The Superfluid Properties of a Bose-Einstein Condensed Gas

Eleanor Hodby

A thesis submitted in partial fulﬁlment of the requirements for the degree of Doctor of Philosophy at the University of Oxford

Christ Church College University of Oxford Trinity Term 2002 Abstract

The Superﬂuid Properties of a Bose-Einstein Condensed Gas

Eleanor Hodby, Christ Church College, Oxford D.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 results. We investigate the superfluid nature of the condensate by observing its response to a variety of torques applied by the trapping potential. Using this dilute, weakly-interacting system the fundamental relation between Bose-Einstein condensation and superfluidity is explored directly, without the complications of strong interatomic interactions. The apparatus and procedure used to achieve quantum degeneracy in our dilute 87Rb vapour is described briefly, with particular emphasis on the modifications that have been necessary for the experiments described in this thesis. Condensates are produced with up to 5 × 104 atoms and at temperatures as low as 150 nK, using laser cooling followed by magnetic trapping and evaporative cooling. Superfluidity imposes the constraint of irrotational flow on a condensate in a rotating potential and leads to the formation of quantized vortices at higher rotation rates. Observation of the scissors mode oscillation and the expansion behaviour of a condensate after release from a slowly rotating potential both confirmed the purely irrotational nature of the condensate flow pattern under conditions where a normal fluid would flow in a rotational manner. The scissors mode oscillation frequency was also measured at higher temperatures and was observed to decrease. This result indicates a reduction in the superfluidity of the condensate fraction close to the critical temperature. A systematic study of the critical trap conditions for vortex nucleation was carried out in a purely magnetic rotating potential. This work provided important data against which the numerous theories of vortex nucleation can be tested. The areas in which our results both agree and disagree with current theories will be discussed. In the final experiment, a superfluid gyroscope was created from a single vortex line and the scissors mode of the condensate. It was used to measure the angular momentum of the vortex line and the results are in good agreement with quantum 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 can Take 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 steam With different approaches but the same aim at heart Group 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 galore if 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 loos No 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 all And 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 fun And 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 had in 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 also like to thank my housemates Carol, Suzanne, Cecily and Onofrio for all the good times that we have had at 12 Oswestry Road. Finally thank you to my family, Mum, Dad, Richard and Katharine, for all your love and encouragement over the last 26 years.

iii 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

iv CONTENTS v

3 BEC Production 45 3.1 Loading the second MOT ...... 46 3.2 Loading the magnetic trap ...... 46 3.2.1 Compression of the cloud in the MOT ...... 46 3.2.2 Optical molasses ...... 47 3.2.3 Optical pumping ...... 47 3.2.4 Loading the TOP trap ...... 47 3.3 Evaporative Cooling ...... 48 3.3.1 Adiabatic compression ...... 48 3.3.2 Evaporation using the magnetic ﬁeld zero ...... 50 3.3.3 Radio-frequency evaporation ...... 50 3.4 Detecting the transition ...... 51

4 Optimizing Condensate Production 53 4.1 Loading the second MOT ...... 53 4.2 Alignment of the second MOT and stray magnetic ﬁeld nulling ...... 54 4.3 Loading the magnetic trap ...... 55 4.3.1 Compression and molasses parameters ...... 56 4.3.2 Optical pumping ...... 56 4.3.3 Initial parameters for the magnetic trap ...... 59 4.4 Evaporative cooling ramps ...... 60 4.4.1 Adiabatic compression ...... 60 4.4.2 Evaporation using the magnetic ﬁeld zero ...... 62 4.4.3 Radio-frequency evaporation ...... 64 4.5 Optimization summary ...... 66

5 Condensate Theory 70 5.1 BEC in a non-interacting gas ...... 70 5.2 The trapped non-interacting Bose gas ...... 72 5.3 Bose-Einstein condensation with interacting particles ...... 73 5.4 The ground state ...... 75 5.5 The hydrodynamic equations ...... 76 5.6 Low-lying collective states ...... 78 5.6.1 Mode frequencies ...... 80

6 Bose-Einstein Condensation and Superfluidity 85 6.1 Introduction ...... 85 6.2 Dissipationless flow and critical velocity ...... 87 6.3 The superfluid response to a torque ...... 88 6.4 Irrotational flow and the reduced moment of inertia ...... 89 6.5 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 our apparatus ...... 144 9.6.1 Evidence for vortex decay mechanisms ...... 146 9.7 Conclusion ...... 148 CONTENTS vii

10 The Superﬂuid Gyroscope 149 10.1 Introduction ...... 149 10.2 The theory of the superﬂuid gyroscope ...... 151 10.3 Exciting and observing the gyroscope ...... 152 10.4 Gyroscope results ...... 154 10.5 How does the vortex core move? ...... 158 10.6 Kelvin waves and the gyroscope experiment ...... 162

11 Conclusion and Future Plans 164 11.1 Summary of results ...... 165 11.2 Future experiments ...... 166 11.2.1 Nucleation of multiply charged vortices ...... 166 11.2.2 Exciting and observing Kelvin waves ...... 169 11.2.3 Damping of the m = 2 modes in the presence of a vortex . . 171 11.2.4 Critical conditions for nucleating a second vortex ...... 171 11.2.5 Anti-vortex production ...... 172 11.2.6 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 87 2.4 The hyperfine structure of the 5S1/2 to 5P3/2 D2 transition in Rb 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 state in 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 only injection-locked slave lasers ...... 21 2.11 The instantaneous magnitude of the TOP trap magnetic field in the x-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 x direction ...... 29 2.16 The horizontal imaging system ...... 31 2.17 The power passing a knife edge as a function of its position in our probe beam ...... 35 2.18 The position of the cloud as it falls under gravity, used to calibrate the magnification of the horizontal imaging system ...... 36 2.19 1D density distributions of expanded condensates at different temperatures ...... 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 system may be calculated ...... 42

viii LIST OF FIGURES ix

3.1 The diﬀerent magnetic substates involved in our optical pumping process ...... 48 3.2 The trap parameters during evaporative cooling ...... 49 3.3 Images of expanded clouds with diﬀerent condensate fractions . . . 52

4.1 The temperature of the cloud as a function of the vertical nulling coil current ...... 55 4.2 The phase-space density, number and temperature of atoms in the magnetic trap as a function of optical pump intensity ...... 58 4.3 The number and temperature of the cloud after loading into traps of different radii but the same stiffness ...... 59 4.4 Phase-space density versus atom number close to condensation for different minimum values of BT ...... 63 4.5 Phase-space density versus B = 0 evaporation time ...... 64 4.6 Dressed atom energy levels in a strong rf field ...... 65 4.7 Typical condensate images taken during the gyroscope experiment, both before and after optimization ...... 67 4.8 Phase-space density versus atom number during the optimized evaporation ramps ...... 68 4.9 The atom number, temperature and phase-space density as a function of time during the optimized evaporation ramps ...... 69

5.1 The density of states as a function of energy in a homogeneous and trapped gas ...... 73 5.2 The six independent quadrupole mode geometries in an axially symmetric 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 symmetric TOP trap ...... 84

6.1 Rotational and irrotational velocity ﬁelds ...... 89 6.2 Vortex core radius as a function of expansion time ...... 93 6.3 The energy of the ﬁrst vortex state in the rotating frame as a function of the vortex position ...... 96 6.4 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 in a non-condensed cloud ...... 106 7.2 The scissors mode excitation procedure ...... 108 7.3 Typical images of the thermal cloud and condensate used for the scissors mode experiment ...... 109 7.4 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 and thermal cloud as a function of temperature ...... 113 7.6 The normalized moment of inertia of the condensate fraction, the thermal cloud and the combined system as a function of temperature116

8.1 The trap conditions (²t, Ω) under which each of the three quadrupole modes of a condensate in a rotating potential exist ...... 120 8.2 The normalized trapping frequencies in an elliptical TOP trap, as a function of E = Bx/By ...... 122 8.3 A schematic diagram of the electronics for the elliptical rotating TOP trap ...... 124 8.4 Absorption images of an initially rotating condensate and a static condensate at diﬀerent expansion times ...... 127 8.5 The angle and aspect ratio of the condensate as a function of expansion time, released from a rotating trap ...... 128 8.6 The asymptotic rotation angle of the condensate after long expansion 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 time after release from a rotating trap ...... 130

9.1 Images of the condensate during nucleation and of lattices of 1-4 vortices ...... 139 9.2 The mean number of vortices as a function of the normalized trap rotation rate ...... 140 9.3 The mean number of vortices as a function of trap deformation . . . 141 9.4 The critical conditions for vortex nucleation ...... 143 9.5 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 ...... 150 10.2 Images of single vortices, illustrating our criterion for a centred vortex153 10.3 Gyroscope data - excitation in xz plane ...... 155 10.4 Gyroscope data - excitation in yz plane ...... 157 10.5 Sideview images of the gyroscope motion in which the vortex line is clearly visible ...... 159 10.6 A mechanical model of the trapped condensate ...... 161

11.1 Radial magnetic ﬁelds in a quadrupole and I-P trap ...... 167 11.2 The scheme for exciting and trapping a vortex with 4 units of circulation ...... 168 11.3 Resonance conditions for exciting the lowest Kelvin wave ...... 170

B.1 The Clebsch-Gordan coeﬃcients for the F=2 to F’=3 hyperﬁne transition ...... 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...... 47 3.2 The evaporative cooling ramps used to obtain BEC ...... 49

4.1 The stages in the optimized TOP trap loading procedure...... 56 4.2 Transition rates for optical pumping with the original and improved schemes...... 58 4.3 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 ...... 173 A.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 gas Bose-Einstein condensate of 87Rb atoms. Bose-Einstein condensation occurs when many identical particles all occupy the same quantum mechanical state. This behaviour results from the quantum statistics of identical bosonic particles and was first predicted by Einstein in 1925 [?]. It lies at the heart of phenomena such as superconductivity and superfluidity of liquid helium [?]. However these systems have strongly interacting particles that make them much more complex than the ideal gas described by Einstein. Realization of a Bose-Einstein condensate (BEC) in an ultra-cold dilute atomic vapour, with properties close to those predicted by Einstein was first achieved by Cornell, Wieman and coworkers in 1995 [?]. This first Bose-Einstein condensate experiment used 87Rb atoms and within a few months condensation of 23Na atoms had also been achieved by Ketterle and his group at MIT [?]. Over the last seven years the field has grown exponentially. There are now over thirty Bose condensate experiments around the world, producing condensates of 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 a superfluid, it was several years before superfluid effects were observed [?, ?, ?, ?]. The most striking signatures of superfluidity involve the response of the system to rotation, since the presence of a single macroscopic wavefunction constrains the flow patterns allowed within its bulk. Thus to probe the superfluid nature of a dilute gas Bose-condensate we have developed a flexible trapping potential that is able to rotate the condensate about any axis. Each experiment described in this thesis shows the superfluid response of the condensate to a different applied torque. The structure of this thesis may be summarized as follows: In chapters 2 to 4 the 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 detailed optimization process that was recently carried out prior to the final gyroscope experiment, which succeeded in tripling the number of atoms in the condensate. Chapters 5 and 6 summarize the BEC theory that underlies the experiments described later in the thesis. Chapter 5 concentrates on the collective excitation spectrum of a weakly interacting Bose condensate in a harmonic trap, whilst chapter 6 discusses the definition of superfluidity and predicts the superfluid response of the condensate to a rotating trapping potential. Both here and in chapter 9 the behaviour of the dilute gas condensate is compared and contrasted to liquid helium II, which to date is the most extensively studied superfluid system. Chapters 7 to 10 present 4 different experiments which each demonstrate the superfluid response of the condensate to a different applied torque. The first two experiments demonstrate the pure irrotational flow pattern characteristic of a simply connected superfluid in different ways. First we investigated the frequency of the small angle oscillations of the condensate relative to the trapping potential known as the scissors mode, whilst chapter 8 describes the distinctive behaviour of an expanding condensate, after it is released from a slowly rotating potential. The original scissors mode experiment provided some of the first evidence that the condensate behaves as a superfluid together with the observation of quantized vortices [?, ?] and a critical velocity for superfluid flow [?]. The scissors mode experiment was later repeated at higher temperatures to investigate the interaction between the condensate and the thermal cloud and how it affects the superfluidity of the condensate fraction. The observation of quantized vortices in a purely magnetic rotating potential is described in chapter 9. The chapter also presents a detailed study of the critical trap conditions (eccentricity and rotation rate) for nucleation, which was used to determine the mechanism by which vortices are nucleated. Finally in chapter 10 we present the results of the superfluid gyroscope experiment, in which simultaneous excitation of the scissors mode and the first vortex state enabled us to measure the quantized circulation associated with the vortex line. In the concluding chapter, these results are summarized and plans for future superfluid experiments are described 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 description of the original set-up is contained in [?], whilst this account focuses on the improvements 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. Atoms are 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 purely magnetic trap for evaporative cooling to the BEC transition temperature. Finally the atoms are imaged on a CCD (charge-coupled device) camera. Figure 2.1 shows these 4 basic elements - the vacuum system, laser system, magnetic trap and imaging system. Precise timing and synchronization of the diﬀerent stages is required to make and image a condensate and so all the diﬀerent components operate under a single control computer.

2.2 Vacuum system

The evaporative cooling process used to make a condensate relies on elastic collisions between atoms in the magnetic trap for rethermalization. The initial collision rate of ∼ 10 s−1 sets the minimum timescale for the evaporative cooling process to around 40 s. Since we desire both a low temperature and high number of atoms to achieve BEC, the lifetime of the magnetic trap must be greater than 40 s. The lifetime of the trap is set by loss from collisions with atoms in the background vapour, 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 rubidium source nearby and so we use a double MOT system to separate the rubidium source from the magnetic trap (fig. 2.2). Other experiments have used different solutions to this problem. The first BEC experiment used a single dark MOT with a very low background pressure of rubidium ∼ 10−11 Torr [?]. Alternatively the use of a pre-cooled beam of atoms from a Zeeman slower to load the experimental MOT 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 is housed inside a 10-way cross with a large (100 mm diameter) front window, and a 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 wire a chemical reaction releases rubidium vapour into the cell. The threshold current for this reaction is 2.6 A, and we currently run it continuously at 3.7 A. Since these 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 be replaced by several dispenser wires so that a new one can be activated without needing 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 diameter of 16 mm. Its long thin shape results in a low conductance of 16 ls−1 and enables us to maintain a pressure differential between the ends of the tube. Thus the experimental MOT is held at < 10−11 mbar by a 40 ls−1 ion pump (Varian VacIon Plus 40). In addition to the ion pump, we also occasionally (roughly once a month) use the titanium sublimation pump (TSP) (Vacuum Generators ST22) which is situated in the pipework behind the quartz cell. A current of 48 A is passed through a wire in the pump for one minute, causing titanium to sublimate and stick to the surrounding walls of the vacuum system. This produces a very thin porous layer which adsorbs background gas molecules. However the efficiency with which the TSP pumps rubidium is unknown and we have no clear evidence that the second MOT 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 operation, it became difficult to produce condensates reliably, although large numbers of atoms could be collected in the second MOT. We suspected that Rb had built up on the walls of the cell, which was liberated when the cell was heated by the surrounding trap coils, producing a high background pressure and inefficient evaporative cooling. A comparison between the original and current magnetic trap lifetimes would have confirmed this idea, however for technical reasons (destructive imaging, heating of cell by trap coils etc) it has never been possible to get a reliable measurement of the lifetime. So we wrapped the cell in an aluminium foil jacket and heated it with a hot air gun overnight, gently raising the surrounding temperature to 80◦C, to investigate whether production could be temporarily improved as some rubidium sublimated from the cell walls and recondensed a little further away on the cold metal pipework. Production did apparently improve for several hours of operation adding weight to our suspicions about the Rb background pressure. We decided to rebake the system, which involved removing the MOT optics, building an oven around the system, attaching the turbo pump and baking at 250◦C for several days. With hindsight I have the following doubts about whether the bakeout was necessary, which could only be resolved with accurate magnetic 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 in good loading but erratic condensate production. This observation is probably 8 Chapter 2. The BEC Apparatus

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 molasses cooling 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 and rate at which condensates are produced. Several of these have been implemented on the second generation BEC apparatus, shown in ﬁg. 2.3. (This apparatus was not used for the experiments described in this thesis).

• A larger pyramidal MOT (base size 6 cm) gives a larger collection region and hence produces an atomic beam of higher flux • The background gas conductance of the tube connecting the 2 MOTs is given by (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 very low pressures. It assumes that the tube size is much smaller than the mean free path and that after collision with a wall, the path of an atom follows a 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 particle has 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 solid angle of the tube D2/L. Comparing this to the expression for background gas conductance, one can increase the flux whilst maintaining the same pressure differential by using a shorter, narrower tube. In the new experiment, the distance between MOTs is reduced to 30 cm, and the pressure differential is maintained 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 or supplemented by a non-evaporable getter (NEG) pump, which removes gas molecules that arrive at the surface of the porous getter cartridge by chemical reaction. 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 increase conductance. In particular the vacuum system is mounted on runners on its own breadboard and so it can be wheeled to the edge of the table to attach the turbo pump for baking out. The original version requires a long pipe reaching to the edge of the table because the vacuum system cannot be moved.

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) and ampliﬁed by further semiconductor devices (a MOPA and a slave laser). The precise control that we require over the intensity and frequency of each beam is achieved using acousto-optic modulators (AOMs), whilst the absolute frequency is set by a saturated absorption locking scheme. In this section I will brieﬂy describe the role and operation of each element in the present laser system and then outline changes that should improve the day-to-day reliability of the system. 10 Chapter 2. The BEC Apparatus

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 the experimental MOT. In summary, a MOT consists of three counter-propagating, red-detuned pairs of laser beams, that intersect at the centre of a quadrupole magnetic field. As atoms move away from the centre of the trap they are Zeeman shifted into resonance with a laser beam that pushes then back again and so trapping is achieved. The polarizations of the beams must be carefully chosen so that the correct Zeeman shifted transition is excited. Cooling occurs because atoms are Doppler shifted into resonance with photons traveling in the opposite direction - preferential absorption of this low energy photon followed by spontaneous emission of a higher energy resonant photon reduces the kinetic energy of the atom. There are also other more subtle cooling mechanisms at work (e.g. Sisyphus cooling, polarization gradient cooling) which are described in detail in [?] and lead to cooling below the Doppler limit. The cooling transition that we use is the 5S1/2 to 5P3/2 transition at 780 nm (the D2-line). The hyperfine structure of this transition is given in fig. 2.4. The master laser generates the more powerful beam for laser cooling, which is red detuned from the closed F=2 to F’=3 transition by 15 MHz. A few atoms (approximately 1 in every 250) are off resonantly excited to the F’=2 state, from where they may decay to the lower hyperfine ground state, F=1, and be lost from the cooling process. To prevent this, a separate laser produces a repumping beam resonant with the F=1 to F’=2 transition, that is mixed with the trapping beam and excites atoms back into the cooling cycle.

2.3.2 The magneto-optical traps We use 2 different designs of magneto-optical trap, both of which operate using the principles described above. The second MOT is a standard design, with 6 circularly polarized beams, each of 0.8 cm waist and containing 3 mW of trapping light and 150 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 optical fibre (OZ optics LPC-02-780-5/125-P-2.2-11AS-40-3A-3-4) - this ensures that the relative powers of the beams remains constant, even though the total power may fluctuate. The fibre also acts as a spatial filter, ensuring that the beams have a smooth Gaussian profile. The first MOT is of a pyramidal design. It is housed inside the 10-way cross described in section 2.2. The quadrupole field of 5 G cm−1 (radial) is produced by a pair of anti-Helmholtz coils, wound around the cross and coaxial with the input beam. They carry 6 A and require water cooling. Four mirrors form the inside of a square based pyramid (of base 3.8 cm), which enclose the cloud. The pyramid is illuminated by a single, circularly polarized, wide diameter input beam (of waist 1.8 cm), which creates the six MOT beams with correct polarizations by reflection 2.3. Lasers and optics 11

87 Figure 2.4: The hyperfine structure of the 5S1/2 to 5P3/2 D2 transition in Rb. Cooling and trapping is done on the closed F=2 to F’=3 transition. The repumping laser is locked to the F=1 to F’=2 transition to recycle atoms that are off-resonantly pumped into the lower hyperfine ground state. 12 Chapter 2. The BEC Apparatus

Figure 2.5: A y-z cross-section of the pyramid MOT. By convention, the polarizations are defined relative to the positive x,y and z axes, rather than the direction of travel of the beam or the local B field. In this convention, pairs of MOT beams have opposite polarization 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 ideal mirror the handedness of polarization relative to the direction of beam travel is reversed). The mirror blanks were custom made by Halbo Optics using BK7 glass and a multilayer dielectric coating was applied by the physics department optical coating facility (courtesy of Chris Goodwin). It is important that the coating produces equal reflectivities and phase shifts for both s and p polarized light so that the reflected beams are still circularly polarized. After 1 reflection, less than 2 % of the light was found to have the wrong handedness, which is sufficient for MOT 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 the second MOT. External bias coils enable us to move the zero of the quadrupole field and hence the atom cloud over the hole to optimize the loading of the experimental MOT. The pyramidal MOT has many advantages over a conventional 6-beam MOT as a large, primary source of cold atoms. With only one input beam, the set 2.3. Lasers and optics 13

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

Laser design For most of the work described in this thesis, both the master and repumping lasers used SDL diodes (SDL-5402-H1). These produce up to 50 mW of power in free running operation at a wavelength of 782 nm. The free running wavelength is lowered towards 780 nm by using the built in peltier and thermistor to hold the diode head at ∼ 14◦C, using a commercial temperature control box (Newport Mod.325). The diodes are mounted in an external cavity which provides optical feedback to improve the spectral properties of the laser, reducing the linewidth to < 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) is mounted in Littrow configuration and reflects the first order back into the diode, whilst the zeroth order forms the output beam. A frequency selective cavity forms between the grating and the back face of the diode. The range of possible lasing frequencies is set by the angle of the grating, whilst the narrow individual mode frequencies are set by the external cavity length. These individual modes may be scanned continuously over ∼ 5 GHz by changing the length of the piezo-electric crystal on which the grating is mounted. Thus large, slow adjustments (up to 1 kHz) may be made to the laser frequency by driving the piezo, whilst small, fast adjustments may be made by varying the laser current. Since the cavity length ultimately determines the laser frequency, we found it necessary to temperature stabilize the whole mount to avoid thermal drifts in the cavity length. This was achieved by placing a peltier element underneath the mount and maintaining it just below room temperature. The blazed gratings that are used reflect over 50% of the incident light back into the grating, providing a strong feedback beam but reducing the maximum output power to ∼ 12 mW. Holographic gratings which reflect only 25% into the first-order feedback were also investigated, in the hope that higher output powers could be achieved. However the smaller feedback reduced the stability of the laser mode 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 the exact power and frequency of each beam is critical, each laser had to be operating 14 Chapter 2. The BEC Apparatus

Figure 2.6: The design of the external cavity diode lasers. The mount was made in house and is made of brass. The vertical adjustment screw is used to direct the first order beam back into the diode - feedback is optimized when the lasing threshold is minimized. The horizontal 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 least 1 GHz either way without mode hopping. Unfortunately the external cavity length was not stable over periods of hours, either due to temperature drifts or relaxation of the mount. This would cause the locking frequency to drift to the edge of the stable mode and the lock to fail. Thus the cavity would have to be realigned several times a day, and the laser temperature and current adjusted to produce a clean single mode again. The horizontal and vertical feedback adjustments worked by flexing a thin piece of brass against a very finely threaded screw (see fig. 2.6). As the mount aged, the elasticity of the brass reduced and no longer flexed smoothly. Finally the current drivers and locking electronics were not well shielded. They picked 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, just seconds 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 reliable laser lock, we would be able to automate the experiment to run without constant supervision, which would dramatically increase the data rate and make more efficient use of our time. Thus we investigated two commercial ECDL systems, the TUI (now Toptica) DL100 and the Laser 2000 TEC500, both of which cost around £6000 for the complete package, including electronics. The Laser 2000 system was not suitable, whilst the Littman mounting was useful because the beam angle does 2.3. Lasers and optics 15

not change with frequency, the output power was low (20 mW), the adjustment screws were awkward to operate and we were unable to produce a pure single mode output at the desired frequency, when viewed on a Fabry-Perot spectrum analyser. The TUI system (which we eventually bought and now operates as the repumping laser) was more promising. It is mechanically very robust and the cavity only requires adjustment every few weeks. It produces up to 50 mW of output power and can be made to go single mode at the desired frequencies. However it also has the confusing ability to lase cleanly in several diﬀerent modes, producing a good saturated absorption signal from the mode at the correct frequency but with most of the power unable to interact with the atoms. This probably occurs due to the lack of AR coating on the Sanyo diodes that are used by TUI. To be sure that the TUI laser is operating single mode, it is necessary to view the output on a Fabry-Perot spectrum analyser. Finally whilst the system does pick up some electronic 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 specific 87Rb hyperfine transitions using a Doppler-free saturated-absorption set up [?] (fig. 2.1). Figure 2.7 shows the saturated-absorption signal obtained from one of 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 derivatives of these peaks provides the error signal for locking. It is generated by dithering the laser current at 100 kHz and feeding the saturated absorption signal into a phase sensitive detector. After integration and phase correction, the error signal is fed to both 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. Further information about the locking circuit may be found in [?, ?, ?]. It can be seen from fig. 2.7 that the repumping saturated absorption lines are weaker than the master lines, resulting in a lower signal/noise ratio in the locking circuit and a less stable lock. We have improved the lock by heating the vapour cell to a steady 40◦C, which increases the rubidium vapour pressure and hence the saturated absorption signal by a factor of five, compared to a cell at room temperature. To provide a stable temperature, the cell is completely enclosed in a metal case. A temperature control circuit (Cebek thermostat module HK00023) with a thermistor and a peltier element attached maintains the temperature at 40 ± 1 ◦C. Figure 2.8 shows that the master laser is locked to the cross-over peak between the F=2 to F’=1 and the F=2 to F’=3 transitions, 214 MHz below the cooling transition. The beam is split between the first and second MOT, amplified and then sent to two double-pass acousto-optic modulators (Crystal-Technology 3110- 16 Chapter 2. The BEC Apparatus

Figure 2.7: The saturated-absorption signal from one of our Rb vapour cells which contains both 87Rb and 85Rb (77% abundant), showing the hyperﬁne transitions of each isotope 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 Doppler curve removed by subtraction. All transitions start from the upper ground hyperﬁne state (F=2) and go to the F’ state indicated. The master laser is locked to the F’= 1-3 cross-over peak. Some power broadening is present in this scan, since the linewidth is apparently greater than the natural width of 5.76 MHz. 2.3. Lasers and optics 17

140), which control the frequency and intensity of light in each MOT. On each pass the frequency changes by 110 ± 24 MHz and so after the double pass we shift the laser light back on resonance with a tuning range of about ± 40 MHz. As well as doubling the tuning range, a well aligned double-pass AOM produces an output beam angle that is independent of frequency. Since the beams are then coupled down optical fibres, this is very important - we do not want large changes in 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 power is independent of frequency over ±10 MHz range and falls off fairly rapidly outside this - 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, with a switching time of ∼ 1 µs, enabling us to generate 5 µs pulses for pumping and probing. Since they can leak light even when off, they are backed up by slower mechanical shutters. The repumping laser is locked directly to the repumping transition 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 a narrow linewidth and good transverse mode structure, However it has only a few mW of power and must be amplified to run the first and second MOTs, which require 50 and 20 mW of trapping light respectively. Originally the experiment used just one amplification device, the master oscillator power amplifier (MOPA), with the output split between the two MOTs. However this arrangement did not produce sufficient power reliably and so the second MOT is now run from a slave laser 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-Perot spectrum 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 the MOPA chip seemed a cheap and simple alternative to a titanium sapphire laser. The only drawback was that it had only just become commercially available and hence its operational lifetime was unknown [?, ?]. A detailed technical description of the MOPA setup is given in [?]. To optimize the output power, the injection beam is carefully matched to the shape of the spontaneous emission beam from the input facet of the MOPA by a row of lenses and prisms (see fig. 2.1). Whilst the MOPA output retains the narrow bandwidth of the master beam, it is highly astigmatic and elliptical. The poor transverse mode quality means that much of the power cannot be coupled through an AOM crystal or down a fibre and so is wasted. In addition, the transverse mode structure and hence the coupling efficiency of the beam drifts with temperature over a period of 18 Chapter 2. The BEC Apparatus

hours. Whilst this problem was temporarily improved by temperature stabilizing the laser mount, the output mode structure deteriorated again after about 1 year of 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 power levels 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 to run the second MOT. This scheme had many advantages:

• The MOPA beam could be spatially filtered and still provide sufficient power to run the first MOT alone. The filtering prevented drifts in the transverse mode 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 [?]. In summary, a master beam is injected into a high power laser diode (analogous to the grating feedback in the ECDL), seeding the lasing process. Whilst much of the master power is used up in the process of controlling its frequency and bandwidth, the slave inherits the master beam qualities and so all of its output power is available for experiments. The quality of the lock is determined by the range of frequencies over which the slave output follows that of the master (which can be tested either using saturated-absorption signals or a Fabry-Perot spectrum analyser) 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 the output beam, but we found that this was not critical.

• The locking range can be increased by raising the injection power above the threshold value of ∼ 0.5 mW. 2.3. Lasers and optics 19

Figure 2.9: A schematic diagram of the optical isolation in front of an injection-locked slave laser. Polarizations are used to ensure that the injected beam can reach the laser diode, but any back reﬂections from the slave beam are stopped at cube 2. Note that the 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 optical length of the cavity can be aﬀected by the slave current and temperature, so for a given current and hence output power the temperature must be scanned until a lock is achieved.

Good temperature stabilization of the slave is essential and was provided by a peltier under the brass mount, a feedback thermistor and temperature control box EW 1251. The optical arrangement for injecting the slave is shown schematically in fig. 2.9. The 45◦ Faraday rotator with polarization beam splitting cubes mounted on either end transmits the injection and output beams, whilst protecting the slave diode from unwanted back reflections. The diode was conveniently mounted inside a Thorlabs (LT230P5) collimation tube, which locks the diode in position and allows 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 fraction of its power was lost passing through the isolation optics, double-pass AOM and fibre and so around 100mW of bare output power was required. We tested 2 different 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- 20 Chapter 2. The BEC Apparatus

spectively, to see whether they could operate and lock reliably above the specified output power. To find the individual maximum power of each laser we added a saw-tooth oscillation of ±10 mA at 100 Hz to the current and looked for a change in the current/output power relationship as the mean current was gradually increased. This change marked the damage threshold of that particular diode, but since the diode spent little time per cycle above this threshold value then damage was unlikely to occur (in theory!) [?]. The first Sanyo diode died when the output power was around 75 mW. It was replaced and a maximum output power of 110 mW was successfully recorded at 130 mA. The laser graphics diode showed no sign of a damage threshold at 100 mW and so it was not tested further since this was sufficient power for the experiment. Both lasers produced similar locking characteristics, shown in table 2.1. The laser graphics diode was eventually chosen and 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 4 Laser 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 the fact that cajoling it to an acceptable power level is one of the most time consuming aspects of the experiment, we do not intend to replace the present chip when it dies. 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 change is in the pyramid MOT beam. The AOM (at which a 40% power loss occurs) is positioned before, rather than after, the amplification stage to obtain sufficient power 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 a new 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 once the experimental MOT has loaded. With the improved transverse beam quality of the slave laser, it might be possible to run the first MOT without a fibre acting as a spatial filter and thus have extra power to increase the loading rate. Whilst this is worth investigating, it is extremely convenient to have the fibres decoupling the alignment of the MOTs from the alignment of beams on the laser table. 2.3. Lasers and optics 21

Figure 2.10: A schematic diagram of the current laser system, and a new laser system which would replace the MOPA with a second injection-locked slave laser. Only those components at which signiﬁcant power loss occurs are shown. 22 Chapter 2. The BEC Apparatus

2.4 The magnetic trap

Until recently, magnetic trapping was essential for the evaporative cooling stages of BEC production. Condensation cannot be achieved in a magneto-optical trap since reabsorption of spontaneously emitted photons limits the maximum density and photon recoil ultimately limits the minimum temperature that can be achieved in a MOT. (The ﬁrst all optically trapped BEC was reported last year, in which atoms are trapped for evaporative cooling using the optical dipole force of a tightly focused, far-detuned CO2 laser beam [?]). Magnetic trapping of neutral atoms is based on the interaction between the atomic magnetic dipole moment µ and a weak inhomogeneous magnetic ﬁeld B. The interaction energy U may be written as:

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 atoms in high-field seeking states since Maxwell’s equations do not allow a local maximum in 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 tight radial confinement, and a pair of Helmholtz pinch coils producing axial confinement. 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 [?] and was used in the first BEC experiment [?]. It is based on a quadrupole field Bq, generated by a pair of anti Helmholtz coils:

0 Bq = Bq (x ˆx + y ˆy − 2z ˆz) (2.2)

Note that the field gradient is twice as strong in the z direction as in the radial directions, to satisfy ∇.B = 0. At first sight, the quadrupole field alone appears to create a suitable confining potential, 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 untrapped mF states known as ‘Majorana spin flips’ and are expelled from the trap. The addition of a static bias field does not solve the spin flip problem, as it would simply displace the centre of the trap to a new position. However if the bias field (or TOP field), BT rotates sufficiently fast, the atoms cannot follow the instantaneous trap centre 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 field of BT at the centre. In our experiment, the bias field rotates in the xy plane and may be represented as: BT = BT (cos ω0t ˆx + sin ω0t ˆy) (2.3) So the total magnetic field is

0 0 0 B = (Bq x + BT cos ω0t) ˆx + (Bq y + BT sin ω0t) ˆy − 2Bq z ˆz (2.4)

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

Z 2π/ω ω0 0 UTOP = mF gF µB |B| dt (2.5) 2π 0 02 ³ ´ µBq 2 2 2 = µBT + x + y + 8z + O(4) (2.6) 4BT

where µ =√gF mF µB. UTOP is harmonic, with trapping frequencies ωx = ωy = ω⊥ and ωz = 8 ω⊥, where r 0 µ Bq ω⊥ = √ (2.7) 2m BT

The rotation rate of the bias field ω0 must be much faster than the trap oscillation frequencies so the translational motion of the atoms is controlled by the time averaged potential. It must also be much slower than the Larmor frequency ωL = µB/h¯, so that the atomic dipole moments can adiabatically follow the direction 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 the Larmor frequency is 1.4 MHz in a minimum field of 1 G and so both of the above conditions are satisfied. If we consider the instantaneous picture rather than the time-averaged one, then the zero of the quadrupole field is moving in a circle of radius: BT r0 = 0 (2.8) Bq This locus of B = 0 (colloquially known as the ‘circle of death’) defines the boundary of the trap in the radial direction. Atoms that reach this radius are flipped into 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 this trap - the bias field rotates around the axis of the quadrupole field and gravity acts along the same axis so that nothing breaks the radial symmetry. The trap is stiffer in the z direction than the radial one, creating√ pancake shaped condensates with an aspect ratio (axial size : radial size) of 1 : 8. (In contrast, most Ioffe Pritchard traps have tighter radial confinement and produce cigar shaped condensates with a typical aspect ratio of 20:1). However, one of the great advantages of the TOP 24 Chapter 2. The BEC Apparatus

Figure 2.11: A diagram showing the instantaneous magnitude of the TOP trap magnetic field in the x-y plane. The total field is an axially symmetric quadrupole field with field 0 0 gradient Bq, displaced in the x-y plane by a distance r0 = BT /Bq. As the bias field rotates, the zero of the total magnetic field follows a circle of radius r0. 2.4. The magnetic trap 25

trap is that it is possible to make physically significant changes to the shape and aspect ratio of the trap simply by changing the symmetry or direction of the TOP bias field. Most of the results in this thesis rely on such modifications, which will be described in detail later. It is worth noting√ that a triaxial TOP trap has also been 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 the atom cloud for minimum heating during loading.

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

• 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 for the second MOT) is generated by a pair of anti-Helmholtz coils of 450 turns each wound onto an aluminium former. A slit along one radius of the former minimizes eddy currents. The large number of turns enables us to create a relatively high field gradient (200 G/cm radially) with a modest current (9.5 A) supplied by a standard bench power supply (Farnell PSU 3510A). Since the total power dissipated is less than 300 W, this low current arrangement can also be cooled easily by pumping water from a beer cooler at 13◦C through the former of the coils. An alternative design with high current and low turns would have 2 advantages. Firstly the coils would be smaller and physically closer to the cell thus increasing the gradient. Secondly the inductance would be lower and so faster switching times could be achieved. 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 the experiments carried out so far. The radial TOP bias field is generated by 2 orthogonal pairs of coils, each of 6 turns, sandwiched between the cell and the quadrupole former. Figure 2.13 shows the circuitry driving each coil to create a rotating bias field. The magnetic field generated by each pair of coils is monitored on single turn pick-up coils wound within the TOP coils. These pick-up coils have negligible impedance and so there 26 Chapter 2. The BEC Apparatus

Figure 2.12: A cross-section diagram and photograph of the TOP trap 2.4. The magnetic trap 27

Figure 2.13: The circuitry driving the TOP coils for a standard TOP trap. Impedance matching the output of the audio amplifier is necessary. It generates a maximum of 600 W into a load of 4 Ω, whilst at 7 kHz the TOP coils only have a reactance of 0.35 Ω. Any unequal 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 the pick up-coil. So far I have described the original, axially symmetric TOP trap apparatus. To perform the experiments described in this thesis, several modifications have been made. The radial TOP circuitry has been modified so that the trap can be made elliptical in the x-y plane and then be rotated. Also a third set of TOP coils (and pick up coil) has been wound around the quadrupole coils (30 turns each), creating an oscillating bias field in the z direction. These enable us both to tilt the trap and change its aspect ratio.

2.4.3 Calibration of the TOP trap

It is important to know the exact trap frequencies during experiments, which from eqn. 2.7 requires an accurate knowledge of the quadrupole and x,y and z TOP ﬁelds at the trap centre. These can only be approximately calculated from the known currents and geometries and so must be measured. Given that there are 4 unknown quantities, then 4 independent measurements must be made. An absolute calibration of the y (and z) TOP ﬁelds may be made using the Zeeman shift. A cloud of cold atoms is released from the trap and imaged with a probe 28 Chapter 2. The BEC Apparatus

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 calibrate By. The data was fitted with a Lorentzian curve, centred on ∆max = 26.0 ± 0.2 MHz, indicating that the By field present had a value of 26.0 / 1.4 = 18.6 ± 0.1 G beam along the y (or z) direction, after all fields have been turned off, except the one under investigation. The detuning of the probe beam is varied and a plot of optical density versus detuning is built up (fig. 2.14). The data has a Lorentzian distribution centred on ∆max. Since the mF = 2 to mF = 3 transition is Zeeman shifted by 1.4 MHz/G, then we have: ∆ B = max G (2.9) TOP 1.4 Secondly, the trap oscillation frequencies in the x and y directions can be measured by ‘kicking’ a small cold cloud of atoms in the harmonic trap to excite the dipole mode. The kick is achieved by suddenly switching off the nulling fields in these 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 (using a probe beam along the z axis) as a function of time and fitted with a sine wave, from which the oscillation frequency is extracted. Typical data is shown in fig. 2.15. To minimize the error on the fitted frequency, the oscillation is sampled for a few cycles over a wide range of evolution times, from 0 to 3100 ms. From the 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 important that the trap is exactly circular in the x-y plane - these dipole measurements determine 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 gravity on the TOP trap. If we include the effect of gravity in the TOP trap potential we 2.4. The magnetic trap 29

Figure 2.15: The x position of the centre of an atom cloud (in pixels) after the x dipole mode has been excited. Each pixel is 24 µm wide and the magniﬁcation is × 1. Only a small 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 ± 0.01 Hz have:

Z 2π/ω ω0 0 UTOP = (µ |B| + mgz) dt (2.10) 2π 0 02 h i 2 1/2 µBq 2 1/2 2 2 2 3/2 2 ≈ µBT (1 − ρ ) + (1 − ρ ) (1 + ρ )r + 8(1 − ρ ) z 4BT

0 where ρ = mg/2µBq. This shows that the ratio of the axial to radial oscillation frequencies is given by: s 2 ωz √ 1 − ρ = 8 × 2 (2.11) ω⊥ 1 + ρ

This ratio does not depend on BT and so measurements of it may be used to 0 determine Bq independently. However this method only provides an accurate cal- 0 ibration for small Bq (≤ 20 G/cm), below which gravity has a significant effect on the trap. Non-linearities in the relationship between the computer output and the quadrupole field produce errors if the calibration is extrapolated to the values of 0 Bq used in the experiment. It is worth noting that gravity provides a simple way of altering the aspect ratio of the TOP trap and even producing a spherical trap (ωz = ω⊥), which holds some very interesting physics [?, ?]. The disadvantage is that these aspect ratios can only be achieved in very weak traps. 30 Chapter 2. The BEC Apparatus

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 the rf radiation that is used for evaporative cooling. They are driven by a frequency generator (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 resonances in the power output that occur at certain driving frequencies, however this was later removed to increase the overall rf power.

2.5 The imaging system

All the information about the size, shape, and temperature of our condensate or thermal cloud is obtained by absorption imaging, essentially looking at the shadow that the atoms cast in the probe beam. The experiment was originally set up with a single ‘horizontal’ imaging system, probing along the y direction, through the side of our pancake-shaped condensate. However following the development of an elliptical trap that rotates in the x-y plane (section 8.3), it became essential to observe the condensate in the plane of rotation. For experiments described in chapter 8 and beyond, a second ‘vertical’ imaging system was available, that probes along the symmetry axis of the cloud (z axis). The optical principles behind both systems are identical although because of the layout of the coils and vacuum system, 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 been allowed to expand freely for a certain time. Thermal clouds are usually imaged in the trap, since the cloud is many pixels in size and the optical density is relatively low. 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 beam path both for the unscattered probe light and also for the image of the shadow created by the condensate. A 5 µs probe pulse is delivered down an optical fibre, using the AOM in the pyramid MOT beam as a fast shutter (see fig. 2.1). The timing of this pulse is synchronized with the rotation of the TOP bias field, so that the pulse is centred on the instant when the field points along the imaging direction (y). The σ+ optimized probe light drives the closed |2, 2i to |3, 3i 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 the unscattered probe light and also for the image of the shadow created by the condensate. 32 Chapter 2. The BEC Apparatus

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 congested area around the cell. The numerical aperture of the first lens sets the resolution of the system. It has a value of D/f = 0.25, where D is the diameter and f is the focal length of the lens. This is the maximum numerical aperture that can 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 object formed 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 important that the two doublet lenses were the correct way round, so that the light was refracted equally at each surface, thus minimizing aberrations on the image. The image is magnified using either a ×4 or ×10 microscope objective, (Comar 04OS10, 10OS25). Microscope objectives are designed with the object to image distance is constant, so that objectives may be exchanged to change the magnification without major refocusing being required - a property which is ideal for our system. The highest magnification that we use is × 10, at which the resolution 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 the signal 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 the resolution limit. The objective is mounted on a vernier translation stage, so that the system may be focused by making fine adjustments to its position along the optical axis.

2.5.2 The camera

The camera used for these experiments is a Princeton Instruments TE/CCD-512SB with an ST-138 controller. It has a CCD array of 512 by 512 pixels, each 24 µm square. The array is 80% efficient around 780 nm and is peltier cooled, to minimize dark noise. Whilst it is an excellent camera it was originally bought with caesium experiments in mind and so some of its specifications are not perfectly optimized for our experiment. The CCD chip is protected by a mechanical shutter. To prevent vibrations from this shutter throwing the lasers off lock, the camera is suspended in a metal cradle from the frame around the experiment. The shutter is opened at the start of each experimental run, but since it is operated in ‘continuous cleans’ mode, any stray MOT light that it detects is wiped from the array well before it is triggered to take an image. After receiving the trigger, the clean CCD array is exposed for 10 ms before being read-out to the controller. The read-out process is slow and limits the time between shots to 1 s. 2.5. The imaging system 33

2.5.3 Data acquisition Each condensate picture is constructed from 3 camera images, which are digitized into 16 bit integers, and passed to our data acquisition package WINVIEW for Microsoft Windows. The ﬁrst shot contains the shadow of the atoms in the probe beam. The second, taken ∼ 1 second later, long after the atoms have dispersed contains just the probe beam. The third shot gives the dark noise, i.e. the charge that is thermally excited onto each pixel between cleans. From these three images, the WINVIEW program constructs a false colour image representing the density of atoms integrated along the imaging direction (y) in the following way: After passing through the atom cloud, the intensity of the probe beam, Ii, arriving at the ith pixel in image 1 is R i i − ni(y)σdy I = I0 e (2.12)

where ni(y) is the number density of atoms along the column of real space imaged th i i i onto the i pixel. The number of counts recorded on this pixel is C1 = α I where αi is the efficiency of the pixel. The intensity at the pixel in the absence of atoms, i i i i I0 is recorded in image 2 so C2 = α I0. Thus we account for variations in intensity across the probe beam and in pixel efficiency. We have assumed that the cloud is optically thin and so the intensity does not change significantly across it. σ is the absorption cross-section given by:

Γ 2I0/Isat hω¯ L σ = × 2 2 × (2.13) 2 1 + 2I0/Isat + 4δ /Γ I0 1 = σ0 2 2 (2.14) 1 + 2I0/Isat + 4δ /Γ

2 2 with Isat = 3.14 mW/cm and σ0, the unsaturated resonant cross-section (6πλ ) = 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 Z i i i i A C1 − C3 N = A n (y)dy = − ln i i (2.15) σ C2 − C3 where A is the area of real space imaged onto each pixel. If M is the magniﬁcation factor, then A is given by: 24 × 24 µm2 A = (2.16) M 2 th i i The dark count of the i pixel (C3) is found from image 3 and subtracted from C1 i and C2 to minimize the eﬀect of dark noise. All further analysis uses this 2D grid of the number of atoms per pixel N i. 34 Chapter 2. The BEC Apparatus

The total number of atoms in the sample is X X i i i A C1 − C3 Ntotal = N = ln i i (2.17) i σ i C2 − C3 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 imaged onto 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 to determine σ (eqn. 2.14), since I0 is comparable to Isat in our apparatus. The magniﬁcation 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 beam proﬁle. The intensity of the probe beam at the cloud I0 can be calculated from a measurement of the total power (Ptot) and the beam waist (w). π P = I w2 (2.18) tot 2 0 To measure the beam waist one places a razor blade mounted on a micrometer in the beam just before the cell and measures the power that gets past (P ) as a function of knife position (x) (positions for the knife and power-meter are indicated in ﬁg. 2.16).

Ã Ã √ !! 2 P = P + P 1 − erf (x − x ) (2.19) 0 1 0 w Figure 2.17 shows our data for the power as a function of knife position, with eqn. 2.19 fitted, from which the value for the beam waist, w = 4.5 ± 0.1 mm, is obtained. Plugging this value into eqn. 2.18 we find that the probe intensity (in mW/cm2) is related to the measured total power (in mW) by: −2 I0 = 3.14 Ptot cm (2.20) The power (and hence intensity) of the probe beam must be checked daily, (remembering that the coupling efficiency down the probe fibre will be a function of probe detuning) if accurate number measurements are required. For most experiments a probe intensity of around Isat was used, since this optimizes the signal to noise of the image formed on the CCD. If a higher power is used then the percentage of light absorbed by the atoms falls and so imperfections in the probe beam become significant. If a lower power is used then few photons are available to be absorbed and so dark noise becomes significant. 2.5. The imaging system 35

Figure 2.17: The power passing a knife edge at position x across our probe beam. The solid 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 ±0.1 mm

Magnification calibration The magnification of the system, M was determined by releasing a cloud from the trap, allowing it to fall under gravity and recording its position on the CCD (s) as a function of time. These data were fitted with the function: µ 1 ¶ s = M − gt2 + V t + S (2.21) 2 0 0

where M is the magnification. The constant V0 accounts for the velocity imparted by any kick when the trap turns off and S0 is the initial position. A typical fit to the 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 image array were added to produce 2 1D density distributions, which were ﬁtted with Gaussian curves. −x2/σ2 n(xj) = n0je j j (2.22)

For the special case of a Gaussian distribution, the same values for σx and σz are obtained as from a 2D Gaussian fit. The summation increases the signal to noise and enables us to use a very fast 1D fitting routine for online analysis. Once the cloud size and number of atoms is extracted, the WINVIEW program is able to use the 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 is particularly useful when optimizing the different stages of the evaporation ramps. 36 Chapter 2. The BEC Apparatus

Figure 2.18: The position of the centre of a cloud as it falls under gravity. Comparing the 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 condensates, all other quantities assume the Gaussian distribution of a thermal cloud, both in the trap and after expansion and so are not accurate for analyzing a condensate in the Thomas-Fermi regime. Condensate images are transferred to a MATLAB routine, developed by Gerald Hechenblaikner, which fits both a 1D and 2D double distribution - a Gaussian for the thermal cloud, with a parabolic condensate distribution 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 calculate the temperature both from the condensate fraction and, more accurately, from the wings of the thermal distribution. Most of the experiments in this thesis tested the predictions of the GP equation, which assumes a pure condensate at T = 0. Thus we used the coldest condensates possible, at T ≤ 0.5 Tc, where the thermal cloud is no longer visible and the temperature cannot be measured. Figure 2.19(b) shows the 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 process of taking a single image, for two reasons. Firstly it uses a pulse of resonant or near resonant light, which heats the condensate 2 orders of magnitude above the critical 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 2.5. The imaging system 37

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

I t σ h¯2k2 ∼ 0 0 × 2 hω¯ 2kBm = 177 × 0.18 µK = 32 µK

Secondly, in a typical trap, the condensate has dimensions ≤ 5 µm, close to the resolution limit of the imaging system. Therefore we must release it from the trap and allow it to expand before imaging. Many of our experiments involve observing the evolution of a condensate excitation, for example plotting a shape oscillation and fitting the oscillation frequency. At present this requires making many condensates with identical starting conditions and varying the evolution time of each one before imaging. This is very time consuming, and inevitably small variations in the starting conditions produce noise on the data. It would be desirable to track the evolution of a single condensate, by taking many non- destructive images. This is possible using phase-contrast or dark ground imaging techniques; these both use the change in phase rather than the change in amplitude of the probe beam, produced by absorption in the condensate [?]. The phase shift 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 resonant light, may be used. Typically 10s of images of the same condensate may be taken before significant heating has occurred. Whilst our imaging system could easily be adapted for non-destructive imaging, until we create much larger condensates, with an in-trap size significantly greater than the resolution limit, we will always have to release and hence destroy the condensate prior to imaging. 38 Chapter 2. The BEC Apparatus

2.5.7 The vertical imaging system The vertical imaging system has the same basic optical design and components as the horizontal system. However the beam path is more complicated, partly because it must share the route up through the cell with the lower MOT beam and partly because the apparatus was not originally designed to allow imaging along a second direction. The scheme is best described pictorially and is shown in the diagram of fig. 2.20 and the photograph of fig. 2.21. After exiting the optical fibre the direction of the probe pulse (horizontal or vertical) is determined by a half-waveplate mounted on a shutter that can swing in and out of the beam, followed by a polarization beam splitting cube. Since the horizontal path must always be used for optical pumping at the start of the evaporation ramps, the shutter operates under computer control. The vertical probe beam is then mixed with the lower MOT path, travels up through the cell and is separated again at a polarization beam splitting cube at the top of the cell. After passing through a hanging 1 to 1 imaging system (Comar lenses 160DQ32), the vertical path rejoins the horizontal one through the microscope objective and onto the camera. The vertical system is focused independently of the horizontal one by moving the second 1 to 1 lens on a vernier mount. Two main considerations guided the design of this imaging system - firstly, is it possible for the MOT and probe beams to coexist with the correct polarizations on the same 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 polarizations of the upper and lower MOT beams. Many standard texts describe the MOT polarizations relative to a fixed axis, typically the positive x, y and z axes and under this convention, beam pairs have circular polarization of opposite handedness (which handedness depends on the direction of current flow in the quadrupole coils). Two quarter-wave plates, one above and one below the cell produce the circularly 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 passing through both, the linear polarization of a beam is unchanged. Given that the polarizations of the MOT beams are fixed, we must now consider if it is possible to use polarization to mix the vertical probe beam with the lower MOT beam and separate it out again after the cell onto another path. The scheme is 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 so have opposite linear polarizations. These polarizations are maintained after passing through the cell and both quarter-waveplates and so the imaging beam is separated from the lower MOT beam by reflection at the polarization beam splitting cube 2.5. The imaging system 39

Figure 2.20: A diagram of the vertical imaging system, showing the polarizations required to mix and separate the vertical probe beam and the lower MOT beam 40 Chapter 2. The BEC Apparatus

Figure 2.21: A photograph of the experimental MOT with the new vertical imaging system in the background. 2.5. The imaging system 41

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

Resolution The vertical imaging system was built primarily to study vortices, which appear as small holes within the condensate, and so optimizing the resolution of the system was an important consideration. From Fourier diffraction theory, small objects of dimension d scatter light within a solid angle ∼ λ/d, and all this light must be gathered onto the camera if the object is to be resolved. Light scattered by the condensate must travel through the hole in the quadrupole coil former, through a polarization beam splitting cube and then into the first imaging lens. The resolution is set by the element with the smallest numerical aperture. The first imaging lens, with a numerical aperture of 0.19, limits the resolution to 4.2 µm. This limit is slightly higher than that for the horizontal system (3.2 µm), since the quadrupole coil former forces the first imaging lens to be further from the condensate but it is still 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 the vertical imaging system) are accelerating under gravity. We can estimate whether the movement of the cloud (∆u) during 10 ms of free expansion is sufficient to cause defocusing by considering the depth of focus (∆vf ) of our system. The depth of focus is the distance from the focal plane at which the first diffraction minimum appears at the centre of the image of a point-like object. Alternatively it is the distance at which the image is twice its diffraction limited size. It provides a crude estimate of the range of image plane positions over which the optical system may be considered to be focused. Consider the lens system in fig. 2.22 (focal length f, diameter D) creating an image of a point-like object. In the focal plane, the diffraction limited radius of the image Wd is: λ W ≈ × v (2.23) d D The size of the image as a function of distance ∆v away from the focal plane is approximately: D/2 W (∆v) ≈ W + × ∆v (2.24) d v The depth of focus is the value of ∆v at which the image has doubled in size: 2λv2 ∆v = (2.25) f D2 42 Chapter 2. The BEC Apparatus

Figure 2.22: A diagram showing how the depth of focus of an imaging system may be calculated. The imaging system is assumed to have a large magniﬁcation factor, so that v À u and u ≈ f. The diﬀraction pattern in the focal plane and a distance ∆vf away from it is shown. 2.5. The imaging system 43

Having found an expression for the depth of focus, it will now be used to ﬁnd the range of positions ∆uf through which the object may move whilst producing a reasonably focused image on a stationary image plane. From the lens equation we have ∆u ∆v = − (2.26) u2 v2 Inserting this into eqn. 2.25 gives:

2λu2 ∆u = (2.27) f D2 2λf 2 2λ ≈ = (2.28) D2 NA2 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. 2.28 gives ∆uf = 43 µm. This is in good agreement with our experimental observation that changing the expansion time by 1 ms required refocusing, for all expansion times 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 increased without compromising resolution by using a smaller magnification and a camera with smaller pixels (Smaller pixels would hold less charge and hence have a smaller signal 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 the diffraction rings around it, at the chosen expansion time. Then vortices were made within the condensate, that have an expanded diameter of ∼ 6 µm, and detailed focusing was done by optimizing the depth and clarity of these small structures. For horizontal focusing we simply minimize the size of a trapped condensate (∼ 3 µm), but this method is not appropriate in the vertical direction because the trapped and expanded condensates are not in the same object planes. Calibrating the vertical magnification can be achieved by measuring the x size of a cloud with both the vertical system and a calibrated horizontal system. This is accurate if the average of many identically prepared clouds is used. The intensity of the vertical probe beam was only approximately known because we did not use the vertical system for accurate number measurements. It would have been difficult to measure the intensity at the cloud because we placed a very small pinhole in the vertical beam, of ∼ 2mm diameter about 1 m from the cell. The size of the hole was adjusted until only the central bright diffraction ring could be observed on the camera. This central ring produced a much smoother beam profile at the condensate 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 condensate, simply by sending light down both paths at once and forming 2 images at 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. 44 Chapter 2. The BEC Apparatus

If probing occurs in a magnetic field, then the image that has the magnetic field correctly aligned for mF = 2 to 3 transition dominates the absorption. One way to improve the system would be to fire two separate probe shots, onto the same CCD screen, allowing just enough time inbetween for the magnetic field to rotate to be optimally aligned for each. The minimum time would be one quarter of the TOP period i.e. 35 µs. After the first shot, the atoms would be heated to a temperature of ∼ 30 µK or a speed of 8 × 104 µm/s. Thus in 35 µs they will have only moved 3 µm, which is small compared to the size of the condensate. Thus the second image would be adequate for observing the shape of the entire condensate although not for resolving structures within it. This system is currently being implemented, using a new liquid crystal polarization rotator (Displaytech LV2500- OEM) to rapidly change the imaging path. Using a specialized driver (Displaytech DR95), this waveplate can change the polarization of a beam in ∼ 30 µs and so the horizontal and vertical probe pulses can be fired separately, when each has the magnetic 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 be found in [?, ?]. It should be read in conjunction with chapter 4, which describes in detail how this procedure was recently optimized to triple the number of atoms in 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 MOT by the trapping light. These cold atoms travel to the second MOT with a velocity of ∼ 10 m/s, which is 100 times greater than their transverse speed. The push beam tends to optically pump the atoms into a low-field seeking state as they leave the pyramidal MOT. Hence magnetic guiding, provided by magnetic strips along the transfer tube, is used increase the transfer efficiency to the experimental MOT by a factor of 1.4. Further investigation of the transfer process may be found in [?, ?]. Conditions in the second MOT are chosen to optimize the number of atoms captured, balancing the need for a large capture velocity vc, with a low central density [?]. The beams have a clean Gaussian profile and are limited to a waist of 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 detuning of −15 MHz or −2.6 Γ. In addition, each beam contains 150 µW of repumping light. The radial quadrupole field is 6.5 G/cm. The number of atoms in the MOT is limited by light-assisted, density dependent collisional losses. Thus the number may be doubled by the addition of a 2 G rotating TOP bias field, that lowers the central density [?]. The relative number of atoms in the second MOT is monitored by the fluorescence collected on a photodiode (fig. 2.1), to ensure that each experimental run starts with the same number of atoms (∼ 6 × 108). Once the required number of atoms is loaded (loading typically takes ∼ 1 minute), the control computer is triggered by the operator and all further steps in the production of 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 magnetic trap at the lowest possible temperature. The important parameters for the transfer of 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 to compress the cloud in the MOT, thus increasing the number of atoms within the spatial boundaries of the magnetic trap. To achieve the compression the bias field is turned off suddenly and the quadrupole field gradient is doubled over a period of 25 ms. Meanwhile the trapping beam detuning is increased to −3.5 Γ and the power reduced slightly to 3 mW/cm2, to reduce the outward radiation pressure from scattered photons at the trap centre. All these changes are made in linear ramps for convenience. 3.2. Loading the magnetic trap 47

0 Stage BT (G) Bq (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 7 Opt. 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, 0 compression and molasses is the intensity per beam. Bq 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 Doppler cooling limit of 138 µK(kB TD =h ¯Γ/2). In zero magnetic field, sub-Doppler cooling mechanisms [?] operate efficiently producing a lower temperature Tmol ∝ I/|δ|. Thus the magnetic trap is turned off, δ is increased and I reduced to the values in table 3.1 over 2 ms and then held there for a further 5 ms. Temperatures as 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 the F=2 manifold and must be optically pumped into the F=2, mF =2 state desired for magnetic trapping. This is achieved by firing three σ+ polarized pulses of light from the probe fibre, each of 5 µs in length, and synchronized with a TOP bias field 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 reduce the transition rate and hence heating for atoms that have reached the desired mF = 2 state (although this can be achieved much more effectively as described in section 4.3.2). During optical pumping the trapping light is off, but the repumping light 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 with 0 BT = 25 G and Bq = 65 G/cm. These values were chosen to create a large locus of B=0 which encloses the entire cloud, and a trap curvature that matches the shape of the cloud to minimize heating. The calculation for mode-matching the trap to the shape of the cloud is given in [?]. 48 Chapter 3. BEC Production

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. The detuning of each transition from resonance in zero field is shown. µBB/h has a value of 1.4 MHz in a magnetic field of 1 G

3.3 Evaporative Cooling

3 Immediately after loading into the trap, the phase-space density (n0 λdB) is ∼ 7 × 10−7. By a process of forced evaporative cooling and trap compression we increase the phase-space density by 7 orders of magnitude to produce a condensate. During forced evaporative cooling, we selectively remove the hottest atoms from the cloud and allow those that remain to rethermalize via elastic collisions at a lower mean temperature. Important parameters during evaporative cooling are the phase-space density, temperature, density and collision rate of the cloud. We aim to achieve a regime of runaway evaporation, where the density and collision rate rises steadily as a result of the decreasing temperature, despite the necessary loss of atoms. The dependencies of these quantities on number, temperature and trap frequency are given in appendix C. The evaporation scheme is shown in table 3.2 and plotted in ﬁg. 3.2.

3.3.1 Adiabatic compression The speed of our evaporation ramps is limited by the elastic collision rate because the cloud must remain thermalized during the evaporation ramps. The initial collision rate during evaporation is 10 s−1, resulting in ramp times of the order of tens 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 so the ﬁrst stage is a compression of the trapped cloud. This may be done quickly 3.3. Evaporative Cooling 49

0 Ramp Stage Dur. Bq 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 29.4 4 B = 0 evap. 16 194 2 123.4 0.1 1.0 2.8 5 Rf evap. 12 194 2 123.4 - 0.24 1.736

Table 3.2: The evaporative cooling ramps used to obtain BEC. The trap conditions quoted 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 cooling 0 ramps; the quadrupole ﬁeld (Bq), TOP bias ﬁeld (BT ), radius of death (r0), radial trap frequency (ω⊥) and rf evaporation frequency (νrf ) . 50 Chapter 3. BEC Production

(2 s) because we keep r0, the locus of the magnetic field zero, constant and so no evaporation occurs. The change is adiabatic (phase-space density constant) provided that the potential energy changes at a lower rate than the kinetic energy of the trapped atoms; this condition can be expressed as: dω ¿ ω2 (3.1) dt and is well satisfied for this ramp. The pure compression stage is limited by the maximum value of BT (45 G) that we can achieve without the temperature of the ◦ 0 coils rising too far (i.e. above 50 C). The next ramp increases Bq to its maximum value, 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 the circular path of the magnetic field zero) since the turning points of their orbits are in the regions of highest potential energy. The initial evaporative cooling method is to gradually reduce the radius of the magnetic field zero by decreasing BT . The magnetic field zero induces Majorana spin flips into untrapped magnetic states, removing atoms at this radius (fig. 2.11). This evaporation method is unique to the TOP trap and is used in ramps 3 and 4. (The process is divided into two separate ramps for a purely technical reason; to enable the rf evaporation field√ to 0 be switched on partway through). Since the trap frequency depends on Bq/ BT it increases throughout both ramps, helping to maintain a high collision rate. At values of BT < 2 G, the trap lifetime is reduced. This occurs because at small TOP fields the transition frequencies between different mF states are reduced and noise on the bias field can drive transitions to untrapped states. Thus this first method 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 shift of 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 transition at the outside edge of the cloud. The resonance condition is: µ g B ν = B F (3.2) rf h As the bias field rotates, the rf cutting region moves in a circle about the centre of the trap on the opposite side to B = 0. The exact size of the region over which atoms 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 the start of ramp 4, cutting outside the locus of B = 0. It moves inwards slightly faster than the magnetic field zero, so that by the end of this ramp both are cutting at the same radius rrf /r0 = 1. During the final rf evaporation ramp, the rf cutting surface moves inside the circle of death to a position of 0.24 r0 at which point a pure BEC is formed. A more detailed discussion of the coupling between trapped and 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 of atom number in fig. 4.8. Bose-Einstein condensation occurs when the phase-space density is equal to 2.61. Using this evaporation procedure, the transition point is 4 reached with ∼ 5×10 atoms and a critical temperature Tc of 300 nK. After further evaporative 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 at the centre of the density distribution, as the occupation of the ground state of the magnetic trap becomes macroscopically large (fig. 3.3(b)). Another clear signature of condensation may be gained from the shape of the cloud after expansion. The internal energy of the condensate is dominated by the repulsive interaction between atoms, whilst in the thermal cloud it is purely kinetic energy. The thermal cloud expands isotropically (from equipartition) becoming spherical at large expansion times (fig. 3.3(a)). The condensate has more energy in the direction that is most tightly confined in the trap (axial) and hence expands faster in this direction. Thus a trapped oblate condensate (pancake-shaped) becomes prolate (sausage-shaped) at long expansion times (fig. 3.3(c)). 52 Chapter 3. BEC Production

Figure 3.3: The atomic cloud with diﬀerent condensate fractions (N0/N) after 15 ms of expansion from a pancake-shaped trap. The top line of pictures are false colour images of the density distribution, whilst the lower line of pictures plots the same density distribution 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 Condensate Production

The gyroscope experiment described in chapter 10 required a long excitation procedure (∼ 2 s) after the condensate had been made, first to produce a vortex and then 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 immediately after the condensate was made, which decayed during excitation to less than 10,000 atoms for imaging. With such a low atom number, fitting the tilted parabolic density distribution to extract an angle√ became very unreliable. The effect of the vortex line (with a core size ∝ 1/ n) on the parabolic density distribution could not be ignored and imperfections in the imaging system became significant against the weak condensate image. We decided to repeat the experiment with larger condensates to ensure that well over 10,000 atoms were always available for imaging. To produce larger condensates it was necessary optimize every step of the production process, from loading the second MOT, through the transfer to the magnetic trap and finally the evaporation ramps. This chapter gives an account of the methods used to optimize each stage of the process and the improvements that were made.

4.1 Loading the second MOT

Poor second MOT loading may be due to a number of factors - misalignment of the pyramid pushing beam, poorly balanced second MOT beams and poor master or repumping laser modes are amongst the most common. Alignment of the second MOT beams will be discussed in section 4.2. The diﬃculties that we have had with 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 suﬀer problems with the mechanical mount, which cause the external cavity to drift whilst the commercial TUI ECDLs have the ability for stable lasing in several modes, which, if undetected, results in only a small fraction of the light interacting with the atoms. The current arrangement uses a homemade master laser and a commercial TUI repumping laser. The weak repumping saturated absorption lines are strengthened by heating the cell (section 2.3.4). The repumping laser mode structure is checked on a weekly basis using a Fabry-Perot spectrum analyser, rather than just relying on the saturated absorption signal. We have checked the output spectrum of our laser ampliﬁers, the MOPA and slave, and these appear to reproduce the master spectrum 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 with maximum phase-space density into the magnetic trap. This requires the MOT to have a regular shape, have the same centre as the magnetic trap and expand evenly and slowly when the quadrupole field is turned off e.g. during the molasses phase. These requirements are affected by several factors including the alignment and 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 in each beam was checked. We use equal power in all of the horizontal beams and approximately 20% extra in the vertical beams. Then we aligned the centre of each beam with the centre of the quadrupole field. This was achieved by placing an iris in the centre of each beam in turn, just before the final alignment mirror and 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 to optimize the current in each of the 3 pairs of nulling coils. The molasses stage uses sub-Doppler cooling mechanisms to reduce the temperature of the cloud below that of the MOT. Sub-Doppler cooling mechanisms require the ground state to have degenerate magnetic sublevels and so only work effectively in zero magnetic field. We measure the temperature of the cloud after it has been loaded into the magnetic trap as a function of one of the nulling coil currents. The data in fig. 4.1 gives a clear minimum in the temperature, indicating the current at which the field is best nulled. We also did a control experiment, in which the molasses stage was removed and the temperature was plotted as a function of one of the nulling currents. This data showed no clear minimum, confirming sub-Doppler cooling was responsible for the reduced temperature in fig. 4.1 and not an improved alignment 4.3. Loading the magnetic trap 55

Figure 4.1: The temperature of the cloud (after 4 s hold in the magnetic trap) as a function 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 MOT beams. Firstly the centre of the MOT cloud must not move if the quadrupole field is suddenly jumped to a large value. This ensures that the MOT is centred on the quadrupole field and hence on the magnetic trap. Secondly the atomic cloud from the MOT should expand slowly and evenly in all directions when the quadrupole field 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 optimize the stages up to and including loading the magnetic trap. Improvements to these stages are all analyzed by looking for an improvement in the phase-space density in the magnetic trap (increased number, reduced temperature). It is important to be sure that the cloud is thermalized before a measurement is made, otherwise the temperature measurement will not be accurate. Thus a detailed study of the behaviour of the cloud after loading was made. No dipole motion was observed, (indicating that the MOT and trap centres were well aligned) and the small amount of quadrupole motion was quickly damped (≤ 1 s). Since we load a spherical MOT cloud into a pancake-shaped magnetic trap, there is an initial size, and hence temperature, mismatch between the x and z directions. This may 56 Chapter 4. Optimizing Condensate Production

be observed in TOF, since the aspect ratio of a non-thermalized cloud will be less than that of a thermalized cloud after expansion: v u u 2 2 X size TOF X size TRAP t1 + ωx t Thermalized AR = = 2 2 (4.1) Z size TOF Z size TRAP 1 + ωz t After holding the cloud in a typical load trap for ∼ 4 s, the aspect ratio reached its thermalized value, in reasonable agreement with the predicted thermalization time of ∼ 1 s (a few collision times). This hold time should always be used between loading 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 discussed below. They may be compared to the original values in table 3.1.

0 Stage BT (G) Bq(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 5 Opt. 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 for the MOT, compression and molasses is the intensity per beam.

Summary of changes • Sudden changes to the trap reduce the phase-space density and should be eliminated. By gradually reducing the MOT bias field during the compression ramp rather than jumping it off at the start, we were able to load a colder cloud into the magnetic trap. We also found that reducing the bias field to zero produces sufficient compression of the cloud in the MOT and so it is not necessary to increase the quadrupole field during this stage. • The molasses stage has been shortened, since the atoms are no longer trapped and are diffusing outwards during this stage. 5 ms is the minimum time for the quadrupole field to die away (1-2 ms) and sub-Doppler cooling to reach a 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 4.3. Loading the magnetic trap 57

are evenly distributed over the 5 magnetic substates of the F=2 ground state manifold (as described in section 3.2.3). Thus optical pumping can theoretically increase the number of trapped atoms by a factor of 5. The original system (described in section 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 pumping power. The fibre alignment was optimized each morning at the pyramid trapping frequency, not the pumping frequency. Since the coupling down the probe fibre varies with detuning (section 2.3.4), the pumping power was actually unknown. Checking the pumping power at the correct detuning every morning solved this problem. 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 a photon it recoils and gains momentumhk ¯ in the direction of the optical pumping beam. If it absorbs and spontaneously emits n photons during the pumping phase, then the total increase in energy will be:

∆E h¯2k2 = (n2 + n) = (n2 + n) × 0.2 × µK (4.2) kB 2 m where the n2 term comes from unidirectional absorption and the n term from isotropic spontaneous emission. Given the n2 dependence, it is important to minimize the number of transitions that each atom undergoes, whilst maintaining efficient pumping. This may be achieved if the atom becomes more and more detuned from the pumping light as it reaches higher mF states. Ideally the mF =2 to mF =3 transition should be so far detuned that the final mF =2 state is effectively dark to σ+ pumping light. Unfortunately, this transition also has the highest Cleb- sch - Gordon coefficient (see appendix B) and so can never be made completely dark whilst maintaining full pumping. The scheme is optimized by pumping in a large B field, so that the different transitions are separated by large Zeeman shifts and by tuning the pump light on resonance with the lowest mF =-2 to mF =-1 transition (fig. 3.1). Two different fields and different pump powers were tested and 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 we use nearer 200 µW, trading a little heating for an increased number of atoms in the correct magnetic substate. A comparison of the different ∆mF =+1 transition rates in the original and improved pumping scheme is given in table 4.2. In the original scheme the transition rate from the final state was 15 times higher than that from the initial state. In the new scheme the rate out of the final mF =+2 state is 4 times smaller than the rate from the lowest mF =-2 state, thus achieving minimum heating and optimal pumping. Using this scheme, optical pumping reliably increases the number of trapped atoms by a factor of ≥ 3.5 and heats the 58 Chapter 4. Optimizing Condensate Production

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 a function of optical pump intensity. 5 × 10 µs pumping pulses were ﬁred in a 30 G ﬁeld and the beam was tuned into resonance with the lowest mF =-2 to mF =-1 transition.

Original BT = 3G Improved BT = 30G mF 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 0.2 0 to 1 6/15 4.93 0.7 14 0.1 1 to 2 2/3 5.17 1.0 21 0.09 2 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 µs pulses at 0.2 Isat. δ is the detuning of the transition from the pump beam frequency. 4.3. Loading the magnetic trap 59

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 recoils from absorbed photons should then cancel, removing the term ∝ n2 in eqn. 4.2. (Since an atom absorbs ∼ 10 photons during the optical pumping pulses, removal of the n2 heating term should be significant). A second beam of the correct polarization can be created simply by reflecting the pump beam and so this scheme was tested, but unfortunately no improvement was observed. This suggests that most of the heating is due to the reabsorption of spontaneously emitted photons within the cloud. We also investigated the number and length of pulses. Experimentally, the maximum pulse length was 15 µs (before the bias field had rotated sufficiently that significant anti-pumping occurred) and a total pumping time of 50 µs was required 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 highest phase-space density and so use these criteria to choose its size and stiﬀness. We should be able to satisfy both conditions simultaneously because the load trap has 0 2 independent parameters, Bq and BT .

Figure 4.3: The number (solid circles) and temperature (open circles) of the cloud after loading 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, 0 which has radius r0 = BT /Bq and defines the radial size of the load trap, falls 60 Chapter 4. Optimizing Condensate Production

outside the radius of the MOT cloud. Figure 4.3 shows the number and temperature in the load trap, as a function of r0. The atoms were held in the trap for 2 seconds before imaging to allow for rethermalization and in each case the fields were adjusted so that each trap had the same stiffness. Thus any changes in the temperature are due to evaporative cooling during the hold time, if the magnetic field zero falls within the radius of the cloud. Both number and temperature begin to fall at r0 < 2.8mm, indicating that this is the radius of the atom cloud at loading. We eventually choose an initial value of r0 of 3.2 mm, which should collect all atoms even from particularly large MOTs, whilst using a relatively small value of BT which is advantageous for the later ramps (see section 4.4.1). Theoretically, the highest phase-space density will be achieved if the curvature 02 or stiffness of the load trap (∝ Bq /BT ) matches the shape of the MOT cloud. Under these conditions the transfer will occur adiabatically and the increase in entropy will be minimized. When the experiment was first set up, much theoretical effort was put into ‘mode-matching’ the transfer and is discussed in [?]. However, perfect mode-matching cannot be achieved since we load a spherically symmetric cloud into a pancake-shaped potential (as described at the start of this section). This agrees with our experimental observation, that the stiffness during loading is not critical. The phase-space density of the cloud is roughly constant when the initial trap has a frequency in the range ω⊥/2π = 10 to 14 Hz, (r0 constant) and falls slowly outside this range. This was checked both after holding the cloud for a few seconds in the load trap and after the first evaporation stage. Using the technique described in [?] it would be possible to create a spherically symmetric TOP trap, into which the MOT cloud could be loaded with maximum phase space density. However this would require major changes to the TOP trap apparatus (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 been investigated further. The final choice of conditions for loading the atom cloud into the magnetic 0 trap was Bq = 44.11 G/cm and BT = 14.1 G. In this trap, the magnetic field zero follows 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 in table 4.3 and may be compared to the original values in table 3.2. Each of the changes is discussed in sections 4.4.1 to 4.4.3 below.

4.4.1 Adiabatic compression

0 The original adiabatic compression stage involved ramping up Bq and BT whilst maintaining a constant ratio between them, thus r0 was held constant whilst the 4.4. Evaporative cooling ramps 61

0 Ramp Stage Dur. Bq 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 29.4 3 B = 0 evap. 18 194 2 123.4 0.1 1 2.8 4 Rf evap. 12 194 2 123.4 - 0.24 1.736

Table 4.3: The optimized evaporative cooling ramps. The trap conditions are quoted at the 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 maximum value of BT (45 G) was reached, the quadrupole field continued to increase alone so that the maximum trap stiffness was achieved. During this stage, r0 decreased and hence some inefficient evaporation (occurring before the collision rate was optimized) could not be avoided. We decided to investigate whether a higher maximum BT would improve the phase-space density, since evaporation would start from a stiffer trap and hence be more efficient. The temperature of the TOP coils was carefully monitored, with a thermistor attached to one of the coils, to ensure 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 evaporation ramp using the original conditions of table 3.2, with BT limited to 45 G. We observed an average of 1.3 ×107 atoms. We then repeated the experiment, contin- 0 uing the adiabatic compression up to BT = 60 G, Bq = 160 G/cm and hence using a shorter ‘compression + evaporation’ stage. The number of atoms immediately dropped to 0.5 ×107, the cloud diameter (after 5 ms of expansion) increased by a factor of 1.13 indicating a higher temperature, and vertical streaks appeared on the image. Over the next hour or so the number of atoms in the cloud gradually deteriorated. All these factors indicate that the cell is being heated significantly at 60 G - an assumption that was backed up by the coil temperature measurements. The TOP coil temperature reached 50◦C in a single run to 60 G, as opposed to 40◦C in a 45 G run. Over a few runs this will increase the background pressure of rubidium in the cell, by sublimating it from the walls and thus reduce the trap lifetime. The streaks on the final image may well be caused by rapid changes in the refractive index of the cell as it cools down, changing the probe beam profile on the camera between the experimental and background images. The final conclusion was 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 adiabatic compression conditions. During the compression we require that no signiﬁcant 62 Chapter 4. Optimizing Condensate Production

evaporation occurs, which can be achieved by holding the cloud size, σ⊥ and the locus of the magnetic ﬁeld zero, r0 (eqn. 2.8) in a constant ratio. The cloud size is given by: s √ 1 2kBT BT √ σ⊥ = ∝ 0 T (4.3) ω⊥ m Bq

Since the cloud size reduces as the trap becomes stiﬀer, r0 need not remain constant during this stage but may reduce by the same factor, allowing a lower BT to be used. √ 0 σ⊥ BT √ Bq ∼ 0 T (4.4) r0 Bq BT

so if σ⊥/r0 remains constant during the adiabatic compression ramp then: T T i = f (4.5) BT i BT f

Now the phase-space density φ is given by Ã ! hω¯ 3 φ = N ho (4.6) kBT

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

0 0 Bq Bq √ i = q f (4.7) BT i Ti BT f Tf

Inserting eqn. 4.7 into eqn. 4.5 we obtain the relationship between the initial and final traps in an adiabatic compression with constant (negligible) evaporation.   0 2/3 B Bq T f =  f  (4.8) B B0 T i qi Using this constant evaporation formula (eqn. 4.8) and the trap loading conditions of table 4.3, it is possible to reach the maximum quadrupole field during the adiabatic compression, at a TOP field of only 38 G. Thus not only does this method reduce the maximum value of BT (minimizing cell heating), but it also removes the 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, can we continue the ramps below 2 G, so that rf evaporation can occur more efficiently in a stiffer trap and secondly can the ramps be shortened to reduce the effect of 4.4. Evaporative cooling ramps 63

Figure 4.4: Phase-space density versus atom number for diﬀerent minimum values of BT . 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, at this point we replaced the ×1 imaging system with a ×10 imaging system, which is 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 range 1.5 G - 4 G) and then continued with a small rf cut, (final rrf > 0.4 r0), stopping above the critical temperature for condensate formation. For each value of BT we plotted phase-space density, φ, versus N. The results plotted in fig. 4.4 indicate an improvement in φ for a given number of atoms down to BT = 1.65 G due to increased evaporation efficiency, and then a sharp reduction below 1.65 G as the decreasing trap lifetime suddenly dominates. Direct measurements of the trap lifetime confirmed that it decreases sharply around this range of BT , from 11 ± 7 s at 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 produce a small condensate fraction. A plot of condensate fraction versus total number showed that with the longer rf evaporation ramp, the trap with BT = 2 G gave the best results. The lifetime of the 1.65 G trap was too short for the longer rf evaporation ramp required to produce a condensate. Finally we investigated the optimal time for the magnetic ﬁeld zero evaporation 64 Chapter 4. Optimizing Condensate Production

ramps. Figure 4.5 shows the phase-space density of the cloud at the end of the second 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 to ineﬃcient evaporation in a non-thermalized cloud, whilst beyond 34 s it falls slowly due 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 rf power and the ramp duration. Originally the coils had produced an oscillating ﬁeld with an amplitude of 6 µT. This was increased by a factor of ∼ 3 by doubling the output voltage of the Stanford rf generator and removing a series resistor of 50 Ω. (The coil reactance at 2 MHz was ∼ 50 Ω). With the higher rf power the number of atoms and hence phase-space density increased by a factor of 2 at a temperature 1.5 Tc. It was useful to consider the rf evaporation process in detail to determine whether even more rf power would result in further improvements. In a weak rf coupling ﬁeld, the evaporation process may be viewed as individual ∆mF =-1 transitions that eventually transfer the atoms to an untrapped state with negative mF . At higher rf power one must consider the ‘dressed atom’ picture. At resonance the 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 4.4. Evaporative cooling ramps 65

Figure 4.6: Dressed atom energy levels in a strong rf ﬁeld. The pattern is periodic is energy, repeating each time an additional photon is added to the dressed state. The avoided crossing relevant to our rf evaporation process in shown in bold. The lowest energy state is seen to evolve continuously from trapped to untrapped as the resonant position 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 example of an avoided crossing. If the atom passes through resonance sufficiently slowly, then it will stay in the lowest energy state and be adiabatically transferred from the mF = 2 to the mF = −2 state. This adiabatic transfer process is desirable since it provides efficient coupling directly to a strongly untrapped state. One can estimate the conditions for adiabatic transfer using eqn. 4.9 (derived from the Landau-Zener formula [?]). 2 µB BT ωo ¿ h¯ ΩR (4.9)

where ωo/2π is the TOP rotation frequency (7 kHz) and ΩR is the Rabi frequency for the ∆mF = 1 transition. Equation 4.9 shows that the rate of change of energy of 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 flopping at ΩR between the different dressed eigenstates at resonance (fig. 4.6). (The energy of the atom also changes because it is moving at some speed v in an inhomogeneous magnetic field, but this effect is less significant than that of the rotating bias field). Sinceh ¯ 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 66 Chapter 4. Optimizing Condensate Production

of 3 moved conditions significantly closer to those for efficient adiabatic transfer, which explains the improvement in phase-space density that we observed. The theory suggests that the rf evaporation process would be even more efficient in the fully 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 to regularly produce ‘pure’ condensates (T < 0.5 Tc, no thermal cloud visible) with 40,000 atoms. This represents an improvement over the previous conditions by a factor of 3. Whilst many small changes contributed to this improvement, the most signiﬁcant factors were:

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

• Reliable power and reduced heating during optical pumping.

• A more adiabatic MOT compression.

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

• Increased power for rf evaporation.

The optimization process was initially motivated by the need for sharper condensate images to improve the signal to noise on our gyroscope experiment data (chapter 10). After the gyroscope excitation procedure we had 19,000 atoms for imaging (rather than 9000) and the improvement in the images can be seen in fig. 4.7. Figure 4.8 shows a plot of phase-space density versus atom number during the optimized evaporation ramp. The gradient of this line indicates the efficiency of the evaporation ramps. A similar plot prior to optimization is also shown for comparison. Finally fig. 4.9 shows how the number, temperature and phase-space density 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 the y direction after 12 ms of expansion. After optimization the condensates are bigger, more regularly shaped and the vortex line has no significant effect on the density profile. 68 Chapter 4. Optimizing Condensate Production

Figure 4.8: Phase-space density versus atom number during the optimized evaporation ramps (data points). The solid line shows the equivalent data prior to optimization for comparison. Note that the apparent change in phase-space density during the adiabatic compression 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 of time 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 quantum statistics of identical bosons, we will start by considering the phase transition in a 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. th In thermal equilibrium the occupation of the i state, ni(Ei,T ), is given by the Bose-Einstein distribution function: 1 ni(Ei,T ) = (5.1) e(Ei−µc)/kB T − 1

where µc is the chemical potential and is determined by the constraint on the total number of particles: X ni(Ei,T ) = N (5.2) i Since V is large, the allowed energy levels are very closely spaced and the sum of eqn. 5.2 can be expressed as an integral: Z g(E) N = N0 + dE (5.3) e(E−µc)/kB T − 1 = N0 + Nex (5.4) where g(E) is the 3D free particle density of states:

V µ2m¶3/2 √ g(E) = E (5.5) 4π2 h¯2

The additional term N0 in eqn. 5.4 must be added because the integral does not contain 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 upper bound on Nex at a given temperature may be found by evaluating the integral with µc = 0. Ã ! 3/2 Z ∞ 1/2 V 2mkBT u Nex < 2 du (5.6) 4π2 h¯ 0 eu − 1 Ã ! mk T 3/2 < 2.612V B (5.7) 2πh¯2

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 [?]. √ Z u1/2 µ3¶ µ3¶ π du = Γ × ζ = × 2.612 (5.8) eu − 1 2 2 2

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

2πh¯2 µ N ¶2/3 Tc = (5.9) mkB 2.612 V

This deﬁnition of Tc is consistent with the picture that condensation occurs when the inter-particle spacing (1/n)1/3 is comparable to the thermal de Broglie wavelength, v u u 2 t 2πh¯ λdB = (5.10) mkBT

Rearrangement of eqns. 5.9 and 5.10 at Tc gives:

3 nλdB = 2.612 (5.11)

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

N µ T ¶3/2 ex = (5.12) N Tc N µ T ¶3/2 0 = 1 − (5.13) N Tc 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 potential: m ³ ´ V = ω2x2 + ω2y2 + ω2z2 (5.14) ext 2 x y z Therefore we must consider the eﬀect of the trapping potential on the condensation process. A trapped ideal gas has energy levels:

µ 1¶ µ 1¶ µ 1¶ E = n + hω¯ + n + hω¯ + n + hω¯ (5.15) x 2 x y 2 y z 2 z

where nx, ny, nz are positive integers. The density of states of the trapped gas is given by: πE2 gt(E) = 3 3 (5.16) 2¯h ωho

1/3 where ωho = (ωxωyωz) 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 controlling the number of particles that can be accommodated in excited states at a given temperature. When the density of states of a trapped gas gt(E) is inserted into eqn. 5.4, we ﬁnd:

Ã !3 kBT Nex = ζ(3) (5.17) hω¯ ho Ã !1/3 hω¯ ho N hω¯ ho 1/3 ⇒ Tc = = 0.941 (N) (5.18) kB ζ(3) kB

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

N µ T ¶3 0 = 1 − (5.19) N Tc

Note that the condensate fraction increases more rapidly as T falls in a trapped gas than in the homogeneous case. This occurs because the trapped gas has a smaller 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 accommodated in excited states at a given temperature and so particles accumulate more rapidly in 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 (solid line) and trapped gas (dashed line)

5.3 Bose-Einstein condensation with interacting particles

The balance between the interaction energy and quantum kinetic energy of the particles in a Bose condensate plays a vital role in determining its physical properties. In our system, the interaction energy depends only on the particle density because the scattering length is fixed. It ranges from negligible in the limit of an ideal gas condensate, to very much greater than the quantum kinetic energy in the Thomas-Fermi limit. The balance between the two energies determines the condensate’s size, shape [?], collective mode frequencies [?, ?], its superfluid properties, expansion behaviour [?] and can even cause it to implode [?]. Although interactions play a critical role in the Bose-condensed gas, it is still considered to be ‘weakly interacting’ because of its very low density. In such a dilute, low temperature gas only binary s-wave interactions need to be considered, which can be accurately modeled by a single parameter, the s-wave (or hard sphere) scattering length, a. In this limit, where only long-range effects are considered, such scattering 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 the same phase change. The 2 body interaction potential can be written as

4πh¯2a V (r − r ) = g δ(r − r ) with g = (5.20) 1 2 1 2 m 74 Chapter 5. Condensate Theory

The constant g is determined by considering the increase in kinetic energy of atom 1, 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 description of the system, is one of the most important and exciting features of this field. 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 the strong interactions cannot be modeled mathematically. Thus the properties of superfluid liquid helium are very difficult to interpret in terms of Bose-Einstein condensation. The mean field description is developed from the many body Hamiltonian describing N bosons in an external potential Vext, interacting via the 2 body contact potential of eqn. 5.20: " # Z h¯2 g Hˆ = d3~r Ψˆ † − ∇2 + V + Ψˆ †Ψˆ Ψˆ . (5.21) 2m ext 2

where the Bose field operators Ψ(ˆ r, t) and Ψˆ †(r, t) annihilate and create a particle at position r respectively. The time evolution of the Bose field is governed by the Heisenberg equation of motion ∂ h i ih¯ Ψˆ = Ψˆ , Hˆ . (5.22) ∂t The mean field approach, originally suggested by Bogoliubov [?], assumes that the system consists of a large ground state population (the condensate) and a small fluctuating population of higher modes (quantum depletion and thermal cloud). Thus the Bose field operator may be written as Ψ(ˆ r, t) = Φ(r, t) + δˆ(r, t) (5.23) Φ(r, t) = hΨ(ˆ r, t)i is a complex number field or wavefunction describing the condensate and the small field operator δˆ(r, t) describes the remaining modes. The equation of motion for the condensate wavefunction is found by replacing the Bose field operator Ψˆ with the mean field Φ in the Heisenberg equation of motion (eqn. 5.22) to give Ã ! ∂ h¯2∇2 ih¯ Φ(r, t) = − + V (r) + g|Φ(r, t)|2 Φ(r, t). (5.24) ∂t 2m ext This is known as the time-dependent Gross-Pitaevskii (GP) equation [?, ?] and has proved an excellent description of condensate behaviour provided that the system is: • Dilute with na3 ¿ 1. First order corrections to mean field theory are of order (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

iµct/¯h Φg(r, t) = φg(r)e (5.25)

where φg is the lowest energy eigenstate of the time-independent GP equation Ã ! h¯2 1 − ∇2 + m(ω2x2 + ω2y2 + ω2z2) + g |φ (r)|2 φ (r) = µ φ (r) (5.26) 2m 2 x y z g g c g

R 2 3 and is normalized to the number of atoms in the condensate, |φg(r)| d r = N0. We have replaced Vext with the 3D harmonic trapping potential used in the majority BEC of experiments. The energy of the ground state is the chemical potential of the condensate µc = µc(Tc). (Recalling the quantum statistics of section 5.1, µc(T ) rises as the thermal cloud is cooled, arriving at the energy of the ground state at Tc. µc cannot exceed this value or the ground state would have a negative population). A small, low density condensate can be described in the ideal gas limit, where interactions are ignored and the wavefunction φg has the Gaussian proﬁle of a harmonic oscillator ground state:

−x2/2a2 φg = φg0 e i i (5.27) q with a harmonic oscillator width ai = h/mω¯ i. In a non-spherical trap useful frequency and length scales for the trapq are the average harmonic oscillator frequency 1/3 ωho = (ωxωyωz) and length aho = h/mω¯ ho. For an interacting gas, the wavefunction φg(r) depends on the ratio of the quantum kinetic energy term to the interaction energy term in the GP equation:

2 2maho Na RTF = ng × 2 = (5.28) h¯ aho

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

1 1 n = |φ|2 = (µ − m(ω2x2 + ω2y2 + ω2z2)) (5.29) g c 2 x y z 76 Chapter 5. Condensate Theory

The widths of the condensate Ri, are determined by the positions on the x, y and z axes at which the density falls to zero: s 2µc Ri = 2 (5.30) m ωi

Normalization leads to an expression for the chemical potential µc in the Thomas- Fermi limit: µ ¶2/5 hω¯ ho 15Na µc = (5.31) 2 aho The parameters used to describe the properties of a condensate are summarized in table 5.1. For the experiments in this thesis condensates of between 10,000 and 45,000 atoms were used and so the values in table 5.1 are based on the smallest condensates that we make. Even for these small condensates the Thomas-Fermi parameter RTF and so all our experiments are well described by the Thomas-Fermi limit.

5.5 The hydrodynamic equations

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

n(r, t) = |Φ|2 (5.32) h¯ v = (Φ∗∇Φ − Φ∇Φ∗) (5.33) 2imn If we write Φ in the form q 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 phase S of the GP wavefunction: h¯ v(r, t) = ∇S(r, t) (5.35) m Substituting eqns. 5.32 and 5.33 for n and v into the GP equation, we ﬁnd that the condensate can be equivalently described by a pair of coupled equations: ∂n + ∇ · (nv) = 0 (5.36) ∂t Ã ! ∂v h¯2 √ mv2 m + ∇ V + gn − √ ∇2 n + = 0. (5.37) ∂t ext 2m n 2 5.5. The hydrodynamic equations 77

Parameter Symbol Formula Typical value

q 0 µ Bq TOP radial frequency ω⊥ √ 2π × 124 Hz 2m BT

q 0 µ Bq TOP axial frequency ωz 8 √ 2π × 351 Hz 2m BT

1/3 Average trap frequency ωho (ωxωyωz) 2π × 175 Hz q Harmonic oscillator width aho h/mω¯ ho 0.81 µm

2 2 ωz −ω⊥ Trap deformation ²t 2 2 7/9 ωz +ω⊥

³ ´2/5 ¯hωho 15Na Chemical potential µc kB × 68nK 2 aho r 2µc BEC radial half-width R⊥ 2 4.6 µm m ω⊥ q 2µc BEC axial half-width Rz 2 1.6 µm m ωz

µc µcm 20 −3 Peak number density n0 g = 4π¯h2a 1.7 × 10 m

3 µc 3 −5 Diluteness parameter n0a g a 3.3 × 10

Na Thomas-Fermi ratio RTF 72 aho

¯hωho 1/3 Ideal gas critical T Tc 0.941 (N) 169 nK kB

Thermal:trap energies kTc 28 ¯hω⊥

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

³ ´2 ³ ´ Lowest Kelvin mode freq. ω ¯h π ln 0.888R⊥ 2π × 161 Hz 1 2m 2Rz ξ

Table 5.1: Useful formulae for condensate parameters. The typical values are based on a 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. 78 Chapter 5. Condensate Theory

Equations 5.36 and 5.37 are known as the hydrodynamic equations for a superfluid because they resemble the equations that govern irrotational flow in a classical hydrodynamic fluid. The former is the continuity equation for particle number conservation. The latter is a force equation which resembles Bernoulli’s equation for irrotational isentropic compressible flow in a hydrodynamic fluid [?, ?]. In the case of a classical fluid, the hydrodynamic regime results when the mean free path is much 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 results from 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 closer to those of a collisionless regime than the hydrodynamic one). It is convenient to convert the GP equation into its hydrodynamic form for several reasons:

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

• It is a convenient form from which to investigate the collective excitations of the condensate in the Thomas-Fermi limit. To ﬁnd the T-F ground state we simply dropped the kinetic energy term proportional toh ¯2 in the GP equation. This approach is not possible for excited states because that term contains both the quantum kinetic energy (negligible) and the kinetic energy associated with the excitation (important). In the second hydrodynamic equation (eqn. 5.37), the quantum kinetic term (∝ h¯2) is separate and can be dropped in the TF limit, without losing the kinetic energy of the excitation, mv2/2. This gives eqn. 5.38, which is used together with eqn. 5.36 to investigate the collective excitations of the condensate in the Thomas-Fermi limit. Ã ! ∂v mv2 m + ∇ V + gn + = 0. (5.38) ∂t ext 2

5.6 Low-lying collective states

Much of the experimental information about the nature of a dilute gas BEC, including the work described in this thesis, comes from the spectroscopy of the excited states of the system. A close analogy can be drawn between the discrete excited states of a condensate and the well defined energy levels of an atom. Spectroscopic measurements of atomic energy levels have guided the development of the quantum mechanical model of the atom and likewise the spectroscopy of the condensate provides crucial information about its nature. For example, the first experiments focused on low energy collective excitations both in a pure condensate close to absolute 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 thermal cloud [?]. The first observation of a transverse collective excitation is described in chapter 7, which provided direct evidence for superfluidity in a condensate. In later experiments we made use of the flexibility of the geometry of the TOP trap to change the energy spectrum of the condensate and observe resonant up and down conversion when one mode had exactly twice the frequency of another [?, ?, ?]. In this section I will describe the geometry of the low-lying collective modes, the nomenclature used to label them and how the energy spectrum may be derived theoretically. The collective mode frequencies for condensates in the Thomas-Fermi regime can be calculated from the hydrodynamic equations for n and v (eqns. 5.36 and 5.38 which ignore the quantum pressure term ∝ h¯2). To linearize the hydrodynamic equations 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 higher state. 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 terms that are first order in small quantities leads to the following wave equation for density fluctuations ∂2δn m = ∇.(c2(r)∇δn) (5.41) ∂t2 2 where c(r) is the local speed of sound and mc (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 size of the condensate cannot be ignored and results in a discrete excitation spectrum. For a spherical, harmonic trapping potential, solutions defined on the interval 0 ≤ r ≤ R have the form

2n l δn(r) = Pl (r/R) r Ylm(θ, φ) (5.42)

2n where Pl (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 momentum), and m (the z component of the angular momentum). The majority of experiments, including those in this thesis, are carried out in traps with cylindrical rather than spherical symmetry. In such axially symmetric traps the azimuthal quantum number, m remains good and is used to describe the modes, but l is no longer a constant of motion. In this thesis we will only consider low-lying modes in an axially symmetric trap that are related to l = 0, 1 and 2 modes in a spherical trap. The term ‘surface modes’ is used to describe modes with no radial nodes (n = 0). 80 Chapter 5. Condensate Theory

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: X Dipole operator : ai xi (5.43) i X Quadrupole operator : aij xixj (5.44) ij X Octupole operator : aijk xixjxk (5.45) ijk This thesis will only consider dipole and quadrupole modes, although the more energetic octupole modes have also been excited e.g. in [?, ?]. The dipole modes correspond to a rigid sloshing of the condensate or thermal cloud from side to side, at the trap oscillation frequency. In a 3D trap there are 3 independent dipole modes in the x, y and z directions. They correspond to the three l = 1 modes in the spherical harmonic nomenclature. There are 6 independent quadrupole modes, related to the 6 different quadratic combinations of x, y and z; x2, y2, z2, xy, yz and xz. The normal modes are linear superpositions of these pairs. Relating this Cartesian description of the modes to spherical harmonic nomenclature, the quadrupole modes correspond to linear superpositions of the five l = 2 modes plus the single l = 0 mode. There are 2 3-dimensional width oscillations (breathing modes), 2 radial width oscillations (radial modes) and 2 modes which involve transverse motion relative to the trap potential with no change of the cloud shape (scissors modes). The geometry of the 6 quadrupole modes plus their relation to the spherical harmonic nomenclature is given in fig. 5.2. For the gyroscope experiment described in chapter 10, it is helpful to visualize the last four modes of fig. 5.2 in the spherical harmonic basis, as the m = ± 2 and m = ± 1 modes rather than a Cartesian basis. The operators and geometries of these 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 spherical 2 and axially symmetric condensates and have the form δn = r Y2m(θ, φ). (Super- positions of these degenerate pairs form the radial breathing and scissors modes respectively, if we wish to use a Cartesian basis). Substituting these expressions for δn into eqn. 5.41 gives the following expressions for the frequencies of these modes: 2 2 ω (l = 2, m = ± 2) = 2ω⊥ (5.46) 2 2 2 ω (l = 2, m = ± 1) = ω⊥ + ωz (5.47) The ﬁnal 2 normal modes in an axially symmetric trap are linear combinations of 2 non-degenerate spherical modes - the l = 0, m = 0 and l = 2, m = 0 modes - 5.6. Low-lying collective states 81

Figure 5.2: A table showing the six independent quadrupole mode operators, their geometry and the related spherical harmonic modes. The diagrams indicate the equilibrium distribution of the condensate (solid line) and the distributions at the extreme points of the oscillation (dotted line). 82 Chapter 5. Condensate Theory

Figure 5.3: A table showing the Cartesian operators and geometries of the m = ± 2 and m = ± 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 modes respectively. Note that the arrows in the bottom right hand box indicate rotation about the z axis in opposite directions for the m = ± 1 modes. 5.6. Low-lying collective states 83

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

2 ¨ 2 ωi(0) bi = ωi (t) bi − (5.48) bibxbybz which describe the width oscillations of a harmonically trapped Thomas-Fermi condensate 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-Dum equations may be written as

β¨(t) + M β(t) = 0 (5.50)

where β(t) = (βx, βy, βz) and   2 2 2 3ω⊥ ω⊥ ω⊥  2 2 2  M =  ω⊥ 3ω⊥ ω⊥  . (5.51) 2 2 2 ωz ωz 3ωz The 3 eigenvalues of M give the eigenfrequencies of the 4 quadrupole modes which involve only width oscillations; the in-phase breathing mode, the anti-phase breathing mode and the two degenerate radial breathing modes. The eigenvectors will 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. 84 Chapter 5. Condensate Theory

Figure 5.4: The spectrum of low-lying, collective√ modes in an axially symmetric TOP trap (ωz/ω⊥ = 8). Chapter 6

Bose-Einstein Condensation and Superﬂuidity

6.1 Introduction

The experiments in this thesis all involve applying a torque to a BEC. The response of the condensate to a torque is one of the signatures of its superfluid nature. In this section I will attempt to define superfluidity, describe its relationship to Bose condensation (a famously knotty problem, which will only be discussed at a basic level) and explain the signatures of superfluidity theoretically. Historically, the theory of superfluids has been developed to explain the re- 4 markable transport properties of liquid He below its critical temperature (Tλ) of 2.17 K, a phase that is usually referred to as Helium II. The phenomena that have been observed include frictionless flow below a critical velocity [?, ?], second sound [?], irrotational flow (and hence a reduced moment of inertia) [?] and the formation of vortices with quantized circulation [?]. Whilst there are many different approaches for defining superfluidity, the following practical definition will be used: A superfluid system displays the same remarkable transport phenomena as He II. A complete theoretical description of these phenomena does not exist, because the strong interparticle interactions in liquid helium cannot be modeled accurately. The theoretical description of the superfluid properties offered below is clearly based 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 theory seems flawed - X-ray and neutron scattering data [?] indicates that at temperatures well below Tλ only ∼ 10% of the atoms are in the ‘condensate’, although several famous experiments have showed that almost the entire system behaves as a superfluid.

85 86 Chapter 6. Bose-Einstein Condensation and Superﬂuidity

Andronikashvili measured the fraction of the total fluid able to flow without friction, as a function of temperature [?]. He suspended a stack of closely spaced metal disks on a torsion wire submerged in liquid He. The spacing was so small that above Tc essentially all the fluid between the disks was dragged round as the disks performed small angle rotational oscillations. Below Tc the oscillation frequency increased sharply, since the superfluid fraction was no longer dragged round by the disks. The change in frequency indicated that as T → 0, the superfluid fraction, ns/n → 1. Hess and Fairbank [?] measured the fraction of the fluid that was constrained to an irrotational flow pattern below Tλ, by measuring the angular momentum transferred from slowly rotating liquid helium to its container, as the temperature is reduced below Tλ. The angular momentum transferred is proportional to the final superfluid fraction, which was measured to be between 70 and 83%, depending on the final temperature. Finally, a value for the superfluid fraction 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 condensate 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’ superfluid component, which can be as large as 90% in He II, may be associated with the ‘quantum depletion’ in most simple situations. (The quantum depletion are those atoms that are not in the single particle ground state at T=0. Population of higher states occurs in the presence of interactions; the ground state of the interacting condensed system is no longer the single particle ground state but contains a superposition of higher states). The normal fraction nN /n, which falls to zero at T = 0 is made of thermally excited atoms. (Whilst the quantum depletion is definitely 100% superfluid in a simple trapping potential at T = 0, there are special situations in which its superfluidity is be reduced. One example is during the transition from a perfectly superfluid condensate to a Mott insulator in an optical lattice at T = 0 [?]). Whilst the quantum depletion is outside the ground state wavefunction, it has been shown experimentally (see above) to obey the superfluid transport properties predicted by this wavefunction. This can be understood (at least in the case of a weakly 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 conserve momentum and so the net momentum of the depletion is the same as that of the condensate. Therefore the depletion follows the flow pattern of the condensate and displays the transport phenomena which define a superfluid. This argument, which justifies the use of the condensate wavefunction to predict the transport properties of the whole superfluid, is reasonable√ in our dilute gas system where 3 the quantum depletion is very small - ndep/n ∼ na = 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 straightforward to define a condensate wavefunction and so the above discussion can only at best provide a qualitative picture of the nature of the superfluid fraction. Although the superfluid nature of dilute gas Bose condensates was always predicted, it was several years after the first observation of BEC that direct experimental evidence for superfluidity was announced. All the superfluid properties at the start of this section have now been observed - second sound [?], critical velocity [?], irrotational flow (and hence a reduced moment of inertia) [?, ?] (chapters. 7 and 8) and the formation of vortices with quantized circulation [?, ?] (chapter 9). This chapter begins with a brief discussion of the theory of dissipationless flow and a superfluid critical velocity since, as the name suggests, this phenomenon has been central to the development of superfluid theory. It will then focus on how a superfluid responds to an applied torque - theory which underlies the experiments described 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 a heavy body can move through the fluid without experiencing a drag force - above this speed superfluid flow breaks down and the system is heated. Thus although the presence of a condensate is a necessary condition for superfluidity, it is not sufficient. The system must also possess a non-zero critical velocity, otherwise any attempt to probe the superfluid system will also destroy it. The critical velocity is determined by the E/p dispersion curve of the low-lying excited states of the system. The lowest velocity at which a moving body can excite any of these states, whilst conserving energy and momentum, is given by the Landau criterion [?, ?]. " # E(p) vL = (6.1) p min The minima of E(p)/p may be found where

E(p) dE(p) = (6.2) p dp 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 Bose condensate was observed in [?, ?] when a blue detuned laser beam was moved through the condensate at diﬀerent velocities. The onset of dissipation was marked both by a distortion of the density distribution around the stirrer and by an increase in the thermal cloud (heating). The measured value of vL, ∼ 1/10 of the speed of sound, is believed to be determined by nucleation of vortices. This discussion of 88 Chapter 6. Bose-Einstein Condensation and Superﬂuidity

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

6.3 The superﬂuid response to a torque

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

h¯ v(r, t) = ∇S(r, t) (6.4) m

Consider the circulation of the velocity field, κ I κ = v.dl (6.5) h¯ I = ∇S(r, t).dl (6.6) m h¯ = × 2πq (6.7) m where q is any integer. For the wavefunction to be single valued, the phase change around any closed loop within the superfluid must be an integer multiple of 2π and so the circulation is quantized into units of h/m. Thus superfluid flow patterns fall into two regimes:

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

• κ = 2πq, with q > 0. If the circulation is non zero, then vortices exist within the superﬂuid. These are lines of zero density associated with quanta of circulation. 6.4. Irrotational ﬂow and the reduced moment of inertia 89

Figure 6.1: Diagrams of two different velocity fields (viewed instantaneously in the lab frame) for a fluid in a very slowly rotating container. (a) shows a rotational or rigid body flow 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 type 2 superconductor. The exclusion of magnetic flux from a superconductor in a weak applied magnetic field (< Hc1) is known as the Meissner effect [?], and is analogous to 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 = 0 inside the superfluid. In stronger applied fields, both systems minimize their energy by allowing quanta of magnetic flux/circulation to penetrate the sample in localized regions (∼ healing length) known as flux lines/vortex lines. Superconductivity breaks down when the applied field exceeds Hc2 and this corresponds to the rotation rate at which vortex cores overlap and superfluid flow ceases. The next two sections discuss 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 be energetically favorable when the container confining the superfluid is rotating very slowly (i.e. a weak applied rotational field). The velocity field within the condensate is irrotational and may be represented as the gradient of the scalar condensate phase S(r, t). Figures 6.1(a) and (b) show a fluid at equilibrium with a slowly rotating ellipsoidal potential. In both cases the ‘shape’ of the fluid follows the potential 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 90 Chapter 6. Bose-Einstein Condensation and Superfluidity

take a continuous range of values. In (b) the flow pattern is irrotational and the circulation is zero. The angular momentum associated with each flow pattern is clearly different, although the density distribution of both fluids is identical and is ‘rotating’ at the same frequency Ω. In (a) the whole velocity field contributes constructively 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 a cylindrical potential, the symmetry would produce perfect cancellation and hence zero angular momentum. Conversely in an extremely elliptical potential, the angular momentum of the irrotational flow pattern tends to that of the rotational flow pattern. Different moments of inertia may be used to account for the different angular momenta associated with each flow pattern. In its most general form, the moment of inertia of a fluid Θ represents the linear response to a rotational field -ΩJz and is given by: hJˆ i Θ = lim z (6.8) Ω→0 Ω The rotational flow pattern has the familiar rigid body moment of inertia for N particles of mass m: 2 2 Θrig = Nmhx + y i (6.9) The irrotational flow pattern has a reduced moment of inertia given by

2 ΘS = ²c Θrig (6.10)

2 where ²c ≤ 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 are present). 2 ²c 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 5.38 as discussed in [?]. Solutions of these equations that are stationary in the rotating frame, have a velocity field of the form:

hx2 − y2i v = Ω ∇(xy) (6.11) hx2 + y2i

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

mhr × vi ΘS = lim (6.12) Ω→0 Ω " # hx2 − y2i 2 = Θ (6.13) hx2 + y2i rig 2 = ²c Θrig (6.14) 6.4. Irrotational ﬂow and the reduced moment of inertia 91

Whilst ²c describes the deformation of the condensate, ²t describes the deformation of the trapping potential in the plane of rotation: 2 2 ωx − ωy ²t = 2 2 (6.15) ωx + ωy For a Thomas-Fermi condensate at equilibrium with a harmonic trapping potential, one can show that ²c = ²t. Note that ²c = 0 in an axially symmetric condensate. Thus in the absence of vortices, a condensate of circular cross-section cannot possess any angular momentum. This is particularly relevant to the experiments described in chapter 8. In [?] it is demonstrated that the same reduced moment of inertia will also be displayed in small angle oscillations of the condensate relative to an elliptical trap potential. These small angle oscillations are the scissors modes. The xz and yz scissors modes are described in section 5.6 for an axially symmetric trap. If the trap is also elliptical in the xy plane, then a third, xy scissors mode will exist. The xy scissors modes is excited by the relevant quadrupole operator XN Q = xiyi (6.16) i=1 In [?], the relationship between the moment of inertia Θ and the imaginary quadrupole 00 response function χQ is derived : R ³ ´ 00 3 ΘS 2 dωχQ(ω)/ω 2 2 R = ωx − ωy 00 (6.17) Θrig dωχQ(ω)ω

where X h i 00 π −βωn −βωm 2 χQ(ω) = e − e |hm|Q|ni| δ(ω − ωmn) (6.18) Z n,m and n, m are the different collective modes of the condensate and β = 1/kT . In the limit of a small selective excitation of the xy-scissors mode in a Thomas-Fermi 2 2 1/2 condensate, with frequency ωsc = (ωx + ωy) , then eqn. 6.17 reduces to ³ ´2 ω2 − ω2 ΘS x y 2 = ³ ´2 = ²t (6.19) Θrig 2 2 ωx + ωy Equations 6.15 and 6.19 shows that the same reduced moment of inertia governs both the complete rotation of the cloud and theq scissors mode oscillations. The condensate will oscillate at a higher frequency ( k/ΘS) than a fluid of the same mass distributionq undergoing small angle oscillations with a rotational flow pattern at frequency k/Θrig. This is the basis of the scissors mode experiment described in chapter 7. Since it is straight forward to measure the frequency of an angular oscillation, the scissors mode is used to measure the moment of inertia as a function of temperature in [?]. It may also provide a method for observing the superfluid BCS transition in a degenerate Fermi gas [?]. 92 Chapter 6. Bose-Einstein Condensation and Superfluidity

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, around which the phase changes by integer multiples of 2π and the circulation is quantized into units of h/m, as show in section 6.3. The phase is totally undeﬁned at the core of the vortex (consider taking a closed loop around which the phase changes by 2π and shrinking it in towards the vortex core at the centre) and hence the density 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 the minimum distance over which the order parameter may heal or alternatively, the minimum distance over which the condensate density may change signiﬁcantly. Consider the condensate density growing from 0 to n over a distance ξ. This corresponds to a quantum kinetic energy per particle of ∼ h¯2/2mξ2. Equating this to the interaction energy per particle gn gives an expression for ξ:

h¯2 gn = (6.20) 2mξ2

where g = 4πh¯2a/m. Rearranging eqn. 6.20 to find an expression for the healing length gives 1 ξ = √ (6.21) 8πna Note that the vortex core shrinks as the interaction energy increases. In Helium II, which is strongly interacting, ξ is of the order of an angstrom and vortices cannot be imaged directly. The arrangement of vortex lines in a lattice has been observed by trapping electrons on the cores and then drawing them off each core and accelerating them to a point on a phosphor screen [?]. In contrast, the vortex cores in our dilute system may be imaged optically. Under the conditions we use to nucleate vortices, ξ has a typical value of 0.3 µm in the trap. During a typical free expansion time of 12 ms, the vortex core increases by a factor of ∼ 10, to a radius of 3 µm, just above the resolution limit for our imaging system. This expansion behaviour has been calculated from the theory of [?] and is plotted in fig. 6.2. The expansion has two regimes; initially the core size adjusts rapidly to the decreasing density and the core expands faster than the condensate. At longer expansion times, the potential energy becomes negligible and the cloud expands as free particles, with ξ/R⊥ constant. Typical images of expanded vortex cores may be 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 core radius is assumed to be the same as the healing length. The plot was calculated for the experimental conditions of chapter 9 (ω⊥/2π = 62 Hz, N = 15, 000) from the theory of [?] 94 Chapter 6. Bose-Einstein Condensation and Superﬂuidity

6.5.2 Vortex energetics and metastability In section 6.3, we described two types of flow for a condensate in equilibrium with a rotating potential. At low rotation rates, the condensate remains in the ground state and follows the potential with an irrotational flow pattern. At higher trap rotation rates, formation of the first vortex state is energetically favorable. To determine the conditions under which formation of the first vortex state is favorable, 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 good quantum number and may be used to label the states. In the rotating frame, the time-independent Gross-Pitaevskii equation (eqn. 5.26) 0 gains an additional term −ΩLz, which lowers the energy (Ei) of those states with non-zero angular momentum: Ã ! h¯2 1 − ∇2 + m(ω2x2 + ω2y2 + ω2z2) + g |φ (r)|2 − ΩL φ (r) = E0 φ (r) (6.22) 2m 2 x y z i z i i i

0 The energy of the ground state in the rotating frame E0, is the same as that in the lab frame E0, since Lz = 0. The first vortex state, with a single centred vortex line, has Lz = Nh¯. It has additional kinetic energy due to the circulating velocity field (E1 > E0) and so at low trap rotation rates, the vortex-free state is energetically favorable. The ‘thermodynamic’ critical angular velocity Ωth occurs when the ground state and the first vortex state have the same energy in the rotating frame. E0 − E Ω = 1 0 (6.23) th Nh¯

An expression for Ωth may be evaluated in the T-F limit, assuming an approximate density proﬁle for the vortex state, [?]: Ã ! 2 5ω⊥ a⊥ 0.67R⊥ Ωth ≈ 2 ln (6.24) 2 R⊥ ξ

In a tri-axial trap, the calculation of Ωth is more complicated because the irrotational flow pattern contains some angular momentum and Lz is no longer a good quantum number. An approximate expression for the critical velocity in a tri-axial trap is also given in [?], but since we use very small eccentricities in the xy plane for our vortex experiments, the expression of eqn. 6.24 is adequate. Evaluating the thermodynamic critical rotation rate for the trap used in chapter 9 (ω⊥/2π = 62 Hz) gives Ωth = 0.14 ω⊥. In practice much higher rotation rates are required to nucleate a vortex because an energy barrier exists between the ground state and the first excited state with a single centred vortex, making both states metastable. The origin of the energy barrier is topological; a 2π phase winding, centred on a point of undefined phase, cannot suddenly appear in the centre of a simply-connected condensate wavefunction. It may only be created at 6.5. Vortex theory 95

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 as the vortex moves from the centre to the edge. First the kinetic energy associated with the vortex core decreases, as the density of the surrounding condensate falls to zero. Secondly the angular momentum associated with the vortex decreases (as described in section 6.5.3) causing the energy of the excited state in the rotating 0 frame E1 = E0 − ΩLz to rise. The sum of these two factors produces a maximum in the energy of the system in the rotating frame (i.e. an energy barrier) when the vortex is part-way between the centre and the edge of the condensate. 0 A plot of the excited state energy E1 versus vortex position (d) is given in [?] for an axially symmetric trap and is shown in fig. 6.3. At low rotation rates the first vortex state is energetically unstable (a). As Ω increases it first becomes metastable (b) and then globally stable (c) as the vortex state becomes the lowest energy state in the rotating frame. The energy barrier reduces as Ω increases beyond 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 which the excess energy may be dissipated. The presence of some thermal cloud will eventually cause the vortex to spiral out to the edge and disappear, but under typical experimental conditions and a temperature of 0.5 Tc the timescale for this process 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 a superfluid. However, so far I have been careful only to discuss quantization of circulation. In an infinite uniform superfluid, quantization of circulation immediately implies quantization of angular momentum. However in a non-infinite superfluid, the angular momentum associated with a vortex line is only quantized into units of Nh¯ when the vortex is centred; as it moves towards the boundary of the system hLzi falls smoothly to zero. Our trapped dilute gas condensate cannot be approximated to an infinite system since vortices are regularly found in all positions, from the centre to the edge. The relationship between hLzi and vortex position is determined by the density profile of the condensate: ZZZ 3 hLzi = ρ(r)[r × v].zˆ d r (6.25)

We will consider an axially symmetric Thomas-Fermi density profile, and so the vortex position may be specified by its scalar distance from the condensate axis, d. Ã ! z2 r2 ρ(r) = ρ0 1 − 2 − 2 for ρ > 0; ρ = 0 elsewhere (6.26) Rz R⊥ 96 Chapter 6. Bose-Einstein Condensation and Superfluidity

Figure 6.3: (Taken from [?]). The energy of the first vortex state (relative to the ground 0 0 state) in the rotating frame ∆E = E1 − E0, as a function of the vortex displacement d from the symmetry axis of the cylindrical trap. ∆E0 is normalized by its value for a centred vortex in a stationary trap. The curves correspond to different trap rotation frequencies (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 has the lowest energy, (d) Ω = 1.5Ωth. Curves (b) to (d) show that an energy barrier exists for 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 the condensate. Since hLzi is a weighted spatial average over the entire condensate we only introduce a negligible error if we use the vortex free density profile to calculate hLzi. Inserting ρ(r) into eqn. 6.25: Ã ! ZZ 2 2 Z 2π 2 z r hLzi = dz d r ρ0 1 − 2 − 2 rvφ dφ (6.27) Rz R⊥ 0 We recognize the last integral in eqn. 6.27 as the circulation around a loop of radius r, which we know to be quantized from section 6.3. If the loop does not enclose the vortex (r < d), the phase change and hence circulation around the loop will be zero. This explains why an off-centre vortex is associated with reduced angular momentum - the area inside the vortex position does not contribute to the total 6.5. Vortex theory 97

angular momentum integral. If the loop does enclose the vortex (r > d) then the value of the integral will be quantized into units of h/m, as given in eqn. 6.7. Thus after performing the integral over φ, eqn. 6.27 may be written as: Ã ! Z Z 2 2 hq R⊥ Rz(r) z r hLzi = ρ0 rdr 1 − 2 − 2 dz (6.28) m d −Rz(r) Rz R⊥ q 2 2 where q is an integer and the limits on the z integral Rz(r) = Rz(0) 1 − r /R⊥ are determined by the ellipsoidal surface on which the condensate density distribution falls to zero. The lower limit on the radial integral is d, because we know that the region within the vortex position does not contribute to the total angular momentum. Evaluating the integral ﬁrst over z and then r gives: Ã ! 2 2 5/2 hq 4Rz R⊥ d hLzi = ρ0 1 − 2 (6.29) m 3 5 R⊥ Ã ! d2 5/2 =hNq ¯ 1 − 2 (6.30) R⊥ where the number normalization condition, ZZZ 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, after the formation of the ﬁrst vortex. If angular momentum, as well as circulation, was quantized in a ﬁnite sized condensate, then we would observe the angular momentum increasing in discreet steps of Nh¯.

6.5.4 Kelvin waves Kelvin waves are helical excitations of the vortex core. Each kelvon or quanta of excitation has an energyhω ¯ , angular momentumh ¯ and linear momentum along the axis of the vortex ±hk¯ . These properties can be derived by considering the wave equation for a vortex line, which can be modeled as a string. We will consider the spectrum of Kelvin modes on a straight vortex with one unit of circulation, along the axis of a cylindrically symmetric, harmonically trapped, Thomas-Fermi condensate. 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 kinetic energy of its flow pattern. Thus it is effectively under tension, because any 98 Chapter 6. Bose-Einstein Condensation and Superfluidity

Figure 6.4: The forces on a vortex line that result in helical Kelvin wave motion. FR is a restoring force and FM is the Magnus force. The circulation vector κ represents the circulating velocity ﬁeld around the vortex core. 6.5. Vortex theory 99

increase in length causes an increase in energy. The tension at position z along the vortex line is given by [?]: Ã ! h¯2 0.888R T (z) = πn(0, z) ln ⊥ (6.32) m ξ

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

d2η(z) F (z) = −T (z) (6.33) R dz2 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 unperturbed number density n, then it experiences a Magnus force perpendicular to the direction 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 singly quantized vortex. We assume that the amplitude of the oscillation is small so that n ≈ n(0, z). The superposition of the linear velocity field and the circulating vortex velocity field produces a net flow rate that is greater on one side of the vortex than the other. From Bernoulli’s equation, for steady flow 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 a transverse force.

The vortex line has zero mass and so the equation of motion contains only these two forces. Resolving the equation of motion into components gives the following pair of coupled equations for Kelvin waves on an axial vortex line with one unit of circulation: d2η dη T (z) x = −hn(0, z) y (6.35) dz2 dt d2η dη T (z) y = +hn(0, z) y (6.36) dz2 dt These coupled equations of motion are satisﬁed by solutions of the form:

i(±kz−ωt) ηx(z, t) = Ae (6.37) i(±kz−ωt) ηy(z, t) = −iAe (6.38) 100 Chapter 6. Bose-Einstein Condensation and Superﬂuidity

representing helical waves traveling in either direction along the z-axis, but always rotating around the axis in the opposite direction to the vortex flow field. Thus if the vortex line has angular momentumh ¯, each kelvon has angular momentum −h¯. One can see from the wave equation that we only expect Kelvin wave solutions with one sense of rotation. It is first order in t and so permits only one angular frequency solution. Waves with opposite senses of rotation correspond to solutions with 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 Ã ! hk¯ 2 0.888R ω(k) = ln ⊥ (6.40) 2m ξ

Note that both FR and FM depend linearly on n(0, z) and so the z dependence cancels out in eqns. 6.35 and 6.36. Thus the dispersion relationship of eqn. 6.40 is exact (in the limit of small amplitude oscillations), even for a condensate in a 3-dimensional harmonic trapping potential. With the application of suitable approximate boundary conditions, the spectrum of Kelvin mode energies may now be estimated. The vortex lines are not constrained or pinned at the boundaries and so these points are antinodes in the vortex wave (analogous to the open end of an organ pipe rather than the closed one). This boundary condition requires that the integer multiples of the Kelvin wave length ﬁt into 4Rz or 2πp k = (6.41) 4Rz and so the mode spectrum as a function of integer p is: Ã ! h¯ µ πp ¶2 0.888R ω(p) = ln ⊥ (6.42) 2m 2Rz ξ

In our trap (ωz/2π = 175 Hz), the frequencies of the 3 lowest Kelvin modes are 0.44 ωz, 1.8 ωz, 4.0 ωz. Note that the frequency and spacing of these modes is comparable to the trap frequencies and hence the collective mode frequencies. Thus in our geometry it may be possible to observe the transfer of energy between an individual collective 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 of the lowest collective mode frequencies (excluding the anti-phase breathing mode, −5 with frequency of 0.7 ωz) and 2×10 kBTc/h¯. In this case, there will be significant thermal excitation of many Kelvin modes and it will be more difficult to observe resonant coupling to an individual Kelvin mode. Further investigation of Kelvin modes 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 discussed in a theoretical paper by Guéry-Odelinand Stringari [?] and which provided some of the first direct evidence that a dilute gas Bose-Einstein condensate behaves as a superfluid. After the first condensate was made in 1995, experimental work focused on verifying the predictions of mean field theory. The excellent agreement of the measured collective oscillation frequencies with the predictions of the Gross- Pitaevskii equation (eqn. 5.24) and the observation of matter-wave interference between two condensates [?, ?] supported the mean-field theory assumption that the condensate is described by a single macroscopic order-parameter or wavefunction. However, the existence of a macroscopic wavefunction is a necessary but not sufficient condition of superfluidity. Three further experiments provided direct proof that the condensate behaved as a superfluid: the measurement of a critical velocity for superfluid flow [?, ?]; the observation of quantized vortices [?, ?]; and finally the scissors mode experiment which demonstrated that the condensate has a purely irrotational flow pattern and measured its reduced superfluid moment of inertia [?]. The name ‘scissors mode’ originated in nuclear physics. It describes the small angle oscillation of superfluid neutron and proton clouds relative to each other within deformed nuclei [?, ?, ?]. In a trapped dilute-gas Bose condensate, the scissors 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 potential are elliptical. In an axially symmetric condensate the xz and yz scissors modes are 2 of the 6 quadrupole modes described in section 5.6. In this experiment we use the 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 not deformed and the density distribution performs a small angle oscillation. Whilst the scissors mode has a purely irrotational flow pattern, small angle oscillations 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 the density distribution of the cloud. A rotational flow pattern is associated with the rigid-body moment of inertia Θrig:

2 2 Θrig = Nmhx + z i (7.1) q and an oscillation frequency of k/Θrig. As explained in section 6.4 an irrotational flow pattern is associated with a reduced moment of inertia ΘS. In the case of a Thomas-Fermi condensate in equilibrium with a harmonic trapping potential ΘS is given by: 2 2 2 (ωz − ωx) 2 ΘS = Θrig = ² Θrig. (7.2) 2 2 2 t (ωz + ωx) q 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 displays both a rotational and an irrotational flow pattern. The lack of rotational flow, that results from the existence of a macroscopic wavefunction, is one of the signatures of a superfluid (as described in section 6.3). First we excite small angle oscillations in a non-condensed thermal cloud, by a sudden rotation of the trapping potential in the xz plane. Under these particular initial conditions both rotational and irrotational flow patterns are simultaneously excited with comparable amplitudes and thus we observe oscillations at two different frequencies. Then we excite a condensate under exactly the same conditions. If the condensate behaves as a superfluid, then the low frequency rotational oscillation should be suppressed and only a single oscillation frequency will be observed. Finally, we must calculate the rotational and irrotational oscillation frequencies of a non-condensed cloud, at the temperature and density of the condensate. We must check that both are excited with significant amplitude in this colder, denser system, so that if only a single frequency 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 the scissors 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 density and velocity distribution of the condensate during the scissors mode, as a function of coupled parameters θ(t) and β(t). When substituted into the hydrodynamic equations, these yield oscillatory solutions at frequency ωsc. Consider the density distribution (with respect to the trap axes) of a T-F condensate after a sudden rotation of the trap through an angle θ. µ m n(r, t) = c − [(ω2 cos2 θ + ω2 sin2 θ)x2 + ω2y2 + g 2g x z y 2 2 2 2 2 2 2 (ωx sin θ + ωz cos θ)z + 2(ωz − ωx) cos θ sin θ xz]. (7.3)

If we now assume that θ ¿ 1, the change in density is proportional to xz: µ m n(r, t) = c − [ω2x2 + ω2y2 + ω2z2 + 4¯ω2² θ(t)xz] (7.4) g 2g x y z t

where s ω2 + ω2 ω¯ = x z (7.5) 2 Hence the subsequent small angle motion may be identified with the pure xz scissors mode, which has quadrupole operator xz, (section 5.6). If a larger excitation angle 2 is used, then the coefficients of xi in eqn. 7.3 can no longer be considered constant and breathing modes are also excited. (Note that the maximum amplitude for pure scissors oscillation is not related to the deformation of the trap ²t). The velocity distribution may be constructed from three constraints on the condensate flow pattern:

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

h¯ v = ∇S(r, t) (7.6) m This constraint ensures that the condensate only displays one oscillation frequency rather than two.

• During the small angle scissors mode, the cloud shape does not deform, but only 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 motion in the y direction. 104 Chapter 7. The Scissors Mode Experiment

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 be solved, using the correct initial conditions, to give:

θ(t) = θ cos(ω t) 0 sc (7.9) β(t) = ²t θ0 ωsc sin(ωsc t)

and √ q 2 2 ωsc = 2ω ¯ = ωx + ωz . (7.10)

In our axially symmetric TOP trap, ωsc = 3ω⊥. An alternative method for calculating the scissors frequency is given in section 5.6, by considering the scissors mode as a superposition of the 2 degenerate l = 2, m = ±1 modes. Hence its frequency, may be determined by substituting the 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 [?, ?]. Prior to reaching an equilibrium state (described by the steady state Boltzmann distribution function) any thermodynamic system (e.g. our trapped dilute gas) will be described by the Boltzmann equation [?]. The method of averages [?] is then used to extract coupled equations for the averages of useful observables from the Boltzmann equations. In this experiment, we are interested in a set of 4 coupled equations involving those observables (or momenta) which couple motion in the x and z directions hxzi, hxvz + zvxi, hvxvzi, hxvz − zvxi. In particular, we know that the quadrupole moment hxzi is proportional to the tilting angle of the cloud in the xz plane θ(t), for θ(t) ¿ 1. If we parameterize the Gaussian density distribution of the cloud as: h ³ ´ i 2 ρ(r, t) = exp − Vext(r) − 2mθ(t)²tω xy /kBT (7.11)

then the following equation for small angle oscillations may be obtained from the coupled equations for xz motion: Ã ! d4θ d2θ 1 d3θ dθ + 4¯ω2 + 4²2ω¯4θ + + 2¯ω2 = 0 (7.12) dt4 dt2 t τ dt3 dt | {z } | {z } collisionless hydrodynamic 7.2. Theory 105

The relaxation time τ is given by 5 5 2 τ = = (7.13) 4γcoll 4 n(0)vthσ

1/2 2 where vth = (8kBT/πm) and σ = 8πa . Equation 7.12 is divided into two parts. The first dominates in the collisionless regime τωho À 1, where a particle oscillates many times in the trap between collisions. The second dominates in the collisional hydrodynamic regime (not to be confused with the apparent hydrodynamic behaviour which results from superfluidity), where many collisions occur per trap period. The relevant parameter 18 −3 is τωho = 75 for the thermal cloud used in this experiment (n0 = 2 × 10 m , T = 1 µK). Thus to describe the thermal cloud oscillations, the hydrodynamic part of eqn. 7.12 may be ignored and the collisionless part solved to find two undamped oscillations at frequencies ω± = |ωz ± ωx|. The lower frequency corresponds to a rotational (rigid body) flow pattern, whilst the higher frequency corresponds to an irrotational flow pattern. The sudden rotation of the trap from equilibrium 00 2 corresponds to the following initial conditions: θ(0) = θ0, θ (0) = −2¯ω θ(0) and θ0(0) = θ000(0) = 0, which excites both frequencies with approximately equal amplitudes. 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 has been calculated from eqn. 7.12 without approximation for our experimental conditions. (a) shows the undamped double frequency angle oscillation as a function of 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 find a possible source of ambiguity in our√ experiment. A non-condensed hydrodynamic fluid oscillates at a single frequency, 2 ω, the same frequency as the condensate. This single frequency results from the collisional properties of a hydrodynamic normal fluid, whilst the single condensate frequency results from superfluid constraints on the flow pattern. To overcome this ambiguity we must show that a normal fluid, at the same temperature and density as the condensate, is not in the hydrodynamic regime and thus will show two clear oscillation frequencies. The value of τωho for such 20 −3 a cloud (n0 = 2 × 10 m , T = 90 nK) would be 2.5, indicating that it is in an intermediate regime and thus will show two clear oscillation frequencies, but significant damping effects may also be observed. This conclusion is confirmed by the 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 from the collisionless values and the lower, rotational oscillation has a slightly reduced amplitude. Thus we can conclude that if the condensate behaved as a normal fluid, its oscillation would be characterized by: 106 Chapter 7. The Scissors Mode Experiment

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 in 12 −3 the experiment, with T = 1 µK and n0 = 2 × 10 cm . The response signal is an undamped two frequency oscillation.

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

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

• (d) The Fourier spectrum of the angular response shown in (b). The frequencies are shifted from the collisionless prediction (dotted lines) and the height of the low frequency component is signiﬁcantly reduced, showing that the 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 ˆx + sin ω0t ˆy) + Bz cos ω0t ˆz. (7.14)

The term Bz (t) = Bz cos ω0t ˆz 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 the −1 plane of the locus of B = 0 by an angle ξ = tan (Bz/BT ) with respect to the xy plane. This causes the symmetry axes of the time-averaged potential to rotate 2 through an angle φ ≈ 7 ξ in the xz plane (this analytic result is only valid for ξ2 ¿ 1). Tilting the locus of B = 0 also reduces the oscillation frequency very slightly in the z direction from its value when Bz = 0. Thus simply switching on Bz(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 trap and then quickly change Bz(t) to −Bz(t) to excite the scissors mode, as shown in fig. 7.2. This procedure rotates the symmetry axes of the trap potential by 2φ without affecting the trap oscillation frequencies. The following experimental procedure was used to excite the scissors mode both in the thermal cloud and in the BEC. Laser-cooled atoms were loaded into the 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 reversing the sign of Bz (t) in less than 100 µs excites the scissors mode, in a trapping potential with its symmetry axes now tilted by −φ, as described above. The initial orientation of the cloud with respect to the new axis is θ0 = 2φ, so this angle is the expected amplitude of the oscillations (fig. 7.2). After allowing the oscillation to evolve in the trap for a variable time t, the cloud is imaged along the y direction. The thermal cloud is imaged in the trap and the condensate after 15 ms of free expansion. Figure 7.3 shows typical absorption images of the thermal cloud (a) and the condensate (b). The angle of the cloud was extracted from a 2-dimensional Gaussian fit of such absorption profiles and plotted (over many experimental runs) as a function of evolution time. 108 Chapter 7. The Scissors Mode Experiment

Figure 7.2: The method of exciting the scissors mode by a sudden rotation of the trapping potential. The solid lines indicate the shape of the atomic cloud and its major axes. The dotted lines indicate the shape of the potential and its major axes. (a) The initial situation after adiabatically ramping on the field in z direction, with cloud and potential aligned, tilted by angle φ. (b) The configuration immediately after suddenly rotating the 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 smaller arrows show the expected quadrupolar flow pattern in the case of a BEC. The cloud is in 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) The thermal cloud in the trap. (b) The condensate after 15 ms of time of ﬂight. The condensate 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 evaporatively cooled to 1 µK which is about 5 times Tc, before the trap was suddenly tilted. At 5 12 this stage there were ∼ 10 atoms remaining, with a peak density of n0 ∼ 2 × 10 cm−3. The results of many runs are presented in ﬁg. 7.4(a) showing the way the thermal cloud angle changes with time. The model used to ﬁt this evolution is the sum 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. These values are in very good agreement with the values 339 ± 3 Hz and 159 ± 2 Hz predicted 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 a thermal cloud in the untilted TOP trap and calculated the modiﬁcation of these frequencies caused by the tilt. The amplitudes at the two frequencies were found to be the same, showing that the energy is shared equally between the two modes of oscillation, as predicted in ﬁg. 7.1(b). 110 Chapter 7. The Scissors Mode Experiment

Figure 7.4: (a) The oscillation of the thermal cloud as a function of time. The solid line is the ﬁtted double cosine function of eqn. 7.15. The temperature and density of our thermal cloud are such that there are few collisions, so no damping of the oscillations is visible. (b) The evolution of the condensate scissors mode oscillation over the same time scale as the data in (a). For the BEC, there is an undamped oscillation at a single frequency 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 out the full evaporative cooling ramp to well below the critical temperature, where no thermal cloud component is observable, leaving more than 104 atoms in a pure condensate. Since the condensate is imaged after free expansion, its aspect ratio is inverted relative to that of trapped thermal cloud (ﬁg. 7.3). The expansion also causes a small increase in the observed amplitude but does not aﬀect the frequency of the oscillation [?]. The scissors mode in the condensate is described by an angle oscillating 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 in the condensate. Consistent data, showing no damping, was recorded for times up to 100 ms. After fitting all the data with the function in eqn. 7.16 we find a frequency of ωsc/2π = 266 ± 2 Hzq which agrees very well with the predicted 2 2 frequency of 265 ± 2 Hz from ωsc = ωx + ωz . The aspect ratio of the time-of- flight distribution is constant throughout the data run confirming that there are no shape oscillations and that the initial velocity of a condensate (proportional to θ˙) does not have a significant effect [?].

7.6 Conclusion

These observations of the scissors mode clearly demonstrate the superfluidity of trapped Bose-Einstein condensed rubidium atoms, in the way predicted by Guéry- Odelin and Stringari [?]. Direct comparison of the thermal cloud and BEC data under the same trapping conditions shows a clear difference in behaviour between the irrotational quantum fluid and a classical gas. The condensate oscillation may also be compared with the simulated data for a normal cloud of the same temperature and density as the condensate. The double oscillation frequency and heavy damping predicted for a normal cloud are clearly different from our experimental condensate data.

7.7 The scissors mode at ﬁnite temperature

There is now a large volume of experimental data confirming that the GP equation provides an excellent description of the condensate as T → 0 (see e.g. [?, ?]). However there is very limited ‘finite temperature’ data, in which the thermal cloud has a significant effect on the collective behaviour of the condensate. The range of interesting temperatures is approximately from 0.6 Tc to Tc, between which the fraction of atoms in the thermal cloud varies from 0.2 to 1 (eqn. 5.19). Prior to 112 Chapter 7. The Scissors Mode Experiment

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

7.7.1 Experimental procedure and results We were able to produce partially condensed clouds, with temperatures ranging from 0.3 Tc to Tc, by varying the depth of the final radio-frequency cut. For each temperature, 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 thermal cloud as a function of time. (Under these conditions the condensate density was 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 the expanded BEC was then extracted by fitting a 2D double distribution (a Gaussian for the thermal cloud and a parabola for the condensate). The condensate oscillation was fitted with an exponentially damped cosine function, whilst a damped two 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 condensate and 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 the total number of atoms (eqn. 5.18), and hence on the rf cut depth [?]. For a given cut, the temperature was extracted from the wings of the thermal cloud distribution 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 temperatures measured, the thermal cloud oscillation appears to be undamped and to occur at the frequencies predicted for a collisionless classical gas. The behaviour of the scissors mode of the condensate as a function of temperature was much more interesting. Below 0.8 Tc the damping rate is well described by Landau damping [?], which is proportional to T/µc [?, ?]. This is a scattering process between quasi-particles in which a low energy collective excitation and a thermal excitation are annihilated and a higher energy thermal excitation is created. Above 0.8 Tc the damping rate appears to decrease, although fitting the angle of such small condensates became increasingly difficult and so the error on the measurements close to Tc is large. The frequency of the condensate oscillation is 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 mode of the condensate (with frequency ω2), calculated in [?] and rescaled by the factor ωsc/ω2. The dotted line is the prediction in [?] for the l = 2 mode in a spherical trap, again rescaled by the scissors frequency. The solid line is from the simulations in [?] for our exact experimental conditions.

• In (b) the condensate scissors frequency shows signiﬁcant negative frequency shifts from the hydrodynamic prediction (dashed line) at higher temperatures. The low temperature frequencies exceed the hydrodynamic value by ∼ 1% because of the ﬁnite number of atoms. The solid line shows the results of simulations in [?]. The thermal cloud frequencies do not appear to be temperature dependent and are in good agreement with collisionless predictions (dotted lines). 114 Chapter 7. The Scissors Mode Experiment

creases as the temperature increases, initially very gradually and then more sharply above ∼ 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 system to be described in terms of a damped simple harmonic oscillator. Above 0.7 Tc, the distinction between the condensate and the thermal cloud gradually become more blurred - the condensate oscillation frequency tends to that of the thermal cloud, whilst the reduced damping rate suggests that relative motion of the two components is decreasing. The analysis of this data is still a matter for debate. Two different models may be used to describe the partially condensed system. In [?], Rusch et al. use a finite temperature field theory, which consistently involves the dynamics of the thermal cloud to second order, to investigate the excitation spectrum of a condensate in a spherical trap. This theory recognizes the discreet nature of the low-lying, thermally populated, excited states. Each condensate excitation is strongly coupled via Landau (and Beliaev) processes to only a few thermally populated states. The individual couplings determine both the damping of the collective condensate excitation and the sign and magnitude of the frequency shift as a function of temperature. Since both the condensate and the thermal cloud are described by the same finite temperature field theory, their identities naturally become blurred as T → Tc (as observed experimentally) and several of the low- lying states become macroscopically occupied. However, whilst this theory has a sound theoretical basis, should be valid up to Tc and may well explain the upwards shift in the frequency of the m = 0 mode observed in [?], it will be computationally demanding to calculate the spectrum of an anisotropic trap as a function of temperature. An alternative model is presented by Jackson and Zaremba in [?]. Instead of describing the whole system by a single finite temperature field theory, the condensate is described by a generalized GP equation, whilst the thermal cloud is represented by a semi-classical Boltzmann kinetic equation. These two components are coupled both by mean fields (giving rise to Landau damping) and by collisional processes, which transfer atoms between the two. Simulations based on this model describe both the damping and frequency shift of the condensate reasonably well up to T = 0.7 Tc but fail at higher temperatures. Above 0.7 Tc, it is suggested that the increasingly massive thermal cloud begins to drive the condensate at its own low rotational oscillation frequency, accounting for the decrease in the condensate oscillation frequency and the observed reduction in damping. In summary, the finite temperature field theory of [?], applied to an axially symmetric trap, might well provide an accurate picture of the scissors mode of a partially condensed system close to Tc. However the two component theory of [?] appears to provide a very useful and tractable approximate theory for the spectrum of the low-lying condensate modes, for temperatures below 0.7 Tc, where the 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 condensate, since its quenching can be regarded as a measure of the superfluidity of the system. The degree of quenching is characterized by the normalized moment of inertia, RΘ = Θ/Θrig. RΘ tends to a steady value of less than 1 as T → 0 and perfect superfluidity is approached, and tends to 1 as superfluidity disappears. It is possible to infer the normalized moment of inertia for a Bose-condensed gas as a function of temperature, from our finite temperature scissors results and hence investigate the superfluidity of the system in this regime. The method used was originally developed by Zambelli and Stringari in [?] and is summarized in section 6.4. They derive an explicit relationship between the moment of inertia of a system and its quadrupole response to a small rotational perturbation i.e. the scissors modes (eqn. 6.17). This relationship may be written in the following useful form [?]: R Θ ³ ´2 Q(ω, T )/ω2 2 2 R RΘ = = ωx − ωy 2 (7.17) Θrig Q(ω, T )ω where Q(ω, T ) is the Fourier transform of the quadrupole moment. It may be split into 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 for small damping) at the measured oscillation frequencies of the system. Information required to determine the height of each spike includes the total atom number, condensate number, temperature and chemical potential and may be extracted from the absorption images of the thermal cloud and condensate.

The natural frequencies of the thermal cloud, ω± are independent of temperature. 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 their spatial distribution. In contrast, the natural frequency of the condensate, ωsc falls as T → Tc and its interaction with low-lying thermally populated states becomes large. The change in condensate oscillation frequency as a function of temperature indicates that the degree to which its moment of inertia is quenched, and hence the degree to which the condensate is superﬂuid, depends on temperature. Thus the condensate contribution to RΘ depends on temperature via both the size of the condensate, and the degree to which it is superﬂuid.

The results for the normalized moment of inertia RΘ of the condensate (solid circles), the thermal cloud (open circles) and the combined system (filled squares) (for calculation see [?]), obtained from our finite temperature scissors data, are given in fig. 7.6. Two important features should be noted from this data. Firstly quenching of the moment of inertia of the condensate fraction (solid circles) occurs 116 Chapter 7. The Scissors Mode Experiment

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 function of temperature. The error on this last data set of the order of 7 %. For low temperatures the normalized moment of inertia of the condensate falls just below the hydrodynamic prediction (dashed line) and is in excellent agreement with the ﬁnite number correction (dotted line). The solid line provides a good approximation to the normalized moment of inertia of the entire gas, and was calculated using the population and widths of the thermal and condensate fractions only. It ignores the dynamical interaction between the 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 superfluid below this temperature. Secondly consider the reduced moment of inertia of the whole system (filled squares). Its value falls gradually below Tc, primarily determined by the population and width of the thermal and condensate fractions, but also influenced close to Tc by the reduced superfluidity of the condensate fraction itself. This quantity indicates the size of the superfluid fraction and could be used to indicate the BCS transition in a dilute degenerate Fermi gas. The solid line, 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 the experimental data, deviating only slightly above 0.6 Tc. Chapter 8

Superﬂuidity and the Expansion of a Rotating BEC

8.1 Introduction

In the previous chapter, we described our investigation of the superfluidity of the condensate by measuring the frequency of its small angle oscillations. The following two chapters will report further direct evidence for the superfluidity of a condensate, by investigating its behaviour in a rotating potential. At high rotation rates, we observe the formation of quantized vortices (chapter 9). In this chapter we find evidence for a pure irrotational flow pattern, one of the signatures of a superfluid, in the expansion behaviour of a slowly rotating, vortex-free condensate. The theoretical work for this experiment was done by Edwards et al. in [?] and the experimental results are published in [?]. The aim of the experiment was to predict the expansion behaviour of a condensate in the Thomas-Fermi limit, assuming that it behaved as a superfluid. If the experimental results agree with these predictions then the original assumption of superfluidity is validated. The important features of the expansion may be understood from a few physical arguments. First consider the expansion of a stationary condensate released from an elliptical trapping potential. The mean field energy causes the condensate to expand most rapidly along those directions in which it was initially most tightly confined [?, ?]. Thus immediately after release from a trap with Rx > Ry, the aspect ratio (Rx/Ry) of the condensate decreases. It becomes instantaneously circular and then at long expansion times it continues to expand with an inverted aspect ratio. Now consider the expansion of an elliptical condensate that is initially in equilibrium in a slowly rotating potential. Assuming that the condensate behaves as a 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 of rotation (see section 6.4). " # hx2 − y2i 2 Θ = Θ (8.1) S hx2 + y2i rig For a Thomas-Fermi condensate eqn. 8.1 becomes:

" 2 2 #2 Rx − Ry ΘS = 2 2 Θrig (8.2) Rx + Ry 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 acts on the system, angular momentum must be conserved and so the rotation rate of the condensate increases ∝ 1/ΘS. Clearly the moment of inertia cannot fall as far as zero because this would create the unphysical situation of infinite rotational kinetic energy: L2 K.E. = (8.3) 2ΘS Thus the aspect ratio of the condensate never becomes unity; during the expansion it reaches a minimum value which is greater than 1, and then the aspect ratio increases again. In the final stages the expansion is most rapid along those directions that were initially least tightly confined in the trap and the rotation rate decreases as the moment of inertia increases. The condensate tends to an asymptotic final angle of between 45◦ and 90◦. This chapter will first outline the theory used to provide a quantitative prediction of the condensate behaviour. Then the experimental procedure is described, including the modifications of the TOP trap required to produce a rotating elliptical potential. Finally the experimental results are presented and compared with the irrotational predictions.

8.2 Theory

In our experiment we have an anisotropic harmonic potential with three angular frequencies ωx < ωy < ωz. The potential rotates about the z-axis with angular frequency Ω. As in chapter 7 it is convenient to deﬁne two parameters which describe the trap in the plane of rotation, which is now the xy plane rather than the xz plane (as in the scissors mode experiment). The trap deformation in the xy plane is ²t: 2 2 ωy − ωx ²t = 2 2 (8.4) ωy + ωx 8.2. Theory 119

The geometric mean of the frequencies in the plane of rotation is ω⊥: s ω2 + ω2 ω = x y (8.5) ⊥ 2 In the hydrodynamic limit, a condensate at equilibrium in this trap displays a purely irrotational or quadrupolar flow pattern, described in fig. 6.1b. The wavefunction corresponding to the quadrupole flow pattern has the form: q i mνxy Ψ(r) = n(r)e h¯ , (8.6)

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

3 ˜ 2 ˜ ν˜ + (1 − 2Ω )˜ν + ²tΩ = 0, (8.7) ˜ where we introduced the dimensionless quantitiesν ˜ = ν/ω⊥ and Ω = Ω/ω⊥. There is one positive root which corresponds to the ‘normal’ quadrupole branch and 2 negative roots which correspond to the ‘overcritical’ branch. The solutions are only physical if |ν| < Ω and so each mode has a different range of trap parameters (²t and Ω) in which it exists. These stability regions are plotted in fig. 8.1. For the work in this chapter, we only need to consider the normal branch. We use ²t = 0.32 and 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 because of the quadrupolar motion of the condensate, so that the condensate density can be written as: µ · m ¸ n(r) = c 1 − (˜ω2x2 +ω ˜2y2 + ω2z2) . (8.8) g 2 x y z

The modiﬁed frequenciesω ˜x andω ˜y are given by:

2 2 2 ω˜x = ωx + ν − 2νΩ 2 2 2 ω˜y = ωy + ν + 2νΩ, (8.9) where ν is a root of eqn. 8.7. Hence, the aspect ratio of the condensate in the trap is s R ω˜ Ω + ν x = y = , (8.10) Ry ω˜x Ω − ν The aspect ratio can be used to identify which quadrupole branch is excited in the condensate, under conditions where more than one mode is stable (ﬁg. 8.1). In particular in the normal mode, which has a positive value of ν, the major axis of the density distribution is aligned with the direction in which the trap is weakest. 120 Chapter 8. Superﬂuidity and the Expansion of a Rotating BEC

Figure 8.1: A plot showing the trap conditions (trap deformation ²t and normalized rotation rate Ω) under which each of the three quadrupole modes of a condensate in a rotating potential exist. The boundaries of the normal mode (dashed line) and 2 branches 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 in which the trap is stiffest. The effective chemical potential and condensate sizes in the trap can also be calculated from eqns. 8.9. To calculate the expansion of a condensate when released from the anisotropic harmonic potential, we use the following ansatz for the condensate number density n(r, t) and the velocity field v(r, t) in the Thomas-Fermi regime (as in [?]):

2 2 2 n(r, t) = n0(t) − nx(t)x − ny(t)y − nz(t)z − nxy(t)xy (8.11) 1 ³ ´ v(r, t) = ∇ v (t)x2 + v (t)y2 + v (t)z2 + v (t)xy . (8.12) 2 x y z xy Inserting this ansatz into the hydrodynamic equations of superfluids (eqns.5.36 and 5.38) yields a set of nine coupled differential equations for the expansion parameters, 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 is assumed to be aligned with the lab frame coordinate axes at the time of release. It is 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) are all zero. vxy is the quadrupole velocity field in the trap and may be found to be equal 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 initial conditions and the nine coupled differential equations for the expansion parameters [?]. Having solved the equations for a given value of t, the angle and aspect ratio of the condensate in the plane of rotation are found by diagonalizing the quadratic density distribution of eqn. 8.11. The code to calculate the expansion behaviour corresponding to our exact experimental conditions was developed by Gerald 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 a rotating trap, that was elliptical in the plane of rotation. This section outlines the modiﬁcations made to our standard axially symmetric TOP trap, ﬁrst to give it a variable 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 a radial bias field rotating at ω0/2π = 7 kHz. The potential experienced by the atoms results from the time-average of these two fields. The radial bias field has 122 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

Figure 8.2: The normalized trapping frequencies in an elliptical TOP trap ωi/ω0i, as a function of the radial bias ﬁeld ratio, E = Bx/By (solid lines). Each frequency is normalized against its value in a standard TOP trap with E = 1. By and the quadrupole 0 gradient Bq are assumed to be constant. Note that the normalized value of ωz is approximately 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 a π/2 phase diﬀerence. Changing the amplitude of one of these components so that Bx/By = E, breaks the radial symmetry and produces an elliptical potential [?] that is the time-average of:

0 U(x, y, z, t) = µBQ|(x + Er0 cos ω0t) ˆx + (y + r0 sin ω0t) ˆy − 2z ˆz| (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 ˆx + sin ω0t ˆy) (8.14)

0 where r0 = BT /Bq. At distances from the trap centre that are small compared to r0, the trap remains harmonic. The trap frequencies must be calculated numerically from eqn. 8.13 and are plotted in fig. 8.2 as a function of the bias field ratio E. We may assume that E ≥ 1 without loss of generality and thus ωx ≤ ωy < ωz. Note that increasing the bias field in the x direction reduces the trapping frequencies in all three directions. The deformation of the trapping potential is described either by e = ωy/ωx or by ²t (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 fields and potential are related by: de 1 ≈ . (8.15) dE 4 The amplitudes of the individual TOP field components Bx and By are controlled from the computer and so the eccentricity of the trap can be changed during an experimental run. This enables us to make a condensate in an axially symmetric trap, in which the evaporative cooling efficiency is optimized, and then deform it. One of the advantages of our purely magnetic rotating potential is that contours of constant energy are well defined ellipses and we have precise control over the eccentricity. After calibrating ωx and ωy using dipole oscillation frequencies, the ratio e is known to an accuracy of at least 0.5 %. We also have access to a wide range of eccentricities corresponding to 1 < ωy/ωx < 1.7 or a trap deformation parameter of 0 < |²t| < 0.5. Other experiments create an elliptical rotating potential using 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 (²t < 0.032 in [?]) and the shape of the potential is less accurately defined (because it depends on the position of the beams 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 rotates at 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: Ã ! Ã !Ã ! x cos Ωt sin Ωt E r cos ω t = 0 0 (8.16) y − sin Ωt cos Ωt r0 sin ω0t 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 a slightly shifted TOP frequency, ω0 − Ω. Since trap rotation rates are in the range 0−100 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 schematically in fig. 8.3. Initially, the slow signals sin Ωt and cos Ωt, were created by a quadrature oscillator. This oscillator is specifically designed to maintain a phase difference of exactly 90◦ between its two outputs, as required by eqn. 8.16. The rotation frequency Ω could be varied manually between 0.8 − 100 Hz. Thus although 124 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

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 fixed for each experimental run. This arrangement was used for the work in chapters 8 and 9. Starting from the 4 basic oscillating signals EBT cos ω0t, BT sin ω0t, cos Ωt and sin Ωt, voltage multipliers produce 4 signals consisting of a rapid oscillation multiplied by a slow one. These pairs are then added and subtracted to produce the signals of eqn. 8.18 that are fed via the audio-amplifier to the TOP coils. The electronics included several low-pass filters to prevent high-frequency noise reaching the amplifier since this has been observed to reduce the lifetime of the magnetic trap. In addition, variable resistors were built into the multiplication and addition stages, so that the signals could be balanced to produce a stable, noise-free elliptical bias field. For the gyroscope experiment described in chapter 10, it became necessary to vary the rotation rate of the trap during an experimental run. Thus a third NI board (MIO16) was fitted inside the control computer to provide two extra analogue 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 frequency noise on the bias field proved unfounded. The output provides 2000 points per cycle which creates a sufficiently smooth signal. The new arrangement also enables us to stop the rotation at any specific angle, jump between different angles and even change the direction of rotation during an experimental run. All this rotational flexibility makes it an ideal apparatus for investigating the superfluid nature of the condensate, 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 temperature of 0.5 Tc. Condensates were produced by evaporative cooling in a standard axially symmetric TOP trap with final frequencies ωx/2π = ωy/2π = 124 Hz and ωz/2π = 350 Hz. Once the condensate had formed we made the trap elliptical by changing 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 of ²t = 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 the trap became deformed, it also began to rotate at the chosen rate Ω. The condensate was left in the rotating trap for another 500 ms before it was released at a fixed reference angle in the trap rotation, from which the angles of the expanding cloud were measured. Column (a) of fig. 8.4 shows typical absorption images of the 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 nucleate vortices, section 9.5) and at the instant of release the long axis of the cloud was along the x-direction. The angle and aspect ratio of the cloud were obtained from a 2D parabolic fit to the density distribution. The cloud reached a minimum 126 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

aspect ratio of 1.31 after about 4 ms time-of-flight and the angle approached its asymptotic value of ∼ 55 degrees after 16 ms. For comparison, the evolution of a condensate released from a non-rotating trap is shown for the same expansion times in column (c) of fig. 8.4. In this case the aspect ratio decreased steadily, becoming circular after about 4 ms and then inverting. Note that the trapping frequencies ωx and ωy were the same in both columns. The enhanced aspect ratio in the left column, both in the trap and at long expansion times, is a result of the irrotational flow pattern in the rotating condensate, eqn. 8.9. It was necessary to go to a high trap eccentricity to observe the deformation of the cloud clearly.

8.5 Results

We investigated the evolution of the condensate after release from a static trap and from traps rotating at both Ω/2π = 20 and 28 Hz. In all three cases the trap 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 after release from these three traps, with the experimental data points superimposed. In the case of the rotating condensates (dashed and solid lines) the angle reaches 45 degrees after about 6 ms and after 18 ms is close to its asymptotic value between 55 and 60 degrees. The angle of the condensate released from a static trap (dotted line) 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 how the 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 continue to expand along its minor axis but the aspect ratio increases again because the condensate cannot become circular under these conditions. However, the condensate released from a static trap has no velocity field which prevents it from becoming circular and hence the aspect ratio passes through 1 at about 6 ms. In fig. 8.5(b) the aspect ratios are plotted in a frame that rotates with the major of the condensate. In the case of a static condensate the situation is ambiguous; we may either plot the aspect ratio falling below one at long expansion times, with no change of angle or we can plot the aspect ratio increasing at long times, with an instantaneous rotation of 90◦ as it passes through 1. We have chosen the latter so that the plot changes smoothly as Ω → 0. Every experimental point displayed is the average of several measurements. To image the plane of rotation we are using the ‘vertical’ (z axis) imaging system described in section 2.5.7. For each expansion time we had to refocus our imaging system 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 minimum value 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 release from 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 and the fitted rotation angle. At the instant of release the major axis of the cloud lay along the x direction. The red arrows indicate the direction and speed of rotation. The black arrows indicate the direction in which the condensate is expanding most rapidly. The rotating 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). 128 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

Figure 8.5: Dashed line and ﬁlled squares, Ω/2π = 28 Hz Solid 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) of the condensate, as a function of expansion time. Note that the theory involves no free parameters. The aspect ratio plot, (b), shows that initially rotating condensates never become circular about the rotation axis (aspect ratio = 1) to ensure 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 have plotted the aspect ratio of the static condensate increasing again at long expansion times, with a corresponding rotation of 90◦. This ensures that we label the 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 = 1.4 ωx, ωz = 1.2 × 8 ωx. consistent with unity. There is good agreement between the experimental data and the theoretical predictions at rotation Ω/2π = 20 Hz. However, we observed a small deviation of the experimental data from the predicted values for the higher rotation frequency of Ω/2π = 28 Hz (dashed curve). This can be accounted for by imperfect focusing. As the rotation rate increases and the condensate deformation becomes more pronounced, focusing becomes more critical The asymptotic rotation angle of the condensate is determined by a balance between two factors, the irrotational velocity field and the mean field expansion. For very small values of Ω, the irrotational velocity field is negligible. The expansion is dominated by the mean field velocity and the angle through which the major axis rotates tends to 90◦. This angle reduces as Ω increases and the irrotational velocity field becomes more significant (see fig. 6.1(b), showing an irrotational velocity field). The minimum rotation angle is 45◦, which would be the result for an almost circular condensate with zero mean field energy. Finally at high rotation rates, just below the lower of the two radial trap frequencies ωx, the mean field energy can be ignored. The cloud becomes elongated along the x direction and the velocity field becomes indistinguishable from a rotational velocity field. In this limit 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 function of 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 thermal cloud for comparison. Both the aspect ratio (solid line) and angle (dotted line) of the cloud may be calculated analytically, assuming a Gaussian distribution and a rotational velocity field at t = 0. (Simulations of a thermal cloud in a rotating 130 Chapter 8. Superfluidity and the Expansion of a Rotating BEC

Figure 8.7: The aspect ratio (solid line) and angle (dashed line) of a thermal cloud as a function of time after release from the trap rotating at Ω/2π = 28 Hz. The cloud is assumed to have a Gaussian distribution and rotational velocity field. The same trapping frequencies are used as in the experiment with a condensate, ωx/2π = 60 Hz, ωy = 1.4 ωx. potential showed that a rotational flow pattern was the equilibrium flow pattern [?]). v u ³ ´ u 2 2 2 2 u(1 + ωx t ) ωy − Ω Thermal aspect ratio = t³ ´ (8.19) 2 2 2 2 1 + ωy t (ωx − Ω ) 1 µ 2Ωt ¶ Thermal Angle = sin−1 (8.20) 2 1 + Ω2t2 The behaviour is strikingly different to that of the condensate due to an isotropic expansion energy and the lack of superfluid constraints on the velocity field. The aspect ratio decreases smoothly to an asymptotic value ≥ 1 and the angle of the cloud always tends to 90◦ at long expansion times, independent of the rotation rate.

8.6 Conclusion

Our results show that an expanding vortex-free Bose-condensate with some angular momentum refuses to become circular about the axis of rotation, as predicted by Edwards et al. [?]. The theoretical curves in both ﬁgs. 8.5(a) and (b) have no free parameters 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 superfluid condensate was correct. This provides direct evidence that Bose-condensed gases have purely irrotational flow and a reduced moment of inertia, as a consequence of their superfluidity. Chapter 9

Vortex Nucleation

The existence of quantized vortices is one of the most striking and fascinating signatures of superfluidity. Vortices were first investigated experimentally in a series of remarkable ‘rotating bucket’ experiments on superfluid liquid helium, that are reviewed at the start of this chapter. The sections that follow outline the methods that have been proposed and used to nucleate and detect vortices in a dilute gas Bose condensate, highlighting both the similarities and differences with the liquid helium work. This thorough introduction motivates the detailed experimental 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 using superfluid liquid helium in a ‘rotating bucket’. The ‘bucket’ was a cylindrical container with rough inner walls mounted on a turntable, the rotation rate of which could be very precisely measured. First it was shown that at very low rotation rates the superfluid fraction does not come into equilibrium with the rotating bucket but remains stationary in the laboratory frame. (Or, more precisely, stationary in the frame of the fixed stars, thus if the lab were on the North Pole the superfluid would rotate once per day with respect to the container! [?].) In contrast the normal 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 cooled through the critical temperature from above in a slowly rotating bucket, the angular momentum of those atoms joining the superfluid fraction must be given up to the bucket; the increase in the rotation rate of the bucket below Tλ was measured and 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 rates was demonstrated by Packard and Sanders [?]. They were able to count vortices by trapping electrons on the cores and then drawing them off onto a detector; the total charge collected was directly proportional to the number of vortices. The frequencies at which each new vortex appears are significantly higher than those predicted from thermodynamics because of the metastability of each state (see section 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 bucket of He II. The circulating velocity field of an axial vortex line changes the spectrum of helical waves on the wire via the Magnus force (section 6.5.4). This change was measured and the circulation of the velocity field inferred. Finally at high rotation rates the meniscus of superfluid helium was observed to have the same curvature as that of a normal fluid [?] although the fountain effect confirmed that the rotating system was still superfluid [?]. A regular ‘Abrikosov’ vortex lattice forms within the superfluid [?]; the net velocity field mimics the rotational field of a normal fluid thus minimizing the energy of the system. This lattice was crudely imaged [?] by trapping electrons on vortex cores and accelerating them off onto a phosphor screen. (The analogous arrangement of flux lines in a type-II superconductor has been more thoroughly investigated using neutron scattering and shown to have the predicted regular hexagonal-lattice structure). Together these experiments demonstrate many of the properties of a superfluid vortex that are predicted in section 6.5; namely a critical rotation rate for nucleation, a zero in the density, a rotational flow pattern around each vortex and quantization of the associated circulation. The only feature that was not directly observed was the 2π phase change around a vortex because the strong atomic interactions in liquid helium mean that a condensate wavefunction and phase cannot be clearly identified.

9.2 Vortices in a dilute gas Bose-Einstein condensate

9.2.1 Nucleation of vortices Following the realization of BEC in a dilute gas, the observation of quantized vortices in this new superfluid system became a matter of great theoretical and experimental interest. A wide variety of different nucleation and detection schemes were proposed and so it was not surprising that the first two groups to observe vortices (at JILA [?] and Paris [?]) used entirely different experimental approaches. Many of the proposed schemes were directly analogous to the liquid helium experiments 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. The rotating eccentricity could either be produced with time-varying magnetic fields as described in section 8.3.2 or by using the dipole force of a far-blue-detuned laser beam ‘stirrer’ [?, ?]. The beam was directed parallel to the axis of a static Ioffe-Pritchard trap, breaking its cylindrical symmetry and then rotated around the axis using a pair of crossed acousto-optic modulators. Whilst experimentalists considered the best method for realizing the rotating bucket, there was lively theoretical debate about the mechanism that would transfer the condensate from the ground state to the first vortex state in such an apparatus (for a review see [?]). As in He II, an energy barrier exists between the two states and hence the ground state is expected to remain metastable even at 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 work described in this chapter focuses on the eccentricity and rotation rate required to nucleate the first vortex and provides clear evidence that surface modes mediate the nucleation process. The energy barrier can be avoided by condensing the system directly into the first vortex state from a spinning thermal cloud. Under such conditions the condensate will form in the state of lowest energy in the rotating frame and thus the critical rotation rate for nucleation can be determined from thermodynamics. However this experiment has many technical difficulties in a dilute gas; the evaporative cooling necessary to achieve condensation cuts only in the xy plane and hence removes those atoms at the outer radius of the cloud that carry the most angular momentum. At Tc the mean angular momentum has been reduced below theh ¯ per particle that is necessary to form the first vortex state. These difficulties were overcome in [?] by using the effect of gravity on a weak TOP trap to move the 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, evaporative cooling increases the mean angular momentum of the cloud. The critical rotation rate was slightly above the thermodynamic value but also significantly lower than the value required to nucleate a vortex in a ground state condensate. The well-defined condensate wavefunction and phase provide an additional approach to vortex nucleation in a dilute gas BEC, that has no analogy in liquid helium: engineering of the condensate phase to produce a 2π phase winding. Phase engineering was originally used to create dark solitons [?, ?] by imprinting a sharp π phase change along a line in the condensate with a far-detuned laser beam. (Dark solitons are lines of minimum density, that propagate without change of shape and are, like vortices, associated with a discontinuity in the condensate phase.) The first vortices were produced in a dilute-gas condensate of 87Rb atoms using a related, but not identical method [?]. Population was transferred from a ground state 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 transitions between the two levels, whilst the AC Stark shift of a rotating far-detuned laser beam provided the spatial and temporal modulation necessary to create a 2π phase winding. A novel approach to nucleation of multiply-charged vortices using phase engineering was recently announced in [?]. This method uses the rotation of the atomic spin vector (Berry’s phase) [?] to create the vortex phase winding and will 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 simplest was the observation of a hole in the condensate density distribution looking along the axis of rotation with a standard absorption imaging system. Whilst the vortex core in a trapped condensate is generally too small to be resolved with such a system (section 6.5.1), calculations in [?] show that the core expands more rapidly than the bulk of the condensate after release from the trap, so that it becomes resolvable 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 distribution along the entire vortex core and so any bends in the vortex line will cause a loss of contrast. High contrast vortex pictures have been made by selective optical pumping of a thin slice of the condensate, so that only atoms within the slice interact with the probe beam. The most striking of these are in [?], showing an array of nearly 200 vortices in an almost perfectly regular hexagonal lattice. A second detection method is analogous to Vinen’s vibrating wire experiment in liquid helium and can be used both to detect the presence of vortices and show that the associated circulation is quantized. Instead of detecting the change in the mode spectrum of a wire as in helium, experiments with a dilute gas investigate the change in the collective mode spectrum of the condensate in the presence of a vortex. The first experiments used the radial breathing mode (m = 2) in a axially symmetric trap [?, ?], which (as described in section 5.6) may be thought of as an equal superposition of 2 counter-rotating m = ± 2 modes. In the presence of a vortex with a definite sense of rotation, these modes are no longer degenerate and so the major axis of the radial breathing mode precesses at a rate proportional to the angular momentum of the vortex line [?, ?]. A analogous effect using the m = ± 1 modes is described in chapter 10. Finally the phase winding associated with a vortex may be used to detect as well as create vortices. If a stationary condensate containing a singly charged vortex overlaps 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. This behaviour was predicted in e.g. [?] and observed in [?, ?]. 136 Chapter 9. Vortex Nucleation

9.3 Vortex nucleation in a rotating potential

The aim of our experimental work was to carry out a detailed investigation of the critical conditions for vortex nucleation in an oblate trapping geometry (ωz/ωx > 1). These data are used to test the theory presented in [?], that surface excitations are necessary to mediate vortex nucleation in a ground state condensate. (Surface modes are collective excitations of the condensate with no radial nodes, described in section 5.6.) Consider a surface mode of multipolarity m driven by a rotating perturbation of the correct symmetry. Above a critical rotation frequency Ωcm , the perturbation imparts energy and angular momentum to the condensate and the amplitude of the mode grows exponentially. Under such conditions the system rapidly goes beyond a linear regime, described by elementary excitations. A wider conﬁguration space is made available and the system can jump into new rotating equilibrium states such 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 the trapping potential [?]. At higher temperatures it may either be driven by the trap or by a rotating thermal cloud [?]. The critical frequency for angular momentum and energy transfer to the condensate is given by an analog of the Landau criterion (section 6.2). We consider the excitation of a surface mode with energyhω ¯ m and angular momentum mh¯ (along the z axis), by a heavy object rotating at frequency Ω. Energy and angular momentum conservation are only satisﬁed in the excitation process if Ω > ωm/m. Thus the minimum rotation frequency at which any surface mode can mediate vortex nucleation is given by: ·ω ¸ Ω = min m (9.1) cL m

Below the critical frequency Ωcm , the condensate adjusts its shape to the rotating potential but does not absorb energy or angular momentum from the perturbation. One of the most important ways to differentiate between the different proposed vortex nucleation mechanisms [?, ?, ?, ?, ?] is to test their predicted critical frequencies against experiment in different trapping geometries. For example, if only

the 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 highest precession frequency (or anomolous mode) of the vortex. For the prolate trap in [?] 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 the work described in this chapter, vortex nucleation data had been published by only two groups in Paris [?, ?] and MIT [?], both using prolate geometries. Their data seemed to support the surface mode theory but further evidence from an oblate geometry was necessary to verify the theory. 9.4. Experimental method 137

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

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

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 diﬀerent multipolarities (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 are also able to investigate nucleation as a function of trap deformation ²t (eqn. 6.15). The magnetic trap oﬀers a wide range of ²t from −0.5 < ²t < 0.5 (as described in section 8.3), and the value of this parameter is known to an accuracy of at least 0.5%.

9.4 Experimental method

Vortices were nucleated using the purely magnetic rotating trap described in section 8.3.2. Evaporative cooling in the static trap, followed by an adiabatic expan- 4 sion, resulted in a condensate of 2 × 10 atoms, at a temperature of 0.5 Tc. At this stage the trap is axially symmetric, with trap frequencies ω⊥/2π = 62 Hz and ωz/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 was elliptical and rotating at a preset value Ω. When we create an eccentric trap by increasing (decreasing) the TOP bias field, all three trap frequencies are reduced 138 Chapter 9. Vortex Nucleation

(increased). Since vortex lifetimes depend on the mean trap frequency [?], we adjusted the quadrupole field during the adiabatic expansion stage, to ensure that all traps had the same average radial trap frequency ω⊥, defined as: s ω2 + ω2 ω = x y . (9.2) ⊥ 2 The condensate was then held in the rotating anisotropic trap for a further 800 ms before being released. After 12 ms of free expansion the cloud was imaged along the axis of rotation. Figure 9.1 shows images of the expanded condensate at different stages during the nucleation process. Initially the cloud elongates, as shown in fig. 9.1(a) providing evidence that nucleation is being mediated by excitation of a quadrupole mode. Then finger-like structures appear on the outside edge of the condensate which eventually close round and produce vortices, ∼ 800 ms after rotation began (fig. 9.1(b)). Approximately 200 ms later, these have moved to equilibrium positions within the bulk of the condensate and appear in symmetric configurations. Figures 9.1(c)-(f) show typical, single-shot images of condensates containing 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 the expansion behaviour predicted in fig. 6.2 for our experimental conditions. We cannot obtain well-formed lattices with more than 4 vortices because of the relatively small number of atoms in the condensate. Figure 9.1(g) shows a condensate on the 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 vortices as a function of the normalized trap rotation rate, Ω = Ω/ω⊥, for a fixed eccentricity. Results for trap deformations ²t = 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 given eccentricity. Increasing ²t increases the range of Ω over which vortices are nucleated, 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 the ellipsoidal 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 resonant excitation of the quadrupole mode as seen previously [?, ?]. The resonance is broader at higher eccentricity, as intuitively expected for stronger driving. Our second study involved holding Ω constant and counting the number of vortices as a function of the trap deformation ²t. Figure 9.3 shows our results for two cases: (a) Ω > Ωc2 and (b) Ω < Ωc2 . Interestingly we were able to nucleate vortices 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 vortex nucleation process, taken along the axis of rotation with Ω = 0.70, ²t = 0.05 and after 12 ms of free expansion. (a) After the 200 ms spinning eccentricity ramp, the condensate is elongated, indicating that a quadrupole mode has been excited. (b) After a further 600 ms in the spinning trap one vortex has just formed near the edge. Approximately 1 s after rotation began, the vortices have reached their equilibrium positions and appear in symmetrical 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. 140 Chapter 9. Vortex Nucleation

Figure 9.2: The mean number of vortices as a function of the normalized trap rotation rate Ω = Ω/ω⊥. Two diﬀerent trap eccentricities were used, ²t = 0.041 (open circles) and ²t = 0.084 (solid circles). Each data point is the mean of 4 runs. 9.5. Nucleation results 141

Figure 9.3: The mean number of vortices as a function of trap deformation ²t at 4 diﬀerent 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) ²t corresponds to ωx < ωy (ωx > ωy). 142 Chapter 9. Vortex Nucleation

Adiabaticity criteria exist for both the change in the eccentricity and rotation rate of the trap that occur during the spin up stage. The former ensures that compressional modes are not excited in the condensate during the spin up process whilst that later ensures that the condensate smoothly follows a particular quadrupole mode and is the most relevant to the vortex nucleation process. The eccentricity ramp is adiabatic if the change in energy of the harmonic oscillator ground state during one trap oscillation period is small compared to the mean energy of that state. This may be expressed mathematically as:

∆ωx 2π ¿ ω⊥ (9.3) ∆t ω⊥

where ∆t is the time for the spin up ramp (200 ms). For small changes in the trap eccentricity from circular, this adiabaticity criterion may be approximated as:

ω 1 − x ω ωy ¿ ⊥ (9.4) ∆t 2π ²t 2π ⇒ αe = ¿ 1 (9.5) ∆t ω⊥

where αe is the eccentricity adiabaticity parameter used to characterize the spin up ramp. All the ramps used in this chapter satisﬁed this condition for adiabaticity. In a typical spin up ramp ²t changes from 0 to 0.1 over 200 ms in a trap with ω⊥/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 and so the criterion for rotational adiabaticity was just satisﬁed. To ensure that it was

satisﬁed at values of Ω closer to Ωc2 we varied the spin up time between 200 ms and 1 s, and detected no diﬀerence in the number of vortices formed.

The critical values of Ω and ²t for nucleation were extracted from plots such as fig. 9.2 and fig. 9.3 and compiled on fig. 9.4. The data points show the minimum eccentricity required for nucleation at a given rotation rate and map out region

B, within which vortices nucleate. Ωc2 appears to be a critical rotation frequency at 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 nucleation, although diﬀerent physical processes control the upper and lower boundaries of region B as explained below. 9.5. Nucleation results 143

Figure 9.4: The critical conditions for vortex nucleation. The data points mark the minimum trap deformation for nucleation at a particular Ω. Vortices may be formed in region B. The solid line shows the theoretical limit of stability of one particular quadrupole mode which is stable in region C. This line is in good agreement with the extreme conditions for vortex formation at Ω > Ωc2 . The dashed line shows the predicted minimum trap deformation for nucleation (under non-adiabatic conditions) for our trap conditions (µc = 13.2 ¯hω⊥) from [?]. 144 Chapter 9. Vortex Nucleation

9.6 Vortex nucleation mechanisms in our apparatus

As suggested throughout this chapter, the nucleation of vortices in our experiment is closely linked to the quadrupole modes of the condensate. Section 8.2 shows that three different modes emerge from the hydrodynamic equations for a superfluid in an elliptical rotating potential, with different aspect ratios and different regions of stability (fig. 8.1). In chapter 8 we considered a condensate in a slowly rotating potential, under conditions where only the normal quadrupole mode was stable. In this chapter we consider conditions where all 3 modes are simultaneously stable. Since they all have the same energy in the rotating frame [?], simulations are required to find out which mode the system will follow under specific experimental conditions [?]. If the eccentricity of the trap is ramped from zero (as in our experiment) at a constant rotation rate Ω > Ωc2 , the condensate follows the lower part of the overcritical branch at small eccentricity, in region C of fig. 9.4. To confirm this we have observed that the condensate has an elliptical density distribution which is orthogonal to the trap potential. It then nucleates vortices when the eccentricity is too 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 region in the ²t versus Ω plot where the overcritical quadrupole mode exists is given by (fig. 8.1):  3/2 2 2Ω2 − 1 ² =   (9.7) t Ω 3 This relation was determined from the solutions of the hydrodynamic equations for 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 for nucleation for Ω > Ωc2 and a wide range of ²t.

Below Ωc2 , the deformation needed to nucleate vortices appears to increase linearly with decreasing Ω. This boundary cannot be explained in terms of the stability limit of a quadrupole mode - the ‘normal branch’ (branch I) is stable on both regions A and B and the ‘overcritical branch’ is stable in neither. Our data appear to be at variance with the results in [?, ?], where no vortices were seen when the eccentricity was increased adiabatically and Ω < Ωc2 . We have observed that on 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 been proposed in [?, ?]. The approach used in [?] is to assume that vortex nucleation will occur (under non-adiabatic conditions) if the vortex can move from the edge of 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 vortex velocity field, the quadrupole velocity field and the interaction between the two. In a trap of fixed deformation ²t and normalized rotation rate Ω, there are two free parameters that may be varied to find a path of decreasing energy, the condensate deformation ²c and the radial vortex position d.

In an axially symmetric condensate (²c = 0) an energy barrier exists between d = R⊥ and d = 0 for all Ω, as shown in fig. 6.3, making vortex nucleation impossible. 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 rotation frequency at which energy and angular momentum can be transferred to the quadrupole mode and hence that spontaneous quadrupole deformation can occur. If the trapping potential is not axially symmetric then the cylindrical symmetry is broken even at low rotation rates. The freedom to vary both d and ²c provides nucleation paths at Ω < Ωc2 . The dashed line in fig. 9.4 shows the minimum Ω at which such a path exists as a function of the trap deformation ²t for a chemical potential close to that used in our experiment [?]. Unfortunately there is not good agreement between this theoretical curve and our experimentally determined nucleation conditions which are also shown in the figure (data points). For a given rotation rate, the trap deformation that was required for nucleation in our experiment was consistently larger than that predicted by this theory. One explanation is that our spin up conditions were adiabatic, whereas the theory requires the rotating 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 in the boundary for vortex nucleation even though αr was in the range 0 − 3 and hence the adiabaticity criterion (eqn. 9.6) was not satisfied.

Another approach for explaining the nucleation of vortices below Ωc2 , which is valid even under adiabatic conditions is given in [?]. Sinha and Castin have shown that there are regions in the plot of ²t versus Ω, both above and below Ωc2 , where the quadrupole solutions of the hydrodynamic equations become dynamically un-

stable. For frequencies above Ωc2 the predicted instability domains coincide with the experimentally observed vortex domains in both [?, ?] and this work, thus indicating a link between their instability analysis and vortices. To make a quantitative prediction for the boundary between regions 1 and 2 shown in fig. 9.4, will require further detailed work for our specific case, possibly looking at high order modes at the condensate surface, where the hydrodynamic approximation is no longer valid [?]. Equation 9.1 shows that higher order surface modes can mediate vortex nucleation at frequencies below Ωc2 and this has been verified experimentally using a laser beam stirrer of the correct symmetry to excite e.g. hexapole modes [?, ?]. However this does not provide an explanation for the observation of vortices below

Ωc2 in our experiment because at the radius of the condensate the only signiﬁcant terms in the trapping potential are quadratic. 146 Chapter 9. Vortex Nucleation

Another possible mechanism for observation of vortices below Ωc2 is that the thermal cloud plays an important role. Transfer of angular momentum to the condensate from the spinning thermal cloud may provide a mechanism for vortices to form at Ω < Ωc2 . However the transfer rate of angular momentum must be greater than any loss rate due to residual trap anisotropy [?]. In [?, ?], gravity produces a small static eccentricity in the trap in the plane of rotation. The ‘spin down’ time for a rotating thermal cloud in the presence of a small deformation parameter ²t = 0.01 is very short, 0.5 s, compared to the spin up time of 15 s, and hence the cloud may never gain significant angular momentum. However in our experiment gravity acts along the rotation axis and hence the trap is symmetric in the plane of rotation, giving a more favourable ratio of spin-up to spin-down times. With this hypothesis in mind, we tested our nucleation curve at lower temperature 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 after condensation and some heating was observed during the nucleation procedure, resulting in a final temperature around 0.8 Tc. To achieve a lower temperature we left on the so-called ‘rf shield’ so as to give an effective trap depth of 800 nK during the nucleation process. This resulted in a temperature of 0.5 Tc. No significant change 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 process in 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 the nucleation conditions, it had a striking effect on the behaviour of vortices after nucleation. Without the rf shield during the nucleation process, vortices were only occasionally found in an equilibrium configuration (i.e. 1 vortex in the centre, 3 vortices in an equilateral triangle as in fig. 9.1) and ∼ 400 ms after forming they had already moved to the edge of the condensate before disappearing. However with the rf shield present holding the temperature at 0.5 Tc, the vortices were normally found in equilibrium positions ∼ 200 ms after formation and could be reliably observed for up to 6 s, limited only by the decay of the condensate itself. Figure 9.5 shows this 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. For these data the condensate and vortex were held in a stationary circular trap; after forming vortices during 1s in an elliptical spinning trap, the trap was ramped back to circular over 200 ms and then the ‘hold time’ began. In a stationary trap, the vortex state is both energetically and dynamically unstable (section 6.5.2) but at 0.5 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 of hold time in a stationary circular trap with ω⊥/2π = 62 Hz. The vortices were created with a 200 ms ramp to a spinning trap, 800 ms in the spinning trap and 200 ms back to circular at which point the ‘hold time’ began. The temperature was held at 0.5 Tc with an rf shield. 148 Chapter 9. Vortex Nucleation

determined instead by the decay of the condensate itself. Similar results for the vortex lifetime were obtained if the condensate was held in a rotating trap for the duration 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 background gas will dominate producing a density independent decay rate, whilst in a dense condensate 3-body recombination will be most important mechanism [?, ?]. Two body 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 know that it is primarily due to 3-body recombination because the condensate lifetime depends 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 conditions for vortex nucleation (after the formation of the condensate) over a wide range of trap eccentricities. For a given eccentricity, we observe both an upper and lower limit to the rotation rate for nucleation. The upper limit confirms the predictions in [?, ?], that vortex nucleation is mediated by the breakdown of a particular quadrupole mode, but over a much wider range of parameters. Thus our results in an oblate trapping geometry compliment previous work in prolate geometries and support the theory that surface modes of the condensate play a central role in vortex nucleation [?]. This surface mode nucleation mechanism could be tested further by finding the critical conditions for nucleation of a second vortex. The presence of the first (centred) vortex will shift the spectrum of surface modes; thus if surface modes play an important role in vortex formation we expect the critical conditions for nucleation of the second vortex to be shifted relative to those of the first. A quantitative calculation of the shift in the critical conditions is given in [?]. This prediction could be readily tested in our apparatus, using the techniques developed in chapter 10 to reliably create the first centred vortex. The lower limit to the rotation rate for nucleation is different to that reported elsewhere. Further theoretical work is required to explain the linear dependence of ²t on Ω shown in fig. 9.4. Finally the thermal cloud is shown to destabilize vortex arrays and so measurements of the vortex lifetime in an oblate geometry were 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 associated with quantized angular momentum, to ensure that the superfluid wavefunction is single valued. If the vortex line is at the centre of an axially symmetric condensate of N atoms, then the associated angular momentum of the whole system is quantized 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 angular momentum is quantized, since classical vortices also exist with a continuous range of angular momenta. The superfluid gyroscope described in this chapter, consists of a condensate with both a single centred vortex line and the scissors mode excited and is used to measure the angular momentum associated with the vortex line. The experiment also raises questions about the motion of the vortex line during the 3-dimensional gyroscope motion and the coupling of the scissors mode components, the m = ± 1, modes to the vortex line, which will be discussed at the end of the chapter. The superfluid gyroscope that was realized in this work, was originally suggested in a theoretical paper by Stringari [?]. It combines a rapid scissors mode oscillation in the xz or yz plane, with the excitation of a singly charged vortex along the z axis. In the presence of the vortex the plane of oscillation of the scissors mode precesses slowly around the z axis. In polar coordinates, the scissors oscillation is in the θ direction and the precession is in the φ direction, as shown in fig. 10.1. From the precession rate we are able to deduce the angular momentum associated with the vortex line, hLzi. It is worth noting that a ‘gyroscope’ is a general term, describing a frictionless system 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 oscillation in 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 a vortex, is an example of such a system, although one must be careful not to draw incorrect analogies to classical gyroscope systems. The nutation or the wobbling motion superimposed on the precession of a spinning top is not analogous to the scissors motion in our system. The nutation frequency depends on the angular momentum of the top whereas the scissors frequency is independent of the vortex angular momentum [?]. It is also interesting to compare and contrast this work with the other superfluid gyroscopic effects that have been observed. Superfluid gyroscopes of liquid helium exhibit persistent currents in toroidal geometries, with many quanta of circulation [?, ?]. In contrast, in our experiment a single vortex of angular momentum hLzi = 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 spectrum of order ξ/R0 and in such a dilute system the vortex core size ξ cannot be ignored with respect to the size of the condensate, R0 [?]. Closely related experiments with vortex lines in dilute trapped gases are described in [?, ?, ?]. The angular momentum of a vortex line was measured in [?] using the precession of a radial breathing mode (a superposition of m = ± 2 quadrupole modes) in the plane perpendicular to the vortex line. The same method has recently been used at MIT to identify the presence of a vortex with multiple units of circulation [?]. In that work, motion is confined to 2 dimensions and the quadrupole oscillation of the condensate does not affect the vortex line. In our experiment we observe a 3-dimensional interaction between the velocity field of the vortex and the scissors mode. In [?, ?] the precession of a vortex line is observed in the absence of any bulk condensate motion, when it is tilted or displaced from the condensate symmetry axes.

10.2 The theory of the superﬂuid gyroscope

The relationship between the precession rate Ωg and hLzi may be derived by considering 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 angle from the horizontal plane rotating around the z axis at the frequency of the scissors oscillation, 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 presence of axial angular momentum hLzi. Provided that the splitting is small compared

1Gyroscope - An instrument designed to illustrate the dynamics of rotating bodies, and consisting essentially of a solid rotating wheel mounted in a ring, and having its axis free to turn in any direction. Oxford English Dictionary online: http://dictionary.oed.com/cgi/entry/00100933 152 Chapter 10. The Superﬂuid Gyroscope

to ωsc, it can be shown (by rearrangement of trigonometric identities) that the precession rate is proportional to the diﬀerence frequency of the two component modes ω − ω Ω = + − . (10.2) g 2 Using a sum rule approach, Stringari was able to express the splitting between the two m = ± 1 modes in terms of the total angular momentum hLzi [?] ω − ω hL i Ω = + − = z (10.3) g 2 2mNhx2 + z2i Where N is the total number of atoms in the condensate and m is the atomic mass. It is interesting to note that the denominator of eqn. 10.3 is twice the rigid body moment of inertia of the condensate for rotational about a radial axis. One would intuitively expect that the reduced moment of inertia would characterize a property of a superﬂuid and this will be discussed later in the chapter. Substituting for hx2 + z2i in eqn. 10.3 for the case of a harmonically trapped condensate in the Thomas-Fermi limit, one obtains [?, ?]

5/3 µ ¶−2/5 7ωsc hLzi λ a Ωg = 2 3/2 15N (10.4) 2 Nh¯ (1 + λ ) aho

2 2 1/2 where λ = ωz/ωx and ωsc = (ωx + ωz ) . This equation was derived independently 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 of the scissors mode Ωg to the total axial angular momentum of the condensate hLzi, provided that the trap frequencies and the number of atoms in the condensate are known.

10.3 Exciting and observing the gyroscope

Observation of the superfluid gyroscope effect requires a complicated experimental procedure. 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 centred 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 rotation rate and the trap eccentricity - we were fortunate that the latter is one of the features of our purely magnetic rotating trap. A detailed discussion of the conditions 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 the nucleation procedure (12 ms 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 spinning trap for a further 1 s we allowed the trap to spin down by ramping both the trap rotation rate and the trap eccentricity to zero over 0.4 s. Using the imaging system that looks along the axis of rotation (z axis), we were able to observe that the nucleation conditions above produced ‘acceptably’ centred vortices over 90% of the time. Variation of the vortex position causes a variation in the precession rate and hence a dephasing between experimental runs. Our criterion for an acceptably centred vortex is based on the maximum dephasing that we can tolerate whilst still observing a clear precession. An acceptably centred vortex 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 of position, 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 to Landau damping), significant dephasing will not build up over the limited duration of the experiment. Figure 10.2 shows images of the condensate taken along the axis of rotation, immediately after the nucleation process. A well centred vortex is pictured in (a) whilst in (b) the vortex is just beyond our limit for acceptable centering. The relationship between the precession rate and hLzi is also affected by variations in the atom number (eqn. 10.4). The measured number of atoms was N = 19, 000 ± 4000 atoms. The variation in N leads to only a 10% variation in the precession rate, which does not cause significant dephasing. Immediately after making a vortex, the TOP trap was suddenly tilted to excite 154 Chapter 10. The Superfluid Gyroscope

either the xz or yz scissors mode. This was achieved by applying an additional magnetic field to the TOP trap in the z direction, oscillating in phase with one of the radial TOP bias-field components, Bx or By, to excite either the xz or yz scissors 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 and hence excited a scissors oscillation of the same amplitude about the new tilted equilibrium position. After allowing the condensate oscillation to evolve for a variable time in the trap, we released the condensate and destructively imaged along the y direction after 12 ms of expansion. By fitting a tilted parabolic density distribution to the image, we extracted the angle of the cloud and thus gradually built up a plot of the scissors oscillation as a function of evolution time. The visibility of the fast scissors oscillation depends on the angle of the cloud projected on the xz plane (the plane perpendicular to the imaging direction) and hence varies at the slow precession frequency Ωg. If the oscillation is in the xz-plane then the projected amplitude is maximum and if it is in the yz-plane then the projected amplitude is zero. By plotting the scissors oscillation as a function of evolution time we observed the slowly oscillating visibility and hence extracted the precession rate. Note that the excitation procedure takes nearly 2 s, (the majority of which is used for vortex nucleation), during which time the number of atoms in the condensate decays by a factor of 2. Thus it is necessary to start with a large condensate, so that after decay there are still sufficient atoms to produce a sharp image, which can be reliably fitted with a parabolic density distribution. In our first attempt at this experiment we had < 10,000 atoms for imaging and typical images are shown in fig. 4.7(a) and (b). The absorption is so weak that fringing on the imaging system gives the condensate√ an irregular shape and the effect of the 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 cloud and so a clear variation in the scissors visibility was not observed. This motivated the optimization procedure described in chapter 4, which increased the condensate size by a factor of 3. Typical gyroscope images after optimization are shown in fig. 4.7(c) and (d). The narrower vortex core is no longer visible and the stronger absorption means that fringing effects are no longer significant.

10.4 Gyroscope results

In the ﬁrst gyroscope experiment, the scissors oscillation was initially excited in the xz plane, perpendicular to the imaging direction. The resulting gyroscope motion is 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 mode is 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 an average of 2 runs; occasionally 5 runs were taken and just for these points the standard error is shown for comparison with (a). 156 Chapter 10. The Superﬂuid Gyroscope

did not contain a vortex. The ﬁtting function used for each was

−γt θ = θeq + θ0 |cos Ωgt| (cos ωsct) e (10.5)

with Ωg set to zero for fig. 10.3(b). In both cases the fast scissors oscillation is clearly visible and the fitted values of ωsc/2π of (a) 179 Hz and (b) 186 Hz agree reasonably well with the theoretical value of 177 Hz. In the presence of a vortex the visibility shrinks rapidly to zero over 30 ms as the oscillation precesses through 90◦ to a plane containing the imaging direction. The oscillation visibility grows again after a further 90◦ precession. In the limit of small tilt angles the variation in oscillation visibility is represented by the |cosΩgt| term in eqn. 10.5. (Consider the projected angle of a rod in the xz plane, which is slightly tilted away from the z axis and precessing around it at frequency Ωg). Note that 2π/Ωg is the time for a full 2π rotation and hence we expect the visibility to fall from maximum to zero in a quarter period, π/2Ωg. The fitted value of Ωg/2π = 8.3 ± 0.7 Hz. From eqn. 10.4 this gives an angular momentum per particle, hlzi, of 1.14h ¯ ± 0.19¯h for N = 19,000 ± 4000 atoms. In fig. 10.3(a), each data point was taken 5 times and the mean and standard deviation is plotted. This averaging was necessary because the slight shot-to-shot variation in the starting conditions, produces slightly different precession rates. The revived amplitude is smaller than the initial amplitude due to Landau damping, which occurs at a rate of γ = 23 ± 7 Hz from the exponential decay term in eqn. 10.5. Damping also occurs at a similar rate of γ = 25 ± 5 Hz in the 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 rotation rate of 35 Hz, just too slow to create vortices. This ensured that in both cases the condensates were at the same temperature and hence had comparable Landau damping rates. The damping rates of approximately 24 Hz at a temperature of 0.5 Tc agree well with the data about the temperature dependence of the scissors mode published in [?]. The control plot also confirmed the theory that an axially symmetric condensate must have hLzi = 0 unless a vortex line is present and hence the vortex is essential for precession. Figure 10.4 shows the same experiment but with the scissors mode initially excited along the imaging direction so that the initial visibility is zero. The appropriate fitting function in this case is

−γt θ = θeq + θ0 |sin Ωgt| (cos ωsct) e (10.6) There was insufficient data to fit the Landau damping rate accurately in this case and 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 oscillation grows as the oscillation plane rotates through 90◦ to the xz plane. This growth of 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 mode is 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 an average of 2 runs, occasionally 5 runs were taken and the standard error is shown for these points for comparison with (a). 158 Chapter 10. The Superﬂuid Gyroscope

decrease of amplitude in fig. 10.3(a), since it cannot be explained by any damping effect. The precession rate from fig. 10.4(a) is 7.2 ± 0.6 Hz, which agrees within the stated errors with the precession rate fig. 10.3(a) and gives hlzi = 0.99h ¯ ± 0.17¯h. 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. This mean angle corresponds to the trap angle (the cloud angle in equilibrium) in the visible xz plane. In fig. 10.3 the trap tilt occurs in the xz plane and so this mean angle is θeq, whereas in fig. 10.4 the tilt is in the yz plane, and so the mean angle in the imaging plane is zero. Combining the results for the xz and yz gyroscope experiments, we measure the angular momentum per particle associated with a vortex line to be 1.07h ¯ ± 0.18h ¯. This is in excellent agreement with the value ofh ¯ per particle predicted by quantum mechanics. It is interesting to note that we deduce an angular momentum per particle slightly greater thanh ¯, even though only one vortex is visible. Given that this vortex is not always perfectly centred, one might expect to observe a value for Ωg and hence hLzi that is slightly lower than the theoretical value. The additional angular momentum must be due to the presence of additional vortices since we are using an axially symmetric condensate with zero moment of inertia about the z-axis. This idea is backed up by the work of Chevy et al. [?], which suggests that under conditions where they could reliably create a single centred vortex, they also created vortices at the edge of the cloud that make a small additional contribution to the angular momentum (eqn. 6.30). These vortices will be in a region of very low density 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 hLzi = 1.2 Nh¯, using the precession of a quadrupole breathing mode.

10.5 How does the vortex core move?

One question that still remains open is ‘How does the vortex core move during the gyroscope motion?’. Does it remain stationary along the z-axis of the trap, or does it oscillate, locked to the axis of the condensate? Our investigations were originally prompted by the suggestion (without a theoretical explanation) in [?] that the core might exactly follow the axis of the condensate and have since followed a variety of routes:

• Further investigation of the precession rate theory to see if it can also be used to predict the vortex motion.

• Looking at images of the vortex during the gyroscope motion taken perpendicular 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 which the vortex line is clearly visible. The images were taken along the y-axis after 12 ms of expansion. 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 [?, ?] does not give immediate information about the two possible motions of the vortex core. For example, Fetter and Svidinsky [?] use the linearized hydrodynamic equations to consider the interaction of the vortex velocity ﬁeld with a very weakly excited m = +1 or m = −1 mode. This interaction lifts the degeneracy of the modes and from this splitting the precession rate is calculated. The theory only includes Lz, 160 Chapter 10. The Superﬂuid Gyroscope

which to first order in the excitation amplitude (or θ [?]) is constant, and so this theory 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 that the vortex line does indeed move with the axis of the condensate. First we tried to observe the vortex line by imaging along the y direction, but after optimizing the experiment we had little success. The imaging beam√ integrates through the 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 be observed. However the gyroscope data that was taken prior to optimizing the experiment, with the atom number a factor of 3 lower, contained a clearly visible vortex in about 50% of the images. Some of the clearest vortex images are shown in fig. 10.5. Despite the small tilt angle and the noise level on the images, they generally confirm the hypothesis that the vortex line tilts with the axis of the condensate, if we assume that the relative angles after expansion reflect the relative angles at the instant of release. Only image (i) in fig. 10.5 appears to show a slightly 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 quantitative calculation of the precession frequency in our superfluid system, proved to be flawed, it provided useful physical insight into the gyroscope motion and so is worth discussing briefly. The model, shown in fig. 10.6 consists of a disk free to spin about its own main axis with an angular momentum Ls (Ls ≈ Lz in the limit of small angle motion) that is analogous to that of the vortex line. The disk is mounted (with frictionless bearings) in the horizontal plane on a vertical wire, so that its main axis and the wire are initially parallel. The wire is elastic, so that any small displacement of the disk from the horizontal plane results in a strong restoring torque, analogous to the torque provided by the magnetic trap. The full equation of motion of this system is: dL = Γ (10.7) dt In the limit of small angles it has a gyroscope-like solution described by the following coupled equations: ¨ 2 θ = −ωrig θ (10.8) ˙ Lz φ = Ωrig = (10.9) 2Θrig

where Θrig is the moment of inertia for oscillation about the x or y axes and   Ã !2 1/2 k Lz ωrig =  +  (10.10) Θrig 2Θrig 10.5. How does the vortex core move? 161

Figure 10.6: A mechanical model of the trapped condensate. The ellipsoidal disk (representing 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 is mounted 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. 162 Chapter 10. The Superﬂuid Gyroscope

Equations 10.8 and 10.9 give the ‘scissors’ frequency and precession rate of the rigid body model under conditions where we know that the spin angular momentum vector follows the axis of the system. At this point we had hoped to predict the precession frequency of a superfluid system in which the vortex core follows the axis of the condensate, by replacing the rigid body moment of inertia Θrig in eqn. 10.9 with the appropriate superfluid one. If the result was in agreement with experiment then this would provide additional evidence that the vortex core does follow the axis of the condensate during the gyroscope motion. This idea works very well for predicting the scissors mode frequency of the condensate, which only depends on the moment of inertiaq 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 q of the condensate is given by ωsc = k/ΘS). However the precession frequency does not arise from a simple torque and moment of inertia but instead from the interaction 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 the condensate will not necessarily be related to the interaction of the vortex with the rotational m = ± 1 velocity field in the rigid body. So in this case simply replacing the rigid body moment of inertia in eqn. 10.9 with the appropriate superfluid one will not predict a valid condensate precession rate. In summary, we have some experimental evidence to suggest that the vortex core moves with the axis of the condensate, but only a full simulation of the system with the correct initial conditions will finally settle the issue. Images of the gyroscope taken perpendicular to the core with a reduced number of atoms (thus under conditions where the gyroscope motion is not necessarily reliable), suggest that the vortex core does follow the condensate axis. The quantum mechanical calculation for the precession rate is insensitive to motion of the vortex core provided that any tilt angle remains small and produces a result which depends, somewhat surpris- ingly, on the rigid body moment of inertia of the condensate. Finally, whilst our rigid body model is helpful for gaining physical insight into the superfluid gyroscope system, it cannot be used to make quantitative predictions of the precession rate.

10.6 Kelvin waves and the gyroscope experiment

Recent experiments in Paris [?] have observed the preferential damping of the m = −2 quadrupole mode, over the m = +2 mode in the presence of a single centred vortex line. The lifetime of the m = +2 mode is 42 ms, while the m = −2 mode has a lifetime of only 18 ms. Two different theories have been suggested to explain this effect and our gyroscope data has provided useful evidence in favour of 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 deformed condensate ‘rotating’ against the flow of the thermal cloud, which is thus damped more rapidly than the m = +2 mode which ‘rotates’ with the thermal cloud. The second suggestion is that the m = −2 mode couples to helical Kelvin waves, which are excitations of the vortex core. As explained in section 6.5.4 each kelvon carries linear momentum ± hk¯ along the z-axis and angular momentum −h¯, since the sense of rotation of the Kelvin wave is always opposite to the flow around the vortex core. To conserve energy, angular and linear momentum (ω, Lz, kz), the decay process will be: ω ω Phonon(ω , −2¯h, 0) → Kelvon( −2 , −h,¯ k) + Kelvon( −2 , −h,¯ −k) (10.11) −2 2 2 Conservation of angular momentum makes this decay mechanism impossible for the m = +2 mode, since kelvons only carry angular momenta opposite to that of the vortex line, explaining why the m = −2 mode damps more quickly. The gyroscope experiment involves an equal superposition of the l = 2, m = ± 1 modes, rather than the m = ± 2 modes discussed above. An unequal damping rate comparable to that observed for the m = ± 2 modes, would cause the gyroscope motion to break down within ∼ 20 ms. Since we observe a well defined gyroscope motion for over 60 ms (fig. 10.3(a)), we conclude that both the m = ± 1 modes are equally damped. This observation has been used to support the theory that coupling to Kelvin waves, rather than interaction with the thermal cloud, is the primary damping mechanism for the m = −2 mode at ENS. If motion against the thermal cloud does not cause significant damping, then we would expect to observe an equal decay rate for the m = ± 1 modes in our experiment; resonant coupling 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 = ± 1 modes 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 a dilute gas 87Rb condensate, exhibited by its response an applied torque. Over the last 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 properties of the dilute gas Bose condensate both from theoreticians within the immediate field and also from those in the liquid helium research. In fact the most important parallels 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 has been carried out on liquid helium over the last 60 years. Another important factor has been the development of a very flexible magnetic trapping potential that is ideally suited to studying the superfluid nature of the condensate. It has two important features:

• A wide variety of well deﬁned torques may be applied to the condensate, by rotating or tilting the potential about any axis. The potential may be rotated at any frequency, both clockwise and anti-clockwise, stopped suddenly at any angle and oscillated between two angles. Many of these features have been used in the experiments described in this thesis and some will be used in future work.

• The response of the condensate to any perturbation is determined not only by its superﬂuid nature but also by the spectrum of low-energy collective modes. This spectrum depends on the geometry of the trapping potential and so we have developed the ability to change all three trapping frequencies independently [?]. Two experiments, which are not directly related to superﬂuidity and have not been included in this thesis, have made use of this 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 two m = 0 breathing modes (section 5.6). In [?] down-conversion was observed between the xz and xy scissors modes.

11.1 Summary of results

In this thesis, four experiments are described, each demonstrating the superfluid response of the condensate to a different applied torque. The first was the scissors mode experiment (theory [?], expt. [?]), which together with the observation of quantized vortices [?, ?] and a critical velocity for superfluid flow [?], provided the first experimental evidence that the condensate behaves as a superfluid. Af- ter a sudden tilt of the trapping potential, the thermal cloud performed a heavily damped oscillation at two frequencies, indicating the presence of both rotational and irrotational flow patterns. Under the same excitation conditions, the condensate oscillated at a single undamped frequency corresponding to the purely irrotational flow pattern predicted for a superfluid. The scissors mode experiment was repeated at a range of temperatures between 0.3 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 large negative shift in the frequency of the condensate oscillation was observed above 0.7 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 in terms of a simple damped harmonic oscillator model and requires an analysis which includes the large thermal occupation of the low-lying excited states. In chapter 8 the pure irrotational flow pattern of a vortex-free condensate is demonstrated by its behaviour after being released from a slowly rotating elliptical potential (theory [?], expt. [?]). During the expansion the condensate never becomes symmetric about the axis of rotation as this would produce a system with zero moment of inertia and thus infinite kinetic energy. A detailed investigation into the critical trap conditions (trap deformation and rotation rate) for vortex nucleation is presented in chapter 9 [?]. Above the critical rotation rate given by Landau theory we confirm that nucleation is mediated by the breakdown of a particular quadrupole mode. Nucleation is also observed at lower 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 circulating velocity field of the vortex breaks the degeneracy of the two counter-rotating components of the xz scissors mode, producing a precession of the scissors mode at a rate proportional to Lz. From the observed precession we measure the angular momentum 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 new modiﬁcations to our trapping potential and the techniques developed for previous experiments (e.g. the production of a single, centred vortex line). These ideas are collected 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) has recently been observed at MIT [?] using ‘Berry’s phase’ [?, ?]. Berry’s phase is an additional phase factor required by the time-dependent Schrodinger equation. It has no time dependence but is revealed by adiabatically varying the parameters that appear in the Hamiltonian of the system around a closed loop, e.g. rotating the individual atomic spin vectors within our condensate. Alternatively if the paths of two different atomic spin vectors may be combined to create a closed loop then a relative phase is acquired between the two atoms. If the path of the spin vectors draws out an area on the surface of a sphere, then the relative phase is equal 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 atomic spin vectors followed this field adiabatically, initially pointing along +z and finally pointing along −z. The plane in which each spin rotates between +z and −z contains the z axis and the direction of the local 2D radial quadrupole field (which dominates when Bz is close to zero), which in turn depends on the azimuthal position 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 the atoms acquire a relative phase of:

φ − φ S = 1 2 × 4π × m (11.1) rel 2π F

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 which results in a single vortex with 4 units of circulation. After producing a vortex by this method, the authors of [?] report the observation of a large hole in the density distribution and measure the associated angular momentum using the precession of the radial breathing mode [?, ?]. How- ever, in the process of reversing the direction of Bz, the Ioﬀe-Pritchard potential is transformed from a trapping potential to an anti-trapping potential with negative curvature. Thus after reversing the ﬁeld the condensate may only be observed for ≤ 50 ms before it is lost. This timespan is not long enough to observe the decay 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 of a 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 and keep 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 geometry in the z = 0 plane from the 2-dimensional quadrupole field of a Ioffe-Pritchard trap (see fig. 11.1). However the local field direction is still a unique function of the azimuthal position within the z = 0 plane, which is the crucial feature for creating a Berry’s phase winding around the z axis. The experimental scheme which has been proposed is shown schematically in fig. 11.2. After evaporative cooling and an adiabatic expansion, the condensate is held in a standard TOP trap, centred on z = 0, with trap frequencies of ω⊥/2π = 60 Hz and ωz/2π = 170 Hz. The rotating radial bias field is turned off and a static bias field Bz turned on, in a time that is short compared to the trap oscillation period but sufficiently long for the spins to follow the changing field adiabatically. This 0 produces a pure quadrupole trap, centred on z0 = Bz/2Bq. The condensate is still at z = 0 and has its spins aligned to the bias field. The condensate is then allowed 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 z direction falls to zero, leaving only the small radial components of the quadrupole field and then increases in the opposite direction, sufficiently slowly for the atomic spins to follow the total field adiabatically. Thus when the condensate arrives at z = 2z0 it has acquired a Berry’s phase winding and contains a vortex with 4 units of circulation. To re-trap the condensate, Bz is doubled and the radial bias field is increased from zero creating a standard TOP trap centred on 2z0 which ‘catches’ the condensate when it is instantaneously stationary. As before the fields are changed rapidly compared to the trap period but slowly enough for the spins 168 Chapter 11. Conclusion and Future Plans

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 static Bz is turned on and BT reduced to zero in a time that is short compared to ωz (b). The condensate is allowed to perform half of a trap oscillation (c). When it instantaneously ◦ stops at 2z0, the spin vectors have rotated with the magnetic ﬁeld through 180 and the 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 the condensate there. 11.2. Future experiments 169

to follow. The decay of the multiply charged vortex can be observed by holding the 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 density distribution. Alternatively, iterating this procedure with an ever increasing axial bias 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 in nine out of ten condensates, providing the reliable initial conditions necessary for a range of experiments on the properties of a single vortex line. One such experiment is the investigation of the spectrum of Kelvin waves on a vortex line. Experimental work in this area has so far been limited to the observation of different damping rates for the m = +2 and m = −2 quadrupole modes at ENS when a vortex line is present; one likely explanation is that one mode couples to a Kelvin wave of the vortex 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 them resonantly to a collective excitation of the condensate. In a prolate trap such as that at ENS, the lowest Kelvin modes and mode spacings are several orders of magnitude lower than the collective mode frequencies and so several high order Kelvin waves are likely to couple to the m = −2 mode rather than one individual one. In contrast, in our oblate geometry the lowest Kelvin modes have similar energies 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 Kelvin modes in a standard TOP trap with ω⊥/2√π = 62 Hz and λ = 8. The m = −2 mode, with frequency 2 ω⊥, is a suitable driving excitation because 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 frequency ω⊥/ 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 standard TOP 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 resonance condition 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 have already achieved using the method described in [?]. A possible driving mode for the second Kelvin wave would be the m = −1 2 1/2 mode which has a frequency of (1 + λ ) ω⊥. However the coupling mechanism is a weaker second order process (involving 2 phonons) and so will require a large amplitude m = −1 mode to observe any significant energy transfer. (Conservation of angular momentum requires each m = −1 phonon to convert to a single kelvon, 170 Chapter 11. Conclusion and Future Plans

Figure 11.3: Plots showing how the lowest Kelvin√ mode can be brought into resonance with 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 (solid line) in a trap of zero stiffness and so coupling√ cannot be achieved. (b) shows the first Kelvin mode frequency (dashed line) and ω⊥/ 2 (solid line) as a function of trap aspect ratio with ω⊥/2π fixed at 62 Hz. Coupling to the m = −2 mode is possible at an aspect ratio of 2.02. but conservation of linear momentum requires 2 kelvons to be generated, hence two phonons are involved in every decay process.) Simulations by Max Kruger [?] show the vortex line behaving very differently when the m = +1 and m = −1 modes are excited. It shows no response to the former but oscillates wildly in the presence of the latter indicating that sufficient energy transfer is theoretically possible even by this second order process. Again it will be necessary to change the trap aspect ratio and resonance can be achieved between the m = −1 mode and the second Kelvin wave in a trap with ω⊥/2π = 62 Hz and λ = 1.60. Such a trap aspect ratio is outside the range available with our current apparatus, but an experimental 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 conditions under which a peak is observed either in the damping rate of the collective mode or in the excitation rate of the vortex line. From the trap conditions at resonance the frequency of the collective mode (of multipolarity m) can be calculated to high accuracy, which will be equal to m times the Kelvon frequency. The excitation rate of the Kelvin mode can be measured in two ways. First we can look along the core of the vortex after a fixed period of driving and record the core visibility as a function of the collective mode frequency. Good visibility implies that the vortex is lying straight along the z axis and few Kelvons are excited. Poor visibility implies that the vortex is highly excited and bent or tilted away from its original 11.2. Future experiments 171

direction. (One would have to ensure that the focusing was consistent from shot to shot otherwise other factors would affect the visibility). Alternatively one could image the condensate from the side as in fig. 10.5 and observe the shape and angle of the vortex core after a fixed driving period as a function of collective mode frequency. The gyroscope work suggests that one would need to use small condensates N ≤ 10000 atoms to obtain clear side-on vortex images. Given that the boundary conditions used to calculate the Kelvin mode frequencies in section 6.5.4 are approximate, the trap frequencies given above only provide an 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 aspect ratios to roughly locate the resonance. Then several measurements of the collective mode damping rate should be taken close to resonance and plotted against the trap aspect ratio. Interpolation of these points should accurately identify the resonant trap conditions and hence the Kelvin frequency.

11.2.3 Damping of the m = 2 modes in the presence of a vortex This brief experiment is closely related to the investigation of the Kelvin wave spectrum discussed above. There is still debate about why diﬀerent damping rates were observed at ENS for the m = −2 and m = +2 modes in the presence of the vortex (see section 10.6). It has been argued that the more rapid damping of the m = −2 mode is not due to the excitation of Kelvons but due to its interaction with the counter-rotating thermal cloud [?]. Repeating the experiment in our oblate trap, under conditions where the m = ± 2 modes are far from any kelvon resonance, should provide conclusive evidence. If the m = −2 mode is also preferentially damped 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 be very small. However if equal damping rates are observed in our experiment, then the additional damping of the m = −2 mode observed at ENS was due to coupling to one of the many near resonant Kelvin waves that exist in a prolate trapping geometry.

11.2.4 Critical conditions for nucleating a second vortex As discussed in section 9.7, the critical trap conditions for nucleating a second vortex (in the presence of a single centred vortex) have been calculated in [?], on the assumption that vortex nucleation is mediated by surface modes of the condensate. These critical conditions are shifted with respect to those of the first vortex because the first vortex modifies the spectrum of surface modes of the condensate. Observation of these shifted nucleation conditions would provide further evidence for the role played by surface modes in the vortex nucleation 172 Chapter 11. Conclusion and Future Plans

process.

11.2.5 Anti-vortex production After nucleation of the ﬁrst vortex, we now have the ability to reverse the direction of the rotating potential and create an anti-vortex. Observation of the dynamics and possible annihilation of the two vortices would be very interesting although a little 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 function of 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 presented for the precession rate as a function of vortex position, one calculated using a sum rule method and the other by considering the velocity field associated with the vortex as a perturbation. Both give the same result when the vortex is centred, 2 2 but differ by a factor (1 − d /R⊥) when the vortex is off centre by a distance d. One would expect the perturbative approach to produce the correct answer but it would 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 breathing mode would be excited along known axes, allowed to evolve for a fixed time ∆t and then imaged. Two quantities may be extracted from the image. The first is the radial position of the vortex d. The second is the angle through which the major axis of the oscillation has precessed ∆φ. Thus over many experimental runs the precession rate ∆φ/∆t could be plotted as a function of vortex position d and compared to the two theories presented in [?]. Appendix A

The Properties of a 87Rb Atom.

Quantity Symbol Value atomic number A 87 nuclear spin I 3/2 mass m 1.45 × 10−25 kg vapour pressure See below 2.34 × 10−7 torr (293K) D1 wavelength 5S1/2 → 5P1/2 795nm D2 wavelength 5S1/2 → 5P3/2 λ 780.026 nm wavenumber k = 2π/λ 8.05 × 106 m−1 natural linewidth Γ 2π × 5.76 MHz Doppler width (300 K) ∆fDoppler 516 MHz −2 saturation intensity Isat 3.14 mW cm −13 2 resonant absorption cross-section σ0 2.9 × 10 m −1 recoil velocity vr =hk/m ¯ 5.85 mm s 2 recoil temperature Tr = mvr /2kB 180 nK gF factor for lower state gF (5S1/2,F = 2) 1/2 gF factor for excited state gF (5P3/2 F = 3) 2/3 −9 s-wave scattering length a 110ao = 5.82 × 10 m interaction parameter g = 4πh¯2a/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 D1 wavelength, 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 15.88253 B 1961.258 4529.635 C -.03771687 0.00058663 D 42.57526 -2.99138

Table A.2: The constants required for calculating the vapour pressure of Rb as a function of temperature. Appendix B

Clebsch-Gordan Coeﬃcients

Figure B.1 shows the relative rates for transitions between diﬀerent magnetic substates within the F=2 to F’=3 hyperﬁne line. The values given relate the absorption cross-section for each transition to that for the stretched transition, σ0. (The stretched transition is |2, 2i to |3, 3i in this case). For example consider resonant + σ polarized light, of intensity I (I ¿ Isat), incident on an atom in the |2, 2i state. The transition rate to the |3, 3i state will be:

15 I R = σ (B.1) 15 0 hω¯ Under the same conditions the transition rate from the |2, 1i state to |3, 2i state would be: 10 I R = σ (B.2) 15 0 hω¯ The stretched transition has the highest individual absorption rate. Thus we probe the atoms with σ+ polarized light in a correctly aligned magnetic field, selectively exciting the stretched transition and maximizing the signal to noise ratio of our images. 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 any mF state. Secondly the sum of coefficients out of any of the lower states is the same. This ensures that the absorption rate of unpolarized light is independent of the 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 and depend only on the angular momentum quantum numbers (F, mF ) of the initial and final states. In general a Clebsch-Gordan coefficient is an angular momentum overlap integral between an initial state |L1, m1,L2, m2i with two independent

175 176 Appendix B. Clebsch-Gordan Coeﬃcients

Figure B.1: A diagram indicating the relative rates for transitions between diﬀerent magnetic substates within the F=2 to F’=3 D2 hyperﬁne line of 87Rb.

angular momenta and a ﬁnal coupled state |L1,L2,L3, m3i, in which the sum of the 2 angular momenta and its z component are constants of motion. Thus a general CG coeﬃcient may be written as:

C(m1, m2,L3, m3) = hL1, m1,L2, m2|L1,L2,L3, m3i (B.3)

In our case, L1 and m1 correspond to the initial |F, mF i state of the atom. L2 and m2 correspond to the angular momentum and polarization of the incident photon. 0 0 L3 and m3 correspond to the ﬁnal |F , mF i 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 the magnetic 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 a classical gas and so the number density is given by a Gaussian distribution: · ³ ´¸ −m 2 2 2 2 2 2 n(x, y, z) = n0 exp ωxx + ωyy + ωz z (C.1) 2kBT The classical gas approximation for the thermal cloud holds provided the thermal energy is much greater than the spacing between the trap energy levels (kBT À hω¯ z) and so the probability of any particular state being occupied remains small. However in our experiment, kBTc is only about 5 times greater thanhω ¯ z. Around and below Tc, the thermal cloud distribution is no longer exactly Gaussian, but becomes more sharply peaked at the centre [?]; this peak is know as Bose en- hancement. At these low temperatures the occupation of the lowest energy states is no longer negligible and the bosonic nature of the identical atoms in the cloud becomes important. Bose-statistics must be used for an accurate description of the density distribution of the thermal cloud around and below Tc. (However the wings of the Bose-enhanced density distribution still have a Gaussian shape, so an accurate value for the temperature of the cloud can be obtained by ﬁtting a Gaussian 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

q 0 µ Bq Radial trap freq. ω⊥ √ 2π × 10.6 Hz 2π × 124 Hz 2m BT

q 0 µ Bq Axial trap freq. ωz 8 √ 2π × 30.0 Hz 2π × 351 Hz 2m BT

1/3 Harmonic osc. freq. ωho (ωxωyωz) 2π × 15.0 Hz 2π × 175 Hz r 2KB T −3 −5 Radial 1/e HW σ⊥ 2 1.1 × 10 m 2.4 × 10 m mωx,y q 2KB T −4 −6 Axial 1/e HW σz 2 3.9 × 10 m 8.4 × 10 m mωz q KB T −4 −5 Axial RMS HW zRMS 2 7.8 × 10 m 1.7 × 10 m mωz

N 17 −3 19 −3 Peak number density n0 3/2 1.1 × 10 m 3.3 × 10 m π x1/ey1/ez1/e · ¸3/2 mω2 N ho 2πkB T

³ ´1/2 h2 −8 −7 de Broglie wavelength λdB 3.5 × 10 m 1.4 × 10 m 2πmkB T

3 n0h −6 −2 Phase-space density φ 3/2 4.7 × 10 8.9 × 10 (2πmkB T )

h i3 N ¯hωho kB T q 8kB T −2 −1 −2 −1 Mean thermal speed vth πm 8.3 × 10 ms 2.1 × 10 ms

1 −1 −1 Collision rate γcoll 2 n0σvth 3.9 s 295 s

2 −2 −5 Mean free path λMFP 2.1 × 10 m 7.1 × 10 m n0σ

Relaxation time τ 5 0.3 s 4.2 × 10−3 s 4γcoll

λMFP Knudsen number K 1/3 27 5 (x1/ey1/ez1/e)

≈ ωhoτ

Table C.1: Useful thermal cloud formulae