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Dynamics of Bose-Einstein Condensation

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Dynamics of Bose-Einstein Condensation DYNAMICS OF BOSE-EINSTEIN CONDENSATION MATTHEW JOHN DAVIS A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Oxford St John's College University of Oxford Hilary Term 2001 Dynamics of Bose-Einstein Condensation Matthew John Davis, St John's College. Thesis submitted for the degree of Doctor of Philosophy at the University of Oxford, Hilary Term 2001. Abstract This thesis is concerned with the dynamics of thermal Bose-Einstein conden- sates with two main areas of emphasis. We summarise the development of the quantum kinetic theory of C. W. Gar- diner, P. Zoller, and co-workers, and in particular its application to the problem of condensate growth. We extend an earlier model of the growth of a Bose-Einstein condensate to include the full dynamical effects of the thermal cloud by numerically solving a modified quantum Boltzmann equation. We find that the results can be easily interpreted by introducing an effective chemical potential for the thermal cloud. Our new results are compared with the earlier model and with the available experimental data. We find that in certain circumstances there is still a discrepancy between theory and experiment. Beginning with the second-quantised many-body Hamiltonian we develop an approximate formalism for calculating the dynamics of a partially condensed Bose gas. We divide the Hilbert space into a highly-occupied coherent region described by a Gross-Pitaevskii equation, and an incoherent region described by a kinetic equation. We discuss the interpretation of the terms in the equations, and their relevance to recent experiments in the field. We numerically solve the projected Gross-Pitaevskii equation we derive, and find that it evolves strongly non-equilibrium states towards equilibrium. We analyse the final distributions in terms of perturbative equilibrium theories, and find that the two approaches are in excellent agreement in their range of validity. We are therefore able to assign a temperature to the numerical simulations. However, the presently available equilibrium theories fail near the critical region, whereas the projected Gross-Pitaevskii equation remains valid throughout the Bose-Einstein condensation phase transition as long as the relevant modes remain highly occupied. This suggests that the equation will be useful in studying the role of vortices in the critical region, and the shift of the transition temperature with the atomic interaction strength. Acknowledgements Over the past three and a bit years many people have helped make my life that little bit easier and more enjoyable. Firstly, my heartfelt thanks go to Prof. Keith Burnett for his encouragement, humour, and emotional support. He always made me feel an important part of the team, and I greatly appreciate his friendship. I have learnt a lot from working with Prof. Rob Ballagh and Prof. Crispin Gardiner, and would like to thank them both for being so patient with me. I hope that I can continue to interact with them in the future. I have benefited greatly from countless discussions with Dr. Sam Morgan, and would like to thank him for putting up with my ignorance. He is the person responsible for all the errors in this thesis, as he was in charge of the proof-reading. (Cheers!) I would like to thank all the stalwarts of the Burnett group during my time in Oxford. Dr. David Hutchinson, Dr. Martin Rusch, Dr. Stephen Choi, Jacob Dunningham and Mark Lee have all made the Clarendon a stimulating environment in which to work. David deserves a special mention for consistently lowering the tone of conversation (along with Dr. John Watson, gone but not forgotten). I would also like to mention all the newcomers, who have ensured that I will miss Oxford when I leave. Thanks to Dr. Thomas Gasenzer, Dr. Vicki Ingamells, Dr. Thorsten Koehler, David Roberts, Alex Rau, Peter Kasprowicz and Karen Braun-Munzinger. I have enjoyed interacting with all the BEC experimentalists, including Dr. Chris Foot, Dr. Jan Arlt, Onofrio Marag`o, Eleanor Hodby, and Gerald Hecken- blaikner. I would also like to thank the atom opticians: Dr. Gil Summy, Dr. Rachel Godun and Dr.(?) Michael D'Arcy. I am grateful to the Clarendon laboratory for the use of facilities; and the iii Commonwealth scholarship and St John's College for financial support. Of course, I greatly appreciate the love and support of family. Thanks to Jonathan, for never letting me forget I am a physics geek, and to Mum and Dad for their belief in me, and for encouraging me to get this far. Finally, I would like to mention my lovely wife Shannon, who has always been at my side emotionally (if not physically) throughout the course of this D. Phil. She has been a great source of motivation, and I would especially like to thank her for looking after me and cheering me up in the final period of writing. Contents 1 Introduction 1 1.1 Indistinguishable particles . 1 1.2 BEC in an ideal gas . 2 1.3 Interacting systems . 5 1.4 BEC in the laboratory . 6 1.4.1 Experimental procedure . 7 1.4.2 Experiments today . 9 1.4.3 Effect of the trapping potential . 10 1.5 Important experiments . 10 1.6 Thesis outline . 11 2 Dynamical quantum field theories 13 2.1 The BEC Hamiltonian . 14 2.1.1 Basis set representation . 15 2.2 Effective low-energy Hamiltonian . 16 2.2.1 The two-body T-matrix . 17 2.2.2 Elimination of high-energy states . 19 2.2.3 Conclusion . 21 2.3 The Gross-Pitaevskii equation at T = 0 . 21 2.3.1 Derivation . 22 2.3.2 Time-independent GPE . 23 2.3.3 Thomas-Fermi solution . 24 2.3.4 Collective excitations . 24 2.3.5 Experimental verification . 26 2.4 Kinetic theory . 29 2.4.1 The Boltzmann transport equation . 29 v vi CONTENTS 2.4.2 The quantum Boltzmann equation . 32 2.4.3 Derivation of the QBE . 32 2.4.4 The GPE kinetic equation . 37 3 Quantum kinetic theory for condensate growth 39 3.1 QKI : Homogeneous Bose gas . 40 3.2 QKIII : Trapped Bose gas . 42 3.2.1 Description of the system . 42 3.2.2 Derivation of the master equation . 43 3.2.3 Bogoliubov transformation for the condensate band . 44 3.2.4 QKV . 46 3.3 A model for condensate growth . 46 3.4 Model A: the first approximation . 49 3.5 Bosonic stimulation experiment . 51 3.6 Model B: inclusion of quasiparticles . 54 3.7 Further development . 57 4 Growth of a trapped Bose-Einstein Condensate 59 4.1 Formalism . 59 4.1.1 The ergodic form of the quantum Boltzmann equation . 60 4.2 Details of the model . 63 4.2.1 Condensate chemical potential µC(n0) . 63 4.2.2 Density of states g¯(") . 64 4.3 Numerical methods . 65 4.3.1 Representation of the distribution function . 65 4.3.2 Solution . 66 4.3.3 Algorithm . 68 4.4 Results . 69 4.4.1 Matching the experimental data . 70 4.4.2 Typical results . 72 4.4.3 Comparison with model B . 74 4.4.4 Effect of final temperature on condensate growth . 75 4.4.5 Effect of size on condensate growth . 78 4.4.6 The appropriate choice of parameters . 80 4.4.7 Comparison with experiment . 80 4.4.8 Outlook . 85 4.5 Conclusions . 86 CONTENTS vii 5 A formalism for BEC dynamics 89 5.1 Decomposition of the field operator . 91 5.2 Equations of motion . 92 5.2.1 The Hamiltonian . 92 5.2.2 Coherent region . 93 5.2.3 Incoherent region . 97 5.3 The individual terms of the FTGPE . 98 5.3.1 The linear terms . 98 5.3.2 The anomalous term . 103 5.3.3 The scattering term . 107 5.3.4 The growth term . 108 5.4 Incoherent region equation of motion . 110 5.5 Outlook . 111 6 The projected GPE 113 6.1 The projected GPE . 113 6.1.1 Conservation of normalisation . 114 6.1.2 Technical aspects of the projector . 116 6.2 Simulations . 118 6.2.1 Parameters . 118 6.2.2 Initial wave functions . 119 6.2.3 Evolution . 121 6.3 Evidence for equilibrium . 124 6.4 Quantitative analysis of distributions . 128 6.4.1 Expected distribution . 128 6.4.2 Bogoliubov theory . 129 6.4.3 Second order theory . 130 6.5 Condensate fraction and temperature . 136 6.6 The role of vortices . 138 6.7 Conclusions . 141 7 Prospects for future development 143 7.1 The PGPE . 143 7.1.1 Homogeneous case . 143 7.1.2 Inhomogeneous case . 144 7.2 The FTGPE . 145 7.3 Limitations . 146 viii CONTENTS 7.4 Final conclusions . 147 A Approximate solution of operator equations 149 B Derivation of the rate W + 151 C Semiclassical density of states 155 C.1 Harmonic oscillator . 156 C.2 Ideal gas . 156 C.3 Thomas-Fermi approximation . 157 D Numerical methods 159 D.1 Preliminaries . 159 D.1.1 Calculation of D^ . 159 D.1.2 Choice of grid . 160 D.2 Symmetrised split-step method (SSM) . 161 D.3 Fourth order Runge-Kutta . 162 D.3.1 RK4IP . 163 D.4 Adaptive step size algorithm (ARK45) . 165 D.4.1 Interaction picture . ..
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