The Superfluid Properties of a Bose-Einstein Condensed Gas Eleanor Hodby A thesis submitted in partial ful¯lment 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 re- sults. We investigate the superfluid nature of the condensate by observing its response to a variety of torques applied by the trapping potential. Using this di- lute, 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 modi¯cations 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 ro- tating 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 con¯rmed 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 fre- quency 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 ¯nal 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 ¯rst 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 ¯ne, 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 di®erent 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 o±ce, 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 o® 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 ampli¯er . 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 ¯eld 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 ¯eld 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 ¯eld 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 ¯nite temperature . 111 7.7.1 Experimental procedure and results . 112 7.7.2 Moment of inertia at ¯nite 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 .