.1gives a clear minimum in the temperature, indicating the current at which thefield is best nulled. We also did a control experiment, in which the molasses stagewas removed and the temperature was plotted as a function of one of the nullingcurrents. This data showed no clear minimum, confirming sub-Doppler cooling wasresponsible for the reduced temperature in fig. 4.1 and not an improved alignment


Figure 4.1: The temperature of the cloud (after 4 s hold in the magnetic trap) as afunction of the vertical nulling coil current. The curve was fitted with a parabola, which

gives the optimum nulling current to be 147 mA.

of the MOT and the magnetic trap.

Two tests may then be used to fine tune the alignment and power of the MOTbeams. Firstly the centre of the MOT cloud must not move if the quadrupole fieldis suddenly jumped to a large value. This ensures that the MOT is centred on thequadrupole field and hence on the magnetic trap. Secondly the atomic cloud fromthe MOT should expand slowly and evenly in all directions when the quadrupolefield is suddenly turned off.

4.3 Loading the magnetic trap

Having devised a reliable way of setting up the second MOT, it is possible to op-timize the stages up to and including loading the magnetic trap. Improvementsto these stages are all analyzed by looking for an improvement in the phase-spacedensity in the magnetic trap (increased number, reduced temperature). It is im-portant to be sure that the cloud is thermalized before a measurement is made,otherwise the temperature measurement will not be accurate. Thus a detailedstudy of the behaviour of the cloud after loading was made. No dipole motion wasobserved, (indicating that the MOT and trap centres were well aligned) and thesmall amount of quadrupole motion was quickly damped (≤ 1 s). Since we loada spherical MOT cloud into a pancake-shaped magnetic trap, there is an initialsize, and hence temperature, mismatch between the x and z directions. This may


be observed in TOF, since the aspect ratio of a non-thermalized cloud will be lessthan that of a thermalized cloud after expansion:

Thermalized AR =X sizeTOF

Z sizeTOF

=X sizeTRAP

Z sizeTRAP

√√√√1 + ω2x t2

1 + ω2z t2

(4.1)

After holding the cloud in a typical load trap for ∼ 4 s, the aspect ratio reachedits thermalized value, in reasonable agreement with the predicted thermalizationtime of ∼ 1 s (a few collision times). This hold time should always be used betweenloading the trap and analyzing the temperature of the cloud.

4.3.1 Compression and molasses parameters

The new molasses and compression parameters are given in table 4.1 and discussedbelow. They may be compared to the original values in table 3.1.

Stage BT (G) B′q(G/cm) δ/Γ I/Isat Duration (ms)

Second MOT 2 6.5 −2.6 1.1 -Compression in MOT 0 5 −4.3 1.0 2.5

Opt. molasses 0 0 −5.2 0.96 5Opt. pump.(5 pulses) 30 0 −4.8 0.2 0.01

TOP trap load 14.1 44.22 - - -

Table 4.1: The stages in the optimized TOP trap loading procedures. The intensity forthe MOT, compression and molasses is the intensity per beam.

Summary of changes

• Sudden changes to the trap reduce the phase-space density and should beeliminated. By gradually reducing the MOT bias field during the compressionramp rather than jumping it off at the start, we were able to load a coldercloud into the magnetic trap. We also found that reducing the bias field tozero produces sufficient compression of the cloud in the MOT and so it is notnecessary to increase the quadrupole field during this stage.

• The molasses stage has been shortened, since the atoms are no longer trappedand are diffusing outwards during this stage. 5 ms is the minimum time forthe quadrupole field to die away (1-2 ms) and sub-Doppler cooling to reacha steady state temperature.

4.3.2 Optical pumping

Optical pumping is used to exclusively populate the magnetically trapped F = 2,mF =2 substate, prior to turning on the magnetic trap. After molasses, the atoms


are evenly distributed over the 5 magnetic substates of the F=2 ground state mani-fold (as described in section 3.2.3). Thus optical pumping can theoretically increasethe number of trapped atoms by a factor of 5. The original system (described insection 3.2.3) was erratic, often producing less than a factor of 2 improvement(over no optical pumping) and also heating the cloud significantly.

The problem of inefficient and erratic pumping was due to varying pumpingpower. The fibre alignment was optimized each morning at the pyramid trappingfrequency, not the pumping frequency. Since the coupling down the probe fibrevaries with detuning (section 2.3.4), the pumping power was actually unknown.Checking the pumping power at the correct detuning every morning solved thisproblem.

The problem of heating was a little more complicated, but given that the phase-space density depends on T−3, it was worth solving. Each time an atom absorbs aphoton it recoils and gains momentum hk in the direction of the optical pumpingbeam. If it absorbs and spontaneously emits n photons during the pumping phase,then the total increase in energy will be:

∆E

kB

= (n2 + n)h2k2

2 m= (n2 + n)× 0.2× µK (4.2)

where the n2 term comes from unidirectional absorption and the n term fromisotropic spontaneous emission. Given the n2 dependence, it is important to min-imize the number of transitions that each atom undergoes, whilst maintainingefficient pumping. This may be achieved if the atom becomes more and more de-tuned from the pumping light as it reaches higher mF states. Ideally the mF =2 tomF =3 transition should be so far detuned that the final mF =2 state is effectivelydark to σ+ pumping light. Unfortunately, this transition also has the highest Cleb-sch - Gordon coefficient (see appendix B) and so can never be made completelydark whilst maintaining full pumping. The scheme is optimized by pumping in alarge B field, so that the different transitions are separated by large Zeeman shiftsand by tuning the pump light on resonance with the lowest mF =-2 to mF =-1transition (fig. 3.1). Two different fields and different pump powers were testedand the results for a 30 G field are given in fig. 4.2.

Pumping in a 30 G bias field, with 5 × 10 µs pulses at I/Isat = 0.1 (100 µW)produced the highest phase-space density in the magnetic trap. In practice weuse nearer 200 µW, trading a little heating for an increased number of atoms inthe correct magnetic substate. A comparison of the different ∆mF =+1 transitionrates in the original and improved pumping scheme is given in table 4.2. In theoriginal scheme the transition rate from the final state was 15 times higher thanthat from the initial state. In the new scheme the rate out of the final mF =+2state is 4 times smaller than the rate from the lowest mF =-2 state, thus achiev-ing minimum heating and optimal pumping. Using this scheme, optical pumpingreliably increases the number of trapped atoms by a factor of ≥ 3.5 and heats the


Figure 4.2: The phase-space density (solid circles, solid line), number (open circles,dotted line) and temperature (crosses, dashed line) of atoms in the magnetic trap as afunction of optical pump intensity. 5 × 10µs pumping pulses were fired in a 30 G fieldand the beam was tuned into resonance with the lowest mF =-2 to mF =-1 transition.

Original BT = 3G Improved BT = 30GmF Relative rate δ/2π Rate δ/2π Rate

transition (∝ CG coeff.2) (MHz) (µ s−1) (MHz) (µ s−1)

-2 to -1 1/15 4.47 0.1 0 0.3-1 to 0 1/5 4.7 0.4 7 0.20 to 1 6/15 4.93 0.7 14 0.11 to 2 2/3 5.17 1.0 21 0.092 to 3 1 5.4 1.5 28 0.07

Table 4.2: Transition rates for optical pumping with the original and improved schemes.The original scheme used 3× 5 µs pulses at ∼ Isat. The improved scheme uses 3× 10 µspulses at 0.2 Isat. δ is the detuning of the transition from the pump beam frequency.


cloud by no more than 15 µK.One would expect that the heating that results from optical pumping could be

reduced if two anti-parallel pump beams were used. The majority of the recoilsfrom absorbed photons should then cancel, removing the term ∝ n2 in eqn. 4.2.(Since an atom absorbs ∼ 10 photons during the optical pumping pulses, removalof the n2 heating term should be significant). A second beam of the correct polar-ization can be created simply by reflecting the pump beam and so this scheme wastested, but unfortunately no improvement was observed. This suggests that mostof the heating is due to the reabsorption of spontaneously emitted photons withinthe cloud. We also investigated the number and length of pulses. Experimentally,the maximum pulse length was 15 µs (before the bias field had rotated sufficientlythat significant anti-pumping occurred) and a total pumping time of 50 µs wasrequired at an intensity of 0.2 Isat.

4.3.3 Initial parameters for the magnetic trap

We wish to load the magnetic trap with the maximum number of atoms and highestphase-space density and so use these criteria to choose its size and stiffness. Weshould be able to satisfy both conditions simultaneously because the load trap has2 independent parameters, B′

q and BT .

Figure 4.3: The number (solid circles) and temperature (open circles) of the cloud afterloading into traps of different radii r0 but the same stiffness, and being held for 2 s. The

solid lines follow the mean of the data points.

The first condition is satisfied by ensuring that the circular locus of B = 0,which has radius r0 = BT /B′

q and defines the radial size of the load trap, falls


outside the radius of the MOT cloud. Figure 4.3 shows the number and temper-ature in the load trap, as a function of r0. The atoms were held in the trap for2 seconds before imaging to allow for rethermalization and in each case the fieldswere adjusted so that each trap had the same stiffness. Thus any changes in thetemperature are due to evaporative cooling during the hold time, if the magneticfield zero falls within the radius of the cloud. Both number and temperature beginto fall at r0 < 2.8mm, indicating that this is the radius of the atom cloud at load-ing. We eventually choose an initial value of r0 of 3.2 mm, which should collect allatoms even from particularly large MOTs, whilst using a relatively small value ofBT which is advantageous for the later ramps (see section 4.4.1).

Theoretically, the highest phase-space density will be achieved if the curvatureor stiffness of the load trap (∝ B′2

q /BT ) matches the shape of the MOT cloud.Under these conditions the transfer will occur adiabatically and the increase inentropy will be minimized. When the experiment was first set up, much theoreticaleffort was put into ‘mode-matching’ the transfer and is discussed in [?]. However,perfect mode-matching cannot be achieved since we load a spherically symmetriccloud into a pancake-shaped potential (as described at the start of this section).This agrees with our experimental observation, that the stiffness during loading isnot critical. The phase-space density of the cloud is roughly constant when theinitial trap has a frequency in the range ω⊥/2π = 10 to 14 Hz, (r0 constant) andfalls slowly outside this range. This was checked both after holding the cloud fora few seconds in the load trap and after the first evaporation stage.

Using the technique described in [?] it would be possible to create a sphericallysymmetric TOP trap, into which the MOT cloud could be loaded with maximumphase space density. However this would require major changes to the TOP trapapparatus (the oscillating bias field in the z direction would require a new wave-form and much larger amplitude than is presently possible) and so has not beeninvestigated further.

The final choice of conditions for loading the atom cloud into the magnetictrap was B′

q = 44.11 G/cm and BT = 14.1 G. In this trap, the magnetic field zerofollows a path of radius r0= 3.2 mm and the radial trap frequency is ω⊥ = 10.6 Hz.

4.4 Evaporative cooling ramps

The optimized conditions for the evaporative cooling ramps are summarized intable 4.3 and may be compared to the original values in table 3.2. Each of thechanges is discussed in sections 4.4.1 to 4.4.3 below.

4.4.1 Adiabatic compression

The original adiabatic compression stage involved ramping up B′q and BT whilst

maintaining a constant ratio between them, thus r0 was held constant whilst the

4.4. Evaporative cooling ramps 61

Ramp Stage Dur. B′q BT ω⊥/2π r0 rrf/r0 νrf

(s) (G/cm) (G) (Hz) (mm) (MHz)

TOP load - 44.11 14.1 10.6 3.2 - -1 Adiab. comp. 4 194 37.8 28.4 1.9 - -2 B = 0 evap. 16 194 20 39.0 1.0 1.1 29.43 B = 0 evap. 18 194 2 123.4 0.1 1 2.84 Rf evap. 12 194 2 123.4 - 0.24 1.736

Table 4.3: The optimized evaporative cooling ramps. The trap conditions are quoted atthe end of the given ramp. The final rf evaporation ramp yields the formation of a pure

condensate of about 40000 atoms.

trap stiffness and collision rate was increased before evaporation. When the max-imum value of BT (45 G) was reached, the quadrupole field continued to increasealone so that the maximum trap stiffness was achieved. During this stage, r0 de-creased and hence some inefficient evaporation (occurring before the collision ratewas optimized) could not be avoided. We decided to investigate whether a highermaximum BT would improve the phase-space density, since evaporation wouldstart from a stiffer trap and hence be more efficient. The temperature of the TOPcoils was carefully monitored, with a thermistor attached to one of the coils, toensure that they did not overheat at the higher currents required.

First we measured the number of atoms at the end of the first B = 0 evapora-tion ramp using the original conditions of table 3.2, with BT limited to 45 G. Weobserved an average of 1.3 ×107 atoms. We then repeated the experiment, contin-uing the adiabatic compression up to BT = 60 G, B′

q = 160 G/cm and hence usinga shorter ‘compression + evaporation’ stage. The number of atoms immediatelydropped to 0.5 ×107, the cloud diameter (after 5 ms of expansion) increased by afactor of 1.13 indicating a higher temperature, and vertical streaks appeared onthe image. Over the next hour or so the number of atoms in the cloud graduallydeteriorated. All these factors indicate that the cell is being heated significantly at60 G - an assumption that was backed up by the coil temperature measurements.The TOP coil temperature reached 50C in a single run to 60 G, as opposed to40C in a 45 G run. Over a few runs this will increase the background pressureof rubidium in the cell, by sublimating it from the walls and thus reduce the traplifetime. The streaks on the final image may well be caused by rapid changes in therefractive index of the cell as it cools down, changing the probe beam profile on thecamera between the experimental and background images. The final conclusionwas that we could improve the experiment by reducing the maximum value of BT ,to minimize heating, rather than increasing it as initially suggested.

The maximum value of BT may be reduced by reconsidering the adiabaticcompression conditions. During the compression we require that no significant


evaporation occurs, which can be achieved by holding the cloud size, σ⊥ and thelocus of the magnetic field zero, r0 (eqn. 2.8) in a constant ratio. The cloud size isgiven by:

σ⊥ =1

ω⊥

√2kBT

m∝√

BT

B′q

√T (4.3)

Since the cloud size reduces as the trap becomes stiffer, r0 need not remain constantduring this stage but may reduce by the same factor, allowing a lower BT to beused.

σ⊥r0

∼√

BT

B′q

√T

B′q

BT

(4.4)

so if σ⊥/r0 remains constant during the adiabatic compression ramp then:

Ti

BT i

=Tf

BTf

(4.5)

Now the phase-space density φ is given by

φ = N

(hωho

kBT

)3

(4.6)

and so for an adiabatic compression ω⊥/T remains constant or

B′qi√

BT i Ti

=B′

qf√BTf Tf

(4.7)

Inserting eqn. 4.7 into eqn. 4.5 we obtain the relationship between the initialand final traps in an adiabatic compression with constant (negligible) evaporation.

BTf

BTi

=

B′

qf

B′qi

2/3

(4.8)

Using this constant evaporation formula (eqn. 4.8) and the trap loading conditionsof table 4.3, it is possible to reach the maximum quadrupole field during theadiabatic compression, at a TOP field of only 38 G. Thus not only does this methodreduce the maximum value of BT (minimizing cell heating), but it also removesthe need for an inefficient ‘evaporation + compression’ stage prior to cooling.

4.4.2 Evaporation using the magnetic field zero

Two improvements to the B = 0 evaporation ramps were considered. Firstly, canwe continue the ramps below 2 G, so that rf evaporation can occur more efficientlyin a stiffer trap and secondly can the ramps be shortened to reduce the effect of


Figure 4.4: Phase-space density versus atom number for different minimum values ofBT . This data was taken after the B = 0 evaporation to the given value of BT and a

short rf evaporation ramp to a variety of depths.

background losses? (Having finished optimizing the early evaporation ramps, atthis point we replaced the ×1 imaging system with a ×10 imaging system, whichis more suitable for imaging the small clouds close to condensation.)

To investigate BT < 2 G, we cut to the chosen value of BT (in the range1.5 G - 4 G) and then continued with a small rf cut, (final rrf > 0.4 r0), stoppingabove the critical temperature for condensate formation. For each value of BT weplotted phase-space density, φ, versus N . The results plotted in fig. 4.4 indicatean improvement in φ for a given number of atoms down to BT = 1.65 G due toincreased evaporation efficiency, and then a sharp reduction below 1.65 G as thedecreasing trap lifetime suddenly dominates. Direct measurements of the traplifetime confirmed that it decreases sharply around this range of BT , from 11± 7 sat 2.5 G to 1.8± 0.8 s at 1.5 G.

Given that the trap lifetime appears to change dramatically around BT = 2 G,[?] we repeated the experiment, using a full length rf cut (∼ 12 s) to producea small condensate fraction. A plot of condensate fraction versus total numbershowed that with the longer rf evaporation ramp, the trap with BT = 2 G gavethe best results. The lifetime of the 1.65 G trap was too short for the longer rfevaporation ramp required to produce a condensate.

Finally we investigated the optimal time for the magnetic field zero evaporation


ramps. Figure 4.5 shows the phase-space density of the cloud at the end of thesecond B = 0 ramp, as a function of the total B = 0 evaporation time (ramp 2+ ramp 3 in table 4.3). Below 34 s, the phase-space density falls sharply due toinefficient evaporation in a non-thermalized cloud, whilst beyond 34 s it falls slowlydue to the finite lifetime of the trap. Thus a total B = 0 evaporation time of 34 s(16 s + 18 s) was chosen.

Figure 4.5: Phase-space density versus total B = 0 evaporation time (ramp 1 + ramp 2).The solid line follows the mean of the data points.

4.4.3 Radio-frequency evaporation

The radio-frequency evaporation ramp had two parameters to optimize - the rfpower and the ramp duration. Originally the coils had produced an oscillatingfield with an amplitude of 6 µT. This was increased by a factor of ∼ 3 by doublingthe output voltage of the Stanford rf generator and removing a series resistor of50 Ω. (The coil reactance at 2 MHz was ∼ 50 Ω). With the higher rf power thenumber of atoms and hence phase-space density increased by a factor of 2 at atemperature 1.5 Tc.

It was useful to consider the rf evaporation process in detail to determinewhether even more rf power would result in further improvements. In a weakrf coupling field, the evaporation process may be viewed as individual ∆mF =-1transitions that eventually transfer the atoms to an untrapped state with negativemF . At higher rf power one must consider the ‘dressed atom’ picture. At resonancethe 5 degenerate states, which consist of an atom dressed with 2 −mF photons,are strongly mixed - the true eigenstates of the system are non-degenerate and


Figure 4.6: Dressed atom energy levels in a strong rf field. The pattern is periodic isenergy, repeating each time an additional photon is added to the dressed state. Theavoided crossing relevant to our rf evaporation process in shown in bold. The lowestenergy state is seen to evolve continuously from trapped to untrapped as the resonantposition is passed. The splitting between the energy eigenstates at resonance is ∼ hΩR

are superpositions of all 5 original mF states (see fig. 4.6). This is an exampleof an avoided crossing. If the atom passes through resonance sufficiently slowly,then it will stay in the lowest energy state and be adiabatically transferred fromthe mF = 2 to the mF = −2 state. This adiabatic transfer process is desirablesince it provides efficient coupling directly to a strongly untrapped state. Onecan estimate the conditions for adiabatic transfer using eqn. 4.9 (derived from theLandau-Zener formula [?]).

µB BT ωo ¿ h Ω2R (4.9)

where ωo/2π is the TOP rotation frequency (7 kHz) and ΩR is the Rabi frequencyfor the ∆mF = 1 transition. Equation 4.9 shows that the rate of change of energyof the atom as the TOP field rotates (bringing it into resonance with the rf field)must be very much less than the rate of change of energy for an atom Rabi floppingat ΩR between the different dressed eigenstates at resonance (fig. 4.6). (The energyof the atom also changes because it is moving at some speed v in an inhomogeneousmagnetic field, but this effect is less significant than that of the rotating bias field).Since h ΩR ∼ µB Brf [?] and Brf ∼ 0.1 BT , we require

ΩR À 10 ω0 (4.10)

for efficient adiabatic transfer.In the original set up the rf field had an amplitude Brf = 6 µT and so ΩR/2π ∼

84 kHz, too small to satisfy the inequality of eqn. 4.10. Increasing Brf by a factor


of 3 moved conditions significantly closer to those for efficient adiabatic transfer,which explains the improvement in phase-space density that we observed. Thetheory suggests that the rf evaporation process would be even more efficient in thefully adiabatic regime and so an rf amplifier has been purchased.

4.5 Optimization summary

Following the optimization procedure described in this chapter we were able toregularly produce ‘pure’ condensates (T < 0.5 Tc, no thermal cloud visible) with40,000 atoms. This represents an improvement over the previous conditions by afactor of 3. Whilst many small changes contributed to this improvement, the mostsignificant factors were:

• Regular checks on the laser mode structures using a Fabry-Perot spectrumanalyser.

• Reliable power and reduced heating during optical pumping.

• A more adiabatic MOT compression.

• Reducing the maximum bias field used during evaporation, thus reducing thebackground rubidium pressure in the cell.

• Increased power for rf evaporation.

The optimization process was initially motivated by the need for sharper con-densate images to improve the signal to noise on our gyroscope experiment data(chapter 10). After the gyroscope excitation procedure we had 19,000 atoms forimaging (rather than 9000) and the improvement in the images can be seen infig. 4.7.

Figure 4.8 shows a plot of phase-space density versus atom number during theoptimized evaporation ramp. The gradient of this line indicates the efficiency ofthe evaporation ramps. A similar plot prior to optimization is also shown forcomparison. Finally fig. 4.9 shows how the number, temperature and phase-spacedensity evolve as a function of time during the optimized evaporation ramps.

4.5. Optimization summary 67

Figure 4.7: Typical condensate images taken during the gyroscope experiment, (a) and(b) before optimization, (c) and (d) after optimization. All pictures are taken along they direction after 12ms of expansion. After optimization the condensates are bigger, more

regularly shaped and the vortex line has no significant effect on the density profile.


Figure 4.8: Phase-space density versus atom number during the optimized evaporationramps (data points). The solid line shows the equivalent data prior to optimization forcomparison. Note that the apparent change in phase-space density during the adiabaticcompression is due to difficulties in accurately measuring the temperature of the cloud

immediately after loading into the magnetic trap.

4.5. Optimization summary 69

Figure 4.9: The atom number, temperature and phase-space density as a function oftime during the optimized evaporation ramps

Chapter 5

Condensate Theory

5.1 BEC in a non-interacting gas

To emphasize that Bose-Einstein condensation results purely from the quantumstatistics of identical bosons, we will start by considering the phase transition ina non-interacting, homogeneous gas of N identical particles in a large volume V .The complications of confinement and interactions will be added in later sections.In thermal equilibrium the occupation of the ith state, ni(Ei, T ), is given by theBose-Einstein distribution function:

ni(Ei, T ) =1

e(Ei−µc)/kBT − 1(5.1)

where µc is the chemical potential and is determined by the constraint on the totalnumber of particles: ∑

i

ni(Ei, T ) = N (5.2)

Since V is large, the allowed energy levels are very closely spaced and the sumof eqn. 5.2 can be expressed as an integral:

N = N0 +∫ g(E)

e(E−µc)/kBT − 1dE (5.3)

= N0 + Nex (5.4)

where g(E) is the 3D free particle density of states:

g(E) =V

4π2

(2m

h2

)3/2√E (5.5)

The additional term N0 in eqn. 5.4 must be added because the integral does notcontain the population of the zero energy ground state (g(0) = 0). The integral

70

5.1. BEC in a non-interacting gas 71

Nex may therefore be interpreted as the population in excited states. An upperbound on Nex at a given temperature may be found by evaluating the integral withµc = 0.

Nex <V

4π2

(2mkBT

h2

)3/2 ∫ ∞

0

u1/2

eu − 1du (5.6)

< 2.612V

(mkBT

2πh2

)3/2

(5.7)

Where we have used the following standard result for the integral in eqn. 5.6,expressed in terms of the gamma-function and the Riemann zeta-function [?].

∫ u1/2

eu − 1du = Γ

(3

2

)× ζ

(3

2

)=

√π

2× 2.612 (5.8)

As the temperature of the system decreases, the number of particles that may beaccommodated in excited states also decreases. Bose-Einstein condensation beginsat a critical temperature Tc, at which Nex = N . Below this temperature it isimpossible to accommodate all the particles in excited states and so a macroscopicnumber of particles are forced to accumulate in the lowest energy level.

Tc =2πh2

mkB

(N

2.612 V

)2/3

(5.9)

This definition of Tc is consistent with the picture that condensation occurs whenthe inter-particle spacing (1/n)1/3 is comparable to the thermal de Broglie wave-length,

λdB =

√√√√ 2πh2

mkBT(5.10)

Rearrangement of eqns. 5.9 and 5.10 at Tc gives:

nλ3dB = 2.612 (5.11)

From eqns. 5.7 and 5.9 we can obtain the following expressions for the fractionof atoms in excited states Nex/N (the thermal cloud) and the ground state N0/N(the condensate) below Tc:

Nex

N=

(T

Tc

)3/2

(5.12)

N0

N= 1−

(T

Tc

)3/2

(5.13)

72 Chapter 5. Condensate Theory

5.2 The trapped non-interacting Bose gas

Our experiments do not occur in free space but in a 3D harmonic trapping poten-tial:

Vext =m

2

(ω2

xx2 + ω2

yy2 + ω2

zz2)

(5.14)

Therefore we must consider the effect of the trapping potential on the condensationprocess. A trapped ideal gas has energy levels:

E =(nx +

1

2

)hωx +

(ny +

1

2

)hωy +

(nz +

1

2

)hωz (5.15)

where nx, ny, nz are positive integers. The density of states of the trapped gas isgiven by:

gt(E) =πE2

2h3ω3ho

(5.16)

where ωho = (ωxωyωz)1/3 is the average trap frequency. The density of states

in the trap is ∝ E2, compared to the√

E dependence of a free gas in eqn. 5.5.Thus trapping has a strong influence on the condensation process by controllingthe number of particles that can be accommodated in excited states at a giventemperature. When the density of states of a trapped gas gt(E) is inserted intoeqn. 5.4, we find:

Nex = ζ(3)

(kBT

hωho

)3

(5.17)

⇒ Tc =hωho

kB

(N

ζ(3)

)1/3

= 0.941hωho

kB

(N)1/3 (5.18)

From eqns. 5.17 and 5.18 we can determine the condensate fraction as a functionof temperature for a trapped gas:

N0

N= 1−

(T

Tc

)3

(5.19)

Note that the condensate fraction increases more rapidly as T falls in a trappedgas than in the homogeneous case. This occurs because the trapped gas has asmaller density of low-energy excited states (the only ones occupied as T → 0)than the free gas, as shown in fig. 5.1. Thus fewer particles can be accommodatedin excited states at a given temperature and so particles accumulate more rapidlyin the ground state as the temperature falls below Tc.

5.3. Bose-Einstein condensation with interacting particles 73

Figure 5.1: The density of states as a function of energy in a homogeneous gas (solidline) and trapped gas (dashed line)

5.3 Bose-Einstein condensation with interacting

particles

The balance between the interaction energy and quantum kinetic energy of theparticles in a Bose condensate plays a vital role in determining its physical prop-erties. In our system, the interaction energy depends only on the particle densitybecause the scattering length is fixed. It ranges from negligible in the limit of anideal gas condensate, to very much greater than the quantum kinetic energy inthe Thomas-Fermi limit. The balance between the two energies determines thecondensate’s size, shape [?], collective mode frequencies [?, ?], its superfluid prop-erties, expansion behaviour [?] and can even cause it to implode [?]. Althoughinteractions play a critical role in the Bose-condensed gas, it is still considered tobe ‘weakly interacting’ because of its very low density. In such a dilute, low tem-perature gas only binary s-wave interactions need to be considered, which can beaccurately modeled by a single parameter, the s-wave (or hard sphere) scatteringlength, a. In this limit, where only long-range effects are considered, such scatter-ing events simply change the phase of the wavefunction of the incoming particle.Therefore a is chosen to be the radius of a hard sphere that would produce thesame phase change. The 2 body interaction potential can be written as

V (r1 − r2) = g δ(r1 − r2) with g =4πh2a

m(5.20)


The constant g is determined by considering the increase in kinetic energy of atom1, as the volume available to it is reduced by the presence of atom 2 [?, ?].

The ability to accurately model the scattering processes in a dilute gas Bose-condensate and then use mean field theory to build a complete mathematical de-scription of the system, is one of the most important and exciting features of thisfield. This approach is valid provided that the system is so dilute that na3 ¿ 1.This is in stark contrast to the situation is liquid helium, where na3 ∼1 and thestrong interactions cannot be modeled mathematically. Thus the properties ofsuperfluid liquid helium are very difficult to interpret in terms of Bose-Einsteincondensation.

The mean field description is developed from the many body Hamiltonian de-scribing N bosons in an external potential Vext, interacting via the 2 body contactpotential of eqn. 5.20:

H =∫

d3~r Ψ†[− h2

2m∇2 + Vext +

g

2Ψ†Ψ

]Ψ. (5.21)

where the Bose field operators Ψ(r, t) and Ψ†(r, t) annihilate and create a particleat position r respectively. The time evolution of the Bose field is governed by theHeisenberg equation of motion

ih∂

∂tΨ =

[Ψ, H

]. (5.22)

The mean field approach, originally suggested by Bogoliubov [?], assumes that thesystem consists of a large ground state population (the condensate) and a smallfluctuating population of higher modes (quantum depletion and thermal cloud).Thus the Bose field operator may be written as

Ψ(r, t) = Φ(r, t) + δ(r, t) (5.23)

Φ(r, t) = 〈Ψ(r, t)〉 is a complex number field or wavefunction describing the con-densate and the small field operator δ(r, t) describes the remaining modes. Theequation of motion for the condensate wavefunction is found by replacing theBose field operator Ψ with the mean field Φ in the Heisenberg equation of motion(eqn. 5.22) to give

ih∂

∂tΦ(r, t) =

(− h2∇2

2m+ Vext(r) + g|Φ(r, t)|2

)Φ(r, t). (5.24)

This is known as the time-dependent Gross-Pitaevskii (GP) equation [?, ?] and hasproved an excellent description of condensate behaviour provided that the systemis:

• Dilute with na3 ¿ 1. First order corrections to mean field theory are oforder (na3)1/2.

5.4. The ground state 75

• Below the critical temperature region around Tc, where the thermal fluctua-tions may be large.

5.4 The ground state

The ground state of the dilute Bose gas system is

Φg(r, t) = φg(r)eiµct/h (5.25)

where φg is the lowest energy eigenstate of the time-independent GP equation

(− h2

2m∇2 +

1

2m(ω2

xx2 + ω2

yy2 + ω2

zz2) + g |φg(r)|2

)φg(r) = µcφg(r) (5.26)

and is normalized to the number of atoms in the condensate,∫ |φg(r)|2 d3r =

N0. We have replaced Vext with the 3D harmonic trapping potential used in themajority BEC of experiments. The energy of the ground state is the chemicalpotential of the condensate µc = µc(Tc). (Recalling the quantum statistics ofsection 5.1, µc(T ) rises as the thermal cloud is cooled, arriving at the energy of theground state at Tc. µc cannot exceed this value or the ground state would have anegative population).

A small, low density condensate can be described in the ideal gas limit, whereinteractions are ignored and the wavefunction φg has the Gaussian profile of aharmonic oscillator ground state:

φg = φg0 e−x2i /2a2

i (5.27)

with a harmonic oscillator width ai =√

h/mωi. In a non-spherical trap useful fre-quency and length scales for the trap are the average harmonic oscillator frequency

ωho = (ωxωyωz)1/3 and length aho =

√h/mωho.

For an interacting gas, the wavefunction φg(r) depends on the ratio of thequantum kinetic energy term to the interaction energy term in the GP equation:

RTF = ng × 2ma2ho

h2 =Na

aho

(5.28)

For values of RTF À 1, the quantum kinetic energy term can be ignored in com-parison to the interaction term, an approximation known as the ‘Thomas-Fermilimit’. Under these conditions, the condensate density profile follows the inverseof the parabolic trapping potential for |φ|2 > 0.

n = |φ|2 =1

g(µc − 1

2m(ω2

xx2 + ω2

yy2 + ω2

zz2)) (5.29)


The widths of the condensate Ri, are determined by the positions on the x, y andz axes at which the density falls to zero:

Ri =

√2µc

m ω2i

(5.30)

Normalization leads to an expression for the chemical potential µc in the Thomas-Fermi limit:

µc =hωho

2

(15Na

aho

)2/5

(5.31)

The parameters used to describe the properties of a condensate are summarizedin table 5.1. For the experiments in this thesis condensates of between 10,000 and45,000 atoms were used and so the values in table 5.1 are based on the smallestcondensates that we make. Even for these small condensates the Thomas-FermiparameterRTF and so all our experiments are well described by the Thomas-Fermilimit.

5.5 The hydrodynamic equations

Two independent parameters are required to fully describe the condensate. Inthe GP equation, the amplitude and the phase of the wavefunction Φ(r, t) areused. However in many situations it is more convenient to use a different pair ofparameters, the number density n(r, t) and the velocity v(r, t):

n(r, t) = |Φ|2 (5.32)

v =h

2imn(Φ∗∇Φ− Φ∇Φ∗) (5.33)

If we write Φ in the form

Φ(r, t) =√

n(r, t) eiS(r,t) (5.34)

then the condensate velocity field v may also be expressed in terms of the phaseS of the GP wavefunction:

v(r, t) =h

m∇S(r, t) (5.35)

Substituting eqns. 5.32 and 5.33 for n and v into the GP equation, we find thatthe condensate can be equivalently described by a pair of coupled equations:

∂n

∂t+∇ · (nv) = 0 (5.36)

m∂v

∂t+∇

(Vext + gn− h2

2m√

n∇2√

n +mv2

2

)= 0. (5.37)

5.5. The hydrodynamic equations 77

Parameter Symbol Formula Typical value

TOP radial frequency ω⊥√

µ2m

B′q√BT

2π × 124 Hz

TOP axial frequency ωz

√8 µ

2m

B′q√BT

2π × 351 Hz

Average trap frequency ωho (ωxωyωz)1/3 2π × 175 Hz

Harmonic oscillator width aho

√h/mωho 0.81 µm

Trap deformation εtω2

z−ω2⊥

ω2z+ω2

⊥7/9

Chemical potential µchωho

2

(15Naaho

)2/5kB × 68nK

BEC radial half-width R⊥√

2µc

m ω2⊥

4.6 µm

BEC axial half-width Rz

√2µc

m ω2z

1.6 µm

Peak number density n0µc

g= µcm

4πh2a1.7× 1020 m−3

Diluteness parameter n0a3 µc

ga3 3.3× 10−5

Thomas-Fermi ratio RTFNaaho

72

Ideal gas critical T Tc 0.941 hωho

kB(N)1/3 169 nK

Thermal:trap energies kTc

hω⊥28

Healing length ξ (8πna)−1/2 2.0× 10−7m

Lowest Kelvin mode freq. ω1h

2m

(π

2Rz

)2ln

(0.888R⊥

ξ

)2π × 161 Hz

Table 5.1: Useful formulae for condensate parameters. The typical values are based ona condensate of 10,000 87Rb atoms, with the trap frequencies given in the table. The

fixed properties of an 87Rb atom are given in appendix A.


Equations 5.36 and 5.37 are known as the hydrodynamic equations for a super-fluid because they resemble the equations that govern irrotational flow in a classicalhydrodynamic fluid. The former is the continuity equation for particle number con-servation. The latter is a force equation which resembles Bernoulli’s equation forirrotational isentropic compressible flow in a hydrodynamic fluid [?, ?]. In the caseof a classical fluid, the hydrodynamic regime results when the mean free path ismuch less than the characteristic length scales of the system (Knudsen number ¿1). In the case of a condensate, the hydrodynamic form of the GP equation resultsfrom the existence of a macroscopic phase and not from its collisional properties.(λMFP /aho ≈ 20 at Tc and so the collisional properties at condensation are closerto those of a collisionless regime than the hydrodynamic one).

It is convenient to convert the GP equation into its hydrodynamic form forseveral reasons:

• The constraint of irrotational flow (∇ × v = 0) within a simply connectedcondensate is demonstrated by taking curl of eqn. 5.37 (remembering thecurl grad (scalar) = 0).

• It is a convenient form from which to investigate the collective excitationsof the condensate in the Thomas-Fermi limit. To find the T-F ground statewe simply dropped the kinetic energy term proportional to h2 in the GPequation. This approach is not possible for excited states because that termcontains both the quantum kinetic energy (negligible) and the kinetic energyassociated with the excitation (important). In the second hydrodynamicequation (eqn. 5.37), the quantum kinetic term (∝ h2) is separate and canbe dropped in the TF limit, without losing the kinetic energy of the excita-tion, mv2/2. This gives eqn. 5.38, which is used together with eqn. 5.36 toinvestigate the collective excitations of the condensate in the Thomas-Fermilimit.

m∂v

∂t+∇

(Vext + gn +

mv2

2

)= 0. (5.38)

5.6 Low-lying collective states

Much of the experimental information about the nature of a dilute gas BEC, includ-ing the work described in this thesis, comes from the spectroscopy of the excitedstates of the system. A close analogy can be drawn between the discrete excitedstates of a condensate and the well defined energy levels of an atom. Spectroscopicmeasurements of atomic energy levels have guided the development of the quan-tum mechanical model of the atom and likewise the spectroscopy of the condensateprovides crucial information about its nature. For example, the first experimentsfocused on low energy collective excitations both in a pure condensate close to ab-solute zero [?, ?] and at finite temperature [?, ?]. These latter experiments led to

5.6. Low-lying collective states 79

a better understanding of the interaction between the condensate and the thermalcloud [?]. The first observation of a transverse collective excitation is describedin chapter 7, which provided direct evidence for superfluidity in a condensate. Inlater experiments we made use of the flexibility of the geometry of the TOP trap tochange the energy spectrum of the condensate and observe resonant up and downconversion when one mode had exactly twice the frequency of another [?, ?, ?].In this section I will describe the geometry of the low-lying collective modes, thenomenclature used to label them and how the energy spectrum may be derivedtheoretically.

The collective mode frequencies for condensates in the Thomas-Fermi regimecan be calculated from the hydrodynamic equations for n and v (eqns. 5.36 and5.38 which ignore the quantum pressure term∝ h2). To linearize the hydrodynamicequations we assume that the system is mainly in the ground state, with a Thomas-Fermi density distribution n0, and has an infinitesimally small excitation of a higherstate. Thus we can write:

n = n0 + δn (5.39)

v = δv (5.40)

Inserting this form for n and v into eqns. 5.36 and 5.38 and considering only termsthat are first order in small quantities leads to the following wave equation fordensity fluctuations

m∂2δn

∂t2= ∇.(c2(r)∇δn) (5.41)

where c(r) is the local speed of sound and mc2(r) = µc − Vext(r).Whilst short wavelength oscillations propagate as sound waves, the low-lying

collective modes correspond to long wavelength oscillations, for which the finite sizeof the condensate cannot be ignored and results in a discrete excitation spectrum.For a spherical, harmonic trapping potential, solutions defined on the interval0 ≤ r ≤ R have the form

δn(r) = P 2nl (r/R) rl Ylm(θ, φ) (5.42)

where P 2nl (r/R) are polynomials of degree 2n containing only even powers. The

modes are labeled by n (the number of radial nodes), l (the total angular momen-tum), and m (the z component of the angular momentum).

The majority of experiments, including those in this thesis, are carried out intraps with cylindrical rather than spherical symmetry. In such axially symmetrictraps the azimuthal quantum number, m remains good and is used to describe themodes, but l is no longer a constant of motion. In this thesis we will only considerlow-lying modes in an axially symmetric trap that are related to l = 0, 1 and 2modes in a spherical trap. The term ‘surface modes’ is used to describe modeswith no radial nodes (n = 0).


Another way of categorizing the low-lying modes is in terms of dipole, quadrupole,octupole etc. This classification results from the operators which excite the modes:

Dipole operator :∑

i

ai xi (5.43)

Quadrupole operator :∑

ij

aij xixj (5.44)

Octupole operator :∑

ijk

aijk xixjxk (5.45)

This thesis will only consider dipole and quadrupole modes, although the moreenergetic octupole modes have also been excited e.g. in [?, ?]. The dipole modescorrespond to a rigid sloshing of the condensate or thermal cloud from side toside, at the trap oscillation frequency. In a 3D trap there are 3 independent dipolemodes in the x, y and z directions. They correspond to the three l = 1 modes inthe spherical harmonic nomenclature.

There are 6 independent quadrupole modes, related to the 6 different quadraticcombinations of x, y and z; x2, y2, z2, xy, yz and xz. The normal modes are linearsuperpositions of these pairs. Relating this Cartesian description of the modesto spherical harmonic nomenclature, the quadrupole modes correspond to linearsuperpositions of the five l = 2 modes plus the single l = 0 mode. There are2 3-dimensional width oscillations (breathing modes), 2 radial width oscillations(radial modes) and 2 modes which involve transverse motion relative to the trappotential with no change of the cloud shape (scissors modes). The geometry of the6 quadrupole modes plus their relation to the spherical harmonic nomenclature isgiven in fig. 5.2.

For the gyroscope experiment described in chapter 10, it is helpful to visualizethe last four modes of fig. 5.2 in the spherical harmonic basis, as the m = ± 2 andm = ± 1 modes rather than a Cartesian basis. The operators and geometries ofthese modes is given in fig. 5.3.

5.6.1 Mode frequencies

The m = ± 2 modes and the m = ± 1 modes are normal modes in both sphericaland axially symmetric condensates and have the form δn = r2 Y2m(θ, φ). (Super-positions of these degenerate pairs form the radial breathing and scissors modesrespectively, if we wish to use a Cartesian basis). Substituting these expressionsfor δn into eqn. 5.41 gives the following expressions for the frequencies of thesemodes:

ω2(l = 2,m = ± 2) = 2ω2⊥ (5.46)

ω2(l = 2,m = ± 1) = ω2⊥ + ω2

z (5.47)

The final 2 normal modes in an axially symmetric trap are linear combinationsof 2 non-degenerate spherical modes - the l = 0, m = 0 and l = 2, m = 0 modes -


Figure 5.2: A table showing the six independent quadrupole mode operators, their geom-etry and the related spherical harmonic modes. The diagrams indicate the equilibriumdistribution of the condensate (solid line) and the distributions at the extreme points of

the oscillation (dotted line).


Figure 5.3: A table showing the Cartesian operators and geometries of the m = ± 2 andm = ± 1 modes in an axially symmetric trap. If we use the quadrupole basis of fig. 5.2,then these modes are superpositions of the radial breathing modes and scissors modesrespectively. Note that the arrows in the bottom right hand box indicate rotation about

the z axis in opposite directions for the m = ± 1 modes.


and so are not represented by pure spherical harmonics. The frequencies may befound by linearizing the Castin-Dum equations [?]:

bi = ω2i (t) bi − ωi(0)2

bibxbybz

(5.48)

which describe the width oscillations of a harmonically trapped Thomas-Fermicondensate in terms of a time-dependent scaling factor bi(t), where i = x, y, z.Thus Ri(t) = bi(t)Ri(0). If we write

bi(t) = 1 + βi(t) (5.49)

and ignore all but the first order terms in βi (βi(t) ¿ 1) , then the 3 Castin-Dumequations may be written as

β(t) + M β(t) = 0 (5.50)

where β(t) = (βx, βy, βz) and

M =

3ω2⊥ ω2

⊥ ω2⊥

ω2⊥ 3ω2

⊥ ω2⊥

ω2z ω2

z 3ω2z

. (5.51)

The 3 eigenvalues of M give the eigenfrequencies of the 4 quadrupole modeswhich involve only width oscillations; the in-phase breathing mode, the anti-phasebreathing mode and the two degenerate radial breathing modes. The eigenvectorswill indicate the geometry of these four modes. Finally the frequencies of the low-lying normal modes of the condensate, in our axially symmetric trap (ωz/ω⊥ =√

8), are plotted in fig. 5.4.


Figure 5.4: The spectrum of low-lying, collective modes in an axially symmetric TOPtrap (ωz/ω⊥ =

√8).

Chapter 6

Bose-Einstein Condensation andSuperfluidity

6.1 Introduction

The experiments in this thesis all involve applying a torque to a BEC. The responseof the condensate to a torque is one of the signatures of its superfluid nature. Inthis section I will attempt to define superfluidity, describe its relationship to Bosecondensation (a famously knotty problem, which will only be discussed at a basiclevel) and explain the signatures of superfluidity theoretically.

Historically, the theory of superfluids has been developed to explain the re-markable transport properties of liquid 4He below its critical temperature (Tλ)of 2.17 K, a phase that is usually referred to as Helium II. The phenomena thathave been observed include frictionless flow below a critical velocity [?, ?], secondsound [?], irrotational flow (and hence a reduced moment of inertia) [?] and theformation of vortices with quantized circulation [?]. Whilst there are many differ-ent approaches for defining superfluidity, the following practical definition will beused: A superfluid system displays the same remarkable transport phenomena asHe II.

A complete theoretical description of these phenomena does not exist, becausethe strong interparticle interactions in liquid helium cannot be modeled accurately.The theoretical description of the superfluid properties offered below is clearlybased on the properties of the ‘condensate’ (the atoms in the lowest energy level),which is governed by a single macroscopic wavefunction. At first sight this theoryseems flawed - X-ray and neutron scattering data [?] indicates that at tempera-tures well below Tλ only ∼ 10% of the atoms are in the ‘condensate’, althoughseveral famous experiments have showed that almost the entire system behaves asa superfluid.

85

86 Chapter 6. Bose-Einstein Condensation and Superfluidity

Andronikashvili measured the fraction of the total fluid able to flow withoutfriction, as a function of temperature [?]. He suspended a stack of closely spacedmetal disks on a torsion wire submerged in liquid He. The spacing was so smallthat above Tc essentially all the fluid between the disks was dragged round asthe disks performed small angle rotational oscillations. Below Tc the oscillationfrequency increased sharply, since the superfluid fraction was no longer draggedround by the disks. The change in frequency indicated that as T → 0, the super-fluid fraction, ns/n → 1. Hess and Fairbank [?] measured the fraction of the fluidthat was constrained to an irrotational flow pattern below Tλ, by measuring theangular momentum transferred from slowly rotating liquid helium to its container,as the temperature is reduced below Tλ. The angular momentum transferred isproportional to the final superfluid fraction, which was measured to be between 70and 83%, depending on the final temperature. Finally, a value for the superfluidfraction can also be inferred from measurements of the velocity of second sound(anti-phase oscillations of the normal and superfluid fractions) [?].

In summary, these experiments show that whilst the wavefunction of the con-densate can be used to predict the superfluid properties of He II, the ‘superfluid’behaviour is exhibited by many atoms that are not part of the condensate (i.e.the superfluid fraction nS/n > the condensate fraction n0/n). This ‘extra’ super-fluid component, which can be as large as 90% in He II, may be associated withthe ‘quantum depletion’ in most simple situations. (The quantum depletion arethose atoms that are not in the single particle ground state at T=0. Population ofhigher states occurs in the presence of interactions; the ground state of the inter-acting condensed system is no longer the single particle ground state but containsa superposition of higher states). The normal fraction nN/n, which falls to zeroat T = 0 is made of thermally excited atoms. (Whilst the quantum depletion isdefinitely 100% superfluid in a simple trapping potential at T = 0, there are spe-cial situations in which its superfluidity is be reduced. One example is during thetransition from a perfectly superfluid condensate to a Mott insulator in an opticallattice at T = 0 [?]).

Whilst the quantum depletion is outside the ground state wavefunction, it hasbeen shown experimentally (see above) to obey the superfluid transport propertiespredicted by this wavefunction. This can be understood (at least in the case of aweakly interacting system with a small quantum depletion) in the following way.The quantum depletion results from scattering events ‘within’ the condensate pop-ulating higher momentum states. However such scattering events must conservemomentum and so the net momentum of the depletion is the same as that of thecondensate. Therefore the depletion follows the flow pattern of the condensateand displays the transport phenomena which define a superfluid. This argument,which justifies the use of the condensate wavefunction to predict the transportproperties of the whole superfluid, is reasonable in our dilute gas system wherethe quantum depletion is very small - ndep/n ∼

√na3 = 0.6% for the experimental

6.2. Dissipationless flow and critical velocity 87

conditions of this thesis [?]. In He II, with strong interactions and a 90% depletion,the microscopic picture is much more complicated - it is not even straightforwardto define a condensate wavefunction and so the above discussion can only at bestprovide a qualitative picture of the nature of the superfluid fraction.

Although the superfluid nature of dilute gas Bose condensates was always pre-dicted, it was several years after the first observation of BEC that direct experi-mental evidence for superfluidity was announced. All the superfluid properties atthe start of this section have now been observed - second sound [?], critical veloc-ity [?], irrotational flow (and hence a reduced moment of inertia) [?, ?] (chapters. 7and 8) and the formation of vortices with quantized circulation [?, ?] (chapter 9).This chapter begins with a brief discussion of the theory of dissipationless flowand a superfluid critical velocity since, as the name suggests, this phenomenon hasbeen central to the development of superfluid theory. It will then focus on how asuperfluid responds to an applied torque - theory which underlies the experimentsdescribed in chapters 7 to 10.

6.2 Dissipationless flow and critical velocity

The critical velocity of a superfluid system is the maximum speed at which aheavy body can move through the fluid without experiencing a drag force - abovethis speed superfluid flow breaks down and the system is heated. Thus althoughthe presence of a condensate is a necessary condition for superfluidity, it is notsufficient. The system must also possess a non-zero critical velocity, otherwise anyattempt to probe the superfluid system will also destroy it.

The critical velocity is determined by the E/p dispersion curve of the low-lyingexcited states of the system. The lowest velocity at which a moving body canexcite any of these states, whilst conserving energy and momentum, is given bythe Landau criterion [?, ?].

vL =

[E(p)

p

]

min

(6.1)

The minima of E(p)/p may be found where

E(p)

p=

dE(p)

dp(6.2)

The critical velocity in Helium II is believed to be determined by its phonon-roton spectrum, as suggested by Landau. The critical velocity of a dilute gas Bosecondensate was observed in [?, ?] when a blue detuned laser beam was movedthrough the condensate at different velocities. The onset of dissipation was markedboth by a distortion of the density distribution around the stirrer and by an increasein the thermal cloud (heating). The measured value of vL, ∼ 1/10 of the speed ofsound, is believed to be determined by nucleation of vortices. This discussion of


the critical velocity emphasizes that superfluidity is a collective effect, dependingon the collective excitation spectrum. A free particle excitation spectrum has acritical velocity of zero and so a non-interacting Bose-condensed gas cannot bedescribed as a superfluid.

6.3 The superfluid response to a torque

The response of a superfluid to an applied torque can be predicted by consideringthe wavefunction of the condensate, and then assuming that the whole superfluidwill follow the same flow pattern, as discussed in section 6.1. Whilst the sametheory may be applied to motion about any axis we will assume that the torque isapplied along the z axis in this chapter. The most general condensate wavefunctionhas the form:

Φ(r, t) =√

n(r, t) eiS(r,t) (6.3)

in which the velocity field is given by the gradient of the phase (eqn. 5.35):

v(r, t) =h

m∇S(r, t) (6.4)

Consider the circulation of the velocity field, κ

κ =∮

v.dl (6.5)

=h

m

∮∇S(r, t).dl (6.6)

=h

m× 2πq (6.7)

where q is any integer. For the wavefunction to be single valued, the phase changearound any closed loop within the superfluid must be an integer multiple of 2π andso the circulation is quantized into units of h/m. Thus superfluid flow patternsfall into two regimes:

• κ = 0. This is known as irrotational flow and is the only flow pattern possiblein a simply connected superfluid. From Stokes theorem, an irrotational flowpattern is also one in which curl v = 0 throughout the entire fluid. This isconsistent with eqn. 6.4 where we express the superfluid velocity field as thegradient of the scalar condensate phase S(r, t).

• κ = 2πq, with q > 0. If the circulation is non zero, then vortices exist withinthe superfluid. These are lines of zero density associated with quanta ofcirculation.

6.4. Irrotational flow and the reduced moment of inertia 89

Figure 6.1: Diagrams of two different velocity fields (viewed instantaneously in the labframe) for a fluid in a very slowly rotating container. (a) shows a rotational or rigid bodyflow pattern, v ∝ Ω× r. (b) shows an irrotational or superfluid flow pattern, v ∝ ∇(xy)

The situation described above is closely related to the behaviour of an extreme type2 superconductor. The exclusion of magnetic flux from a superconductor in a weakapplied magnetic field (< Hc1) is known as the Meissner effect [?], and is analogousto the exclusion of circulation from a superfluid in a weak applied rotational field.In the former ∇×A = 0 inside the superconductor, whilst in the latter ∇× v = 0inside the superfluid. In stronger applied fields, both systems minimize their energyby allowing quanta of magnetic flux/circulation to penetrate the sample in localizedregions (∼ healing length) known as flux lines/vortex lines. Superconductivitybreaks down when the applied field exceeds Hc2 and this corresponds to the rotationrate at which vortex cores overlap and superfluid flow ceases. The next two sectionsdiscuss the irrotational and vortex regimes in more detail.

6.4 Irrotational flow and the reduced moment of

inertia

In this section we will consider a superfluid with zero circulation, which will beenergetically favorable when the container confining the superfluid is rotating veryslowly (i.e. a weak applied rotational field). The velocity field within the conden-sate is irrotational and may be represented as the gradient of the scalar condensatephase S(r, t). Figures 6.1(a) and (b) show a fluid at equilibrium with a slowlyrotating ellipsoidal potential. In both cases the ‘shape’ of the fluid follows thepotential but the flow pattern within the fluid is very different in each case. In(a) the flow pattern is rotational, like that of a rigid body and the circulation can


take a continuous range of values. In (b) the flow pattern is irrotational and thecirculation is zero. The angular momentum associated with each flow pattern isclearly different, although the density distribution of both fluids is identical andis ‘rotating’ at the same frequency Ω. In (a) the whole velocity field contributesconstructively towards the total angular momentum Lz, whilst in (b) Lz is smaller,because contributions from different parts of the fluid cancel out. Note that in acylindrical potential, the symmetry would produce perfect cancellation and hencezero angular momentum. Conversely in an extremely elliptical potential, the an-gular momentum of the irrotational flow pattern tends to that of the rotationalflow pattern.

Different moments of inertia may be used to account for the different angularmomenta associated with each flow pattern. In its most general form, the momentof inertia of a fluid Θ represents the linear response to a rotational field -ΩJz andis given by:

Θ = limΩ→0

〈Jz〉Ω

(6.8)

The rotational flow pattern has the familiar rigid body moment of inertia forN particles of mass m:

Θrig = Nm〈x2 + y2〉 (6.9)

The irrotational flow pattern has a reduced moment of inertia given by

ΘS = ε2cΘrig (6.10)

where ε2c ≤ 1. ΘS is often referred to as the ‘superfluid’ moment of inertia, since

a superfluid may only flow in an irrotational manner (assuming no vortices arepresent).

ε2c is related to the elliptical shape of the fluid in the plane of rotation and

can be evaluated using the hydrodynamic equations of superfluids 5.36 and 5.38as discussed in [?]. Solutions of these equations that are stationary in the rotatingframe, have a velocity field of the form:

v = Ω〈x2 − y2〉〈x2 + y2〉∇(xy) (6.11)

Then using the definition of Θ given in eqn. 6.8,

ΘS = limΩ→0

m〈r× v〉Ω

(6.12)

=

[〈x2 − y2〉〈x2 + y2〉

]2

Θrig (6.13)

= ε2cΘrig (6.14)

6.4. Irrotational flow and the reduced moment of inertia 91

Whilst εc describes the deformation of the condensate, εt describes the deformationof the trapping potential in the plane of rotation:

εt =ω2

x − ω2y

ω2x + ω2

y

(6.15)

For a Thomas-Fermi condensate at equilibrium with a harmonic trapping poten-tial, one can show that εc = εt. Note that εc = 0 in an axially symmetric conden-sate. Thus in the absence of vortices, a condensate of circular cross-section cannotpossess any angular momentum. This is particularly relevant to the experimentsdescribed in chapter 8.

In [?] it is demonstrated that the same reduced moment of inertia will also bedisplayed in small angle oscillations of the condensate relative to an elliptical trappotential. These small angle oscillations are the scissors modes. The xz and yzscissors modes are described in section 5.6 for an axially symmetric trap. If thetrap is also elliptical in the xy plane, then a third, xy scissors mode will exist. Thexy scissors modes is excited by the relevant quadrupole operator

Q =N∑

i=1

xiyi (6.16)

In [?], the relationship between the moment of inertia Θ and the imaginary quadrupoleresponse function χ′′Q is derived :

ΘS

Θrig

=(ω2

x − ω2y

)2∫

dωχ′′Q(ω)/ω3

∫dωχ′′Q(ω)ω

(6.17)

whereχ′′Q(ω) =

π

Z

∑n,m

[e−βωn − e−βωm

]|〈m|Q|n〉|2δ(ω − ωmn) (6.18)

and n,m are the different collective modes of the condensate and β = 1/kT . Inthe limit of a small selective excitation of the xy-scissors mode in a Thomas-Fermicondensate, with frequency ωsc = (ω2

x + ω2y)

1/2, then eqn. 6.17 reduces to

ΘS

Θrig

=

(ω2

x − ω2y

)2

(ω2

x + ω2y

)2 = ε2t (6.19)

Equations 6.15 and 6.19 shows that the same reduced moment of inertia governsboth the complete rotation of the cloud and the scissors mode oscillations. The

condensate will oscillate at a higher frequency (√

k/ΘS) than a fluid of the samemass distribution undergoing small angle oscillations with a rotational flow pattern

at frequency√

k/Θrig. This is the basis of the scissors mode experiment describedin chapter 7. Since it is straight forward to measure the frequency of an angularoscillation, the scissors mode is used to measure the moment of inertia as a functionof temperature in [?]. It may also provide a method for observing the superfluidBCS transition in a degenerate Fermi gas [?].


6.5 Vortex theory

If the circulation within the condensate is non-zero, then vortices are present.Vortices are lines of zero density, perpendicular to the plane of rotation, aroundwhich the phase changes by integer multiples of 2π and the circulation is quantizedinto units of h/m, as show in section 6.3. The phase is totally undefined at thecore of the vortex (consider taking a closed loop around which the phase changesby 2π and shrinking it in towards the vortex core at the centre) and hence thedensity must fall to zero at this point, for the wavefunction to remain well defined.

6.5.1 Core size

The radial size of the vortex core is of the order of the healing length ξ; this is theminimum distance over which the order parameter may heal or alternatively, theminimum distance over which the condensate density may change significantly.Consider the condensate density growing from 0 to n over a distance ξ. Thiscorresponds to a quantum kinetic energy per particle of ∼ h2/2mξ2. Equating thisto the interaction energy per particle gn gives an expression for ξ:

gn =h2

2mξ2(6.20)

where g = 4πh2a/m. Rearranging eqn. 6.20 to find an expression for the healinglength gives

ξ =1√

8πna(6.21)

Note that the vortex core shrinks as the interaction energy increases. In HeliumII, which is strongly interacting, ξ is of the order of an angstrom and vorticescannot be imaged directly. The arrangement of vortex lines in a lattice has beenobserved by trapping electrons on the cores and then drawing them off each coreand accelerating them to a point on a phosphor screen [?]. In contrast, the vortexcores in our dilute system may be imaged optically. Under the conditions we useto nucleate vortices, ξ has a typical value of 0.3 µm in the trap. During a typicalfree expansion time of 12 ms, the vortex core increases by a factor of ∼ 10, toa radius of 3 µm, just above the resolution limit for our imaging system. Thisexpansion behaviour has been calculated from the theory of [?] and is plotted infig. 6.2. The expansion has two regimes; initially the core size adjusts rapidly tothe decreasing density and the core expands faster than the condensate. At longerexpansion times, the potential energy becomes negligible and the cloud expands asfree particles, with ξ/R⊥ constant. Typical images of expanded vortex cores maybe seen in fig. 9.1.

6.5. Vortex theory 93

Figure 6.2: The vortex core radius ξ (dotted line), condensate radius R⊥ (dashed line)and the ratio of the two (solid line) as a function of expansion time. The vortex coreradius is assumed to be the same as the healing length. The plot was calculated for theexperimental conditions of chapter 9 (ω⊥/2π = 62Hz, N = 15, 000) from the theory of

[?]


6.5.2 Vortex energetics and metastability

In section 6.3, we described two types of flow for a condensate in equilibriumwith a rotating potential. At low rotation rates, the condensate remains in theground state and follows the potential with an irrotational flow pattern. At highertrap rotation rates, formation of the first vortex state is energetically favorable.To determine the conditions under which formation of the first vortex state isfavorable, we must consider the energy of the condensate in the rotating frame.For simplicity we will consider an axially symmetric system, in which Lz is a goodquantum number and may be used to label the states.

In the rotating frame, the time-independent Gross-Pitaevskii equation (eqn. 5.26)gains an additional term −ΩLz, which lowers the energy (E ′

i) of those states withnon-zero angular momentum:(− h2

2m∇2 +

1

2m(ω2

xx2 + ω2

yy2 + ω2

zz2) + g |φi(r)|2 − ΩLz

)φi(r) = E ′

i φi(r) (6.22)

The energy of the ground state in the rotating frame E ′0, is the same as that

in the lab frame E0, since Lz = 0. The first vortex state, with a single centredvortex line, has Lz = Nh. It has additional kinetic energy due to the circulatingvelocity field (E1 > E0) and so at low trap rotation rates, the vortex-free state isenergetically favorable. The ‘thermodynamic’ critical angular velocity Ωth occurswhen the ground state and the first vortex state have the same energy in therotating frame.

Ωth =E ′

1 − E0

Nh(6.23)

An expression for Ωth may be evaluated in the T-F limit, assuming an approximatedensity profile for the vortex state, [?]:

Ωth ≈ 5ω⊥2

a2⊥

R2⊥

ln

(0.67R⊥

ξ

)(6.24)

In a tri-axial trap, the calculation of Ωth is more complicated because theirrotational flow pattern contains some angular momentum and Lz is no longer agood quantum number. An approximate expression for the critical velocity in atri-axial trap is also given in [?], but since we use very small eccentricities in thexy plane for our vortex experiments, the expression of eqn. 6.24 is adequate.

Evaluating the thermodynamic critical rotation rate for the trap used in chap-ter 9 (ω⊥/2π = 62 Hz) gives Ωth = 0.14 ω⊥. In practice much higher rotationrates are required to nucleate a vortex because an energy barrier exists betweenthe ground state and the first excited state with a single centred vortex, makingboth states metastable. The origin of the energy barrier is topological; a 2π phasewinding, centred on a point of undefined phase, cannot suddenly appear in thecentre of a simply-connected condensate wavefunction. It may only be created at


the edge of the condensate where the density → 0 and then travel to the centre.Similarly, once a vortex exists at the centre of the condensate it may only be de-stroyed by travelling to the edge. Two factors affect the energy of the system asthe vortex moves from the centre to the edge. First the kinetic energy associatedwith the vortex core decreases, as the density of the surrounding condensate fallsto zero. Secondly the angular momentum associated with the vortex decreases (asdescribed in section 6.5.3) causing the energy of the excited state in the rotatingframe E ′

1 = E0 − ΩLz to rise. The sum of these two factors produces a maximumin the energy of the system in the rotating frame (i.e. an energy barrier) when thevortex is part-way between the centre and the edge of the condensate.

A plot of the excited state energy E ′1 versus vortex position (d) is given in

[?] for an axially symmetric trap and is shown in fig. 6.3. At low rotation ratesthe first vortex state is energetically unstable (a). As Ω increases it first becomesmetastable (b) and then globally stable (c) as the vortex state becomes the lowestenergy state in the rotating frame. The energy barrier reduces as Ω increasesbeyond Ωth (d).

A vortex state in a pure (superfluid) condensate will persist in a static trap,even though it is energetically unstable, because there is no mechanism by whichthe excess energy may be dissipated. The presence of some thermal cloud willeventually cause the vortex to spiral out to the edge and disappear, but undertypical experimental conditions and a temperature of 0.5 Tc the timescale for thisprocess exceeds the lifetime of our condensate (section 10.11).

6.5.3 Quantization of angular momentum

Quantization of angular momentum is often discussed in relation to vortices in asuperfluid. However, so far I have been careful only to discuss quantization of cir-culation. In an infinite uniform superfluid, quantization of circulation immediatelyimplies quantization of angular momentum. However in a non-infinite superfluid,the angular momentum associated with a vortex line is only quantized into unitsof Nh when the vortex is centred; as it moves towards the boundary of the system〈Lz〉 falls smoothly to zero. Our trapped dilute gas condensate cannot be approx-imated to an infinite system since vortices are regularly found in all positions,from the centre to the edge. The relationship between 〈Lz〉 and vortex position isdetermined by the density profile of the condensate:

〈Lz〉 =∫ ∫ ∫

ρ(r) [r× v].z d3r (6.25)

We will consider an axially symmetric Thomas-Fermi density profile, and so thevortex position may be specified by its scalar distance from the condensate axis,d.

ρ(r) = ρ0

(1− z2

R2z

− r2

R2⊥

)for ρ > 0; ρ = 0 elsewhere (6.26)


Figure 6.3: (Taken from [?]). The energy of the first vortex state (relative to the groundstate) in the rotating frame ∆E′ = E′

1 − E0, as a function of the vortex displacementd from the symmetry axis of the cylindrical trap. ∆E′ is normalized by its value for acentred vortex in a stationary trap. The curves correspond to different trap rotation fre-quencies (a) Ω = 0, (b) Ω = Ωm, the rate at which a centred vortex becomes metastable.(c) Ω = Ωth the thermodynamic critical frequency above which the first vortex state hasthe lowest energy, (d) Ω = 1.5Ωth. Curves (b) to (d) show that an energy barrier existsfor a vortex entering or leaving the condensate even when a state with a centred vortex

is locally or globally stable.

The small vortex core modifies the density profile in a very limited region of thecondensate. Since 〈Lz〉 is a weighted spatial average over the entire condensate weonly introduce a negligible error if we use the vortex free density profile to calculate〈Lz〉. Inserting ρ(r) into eqn. 6.25:

〈Lz〉 =∫ ∫

dz d2r ρ0

(1− z2

R2z

− r2

R2⊥

) ∫ 2π

0rvφ dφ (6.27)

We recognize the last integral in eqn. 6.27 as the circulation around a loop of radiusr, which we know to be quantized from section 6.3. If the loop does not enclosethe vortex (r < d), the phase change and hence circulation around the loop willbe zero. This explains why an off-centre vortex is associated with reduced angularmomentum - the area inside the vortex position does not contribute to the total


angular momentum integral. If the loop does enclose the vortex (r > d) then thevalue of the integral will be quantized into units of h/m, as given in eqn. 6.7. Thusafter performing the integral over φ, eqn. 6.27 may be written as:

〈Lz〉 = ρ0hq

m

∫ R⊥

drdr

∫ Rz(r)

−Rz(r)

(1− z2

R2z

− r2

R2⊥

)dz (6.28)

where q is an integer and the limits on the z integral Rz(r) = Rz(0)√

1− r2/R2⊥ are

determined by the ellipsoidal surface on which the condensate density distributionfalls to zero. The lower limit on the radial integral is d, because we know thatthe region within the vortex position does not contribute to the total angularmomentum. Evaluating the integral first over z and then r gives:

〈Lz〉 = ρ0hq

m

4Rz

3

R2⊥5

(1− d2

R2⊥

)5/2

(6.29)

= hNq

(1− d2

R2⊥

)5/2

(6.30)

where the number normalization condition,

N =∫ ∫ ∫

ρ(r) d3r (6.31)

has been used in eqn. 6.30.This calculation explains the results of [?], in which the angular momentum

of a condensate is observed to increase linearly with the trap rotation rate, afterthe formation of the first vortex. If angular momentum, as well as circulation,was quantized in a finite sized condensate, then we would observe the angularmomentum increasing in discreet steps of Nh.

6.5.4 Kelvin waves

Kelvin waves are helical excitations of the vortex core. Each kelvon or quanta ofexcitation has an energy hω, angular momentum h and linear momentum alongthe axis of the vortex ±hk. These properties can be derived by considering thewave equation for a vortex line, which can be modeled as a string. We will considerthe spectrum of Kelvin modes on a straight vortex with one unit of circulation,along the axis of a cylindrically symmetric, harmonically trapped, Thomas-Fermicondensate.

Two forces act on the vortex line, as indicated in fig. 6.4:

• The restoring force, FR

The vortex line has an energy per unit length, associated with the kineticenergy of its flow pattern. Thus it is effectively under tension, because any


Figure 6.4: The forces on a vortex line that result in helical Kelvin wave motion. FR isa restoring force and FM is the Magnus force. The circulation vector κ represents the

circulating velocity field around the vortex core.


increase in length causes an increase in energy. The tension at position zalong the vortex line is given by [?]:

T (z) = πn(0, z)h2

mln

(0.888R⊥

ξ

)(6.32)

where n(0, z) is the number density on axis at position z. Thus the restoringforce FR(z) is given by:

FR(z) = −T (z)d2η(z)

dz2(6.33)

where η(z) is the radial displacement of a the core from the axis.

• The Magnus force, FM

If a vortex line moves with velocity v through a superfluid with unperturbednumber density n, then it experiences a Magnus force perpendicular to thedirection of motion and the vortex axis

FM(z) = n(0, z)m κ× v (6.34)

where κ is the circulation vector, which has a magnitude of h/m for a singlyquantized vortex. We assume that the amplitude of the oscillation is smallso that n ≈ n(0, z). The superposition of the linear velocity field and thecirculating vortex velocity field produces a net flow rate that is greater onone side of the vortex than the other. From Bernoulli’s equation, for steadyflow in an incompressible superfluid (which can be derived from eqn. 5.38),this results in a pressure imbalance on either side of the vortex and hence atransverse force.

The vortex line has zero mass and so the equation of motion contains only thesetwo forces. Resolving the equation of motion into components gives the followingpair of coupled equations for Kelvin waves on an axial vortex line with one unit ofcirculation:

T (z)d2ηx

dz2= −hn(0, z)

dηy

dt(6.35)

T (z)d2ηy

dz2= +hn(0, z)

dηy

dt(6.36)

These coupled equations of motion are satisfied by solutions of the form:

ηx(z, t) = Aei(±kz−ωt) (6.37)

ηy(z, t) = −iAei(±kz−ωt) (6.38)


representing helical waves traveling in either direction along the z-axis, but alwaysrotating around the axis in the opposite direction to the vortex flow field. Thus ifthe vortex line has angular momentum h, each kelvon has angular momentum −h.One can see from the wave equation that we only expect Kelvin wave solutionswith one sense of rotation. It is first order in t and so permits only one angularfrequency solution. Waves with opposite senses of rotation correspond to solutionswith angular frequency ±ω.

The dispersion relation is:

T (z)k2 = hn(0, z) ω(k) (6.39)

Substituting for T (z) from eqn. 6.32 gives

ω(k) =hk2

2mln

(0.888R⊥

ξ

)(6.40)

Note that both FR and FM depend linearly on n(0, z) and so the z dependencecancels out in eqns. 6.35 and 6.36. Thus the dispersion relationship of eqn. 6.40is exact (in the limit of small amplitude oscillations), even for a condensate in a3-dimensional harmonic trapping potential.

With the application of suitable approximate boundary conditions, the spec-trum of Kelvin mode energies may now be estimated. The vortex lines are notconstrained or pinned at the boundaries and so these points are antinodes in thevortex wave (analogous to the open end of an organ pipe rather than the closedone). This boundary condition requires that the integer multiples of the Kelvinwave length fit into 4Rz or

k =2πp

4Rz

(6.41)

and so the mode spectrum as a function of integer p is:

ω(p) =h

2m

(πp

2Rz

)2

ln

(0.888R⊥

ξ

)(6.42)

In our trap (ωz/2π = 175 Hz), the frequencies of the 3 lowest Kelvin modes are 0.44ωz, 1.8 ωz, 4.0 ωz. Note that the frequency and spacing of these modes is compara-ble to the trap frequencies and hence the collective mode frequencies. Thus in ourgeometry it may be possible to observe the transfer of energy between an individualcollective mode and an individual Kelvin mode (appendix 11.2.2). For comparison,the situation is very different in the elongated trap used in Paris [?] (ωz/ω⊥ = 0.05),where some investigation of Kelvin modes is already underway [?] (section 10.6).The lowest Kelvin mode has frequency 0.005 ωz, which corresponds < 1/5000 ofthe lowest collective mode frequencies (excluding the anti-phase breathing mode,with frequency of 0.7 ωz) and 2×10−5 kBTc/h. In this case, there will be significantthermal excitation of many Kelvin modes and it will be more difficult to observeresonant coupling to an individual Kelvin mode. Further investigation of Kelvinmodes in both geometries would be of great interest.

Chapter 7

The Scissors Mode Experiment

7.1 Introduction

This chapter describes the scissors mode experiment, which was initially discussedin a theoretical paper by Guery-Odelin and Stringari [?] and which provided someof the first direct evidence that a dilute gas Bose-Einstein condensate behaves asa superfluid. After the first condensate was made in 1995, experimental work fo-cused on verifying the predictions of mean field theory. The excellent agreementof the measured collective oscillation frequencies with the predictions of the Gross-Pitaevskii equation (eqn. 5.24) and the observation of matter-wave interferencebetween two condensates [?, ?] supported the mean-field theory assumption thatthe condensate is described by a single macroscopic order-parameter or wavefunc-tion. However, the existence of a macroscopic wavefunction is a necessary butnot sufficient condition of superfluidity. Three further experiments provided directproof that the condensate behaved as a superfluid: the measurement of a criticalvelocity for superfluid flow [?, ?]; the observation of quantized vortices [?, ?]; andfinally the scissors mode experiment which demonstrated that the condensate hasa purely irrotational flow pattern and measured its reduced superfluid moment ofinertia [?].

The name ‘scissors mode’ originated in nuclear physics. It describes the smallangle oscillation of superfluid neutron and proton clouds relative to each otherwithin deformed nuclei [?, ?, ?]. In a trapped dilute-gas Bose condensate, thescissors mode is a small angle oscillation of the cloud relative to the trap potential,that can be excited in any plane in which the contours of the trapping potentialare elliptical. In an axially symmetric condensate the xz and yz scissors modes are2 of the 6 quadrupole modes described in section 5.6. In this experiment we usethe xz scissors mode, excited by a sudden rotation of the trap in the xz plane as

101

102 Chapter 7. The Scissors Mode Experiment

shown in fig. 7.2. The excitation amplitude must be small, so that the cloud is notdeformed and the density distribution performs a small angle oscillation.

Whilst the scissors mode has a purely irrotational flow pattern, small angleoscillations in a normal fluid may also result from a rotational flow pattern (fig. 6.1).The type of flow pattern can be identified from the oscillation frequency and thedensity distribution of the cloud. A rotational flow pattern is associated with therigid-body moment of inertia Θrig:

Θrig = Nm〈x2 + z2〉 (7.1)

and an oscillation frequency of√

k/Θrig. As explained in section 6.4 an irrotationalflow pattern is associated with a reduced moment of inertia ΘS. In the case of aThomas-Fermi condensate in equilibrium with a harmonic trapping potential ΘS

is given by:

ΘS =(ω2

z − ω2x)

2

(ω2z + ω2

x)2 Θrig = ε2

t Θrig. (7.2)

Since ΘS < Θrig the oscillation frequency√

k/ΘS is higher than that for rigid-body

motion. (For a more detailed discussion see section 6.4).The aim of this experiment is to show that the condensate displays a purely

irrotational flow pattern, under excitation conditions where a normal fluid displaysboth a rotational and an irrotational flow pattern. The lack of rotational flow, thatresults from the existence of a macroscopic wavefunction, is one of the signaturesof a superfluid (as described in section 6.3). First we excite small angle oscillationsin a non-condensed thermal cloud, by a sudden rotation of the trapping potentialin the xz plane. Under these particular initial conditions both rotational andirrotational flow patterns are simultaneously excited with comparable amplitudesand thus we observe oscillations at two different frequencies. Then we excite acondensate under exactly the same conditions. If the condensate behaves as asuperfluid, then the low frequency rotational oscillation should be suppressed andonly a single oscillation frequency will be observed. Finally, we must calculate therotational and irrotational oscillation frequencies of a non-condensed cloud, at thetemperature and density of the condensate. We must check that both are excitedwith significant amplitude in this colder, denser system, so that if only a singlefrequency is observed in the condensate, then it can only be due to superfluidity.

7.2 Theory

7.2.1 The scissors mode oscillation of the condensate

There are several complementary ways of finding the oscillation frequency of thescissors mode of the condensate [?], however the following method gives some useful

7.2. Theory 103

physical insight into the nature of the mode. We find expressions for the densityand velocity distribution of the condensate during the scissors mode, as a functionof coupled parameters θ(t) and β(t). When substituted into the hydrodynamicequations, these yield oscillatory solutions at frequency ωsc.

Consider the density distribution (with respect to the trap axes) of a T-Fcondensate after a sudden rotation of the trap through an angle θ.

n(r, t) =µc

g− m

2g[(ω2

x cos2 θ + ω2z sin2 θ)x2 + ω2

yy2 +

(ω2x sin2 θ + ω2

z cos2 θ)z2 + 2(ω2z − ω2

x) cos θ sin θ xz]. (7.3)

If we now assume that θ ¿ 1, the change in density is proportional to xz:

n(r, t) =µc

g− m

2g[ω2

xx2 + ω2

yy2 + ω2

zz2 + 4ω2εtθ(t)xz] (7.4)

where

ω =

√ω2

x + ω2z

2(7.5)

Hence the subsequent small angle motion may be identified with the pure xz scissorsmode, which has quadrupole operator xz, (section 5.6). If a larger excitation angleis used, then the coefficients of x2

i in eqn. 7.3 can no longer be considered constantand breathing modes are also excited. (Note that the maximum amplitude forpure scissors oscillation is not related to the deformation of the trap εt).

The velocity distribution may be constructed from three constraints on thecondensate flow pattern:

• The velocity field is purely irrotational and may be expressed as the gradientof the condensate phase S(r, t):

v =h

m∇S(r, t) (7.6)

This constraint ensures that the condensate only displays one oscillationfrequency rather than two.

• During the small angle scissors mode, the cloud shape does not deform, butonly rotate. Thus the velocity field must satisfy:

∇.v = 0 (7.7)

• The torque applied by the trap is in the xz plane only, so there is no motionin the y direction.


These three constraints determine that the velocity field has the form:

v = ∇ (β(t)xz) (7.8)

Substituting eqns. 7.4 and 7.8 for n and v into the hydrodynamic equations(eqns. 5.36 and 5.38) gives coupled equations for θ(t) and β(t), which may besolved, using the correct initial conditions, to give:

θ(t) = θ0 cos(ωsc t)β(t) = εt θ0 ωsc sin(ωsc t)

(7.9)

andωsc =

√2 ω =

√ω2

x + ω2z . (7.10)

In our axially symmetric TOP trap, ωsc = 3ω⊥.An alternative method for calculating the scissors frequency is given in sec-

tion 5.6, by considering the scissors mode as a superposition of the 2 degeneratel = 2,m = ±1 modes. Hence its frequency, may be determined by substitutingthe l = 2,m = ±1 wavefunctions into the linearized hydrodynamic wave equation(eqn. 5.41).

7.2.2 Oscillation frequencies of the thermal cloud

In this section I will summarize the method used to investigate the ‘scissors-like’oscillations of an uncondensed thermal cloud, for further details see [?, ?]. Priorto reaching an equilibrium state (described by the steady state Boltzmann dis-tribution function) any thermodynamic system (e.g. our trapped dilute gas) willbe described by the Boltzmann equation [?]. The method of averages [?] is thenused to extract coupled equations for the averages of useful observables from theBoltzmann equations. In this experiment, we are interested in a set of 4 coupledequations involving those observables (or momenta) which couple motion in the xand z directions 〈xz〉, 〈xvz + zvx〉, 〈vxvz〉, 〈xvz − zvx〉. In particular, we know thatthe quadrupole moment 〈xz〉 is proportional to the tilting angle of the cloud in thexz plane θ(t), for θ(t) ¿ 1. If we parameterize the Gaussian density distributionof the cloud as:

ρ(r, t) = exp[−

(Vext(r)− 2mθ(t)εtω

2xy)/kBT

](7.11)

then the following equation for small angle oscillations may be obtained from thecoupled equations for xz motion:

d4θ

dt4+ 4ω2d2θ

dt2+ 4ε2

t ω4θ

︸︷︷︸collisionless

+1

τ

(d3θ

dt3+ 2ω2dθ

dt

)

︸︷︷︸hydrodynamic

= 0 (7.12)

7.2. Theory 105

The relaxation time τ is given by

τ =5

4γcoll

=5

4

2

n(0)vthσ(7.13)

where vth = (8kBT/πm)1/2 and σ = 8πa2.Equation 7.12 is divided into two parts. The first dominates in the collision-

less regime τωho À 1, where a particle oscillates many times in the trap betweencollisions. The second dominates in the collisional hydrodynamic regime (not tobe confused with the apparent hydrodynamic behaviour which results from super-fluidity), where many collisions occur per trap period. The relevant parameteris τωho = 75 for the thermal cloud used in this experiment (n0 = 2 × 1018 m−3,T = 1 µK). Thus to describe the thermal cloud oscillations, the hydrodynamic partof eqn. 7.12 may be ignored and the collisionless part solved to find two undampedoscillations at frequencies ω± = |ωz ± ωx|. The lower frequency corresponds toa rotational (rigid body) flow pattern, whilst the higher frequency corresponds toan irrotational flow pattern. The sudden rotation of the trap from equilibriumcorresponds to the following initial conditions: θ(0) = θ0, θ′′(0) = −2ω2θ(0) andθ′(0) = θ′′′(0) = 0, which excites both frequencies with approximately equal ampli-tudes. The accuracy of the collisionless approximation is confirmed by the Monte-Carlo simulation of the thermal cloud oscillation in fig. 7.1(a) and (b), which hasbeen calculated from eqn. 7.12 without approximation for our experimental con-ditions. (a) shows the undamped double frequency angle oscillation as a functionof time, whilst the Fourier transform shows two peaks of almost equal amplitude,in excellent agreement with the predicted collisionless frequencies.

If we now consider the solution of eqn. 7.12 in the hydrodynamic limit, we finda possible source of ambiguity in our experiment. A non-condensed hydrodynamicfluid oscillates at a single frequency,

√2 ω, the same frequency as the condensate.

This single frequency results from the collisional properties of a hydrodynamic nor-mal fluid, whilst the single condensate frequency results from superfluid constraintson the flow pattern.

To overcome this ambiguity we must show that a normal fluid, at the sametemperature and density as the condensate, is not in the hydrodynamic regimeand thus will show two clear oscillation frequencies. The value of τωho for sucha cloud (n0 = 2 × 1020 m−3, T = 90 nK) would be 2.5, indicating that it is inan intermediate regime and thus will show two clear oscillation frequencies, butsignificant damping effects may also be observed. This conclusion is confirmed bythe Monte-Carlo simulation of such a system (from eqn. 7.12) in fig. 7.1(c) and (d).In (c) a double frequency oscillation is observed with a damping time of ∼ 15 ms.The Fourier transform shows that both frequencies are significantly shifted fromthe collisionless values and the lower, rotational oscillation has a slightly reducedamplitude. Thus we can conclude that if the condensate behaved as a normal fluid,its oscillation would be characterized by:


Figure 7.1:Direct Monte-Carlo simulations of the ‘scissors’ type oscillation in a

non-condensed cloud.

• (a) shows the oscillation of a thermal cloud, close to the conditions used inthe experiment, with T = 1 µK and n0 = 2× 1012 cm−3. The response signalis an undamped two frequency oscillation.

• (b) The Fourier spectrum of the angular response shown in (a). The fre-quency of the peaks agrees with the collisionless prediction (dotted lines).The height of the peaks is the same, indicating that the energy is sharedequally between the two modes.

• (c) shows the oscillation of a thermal cloud, with approximately the sametemperature and density as the condensate, T = 90 nK and density n0 =2×1014 cm−3. Both frequency components are still present and the dampingtime is about 15 ms.

• (d) The Fourier spectrum of the angular response shown in (b). The fre-quencies are shifted from the collisionless prediction (dotted lines) and theheight of the low frequency component is significantly reduced, showing thatthe hydrodynamic regime is approached.

7.3. Experimental procedure 107

• Two clear oscillation frequencies with comparable but not equal amplitude

• A rapid damping of the oscillation over ∼ 15 ms.

7.3 Experimental procedure

The trap used for the scissors mode experiment was a modified TOP trap [?],consisting of a static quadrupole field and a bias field oscillating at ω0 = 7 kHz.The bias field is given by:

B (t) = BT (cos ω0t x + sin ω0t y) + Bz cos ω0t z. (7.14)

The term Bz (t) = Bz cos ω0t z is additional to the usual field of amplitude BT

rotating in the xy plane. The effect of the additional term Bz (t) is to tilt theplane of the locus of B = 0 by an angle ξ = tan−1(Bz/BT ) with respect to thexy plane. This causes the symmetry axes of the time-averaged potential to rotatethrough an angle φ ≈ 2

7ξ in the xz plane (this analytic result is only valid for

ξ2 ¿ 1). Tilting the locus of B = 0 also reduces the oscillation frequency veryslightly in the z direction from its value when Bz = 0. Thus simply switching onBz(t) also changes the cloud shape and so excites quadrupole mode oscillations.To avoid this, we first adiabatically modify the usual TOP trap to a tilted trapand then quickly change Bz(t) to −Bz(t) to excite the scissors mode, as shownin fig. 7.2. This procedure rotates the symmetry axes of the trap potential by 2φwithout affecting the trap oscillation frequencies.

The following experimental procedure was used to excite the scissors modeboth in the thermal cloud and in the BEC. Laser-cooled atoms were loaded intothe magnetic trap and after evaporative cooling the trap frequencies were ω⊥ =90 ± 0.2 Hz and ωz =

√8 ω⊥. The trap was then adiabatically tilted by an angle

of φ = 3.6 by linearly ramping Bz (t) over a period of 1 s. The increase in Bz

resulted in a reduction of the axial trap frequency ωz by 2%. Suddenly reversingthe sign of Bz (t) in less than 100 µs excites the scissors mode, in a trappingpotential with its symmetry axes now tilted by −φ, as described above. Theinitial orientation of the cloud with respect to the new axis is θ0 = 2φ, so thisangle is the expected amplitude of the oscillations (fig. 7.2). After allowing theoscillation to evolve in the trap for a variable time t, the cloud is imaged alongthe y direction. The thermal cloud is imaged in the trap and the condensate after15 ms of free expansion. Figure 7.3 shows typical absorption images of the thermalcloud (a) and the condensate (b). The angle of the cloud was extracted froma 2-dimensional Gaussian fit of such absorption profiles and plotted (over manyexperimental runs) as a function of evolution time.


Figure 7.2: The method of exciting the scissors mode by a sudden rotation of the trappingpotential. The solid lines indicate the shape of the atomic cloud and its major axes. Thedotted lines indicate the shape of the potential and its major axes. (a) The initialsituation after adiabatically ramping on the field in z direction, with cloud and potentialaligned, tilted by angle φ. (b) The configuration immediately after suddenly rotatingthe potential, with the cloud displaced from its equilibrium position by angle 2φ. (c)The large arrow indicates the direction of the scissors mode oscillation and the smallerarrows show the expected quadrupolar flow pattern in the case of a BEC. The cloud isin the middle of an oscillation period. (The angles have been exaggerated for clarity.)

7.4. Thermal cloud results 109

Figure 7.3: Typical absorption images used for the scissors mode experiment. (a) Thethermal cloud in the trap. (b) The condensate after 15ms of time of flight. The conden-sate expands most rapidly along the direction which is initially most tightly confined,and hence its major and minor axes appear to be inverted relative to those of the thermal

cloud.

7.4 Thermal cloud results

For the observation of the thermal cloud oscillation, the atoms were evaporativelycooled to 1 µK which is about 5 times Tc, before the trap was suddenly tilted. Atthis stage there were ∼ 105 atoms remaining, with a peak density of n0 ∼ 2× 1012

cm−3. The results of many runs are presented in fig. 7.4(a) showing the way thethermal cloud angle changes with time. The model used to fit this evolution is thesum of two cosines, oscillating at frequencies ω1 and ω2.

θ (t) = −φ + θ1 cos(ω1t) + θ2 cos(ω2t) (7.15)

From the data we deduce ω1/2π = 339± 2 Hz and ω2/2π = 159± 4 Hz. Thesevalues are in very good agreement with the values 339 ± 3 Hz and 159 ± 2 Hzpredicted by theory [?]; which correspond to ω1 = ωz + ωx and ω2 = ωz − ωx.We measured ωx and ωz by observing the center of mass (dipole) oscillations of athermal cloud in the untilted TOP trap and calculated the modification of thesefrequencies caused by the tilt. The amplitudes at the two frequencies were foundto be the same, showing that the energy is shared equally between the two modesof oscillation, as predicted in fig. 7.1(b).


Figure 7.4: (a) The oscillation of the thermal cloud as a function of time. The solid lineis the fitted double cosine function of eqn. 7.15. The temperature and density of ourthermal cloud are such that there are few collisions, so no damping of the oscillationsis visible. (b) The evolution of the condensate scissors mode oscillation over the sametime scale as the data in (a). For the BEC, there is an undamped oscillation at a singlefrequency ωsc, which is fitted with the formula of eqn. 7.16. This frequency is not the

same as either of the thermal cloud frequencies.

7.5. Scissors mode results for the condensate 111

7.5 Scissors mode results for the condensate

To observe the scissors mode in a Bose-Einstein condensed gas, we carried outthe full evaporative cooling ramp to well below the critical temperature, where nothermal cloud component is observable, leaving more than 104 atoms in a purecondensate. Since the condensate is imaged after free expansion, its aspect ratiois inverted relative to that of trapped thermal cloud (fig. 7.3). The expansion alsocauses a small increase in the observed amplitude but does not affect the frequencyof the oscillation [?]. The scissors mode in the condensate is described by an angleoscillating at a single frequency ωsc:

θ (t) = −φ + θ0 cos (ωsct) (7.16)

Figure 7.4(b) shows some of the data obtained by exciting the scissors mode inthe condensate. Consistent data, showing no damping, was recorded for timesup to 100 ms. After fitting all the data with the function in eqn. 7.16 we finda frequency of ωsc/2π = 266 ± 2 Hz which agrees very well with the predicted

frequency of 265 ± 2 Hz from ωsc =√

ω2x + ω2

z . The aspect ratio of the time-of-flight distribution is constant throughout the data run confirming that there areno shape oscillations and that the initial velocity of a condensate (proportional toθ) does not have a significant effect [?].

7.6 Conclusion

These observations of the scissors mode clearly demonstrate the superfluidity oftrapped Bose-Einstein condensed rubidium atoms, in the way predicted by Guery-Odelin and Stringari [?]. Direct comparison of the thermal cloud and BEC dataunder the same trapping conditions shows a clear difference in behaviour betweenthe irrotational quantum fluid and a classical gas. The condensate oscillation mayalso be compared with the simulated data for a normal cloud of the same temper-ature and density as the condensate. The double oscillation frequency and heavydamping predicted for a normal cloud are clearly different from our experimentalcondensate data.

7.7 The scissors mode at finite temperature

There is now a large volume of experimental data confirming that the GP equationprovides an excellent description of the condensate as T → 0 (see e.g. [?, ?]).However there is very limited ‘finite temperature’ data, in which the thermal cloudhas a significant effect on the collective behaviour of the condensate. The rangeof interesting temperatures is approximately from 0.6 Tc to Tc, between which thefraction of atoms in the thermal cloud varies from 0.2 to 1 (eqn. 5.19). Prior to


the work described in this section, only two other groups had investigated thecollective excitations of the condensate at higher temperature, the m = 0 modein [?] and both the m = 0 and m = 2 modes in [?]. However, analyzing thedata from these compressional modes is difficult. In contrast, the scissors mode isideally suited for investigating our condensed system at finite temperature. Theangle fitting procedure is robust, model independent, not sensitive to shot to shotnumber fluctuations and gives accurate results for very small condensates close toTc. Our results are given in detail in [?, ?], and will only be summarized in thissection.

7.7.1 Experimental procedure and results

We were able to produce partially condensed clouds, with temperatures rangingfrom 0.3 Tc to Tc, by varying the depth of the final radio-frequency cut. For eachtemperature, two scissors runs were made. The first imaged the cloud in the trap,and then by fitting a 2D Gaussian we were able to plot the oscillation of the ther-mal cloud as a function of time. (Under these conditions the condensate densitywas imaged on a single pixel and thus did not affect the angle of the distribution).The second run used 14 ms of free expansion prior to imaging. The angle of theexpanded BEC was then extracted by fitting a 2D double distribution (a Gaussianfor the thermal cloud and a parabola for the condensate). The condensate oscil-lation was fitted with an exponentially damped cosine function, whilst a dampedtwo frequency fit was used for the thermal cloud, as explained in section 7.2.2.

The results for the damping rates and oscillation frequencies of the condensateand thermal cloud as a function of scaled temperature T/Tc are given in fig. 7.5.Temperature scaling is necessary because the critical temperature depends on thetotal number of atoms (eqn. 5.18), and hence on the rf cut depth [?]. For agiven cut, the temperature was extracted from the wings of the thermal clouddistribution as described in section 2.5.5. The thermal cloud data extends to∼ 0.8 Tc, below which it was too small to detect. Over the range of temperaturesmeasured, the thermal cloud oscillation appears to be undamped and to occur atthe frequencies predicted for a collisionless classical gas.

The behaviour of the scissors mode of the condensate as a function of temper-ature was much more interesting. Below 0.8 Tc the damping rate is well describedby Landau damping [?], which is proportional to T/µc [?, ?]. This is a scatteringprocess between quasi-particles in which a low energy collective excitation and athermal excitation are annihilated and a higher energy thermal excitation is cre-ated. Above 0.8 Tc the damping rate appears to decrease, although fitting theangle of such small condensates became increasingly difficult and so the error onthe measurements close to Tc is large. The frequency of the condensate oscillationis in excellent agreement with the hydrodynamic prediction at low temperature,when corrections are made for a finite number condensate [?]. The frequency de-

7.7. The scissors mode at finite temperature 113

Figure 7.5:The damping rate (a) and ‘scissors’ frequencies (b) of the condensate (solid

circles) and thermal cloud (open circles) as a function of temperature.

• In (a) the dot-dash line is the Landau damping rate for the m = 2 modeof the condensate (with frequency ω2), calculated in [?] and rescaled by thefactor ωsc/ω2. The dotted line is the prediction in [?] for the l = 2 mode ina spherical trap, again rescaled by the scissors frequency. The solid line isfrom the simulations in [?] for our exact experimental conditions.

• In (b) the condensate scissors frequency shows significant negative frequencyshifts from the hydrodynamic prediction (dashed line) at higher tempera-tures. The low temperature frequencies exceed the hydrodynamic value by∼ 1% because of the finite number of atoms. The solid line shows the resultsof simulations in [?]. The thermal cloud frequencies do not appear to be tem-perature dependent and are in good agreement with collisionless predictions(dotted lines).


creases as the temperature increases, initially very gradually and then more sharplyabove ∼ 0.7 Tc. It is worth noting that whilst the frequency shift has the right sign(negative), it is too large, relative to the observed damping rate, for the systemto be described in terms of a damped simple harmonic oscillator. Above 0.7 Tc,the distinction between the condensate and the thermal cloud gradually becomemore blurred - the condensate oscillation frequency tends to that of the thermalcloud, whilst the reduced damping rate suggests that relative motion of the twocomponents is decreasing.

The analysis of this data is still a matter for debate. Two different modelsmay be used to describe the partially condensed system. In [?], Rusch et al.use a finite temperature field theory, which consistently involves the dynamicsof the thermal cloud to second order, to investigate the excitation spectrum ofa condensate in a spherical trap. This theory recognizes the discreet nature ofthe low-lying, thermally populated, excited states. Each condensate excitationis strongly coupled via Landau (and Beliaev) processes to only a few thermallypopulated states. The individual couplings determine both the damping of thecollective condensate excitation and the sign and magnitude of the frequency shiftas a function of temperature. Since both the condensate and the thermal cloudare described by the same finite temperature field theory, their identities naturallybecome blurred as T → Tc (as observed experimentally) and several of the low-lying states become macroscopically occupied. However, whilst this theory hasa sound theoretical basis, should be valid up to Tc and may well explain theupwards shift in the frequency of the m = 0 mode observed in [?], it will becomputationally demanding to calculate the spectrum of an anisotropic trap as afunction of temperature.

An alternative model is presented by Jackson and Zaremba in [?]. Insteadof describing the whole system by a single finite temperature field theory, thecondensate is described by a generalized GP equation, whilst the thermal cloud isrepresented by a semi-classical Boltzmann kinetic equation. These two componentsare coupled both by mean fields (giving rise to Landau damping) and by collisionalprocesses, which transfer atoms between the two. Simulations based on this modeldescribe both the damping and frequency shift of the condensate reasonably wellup to T = 0.7 Tc but fail at higher temperatures. Above 0.7 Tc, it is suggested thatthe increasingly massive thermal cloud begins to drive the condensate at its ownlow rotational oscillation frequency, accounting for the decrease in the condensateoscillation frequency and the observed reduction in damping.

In summary, the finite temperature field theory of [?], applied to an axiallysymmetric trap, might well provide an accurate picture of the scissors mode ofa partially condensed system close to Tc. However the two component theoryof [?] appears to provide a very useful and tractable approximate theory for thespectrum of the low-lying condensate modes, for temperatures below 0.7 Tc, wherethe thermal occupation of other low-lying states is limited.

7.7. The scissors mode at finite temperature 115

7.7.2 Moment of inertia at finite temperature

The moment of inertia is an important property of a finite temperature Bose con-densate, since its quenching can be regarded as a measure of the superfluidity ofthe system. The degree of quenching is characterized by the normalized momentof inertia, RΘ = Θ/Θrig. RΘ tends to a steady value of less than 1 as T → 0 andperfect superfluidity is approached, and tends to 1 as superfluidity disappears. Itis possible to infer the normalized moment of inertia for a Bose-condensed gas asa function of temperature, from our finite temperature scissors results and henceinvestigate the superfluidity of the system in this regime. The method used wasoriginally developed by Zambelli and Stringari in [?] and is summarized in sec-tion 6.4. They derive an explicit relationship between the moment of inertia ofa system and its quadrupole response to a small rotational perturbation i.e. thescissors modes (eqn. 6.17). This relationship may be written in the following usefulform [?]:

RΘ =Θ

Θrig

=(ω2

x − ω2y

)2∫

Q(ω, T )/ω2

∫Q(ω, T )ω2

(7.17)

where Q(ω, T ) is the Fourier transform of the quadrupole moment. It may be splitinto contributions from the condensate and the thermal cloud.

Q(ω, T ) = Qc(ω, T ) + Qth(ω, T ) (7.18)

Q(ω, T ) is a series of spikes (which may be approximated to delta functions forsmall damping) at the measured oscillation frequencies of the system. Informationrequired to determine the height of each spike includes the total atom number,condensate number, temperature and chemical potential and may be extractedfrom the absorption images of the thermal cloud and condensate.

The natural frequencies of the thermal cloud, ω± are independent of temper-ature. Thus Θth/Θrig is always equal to 1 and its contribution to the total RΘ

(via Q(ω, T )) depends only on the fraction of atoms in the thermal cloud and theirspatial distribution. In contrast, the natural frequency of the condensate, ωsc fallsas T → Tc and its interaction with low-lying thermally populated states becomeslarge. The change in condensate oscillation frequency as a function of temperatureindicates that the degree to which its moment of inertia is quenched, and hencethe degree to which the condensate is superfluid, depends on temperature. Thusthe condensate contribution to RΘ depends on temperature via both the size ofthe condensate, and the degree to which it is superfluid.

The results for the normalized moment of inertia RΘ of the condensate (solidcircles), the thermal cloud (open circles) and the combined system (filled squares)(for calculation see [?]), obtained from our finite temperature scissors data, aregiven in fig. 7.6. Two important features should be noted from this data. Firstlyquenching of the moment of inertia of the condensate fraction (solid circles) occurs


Figure 7.6: The normalized moment of inertia of the condensate fraction (solid circles),the thermal cloud (open circles) and the combined system (solid squares) [?] as a functionof temperature. The error on this last data set of the order of 7%. For low temperaturesthe normalized moment of inertia of the condensate falls just below the hydrodynamicprediction (dashed line) and is in excellent agreement with the finite number correction(dotted line). The solid line provides a good approximation to the normalized momentof inertia of the entire gas, and was calculated using the population and widths of thethermal and condensate fractions only. It ignores the dynamical interaction betweenthe condensate and the thermal cloud and hence the fact that the quenching of the

condensate moment of inertia depends on temperature.

between Tc and ∼ 0.7 Tc and the condensate fraction appears to be almost fully su-perfluid below this temperature. Secondly consider the reduced moment of inertiaof the whole system (filled squares). Its value falls gradually below Tc, primarilydetermined by the population and width of the thermal and condensate fractions,but also influenced close to Tc by the reduced superfluidity of the condensate frac-tion itself. This quantity indicates the size of the superfluid fraction and could beused to indicate the BCS transition in a dilute degenerate Fermi gas. The solidline, which calculates the normalized moment of inertia of the entire system ignor-ing the frequency shift of the condensate fraction, is a good approximation to theexperimental data, deviating only slightly above 0.6 Tc.

Chapter 8

Superfluidity and the Expansionof a Rotating BEC

8.1 Introduction

In the previous chapter, we described our investigation of the superfluidity ofthe condensate by measuring the frequency of its small angle oscillations. Thefollowing two chapters will report further direct evidence for the superfluidity of acondensate, by investigating its behaviour in a rotating potential. At high rotationrates, we observe the formation of quantized vortices (chapter 9). In this chapterwe find evidence for a pure irrotational flow pattern, one of the signatures of asuperfluid, in the expansion behaviour of a slowly rotating, vortex-free condensate.The theoretical work for this experiment was done by Edwards et al. in [?] andthe experimental results are published in [?].

The aim of the experiment was to predict the expansion behaviour of a conden-sate in the Thomas-Fermi limit, assuming that it behaved as a superfluid. If theexperimental results agree with these predictions then the original assumption ofsuperfluidity is validated. The important features of the expansion may be under-stood from a few physical arguments. First consider the expansion of a stationarycondensate released from an elliptical trapping potential. The mean field energycauses the condensate to expand most rapidly along those directions in which itwas initially most tightly confined [?, ?]. Thus immediately after release from atrap with Rx > Ry, the aspect ratio (Rx/Ry) of the condensate decreases. It be-comes instantaneously circular and then at long expansion times it continues toexpand with an inverted aspect ratio.

Now consider the expansion of an elliptical condensate that is initially in equi-librium in a slowly rotating potential. Assuming that the condensate behaves asa superfluid then the flow pattern will be purely irrotational and the moment of

117

118 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

inertia, ΘS will be related to the cross-section of the condensate in the plane ofrotation (see section 6.4).

ΘS =

[〈x2 − y2〉〈x2 + y2〉

]2

Θrig (8.1)

For a Thomas-Fermi condensate eqn. 8.1 becomes:

ΘS =

[R2

x −R2y

R2x + R2

y

]2

Θrig (8.2)

Immediately after release, the condensate follows the mean field expansion pattern,with the smallest dimension expanding most rapidly and the aspect ratio, Rx/Ry,decreasing towards 1. As the aspect ratio becomes less elliptical and Rx → Ry,eqn. 8.2 shows that the moment of inertia falls towards zero. Since no torque actson the system, angular momentum must be conserved and so the rotation rate ofthe condensate increases ∝ 1/ΘS. Clearly the moment of inertia cannot fall asfar as zero because this would create the unphysical situation of infinite rotationalkinetic energy:

K.E. =L2

2ΘS

(8.3)

Thus the aspect ratio of the condensate never becomes unity; during the expansionit reaches a minimum value which is greater than 1, and then the aspect ratio in-creases again. In the final stages the expansion is most rapid along those directionsthat were initially least tightly confined in the trap and the rotation rate decreasesas the moment of inertia increases. The condensate tends to an asymptotic finalangle of between 45 and 90.

This chapter will first outline the theory used to provide a quantitative predic-tion of the condensate behaviour. Then the experimental procedure is described,including the modifications of the TOP trap required to produce a rotating ellip-tical potential. Finally the experimental results are presented and compared withthe irrotational predictions.

8.2 Theory

In our experiment we have an anisotropic harmonic potential with three angularfrequencies ωx < ωy < ωz. The potential rotates about the z-axis with angularfrequency Ω. As in chapter 7 it is convenient to define two parameters whichdescribe the trap in the plane of rotation, which is now the xy plane rather thanthe xz plane (as in the scissors mode experiment). The trap deformation in the xyplane is εt:

εt =ω2

y − ω2x

ω2y + ω2

x

(8.4)

8.2. Theory 119

The geometric mean of the frequencies in the plane of rotation is ω⊥:

ω⊥ =

√ω2

x + ω2y

2(8.5)

In the hydrodynamic limit, a condensate at equilibrium in this trap displays apurely irrotational or quadrupolar flow pattern, described in fig. 6.1b. The wave-function corresponding to the quadrupole flow pattern has the form:

Ψ(r) =√

n(r)ei mνxyh , (8.6)

where n(r) is the condensate number density given in eqn. 8.8 below. Three dif-ferent quadrupole modes exist, each characterized by a quadrupole frequency ν,which is a solution of the following cubic equation [?, ?, ?]:

ν3 + (1− 2Ω2)ν + εtΩ = 0, (8.7)

where we introduced the dimensionless quantities ν = ν/ω⊥ and Ω = Ω/ω⊥. Thereis one positive root which corresponds to the ‘normal’ quadrupole branch and 2negative roots which correspond to the ‘overcritical’ branch. The solutions areonly physical if |ν| < Ω and so each mode has a different range of trap parameters(εt and Ω) in which it exists. These stability regions are plotted in fig. 8.1. For thework in this chapter, we only need to consider the normal branch. We use εt = 0.32and Ω < 0.38 and so are in range of parameters where only this one mode exists.The overcritical branch is important at higher rotation rates for vortex nucleation,as discussed in chapter 9.

In the rotating frame, the effective trapping frequencies are modified becauseof the quadrupolar motion of the condensate, so that the condensate density canbe written as:

n(r) =µc

g

[1− m

2(ω2

xx2 + ω2

yy2 + ω2

zz2)

]. (8.8)

The modified frequencies ωx and ωy are given by:

ω2x = ω2

x + ν2 − 2νΩ

ω2y = ω2

y + ν2 + 2νΩ, (8.9)

where ν is a root of eqn. 8.7. Hence, the aspect ratio of the condensate in the trapis

Rx

Ry

=ωy

ωx

=

√Ω + ν

Ω− ν, (8.10)

The aspect ratio can be used to identify which quadrupole branch is excited in thecondensate, under conditions where more than one mode is stable (fig. 8.1). Inparticular in the normal mode, which has a positive value of ν, the major axis ofthe density distribution is aligned with the direction in which the trap is weakest.


Figure 8.1: A plot showing the trap conditions (trap deformation εt and normalizedrotation rate Ω) under which each of the three quadrupole modes of a condensate ina rotating potential exist. The boundaries of the normal mode (dashed line) and 2branches of the overcritical mode (solid and dotted lines) are shown. The number of

modes that are stable in each region is indicated on the plot.

8.3. The rotating anisotropic trap 121

However eqns. 8.9 and 8.10 show that for the overcritical branch, with negative ν,the density distribution is inverted and has its major axis along the direction inwhich the trap is stiffest. The effective chemical potential and condensate sizes inthe trap can also be calculated from eqns. 8.9.

To calculate the expansion of a condensate when released from the anisotropicharmonic potential, we use the following ansatz for the condensate number densityn(r, t) and the velocity field v(r, t) in the Thomas-Fermi regime (as in [?]):

n(r, t) = n0(t)− nx(t)x2 − ny(t)y

2 − nz(t)z2 − nxy(t)xy (8.11)

v(r, t) =1

2∇

(vx(t)x

2 + vy(t)y2 + vz(t)z

2 + vxy(t)xy). (8.12)

Inserting this ansatz into the hydrodynamic equations of superfluids (eqns.5.36 and5.38) yields a set of nine coupled differential equations for the expansion parame-ters, that were integrated numerically. At t = 0, the instant of release, n0(0), nx(0),ny(0) and nz(0) may be determined from eqn. 8.8. nxy(0) = 0 as the condensate isassumed to be aligned with the lab frame coordinate axes at the time of release. Itis the growth of this term during expansion that causes the condensate to rotate.Since no compressional modes are excited in the trap vx(0), vy(0) and vz(0) areall zero. vxy is the quadrupole velocity field in the trap and may be found to beequal to 2ν, from the condensate wavefunction (eqn. 8.6) and velocity (eqn. 7.8).The behavior of the condensate is thus completely determined by the above initialconditions and the nine coupled differential equations for the expansion parame-ters [?]. Having solved the equations for a given value of t, the angle and aspectratio of the condensate in the plane of rotation are found by diagonalizing thequadratic density distribution of eqn. 8.11. The code to calculate the expansionbehaviour corresponding to our exact experimental conditions was developed byGerald Hechenblaikner and the results are plotted in fig. 8.5.

8.3 The rotating anisotropic trap

This experiment and the vortex experiments described in chapter 9 required arotating trap, that was elliptical in the plane of rotation. This section outlines themodifications made to our standard axially symmetric TOP trap, first to give it avariable ellipticity and then to allow it rotate at a chosen rate Ω in the xy plane[?].

8.3.1 The elliptical TOP trap

The standard TOP trap consists of an axially symmetric quadrupole field and aradial bias field rotating at ω0/2π = 7 kHz. The potential experienced by theatoms results from the time-average of these two fields. The radial bias field has


Figure 8.2: The normalized trapping frequencies in an elliptical TOP trap ωi/ω0i, as afunction of the radial bias field ratio, E = Bx/By (solid lines). Each frequency is nor-malized against its value in a standard TOP trap with E = 1. By and the quadrupolegradient B′

q are assumed to be constant. Note that the normalized value of ωz is approx-imately equal to the mean of the other two normalized frequencies (dotted line). The

dashed line shows the ratio of the radial trapping frequencies, e = ωy/ωx.

two components of equal amplitude BT , oscillating in the x and y directions with aπ/2 phase difference. Changing the amplitude of one of these components so thatBx/By = E, breaks the radial symmetry and produces an elliptical potential [?]that is the time-average of:

U(x, y, z, t) = µB′Q|(x + Er0 cos ω0t) x + (y + r0 sin ω0t) y − 2z z| (8.13)

The path followed by the locus of B = 0, is now an ellipse (rather than a circle)and follows the curve:

r = r0 (E cos ω0t x + sin ω0t y) (8.14)

where r0 = BT /B′q.

At distances from the trap centre that are small compared to r0, the trapremains harmonic. The trap frequencies must be calculated numerically fromeqn. 8.13 and are plotted in fig. 8.2 as a function of the bias field ratio E. Wemay assume that E ≥ 1 without loss of generality and thus ωx ≤ ωy < ωz. Notethat increasing the bias field in the x direction reduces the trapping frequencies inall three directions. The deformation of the trapping potential is described eitherby e = ωy/ωx or by εt (eqn. 8.4). The eccentricity of the trapping potential, e is

8.3. The rotating anisotropic trap 123

much less than the eccentricity of the fields E. For small deformations the fieldsand potential are related by:

de

dE≈ 1

4. (8.15)

The amplitudes of the individual TOP field components Bx and By are con-trolled from the computer and so the eccentricity of the trap can be changed duringan experimental run. This enables us to make a condensate in an axially symmetrictrap, in which the evaporative cooling efficiency is optimized, and then deform it.One of the advantages of our purely magnetic rotating potential is that contoursof constant energy are well defined ellipses and we have precise control over theeccentricity. After calibrating ωx and ωy using dipole oscillation frequencies, theratio e is known to an accuracy of at least 0.5 %. We also have access to a widerange of eccentricities corresponding to 1 < ωy/ωx < 1.7 or a trap deformation pa-rameter of 0 < |εt| < 0.5. Other experiments create an elliptical rotating potentialusing the dipole force from a far-detuned laser beam to break the xy symmetry.This arrangement offers more flexibility in the shape of the rotating potential [?],but the maximum trap deformation is small (εt < 0.032 in [?]) and the shape ofthe potential is less accurately defined (because it depends on the position of thebeams relative to the magnetic trap).

8.3.2 Rotating the trap

The elliptical locus of B = 0 (eqn. 8.14) and the elliptical contours of the time-averaged potential share the same major and minor axes. Thus if the former rotatesat angular frequency Ω, the trapping potential will also rotate at the same rate.Mathematically, rotation through an angle Ωt is described by a rotation matrix.Applying this to the elliptical locus of B = 0 gives:

(xy

)=

(cos Ωt sin Ωt− sin Ωt cos Ωt

) (E r0 cos ω0tr0 sin ω0t

)(8.16)

To create this elliptical locus of B = 0, the rotating bias field must have the form:

Bx = BT (E cos Ωt cos ω0t + sin Ωt sin ω0t) (8.17)

By = BT (−E sin Ωt cos ω0t + cos Ωt sin ω0t) . (8.18)

Note that for E = 1, we return to a standard axially symmetric TOP trap, with aslightly shifted TOP frequency, ω0 − Ω. Since trap rotation rates are in the range0−100 Hz, and ω0/2π = 7 kHz, this shift does not affect the properties of the trap.

The electronics for creating Bx and By in a rotating trap are shown schemat-ically in fig. 8.3. Initially, the slow signals sin Ωt and cos Ωt, were created by aquadrature oscillator. This oscillator is specifically designed to maintain a phasedifference of exactly 90 between its two outputs, as required by eqn. 8.16. The ro-tation frequency Ω could be varied manually between 0.8−100 Hz. Thus although


Figure 8.3: A schematic diagram of the electronics for the elliptical rotating TOP trap.

8.4. Experimental procedure 125

the trap ellipticity was under computer control, the rotation rate remained fixedfor each experimental run. This arrangement was used for the work in chapters 8and 9. Starting from the 4 basic oscillating signals EBT cos ω0t, BT sin ω0t, cos Ωtand sin Ωt, voltage multipliers produce 4 signals consisting of a rapid oscillationmultiplied by a slow one. These pairs are then added and subtracted to producethe signals of eqn. 8.18 that are fed via the audio-amplifier to the TOP coils. Theelectronics included several low-pass filters to prevent high-frequency noise reach-ing the amplifier since this has been observed to reduce the lifetime of the magnetictrap. In addition, variable resistors were built into the multiplication and addi-tion stages, so that the signals could be balanced to produce a stable, noise-freeelliptical bias field.

For the gyroscope experiment described in chapter 10, it became necessaryto vary the rotation rate of the trap during an experimental run. Thus a thirdNI board (MIO16) was fitted inside the control computer to provide two extraanalogue outputs that were used to generate the slow sin Ωt and cos Ωt signals.Our initial concern that the output of the boards might produce high frequencynoise on the bias field proved unfounded. The output provides 2000 points per cyclewhich creates a sufficiently smooth signal. The new arrangement also enables usto stop the rotation at any specific angle, jump between different angles and evenchange the direction of rotation during an experimental run. All this rotationalflexibility makes it an ideal apparatus for investigating the superfluid nature of thecondensate, which is most strikingly displayed in its response to an applied torque.

8.4 Experimental procedure

In this experiment the condensate contained ∼ 1.5× 104 atoms and had a temper-ature of 0.5 Tc. Condensates were produced by evaporative cooling in a standardaxially symmetric TOP trap with final frequencies ωx/2π = ωy/2π = 124 Hz andωz/2π = 350 Hz. Once the condensate had formed we made the trap elliptical bychanging the ratio of the two TOP-field components to Bx/By = 4.2 over 500 ms.This corresponds to a frequency ratio of ωy/ωx = 1.4 or a trap deformation ofεt = 0.32. The eccentricity was ramped up from zero to its final value in 500 ms.This experiment was done using the quadrature oscillator and so as soon as thetrap became deformed, it also began to rotate at the chosen rate Ω. The conden-sate was left in the rotating trap for another 500 ms before it was released at afixed reference angle in the trap rotation, from which the angles of the expandingcloud were measured. Column (a) of fig. 8.4 shows typical absorption images ofthe condensate, taken along the axis of rotation, after different expansion times.For these pictures the trap was rotating at 28 Hz (well below the threshold to nu-cleate vortices, section 9.5) and at the instant of release the long axis of the cloudwas along the x-direction. The angle and aspect ratio of the cloud were obtainedfrom a 2D parabolic fit to the density distribution. The cloud reached a minimum


aspect ratio of 1.31 after about 4 ms time-of-flight and the angle approached itsasymptotic value of ∼ 55 degrees after 16 ms.

For comparison, the evolution of a condensate released from a non-rotatingtrap is shown for the same expansion times in column (c) of fig. 8.4. In this casethe aspect ratio decreased steadily, becoming circular after about 4 ms and theninverting. Note that the trapping frequencies ωx and ωy were the same in bothcolumns. The enhanced aspect ratio in the left column, both in the trap and atlong expansion times, is a result of the irrotational flow pattern in the rotatingcondensate, eqn. 8.9. It was necessary to go to a high trap eccentricity to observethe deformation of the cloud clearly.

8.5 Results

We investigated the evolution of the condensate after release from a static trapand from traps rotating at both Ω/2π = 20 and 28 Hz. In all three cases thetrap frequencies were ωx/2π = 60 Hz, ωy/2π = 1.4 × 60 Hz and ωz/2π = 206 Hz.Figure 8.5(a) shows the calculated angle of the condensate density distribution afterrelease from these three traps, with the experimental data points superimposed.In the case of the rotating condensates (dashed and solid lines) the angle reaches45 degrees after about 6 ms and after 18 ms is close to its asymptotic value between55 and 60 degrees. The angle of the condensate released from a static trap (dottedline) changes from 0 to 90 instantaneously, as the aspect ratio passes through 1.

Figure 8.5(b) shows the aspect ratios extracted from the same images as fig. 8.5(a),with the theoretical predictions superimposed. The data clearly demonstrate howthe aspect ratio of an initially rotating condensate decreases up to a critical point,which is reached after approximately 4 ms. From that point on it does not con-tinue to expand along its minor axis but the aspect ratio increases again becausethe condensate cannot become circular under these conditions. However, the con-densate released from a static trap has no velocity field which prevents it frombecoming circular and hence the aspect ratio passes through 1 at about 6 ms. Infig. 8.5(b) the aspect ratios are plotted in a frame that rotates with the major ofthe condensate. In the case of a static condensate the situation is ambiguous; wemay either plot the aspect ratio falling below one at long expansion times, withno change of angle or we can plot the aspect ratio increasing at long times, withan instantaneous rotation of 90 as it passes through 1. We have chosen the latterso that the plot changes smoothly as Ω → 0.

Every experimental point displayed is the average of several measurements. Toimage the plane of rotation we are using the ‘vertical’ (z axis) imaging systemdescribed in section 2.5.7. For each expansion time we had to refocus our imagingsystem as the atoms move out of focus under gravity when released from the trap.Incorrect focusing would only result in a more circular image and the minimumvalue for the measurement of the aspect ratio of rotating condensates was never

8.5. Results 127

Figure 8.4: Typical absorption images of the condensate, taken along the axis of rotation,at different times after release from a trap rotating at 28 Hz (column (a)) and after releasefrom a non-rotating trap (column (c)). Columns (b) and (d) show the data of columns(a) and (c) respectively in a simplified form, identifying the major and minor axes andthe fitted rotation angle. At the instant of release the major axis of the cloud lay alongthe x direction. The red arrows indicate the direction and speed of rotation. The blackarrows indicate the direction in which the condensate is expanding most rapidly. Therotating condensate is observed to have a much larger asymptotic aspect ratio than the

static one, as predicted by the upper and lower theoretical curves of fig. 8.5(b).


Figure 8.5:

Dashed line and filled squares, Ω/2π = 28 HzSolid line and open circles, Ω/2π = 20 Hz

Dotted line and filled triangles, non-rotating trap.

The theoretical and experimental results for the angle (a) and aspect ratio (b) ofthe condensate, as a function of expansion time. Note that the theory involves no

free parameters. The aspect ratio plot, (b), shows that initially rotatingcondensates never become circular about the rotation axis (aspect ratio = 1) toensure that angular momentum is conserved. However, after release from the

non-rotating trap, the aspect ratio does pass through 1 at about 6 ms. (We haveplotted the aspect ratio of the static condensate increasing again at long

expansion times, with a corresponding rotation of 90. This ensures that we labelthe major and minor axes consistently and the apparent behaviour changes

smoothly as Ω → 0).

8.5. Results 129

Figure 8.6: The asymptotic rotation angle of the condensate after a long expansion time(10 s), as a function of the initial trap rotation rate, Ω/2π. The condensate was released

from a trap with ωx/2π = 60 Hz, ωy = 1.4ωx, ωz = 1.2×√8ωx.

consistent with unity. There is good agreement between the experimental dataand the theoretical predictions at rotation Ω/2π = 20 Hz. However, we observed asmall deviation of the experimental data from the predicted values for the higherrotation frequency of Ω/2π = 28 Hz (dashed curve). This can be accounted for byimperfect focusing. As the rotation rate increases and the condensate deformationbecomes more pronounced, focusing becomes more critical

The asymptotic rotation angle of the condensate is determined by a balance be-tween two factors, the irrotational velocity field and the mean field expansion. Forvery small values of Ω, the irrotational velocity field is negligible. The expansionis dominated by the mean field velocity and the angle through which the majoraxis rotates tends to 90. This angle reduces as Ω increases and the irrotationalvelocity field becomes more significant (see fig. 6.1(b), showing an irrotational ve-locity field). The minimum rotation angle is 45, which would be the result for analmost circular condensate with zero mean field energy. Finally at high rotationrates, just below the lower of the two radial trap frequencies ωx, the mean fieldenergy can be ignored. The cloud becomes elongated along the x direction andthe velocity field becomes indistinguishable from a rotational velocity field. In thislimit the asymptotic rotation angle increases again to its maximum value of 90.The asymptotic rotation angle of the condensate is plotted in fig. 8.6 as a functionof trap rotation rate Ω/2π, for the trap conditions used in the experiment.

Finally in fig. 8.7 we plot the expansion behaviour of a non-condensed thermalcloud for comparison. Both the aspect ratio (solid line) and angle (dotted line)of the cloud may be calculated analytically, assuming a Gaussian distribution anda rotational velocity field at t = 0. (Simulations of a thermal cloud in a rotating


Figure 8.7: The aspect ratio (solid line) and angle (dashed line) of a thermal cloud asa function of time after release from the trap rotating at Ω/2π = 28 Hz. The cloud isassumed to have a Gaussian distribution and rotational velocity field. The same trappingfrequencies are used as in the experiment with a condensate, ωx/2π = 60Hz, ωy = 1.4ωx.

potential showed that a rotational flow pattern was the equilibrium flow pattern[?]).

Thermal aspect ratio =

√√√√√(1 + ω2

x t2)(ω2

y − Ω2)

(1 + ω2

y t2)

(ω2x − Ω2)

(8.19)

Thermal Angle =1

2sin−1

(2Ωt

1 + Ω2t2

)(8.20)

The behaviour is strikingly different to that of the condensate due to an isotropicexpansion energy and the lack of superfluid constraints on the velocity field. Theaspect ratio decreases smoothly to an asymptotic value ≥ 1 and the angle of thecloud always tends to 90 at long expansion times, independent of the rotationrate.

8.6 Conclusion

Our results show that an expanding vortex-free Bose-condensate with some angularmomentum refuses to become circular about the axis of rotation, as predicted byEdwards et al. [?]. The theoretical curves in both figs. 8.5(a) and (b) have no freeparameters and are not fitted to the data. Thus the excellent agreement between

8.6. Conclusion 131

theory and experiment confirms that the assumption of an irrotational superfluidcondensate was correct. This provides direct evidence that Bose-condensed gaseshave purely irrotational flow and a reduced moment of inertia, as a consequenceof their superfluidity.

Chapter 9

Vortex Nucleation

The existence of quantized vortices is one of the most striking and fascinatingsignatures of superfluidity. Vortices were first investigated experimentally in aseries of remarkable ‘rotating bucket’ experiments on superfluid liquid helium,that are reviewed at the start of this chapter. The sections that follow outlinethe methods that have been proposed and used to nucleate and detect vorticesin a dilute gas Bose condensate, highlighting both the similarities and differenceswith the liquid helium work. This thorough introduction motivates the detailedexperimental work that we have done on vortex nucleation in a rotating potential,that has been published in [?] and will be discussed at the end of the chapter.

9.1 Vortices in He II

The first experiments on superfluid vortices were done during the 1960s usingsuperfluid liquid helium in a ‘rotating bucket’. The ‘bucket’ was a cylindrical con-tainer with rough inner walls mounted on a turntable, the rotation rate of whichcould be very precisely measured. First it was shown that at very low rotation ratesthe superfluid fraction does not come into equilibrium with the rotating bucket butremains stationary in the laboratory frame. (Or, more precisely, stationary in theframe of the fixed stars, thus if the lab were on the North Pole the superfluidwould rotate once per day with respect to the container! [?].) In contrast the nor-mal fraction rapidly reaches an equilibrium state (with a rotational flow pattern)in which it is stationary in the rotating frame. Thus as liquid helium is cooledthrough the critical temperature from above in a slowly rotating bucket, the angu-lar momentum of those atoms joining the superfluid fraction must be given up tothe bucket; the increase in the rotation rate of the bucket below Tλ was measuredand confirmed that the superfluid fraction indeed had zero angular momentum.

132

9.2. Vortices in a dilute gas Bose-Einstein condensate 133

The appearance of 1,2 and 3 vortex lines at well-defined critical rotation rateswas demonstrated by Packard and Sanders [?]. They were able to count vorticesby trapping electrons on the cores and then drawing them off onto a detector; thetotal charge collected was directly proportional to the number of vortices. Thefrequencies at which each new vortex appears are significantly higher than thosepredicted from thermodynamics because of the metastability of each state (seesection 6.5.2).

The quantization of circulation into units of h/m was first measured by Vinen[?] using the normal mode frequencies of a wire tethered along the axis of a bucketof He II. The circulating velocity field of an axial vortex line changes the spectrumof helical waves on the wire via the Magnus force (section 6.5.4). This change wasmeasured and the circulation of the velocity field inferred.

Finally at high rotation rates the meniscus of superfluid helium was observed tohave the same curvature as that of a normal fluid [?] although the fountain effectconfirmed that the rotating system was still superfluid [?]. A regular ‘Abrikosov’vortex lattice forms within the superfluid [?]; the net velocity field mimics therotational field of a normal fluid thus minimizing the energy of the system. Thislattice was crudely imaged [?] by trapping electrons on vortex cores and acceler-ating them off onto a phosphor screen. (The analogous arrangement of flux linesin a type-II superconductor has been more thoroughly investigated using neutronscattering and shown to have the predicted regular hexagonal-lattice structure).

Together these experiments demonstrate many of the properties of a superfluidvortex that are predicted in section 6.5; namely a critical rotation rate for nu-cleation, a zero in the density, a rotational flow pattern around each vortex andquantization of the associated circulation. The only feature that was not directlyobserved was the 2π phase change around a vortex because the strong atomic in-teractions in liquid helium mean that a condensate wavefunction and phase cannotbe clearly identified.

9.2 Vortices in a dilute gas Bose-Einstein con-

densate

9.2.1 Nucleation of vortices

Following the realization of BEC in a dilute gas, the observation of quantizedvortices in this new superfluid system became a matter of great theoretical andexperimental interest. A wide variety of different nucleation and detection schemeswere proposed and so it was not surprising that the first two groups to observevortices (at JILA [?] and Paris [?]) used entirely different experimental approaches.

Many of the proposed schemes were directly analogous to the liquid heliumexperiments described above. The rotating bucket with rough inner walls was

134 Chapter 9. Vortex Nucleation

replaced by a magnetic trapping potential with a small rotating eccentricity. Therotating eccentricity could either be produced with time-varying magnetic fieldsas described in section 8.3.2 or by using the dipole force of a far-blue-detunedlaser beam ‘stirrer’ [?, ?]. The beam was directed parallel to the axis of a staticIoffe-Pritchard trap, breaking its cylindrical symmetry and then rotated aroundthe axis using a pair of crossed acousto-optic modulators.

Whilst experimentalists considered the best method for realizing the rotat-ing bucket, there was lively theoretical debate about the mechanism that wouldtransfer the condensate from the ground state to the first vortex state in such anapparatus (for a review see [?]). As in He II, an energy barrier exists betweenthe two states and hence the ground state is expected to remain metastable evenat rotation rates where it is no longer the state of lowest energy (section 6.5.2).Each different nucleation mechanism is characterized by a critical rotation rate Ωc,at which the energy barrier is overcome and vortex nucleation occurs. The workdescribed in this chapter focuses on the eccentricity and rotation rate required tonucleate the first vortex and provides clear evidence that surface modes mediatethe nucleation process.

The energy barrier can be avoided by condensing the system directly into thefirst vortex state from a spinning thermal cloud. Under such conditions the con-densate will form in the state of lowest energy in the rotating frame and thusthe critical rotation rate for nucleation can be determined from thermodynamics.However this experiment has many technical difficulties in a dilute gas; the evap-orative cooling necessary to achieve condensation cuts only in the xy plane andhence removes those atoms at the outer radius of the cloud that carry the mostangular momentum. At Tc the mean angular momentum has been reduced belowthe h per particle that is necessary to form the first vortex state. These difficultieswere overcome in [?] by using the effect of gravity on a weak TOP trap to movethe rf cutting surface to an area on the bottom of the cloud, close to the z axis,where the atoms have little angular momentum. Using this method, evaporativecooling increases the mean angular momentum of the cloud. The critical rotationrate was slightly above the thermodynamic value but also significantly lower thanthe value required to nucleate a vortex in a ground state condensate.

The well-defined condensate wavefunction and phase provide an additional ap-proach to vortex nucleation in a dilute gas BEC, that has no analogy in liquid he-lium: engineering of the condensate phase to produce a 2π phase winding. Phaseengineering was originally used to create dark solitons [?, ?] by imprinting a sharpπ phase change along a line in the condensate with a far-detuned laser beam. (Darksolitons are lines of minimum density, that propagate without change of shape andare, like vortices, associated with a discontinuity in the condensate phase.) Thefirst vortices were produced in a dilute-gas condensate of 87Rb atoms using a re-lated, but not identical method [?]. Population was transferred from a groundstate wavefunction in the lower hyperfine level (F=1), directly into a vortex state

9.2. Vortices in a dilute gas Bose-Einstein condensate 135

in the upper hyperfine level (F=2). A two-photon microwave field induced transi-tions between the two levels, whilst the AC Stark shift of a rotating far-detunedlaser beam provided the spatial and temporal modulation necessary to create a 2πphase winding. A novel approach to nucleation of multiply-charged vortices usingphase engineering was recently announced in [?]. This method uses the rotation ofthe atomic spin vector (Berry’s phase) [?] to create the vortex phase winding andwill be discussed in section 11.2.1.

9.2.2 Detection of vortices

A variety of methods were proposed for the detection of vortices. The simplestwas the observation of a hole in the condensate density distribution looking alongthe axis of rotation with a standard absorption imaging system. Whilst the vortexcore in a trapped condensate is generally too small to be resolved with such asystem (section 6.5.1), calculations in [?] show that the core expands more rapidlythan the bulk of the condensate after release from the trap, so that it becomesresolvable after 12 ms of expansion e.g. the core has a 6 µm diameter. This method,whilst adequate for most experiments, involves integrating the density distributionalong the entire vortex core and so any bends in the vortex line will cause a lossof contrast. High contrast vortex pictures have been made by selective opticalpumping of a thin slice of the condensate, so that only atoms within the sliceinteract with the probe beam. The most striking of these are in [?], showing anarray of nearly 200 vortices in an almost perfectly regular hexagonal lattice.

A second detection method is analogous to Vinen’s vibrating wire experimentin liquid helium and can be used both to detect the presence of vortices and showthat the associated circulation is quantized. Instead of detecting the change in themode spectrum of a wire as in helium, experiments with a dilute gas investigatethe change in the collective mode spectrum of the condensate in the presence of avortex. The first experiments used the radial breathing mode (m = 2) in a axiallysymmetric trap [?, ?], which (as described in section 5.6) may be thought of asan equal superposition of 2 counter-rotating m = ± 2 modes. In the presence of avortex with a definite sense of rotation, these modes are no longer degenerate andso the major axis of the radial breathing mode precesses at a rate proportionalto the angular momentum of the vortex line [?, ?]. A analogous effect using them = ± 1 modes is described in chapter 10.

Finally the phase winding associated with a vortex may be used to detect as wellas create vortices. If a stationary condensate containing a singly charged vortexoverlaps with a condensate with linear momentum k, the standard ‘Young’s fringes’interference pattern has an extra half fringe inserted at the core of the vortex. Thisbehaviour was predicted in e.g. [?] and observed in [?, ?].


9.3 Vortex nucleation in a rotating potential

The aim of our experimental work was to carry out a detailed investigation of thecritical conditions for vortex nucleation in an oblate trapping geometry (ωz/ωx > 1).These data are used to test the theory presented in [?], that surface excitationsare necessary to mediate vortex nucleation in a ground state condensate. (Surfacemodes are collective excitations of the condensate with no radial nodes, describedin section 5.6.)

Consider a surface mode of multipolarity m driven by a rotating perturbationof the correct symmetry. Above a critical rotation frequency Ωcm , the perturbationimparts energy and angular momentum to the condensate and the amplitude of themode grows exponentially. Under such conditions the system rapidly goes beyonda linear regime, described by elementary excitations. A wider configuration spaceis made available and the system can jump into new rotating equilibrium statessuch as those containing quantized vortices. At very low temperatures (T → 0)the surface mode of the condensate may be driven by a rotating anisotropy in thetrapping potential [?]. At higher temperatures it may either be driven by the trapor by a rotating thermal cloud [?].

The critical frequency for angular momentum and energy transfer to the con-densate is given by an analog of the Landau criterion (section 6.2). We considerthe excitation of a surface mode with energy hωm and angular momentum mh(along the z axis), by a heavy object rotating at frequency Ω. Energy and angularmomentum conservation are only satisfied in the excitation process if Ω > ωm/m.Thus the minimum rotation frequency at which any surface mode can mediatevortex nucleation is given by:

ΩcL= min

[ωm

m

](9.1)

Below the critical frequency Ωcm , the condensate adjusts its shape to the rotatingpotential but does not absorb energy or angular momentum from the perturbation.

One of the most important ways to differentiate between the different proposedvortex nucleation mechanisms [?, ?, ?, ?, ?] is to test their predicted critical fre-quencies against experiment in different trapping geometries. For example, if onlythe m = 2 mode is excited then the surface mode theory [?] predicts Ωc2 = 0.71 ω⊥in all geometries. The theory of [?] links the critical frequency to the highestprecession frequency (or anomolous mode) of the vortex. For the prolate trapin [?] the numerical prediction of 0.73 ω⊥ is in good agreement with experiment.However applied to an oblate trap like ours the same theory gives Ωc < 0.1 ω⊥,significantly lower than the critical frequencies observed in section 9.5. Prior to thework described in this chapter, vortex nucleation data had been published by onlytwo groups in Paris [?, ?] and MIT [?], both using prolate geometries. Their dataseemed to support the surface mode theory but further evidence from an oblategeometry was necessary to verify the theory.

9.4. Experimental method 137

Other factors also ensured that our apparatus was well suited to a vortex nu-cleation experiment. The critical frequency depends on the multipolarity of thesurface mode that has been excited. In [?], many different modes are excited mak-ing it difficult to link the observed critical frequency to a particular mechanism. Inour apparatus, the rotating magnetic potential has a very well defined quadraticsymmetry, with no significant higher order terms. Therefore we expect the m = 2mode (with energy hω2) to be exclusively excited and to observe a critical fre-quency of ω2/2 = 0.71 ω⊥, if the surface mode theory is correct. Table 9.1 (takenfrom [?]) shows that in an oblate geometry the critical frequencies for differentsurface modes are more widely spaced than in other geometries, making it easierto identify the observed critical frequency with a particular mode.

Condensate λ ω⊥/2π N ω2/2 ω3/3 ω4/4 ΩcLΩth

Prolate 0.0058 175 Hz 2.5× 105 0.72 0.61 0.56 0.53 0.35

Spherical 1 7.8 Hz 3× 105 0.71 0.59 0.53 0.44 0.24

Oblate 10 10 Hz 3× 105 0.71 0.58 0.50 0.33 0.12

Table 9.1: Critical rotation rates for vortex nucleation via surface modes of differentmultipolarities (taken from [?]). All frequencies are in units of the radial trap frequency

ω⊥. ΩcL gives the minimum rotation rate for nucleation via any surface mode.

In addition to investigating nucleation as a function of rotation rate, we arealso able to investigate nucleation as a function of trap deformation εt (eqn. 6.15).The magnetic trap offers a wide range of εt from −0.5 < εt < 0.5 (as described insection 8.3), and the value of this parameter is known to an accuracy of at least0.5%.

9.4 Experimental method

Vortices were nucleated using the purely magnetic rotating trap described in sec-tion 8.3.2. Evaporative cooling in the static trap, followed by an adiabatic expan-sion, resulted in a condensate of 2 × 104 atoms, at a temperature of 0.5 Tc. Atthis stage the trap is axially symmetric, with trap frequencies ω⊥/2π = 62 Hz andωz/2π = 175 Hz.

To create vortices, the ratio of the TOP bias field components E = Bx/By

was ramped linearly over 200 ms from 1 to its final value, to give a trap that waselliptical and rotating at a preset value Ω. When we create an eccentric trap byincreasing (decreasing) the TOP bias field, all three trap frequencies are reduced


(increased). Since vortex lifetimes depend on the mean trap frequency [?], weadjusted the quadrupole field during the adiabatic expansion stage, to ensure thatall traps had the same average radial trap frequency ω⊥, defined as:

ω⊥ =

√ω2

x + ω2y

2. (9.2)

The condensate was then held in the rotating anisotropic trap for a further 800 msbefore being released. After 12 ms of free expansion the cloud was imaged alongthe axis of rotation. Figure 9.1 shows images of the expanded condensate at dif-ferent stages during the nucleation process. Initially the cloud elongates, as shownin fig. 9.1(a) providing evidence that nucleation is being mediated by excitationof a quadrupole mode. Then finger-like structures appear on the outside edge ofthe condensate which eventually close round and produce vortices, ∼ 800 ms af-ter rotation began (fig. 9.1(b)). Approximately 200 ms later, these have moved toequilibrium positions within the bulk of the condensate and appear in symmetricconfigurations. Figures 9.1(c)-(f) show typical, single-shot images of condensatescontaining 1-4 vortices in their equilibrium arrangement. The depth of each vortex(in the integrated absorption profile) is up to 95% of the surrounding condensate.The core diameters of ∼ 6 µm after 12 ms time-of-flight are consistent with theexpansion behaviour predicted in fig. 6.2 for our experimental conditions. We can-not obtain well-formed lattices with more than 4 vortices because of the relativelysmall number of atoms in the condensate. Figure 9.1(g) shows a condensate onthe verge of breaking up after the nucleation of ∼ 7 vortices.

9.5 Nucleation results

Our first study of the nucleation process involved counting the number of vorticesas a function of the normalized trap rotation rate, Ω = Ω/ω⊥, for a fixed eccen-tricity. Results for trap deformations εt = 0.084 and 0.041 are given in fig. 9.2.These graphs show a maximum and minimum value of Ω for nucleation at a giveneccentricity. Increasing εt increases the range of Ω over which vortices are nucle-ated, both by lowering Ωmin and raising Ωmax. In the limit of small eccentricities,the frequency of the m = 2 quadrupole mode is

√2 ω⊥, and this has been shown

elsewhere to play a critical role in the nucleation process [?, ?]. Rotation of theellipsoidal trap at half this frequency, i.e. Ωc2 = 1/

√2 ' 0.71, resonantly excites

this mode. The plots in fig. 9.2 confirm that the nucleation depends on resonantexcitation of the quadrupole mode as seen previously [?, ?]. The resonance isbroader at higher eccentricity, as intuitively expected for stronger driving.

Our second study involved holding Ω constant and counting the number ofvortices as a function of the trap deformation εt. Figure 9.3 shows our results fortwo cases: (a) Ω > Ωc2 and (b) Ω < Ωc2 . Interestingly we were able to nucleatevortices under adiabatic conditions when Ω < Ωc2 .

9.5. Nucleation results 139

Figure 9.1: (a) and (b) show images of the condensate at different stages during the vortexnucleation process, taken along the axis of rotation with Ω = 0.70, εt = 0.05 and after12 ms of free expansion. (a) After the 200 ms spinning eccentricity ramp, the condensateis elongated, indicating that a quadrupole mode has been excited. (b) After a further600 ms in the spinning trap one vortex has just formed near the edge. Approximately 1 safter rotation began, the vortices have reached their equilibrium positions and appear insymmetrical configurations as shown in figures (c)-(f). (g) shows a condensate containing

∼ 7 vortices - too many to form a stable lattice in our small condensates.


Figure 9.2: The mean number of vortices as a function of the normalized trap rotationrate Ω = Ω/ω⊥. Two different trap eccentricities were used, εt = 0.041 (open circles)

and εt = 0.084 (solid circles). Each data point is the mean of 4 runs.


Figure 9.3: The mean number of vortices as a function of trap deformation εt at 4different trap rotation rates: (a) above and (b) below the critical value Ωc2 = 0.71. In(a), Ω = 0.74 (solid circles) and 0.81 (open circles). In (b) Ω = 0.61 (solid circles) and

0.70 (open circles). Positive (negative) εt corresponds to ωx < ωy (ωx > ωy).


Adiabaticity criteria exist for both the change in the eccentricity and rotationrate of the trap that occur during the spin up stage. The former ensures thatcompressional modes are not excited in the condensate during the spin up pro-cess whilst that later ensures that the condensate smoothly follows a particularquadrupole mode and is the most relevant to the vortex nucleation process. Theeccentricity ramp is adiabatic if the change in energy of the harmonic oscillatorground state during one trap oscillation period is small compared to the meanenergy of that state. This may be expressed mathematically as:

∆ωx

∆t

2π

ω⊥¿ ω⊥ (9.3)

where ∆t is the time for the spin up ramp (200 ms). For small changes in the trapeccentricity from circular, this adiabaticity criterion may be approximated as:

1− ωx

ωy

∆t¿ ω⊥

2π(9.4)

⇒ αe =εt

∆t

2π

ω⊥¿ 1 (9.5)

where αe is the eccentricity adiabaticity parameter used to characterize the spin upramp. All the ramps used in this chapter satisfied this condition for adiabaticity.In a typical spin up ramp εt changes from 0 to 0.1 over 200 ms in a trap withω⊥/2π = 62 Hz giving αe = 0.01.

For the change in rotation rate to be adiabatic we require [?]:

αr = |∆t ω⊥ (Ω− Ωc2)| À 1 (9.6)

During spin up ramp described above with Ω = 0.56 ω⊥, αr had a value of 11 andso the criterion for rotational adiabaticity was just satisfied. To ensure that it wassatisfied at values of Ω closer to Ωc2 we varied the spin up time between 200 msand 1 s, and detected no difference in the number of vortices formed.

The critical values of Ω and εt for nucleation were extracted from plots such asfig. 9.2 and fig. 9.3 and compiled on fig. 9.4. The data points show the minimumeccentricity required for nucleation at a given rotation rate and map out regionB, within which vortices nucleate. Ωc2 appears to be a critical rotation frequencyat which vortices can be nucleated with minimum eccentricity, as predicted in [?].Changing Ω from 0.71 in either direction requires a more elliptical trap for nucle-ation, although different physical processes control the upper and lower boundariesof region B as explained below.


Figure 9.4: The critical conditions for vortex nucleation. The data points mark theminimum trap deformation for nucleation at a particular Ω. Vortices may be formedin region B. The solid line shows the theoretical limit of stability of one particularquadrupole mode which is stable in region C. This line is in good agreement with theextreme conditions for vortex formation at Ω > Ωc2 . The dashed line shows the predictedminimum trap deformation for nucleation (under non-adiabatic conditions) for our trap

conditions (µc = 13.2 hω⊥) from [?].


9.6 Vortex nucleation mechanisms in our

apparatus

As suggested throughout this chapter, the nucleation of vortices in our experimentis closely linked to the quadrupole modes of the condensate. Section 8.2 shows thatthree different modes emerge from the hydrodynamic equations for a superfluid inan elliptical rotating potential, with different aspect ratios and different regions ofstability (fig. 8.1). In chapter 8 we considered a condensate in a slowly rotatingpotential, under conditions where only the normal quadrupole mode was stable. Inthis chapter we consider conditions where all 3 modes are simultaneously stable.Since they all have the same energy in the rotating frame [?], simulations arerequired to find out which mode the system will follow under specific experimentalconditions [?].

If the eccentricity of the trap is ramped from zero (as in our experiment) ata constant rotation rate Ω > Ωc2 , the condensate follows the lower part of theovercritical branch at small eccentricity, in region C of fig. 9.4. To confirm this wehave observed that the condensate has an elliptical density distribution which isorthogonal to the trap potential. It then nucleates vortices when the eccentricity istoo large for this quadrupole mode to be a solution of the hydrodynamic equations.After nucleation the density distribution is observed to be parallel to the trap axes,confirming that the quadrupole mode has changed. The boundary of the regionin the εt versus Ω plot where the overcritical quadrupole mode exists is given by(fig. 8.1):

εt =2

Ω

2Ω

2 − 1

3

3/2

(9.7)

This relation was determined from the solutions of the hydrodynamic equationsfor superfluids as explained in section 8.2 and it is plotted as a solid line in fig. 9.4.This line agrees well with the experimental data for the critical conditions fornucleation for Ω > Ωc2 and a wide range of εt.

Below Ωc2 , the deformation needed to nucleate vortices appears to increaselinearly with decreasing Ω. This boundary cannot be explained in terms of thestability limit of a quadrupole mode - the ‘normal branch’ (branch I) is stable onboth regions A and B and the ‘overcritical branch’ is stable in neither. Our dataappear to be at variance with the results in [?, ?], where no vortices were seen whenthe eccentricity was increased adiabatically and Ω < Ωc2 . We have observed thaton this branch the elliptical density distribution is parallel to the trap potential,again in agreement with [?].

Mechanisms for the creation of vortices at frequencies below Ωc2 have beenproposed in [?, ?]. The approach used in [?] is to assume that vortex nucleationwill occur (under non-adiabatic conditions) if the vortex can move from the edgeof the condensate to the centre along a path of continuously decreasing energy.

9.6. Vortex nucleation mechanisms in ourapparatus 145

Kinetic energy terms from three different factors must be considered; the vortexvelocity field, the quadrupole velocity field and the interaction between the two.In a trap of fixed deformation εt and normalized rotation rate Ω, there are two freeparameters that may be varied to find a path of decreasing energy, the condensatedeformation εc and the radial vortex position d.

In an axially symmetric condensate (εc = 0) an energy barrier exists betweend = R⊥ and d = 0 for all Ω, as shown in fig. 6.3, making vortex nucleation impossi-ble. The energy barrier may only be overcome if the condensate is free to deform.In an axially symmetric trap Ωc2 = 1/

√2 is significant because it is the lowest ro-

tation frequency at which energy and angular momentum can be transferred to thequadrupole mode and hence that spontaneous quadrupole deformation can occur.If the trapping potential is not axially symmetric then the cylindrical symmetryis broken even at low rotation rates. The freedom to vary both d and εc providesnucleation paths at Ω < Ωc2 . The dashed line in fig. 9.4 shows the minimum Ω atwhich such a path exists as a function of the trap deformation εt for a chemicalpotential close to that used in our experiment [?]. Unfortunately there is not goodagreement between this theoretical curve and our experimentally determined nu-cleation conditions which are also shown in the figure (data points). For a givenrotation rate, the trap deformation that was required for nucleation in our experi-ment was consistently larger than that predicted by this theory. One explanation isthat our spin up conditions were adiabatic, whereas the theory requires the rotat-ing trap to switch on non-adiabatically to create a non-equilibrium configuration.However using the shortest spin up ramp available (50 ms), we saw no change inthe boundary for vortex nucleation even though αr was in the range 0 − 3 andhence the adiabaticity criterion (eqn. 9.6) was not satisfied.

Another approach for explaining the nucleation of vortices below Ωc2 , which isvalid even under adiabatic conditions is given in [?]. Sinha and Castin have shownthat there are regions in the plot of εt versus Ω, both above and below Ωc2 , wherethe quadrupole solutions of the hydrodynamic equations become dynamically un-stable. For frequencies above Ωc2 the predicted instability domains coincide withthe experimentally observed vortex domains in both [?, ?] and this work, thusindicating a link between their instability analysis and vortices. To make a quan-titative prediction for the boundary between regions 1 and 2 shown in fig. 9.4, willrequire further detailed work for our specific case, possibly looking at high ordermodes at the condensate surface, where the hydrodynamic approximation is nolonger valid [?].

Equation 9.1 shows that higher order surface modes can mediate vortex nucle-ation at frequencies below Ωc2 and this has been verified experimentally using alaser beam stirrer of the correct symmetry to excite e.g. hexapole modes [?, ?].However this does not provide an explanation for the observation of vortices belowΩc2 in our experiment because at the radius of the condensate the only significantterms in the trapping potential are quadratic.


Another possible mechanism for observation of vortices below Ωc2 is that thethermal cloud plays an important role. Transfer of angular momentum to thecondensate from the spinning thermal cloud may provide a mechanism for vorticesto form at Ω < Ωc2 . However the transfer rate of angular momentum must begreater than any loss rate due to residual trap anisotropy [?]. In [?, ?], gravityproduces a small static eccentricity in the trap in the plane of rotation. The ‘spindown’ time for a rotating thermal cloud in the presence of a small deformationparameter εt = 0.01 is very short, 0.5 s, compared to the spin up time of 15 s, andhence the cloud may never gain significant angular momentum. However in ourexperiment gravity acts along the rotation axis and hence the trap is symmetricin the plane of rotation, giving a more favourable ratio of spin-up to spin-downtimes.

With this hypothesis in mind, we tested our nucleation curve at lower tempera-ture to see if there was any change when the amount of thermal cloud was reduced.When acquiring the data of fig. 9.4 the rf evaporation field was turned off aftercondensation and some heating was observed during the nucleation procedure, re-sulting in a final temperature around 0.8 Tc. To achieve a lower temperature weleft on the so-called ‘rf shield’ so as to give an effective trap depth of 800 nK dur-ing the nucleation process. This resulted in a temperature of 0.5 Tc. No significantchange was observed in the nucleation curve at this lower temperature. However,this does not totally rule out a role for the thermal cloud in the nucleation processin our experiment since even at 0.5 Tc, there was still a significant thermal cloud(15% of the total number of atoms).

9.6.1 Evidence for vortex decay mechanisms

Although the exact amount of thermal cloud seemed to have little effect on thenucleation conditions, it had a striking effect on the behaviour of vortices afternucleation. Without the rf shield during the nucleation process, vortices were onlyoccasionally found in an equilibrium configuration (i.e. 1 vortex in the centre, 3vortices in an equilateral triangle as in fig. 9.1) and ∼ 400 ms after forming they hadalready moved to the edge of the condensate before disappearing. However with therf shield present holding the temperature at 0.5 Tc, the vortices were normally foundin equilibrium positions ∼ 200 ms after formation and could be reliably observedfor up to 6 s, limited only by the decay of the condensate itself. Figure 9.5 showsthis behaviour clearly; the figure shows that both the mean number of vortices (a)and the number of atoms (b) in the condensate decay at a comparable rate. Forthese data the condensate and vortex were held in a stationary circular trap; afterforming vortices during 1s in an elliptical spinning trap, the trap was ramped backto circular over 200 ms and then the ‘hold time’ began. In a stationary trap, thevortex state is both energetically and dynamically unstable (section 6.5.2) but at0.5 Tc the dissipation mechanisms are sufficiently slow for the vortex lifetime to be

9.6. Vortex nucleation mechanisms in ourapparatus 147

Figure 9.5: The mean number of vortices (a) and number of atoms (b) as a function ofhold time in a stationary circular trap with ω⊥/2π = 62Hz. The vortices were createdwith a 200 ms ramp to a spinning trap, 800ms in the spinning trap and 200 ms back tocircular at which point the ‘hold time’ began. The temperature was held at 0.5Tc with

an rf shield.


determined instead by the decay of the condensate itself. Similar results for thevortex lifetime were obtained if the condensate was held in a rotating trap for theduration of the hold time.

There are two important decay mechanisms for a condensate of 87Rb atoms.At low condensate densities or in a poor vacuum, collisions with the backgroundgas will dominate producing a density independent decay rate, whilst in a densecondensate 3-body recombination will be most important mechanism [?, ?]. Twobody dipolar relaxation is not significant for this isotope. The data of fig. 9.5(b)is not sufficient to determine the density dependence of the decay but we knowthat it is primarily due to 3-body recombination because the condensate lifetimedepends strongly on the trap stiffness and hence the condensate density.

9.7 Conclusion

In summary, we have used a purely magnetic rotating trap to investigate conditionsfor vortex nucleation (after the formation of the condensate) over a wide range oftrap eccentricities. For a given eccentricity, we observe both an upper and lowerlimit to the rotation rate for nucleation. The upper limit confirms the predictionsin [?, ?], that vortex nucleation is mediated by the breakdown of a particularquadrupole mode, but over a much wider range of parameters. Thus our resultsin an oblate trapping geometry compliment previous work in prolate geometriesand support the theory that surface modes of the condensate play a central rolein vortex nucleation [?]. This surface mode nucleation mechanism could be testedfurther by finding the critical conditions for nucleation of a second vortex. Thepresence of the first (centred) vortex will shift the spectrum of surface modes; thusif surface modes play an important role in vortex formation we expect the criticalconditions for nucleation of the second vortex to be shifted relative to those of thefirst. A quantitative calculation of the shift in the critical conditions is given in[?]. This prediction could be readily tested in our apparatus, using the techniquesdeveloped in chapter 10 to reliably create the first centred vortex.

The lower limit to the rotation rate for nucleation is different to that reportedelsewhere. Further theoretical work is required to explain the linear dependenceof εt on Ω shown in fig. 9.4. Finally the thermal cloud is shown to destabilizevortex arrays and so measurements of the vortex lifetime in an oblate geometrywere made with an rf shield to prevent heating.

Chapter 10

The Superfluid Gyroscope

10.1 Introduction

In chapter 5, we demonstrated that a vortex line in a superfluid must be associatedwith quantized angular momentum, to ensure that the superfluid wavefunction issingle valued. If the vortex line is at the centre of an axially symmetric conden-sate of N atoms, then the associated angular momentum of the whole system isquantized into units of Nh. To show that the condensate behaves as a superfluid,we must not only observe the vortices, but also show that the associated angularmomentum is quantized, since classical vortices also exist with a continuous rangeof angular momenta. The superfluid gyroscope described in this chapter, consistsof a condensate with both a single centred vortex line and the scissors mode ex-cited and is used to measure the angular momentum associated with the vortexline. The experiment also raises questions about the motion of the vortex lineduring the 3-dimensional gyroscope motion and the coupling of the scissors modecomponents, the m = ± 1, modes to the vortex line, which will be discussed at theend of the chapter.

The superfluid gyroscope that was realized in this work, was originally sug-gested in a theoretical paper by Stringari [?]. It combines a rapid scissors modeoscillation in the xz or yz plane, with the excitation of a singly charged vortexalong the z axis. In the presence of the vortex the plane of oscillation of the scis-sors mode precesses slowly around the z axis. In polar coordinates, the scissorsoscillation is in the θ direction and the precession is in the φ direction, as shown infig. 10.1. From the precession rate we are able to deduce the angular momentumassociated with the vortex line, 〈Lz〉.

It is worth noting that a ‘gyroscope’ is a general term, describing a frictionlesssystem with a large angular momentum vector, that is able to rotate about any

149

150 Chapter 10. The Superfluid Gyroscope

Figure 10.1: The gyroscope motion. The condensate performs a fast scissors oscillationin the θ direction at ωsc, whilst the plane of this oscillation slowly precesses in the φ

direction at frequency Ωg. The vortex core is not shown in this diagram.

10.2. The theory of the superfluid gyroscope 151

axis1. Our condensate, supported in a frictionless magnetic trap and containing avortex, is an example of such a system, although one must be careful not to drawincorrect analogies to classical gyroscope systems. The nutation or the wobblingmotion superimposed on the precession of a spinning top is not analogous to thescissors motion in our system. The nutation frequency depends on the angularmomentum of the top whereas the scissors frequency is independent of the vortexangular momentum [?].

It is also interesting to compare and contrast this work with the other superfluidgyroscopic effects that have been observed. Superfluid gyroscopes of liquid heliumexhibit persistent currents in toroidal geometries, with many quanta of circulation[?, ?]. In contrast, in our experiment a single vortex of angular momentum 〈Lz〉 =Nh significantly modifies the motion of a trapped BEC gas in an excited state.This possible because the vortex produces relative shifts in the excitation spectrumof order ξ/R0 and in such a dilute system the vortex core size ξ cannot be ignoredwith respect to the size of the condensate, R0 [?]. Closely related experimentswith vortex lines in dilute trapped gases are described in [?, ?, ?]. The angularmomentum of a vortex line was measured in [?] using the precession of a radialbreathing mode (a superposition of m = ± 2 quadrupole modes) in the planeperpendicular to the vortex line. The same method has recently been used atMIT to identify the presence of a vortex with multiple units of circulation [?]. Inthat work, motion is confined to 2 dimensions and the quadrupole oscillation ofthe condensate does not affect the vortex line. In our experiment we observe a3-dimensional interaction between the velocity field of the vortex and the scissorsmode. In [?, ?] the precession of a vortex line is observed in the absence of any bulkcondensate motion, when it is tilted or displaced from the condensate symmetryaxes.

10.2 The theory of the superfluid gyroscope

The relationship between the precession rate Ωg and 〈Lz〉 may be derived by con-sidering the scissors mode as an equal superposition of two counter-rotating l = 2,m = ± 1 modes [?]. These modes represent a condensate tilted by a small anglefrom the horizontal plane rotating around the z axis at the frequency of the scissorsoscillation, ω± = ωsc. The operators for these excitations are:

f± = (x± iy)z (10.1)

The symmetry and hence degeneracy of these modes is broken by the presenceof axial angular momentum 〈Lz〉. Provided that the splitting is small compared

1Gyroscope - An instrument designed to illustrate the dynamics of rotating bodies, and con-sisting essentially of a solid rotating wheel mounted in a ring, and having its axis free to turn inany direction. Oxford English Dictionary online: http://dictionary.oed.com/cgi/entry/00100933


to ωsc, it can be shown (by rearrangement of trigonometric identities) that theprecession rate is proportional to the difference frequency of the two componentmodes

Ωg =ω+ − ω−

2. (10.2)

Using a sum rule approach, Stringari was able to express the splitting between thetwo m = ± 1 modes in terms of the total angular momentum 〈Lz〉 [?]

Ωg =ω+ − ω−

2=

〈Lz〉2mN〈x2 + z2〉 (10.3)

Where N is the total number of atoms in the condensate and m is the atomicmass. It is interesting to note that the denominator of eqn. 10.3 is twice the rigidbody moment of inertia of the condensate for rotational about a radial axis. Onewould intuitively expect that the reduced moment of inertia would characterize aproperty of a superfluid and this will be discussed later in the chapter.

Substituting for 〈x2 + z2〉 in eqn. 10.3 for the case of a harmonically trappedcondensate in the Thomas-Fermi limit, one obtains [?, ?]

Ωg =7ωsc

2

〈Lz〉Nh

λ5/3

(1 + λ2)3/2

(15N

a

aho

)−2/5

(10.4)

where λ = ωz/ωx and ωsc = (ω2x + ω2

z)1/2. This equation was derived indepen-

dently by Fetter and Svidinsky in [?]. They used the hydrodynamic equations(section 5.5), to calculate the splitting of the normal mode eigenfrequencies ω±due to the circulating velocity field associated with a vortex line.

Equation 10.4 links the observed precession rate of the plane of oscillation ofthe scissors mode Ωg to the total axial angular momentum of the condensate 〈Lz〉,provided that the trap frequencies and the number of atoms in the condensate areknown.

10.3 Exciting and observing the gyroscope

Observation of the superfluid gyroscope effect requires a complicated experimentalprocedure. We wish to measure the angular momentum associated with a single,centred vortex line and so the first requirement is to reliably produce a single cen-tred vortices. (The precession data must be built up over many experimental runs,from identical starting conditions, since we are limited to destructive imaging).This requires precise control of the time spent in the spinning trap, the trap rota-tion rate and the trap eccentricity - we were fortunate that the latter is one of thefeatures of our purely magnetic rotating trap. A detailed discussion of the condi-tions for vortex nucleation in our apparatus is given in chapter 9, but in summary,the excitation procedure used for this experiment was as follows: First we produced

10.3. Exciting and observing the gyroscope 153

Figure 10.2: Images of vortices, taken along the axis of rotation, immediately after thenucleation procedure (12ms of expansion used). (a) shows a nicely centred vortex. (b)

shows a vortex that is just outside our criterion for an acceptably centred vortex.

a condensate in a circular TOP trap with ω⊥/2π = 62 Hz and ωz/2π = 175 Hz.To spin up the condensate we made the trap eccentric (ωx/ωy = 1.04) over 0.2 s,with a trap rotation rate of 44 Hz. After holding the condensate in the spinningtrap for a further 1 s we allowed the trap to spin down by ramping both the traprotation rate and the trap eccentricity to zero over 0.4 s.

Using the imaging system that looks along the axis of rotation (z axis), wewere able to observe that the nucleation conditions above produced ‘acceptably’centred vortices over 90% of the time. Variation of the vortex position causes avariation in the precession rate and hence a dephasing between experimental runs.Our criterion for an acceptably centred vortex is based on the maximum dephasingthat we can tolerate whilst still observing a clear precession. An acceptably centredvortex was one that lay within a third of the condensate radius from the centre.From eqn. 6.30, which gives the angular momentum of a vortex line as a function ofposition, this corresponds to a maximum reduction in the precession rate of 25%.Given that we are only able to observe around ∼ 0.5 of a precession period (due toLandau damping), significant dephasing will not build up over the limited durationof the experiment. Figure 10.2 shows images of the condensate taken along theaxis of rotation, immediately after the nucleation process. A well centred vortexis pictured in (a) whilst in (b) the vortex is just beyond our limit for acceptablecentering. The relationship between the precession rate and 〈Lz〉 is also affectedby variations in the atom number (eqn. 10.4). The measured number of atoms wasN = 19, 000 ± 4000 atoms. The variation in N leads to only a 10% variation inthe precession rate, which does not cause significant dephasing.

Immediately after making a vortex, the TOP trap was suddenly tilted to excite


either the xz or yz scissors mode. This was achieved by applying an additionalmagnetic field to the TOP trap in the z direction, oscillating in phase with oneof the radial TOP bias-field components, Bx or By, to excite either the xz or yzscissors mode respectively (chapter 7). The amplitude of this field was 0.55 G,which combined with a 2 G radial bias field tilted the trap by 4.4 degrees andhence excited a scissors oscillation of the same amplitude about the new tiltedequilibrium position.

After allowing the condensate oscillation to evolve for a variable time in thetrap, we released the condensate and destructively imaged along the y directionafter 12 ms of expansion. By fitting a tilted parabolic density distribution to theimage, we extracted the angle of the cloud and thus gradually built up a plot of thescissors oscillation as a function of evolution time. The visibility of the fast scissorsoscillation depends on the angle of the cloud projected on the xz plane (the planeperpendicular to the imaging direction) and hence varies at the slow precessionfrequency Ωg. If the oscillation is in the xz-plane then the projected amplitudeis maximum and if it is in the yz-plane then the projected amplitude is zero. Byplotting the scissors oscillation as a function of evolution time we observed theslowly oscillating visibility and hence extracted the precession rate.

Note that the excitation procedure takes nearly 2 s, (the majority of whichis used for vortex nucleation), during which time the number of atoms in thecondensate decays by a factor of 2. Thus it is necessary to start with a largecondensate, so that after decay there are still sufficient atoms to produce a sharpimage, which can be reliably fitted with a parabolic density distribution. In ourfirst attempt at this experiment we had < 10,000 atoms for imaging and typicalimages are shown in fig. 4.7(a) and (b). The absorption is so weak that fringingon the imaging system gives the condensate an irregular shape and the effect ofthe large vortex core (radius ∝ 1/

√n) on the density profile cannot be ignored.

The fitting program was unable to accurately determine the angle of the cloudand so a clear variation in the scissors visibility was not observed. This motivatedthe optimization procedure described in chapter 4, which increased the condensatesize by a factor of 3. Typical gyroscope images after optimization are shown infig. 4.7(c) and (d). The narrower vortex core is no longer visible and the strongerabsorption means that fringing effects are no longer significant.

10.4 Gyroscope results

In the first gyroscope experiment, the scissors oscillation was initially excited in thexz plane, perpendicular to the imaging direction. The resulting gyroscope motionis plotted as a function of time, projected along the y-direction, in fig. 10.3(a). Fig-ure 10.3(b) is a control run under identical conditions, except that the condensates

10.4. Gyroscope results 155

Figure 10.3: The angle of the cloud projected on the xz plane when the scissors modeis initially excited in the xz plane, in (a) with a vortex and in (b) without a vortex. In(a) each data point is the mean of 5 runs and the standard error on each point is shown.The solid line is the fitted function given in eqn. 10.5. In (b) most data points are anaverage of 2 runs; occasionally 5 runs were taken and just for these points the standard

error is shown for comparison with (a).


did not contain a vortex. The fitting function used for each was

θ = θeq + θ0 |cos Ωgt| (cos ωsct) e−γt (10.5)

with Ωg set to zero for fig. 10.3(b). In both cases the fast scissors oscillation isclearly visible and the fitted values of ωsc/2π of (a) 179 Hz and (b) 186 Hz agreereasonably well with the theoretical value of 177 Hz. In the presence of a vortexthe visibility shrinks rapidly to zero over 30 ms as the oscillation precesses through90 to a plane containing the imaging direction. The oscillation visibility growsagain after a further 90 precession. In the limit of small tilt angles the variationin oscillation visibility is represented by the |cosΩgt| term in eqn. 10.5. (Considerthe projected angle of a rod in the xz plane, which is slightly tilted away from thez axis and precessing around it at frequency Ωg). Note that 2π/Ωg is the time for afull 2π rotation and hence we expect the visibility to fall from maximum to zero ina quarter period, π/2Ωg. The fitted value of Ωg/2π = 8.3± 0.7 Hz. From eqn. 10.4this gives an angular momentum per particle, 〈lz〉, of 1.14 h ± 0.19h for N =19,000 ± 4000 atoms. In fig. 10.3(a), each data point was taken 5 times and themean and standard deviation is plotted. This averaging was necessary because theslight shot-to-shot variation in the starting conditions, produces slightly differentprecession rates.

The revived amplitude is smaller than the initial amplitude due to Landaudamping, which occurs at a rate of γ = 23 ± 7 Hz from the exponential decayterm in eqn. 10.5. Damping also occurs at a similar rate of γ = 25 ± 5 Hz inthe control run, fig. 10.3(b), without the presence of a vortex. Note that in (b)the condensate underwent the same spinning up procedure but at a trap rotationrate of 35 Hz, just too slow to create vortices. This ensured that in both casesthe condensates were at the same temperature and hence had comparable Landaudamping rates. The damping rates of approximately 24 Hz at a temperature of0.5 Tc agree well with the data about the temperature dependence of the scissorsmode published in [?]. The control plot also confirmed the theory that an axiallysymmetric condensate must have 〈Lz〉 = 0 unless a vortex line is present and hencethe vortex is essential for precession.

Figure 10.4 shows the same experiment but with the scissors mode initiallyexcited along the imaging direction so that the initial visibility is zero. The ap-propriate fitting function in this case is

θ = θeq + θ0 |sin Ωgt| (cos ωsct) e−γt (10.6)

There was insufficient data to fit the Landau damping rate accurately in this caseand so the value of γ was fixed at 24.2 Hz, as determined from the data of fig. 10.3.In the presence of a vortex (fig. 10.4(a)) the visibility of the scissors oscillationgrows as the oscillation plane rotates through 90 to the xz plane. This growthof an oscillation is perhaps a more significant proof of precession than the initial

10.4. Gyroscope results 157

Figure 10.4: The angle of the cloud projected on the xz plane when the scissors modeis initially excited in the yz plane, in (a) with a vortex and in (b) without a vortex. In(a) each data point is the mean of 5 runs, with the standard error on each point shown.The solid line is the fitted function given in eqn. 10.6. In (b) most data points are anaverage of 2 runs, occasionally 5 runs were taken and the standard error is shown for

these points for comparison with (a).


decrease of amplitude in fig. 10.3(a), since it cannot be explained by any dampingeffect. The precession rate from fig. 10.4(a) is 7.2 ± 0.6 Hz, which agrees within thestated errors with the precession rate fig. 10.3(a) and gives 〈lz〉 = 0.99 h ± 0.17h.In fig. 10.4(b) there was no vortex and so the oscillation remained in the yz plane,with zero angle projected onto the xz direction.

Note that the mean angles in fig. 10.3(a) and fig. 10.4(a) are different. Thismean angle corresponds to the trap angle (the cloud angle in equilibrium) in thevisible xz plane. In fig. 10.3 the trap tilt occurs in the xz plane and so this meanangle is θeq, whereas in fig. 10.4 the tilt is in the yz plane, and so the mean anglein the imaging plane is zero.

Combining the results for the xz and yz gyroscope experiments, we measure theangular momentum per particle associated with a vortex line to be 1.07 h ± 0.18 h.This is in excellent agreement with the value of h per particle predicted by quantummechanics. It is interesting to note that we deduce an angular momentum perparticle slightly greater than h, even though only one vortex is visible. Given thatthis vortex is not always perfectly centred, one might expect to observe a value forΩg and hence 〈Lz〉 that is slightly lower than the theoretical value. The additionalangular momentum must be due to the presence of additional vortices since weare using an axially symmetric condensate with zero moment of inertia about thez-axis. This idea is backed up by the work of Chevy et al. [?], which suggests thatunder conditions where they could reliably create a single centred vortex, they alsocreated vortices at the edge of the cloud that make a small additional contributionto the angular momentum (eqn. 6.30). These vortices will be in a region of very lowdensity and so may not create sufficient contrast to be observed after expansion.Under rotation conditions which reliably produced a single centred vortex line,they measured an angular momentum of 〈Lz〉 = 1.2 Nh, using the precession of aquadrupole breathing mode.

10.5 How does the vortex core move?

One question that still remains open is ‘How does the vortex core move during thegyroscope motion?’. Does it remain stationary along the z-axis of the trap, or doesit oscillate, locked to the axis of the condensate? Our investigations were originallyprompted by the suggestion (without a theoretical explanation) in [?] that the coremight exactly follow the axis of the condensate and have since followed a varietyof routes:

• Further investigation of the precession rate theory to see if it can also beused to predict the vortex motion.

• Looking at images of the vortex during the gyroscope motion taken perpen-dicular to the axis of rotation.

10.5. How does the vortex core move? 159

Figure 10.5: Sideview images of the gyroscope motion with < 10,000 atoms, in whichthe vortex line is clearly visible. The images were taken along the y-axis after 12 ms ofexpansion. All the images support the hypothesis that the vortex line moves with the

axis of the condensate apart from (i)

• Developing a mechanical rigid body model of our experiment

The theory that was used to predict the gyroscope precession rate [?, ?] doesnot give immediate information about the two possible motions of the vortex core.For example, Fetter and Svidinsky [?] use the linearized hydrodynamic equationsto consider the interaction of the vortex velocity field with a very weakly excitedm = +1 or m = −1 mode. This interaction lifts the degeneracy of the modes andfrom this splitting the precession rate is calculated. The theory only includes Lz,


which to first order in the excitation amplitude (or θ [?]) is constant, and so thistheory is insensitive to any small angle motion of the vortex core.

Observing the motion of the vortex line relative to the condensate was tech-nically very difficult but the data that we have generally confirms the idea thatthe vortex line does indeed move with the axis of the condensate. First we triedto observe the vortex line by imaging along the y direction, but after optimiz-ing the experiment we had little success. The imaging beam integrates throughthe entire condensate and thus a narrow vortex core (∝ 1/

√n) at the centre of

a large condensate will not modify the parabolic density profile sufficiently to beobserved. However the gyroscope data that was taken prior to optimizing the ex-periment, with the atom number a factor of 3 lower, contained a clearly visiblevortex in about 50% of the images. Some of the clearest vortex images are shownin fig. 10.5. Despite the small tilt angle and the noise level on the images, theygenerally confirm the hypothesis that the vortex line tilts with the axis of the con-densate, if we assume that the relative angles after expansion reflect the relativeangles at the instant of release. Only image (i) in fig. 10.5 appears to show aslightly tilted condensate but a vortex line along the z-axis.

Finally we studied a mechanical rigid body model of our condensate system.Although our initial hope, that we could use the model to make a quantitativecalculation of the precession frequency in our superfluid system, proved to beflawed, it provided useful physical insight into the gyroscope motion and so isworth discussing briefly. The model, shown in fig. 10.6 consists of a disk free tospin about its own main axis with an angular momentum Ls (Ls ≈ Lz in the limitof small angle motion) that is analogous to that of the vortex line. The disk ismounted (with frictionless bearings) in the horizontal plane on a vertical wire, sothat its main axis and the wire are initially parallel. The wire is elastic, so thatany small displacement of the disk from the horizontal plane results in a strongrestoring torque, analogous to the torque provided by the magnetic trap.

The full equation of motion of this system is:

dL

dt= Γ (10.7)

In the limit of small angles it has a gyroscope-like solution described by the fol-lowing coupled equations:

θ = −ωrig2θ (10.8)

φ = Ωrig =Lz

2Θrig

(10.9)

where Θrig is the moment of inertia for oscillation about the x or y axes and

ωrig =

k

Θrig

+

(Lz

2Θrig

)2

1/2

(10.10)

10.5. How does the vortex core move? 161

Figure 10.6: A mechanical model of the trapped condensate. The ellipsoidal disk (rep-resenting the condensate) is free to rotate about its axis (indicated with dotted lines)with angular momentum Ls, analogous to the vortex line in the condensate. The disk ismounted on elastic strings that provide a restoring torque analogous to that exerted by

the magnetic trap on a condensate tilted out of the horizontal plane.


Equations 10.8 and 10.9 give the ‘scissors’ frequency and precession rate of therigid body model under conditions where we know that the spin angular momentumvector follows the axis of the system. At this point we had hoped to predict theprecession frequency of a superfluid system in which the vortex core follows theaxis of the condensate, by replacing the rigid body moment of inertia Θrig ineqn. 10.9 with the appropriate superfluid one. If the result was in agreement withexperiment then this would provide additional evidence that the vortex core doesfollow the axis of the condensate during the gyroscope motion. This idea worksvery well for predicting the scissors mode frequency of the condensate, which onlydepends on the moment of inertia of the system and the torque applied by the trap.

(From eqn. 10.10 we have ωrig ≈√

k/Θrig and from chapter 7 the scissors frequency

of the condensate is given by ωsc =√

k/ΘS). However the precession frequencydoes not arise from a simple torque and moment of inertia but instead from theinteraction between the velocity field of the vortex and that of the m = ± 1 modes.The interaction of the vortex with the irrotational m = ± 1 velocity field in thecondensate will not necessarily be related to the interaction of the vortex with therotational m = ± 1 velocity field in the rigid body. So in this case simply replacingthe rigid body moment of inertia in eqn. 10.9 with the appropriate superfluid onewill not predict a valid condensate precession rate.

In summary, we have some experimental evidence to suggest that the vortexcore moves with the axis of the condensate, but only a full simulation of the systemwith the correct initial conditions will finally settle the issue. Images of the gyro-scope taken perpendicular to the core with a reduced number of atoms (thus underconditions where the gyroscope motion is not necessarily reliable), suggest that thevortex core does follow the condensate axis. The quantum mechanical calculationfor the precession rate is insensitive to motion of the vortex core provided that anytilt angle remains small and produces a result which depends, somewhat surpris-ingly, on the rigid body moment of inertia of the condensate. Finally, whilst ourrigid body model is helpful for gaining physical insight into the superfluid gyro-scope system, it cannot be used to make quantitative predictions of the precessionrate.

10.6 Kelvin waves and the gyroscope experiment

Recent experiments in Paris [?] have observed the preferential damping of them = −2 quadrupole mode, over the m = +2 mode in the presence of a singlecentred vortex line. The lifetime of the m = +2 mode is 42 ms, while the m = −2mode has a lifetime of only 18 ms. Two different theories have been suggested toexplain this effect and our gyroscope data has provided useful evidence in favourof one of them.

The first suggestion [?] is that the thermal cloud is spinning in the same sense as

10.6. Kelvin waves and the gyroscope experiment 163

the condensate around the vortex core. The m = −2 mode represents a deformedcondensate ‘rotating’ against the flow of the thermal cloud, which is thus dampedmore rapidly than the m = +2 mode which ‘rotates’ with the thermal cloud. Thesecond suggestion is that the m = −2 mode couples to helical Kelvin waves, whichare excitations of the vortex core. As explained in section 6.5.4 each kelvon carrieslinear momentum ± hk along the z-axis and angular momentum −h, since thesense of rotation of the Kelvin wave is always opposite to the flow around thevortex core. To conserve energy, angular and linear momentum (ω, Lz, kz), thedecay process will be:

Phonon(ω−2,−2h, 0) → Kelvon(ω−2

2,−h, k) + Kelvon(

ω−2

2,−h,−k) (10.11)

Conservation of angular momentum makes this decay mechanism impossible forthe m = +2 mode, since kelvons only carry angular momenta opposite to that ofthe vortex line, explaining why the m = −2 mode damps more quickly.

The gyroscope experiment involves an equal superposition of the l = 2,m = ± 1modes, rather than the m = ± 2 modes discussed above. An unequal damping ratecomparable to that observed for the m = ± 2 modes, would cause the gyroscopemotion to break down within ∼ 20 ms. Since we observe a well defined gyroscopemotion for over 60 ms (fig. 10.3(a)), we conclude that both the m = ± 1 modesare equally damped. This observation has been used to support the theory thatcoupling to Kelvin waves, rather than interaction with the thermal cloud, is theprimary damping mechanism for the m = −2 mode at ENS. If motion againstthe thermal cloud does not cause significant damping, then we would expect toobserve an equal decay rate for the m = ± 1 modes in our experiment; resonantcoupling to a Kelvin mode is unlikely in an oblate trapping potential (section 6.5.4)and conservation of linear and angular momentum prevents either of the m = ± 1modes coupling Kelvin waves by a first order process.

Chapter 11

Conclusion and Future Plans

This thesis provides an extensive experimental study of the superfluid nature of adilute gas 87Rb condensate, exhibited by its response an applied torque. Over thelast four years the choice of experiments has been motivated by various factors.Firstly there has been a high level of theoretical interest in the superfluid propertiesof the dilute gas Bose condensate both from theoreticians within the immediatefield and also from those in the liquid helium research. In fact the most importantparallels and comparisons between these two well studied Bose-condensed systems,liquid helium II and dilute alkali condensates, concern their superfluid properties.Many of the experiments in this thesis have close analogies in the work that hasbeen carried out on liquid helium over the last 60 years.

Another important factor has been the development of a very flexible magnetictrapping potential that is ideally suited to studying the superfluid nature of thecondensate. It has two important features:

• A wide variety of well defined torques may be applied to the condensate, byrotating or tilting the potential about any axis. The potential may be rotatedat any frequency, both clockwise and anti-clockwise, stopped suddenly at anyangle and oscillated between two angles. Many of these features have beenused in the experiments described in this thesis and some will be used infuture work.

• The response of the condensate to any perturbation is determined not onlyby its superfluid nature but also by the spectrum of low-energy collectivemodes. This spectrum depends on the geometry of the trapping potentialand so we have developed the ability to change all three trapping frequen-cies independently [?]. Two experiments, which are not directly related tosuperfluidity and have not been included in this thesis, have made use ofthis technique to bring one collective mode into resonance with twice the

164

11.1. Summary of results 165

frequency of the other. In [?], up-conversion was observed between the twom = 0 breathing modes (section 5.6). In [?] down-conversion was observedbetween the xz and xy scissors modes.

11.1 Summary of results

In this thesis, four experiments are described, each demonstrating the superfluidresponse of the condensate to a different applied torque. The first was the scissorsmode experiment (theory [?], expt. [?]), which together with the observation ofquantized vortices [?, ?] and a critical velocity for superfluid flow [?], providedthe first experimental evidence that the condensate behaves as a superfluid. Af-ter a sudden tilt of the trapping potential, the thermal cloud performed a heavilydamped oscillation at two frequencies, indicating the presence of both rotationaland irrotational flow patterns. Under the same excitation conditions, the con-densate oscillated at a single undamped frequency corresponding to the purelyirrotational flow pattern predicted for a superfluid.

The scissors mode experiment was repeated at a range of temperatures between0.3 Tc and Tc so that both condensate and thermal cloud were excited together [?].The oscillation frequency and damping rate of each fraction was measured. A largenegative shift in the frequency of the condensate oscillation was observed above0.7 Tc which signaled the reduction in the superfluidity of the condensate fraction.The effect of the thermal cloud on the condensate cannot be fully explained interms of a simple damped harmonic oscillator model and requires an analysis whichincludes the large thermal occupation of the low-lying excited states.

In chapter 8 the pure irrotational flow pattern of a vortex-free condensate isdemonstrated by its behaviour after being released from a slowly rotating ellipticalpotential (theory [?], expt. [?]). During the expansion the condensate never be-comes symmetric about the axis of rotation as this would produce a system withzero moment of inertia and thus infinite kinetic energy.

A detailed investigation into the critical trap conditions (trap deformation androtation rate) for vortex nucleation is presented in chapter 9 [?]. Above the criticalrotation rate given by Landau theory we confirm that nucleation is mediated bythe breakdown of a particular quadrupole mode. Nucleation is also observed atlower rotation rates but the mechanism has yet to be understood.

Finally the gyroscope experiment measures the angular momentum of a single,centred vortex line, using a method analogous to the Sagnac effect. The circulat-ing velocity field of the vortex breaks the degeneracy of the two counter-rotatingcomponents of the xz scissors mode, producing a precession of the scissors mode ata rate proportional to Lz. From the observed precession we measure the angularmomentum associated with the vortex line to be 1.07(±0.18)Nh.

166 Chapter 11. Conclusion and Future Plans

11.2 Future experiments

Several new experiments are ready to be implemented, making use of the newmodifications to our trapping potential and the techniques developed for previousexperiments (e.g. the production of a single, centred vortex line). These ideas arecollected together in this section for future reference.

11.2.1 Nucleation of multiply charged vortices

Nucleation of multiply charged vortices (with a phase winding of 4π and 8π) hasrecently been observed at MIT [?] using ‘Berry’s phase’ [?, ?]. Berry’s phase is anadditional phase factor required by the time-dependent Schrodinger equation. Ithas no time dependence but is revealed by adiabatically varying the parametersthat appear in the Hamiltonian of the system around a closed loop, e.g. rotatingthe individual atomic spin vectors within our condensate. Alternatively if thepaths of two different atomic spin vectors may be combined to create a closed loopthen a relative phase is acquired between the two atoms. If the path of the spinvectors draws out an area on the surface of a sphere, then the relative phase isequal to the solid angle subtended, multiplied by the magnitude of the spin vector.

In the experiment described in [?], the bias field of a Ioffe-Pritchard trap Bz

was gradually brought to zero and reversed over a period of ∼ 10 ms. The atomicspin vectors followed this field adiabatically, initially pointing along +z and finallypointing along −z. The plane in which each spin rotates between +z and −zcontains the z axis and the direction of the local 2D radial quadrupole field (whichdominates when Bz is close to zero), which in turn depends on the azimuthalposition of the atom (fig. 11.1(a)). Thus spin vectors at two different positions(φ1, φ2) around the z axis, mark out a segment on the surface of a sphere and theatoms acquire a relative phase of:

Srel =φ1 − φ2

2π× 4π ×mF (11.1)

If Bz = 0 passes through the condensate once, a condensate in the mF = 2 state(as in our experiment) acquires a phase winding of 8π around the z axis whichresults in a single vortex with 4 units of circulation.

After producing a vortex by this method, the authors of [?] report the ob-servation of a large hole in the density distribution and measure the associatedangular momentum using the precession of the radial breathing mode [?, ?]. How-ever, in the process of reversing the direction of Bz, the Ioffe-Pritchard potential istransformed from a trapping potential to an anti-trapping potential with negativecurvature. Thus after reversing the field the condensate may only be observedfor ≤ 50 ms before it is lost. This timespan is not long enough to observe thedecay of the energetically unstable multiply charged vortex into an array of singly

11.2. Future experiments 167

Figure 11.1: The magnetic field in the z = 0 plane for (a) the 2-d quadrupole field ofa Ioffe-Pritchard trap, (b) the 3-d quadrupole field of the TOP trap. In both cases the

direction of the local field is determined the azimuthal position.

charged vortices. Nor is it possible to cause the spin vectors to rotate several times,generating a vortex with even higher circulation.

Using a TOP trap it should be possible to make a multiply charged vortex andkeep the condensate trapped, so that the decay to single vortices may be observed.The TOP trap uses a 3-dimensional quadrupole field, which has a different geome-try in the z = 0 plane from the 2-dimensional quadrupole field of a Ioffe-Pritchardtrap (see fig. 11.1). However the local field direction is still a unique function of theazimuthal position within the z = 0 plane, which is the crucial feature for creatinga Berry’s phase winding around the z axis. The experimental scheme which hasbeen proposed is shown schematically in fig. 11.2.

After evaporative cooling and an adiabatic expansion, the condensate is heldin a standard TOP trap, centred on z = 0, with trap frequencies of ω⊥/2π = 60 Hzand ωz/2π = 170 Hz. The rotating radial bias field is turned off and a static biasfield Bz turned on, in a time that is short compared to the trap oscillation periodbut sufficiently long for the spins to follow the changing field adiabatically. Thisproduces a pure quadrupole trap, centred on z0 = Bz/2B

′q. The condensate is

still at z = 0 and has its spins aligned to the bias field. The condensate is thenallowed to complete half of a dipole oscillation, moving from z = 0 to z = 2z0.Viewing this motion from the frame of the cloud, the magnetic field in the zdirection falls to zero, leaving only the small radial components of the quadrupolefield and then increases in the opposite direction, sufficiently slowly for the atomicspins to follow the total field adiabatically. Thus when the condensate arrives atz = 2z0 it has acquired a Berry’s phase winding and contains a vortex with 4units of circulation. To re-trap the condensate, Bz is doubled and the radial biasfield is increased from zero creating a standard TOP trap centred on 2z0 which‘catches’ the condensate when it is instantaneously stationary. As before the fieldsare changed rapidly compared to the trap period but slowly enough for the spins


Figure 11.2: The scheme for exciting and trapping a vortex with 4 units of circulation.Initially (a) the condensate is in equilibrium with a TOP trap centred on z = 0. A staticBz is turned on and BT reduced to zero in a time that is short compared to ωz (b). Thecondensate is allowed to perform half of a trap oscillation (c). When it instantaneouslystops at 2z0, the spin vectors have rotated with the magnetic field through 180 andthe vortex has formed. As the condensate is briefly stationary, Bz is doubled and BT

is increased to its original value, moving the TOP trap centre to 2z0 and catching thecondensate there.


to follow. The decay of the multiply charged vortex can be observed by holdingthe condensate in the trap for a varying time, releasing it, allowing it to expand,imaging along the axis of rotation and counting the number of holes in the densitydistribution. Alternatively, iterating this procedure with an ever increasing axialbias field would produce vortices with 8, 12, 16 etc. units of circulation.

11.2.2 Exciting and observing Kelvin waves

The gyroscope work demonstrated that we can produce a single centred vortex innine out of ten condensates, providing the reliable initial conditions necessary for arange of experiments on the properties of a single vortex line. One such experimentis the investigation of the spectrum of Kelvin waves on a vortex line. Experimentalwork in this area has so far been limited to the observation of different dampingrates for the m = +2 and m = −2 quadrupole modes at ENS when a vortex lineis present; one likely explanation is that one mode couples to a Kelvin wave of thevortex line whereas the other does not (as described in section 10.6).

We hope to measure the energies of the lowest Kelvin waves by coupling themresonantly to a collective excitation of the condensate. In a prolate trap such asthat at ENS, the lowest Kelvin modes and mode spacings are several orders ofmagnitude lower than the collective mode frequencies and so several high orderKelvin waves are likely to couple to the m = −2 mode rather than one individualone. In contrast, in our oblate geometry the lowest Kelvin modes have similarenergies and energy spacings to the low-lying collective modes of the condensate.In chapter 6 frequencies of 0.44 and 1.8 ωz are derived for the two lowest Kelvinmodes in a standard TOP trap with ω⊥/2π = 62 Hz and λ =

√8.

The m = −2 mode, with frequency√

2 ω⊥, is a suitable driving excitationbecause it couples to Kelvin waves in a first order process described by eqn. 10.11.Each m = −2 phonon decays into 2 kelvons of equal energy to conserve momentum,thus resonant coupling should be possible when a Kelvin mode has frequencyω⊥/

√2. Figure 11.3(a) shows the frequencies of the two lowest Kelvin modes

(eqn. 6.42) and ω⊥/√

2 plotted as a function of radial trap frequency in a standardTOP trap, with λ =

√8. All the modes depend on the overall trap stiffness in a

similar manner and so the curves only intersect at ω⊥ = 0. Thus the resonancecondition cannot be satisfied with λ =

√8. However fig. 11.3(b) shows that the

resonance condition can be met for the lowest Kelvin mode in a trap with ω⊥/2π =62 Hz and λ = 2.02. This is within the range of trap aspect ratios that we havealready achieved using the method described in [?].

A possible driving mode for the second Kelvin wave would be the m = −1mode which has a frequency of (1 + λ2)1/2 ω⊥. However the coupling mechanismis a weaker second order process (involving 2 phonons) and so will require a largeamplitude m = −1 mode to observe any significant energy transfer. (Conservationof angular momentum requires each m = −1 phonon to convert to a single kelvon,


Figure 11.3: Plots showing how the lowest Kelvin mode can be brought into resonancewith half of the m = −2 mode frequency, ω⊥/

√2. Both plots use N = 15, 000. (a) shows

the first and second Kelvin mode frequencies (dashed and dotted lines respectively)plotted from eqn. 6.42 against radial trap frequency, in a standard TOP trap with λ =√

8. With this trap aspect ratio, the Kelvin waves are only resonant with ω⊥/√

2 (solidline) in a trap of zero stiffness and so coupling cannot be achieved. (b) shows the firstKelvin mode frequency (dashed line) and ω⊥/

√2 (solid line) as a function of trap aspect

ratio with ω⊥/2π fixed at 62Hz. Coupling to the m = −2 mode is possible at an aspectratio of 2.02.

but conservation of linear momentum requires 2 kelvons to be generated, hencetwo phonons are involved in every decay process.) Simulations by Max Kruger [?]show the vortex line behaving very differently when the m = +1 and m = −1modes are excited. It shows no response to the former but oscillates wildly inthe presence of the latter indicating that sufficient energy transfer is theoreticallypossible even by this second order process. Again it will be necessary to changethe trap aspect ratio and resonance can be achieved between the m = −1 modeand the second Kelvin wave in a trap with ω⊥/2π = 62 Hz and λ = 1.60. Such atrap aspect ratio is outside the range available with our current apparatus, but anexperimental scheme for achieving aspect ratios as low as 1 is outlined in [?].

To measure the frequency of a Kelvin wave, we wish to find the trap conditionsunder which a peak is observed either in the damping rate of the collective modeor in the excitation rate of the vortex line. From the trap conditions at resonancethe frequency of the collective mode (of multipolarity m) can be calculated to highaccuracy, which will be equal to m times the Kelvon frequency. The excitationrate of the Kelvin mode can be measured in two ways. First we can look along thecore of the vortex after a fixed period of driving and record the core visibility as afunction of the collective mode frequency. Good visibility implies that the vortexis lying straight along the z axis and few Kelvons are excited. Poor visibilityimplies that the vortex is highly excited and bent or tilted away from its original


direction. (One would have to ensure that the focusing was consistent from shotto shot otherwise other factors would affect the visibility). Alternatively one couldimage the condensate from the side as in fig. 10.5 and observe the shape andangle of the vortex core after a fixed driving period as a function of collectivemode frequency. The gyroscope work suggests that one would need to use smallcondensates N ≤ 10000 atoms to obtain clear side-on vortex images.

Given that the boundary conditions used to calculate the Kelvin mode frequen-cies in section 6.5.4 are approximate, the trap frequencies given above only providean estimate of the conditions under which coupling will occur. I expect that side-on imaging will provide a quick method for scanning a large range of trap aspectratios to roughly locate the resonance. Then several measurements of the collec-tive mode damping rate should be taken close to resonance and plotted againstthe trap aspect ratio. Interpolation of these points should accurately identify theresonant trap conditions and hence the Kelvin frequency.

11.2.3 Damping of the m = 2 modes in the presence of avortex

This brief experiment is closely related to the investigation of the Kelvin wavespectrum discussed above. There is still debate about why different damping rateswere observed at ENS for the m = −2 and m = +2 modes in the presence of thevortex (see section 10.6). It has been argued that the more rapid damping of them = −2 mode is not due to the excitation of Kelvons but due to its interaction withthe counter-rotating thermal cloud [?]. Repeating the experiment in our oblatetrap, under conditions where the m = ± 2 modes are far from any kelvon resonance,should provide conclusive evidence. If the m = −2 mode is also preferentiallydamped in our experiment then this must be due to its interaction with the counter-rotating thermal cloud because the off-resonant coupling to Kelvin waves will bevery small. However if equal damping rates are observed in our experiment, thenthe additional damping of the m = −2 mode observed at ENS was due to couplingto one of the many near resonant Kelvin waves that exist in a prolate trappinggeometry.

11.2.4 Critical conditions for nucleating a second vortex

As discussed in section 9.7, the critical trap conditions for nucleating a secondvortex (in the presence of a single centred vortex) have been calculated in [?],on the assumption that vortex nucleation is mediated by surface modes of thecondensate. These critical conditions are shifted with respect to those of thefirst vortex because the first vortex modifies the spectrum of surface modes ofthe condensate. Observation of these shifted nucleation conditions would providefurther evidence for the role played by surface modes in the vortex nucleation


process.

11.2.5 Anti-vortex production

After nucleation of the first vortex, we now have the ability to reverse the directionof the rotating potential and create an anti-vortex. Observation of the dynamicsand possible annihilation of the two vortices would be very interesting although alittle difficult with destructive imaging.

11.2.6 Precession with an off-centred vortex line

In section 6.5.3 we show that the angular momentum of a vortex line is a functionof its radial position. Thus the precession rate of the radial breathing mode [?, ?]will also be a function of vortex position. In [?] two different formulae are presentedfor the precession rate as a function of vortex position, one calculated using a sumrule method and the other by considering the velocity field associated with thevortex as a perturbation. Both give the same result when the vortex is centred,but differ by a factor (1 − d2/R2

⊥) when the vortex is off centre by a distance d.One would expect the perturbative approach to produce the correct answer but itwould be interesting to verify this experimentally.

First a vortex would be nucleated using a reduced time in the spinning trap,so that its radial position varies between experimental runs. The radial breathingmode would be excited along known axes, allowed to evolve for a fixed time ∆tand then imaged. Two quantities may be extracted from the image. The first isthe radial position of the vortex d. The second is the angle through which themajor axis of the oscillation has precessed ∆φ. Thus over many experimental runsthe precession rate ∆φ/∆t could be plotted as a function of vortex position d andcompared to the two theories presented in [?].

Appendix A

The Properties of a 87Rb Atom.

Quantity Symbol Value

atomic number A 87nuclear spin I 3/2

mass m 1.45× 10−25 kgvapour pressure See below 2.34× 10−7 torr (293K)

D1 wavelength 5S1/2 → 5P1/2 795nmD2 wavelength 5S1/2 → 5P3/2 λ 780.026 nm

wavenumber k = 2π/λ 8.05× 106 m−1

natural linewidth Γ 2π × 5.76 MHzDoppler width (300 K) ∆fDoppler 516 MHz

saturation intensity Isat 3.14 mW cm−2

resonant absorption cross-section σ0 2.9× 10−13 m2

recoil velocity vr = hk/m 5.85 mm s−1

recoil temperature Tr = mv2r/2kB 180 nK

gF factor for lower state gF (5S1/2, F = 2) 1/2gF factor for excited state gF (5P3/2 F = 3) 2/3s-wave scattering length a 110ao = 5.82× 10−9m

interaction parameter g = 4πh2a/m 5.66× 10−51Jm3

s-wave collision cross-section 8πa2 8.51× 10−16m2

Table A.1: The properties of 87Rb relevant to this experiment. Apart from the D1wavelength, all the spectroscopic properties are quoted for the D2 transition, which is

the only transition used in this experiment.

Vapour pressure formula:

Log10(P ) = A−B/T + CT + Dlog10T (A.1)

173

174 Appendix A. The Properties of a 87Rb Atom.

where T is in K and P is in torr. The values for the constants are given by:

Constant T < 312 K (solid) T > 312 K (Liquid)

A -94.04826 15.88253B 1961.258 4529.635C -.03771687 0.00058663D 42.57526 -2.99138

Table A.2: The constants required for calculating the vapour pressure of Rb as a functionof temperature.

Appendix B

Clebsch-Gordan Coefficients

Figure B.1 shows the relative rates for transitions between different magnetic sub-states within the F=2 to F’=3 hyperfine line. The values given relate the absorp-tion cross-section for each transition to that for the stretched transition, σ0. (Thestretched transition is |2, 2〉 to |3, 3〉 in this case). For example consider resonantσ+ polarized light, of intensity I (I ¿ Isat), incident on an atom in the |2, 2〉 state.The transition rate to the |3, 3〉 state will be:

R =15

15σ0

I

hω(B.1)

Under the same conditions the transition rate from the |2, 1〉 state to |3, 2〉 statewould be:

R =10

15σ0

I

hω(B.2)

The stretched transition has the highest individual absorption rate. Thus we probethe atoms with σ+ polarized light in a correctly aligned magnetic field, selectivelyexciting the stretched transition and maximizing the signal to noise ratio of ourimages.

Note that the sum of coefficients out of each of the upper states is the same.This ensures that the spontaneous emission rate is the same for an atom in anymF state. Secondly the sum of coefficients out of any of the lower states is thesame. This ensures that the absorption rate of unpolarized light is independent ofthe mF state of the atom.

The normalized transition rates are the squares of the Clebsch-Gordan (CG)coefficients for each transition, which are found tabulated in many text-books anddepend only on the angular momentum quantum numbers (F, mF ) of the initialand final states. In general a Clebsch-Gordan coefficient is an angular momentumoverlap integral between an initial state |L1,m1, L2,m2〉 with two independent

175

176 Appendix B. Clebsch-Gordan Coefficients

Figure B.1: A diagram indicating the relative rates for transitions between differentmagnetic substates within the F=2 to F’=3 D2 hyperfine line of 87Rb.

angular momenta and a final coupled state |L1, L2, L3,m3〉, in which the sum ofthe 2 angular momenta and its z component are constants of motion. Thus ageneral CG coefficient may be written as:

C(m1,m2, L3,m3) = 〈L1,m1, L2,m2|L1, L2, L3,m3〉 (B.3)

In our case, L1 and m1 correspond to the initial |F, mF 〉 state of the atom. L2 andm2 correspond to the angular momentum and polarization of the incident photon.L3 and m3 correspond to the final |F′,m′

F 〉 state of the excited atom.

Appendix C

Thermal Cloud Formulae

Table C.1 contains formulae for the parameters used to describe the thermal cloud.It also contains typical values for these parameters immediately after loading themagnetic trap and towards the end of the evaporation ramps when T = 2.5 Tc.Over this range of conditions the thermal cloud may be accurately described as aclassical gas and so the number density is given by a Gaussian distribution:

n(x, y, z) = n0 exp[ −m

2kBT

(ω2

xx2 + ω2

yy2 + ω2

zz2)]

(C.1)

The classical gas approximation for the thermal cloud holds provided the thermalenergy is much greater than the spacing between the trap energy levels (kBT Àhωz) and so the probability of any particular state being occupied remains small.However in our experiment, kBTc is only about 5 times greater than hωz. Aroundand below Tc, the thermal cloud distribution is no longer exactly Gaussian, butbecomes more sharply peaked at the centre [?]; this peak is know as Bose en-hancement. At these low temperatures the occupation of the lowest energy statesis no longer negligible and the bosonic nature of the identical atoms in the cloudbecomes important. Bose-statistics must be used for an accurate description ofthe density distribution of the thermal cloud around and below Tc. (However thewings of the Bose-enhanced density distribution still have a Gaussian shape, soan accurate value for the temperature of the cloud can be obtained by fitting aGaussian distribution just to the outer region of the cloud (section 2.5.5)).

177

178 Appendix C. Thermal Cloud Formulae

Parameter Symbol Formula Typical values

Load trap T ≈ 2.5 Tc

Number of atoms N 3× 108 9× 105

Temperature T 2.86× 10−5 K 1.8× 10−6 K

Radial trap freq. ω⊥√

µ2m

B′q√BT

2π × 10.6 Hz 2π × 124 Hz

Axial trap freq. ωz

√8 µ

2m

B′q√BT

2π × 30.0 Hz 2π × 351 Hz

Harmonic osc. freq. ωho (ωxωyωz)1/3 2π × 15.0 Hz 2π × 175 Hz

Radial 1/e HW σ⊥√

2KBTmω2

x,y1.1× 10−3 m 2.4× 10−5 m

Axial 1/e HW σz

√2KBTmω2

z3.9× 10−4 m 8.4× 10−6 m

Axial RMS HW zRMS

√KBTmω2

z7.8× 10−4 m 1.7× 10−5 m

Peak number density n0N

π3/2x1/ey1/ez1/e1.1× 1017 m−3 3.3× 1019 m−3

N[

mω2ho

2πkBT

]3/2

de Broglie wavelength λdB

(h2

2πmkBT

)1/23.5× 10−8 m 1.4× 10−7 m

Phase-space density φ n0h3

(2πmkBT )3/2 4.7× 10−6 8.9× 10−2

N[

hωho

kBT

]3

Mean thermal speed vth

√8kBTπm

8.3× 10−2 ms−1 2.1× 10−2 ms−1

Collision rate γcoll12n0σvth 3.9 s−1 295 s−1

Mean free path λMFP2

n0σ2.1× 10−2 m 7.1× 10−5 m

Relaxation time τ 54γcoll

0.3 s 4.2× 10−3 s

Knudsen number K λMFP

(x1/ey1/ez1/e)1/3 27 5

≈ ωhoτ

Table C.1: Useful thermal cloud formulae

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