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Ultra-Cold Collisions and Evaporative Cooling of Caesium in a Magnetic Trap

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Ultra-Cold Collisions and Evaporative Cooling of Caesium in a Magnetic Trap Ultra-cold Collisions and Evaporative Cooling of Caesium in a Magnetic Trap Angharad Mair Thomas A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Oxford Jesus College University of Oxford Hilary Term 2004 Abstract Ultra-cold Collisions and Evaporative Cooling of Caesium in a Magnetic Trap Angharad Mair Thomas, Jesus College, Oxford University DPhil Thesis, Hilary Term 2004 Caesium-133 atoms have been evaporatively cooled in a magnetic trap to tem- peratures as low as 8 nK producing a final phase-space density within a factor of 4 of that required for the onset of Bose-Einstein condensation. At the end of the forced radio-frequency evaporation, 1500 atoms in the F = 3; mF = ¡3 state remain in the magnetic trap. A decrease in the one-dimensional evaporative cool- ing efficiency at very low temperatures is observed as the trapped sample enters the collisionally thick (hydrodynamic) regime. Different evaporation trajectories are experimentally studied leading to a greater understanding of one-dimensional evaporation, inelastic collisions and the hydrodynamic regime. A theoretical sim- ulation accurately reproduces the experimental data and indicates that the reduc- tion in the evaporative cooling efficiency as the cloud enters the hydrodynamic regime is the main obstacle to the realization of Bose-Einstein condensation in the F = 3; mF = ¡3 state. In addition, we report measurements of the two-body inelastic collision rate coefficient for caesium atoms as a function of magnetic field and collision energy. The positions of three previously identified resonances are confirmed, with reduced uncertainties, at magnetic fields of 108.87(6), 118.46(3) and 133.52(3) Gauss. The resonance centred at 118.46 Gauss is also investigated as a function of temperature, thus demonstrating the dependence of the inelastic Feshbach resonance on collision energy. The importance of including partial waves of a higher basis in theoretical i closed channel calculations in order to accurately determine the line shape of a Feshbach resonance is also demonstrated. ii Acknowledgements This work could not have been carried out without the guidance and support of a vast number of people. My first thanks must go to my supervisor Prof. Christopher Foot who provided me with the opportunity to work in his group and to study BEC which had first intrigued me as an undergraduate. I greatly appreciate the work he has done in reading through the various drafts of this thesis and for the suggestions and comments which have crystallized my understanding. Where would I be without Dr. Simon Cornish? Probably still taking slosh data! His enthusiasm over the last three years has been boundless and his patience limitless. My enjoyment of my time here in Oxford has been in no small part due to his guidance and his friendship. He has also read my thesis carefully and offered advice and corrections in making the thesis intelligible. I must also thank Prof. Paul Julienne (NIST) for the theoretical data he produced. If I absorb a hundredth of his understanding of ultra–cold collisions, it would then become my chosen specialist subject in Mastermind. Thank you to the present and past members of the group for not only being willing to answer my questions, but actually answering them: Vincent, Gerald, Onofrio, Donatella, Rachel and Eleanor. Thanks also to all the other members of the group for providing some light in the dark basement. The Clarendon staff must also be thanked for making my life easy: Sue, Gra- ham, Terry, Dave and Alan in Stores, George and Rob in the Research Workshop, Reg and all in the Main Workshop, D.T. Smith and all at Central Electronics, Graham, Alan and Casper and everyone in maintenance. iii Over the past three years I have been very fortunate to have many good friends. Special mention must go to Emma. Unsuccessful days in the lab were soon forgot- ten over a glass of wine and an episode of ‘Friends’. I must also thank Edward for his friendship and for proof–reading. My last but by no means least thanks must go to my parents for their love and continual support. Diolch yn fawr iawn am bopeth. I must finish these acknowledgments with some words taken from the lab song: “No one said it would be easy” Sheryl Crow Tuesday Night Music Club (1993). iv Contents 1 Introduction 1 1.1 Bose-Einstein Condensation . 1 1.1.1 Bosons and Fermions . 2 1.1.2 Bose-Einstein Condensation . 3 1.1.3 Experimental Observation . 4 1.1.4 Current Experiments . 5 1.2 Caesium: State of Play – October 2000 . 6 1.2.1 F = 4, mF =+4 ........................ 7 1.2.2 F = 3, mF = ¡3 ........................ 9 1.2.3 F = 3, mF = +3 . 11 1.2.4 Recent Collisional Studies . 13 1.3 Overview of Thesis . 17 1.3.1 Notation . 17 1.3.2 Abbreviations Used in Thesis . 18 1.3.3 Constants and Conversion Factors . 18 1.3.4 Prior Publication of Work in this Thesis . 18 2 BEC: An Experimentalist’s Guide 21 2.1 The Recipe for Creating a BEC . 21 2.2 Laser Cooling . 23 2.2.1 Alkali Metals . 23 2.2.2 Doppler Cooling . 26 v CONTENTS vi 2.2.3 Sub-Doppler Cooling . 27 2.3 Magneto-Optical Traps . 31 2.3.1 Six-Beam MOT . 31 2.3.2 Pyramidal MOT . 33 2.4 Magnetic Trapping . 34 2.4.1 Zeeman Effect on the Hyperfine Ground States . 34 2.4.2 Simple Magnetic Trap . 35 2.4.3 Majorana Transitions . 36 2.4.4 Ioffe-Pritchard Trap . 37 2.4.5 Baseball Trap . 39 2.4.6 Experimental Considerations . 40 2.5 Evaporative Cooling . 41 2.5.1 Radio-Frequency Evaporation . 41 2.5.2 Effects of Gravity on Evaporation . 42 2.5.3 Dimensionality of Evaporation . 44 3 Scattering Theory and Feshbach Resonances 46 3.1 Atomic Interactions . 46 3.1.1 Unperturbed Hamiltonian . 47 3.1.2 van der Waals Interaction and Exchange Terms . 47 3.1.3 Dipolar Interaction . 50 3.2 Scattering . 52 3.2.1 Partial Wave Analysis . 52 3.2.2 Scattering Length . 54 3.2.3 Scattering at Low Temperatures . 55 3.3 Resonances . 56 3.3.1 Potential Resonances . 57 3.3.2 Feshbach Resonances . 59 4 Evaporative Cooling 62 4.1 Elastic and Inelastic Collisions . 63 CONTENTS vii 4.1.1 Elastic Collisions . 63 4.1.2 Inelastic Collisions . 63 4.2 Efficiency of Evaporation . 65 4.2.1 Hydrodynamic (Collisionally thick) regime . 67 4.3 Evaporative Cooling Models . 69 4.3.1 Simple Model of Evaporative Cooling . 70 4.3.2 Monte-Carlo Simulation . 71 4.4 Evaporation Strategy . 76 5 Experimental Apparatus 77 5.1 Apparatus . 77 5.1.1 Lasers . 77 5.1.2 Frequency Stabilization - DAVLL . 81 5.1.3 The Optical Table . 88 5.1.4 Vacuum System . 91 5.1.5 Pyramidal MOT . 92 5.1.6 2nd MOT – Experimental MOT . 95 5.1.7 Magnetic Trap . 97 5.1.8 Evaporation . 104 5.2 Control . 104 5.2.1 Experimental Synchronization . 104 5.2.2 Analogue Voltages . 105 5.3 Diagnostics . 107 5.3.1 Recapture . 107 5.3.2 Absorption Imaging . 108 5.3.3 Comparison of Recapture and Imaging . 112 5.4 Optimization and Characterization . 113 5.4.1 Pyramidal MOT . 113 5.4.2 Experimental MOT . 114 5.4.3 Loading the Experimental MOT . 114 CONTENTS viii 5.4.4 Compressed MOT and Molasses . 117 5.4.5 Optical Pumping . 119 5.4.6 Calibration of Magnetic Trapping Parameters . 120 6 Loss Measurements 124 6.1 Inelastic Feshbach Resonances . 125 6.1.1 Comparison of Theoretical and Experimental Data . 125 6.2 Feshbach Resonances in the F = 3; mF = ¡3 State . 126 6.2.1 Determination of the Two-Body Inelastic Collision Rate Co- efficient . 126 6.2.2 Experimental Results . 127 6.3 Temperature Dependence of the 118.5 G Resonance . 130 6.3.1 Experimental Results . 130 6.3.2 Theoretical Results . 133 6.3.3 Experimental and Theoretical Results . 135 6.3.4 Discussion . 139 7 Evaporation Results 141 7.1 Initial Evaporative Cooling Attempt . 142 7.1.1 Experimental Procedure . 142 7.1.2 Results . 144 7.2 Two-Frequency Evaporation . 148 7.2.1 Experimental Procedure and Results . 149 7.2.2 Discussion . 150 7.2.3 Heating . 153 7.3 Very Weak Trap . 155 7.3.1 Experimental Procedure . 155 7.3.2 Data Analysis . 157 7.3.3 Results . 159 7.3.4 Discussion . 161 CONTENTS ix 8 Conclusion 163 8.1 Outlook . 164 A Bose-Einstein Condensation in Caesium 167 A.1 Experimental Setup . 167 A.2 Evaporation . 168 A.3 Comparison . 171 B Experimental Equipment Details 173 B.1 Computer Control Boards . 173 B.2 Electronic Equipment Details . 174 C Caesium Atomic Data and Experimental Equations 175 C.1 Atomic Parameters . 175 C.2 Magnetic Field Calculations . ..
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