ABSTRACT Title of dissertation: A RING WITH A SPIN: SUPERFLUIDITY IN A TOROIDAL BOSE-EINSTEIN CONDENSATE Anand Krishnan Ramanathan, Doctor of Philosophy, 2011 Dissertation directed by: Steve Rolston Chemical Physics Program Superfluidity is a remarkable phenomenon. Superfluidity was initially charac- terized by flow without friction, first seen in liquid helium in 1938, and has been studied extensively since. Superfluidity is believed to be related to, but not iden- tical to Bose-Einstein condensation, a statistical mechanical phenomena predicted by Albert Einstein in 1924 based on the statistics of Satyendra Nath Bose, where bosonic atoms make a phase transition to form a Bose-Einstein condensate (BEC), a gas which has macroscopic occupation of a single quantum state. Developments in laser cooling of neutral atoms and the subsequent realization of Bose-Einstein condensates in ultracold gases have opened a new window into the study of superfluidity and its relation to Bose-Einstein condensation. In our atomic sodium BEC experiment, we studied superfluidity and dissipationless flow in an all-optical toroidal trap, constructed using the combination of a horizontal ?sheet?-like beam and vertical ?ring?-like beam, which, like a circuit loop, allows flow around the ring. On inducing a single quantum of circulation in the condensate, the smoothness and uniformity of the toroidal BEC enabled the sustaining of a persistent current lasting 40 seconds, limited by the lifetime of the BEC due to background gas pressure. This success set the stage for further experiments studying superfluidity. In a first set of experiments, we studied the stability of the persistent current by inserting a barrier in the flow path of the ring. The superflow stopped abruptly at a barrier strength such that the local flow velocity at the barrier exceeded a critical velocity, which supported decay via the creation of a vortex-antivortex pair. Our precise control in inducing and arresting superflow in the BEC is a first step toward studying other aspects of superfluidity, such as the effect of temperature and dimensionality. This thesis discusses these experiments and also details partial-transfer ab- sorption imaging, an imaging technique developed in the course of this work. A Ring with a Spin : Superfluidity in a toroidal Bose-Einstein condensate by Anand Krishnan Ramanathan Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2011 Advisory Committee: Professor Steven Rolston, Chair Doctor Gretchen Campbell Professor Kristian Helmerson Professor Wendell Hill Professor Christopher Lobb, Deans?s Representative Professor William Phillips c? Copyright by Anand Ramanathan 2011 To my Father, who has been a source of inspiration and strength throughout my education and my life so far. Acknowledgments I owe my gratitude to all the people who have made this thesis possible and because of whom my graduate experience has been one that I will cherish forever. The Laser Cooling and Trapping (LC) Group at NIST has been a fantastic place to work at, and I am indebted for the extremely friendly and helpful atmosphere, and all the people whom I have had a chance to interact with. First, I would like to thank my advisors Kris Helmerson and Gretchen Camp- bell. I feel very fortunate to have had the opportunity to interact and to learn from both of them. The support and encouragement shown by Kris was crucial during my first few months at NIST. Kris has always shown the readiness to help right from choosing the right conference to attend to setting things right in the lab. I have admired Kris for his diverse interests, from BEC physics to biophysics, and for the many collaborators he has around the world. Gretchen, who joined the group in my fourth year, has been immensely helpful during the later part of my PhD, and more than once, has prevented me from making a mistake that would have been detrimental to my career. She has always gone out of the way to make sure that I got the help and advice that I needed. She has been key in checking on the progress of my thesis and setting goals for completion. She has given prompt feedback on anything I have sent her. I?ve always admired how she has been on top of things, and follows up on even the smallest thing such as a question about Latex, despite her several commitments at NIST and UMD and her new Strontium lab. iii Bill Phillips has been the biggest inspiration through my PhD. His enthusiasm for physics has always been infectious, and kept me going when nothing seemed to work int the lab. He has asked some of the best questions, which have stimulated many thought-provoking discussions. Bill started our group?s Wednesday meetings, where we met with Chris Lobb and Wendell Hill and others from UMD. Although I?ve often felt that the discussions in such meetings led to nowhere, they fostered our best collaboration, without which a lot of this work would not have happened. Bill has the amazing ability to reach out to people from different backgrounds, and stimulate a discussion that brings about interesting physics. As a group leader, Bill has also been a role model in many ways. He is always approachable even when his schedule is busy. He takes time to give some carefully thought out advice. He is encouraging of everyone. I feel very fortunate to have discussed physics with Chris Lobb. I started reading about superconductivity during my third year under his guidance, and en- joyed our discussions on the subject. He always found time to guide me through Tinkham?s book, despite his hectic schedule, and was very patient through my slow learning of the subject. The little I learned from that time has gone a long way in giving me a different perspective on the physics we have seen. I have also enjoyed the many discussions I have had with Chris at his office. He has been remarkable in the number of wonderful insights he has come up with. Despite that, he has always been keenly aware of any small contribution by someone else and has been very generous in giving credit for it. I have also benefited from Wendell Hill and Steve Rolston. Wendell has always iv kept in touch with what I am doing, and has given me useful advice along the way. Steve has had an open door for discussing physics or my thesis and has provided me invaluable administrative support to ensure that I have had no problems in working at NIST, which is off-campus from UMD. During my time at NIST, I have had the fortune of working with several ex- tremely talented post-docs, Mikkel Andersen, Changhyun Ryu, Pierre Clad?e, S?ergio Muniz and Kevin Wright, who have helped me along the way. Mikkel got me started into the lab, helping me with my initial experiment. Chang helped me greatly in understanding the complicated BEC apparatus. He also shared with me, what he had learned about the field, to try and give me a broader perspective. Pierre got me started on analyzing data, which has stood me good right through my PhD. I cannot forget our discussions on BKT physics, which got me enthusiastic about the physics, and along with that, integrated into the science of the experiment. I am grateful to Pierre and Chang into taking care to see that I had a defined PhD project before they left. I have worked the longest with S?ergio, and we have been through many battles together. S?ergio is one of the warmest and most helpful people I?ve known. He has taught me a lot of electronics, and an amazing array of other miscellaneous lab- related things. A lot of my skills as an experimentalist were built with him looking over me. I probably would not be writing this thesis at this point if not for Kevin. Kevin gave the lab its critical mass, and along with S?ergio, we had an extremely productive year, which brought out the best in all of us. From Kevin, just when I v thought I knew everything that was there in the lab, I ever so often learned how something could be done better. I have admired his meticulousness and tidiness in working. His knowledge of optics, and his ability to explain it is phenomenal. I have also learned from him, how to be generous with credit. Apart from post-docs, our lab had several visitors who helped along the way. Of them, I learned a lot from fellow graduate students Russ Anderson and Martin Zelan, who showed me that a lot could be done even in a short visit. I also had the fortune of working with Vasant Natarajan, whom I had known since my undergrad- uate days. He has provided excellent mentorship right through, and has played an important role in shaping my career. There are many other people in the LC Group who have helped me along the way. Most notable is the other sodium BEC lab, with Paul Lett, Lincoln Turner, Yingmei Liu, Jianing Han and Jonathan Wrubel. Rani Kishore of the Tweezer lab has also helped me in many ways. I greatly cherish sharing my office with Brooke Hester (previously Cranswick), who provided great company through the years. She was quick to tell me about any opportunity that could be of use to me, and got me involved with the OSA and other activities at UMD, which broadened my horizon beyond the confines of my research. She always there when I had a bad day or when I was worried about something. I can never forget the administrative help I have received from Gail Griffin- Ferris and Kim Emswiler. They have been extremely prompt with any thing that I requested for or any help that I needed. Gail has, more than once, gone out of her way to cover up for my being late on a poster printing form or a purchase order. vi The daily good mornings passing by their office have brightened up my days at the LC group. At NIST, there have been other wonderful people I have had the opportunity to interact with, most notably Charles Clark and Ludwig Mathey. Charles was instrumental in my coming to NIST, and has helped me throughout my PhD. The discussions I have had with Ludwig have been fascinating, with his remarkable ability to bridge the gap between theorists and experimentalists. At UMD, Mike Coplan has been a mentor right through my PhD. As program director of the Chemical Physics program, he has always kept in touch with me, and been there to provide guidance in the PhD process when I have been completely lost. I am also indebted to Debbie Jenkins, who has handled the administrative side flawlessly through everything, from my Chemical Physics office space to my qualifier exam to my PhD defense date. Paulina Rirksopa of Physics was always there for help regarding my RA. I can never forget her smiling face. The PhD process would be a lot duller if not for fellow grad students. I have enjoyed the company of Anzi Hu, Jeremy Clark, Roger Brown, Sam Brewer, Mingdong Li, Kai Li, Suri Vaikuntanathan, Andy Ballard, Baladitya Suri, Sandeep Somani, Debjani Roy, Abigail Perry and a lot of the other JQI grad students. Some people are placed in the right place and the right time. My short associ- ation with Sameer Rahatekar came when I most needed it, and thanks to his help, I was able to navigate through a crisis in my PhD. Sukumar Rajauria, who joined NIST during my fourth year, has guided me through the later part of my PhD. I have gone to him several times when I was unable to figure out what a PhD meant vii or what was expected of me. He has always been clear-sighted through all this and has been someone whose advice I could blindly trust when I was most confused. Lastly, my family has been extremely supportive through my PhD. My wife, Shilpa Viswanathan, has stood by me despite all the delays in the completion of my PhD, and all the mistakes that I made. She has helped me through my frustrations in writing proposals and seeing them being rejected, until something finally came through. She always took care to do what was best for my dissertation, even if it meant us spending less time together. I can?t thank her enough for all that she has done. I am indebted to my parents for all that they have done for me. My father, R K Ramanathan, has kept a watchful eye on my PhD, often without my realizing it. He has been far-sighted in so many things, and has taught me to look at the bigger picture, and not be fooled by what I see around me. At all the critical junctures, when I have felt my worst, he has been the pillar of strength to help me get through. He has done his best, despite being more than 10,000 km away, in making sure that despite the many bumps, my PhD went in the right direction, and to him I dedicate my thesis. It is impossible to remember all, and I apologize to those I?ve inadvertently left out. viii Table of Contents List of Tables xiii List of Figures xiv List of Abbreviations xvi 1 Introduction 1 1.1 Liquid helium and the ?old? superfluidity . . . . . . . . . . . . . . . . 2 1.1.a Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.b Landau theory: quantitative description of superfluidity . . . . 4 1.1.c Rotation and vortices in helium II . . . . . . . . . . . . . . . . 6 1.2 Ultracold gases and superfluidity . . . . . . . . . . . . . . . . . . . . 7 1.3 In this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Theory of superfluidity and Bose-Einstein condensates 12 2.1 Bose-Einstein condensation in an ideal Bose gas . . . . . . . . . . . . 12 2.2 The weakly interacting Bose gas . . . . . . . . . . . . . . . . . . . . . 15 2.2.a Atom-atom interactions . . . . . . . . . . . . . . . . . . . . . 16 2.2.b The ground state . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.c Low-energy excitation spectrum . . . . . . . . . . . . . . . . . 19 2.3 Bose gas in an inhomogeneous system: The GP equation . . . . . . . 22 2.4 The Landau critical velocity . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Irrotational flow and quantized vortices . . . . . . . . . . . . . . . . . 26 2.5.a Energy of single vortex . . . . . . . . . . . . . . . . . . . . . . 29 2.5.b Decay of flow by vortex excitations . . . . . . . . . . . . . . . 31 3 Sodium Bose-Einstein condensate apparatus 33 3.1 Atomic physics and the two-level atom . . . . . . . . . . . . . . . . . 33 3.1.a Resonant interaction and Rabi flopping . . . . . . . . . . . . . 36 3.1.b Far off-resonant interaction and energy level shifts . . . . . . . 38 3.1.c Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.d Light forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Atomic structure in real world atoms . . . . . . . . . . . . . . . . . . 44 3.3 The Sodium atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Creating a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.a Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.b Oven and Zeeman slower . . . . . . . . . . . . . . . . . . . . . 51 3.4.c Magneto-optical trap (MOT) . . . . . . . . . . . . . . . . . . 51 3.4.d Magnetic trapping and evaporation . . . . . . . . . . . . . . . 52 3.5 Laguerre-Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.6 The optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.a The sheet trap . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6.b The ring trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ix 3.7 The microwave system . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.8 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Partial-transfer absorption imaging 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.a Limit of Low Intensity . . . . . . . . . . . . . . . . . . . . . . 69 4.2.b Effect of saturation . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Imaging with PTAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.a Requirements for implementing PTAI . . . . . . . . . . . . . . 71 4.3.b Using PTAI with alkali atoms . . . . . . . . . . . . . . . . . . 72 4.4 Practical examples using PTAI . . . . . . . . . . . . . . . . . . . . . 73 4.4.a In situ imaging of an optically thick BEC . . . . . . . . . . . 74 4.4.b Eliminating trap drift . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.c Accounting for number fluctuations . . . . . . . . . . . . . . . 75 4.5 Uncertainty of measurement . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.a Setting up the formalism . . . . . . . . . . . . . . . . . . . . . 78 4.5.b Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5.c Uncertainty in measured optical depth . . . . . . . . . . . . . 81 4.5.d Comparison with phase-contrast imaging . . . . . . . . . . . . 83 4.5.e Scattering of many photons per atom . . . . . . . . . . . . . . 86 4.6 Correcting for optical pumping . . . . . . . . . . . . . . . . . . . . . 86 4.6.a Solving for integraltextzfzi dzI(z,?)?(z,?) . . . . . . . . . . . . . . . . . . 88 4.6.b Obtaining the final expression . . . . . . . . . . . . . . . . . . 88 4.6.c Analytical solutions in special cases . . . . . . . . . . . . . . . 89 4.7 Practical considerations in using PTAI . . . . . . . . . . . . . . . . . 90 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5 The ring trap 94 5.1 Applications of a toroidal potential . . . . . . . . . . . . . . . . . . . 94 5.1.a Atom interferometry . . . . . . . . . . . . . . . . . . . . . . . 95 5.1.b Quasi-1D physics . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Previous ring trap experiments . . . . . . . . . . . . . . . . . . . . . 97 5.3 The ring trap potential . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3.a The trap depth . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.b The trap frequencies . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.c Vertical confinement . . . . . . . . . . . . . . . . . . . . . . . 105 5.3.d Sample calculation of trapping parameters . . . . . . . . . . . 106 5.4 Condensate wavefunction in the ring trap . . . . . . . . . . . . . . . . 106 5.4.a Harmonic oscillator ground state regime . . . . . . . . . . . . 109 5.4.b Radial Thomas-Fermi regime . . . . . . . . . . . . . . . . . . 109 5.4.c Fully Thomas-Fermi regime . . . . . . . . . . . . . . . . . . . 110 5.4.d Sample BEC parameters for our trap . . . . . . . . . . . . . . 112 5.5 Characterizing the ring BEC from time-of-flight expansion . . . . . . 113 x 5.5.a The thermal cloud . . . . . . . . . . . . . . . . . . . . . . . . 115 5.5.b The condensate . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.6 Azimuthal smoothness of the ring . . . . . . . . . . . . . . . . . . . . 118 5.7 Sloshing and breathing oscillations . . . . . . . . . . . . . . . . . . . 122 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6 Persistent currents in a BEC 126 6.1 Flow in a toroidal BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 Transferring orbital angular momentum to atoms . . . . . . . . . . . 128 6.3 Transferring orbital angular momentum using a LG beam . . . . . . . 130 6.4 Changes from the previous experiment . . . . . . . . . . . . . . . . . 132 6.5 Practical considerations for the transfer of circulation . . . . . . . . . 134 6.5.a Single photon scattering . . . . . . . . . . . . . . . . . . . . . 135 6.5.b Dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.5.c Cleaning up the remnant . . . . . . . . . . . . . . . . . . . . . 137 6.5.d Our optimized scheme . . . . . . . . . . . . . . . . . . . . . . 138 6.6 Detecting the circulation . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.7 Lifetime of persistent currents . . . . . . . . . . . . . . . . . . . . . . 144 6.8 Multiply-charged circulation . . . . . . . . . . . . . . . . . . . . . . . 145 6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7 Superfluid flow in the presence of a barrier 149 7.1 Superfluid flow in a toroidal condensate . . . . . . . . . . . . . . . . . 151 7.1.a Effect of a barrier . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.1.b Stability of flow in a ring . . . . . . . . . . . . . . . . . . . . . 153 7.1.c Phase slips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.2 Dimensionality and quasi-2D physics . . . . . . . . . . . . . . . . . . 157 7.2.a Superfluidity and the BKT transition . . . . . . . . . . . . . . 158 7.2.b Case of a quasi-2D Bose gas . . . . . . . . . . . . . . . . . . . 161 7.2.c Previous experiments studying the BKT transition . . . . . . 163 7.2.d NIST experiments in the quasi-2D trap . . . . . . . . . . . . . 165 7.2.e Effects of BKT physics and quasi-2D dimensionality . . . . . . 168 7.3 Experimental setup and procedure . . . . . . . . . . . . . . . . . . . 170 7.4 Observation of the breakdown of flow . . . . . . . . . . . . . . . . . . 173 7.5 Flow velocity of a circulating ring . . . . . . . . . . . . . . . . . . . . 176 7.6 The critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.7 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . 181 Appendices 183 A Analyzing PTAI images 184 A.1 Calculating the uncertainty of measurement using PTAI . . . . . . . 184 A.2 Calculating the uncertainty in measurement using PCI . . . . . . . . 185 A.2.a Photon shot noise . . . . . . . . . . . . . . . . . . . . . . . . . 186 A.2.b Uncertainty in the measured optical depth . . . . . . . . . . . 187 xi A.3 Change in transmitted intensity due to optical pumping . . . . . . . . 188 B Fully Thomas-Fermi condensate in the ring trap 190 C Calibrations for critical velocity 193 C.1 Time calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 C.2 Length calibration and pixel size . . . . . . . . . . . . . . . . . . . . . 193 C.2.a Calibrating the horizontal imaging system using gravity . . . . 193 C.2.b Calibrating the vertical imaging system by matching cloud size195 C.3 Measuring the trap frequencies . . . . . . . . . . . . . . . . . . . . . . 195 C.3.a Vertical trapping frequency . . . . . . . . . . . . . . . . . . . 196 C.3.b Radial trapping frequency . . . . . . . . . . . . . . . . . . . . 196 C.4 Measuring imaging parameters . . . . . . . . . . . . . . . . . . . . . . 197 C.4.a Measuring saturation intensity . . . . . . . . . . . . . . . . . . 198 C.4.b Calibrating optical pumping for PTAI images . . . . . . . . . 198 C.4.c Measuring the scattering cross-section . . . . . . . . . . . . . 200 C.5 Calibrating the mean field . . . . . . . . . . . . . . . . . . . . . . . . 201 C.6 Using ?0 to calibrate absorption scattering cross-section . . . . . . . . 204 C.7 Calibrating the effect of the barrier . . . . . . . . . . . . . . . . . . . 205 Bibliography 211 xii List of Tables 5.1 Ring trap parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Ring BEC relations for the simplified treatments . . . . . . . . . . . . 111 5.3 Ring BEC parameter values for the 3 simplified treatments . . . . . . 113 7.1 Phases of a quasi-2D Bose gas . . . . . . . . . . . . . . . . . . . . . . 163 xiii List of Figures 1.1 Observation of superflow in helium . . . . . . . . . . . . . . . . . . . 3 1.2 Vortex lattices in a BEC . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Solution to GP equation in certain limits . . . . . . . . . . . . . . . . 24 2.2 Multiply connected geometry . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Quantized vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Flow decay via formation of vortex ring . . . . . . . . . . . . . . . . . 31 3.1 Two-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Resonant Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Light shift on a two-level atom . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Light force on a two-level atom . . . . . . . . . . . . . . . . . . . . . 42 3.5 Velocity sensitive light forces . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Sodium D2 transition hyperfine structure . . . . . . . . . . . . . . . . 47 3.7 Sodium ground state Zeeman shift . . . . . . . . . . . . . . . . . . . 48 3.8 Schematic and photos of the BEC setup . . . . . . . . . . . . . . . . 50 3.9 Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.10 Magnetic quadrupole trap . . . . . . . . . . . . . . . . . . . . . . . . 53 3.11 Laguerre-Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.12 Schematic of the optical system for the sheet trapping beam . . . . . 58 3.13 Arbitrary shaped BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.14 Schematic of the optical system for the ring trapping beam . . . . . . 59 3.15 Schematic of the microwave system . . . . . . . . . . . . . . . . . . . 61 3.16 Microwave Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.17 Schematic of imaging system . . . . . . . . . . . . . . . . . . . . . . . 63 4.1 PTAI energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Imaging an optically thick cloud . . . . . . . . . . . . . . . . . . . . . 75 4.3 Eliminating trap drift . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 Using PTAI to lower number fluctuations . . . . . . . . . . . . . . . . 77 4.5 Image analysis setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6 Comparing PTAI with PCI . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Lower sensitivity of PTAI to imaging losses . . . . . . . . . . . . . . . 85 4.8 Effect of optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1 Basic scheme of an atom interferometer . . . . . . . . . . . . . . . . . 95 5.2 Sagnac interferometer using toroidal confinement . . . . . . . . . . . 96 5.3 Previous ring confinement schemes . . . . . . . . . . . . . . . . . . . 100 5.4 Schematic of ring trap . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.5 LG10 beam profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.6 Time-of-flight expansion of the ring . . . . . . . . . . . . . . . . . . . 114 5.7 Obtaining temperature and mean field . . . . . . . . . . . . . . . . . 117 5.8 Effects of azimuthal bumpiness . . . . . . . . . . . . . . . . . . . . . 120 xiv 5.9 Azimuthal density profile of a ring BEC . . . . . . . . . . . . . . . . 121 5.10 Radial sloshing oscillations in a ring BEC . . . . . . . . . . . . . . . . 123 5.11 Radial breathing oscillations in a ring BEC . . . . . . . . . . . . . . . 124 6.1 OAM transfer via counterpropagating beams . . . . . . . . . . . . . . 132 6.2 OAM transfer scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3 Plugged TOP trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.4 OAM transfer process . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.5 OAM transfer efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.6 Time-of-flight sequence after relaxing radial confinement . . . . . . . 142 6.7 Distinguishing between circulation and no circulation . . . . . . . . . 144 6.8 40 second persistent current . . . . . . . . . . . . . . . . . . . . . . . 145 6.9 Multiply-charged circulation . . . . . . . . . . . . . . . . . . . . . . . 146 7.1 Effect of a barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.2 Flow stability and phase slips . . . . . . . . . . . . . . . . . . . . . . 155 7.3 Phase profile of a 2D superfluid . . . . . . . . . . . . . . . . . . . . . 159 7.4 Phases of a quasi-2D Bose gas . . . . . . . . . . . . . . . . . . . . . . 162 7.5 Excitations in a 2D BEC . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.6 Intentionally created excitations in a 2D BEC . . . . . . . . . . . . . 166 7.7 Critical density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.8 Schematic of barrier beam . . . . . . . . . . . . . . . . . . . . . . . . 171 7.9 Experimental sequence for studying critical flow . . . . . . . . . . . . 172 7.10 Flow survival in the presence of a barrier . . . . . . . . . . . . . . . . 174 7.11 Critical chemical potential . . . . . . . . . . . . . . . . . . . . . . . . 175 7.12 Azimuthal flow profile . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.13 Critical velocity comparison . . . . . . . . . . . . . . . . . . . . . . . 180 C.1 Calibrating length scale using gravity . . . . . . . . . . . . . . . . . . 194 C.2 Matching the line profiles from the two cameras . . . . . . . . . . . . 195 C.3 Calibrating the saturation intensity . . . . . . . . . . . . . . . . . . . 197 C.4 Calibrating the optical pumping . . . . . . . . . . . . . . . . . . . . . 199 C.5 Verifying the optical pumping calibration . . . . . . . . . . . . . . . 200 C.6 Comparing the uncorrected chemical potential . . . . . . . . . . . . . 204 C.7 Correcting for resolution and smearing . . . . . . . . . . . . . . . . . 206 C.8 Final corrected and calibrated chemical potential . . . . . . . . . . . 207 C.9 Azimuthal mean field profile of the cloud . . . . . . . . . . . . . . . . 208 C.10 Calibration of barrier beam . . . . . . . . . . . . . . . . . . . . . . . 210 xv List of Abbreviations and important Symbols BEC Bose-Einstein Condensate BKT Berezinskii-Kosterlitz-Thouless GL Ginzburg-Landau JQI Joint Quantum Institute MOT Magneto-Optical Trap NIST National Institute of Standards and Technology LG Laguerre-Gaussian OD Optical Depth PCI Phase-Contrast Imaging PTAI Partial-Transfer Absorption Imaging rf radio frequency SQUID Superconducting QUantum Interference Device S/N Signal-to-Noise ratio TF Thomas-Fermi TOF time-of-flight TOP Time-averaged Orbiting Potential UMD University of Maryland a s-wave scattering length of a Bose gas ? optical depth c speed of sound in BEC/superfluid or speed of light E Energy or electric field ? phase of a wave or wavefunction g interaction constant, 4?planckover2pi12a/m ? natural linewidth of a transition ? detuning of a laser beam from an atomic transition in Hz ?? ? normalized by ?/2 h Planck?s constant 6.63?10?34 Js planckover2pi1 h/2? I light intensity Isat saturation intensity for an optical transition k wave vector of light or matter wave kB Boltzmann?s constant 1.38?10?23 J/K log natural logarithm (base e, not 10) m mass of bosonic atom ? mean field ?0 chemical potential n3D 3D density of a BEC p momentum ? scattering cross-section v velocity V trapping potential ? Rabi frequency xvi Chapter 1 Introduction The theory of quantum mechanics, which developed in the early 20th cen- tury, brought in several new concepts that changed physics. Several new fields in physics, such as atomic physics, nuclear physics, low-temperature physics and quan- tum electrodynamics were all born during that time. Quantum mechanics has been successfully applied to nearly every field of physics, and is an integral part of a physi- cist?s education today. On the practical side, quantum mechanics has contributed to many technological advances such as the transistor, the laser, nuclear power and MRI machines. The liquefaction of helium at 4 K by Kamerlingh Onnes at the University of Leiden led to the field of low-temperature physics, which led to the observation of superconductivity and superfluidity. Both phenomena were demonstrated by con- ceptually simple experiments that could not be explained by any classical theory. They have played a significant role in the development of some of the formalism for quantum mechanics related to condensed matter physics. Nevertheless, even a century later, many experts feel that phenomena like the origins of high tempera- ture superconductivity [1] or the basic nature of superfluidity [2] are still not fully understood. The observation of Bose-Einstein condensates (BECs) in alkali gases in 1995 opened a new window into the study of low-temperature physics. Studying superfluidity in BECs is now an active area of research with interesting possibilities, such as using BECs for navigation as detectors of rotation. This thesis is primarily about Bose-Einstein condensates and superfluidity. However, since superfluidity and superconductivity are related, I will mention, wher- ever applicable, superconducting analogies to phenomena seen in Bose condensates. 1 1.1 Liquid helium and the ?old? superfluidity Prior to the creation of atomic BECs in 1995, superfluidity was associated exclusively with helium. Almost all theory and modeling of superfluids until then was based solely on liquid helium, despite the fact that the strong interaction in liquid helium makes the theory much more complicated. Given that, it is quite remarkable that the theory could not only model liquid helium well, but also be directly applied to BECs. In this section I will discuss the developments in the field of superfluidity in liquid helium. 1.1.a Origins The concept of superflow, or dissipationless flow, first arose in the context of superconductivity in 1911, when Onnes [3] found that the resistance of solid mercury went to zero below 4 K. While it was known that resistance decreased with temperature, the sudden drop to zero at non-zero temperature was unexpected. It was reasoned that if the resistance of a substance were truly zero, any current set in it would last forever, leading to a concept of a persistent current. Careful experiments in the many decades since, have shown, with increasing precision, that persistent currents can, in principle, last longer than the age of the universe [4, 5]. While superfluidity in liquid helium was not observed until 1938, some im- portant developments in bosonic statistics occurred earlier. In 1924, Satyendranath Bose [6] explained Max Planck?s formula for black body radiation based on simple combinatorics. Einstein [7, 8] applied Bose statistics to material particles, and pre- dicted a special condensation to occur at low temperatures. The condensation did not require interactions, and was characterized by a macroscopic occupation of the single-particle ground state. After the development of wave mechanics, the basis of Bose-Einstein statistics could be understood in terms of a wavefunction symmetry under exchange. Although Bose-Einstein condensation itself was not realized until much later, the extensive theoretical work in the first half of the 20th century set up the framework for understanding bosonic particles. 2 glass disks change in height narrow channel for He flow Figure 1.1: Observation of superflow in helium (from Kapitza [9]): By observing the rate of flow of liquid helium from the inner column to the outer reservoir via the narrow gap (about 0.5 micron) between large (3 cm diameter) glass disks, Kapitza found that the viscosity of helium dropped by a factor of at least 1500 as the temperature dropped below the ?-point. The rate of flow was determined by the change in height in the inner column as a function of time. In the years following the liquefaction of helium in 1908, strange properties of helium below 2.2 K (known as the ?-point) were noted, such as an abrupt change in specific heat and the sudden ability of the liquid to pass through small leaks. In the second issue of the journal Nature in January 1938, two groups showed evidence of superflow in helium II (helium below the ?-point). Kapitza [9] measured the rate of flow of helium through a narrow gap between two large disks (schematic shown in figure 1.1) and found that the viscosity was lower by a factor of at least 1500 below the ?-point. As an analogy to superconductivity, Kapitza termed helium below the ?-point as ?superfluid?. Simultaneously, Allen and Misener [10] measured the rate of flow of liquid helium through narrow capillaries, and found that the viscosity of helium II was below 10?9 Pa s, consistent with the measurements of Kapitza. In April 1938, London [11] came up with a qualitative explanation for the phenomena of superfluidity, relating it to Bose-Einstein condensation. Since 4He has an even number of protons(2), neutrons(2) and electrons(2), it would come under the class of particles which obey Bose-Einstein statistics. Using a simple ideal gas 3 model for helium (he states ?...model which is so far away from reality...?), London obtained a transition temperature of 3.09 K, which was reasonably close to the ?- point. Shortly after, Tisza [12] extended London?s idea and described the superfluid as having two components, a ?condensed? fraction which is responsible for the zero viscosity and the remaining excited state fraction which can undergo dissipation. Tisza?s formalism helped understand superfluidity in a framework based on Bose- Einstein condensation. 1.1.b Landau theory: quantitative description of superfluidity While London and Tisza had an intuitive and qualitative explanation for su- perfluidity, there was no quantitative model with experimentally testable predictions until Landau. In a landmark paper, Landau [13] came up a with a quantitative de- scription of superfluidity that introduced several key concepts, which still remain the basis for understanding superfluidity. The theory could explain the transition temperature and flow without viscosity. While Landau rejected Tisza?s idea of Bose-Einstein condensation, he nevertheless used a two-fluid model, consisting of a ground state ?superfluid? and an excited state ?normal? fluid. He clarified that the two-fluid model was only a convenient way of expressing the formalism, and that there were no separate ground state and excited state atoms. The excited state spectrum consisted of phonons (analogous to the lattice excitations in a solid) and rotons1 (Landau referred to these as the vortex spectrum), and the occupation level of these states depended on the temperature. Landau theory also predicted second sound, which consists of waves where the superfluid and normal fluid oscillate out of phase. Second sound was observed by Peshkov [14]. At this point, it is relevant to point out that a similar formalism was used to de- scribe superconductivity. The Ginzburg-Landau (GL) theory [15] used the concept of a superconducting ground state with an excitation spectrum to treat the prob- lem of superconductivity. Despite the fact that superconductivity and superfluidity 1See section 2.4 for a brief discussion on rotons. Landau had assumed that rotons came from the vortex spectrum. However, this is not believed to be true anymore. 4 involved fundamentally different particles (the former involved electrons, which are fermions, and the latter involved bosons), Landau?s generalized formalism based on the second-order phase transition was successful in quantitatively describing both phenomena. While Landau?s theory was largely successful in many ways, it still lacked a microscopic explanation. Bogoliubov [16] bridged some of the gap between Lan- dau?s model and Bose-Einstein condensation. By considering the case of a weakly interacting BEC, he showed that the excitations needed to be transformed into col- lective modes. The excitation spectra of these modes had a linear dispersion relation similar to that of the phonon modes. It is unfortunate that helium was the only su- perfluid available back then. While the Bogoliubov approach was not very popular with helium II, it has been hugely successful with ultracold gas BECs and accurately models the excitation spectrum and dynamics. Still, even in the context of liquid helium, Bogoliubov made the connection between Landau?s two-fluid approach to a microscopic theory of Bose-Einstein condensation. Further research on the connection between helium II and Bose-Einstein con- densation [17] made the distinction between the BEC fraction and superfluid frac- tion. The former refers to the fraction of atoms in a single quantum state. Penrose and Onsager [17] estimated that, at absolute zero, ? 8% of the atoms were Bose condensed, while according to the Landau theory, at absolute zero, the superfluid fraction should approach unity. Neutron scattering experiments [18] have more or less confirmed the predicted (the ?10%) Bose condensed fraction in nearly zero temperature liquid helium (see Griffin [19] for an extensive discussion). Before we move on, there is one final point of Landau?s theory relevant to our experiment. The theory predicts the existence of a critical velocity that de- pends on the energy-momentum dispersion of the low energy excitations, above which superfluid flow can dissipate into other excitations. However, experiments of that time showed that superfluid critical velocity was about 2 orders of magnitude less than what one would expect from the superfluid phonon and roton excitation spectrum [20]. The discrepancy was explained later in terms of quantized vortices, 5 which could also contribute to dissipation. Nevertheless, the Landau theory set the framework for understanding superfluidity, upon which further developments could take place. 1.1.c Rotation and vortices in helium II Landau stated that the superfluid fraction was irrotational (??v = 0, where v is the velocity field), and that only the normal component would move if helium II were placed in a rotating vessel. However, experiments at the time indicated that the superfluid was not irrotational [21, 22]. Onsager [23] and Feynman [24] made an important addition to the superfluid theory, introducing the notion of quantized vortex lines2 in a superfluid, which could allow the superfluid to rotate. Since the su- perfluid density is zero at the vortex line, the geometry becomes multiply connected and the irrotationality condition of Landau?s theory is not violated. Feynman also suggested that a vortex line closed into itself (vortex-ring) of appropriate size could be the roton excitation of the Landau theory. Evidence of quantized vortex lines in liquid helium was seen in various ways. Vinen [25, 26] studied the oscillation modes of a fine wire in rotating helium super- fluid, and observed a stable single quantum of circulation around the wire. Steyert et al. [27] used hydrogen-deuterium mixture ?snow flakes? (H-D mixture adjusted to give particles of the same density as He) to track the rotation of helium, and saw evidence of quantized flow. Other experiments [28?30] have studied quantized vortices using charged ions trapped in the vortex core. Of them, Yarmchuk and Packard [30] obtained a lattice pattern for the arrangement of quantized vortices that matched theoretical predictions. Another consequence of quantized vortex lines [24] is a modification of the superfluid excited state spectrum, particularly in finite geometries. For flow in a cylindrical channel, dissipation occurs at lower flow velocities via a vortex-ring 2While the standard definition of a vortex line is something that is straight with circulation around it, vortex lines may be curved. Vortex lines can either end at the superfluid boundary or can close upon themselves. 6 mechanism rather than by phonon or roton mechanisms. The reduced critical ve- locity is referred to as the Feynman critical velocity (see Wagner [31] or Varoquaux [32]). Evidence of quantized vortex-ring dissipation events has been observed in flow of helium II through a channel [33?35]. The critical flow velocities for such experiments were found to be consistent with estimates of the Feynman critical velocity [32]. At this point, I need to emphasize that this introduction covers only a tiny fraction of the literature on superfluid helium. There have been a lot of studies on various other aspects, such as the specific heat of helium II (discussed by Landau [13]), superfluidity in 3He [36, 37], the third sound [38], helium gyroscopes [39, 40], etc., which are beyond the scope of this thesis. We will now turn to the more recent history of superfluidity in atomic Bose gases. 1.2 Ultracold gases and superfluidity Advances in laser cooling of atoms in the 1980s opened up the field of ultracold gases and led to the observation of Bose-Einstein condensation in alkali gases [41?43] in 1995. Atomic Bose-Einstein condensates (BECs) are very different from liquid helium, typically involving 106 atoms or less, which amounts to ? 10?17 grams of atoms (densities < 1020 atoms/m?3), compared to the greaterorsimilar 100 grams (density ? 1028 atoms/m?3) for liquid helium. Ultracold gases also have weak interactions compared to the case of liquid helium, which has made observing Bose-Einstein condensation relatively easier. Some of the aspects of superfluidity such as quantized vortices have been clearly observed [44?47], but others such as dissipationless flow are more involved. Ironically, some of the first experiments showing superfluidity in liquid helium [9, 10] are nearly impossible to replicate in ultracold gases owing to constraints of the system. However, it is still possible to observe dissipationless flow in a BEC, which is an important part of this thesis. There are several advantages to using ultracold gases for studying superfluidity, compared to the superfluid helium experiments. 7 ? Pure ground state : The ability to attain a nearly pure BEC enables one to reach the ground state of the system with almost no thermal excitations or quantum depletion. This allows a better study of the condensed phase, with thermal excitations being introduced as and when needed. ? Arbitrary, dynamic potentials : With cold atoms, one can create almost arbi- trary potentials with light fields and magnetic gradients. This is particularly advantageous when trying to produce strong confinement for atoms. One can also create time varying potentials, both adiabatic and sudden, and so can conveniently switch between one potential and another. ? Direct imaging : Exploiting the strong interaction of ultracold gases with light, one can directly image the atom density. In addition, one can make interferometric measurements to obtain the coherence length and phase. ? Simpler system : Atomic BECs have low densities and atom-atom interactions can be completely determined by the s-wave scattering length, in contrast with helium, where both the attractive van der Waal?s interactions and the short range repulsion play a role. Hence, one can more exactly model atomic BECs from first principles. Atomic BECs are typically held in traps having dimensions of around 100 ?m. Creating flow in such a system is not easy, and observing superflow even more tricky. A first step in that direction is to look for vortices. While experiments at JILA [44] were the first to create a vortex in a BEC, experiments at ENS, Paris [45, 46] were the first to observe the hydrodynamic nature of superfluidity, which was done by stirring a BEC with a focused laser beam. For low stirring speeds, there were no excitations seen. As the speed was increased, vortices began to appear. The experiments tell us two things. First, the appearance of vortices in these experiments indicates hydrodynamic behavior as one would expect for a fluid (as opposed to a gas) and evidence of quantized vortices, which is an indication of superfluidity. Second, the lack of excitations at low speeds for the Paris experiment 8 Figure 1.2: Observation of vortex lattices (from Abo-Shaeer et al. [47]): The figure show density distributions of an image of a circular cloud. The density holes in the cloud are vortices. The images show a BEC with approximately (A) 16, (B) 32, (C) 80, and (D) 130 vortices. The triangular lattice pattern minimizes the energy of the rotating superfluid. is an indication of superflow, where there is dissipationless flow in the BEC around the laser beam. In further experiments in stirring a BEC, Abo-Shaeer et al. [47] observed the formation of a stable vortex lattice (see figure 1.2). These experiments made the connection between angular momentum in a BEC to magnetic field in a type-II superconductor. When the magnetic field exceeds a certain critical value, the mag- netic flux penetrates through singly quantized lines of flux with non-superconducting cores. Analogously, in a BEC, the rotation penetrates through quantized vortices of rotation flux, which have cores where the BEC density goes to zero. There have been other experiments indirectly looking for superflow by studying the effect of dragging an object (a focused laser beam, which serves as a repulsive potential) through a BEC [48?51]. In all these experiments, there was an onset of excitations only when the speed of the moving object crossed a certain threshold, indicating dissipationless flow for low speeds. This was taken further in experiments observing persistent currents. Ryu et al. [52] observed stable persistent flow in a multiply connected BEC. The flow was found to persist without decay for up to 10 s, limited only by experimental factors such as disortion of the trap geometry due to relative drift between trapping fields, and the BEC lifetime in the trap. Since there was only one quantum of flow, any dissipation would lead to a complete loss 9 of circulation. While 10 seconds of dissipationless flow may seem a short time for something that is supposed to be persistent, it needs to be seen in context of the timescales on which BEC dynamics (based on the kinetic and interaction energies) occur, which are of the order of tens of milliseconds. This experiment makes the connection back to the original superfluidity experiments of Kapitza [9] and Allen and Misener [10]. Having realized persistent flow, the question may arise: What is left to study in superfluidity of atomic BECs besides pushing the flow lifetime to longer timescales? While superfluidity may have been observed in BECs, there are still several unanswered questions. The observed critical velocities for the breakdown of super- flow have been found to be lower than simple predictions. The mechanisms for decay have not been fully understood. The role of temperature in the decay of superflow has not been fully explored. The relation of the dimensionality of the cloud to superfluidity opens several questions. While a three-dimensional (3D) Bose gas is expected to Bose condense with the BEC being superfluid, a two-dimensional (2D) Bose gas is expected to undergo a superfluid transition (but not BEC). Is it possible to observe superfluidity in a 2D Bose gas? In the case of a highly anisotropic trap, will reduced dimensional effects play a role? What kind of transition will a quasi-2D Bose gas undergo? Several of these questions boil down to a single fundamental question. What is the relation between Bose-Einstein condensation and superfluidity? Both phe- nomena involve Bose gases at low temperatures. However, superfluidity requires interaction, while BEC does not [7, 8]. While theoretical frameworks, such as the Bogoliubov formulation [16] do bridge some of the gap, it is nevertheless impor- tant for experiments to validate the theoretical predictions, and to probe interesting dimensional cross-overs which are not easily understood by theory. As an extension, there is also a need to better understand the connection between bosonic superfluidity (and BECs) and superconductivity. Superconductors have been used in several practical applications. Principles of superconductivity are used in making SQUIDs for sensitive detection of magnetic fields. Since magnetic 10 field in a superconductor is analogous to rotation in a BEC, building an analogous device using a BEC could make sensitive measurements of angular velocity. Hence, while superconductors may be fundamentally different from BECs, it is important to make connections wherever possible. 1.3 In this thesis In this thesis, I will present my research work at NIST. The work was done working with other researchers at NIST, Gaithersburg, during the period 2006 to 2010. In a first set of experiments [53], we studied the 2D superfluid transition in a sheet-like trap, which I will only briefly talk about. In a second set of experiments, we studied the breakdown of superflow [54] in a toroidal trap. During this time, we developed a new imaging technique [55] and an extensive set of tools for creating and manipulating a BEC in a toroidal trap. In chapter 2, I will give an introduction to the theory of BECs and superflu- idity. Chapter 3 gives an overview of the experimental setup and introduces the concepts of atomic physics that are crucial for the creation and manipulation of BECs. Chapter 4 deals with partial transfer absorption imaging, the new technique developed to image optically thick clouds, largely based on Ramanathan et al. [55]. In chapters 5 and 6, I will go over details about the ring trap, and persistent currents respectively. In chapter 7, I will discuss our experiments studying the breakdown of superflow, before concluding in chapter 8. 11 Chapter 2 Theory of superfluidity and Bose-Einstein condensates Historically, the study Bose-Einstein condensation was ruled by liquid helium, and was understood in terms of the Landau two-fluid model (see Landau and Lifshitz [56], Daunt and Smith [57], Khalatnikov [58] or Tilley and Tilley [59] for example). However, with the advent of atomic Bose-Einstein condensates (BECs), the phe- nomenon is now understood in terms of the weakly interacting Bose gas model, using the formalism first developed by Bogoliubov [16]. Since the formalism has a microscopic basis, derived by adding a perturbation to the non-interacting Bose gas, it is more useful as a starting point for understanding superfluidity in BECs than the original Landau theory. Nevertheless, it is important to remember that a lot of the theory for superfluidity and BECs, such as coherence, excitations, vortices and the critical velocity had been developed prior to atomic BECs, despite the fact that liquid helium was the only available superfluid. For a more complete review on superfluidity in liquid helium, see Khalatnikov [58] or Tilley and Tilley [59]. In this chapter, I will introduce the concepts of Bose-Einstein condensation and superfluidity relevant to this thesis, starting from the ideal Bose gas, and then moving to the weakly interacting gas. I will discuss the excitations of a BEC, rotation and the critical velocity of flow. The approach used is similar to that of Pitaevskii and Stringari [60]. 2.1 Bose-Einstein condensation in an ideal Bose gas Before jumping to Bose statistics, we first review the familiar expression for the occupancy of an energy level, ? of a classical gas, given by the Maxwell-Boltzmann statistics: n(?) = exp parenleftbigg??? kBT parenrightbigg , (2.1) 12 where ? is the chemical potential, kB is the Boltzmann constant and T is the tem- perature. At a given chemical potential and temperature, states at lower energies tend to have higher occupancies than states at higher energies. Looking at equation (2.1), we see that on raising the chemical potential, the occupancy of all states in- crease. On lowering the temperature, the occupancy of high energy states sharply decreases. This characterizes the general description of Maxwell-Boltzmann statis- tics. The reader should note that in practice (for example, in 3D space), the density of states at higher energies is larger, which leads to a peak in the density distribution at a non-zero energy (?kBT for a 3D gas with only translation energy). For bosons, the indistinguishability of particles has to be factored in when ob- taining the occupancy of a given energy level. The effect of indistinguishability has been explained in Cornell et al.[61] and illustrated beautifully by Stamper-Kurn [62]. At low occupancies (equivalently low phase-space density), where the probability of two particles being in the same state is nearly zero, the indistinguishability plays lit- tle role and one can assume Maxwell-Boltzmann statistics for the system. However, at higher occupancies (which tend to occur at low temperatures and high density), one needs to account for Bose-Einstein statistics, which leads to (see Pitaevskii and Stringari [60] for a derivation): n(?) = 1 exp parenleftBig ??? kBT parenrightBig ?1 . (2.2) As one can see, at a low chemical potential (for a given temperature) such that ??? kBT greaterorsimilar 4 (e ??? kBT greaterorsimilar 50), equation (2.2) can be approximated by equation (2.1), which corresponds to the case of low occupancy. The expression is valid for an ideal Bose gas only when ? < ?min, or equivalently when the chemical potential is less than the ground state energy. As the chemical potential approaches the ground state energy (???min), the occupancy of the ground state, n(?min) becomes large, which corresponds to a macroscopic occupation of the ground-state. For a derivation of Bose-Einstein condensation, one needs to include the den- sity of states, ??(?). The density of states determines the scaling of the number of particles (or density of particles in space) with ?. Although the effects of indis- 13 tinguishability do play a role at low temperatures (and ? approaching ?min), those effects may or may not translate to Bose-Einstein condition, depending on ??(?). A full discussion of this can be found in several standard textbooks (Pitaevskii and Stringari [60] for example). For this chapter, it suffices to say that for a gas in free space (homogeneous infinite system), only in 3D (or higher dimensions), Bose- Einstein condensation occurs at non-zero temperatures. To obtain the conditions for Bose-Einstein condensation, one can take a semi- classical approach (see Dalfovo et al. [63]) and look at the 3D phase-space density ??3T, where ? = N/V is the spatial density (number of particles divided by the volume ofthe sample) and?T =radicalbig2?planckover2pi12/mkBT is the thermal de Broglie wavelength, which defines a characteristic length scale. When ??3T ? 1, the phase-space density is low and the gas behaves like a classical gas. When ??3T greaterorsimilar 1 1, Bose-Einstein condensation occurs. This can be seen as the overlap of de Broglie waves of different particles to form a macroscopic wave. For Bose-Einstein condensation, the critical density as a function of temperature, from the expression of phase-space density, is ?? parenleftbiggmk BT 2?planckover2pi12 parenrightbigg3/2 . (2.3) In practice, rather than ?, it is easier to measure N, the total number of atoms, which can be directly measured by absorption or other imaging techniques. One can express the equation (2.3) in terms of the number of atoms to obtain the transition temperature at which atoms are expected to Bose condense [64]. For a 3D (harmonically) trapped Bose gas, the critical number is given by, Nc ? parenleftbiggk BT planckover2pi1?0 parenrightbigg3 , (2.4) where ?0 is the harmonic oscillator frequency of the trap (assuming it to be isotropic in 3D). As can be seen in equations (2.3) and (2.4), for a non-interacting Bose gas, the critical number of atoms (or density) for Bose-Einstein condensation has a direct relationship with the temperature. As the temperature increases, the critical number also increases. 1The exact value is a number that can obtained by a rigorous calculation involving the zeta function integral [56]. 14 The non-interacting ideal Bose gas model is very useful to illustrate the Bose- Einstein condensation process, obtain the temperature-number relation and estimate the transition temperature. However, it does not account for superfluidity and so has limited value. For the interested reader, there are several sources which contain a more detailed and rigorous derivation of the above relations (Dalfovo et al. [63], Pitaevskii and Stringari [60], Pethick and Smith [65]). While in theory a Bose condensate can be obtained at any temperature, there are some obvious practical considerations. For example, consider a room temper- ature gas of sodium, which is bosonic: The critical density for Bose condensation is ?c ? 1032 atoms/ m3 = 106 kg/m3, which is 3 orders of magnitude denser than sodium metal. Apart from the fact that such densities are impossible to attain at room temperature, a gas under such conditions would be solid. To get away from the problem of liquefaction or solidification, one needs to work with gases at low densities2. In our sodium BEC experiment, we work at temperatures around 40 nK, and densities of ? ? 1019 atoms/m3 > ?c ? 1017 atoms/m3. In terms of atom numbers, we obtain N ? 105 >Nc ? 100 (using ?0 ? 70 Hz). 2.2 The weakly interacting Bose gas The previous section described an ideal non-interacting Bose gas. Just to reiterate, Bose-Einstein condensation occurs as a consequence of indistinguishability and Bose-Einstein statistics, and occurs even in the case of a non-interacting gas. We will now deal with the practical case of an interacting Bose gas. Taking the general situation, interaction in gases can cause a variety of effects, such as condensation to liquids or solids or forming molecules. However, with ultra- cold gases, we work at low densities and ultra-high vacuum. From an experimental point of view, such conditions prevent any of those processes from taking place and 2In fact, even at BEC temperatures, the equilibrium state of sodium, or any other Bose gases is solid. The BEC is a metastable state. The low densities which we work with make the time constant for forming molecules much longer than experimental timescales and hence the Bose condensate behaves as an equilibrium state [61]. 15 allow atoms to be cooled to the nanokelvin BEC temperatures. From a theoretical point of view, such conditions greatly simplify the picture and allow us to character- ize the interaction of the cloud based on the atom-atom low-energy scattering. The experimental setup and techniques to reach condensation are discussed in chapter 3. We will now talk about the treatment of interactions. 2.2.a Atom-atom interactions Ultracold gas experiments are typically performed under ultra-high vacuum (10?14 atm for background gas), to ensure minimal collisions and consequently heat- ing from background gas. For the atomic species under considerations, we work at low densities, 5-10 orders of magnitude below a gas at STP (Standard Temperature and Pressure: 20 Celsius, 1 atmosphere). Under such conditions, simultaneous in- teraction between three or more atoms can be safely neglected, and the formation of molecules is unlikely. This leaves us with just the two-body interaction. The two-body interaction is normally characterized by the van der Waals potential (see Pethick and Smith [65] for a detailed description of how the potential plays a role), the long-range attraction that arises from the (induced) electric dipole-dipole inter- action, and the strong repulsive core due to the repulsion from the overlap of the electron clouds. However, given the low temperature of the sample, we can make a further simplifying assumption. Atoms scatter at low momenta planckover2pi1k (k is the mag- nitude of the wave-vector of the atom-atom scattering wavefunction), such that the kinetic energy is much below the angular momentum barrier of p-wave (scattering process that has one unit of angular momentum, l = 1) and higher modes (l> 1): planckover2pi12k2 2m ? planckover2pi12 2mr20 ? k ? 1 r0, (2.5) where m is the mass of the atom, and r0 is the range of the potential. For any scattering process that has l? 1, the interacting atoms do not get within the range of the inter-atomic potential. To give an estimate of a typical scenario, for sodium atoms at a temperature T=10 ?K, planckover2pi12k2/2m?kBT ? 10?5 ?planckover2pi12/2mr20 (r0 < 1 nm, or twenty Bohr radii [65]), and hence the interaction can be characterized by the 16 s-wave scattering length, which arises from the l = 0 scattering process. For atom-atom collision processes, it is impossible from theory alone to evalu- ate scattering properties of cold atoms because the atom-atom interaction potentials cannot be calculated with sufficient accuracy [65]. However, using photoassociation spectroscopy and the study of Feshbach resonances (see Weiner et al. [66]), infor- mation about the interaction potentials can be obtained, which could then be used to characterize the scattering process and make further experimental predictions. Hence, the scattering process is determined, based on experiments. In an elastic scattering process, there is an incoming wave and an outgoing wave (both with wave vector magnitude k). The phase-shift of the outgoing wave with respect to the incoming wave determines the strength of the scattering process. The phase-shift, ?0, in the long wavelenght limit of k, k ? 0, can be expressed as ?0 = ?ka, where a is the scattering length. Consequently, the scattering length can be expressed as a = ?0k vextendsinglevextendsingle vextendsinglevextendsingle k?0 . (2.6) Nearly all the properties of low-energy scattering can be obtained from the s-wave scattering length a. As for the nature of the interaction, a > 0 implies repulsive interaction, while a< 0 implies attractive interaction. The strength of the scattering process is given by the total scattering cross-section, 4?a2. The scattering cross-section is used in calculating the elastic collision rate, which is important to model evaporative cooling. The scattering length also plays a role in BECs, where the interaction energy term of the hamiltonian is proportional to a. 2.2.b The ground state The problem of the weakly-interacting Bose gaswas first treated by Bogoliubov [16], using a mean-field based approach, where the many-body wavefunction (or field operator) was expressed in terms of a single particle wavefunction basis. In this section and the next, we will discuss the ground state properties of a Bose gas and then solve for the low-excitation spectrum, based on the Bogoliubov theory of 17 a weakly-interacting Bose gas. In a uniform system, the density is constant in space and it is useful to go into a momentum basis, ?(p). Such a basis also separates the zero-momentum Bose- condensed ground state from excited states. The normalization condition gives integraltext ??(p)?(p)dp = N, where N is the number of particles. The energy of the system can be written as: E = ? ?H? = integraldisplay ??(p) p 2 2m?(p)dp + 12V integraldisplay dp1dp2??(p1)??(p2)?(p1 +q)?(p2 ?q)Vqdq, (2.7) where p1+q, p2?q are the incoming wave vectors, p1 and p2 are the outgoing wave vectors, Vq is the scattering angle-dependent time-averaged scattering potential, and V is the volume of the system. As discussed in the previous section, at low temperatures, only the spherically symmetric s-wave scattering plays a role, and Vq can be integrated over q to give 4?planckover2pi12a/m. The energy functional can then be simplified to E = integraldisplay ??(p) p 2 2m?(p)dp+ g 2V integraldisplay dp1dp2??(p1)??(p2)?(p1)?(p2), (2.8) where g = 4?planckover2pi12a/m. At T = 0, all the atoms are in the ground state, p = 0, giving |?(p)|2 = |?0|2 = N?(p). (2.9) The ?(p) expresses the fact that for a non-interacting Bose gas, the ground state is the zero momentum state. The ground state energy E0 is given by, E0 = N 2g 2V = Ng? 2 , (2.10) where ? = N/V. The ground state energy per particle E0/N is proportional to the particle density and the interaction strength. Using equation (2.10), we can now go ahead and calculate the sound speed, c, which is the speed of propagation of compression waves, for the system. The thermodynamic pressure of the weakly interacting Bose gas can be obtained using P = ??E0?V = g? 2 2 . (2.11) 18 The compressibility of the system can be expressed as ?? ?P = 1 g?. (2.12) The sound speed is given by the ratio of the square root of the elasticity, ??P?? , divided by the (mass) density m?, giving c = radicalbiggg? m. (2.13) In taking the square-root, we assume that g, and hence the s-wave scattering length (a) are positive, implying repulsive interaction3. 2.2.c Low-energy excitation spectrum The physics of a weakly interacting Bose gas depends strongly on the low- energy excitations. For example, the decay of superflow occurs via the transfer of energy to such excitations. The low-energy excitation spectrum is derived using the Bogoliubov-de Gennes method [67]. While this is generally worked out using the second-quantization operator notation [60], I will write it in terms of the wavefunc- tions. We begin with writing out the time-dependent Schr?odinger equation4 for the energy functional in equation (2.8), using iplanckover2pi1??/?t = ?E/??? (see Pitaevskii and Stringari [60] for more details) to get p2 2m?(p) +g??(p) = iplanckover2pi1 ??(p) ?t . (2.14) The above equation is valid only when ? is the order parameter. The or- der parameter characterizes the macroscopically occupied states. For a degenerate, weakly-interacting Bose gas in thermal equilibrium, the order parameter refers to the ground state wavefunction (for a non-interacting gas, this is identical to the sin- gle particle ground state). However, often the superfluid has been perturbed (and hence it is not in thermal equilibrium), in which case the order parameter need not be the ground state. 3In general, interactions cause an attractive BEC having a< 0 to collapse upon itself [63]. 4If written in a position basis, equation (2.14) would be a variant of the Gross-Pitaevskii equation, which is discussed in section 2.3. 19 We continue to use the momentum basis (will not explicitly show p depen- dence), and set a trial wavefunction, that consists of the ground state ?0, with a small additional perturbation ?? added to it: ? = ?0 +u?? +v???, (2.15) where u and v are complex numbers. This is known as the Bogoliubov transforma- tion, where ??? is the Hermitian conjugate of ??. Expressing ?0 explicitly in terms of the condensate number and phase, ?0(p) = radicalbig N0?(p)ei?, (2.16) we assume that the amplitude, the square-root of the condensate number, is time independent, while the phase, ?, can evolve in time. In the formalism we are using, the wavefunctions ?? and ??? are constant, and all the variation lies in the complex amplitudes u and v. We now substitute ? into equation (2.14) to obtain p2 2m( radicalbig N0?(p)ei? +u?? +v???) + g( radicalbig N0?(p)ei? +u?? +v???)2 ?( radicalbig N0?(p)e?i? +u???? +v???) = iplanckover2pi1 ??t( radicalbig N0?(p)ei? +u?? +v???). (2.17) We now make the assumption that the excitation,?? has a definite momentum, p?, and correspondingly, ??? has a momentum, ?p?. The ground state corresponds to a zero momentum eigenstate. Multiplying through, we keep terms to first order in u and v: p?2 2m(u? ? +v???) + g?0(radicalbigN0?(p)ei? + 2u?? + 2v??? +ei?u???? +ei?v???) = iplanckover2pi1 ??t( radicalbig N0?(p)ei? +u?? +v???). (2.18) We can now separate terms containing individual wavefunctions in order to solve for u, v and ?: p?2 2mu? ? +g?0(2u?? +e2i?v???) = iplanckover2pi1?u ?t? ?, (2.19) p?2 2mv? ?? +g?0(2v??? +e2i?u????) = iplanckover2pi1?v ?t? ??, (2.20) g?0 radicalbig N0ei? = iplanckover2pi1 radicalbig N0?e i? ?t . (2.21) 20 Solving for ? from equation (2.21), we obtain ? = ?g?0tplanckover2pi1 . (2.22) We now multiply e?i? through equations (2.19) and (2.20) and substitute, u = u?ei? and v = v?ei? to obtain the matrix differential equation ? ? p?2 2m +g?0 ?g?0 g?0 ?p?22m ?g?0 ? ? ? ? u ? ?v?? ? ?= iplanckover2pi1 ??t ? ? u ? ?v?? ? ?. (2.23) By taking the complex conjugate of equation (2.20), we have obtained two linear first order differential equations, shown in equation (2.23). The stationary solutions of the excitation (amplitudes of u? and v? are constant in time) are the eigenmodes of the matrix, of the form?1??+?2??? which have energies corresponding to the eigenvalues of the matrix. The energy of the excitation, ?(p?) from the positive eigenvalue is: ?(p?) = radicalBiggparenleftbigg p?2 2m parenrightbigg2 + p ?2g?0 m . (2.24) This is the fundamental expression for Bogoliubov excitation modes, containing the energy-momentum dispersion relation. At low momenta, p?2 ? g?0m, the modes resemble a phononic excitation spectrum, which can be expressed in terms of the sound speed ?(p?) = cp?. (2.25) In this derivation, we have assumed a uniform, infinite system, and the excita- tion spectrum is phononic at low momenta and single particle-like at high momenta (p?2 ? g?0m). In a finite and therefore nonuniform system, however, excitations such as vortex rings have a lower energy [24]. As mentioned previously, Landau predicted the decay of superflow to occur when the flow velocity exceeds a critical velocity set by the energy-momentum dis- persion of the lowest energy excitations. Given the Bogoliubov energy-momentum relation, shown in equation (2.25), the Landau critical velocity is the speed of sound c (derived in the section 2.4). 21 2.3 Bose gas in an inhomogeneous system: The GP equation Until now, we have talked about a Bose condensate in a uniform system. The assumption of uniformity holds better for liquid helium (although the weakly- interacting Bose gas model does not apply), since it is produced in large quantities and is reasonably incompressible. As mentioned previously, in contrast to helium, atomic BECs consist of relatively very few atoms. In addition, they are held at very low temperatures (<1 ?K as opposed to ?0.1 K of superfluid helium) and so any heat absorbed from the surroundings would be catastrophic to the sample. For this reason, atomic BECs are held in traps created by electromagnetic fields and are suspended in ultra-high vacuum so that there is no contact with the surroundings. A consequence of such traps is that atomic BECs are always in an inhomogeneous trapping potential. To model atomic BECs accurately, which due to their diluteness are also much more compressible, we modify equation (2.14) to include a spatially varying trapping potential. Also, we use the wavefunction defined in the position basis. This non- linear Schr?odinger equation, known as the Gross-Pitaevskii (GP) [68?70] equation, is given by iplanckover2pi1 ??t?(r,t) = parenleftbigg ?planckover2pi1 2?2 2m +V(r) +g|?(r,t)| 2 parenrightbigg ?(r,t), (2.26) where ?(r,t) is the order parameter and V(r) is the externally applied potential. The condensate density is given by ?(r,t) = |?(r,t)|2. The stationary solution of the GP equation gives the ground state wavefunction, and its eigenvalue, ?, corresponds to the ground state energy (per particle). ? can also be taken as the chemical potential of the system, which is the energy cost of adding a particle to the BEC. The stationary GP equation is ??(r) = parenleftbigg ?planckover2pi1 2?2 2m +V(r) +g|?(r)| 2 parenrightbigg ?(r). (2.27) Once the ground state wavefunction and ground state energy are obtained, the low-energy excitations can be perturbatively obtained by the Bogoliubov-de Gennes method, using a trial wavefunction similar to equation (2.15). 22 While the full GP equation is not easy to solve, there are simple analytic solutions in certain limits. We start by looking at the terms contained in the GP Hamiltonian, namely the kinetic energy, the potential energy and the interaction energy, and consider the cases where one of the terms can be ignored. The potential energy term is always present, except for when the cloud is released in time-of-flight, and hence cannot be ignored for the trapped cloud. In the absence of interaction (g = 0) and hence zero interaction energy, the GP equation becomes a linear Schr?odinger equation, and one can use the sin- gle particle solution. For a harmonic confinement potential along one dimension (V(x) = 12m?2x2, with ? being the oscillator angular frequency), using the single particle ground state, ?0 in that dimension is given by (see, for example, Sakurai [71]) |?0(x)|2 ? exp parenleftbigg ?x 2 x20 parenrightbigg , (2.28) where x0 = radicalbigplanckover2pi1/m? is the harmonic oscillator length (plotted in figure 2.1 (left)). Even for traps that are not perfectly harmonic (have small quartic or higher order terms), the harmonic oscillator approximation can be used. Corrections to the ground state are typically small, and usually do not affect the physics. For a 3D trap in the absence of interaction, one gets solutions of the form of equation (2.28) for all 3 directions. This harmonic oscillator limit solution is valid at low densities, where the mean-field is small compared to the kinetic energy. In the opposite limit, where the interaction energy is much larger than the kinetic energy, the GP equation is no longer a differential equation and the ground state of the GP equation can be easily solved, given by g|?0(r)|2 = ??V(r) for ?>V(r) = 0 for ?x0 (2.30) wherex0 =radicalbig2?/m?2. The TF solution is not valid for a small region (discontinuity of the slope seen in the plot at x = ?1), where x is close to x0. However, so long as we are looking at the overall shape and dynamics, and not focusing on that region, the TF approximation works very well. In this section, we have seen the solutions of the GP equation in both the non- interacting and the TF regimes, for a sample problem, the 1D harmonic oscillator. While such solutions are not complete, they are nevertheless instructive about what physics to expect. We will be discussing the BEC in a ring in chapter 5. 24 Before we move on, the reader should note that the GP equation is very important in characterizing a BEC. While most of the theory in this chapter has been presented assuming a uniform system so that the physics is more apparent, nearly all the theory from the low-energy excitations to the critical velocities have to be modified suitably for a non-uniform BEC, using the GP equation. We will now move on to other properties of a Bose superfluid, while still using the weakly- interacting Bose gas model to illustrate the physics. 2.4 The Landau critical velocity In Landau?s 1941 paper [13], Landau made the argument that ?... If such a liquid is considered when flowing as a whole along a capillary, it can be easily shown that the interactions between it and the walls of the capillary cannot lead (when the velocity of flow is not too great) to an excitation of internal motion, i.e. to an energy dissipation;? The Landau critical velocity is based on an energetic argument, which I will derive here. We consider an excitation which is characterized by a velocity field v(r), and a number density field ?(r). The energy of the (v(r), ?(r)) excitation in an otherwise stationary fluid is (from ? = 12mv2) ? = m integraldisplay dr parenleftbigg1 2?(r)v(r)?v(r) parenrightbigg , (2.31) where the integral is over the superfluid volume. We now consider the same excita- tion in a superfluid moving with velocity vs. By Galilean invariance, the energy of the same excitation with respect to the moving fluid is ?? = m integraldisplay dr parenleftbigg1 2?(r)(v(r) +vs)?(v(r) +vs)? 1 2?(r)vs ?vs parenrightbigg . (2.32) Substituting p = mintegraltext dr?(r)v(r), we can express ?? as ?? = ?+p?vs. (2.33) In the rest (lab) frame, there is always an energy cost to create the (v(r), ?(r)) excitation, which is ?. However, in the moving frame, for a velocity vs, such that, p?vs +?< 0, 25 or equivalently, for an appropriately chosen direction for vs, and vs > ?p, (2.34) it will be energetically favorable to create the excitation. In creating the excitation, energy from the flow is transferred to the excitation, and the flow dissipates. From this, we can deduce that the critical velocity at which superflow breaks down is given by vc = min braceleftbigg? p bracerightbigg , (2.35) where the minimum refers to the excitation with the lowest ?/p ratio. Below this critical velocity, it is always energetically unfavorable to create any excitation. For a phononic excitation spectrum, with a dispersion relation approximated by equation (2.25), the critical velocity is given by the speed of sound, vc = c. (2.36) In Landau?s original paper, he postulated the existence of roton excitations in liquid helium, which modify the critical flow velocity. Landau referred to rotons as the vortex spectrum, and computed the exact energy of rotons on the basis of experimental measurements of specific heat. While the exact nature of rotons are still not fully understood, they are a consequence of the strong inter-particle interaction in liquid helium, and do not play a role in weakly interacting Bose gases. The excitation spectra of atomic Bose gases have been found to agree with Bogoliubov theory (see Dalfovo et al. [63]). 2.5 Irrotational flow and quantized vortices To treat vortices in Bose gas, as with the Bose gas in an inhomogeneous system (section 2.3), we use the wavefunction in spatial coordinates, ?(r) = |?(r)|ei?(r), where the density is given by ?(r) = |?(r)|2. The flow velocity, vs(r) for such a wavefunction is given by vs(r)?(r) = ?i planckover2pi1m??(r). (2.37) 26 If the fluid is uniform, |?(r)| = constant, we get vs(r) = planckover2pi1m??(r). (2.38) From equation (2.38), we can see that, ??vs(r) ? ????(r) = 0, (2.39) and hence superfluid flow for a uniform system is irrotational6. We now look at the more general case of a simply-connected geometry (see Fig 2.2 (left))7 and the integral of the flow velocity along a path from r1 to r2, using equation (2.37), we obtain m planckover2pi1 integraldisplay r2 r1 dl?vs(r) = ?(r2)??(r1). (2.40) If the path were closed, r1 = r2, the integral goes to zero8, and hence for a simply connected geometry (see figure 2.2 left), m planckover2pi1 contintegraldisplay dl?vs = 0, (2.41) which implies that there cannot be any flow along a closed path. This is a larger statement of no rotation than the local irrotational flow condition of equation (2.39). In essence, the single-valuedness of ?(r) implies that there is no closed loop flow in any simply-connected geometry. In multiply-connected geometry9 (where the condition of simply-connected geometry does not hold, see Fig 2.2 (right)), however, if one were to integrate the 6This is in general true for any superfluid [2], but easier to show for the case of the uniform system. 7A simply-connected geometry is defined as a geometry, where given two points r1 and r2 connected by a path l (that stays within the geometry), one can continuously deform the path to obtain all possible paths between r1 and r2. 8This can also be arrived at by applying Stoke?s theorem to equation (2.39 [2].) 9Multiply-connected geometry can arise in two ways. The trap or the container itself could be multiply-connected, or, there maybe a vortex in the system which causes a density zero at its core, leading to a multiply-connected geometry. Here, I am using multiply-connected in the general sense. 27 dl ?=0 dl ?=2pi n Figure 2.2: In the case of simply-connected geometry (left), the phase winding around any closed path is zero implying no closed loop flow, while in the case of multiply-connected geometry (right), by taking the path shown in the diagram, the phase winding can be any integral multiple of 2pi, and there can be closed loop flow. phase over a path which encloses an area that contains the region of no superfluid (shown in figure 2.2 right), one obtains m planckover2pi1 contintegraldisplay dl?vs = 2?n, (2.42) where n is an integer. Equation (2.42)10 shows two things. Superfluid flow in multiply-connected geometry can have closed loop flow; and circulation is quantized in units of 2?. In addition to the concept of circulation in multiply-connected geometry, On- sager [23] and Feynman [24] also introduced the concept of quantized vortex lines, which can carry circulation in a superfluid. At the center of each vortex line is a vortex core, where the superfluid density goes to zero and the superfluid phase has a singularity. A vortex line in a simply-connected geometry breaks the simply- connected condition (see figure 2.3 (left)), since the superfluid density goes to zero at its core. Hence, even a simply-connected (or what appears to be simply-connected) 10While equation (2.42) looks identical to equation (2.41), the non-superfluid region enclosed by the path prevents the application of Stoke?s theorem. Given that ?(r) is a complex field, the integral need not be zero and can be any integer multiple of 2pi [2]. 28 superfluid can sustain rotation (closed loop flow) via quantized vortex lines. A ro- tating superfluid is expected to relax to a state with quantized vortex lines, each having a 2? phase winding. A path enclosing N lines will have a phase winding of 2?N. For the interested reader, the structure of the vortex core as well as the lattice-like arrangement of quantized vortices have been studied extensively, and are reviewed by Glaberson and Donnelly [74]. circulation quantized vortices L R Figure 2.3: Quantized vortices: (left) A single quantized vortex in a large cylinder. The vortex line extends all the way from the bottom of the cylinder to the top. Quantized vortices, by virtue of the density zero at the center of the vortex core, allow even simply- connected superfluids to sustain rotation. (right) An array of quantized vortices carrying a circulation of 7pi is shown. While the phase winding around each vortex is 2pi, the phase winding along the dashed arrow adds up to 14pi. 2.5.a Energy of single vortex We now consider the case of a single quantized vortex in a large uniform system. For the sake of simplicity, we assume the system to be a large cylinder of height L and radius R, shown in figure 2.3 (left). The vortex line is at the center of 29 the cylinder and extends from the bottom to top11. As mentioned previously, the superfluid density goes to zero at the center of the vortex line. Owing to cylindrical symmetry, the phase of the superfluid can be written in terms ofr(r,?,z) coordinates as ? = ?. (2.43) The flow velocity is given by vs = planckover2pi1mr??. (2.44) The 1/r dependence of the velocity means that atoms close to the core move the fastest, while those furthest away move the slowest. This is the exact opposite of rigid body rotation, where the velocity has an r dependence. The energy of a single vortex is given by ?v ? integraldisplay dr12?0mv2s ? ?planckover2pi1 2?0L m ln R rv, (2.45) where rv is the size of the vortex core. While the exact calculation has additional correction terms [60] for small regions of smallr, equation (2.45) captures the scaling necessary for understanding and predicting the physics. The vortex core size rv is an artificial cut off, put in to avoid the singularity at the center. The value of rv is obtained by finding the distance at which the kinetic energy of high velocity field (close to the center of rotation) equals the repulsive mean-field potential of the superfluid. The vortex-core size (< 1 nm for liquid helium, ?1 ?m for ultracold gases) is usually much smaller than the size of the system and is often approximated to be the healing length [63], which is given by ? = (8??a)?1/2. (2.46) The healing length is the characteristic length at which the superfluid density changes. For example, for a superfluid in a container with hard walls, the su- perfluid density has to go to zero at the walls. The superfluid density cannot change 11The reader should note that a vortex line can only terminate at the superfluid boundary or with itself, as in a vortex ring 30 Flow vortex ring2R Figure 2.4: Flow decay via formation of vortex ring: The lowest energy excitation is a vortex ring the size of the channel. discontinuously. The distance from the hard wall, at which the superfluid density approaches the mean density in the container scales as the healing length. Having discussed the case of a simple vortex excitation, we will now talk about other vortex-like excitations and their role in superfluid dynamics. 2.5.b Decay of flow by vortex excitations Equation (2.45) also shows that the energy of a vortex diverges with system size, with both L and R. Consequently, a vortex ring in 3D or a vortex-antivortex pair in 2D have a lower energy, since their energies do not diverge with system size. The energy of a vortex ring of radius Rv is proportional to Rv log(Rv/rv) [75]. The energy of a vortex-antivortex pair in 2D separated byrs is proportional to log(rs/rv). Hence, these excitations are more likely to play a role in any decay of flow compared to just a simple vortex line. We shall now discuss the breakdown of superflow in a cylindrical channel via the formation of a vortex ring. We consider superfluid flow through a cylinder of radius R (see figure 2.4). The energy and momentum of a vortex ring excitation are given by [32, 75] ?vr ? 2? 2planckover2pi12?Rv m parenleftbigg log Rvr v ? 74 parenrightbigg , (2.47) pvr = 2?2planckover2pi1?R2v. (2.48) 31 Using equation (2.35), we can find the critical velocity forcreating such an excitation: vc ? min braceleftbiggplanckover2pi1log(R v/rv) mRv bracerightbigg (2.49) The vortex ring with the largest radius, Rv has the lowest critical velocity (shown in figure 2.4), and hence Rv ?R, giving the Feynman critical velocity [24], vc ? planckover2pi1log(R/rv)mR . (2.50) The above expression is only an approximation and can only give an estimate of what the critical velocity should be. The expression shows a decreasing trend with increasing channel size, R. This is counter-intuitive to the notion that flow is most likely to break at the ?weakest? connection or narrowest part of the tubing. In fact, superflow is expected to decay more easily in wider channels. This could explain why the original experiments of Kapitza and Allen and Misener had to use extremely narrow channels in order to observe dissipationless flow. Experiments studying the critical flow velocity in superfluid helium have been more consistent with the Feynman critical velocity than the critical velocity for phonon or roton excitations [76]. 32 Chapter 3 Sodium Bose-Einstein condensate apparatus A typical Bose-Einstein condensate (BEC) in our lab consists of 105 sodium atoms at a temperature of around 40 nK. Such temperatures are orders of magni- tude colder than liquid nitrogen (?80 K), liquid helium (?1 K) and even the most sophisticated refrigeration techniques (dilution refrigerators go down to lessorsimilar10 mK). Getting atoms to such low temperatures requires a specialized setup and several steps of cooling and trapping. These cooling and trapping techniques are based on certain properties of the atom. Although the physics of BECs does not require many of these atomic properties, they are nevertheless important for cooling, trap- ping, manipulating and detecting the atomic cloud. The choice of atom for making a BEC is limited to atoms that can be laser cooled (or in some cases helium buffer gas cooled). The electronic structure of alkali atoms makes them easy to laser cool and hence the first atoms laser cooled and the first BECs were all alkali atoms. The apparatus I worked on in the Laser Cooling and Trapping group was the first BEC apparatus (BEC achieved in 1998 [77]) of the group, and is one of the older surviving BEC apparatuses. For this reason, the choice of using sodium atoms is purely due to historic reasons. Since most of the apparatus was built before my time, I will focus on explaining only their principles of operation. I will go over the more recent developments, such as the optical trap, in more detail. 3.1 Atomic physics and the two-level atom While there are no two-level atoms, the two-level atom model can nonetheless explain several aspects of the physics relevant to the cooling, trapping and manipu- lation of atom clouds (the atomic physics tool set). The two-level atom consists of an atom interacting with an oscillating electromagnetic field, which causes coupling 33 between the two levels. To begin, we consider a simple atom-field interaction, which can be treated in the following ways: 1. Fully classically : Here both the atom and field are treated classically. The atom is modeled as a charge dipole with a spring constant and the field is modeled as a classical Maxwellian electromagnetic field. 2. Semiclassically : Here the atom is treated as a two-level quantum mechan- ical system with a dipole coupling between the levels. The field is treated classically. 3. Fully quantum mechanically : Here, along with the atom, the field is also treated quantum-mechanically with discrete energy levels, and can exchange energy with the atom. While the fully quantum treatment is required for explaining cavity quantum electrodynamic and other experiments, the semiclassical treatment is illustrative and sufficient for the purposes of this thesis. The treatment shown in this section is quite standard and is presented in several textbooks (Scully and Zubairy [78], Metcalf and van der Straten [79]), which also discuss the subtleties and details of the various approximations involved. I will take the case of a two-level atom with an electric dipole transition, which is a good approximation for the D-line transitions of an alkali atom. Although magnetic dipole transitions arise from coupling to the magnetic dipole moment of the atom, some of the most general spin selection rules and phenomena such as Rabi flopping can nevertheless be treated analogously. The electromagnetic field (or light field) is assumed to be homogeneous and monochromatic. The electric field can be written as E(t) = E0 cos(?t), (3.1) whereE0 is the amplitude of the field,?is the frequency andtis time. For simplicity, I will only be considering the electric dipole moment of the atom in the direction of the field and hence the electric field can be written as a scalar. The cosine term in 34 the expression can be split up to give E(t) = E02 (ei?t +e?i?t). (3.2) The atom is modeled as having two levels, the ground state |0? and the excited state |1?, with an energy difference of ?0 (in frequency units). The detuning of the light field from the atom is given by ? = ???0 (see figure 3.1). |0> |1> ? 0 ? ? Figure 3.1: Two-level atom : The light field of frequency ? is detuned by ? from the atomic resonance. The atom and light field interact via the electric dipole transition moment of the atom, d, giving a cross-coupling energy term d?E(t). The transition dipole momentdresults from the dipole operator connecting the ground and excited states. The hamiltonian of the atom in the presence of the field can be written as H = H0 +HE, (3.3) where H0 is the unperturbed hamiltonian, and HE = d?E(t). In matrix form in terms of states |0? and |1?, the hamiltonian can be written as H = ? ? 0 0 0 planckover2pi1?0 ? ?+ ? ? 0 d?E(t) d?E(t) 0 ? ?, (3.4) using a |0? = ? ? 1 0 ? ?, |1? = ? ? 0 1 ? ? basis. Equation (3.4) is the basic two-level atom-light hamiltonian. I will now discuss two specific solutions of the equation, the resonant case and the far off-resonant case. 35 In the resonant case, ? = 0, and the primary effect of the light field is Rabi flopping, transferring atoms from |0? to |1? and back. In the far off-resonant case, the primary effect is to cause a shift in the energy levels. 3.1.a Resonant interaction and Rabi flopping We begin with equation (3.4)and use the time-dependent Schr?odinger equation H?(t) = iplanckover2pi1??(t)?t , (3.5) where the time-dependent two-level atom wavefunction is given by ?(t) = c0(t)|0?+ c1(t)|1?, or ?(t) = ? ? c0(t) c1(t) ? ?, with the normalization condition |c0|2 + |c1|2 = 1. Substituting H and ?, we obtain ? ? 0 d?E0 cos(?t) d?E0 cos(?t) planckover2pi1?0 ? ? ? ? c0(t) c1(t) ? ?= iplanckover2pi1 ? ? ?c0(t) ?c1(t) ? ?, (3.6) where the dot abovecindicates derivative with respect to time. This matrixequation gives us two linear differential equations, planckover2pi1?0c1(t) +d? E02 (ei?t +e?i?t)c0(t) = iplanckover2pi1?c1(t) (3.7) d? E02 (ei?t +e?i?t)c1(t) = iplanckover2pi1?c0(t), (3.8) where the cosine terms have been expanded out. In order to solve these equations, one can make a simplifying substitutions b0(t) = c0(t) and b1(t) = c1(t)ei?t. For the resonant case, ? = 0 and hence ? = ?0, giving d? E02 (ei?t +e?i?t)b0(t) = iplanckover2pi1?b1(t)e?i?t (3.9) d? E02 (ei?t +e?i?t)b1(t)e?i?t = iplanckover2pi1?b0(t). (3.10) We now use the rotating-wave approximation, where we average over fast oscillations of the order of the carrier frequency ?. These oscillations are typically too fast (particularly for optical transitions) for the atoms to follow or for any detector to measure and most of the relevant dynamics happen on much slower 36 timescales. After multiplying through with ei?t for the first equation and removing terms containing e?2i?t, we are left with d? E02 b0(t) = iplanckover2pi1?b1(t) (3.11) d? E02 b1(t) = iplanckover2pi1?b0(t), (3.12) which has solutions of the form b0(t) = b0(0)cos(?t/2)?ib1(0)sin(?t/2) (3.13) b1(t) = b1(0)cos(?t/2)?ib0(0)sin(?t/2), (3.14) where planckover2pi1? = d?E0 (3.15) defines the Rabi frequency ?. If one started with an atom in state |0? at time t = 0, the state probability coefficients are given by |c0(t)|2 = |b0(t)|2 = cos2 parenleftbigg?t 2 parenrightbigg , (3.16) |c1(t)|2 = |b1(t)|2 = sin2 parenleftbigg?t 2 parenrightbigg . (3.17) Figure 3.2: Resonant Rabi flopping : The figure shows the |0? probability coefficient |c0(t)|2 (solid) and the |1? probability coefficient |c1(t)|2 (dashed) for a resonant light field incident on the two-level atom. The time units are in 2pi/?, where ? is the Rabi frequency. As one can see in figure 3.2, the atom flops between states |0? and |1?. The time 2?/? is known as the 2?-pulse time. If the field is turned on for the duration 37 of a ?-pulse, the atom is shifted from state |0? to state |1?. A ?/2-pulse puts the atom in an equal superposition of |0? and |1?. 3.1.b Far off-resonant interaction and energy level shifts When the light field is significantly far detuned from the energy separation of the two levels of the atom, the light causes little transfer from the ground to the excited state. Rather, it causes a light-shift in the energies of the two levels. In order to eliminate the time dependence of the electromagnetic field, we go into the rotating frame e?i?t of the field, and just like in the resonant case, apply the rotating-wave approximation1 eliminating terms of the order e?i?t. We start with equation (3.6), and set b0 = c0(t) and b1 = c1(t)ei?t to obtain? ? 0 d? E0 2 (e i?t+e?i?t) d? E02 (ei?t+e?i?t) planckover2pi1?0 ? ? ? ? b0 b1e?i?t ? ?= iplanckover2pi1 ? ? ?b0 e?i?t(?b1 ?i?b1) ? ?. Applying the rotating wave approximation and expressing the equation in terms of ? ? b0 b1 ? ?, we get a time-dependent Schr?odinger equation of the form ? ? 0 d? E0 2 d? E02 ?planckover2pi1? ? ? ? ? b0 b1 ? ?= iplanckover2pi1 ? ? ?b0 ?b1 ? ?, (3.18) which can be written as a hamiltonian in the (|0?,e?i?t|1?) basis: H? = planckover2pi1 ? ? 0 ?/2 ?/2 ?? ? ?= planckover2pi1 ? ? ??/2 0 0 ??/2 ? ?+ planckover2pi1 ? ? ?/2 ?/2 ?/2 ??/2 ? ?. (3.19) By solving for the eigenvalues of the new hamiltonian, the modified energy levels come out to be ?? = parenleftbigg ??2 ? ?? 2 parenrightbigg planckover2pi1, (3.20) where ?? = ??2 + ?2. In the limit of large detunings, ? ? ?, ?? simplifies to ?(1 + ?22?2), which gives a ground state energy shift of ??0 = planckover2pi1? 2 4?, (3.21) 1The rotating-wave approximation is valid so long as one is not too far detuned. If ? ? ?0, then there is not much difference between |?+?0| and |???0?. 38 and the excited state energy shift of ??1 = ?planckover2pi1? 2 4?. (3.22) The energy shifts are proportional to the intensity of the radiation (E20). When the applied field is blue-detuned, ? > 0, the ground state energy is raised and the excited state energy is lowered, whereas when the applied field is red-detuned, ? < 0, the ground state energy is lowered and the excited state energy is raised (see figure 3.3). If one were to apply a red-detuned light field, an atom in the ground state will be attracted towards regions of higher intensity, due to the lower potential there. Hence, red-detuned fields with localized maxima can be used to trap atoms via the dipole force. Blue-detuned light fields have the opposite effect and can be used to create a repulsive potential. |0> |1> ? ? |0> |1> ? ? Figure 3.3: Light shift on a two-level atom: The effect of blue-detuned (left) and red- detuned (right) light on the two level atom. The effect shown is exaggerated for clarity. The light shift is typically orders of magnitude smaller than the detuning. 3.1.c Spontaneous emission In practice, an atom in the excited state will emit a photon and decay to the ground state even in the absence of an applied coupling field2, via a process known as spontaneous emission. The rate of spontaneous emission depends on the strength 2Spontaneous emission can also be viewed as the coupling of an atom to the vacuum field, which according to quantum electrodynamics is nonzero. 39 of the transition and the resonance frequency, and is given by [80] ? = ? 3 0 3??0planckover2pi1c3|d| 2, (3.23) where ?0 is the permittivity of vacuum, c is the speed of light and d is the total dipole moment of the atom (in calculating the Rabi frequency, we used only the component in the direction of the electric field of the light). The expression for spontaneous emission can be derived quantum mechanically from the coupling of an atom to the free space vacuum modes (originally in Weisskopf and Wigner [81], also shown, for example, in Scully and Zubairy [78] and Steck [80]). Equivalently, it can also be derived by comparing the thermodynamic steady state of an atom in a light field with the Planck blackbody distribution [82]. The?30 dependence of the spontaneous emission decay rate causes optically ex- cited states with electric dipole moment transitions to decay at rates of 107 s?1. Mi- crowave and rf transitions, which are typically magnetic dipole and have a lower tran- sition frequency, tend to have decay times much longer than experimental timescales and hence can be neglected. Spontaneous emission causes an atom in a light field to reach a steady state, as opposed to undergoing Rabi oscillations. In contrast to Rabi oscillations where the photon is emitted coherently into the driving field, spontaneous emission causes photons to be emitted incoherently in all directions. This process is key to imaging techniques, which either measure the loss of photons as they pass through an atomic sample (absorption imaging, discussed in 3.8), or the number photons scattered in other directions (fluorescence imaging). The steady state excited population fraction of atoms in a radiation field depends on the intensity and detuning of the radiation, and is given by [80] |c1(t??)|2 = (?/?) 2 1 + 4(?/?)2 + 2(?/?)2. (3.24) The spontaneous scattering rate per atom is Rs = ?|c1(t??)|2, leading to Rs = parenleftbigg? 2 parenrightbigg (I/I sat) 1 + 4(?/?)2 + (I/Isat), (3.25) 40 where we have defined a saturation intensityIsat, such thatI/Isat = 2(?/?)2, andI is the intensity of the light field (in energy per area per time), given byI = (1/2)c?0E20. The saturation intensity can be expressed in terms of the dipole interaction: Isat = c?0? 2planckover2pi12 4|???d|2, (3.26) where, to remind the reader, ??? d is the component of the dipole moment in the direction of the electric field. The effect of detuning is to lower the steady state population in the excited state and hence the spontaneous scattering rate. However, this can be overcome by using a higher intensity of light. For a fixed detuning, in the limit of low intensity, the population fraction of the excited state is linearly proportional to the intensity (or ?2) of the radiation. In the limit of high intensity, the population saturates to 0.5 (50 %), and the scattering rate saturates to ?/2. 3.1.d Light forces In the previous section, we discussed the steady state of an atom in a light field. If such a light field were unidirectional as is typical of an optical beam (see figure 3.4), there is a net force on the atom due to the difference in momenta between the photons absorbed from the incident beam and the photons spontaneously emitted in all directions3. For an atom in free space (assuming no collisions with other atoms), the atom undergoes a momentum change due to the net force. The velocity shift due to the momentum change in turn causes a Doppler shift in the light frequency that the atom sees, which eventually takes the light field off-resonance, and the atom is no longer affected by the field. The Doppler shift of a moving atom can also be used to slow it down. This was first proposed independently by H?ansch and Schawlow [85] and by Wineland and Dehmelt [86] and later demonstrated in ions by Wineland 3The effect of light on an atom also depends on the state of matter in which it is in. For example, for atoms in a conducting solid, light can cause pairing of electrons [83]. In this thesis, we only consider the effect of light on a pure sample of neutral atoms in the gaseous state. 41 spontaneous emission Incident Field net force on atom Figure 3.4: Light force on a two-level atom: An atom in a light field in steady state absorbs radiation from the incident light field, and emits radiation in nearly all directions (dipole radiation pattern [84]) via spontaneous emission. Through this process, the atom undergoes a net force in the direction of incidence of the light field. et al. [87] and Neuhauser et al. [88]. For a gas of atoms, where the energy is primarily translational, slowing down atoms corresponds to cooling, and hence the technique is known as laser cooling, which enables nearly all BEC experiments. We will now see a simple example of laser cooling. In figure 3.5, the effect of a 1D pair of counter-propagating beams is shown. The beams are near red-detuned. For an atom moving towards one of the beams, the atom sees the beam with a blue Doppler shift (closer to resonance) and experiences a stronger light force from that beam. Since it is moving away from the other beam, it sees the beam with a red Doppler shift (further red-shifted) and experiences a weaker force from that beam. The net result of the two beams is to damp the velocity of the atom along the beams? axis. An atom that is not moving (in the axis of the beams) feels no net force. This velocity-sensitive damping force can be used to slow and hence cool the atoms down. 42 Figure 3.5: Velocity sensitive light forces: An atom in a field created by a pair of near red-detuned counterpropagating beams undergoes a light force opposing the direction of its velocity, shown by black arrows. An atom at rest (center) feels no net force. This acts as a damping force and can be used slow atoms down. Using the scheme shown in figure 3.5 along all 3 axes, atoms can be cooled down to millikelvin temperatures. This scheme is known as optical molasses and has been studied extensively [89?94]. The simple two-level atom picture described here cannot fully explain laser cooling from optical molasses. In fact, the magnetic sublevel structure of alkali atoms causes further cooling of the atoms, known as sub- doppler cooling [90]. The exact effect of light forces due to sub-doppler cooling is complicated and beyond the scope of this thesis. For a good review on the subject, see Cohen-Tannoudji [89], Cohen-Tannoudji [95] or Phillips [96]. Having described the various interactions of the two-level atom with laser light, we can now focus our attention towards the more practical aspects of atomic physics and our system. 43 3.2 Atomic structure in real world atoms Unlike their imaginary counterparts, real world atoms have a more complicated energy level structure. As mentioned before, the energy level properties of the atom are important mainly for cooling, trapping, manipulating and imaging the atoms, and function as the basis of ?engineering tools? for the atom. So long as we are able to cool and trap atoms, we need not delve into details of the internal structure of the atom. In that respect, as engineers, all we need to know are the energy level ?specifications? of the atom. Nevertheless, I feel it is instructive to go through some background as to how the energy level structure of the atom arises. The basic energy level structure, namely the principal quantum numbern(n? {1,2,...}), and the orbital angular momentum quantum numbers l and ml (l, ml: 0 ?l200 ?s), fluctuations in the magnetic bias field dephase the oscillations. degrees to the Zeeman slower axis, in the horizontal plane. The magnetic field of the microwaves are oriented along the vertical, perpendicular to the magnetic bias field direction, allowing hyperfine transitions that change mF by ?1 (see Jackson [84]). We use our microwave system primarily to couple from the initial F = 1, mF = ?1 state to the F = 2, mF = ?2 state. Given our typical magnetic bias fields (B < 5G), the F = 1, mF = ?1 to F = 2, mF = ?2 transition frequency ranges from 1.761 to 1.771 GHz. By varying the pulse duration, we can choose the fraction we want to transfer to the F = 2, mF = ?2 state. The transfer fraction 62 follows Rabi flopping and sinusoidally oscillates (Figure. 3.16). The variability of the transfer fraction was the key to partial transfer absorption imaging, which is discussed in chapter 4. 3.8 Imaging Slit Relay Image Plane 16 cm CVI Triplet BEC Mirror 75 cm doublet Zoom LensCamera 5x magnification Sample Image Probe Beam Relay image of BEC Figure 3.17: Schematic of the imaging system: Our image system images the plane of the sheet and has a resolution of about 4 ?m. We use a rectangular slit in the relay image plane to reduce ambient light falling on the camera. While a hole would be more appropriate, using a slit allowed us to independently vary the x and y dimensions of the image. For our experiment, we use simple absorption imaging. Chapter 4 discusses partial-transfer absorption imaging, which, while a different technique, is still based on absorption imaging. We have the capability to image along both the horizontal and the vertical axes. For most of the data, we image along the vertical axis (see figure 3.17), which images the plane of the sheet yielding more spatial information. Our imaging system uses two steps of relay imaging, each providing magnification. In the first step, which uses the same optics that are used to focus down the LG 63 trapping beam, using a lens pair that consists of a 16 cm triplet (5.08 cm diame- ter, ?f/3) and a 75 cm doublet (5.08 cm diameter), a 3x magnified relay image is produced. The optical resolution of our system is limited by the diffraction limit of the triplet (diffraction limited spot size = 0.61 ?/NA ?2 ?m, NA = numerical aperture). Aberrations due to the triplet further reduce it. The relay image is then magnified using a 5x zoom lens on to the Andor I?on camera. While the total magnification is 15x, with each camera pixel corresponding to 1.2 ?m in the plane of the atoms, the actual 1/e optical resolution is around 4 ?m. We will now calculate the effect of a certain column density of atoms on res- onant (? = 0) probe light. The scattering cross-section, ?, is defined as the ratio of the scattered power, Rs ?planckover2pi1? to the incident intensity (or energy flux), I. Using equation (3.25), we obtain the scattering cross-section ? = ?planckover2pi1?2I sat(1 +I/Isat) , (3.31) where the saturation intensity Isat is defined in equation (3.26). The next chapter will discuss absorption imaging in detail and exactly solve the absorption imaging problem. For the sake of completeness, I will briefly mention how the basic imaging process works. In the limit of low intensity (Ii ?Isat), for a given incident intensity Ii, the transmitted intensity is given by It(x,y) = Iie??0??(x,y), (3.32) where ??(x,y) is the atom number density, and ?0 is the scattering cross-section in the low intensity limit (? in the limit of Ii/Isat ? 1. ?0 is the largest scattering cross-section). Using equation (3.32), we can obtain the column density of the atoms from the knowledge of the incident and transmitted light intensity. In practice, this is done by taking 3 images per run; an absorption image in the presence of atoms I1(x,y), a reference image of the probe in the absence of atoms I2(x,y), and an image of the background in the absence of probe light I3(x,y). The column density of atoms (at low probe intensities) is given by ??(x,y) = ?0 log parenleftbiggI 2(x,y)?I3(x,y) I1(x,y)?I3(x,y) parenrightbigg . (3.33) 64 3.9 Conclusion In this chapter, I hope to have given the reader an introduction to atomic physics and some idea of how experiments are performed. Since atomic physics is a large field in itself on which several textbooks have been written, I have been brief, only introducing concepts that are relevant to this thesis. Likewise, a complete description of the details of the experimental apparatus is beyond the scope of this thesis, and I have only given a general overview of our setup. There are several theses, such as Stamper-Kurn [62], that provide details of the apparatuses and experimental techniques. Having described our experimental apparatus, and given some background on atomic physics and the sodium atom, which we use, we are ready to delve into the experiments and science that constitute this thesis. In chapter 5, I will discuss our experiments on the toroidal BEC. However, before that, I wish to take the reader on a slight detour and discuss partial-transfer absorption imaging, a technique developed during the course of this work. The reader can feel free to skip chapter 4 and get to the ring trap experiments in chapter 5. 65 Chapter 4 Partial-transfer absorption imaging Most BEC experiments take images in time-of-flight as opposed to in situ, largely because features in the BEC are too small to image in situ and so the cloud of atoms is allowed to expand before imaging1. In addition, the time-of-flight expansion provides information about the momentum distribution, which often is more useful than the spatial distribution2. In contrast, when we began experiments using the toroidal optical dipole trap, it was important to obtain in situ density profiles to be able to create a smooth, uniform ring BEC. When performing simple absorption imaging on the ring BEC, we found that the probe beam was completely extinguished in regions of high density, preventing us from making accurate quantitative measurements of uniformity. That informa- tion was also lost in time-of-flight because of the ring closing on itself. While we could have used an off-resonant probe and tried to deconvolve the distortion from its refraction through the cloud, we developed a new technique, partial-transfer absorp- tion imaging. As it turned out, the technique could also be used to take multiple images of the same cloud. In this chapter, I will describe the technique in the context of other imaging techniques and compare its performance to the more established phase-contrast imaging. 4.1 Introduction In choosing an imaging technique for an ultracold gas cloud, one tradition- ally has three choices: absorption imaging, fluorescence imaging or phase-contrast 1Also, in many experiments, the optical depth of the cloud in situ is too high to perform absorption imaging, and hence the cloud is imaged after time-of-flight. 2Since most of the early BEC experiments were performed in magnetic traps, magnetic field gradients would distort the in situ images, requiring, at the very least, a short time-of-flight before imaging to allow the magnetic fields to shut off. 66 imaging. Absorption imaging exploits the strong interaction of ultracold gases with resonant laser light. However, the scattering of photons by the ultracold gas cloud leads to heating and destruction of the sample, allowing only one image to be taken per sample. In addition, on-resonant absorption imaging cannot be used for opti- cally thick condensates, due to attenuation of the probe beam. Modifications such as imaging off-resonance, or imaging after expanding the cloud in time-of-flight, can overcome this limitation, but make reconstructing the original cloud profile more difficult. Fluorescence imaging works similarly to absorption imaging, detecting the scattered photons from the cloud instead of the transmitted probe beam. However, the small solid angles for capturing fluorescent light typically make it inferior to absorption imaging for ultracold gas clouds. Phase-contrast imaging (PCI) [136, 137] and other dispersive techniques [138, 139] are typically used to image clouds of high optical depth (OD). Such techniques use off-resonant probe light, where a relative phase-shift, imparted by the cloud, is detected on the transmitted probe beam. Since the scattering cross-section is severely reduced by going off-resonance, the perturbation to the sample is small, allowing the cloud to be imaged multiple times. PCI is unsuitable for imaging clouds of low OD as it gives a weak signal3. Using partial-transfer absorption imaging (PTAI), one retains the advantages of absorption imaging for optically thin clouds while also being able to image op- tically thick clouds. In this technique, a portion of the cloud from the initial state is first coherently transferred to an auxiliary internal state that has a cycling tran- sition. The transferred cloud is then resonantly imaged on the cycling transition. The recoil energy imparted by scattered photons from the imaging light cause the transferred cloud to be ejected from the trap, while the cloud remaining in the initial state is almost unperturbed, and can subsequently be re-imaged. In any measurement process, there is a certain amount of perturbation to the 3One could use an intense probe beam (or a long probe pulse) to overcome the problem of a weak signal. However, even in such situations, the small relative change in the intensity of the probe beam on the detector uses only a small part of the dynamic range of the detector. 67 sample. In a minimally-destructive imaging technique, that perturbation depends on the desired information, quantified as a signal-to-noise ratio (S/N), leading to a trade-off between the obtained S/N and the sample perturbation. For example, in PCI, reducing the probe light detuning improves the S/N but also causes a greater perturbation to the system [140]. PTAI extends absorption imaging to a minimally- destructive regime, where, by transferring a larger fraction to the auxiliary state for imaging, one obtains a better S/N at the cost of a larger perturbation to the initial cloud. A key advantage of PTAI over PCI is the ability to easily vary the degree of perturbation to the sample, even for different images within the same experimental run, which is useful in situations where the final cloud is optically much thinner than the initial cloud, such as in characterizing Feshbach resonances [141?143] or Efimov states [144, 145]. PTAI is also useful for studying in situ dynamics of quasi two-dimensional condensates where the cloud optical depth in the tightly confined direction may not be high enough to obtain a good phase-contrast image [53, 146, 147]. There have been several techniques used to overcome the problem of high OD [148?150]. However, such schemes have been destructive and cannot be used to take multiple images. Ref. [151] used a partial microwave out-coupling technique, similar to PTAI, in a magnetically trapped BEC. However, since the focus of the work was to study vortex dynamics, the atoms were imaged in time-of-flight. In this chapter, I will focus on using PTAI to obtain an accurate minimally-destructive image of the in situ density profile. The structure of this chapter is as follows: In section 4.2, I will go over the basics of absorption imaging. In section 4.3, I will discuss the PTAI technique and its applicability. In section 4.4, I will present some practical uses of the technique in our experiment. In section 4.5, I will discuss the noise in the measurement process and compare the technique to PCI as a minimally-destructive imaging technique. In section 4.6, I will derive expressions to correct for optical pumping. In section 4.7, I will discuss some of the practical aspects before concluding. 68 4.2 Absorption imaging Before we go on to PTAI, let us quickly go over the basics of absorption imaging. Absorption imaging is a standard technique used in many fields to image a sample. The fundamental scatterer is an atom (or molecule), which has a (on- resonant) scattering cross-section ? = ?planckover2pi1?/[2Isat(1+I/Isat)] (from equation (3.31)), where, to remind the reader, planckover2pi1 isPlanck?s constant,?isthe frequency ofthe resonant light, ? is the natural linewidth of the transition andIsat is the saturation intensity of the transition (defined in equation (3.26)). We factor out the intensity dependence of the (on-resonant) scattering cross-section, expressing it as ? = ?01 +I/I sat , (4.1) where ?0 is the scattering cross-section in the low intensity limit (Ii ? Isat), given by ?0 = planckover2pi1??2I sat . (4.2) For a cloud density of ?(x,y,z), where (x,y,z) is the position in space, the change in light intensity, I, for a resonant light beam propagating along the z- direction is given by [80] dI dz = ?I??(x,y,z) = ? I?0?(x,y,z) 1 +I/Isat . (4.3) 4.2.a Limit of Low Intensity In the limit of low intensity, I ?Isat, equation (4.3) simplifies to dI dz = ?I?0?(x,y,z), (4.4) where one can see that the absorption is linearly proportional to the intensity of the light. Equation (4.4) can be integrated along z, giving an expression for the transmitted light intensity, It as a function of the incident light intensity Ii and the column density of the atoms. We will assume that the cloud extends from zi to zf. This implies that I(zi) = Ii and I(zf) = It. The column density of the atoms is 69 given by, ?n(x,y) =integraltextzfzi dz?(x,y,z). Integrating equation (4.4), we obtain: integraldisplay It I0 dI I = ? integraldisplay zf zi ?0?(x,y,z)dz log ItI i = ??0?n(x,y) = ?? (4.5) It = Ii exp(??(x,y)), (4.6) where ?(x,y), the (on-resonance) optical depth, is defined as ?(x,y) = ?0?n(x,y) (see appendix C for a sample calibration of ?0). The optical depth (OD) is a dimensionless quantity which is useful for parameterizing the absorption through the sample. An OD of less than 0.5 corresponds to little absorption. An OD of 1 corresponds to a 1/e (37%) transmission. An OD of 4 corresponds to almost complete absorption (nearly 99%). As one can see in equation (4.6), the transmitted intensity is related to the OD by a simple exponential function. Similarly, the optical depth can be expressed analytically in terms of the transmitted intensity (equation (4.5)). In practice, analyzing an image using equation (4.5) underestimates the col- umn density. For good image quality, one typically uses an intensity of I ? Isat/2, which underestimates the column density by about 30%. If one were to weaken the probe down to I = Isat/5, there is still a 20% correction. Hence, while simple anal- ysis of an absorption image can give good qualitative information about an atom cloud, it needs to be corrected for saturation to obtain a quantitatively accurate picture. Equation (4.5) will be used as a basis for getting a naive estimate of the optical depth and subsequently adding corrections to get the true optical depth. 4.2.b Effect of saturation The first correction to the analysis of an absorption imaging is saturation. We can rewrite equation (4.3) to separate the variables: parenleftbigg1 I + 1 Isat parenrightbigg dI = ??0?(x,y,z)dz. (4.7) 70 Integrating over the the propagation direction, z, similar to the simple case, we get: integraldisplay It Ii parenleftbigg1 I + 1 Isat parenrightbigg dI = ? integraldisplay zf zi ?0?(x,y,z)dz (4.8) log parenleftbiggI t Ii parenrightbigg + It ?IiI sat = ??0?n(x,y) = ?? (4.9) Equation (4.9) is identical to equation (4.5) except for a correction term due to the saturation given by (It ?Ii)/Isat (see appendix C for a sample calibration of Isat). In the limit of low incident intensity Ii,It ?Isat, the correction term goes to zero. For a given incident and transmitted intensity, equation 4.9 gives an expression for the optical depth. From a practical standpoint, this suffices for a calculating the optical depth. However, if one were to try and invert the expression (i.e. It = f(?)), there is no simple analytic way (using commonly used functions) of obtaining the transmitted intensity for a given optical depth. This makes incorporating further corrections messier, even though the fundamental differential equation is simple. Having gone through the basics of absorption imaging, we are now set to introduce PTAI. 4.3 Imaging with PTAI In this section, I will first discuss the requirements for implementing PTAI in the general case before applying it to alkali atoms, and, as is relevant to this thesis, sodium. 4.3.a Requirements for implementing PTAI PTAI can be used for any atomic species with the following properties (see figure 4.1a). In addition to the initial atomic ground state, |g?, the species needs to have an auxiliary state, |i? to which the atoms can be coherently transferred. The state |i? should have access to a cycling transition to a state |c? for optical imaging, where ideally the species only decays from |c? to |i?. The lifetime of |i? must exceed imaging timescales. The energy splitting between the two states, planckover2pi1?ig 71 should be large enough so that the imaging light does not affect the initial state, i.e., ?ig ? ?c, where ?c is the natural linewidth of the imaging transition. In most cases, there is some off-resonant excitation from |i? that causes the transferred fraction, which is being imaged, to eventually be optically pumped out of the cycling transition to another state. In certain circumstances, one could use a second optical repumping beam to bring that fraction back to the cycling transition. However, in situations where the transferred fraction is optically pumped back to |g?, one cannot use repumping light, but can nevertheless still use PTAI if the optical pumping due to off-resonant excitation is sufficiently low (discussed in section 4.5.c). PTAI is well suited (but not limited) to optical dipole traps, which have no spatially varying Zeeman shift that could affect the uniformity of the transfer. For most optical traps, the scattering of several photons transfers a sufficient amount of energy and momentum such that the atoms leave the trap in the direction of propagation of the probe beam. Heating due to collisions between atoms leaving the cloud and the remaining atoms can typically be neglected due to the combination of weak atom-atom interactions and the low densities of ultracold gases as was also observed by Ref.[151]. 4.3.b Using PTAI with alkali atoms The hyperfine structure of alkali metal atoms make them suitable for employ- ing PTAI. The S1/2 ground state of alkali atoms has two hyperfine levels between which atoms can be coherently transferred. The condition ?ig ? ?c is satisfied for the ground state hyperfine splitting of all alkali atoms. Alkali atoms have the D2 optical transition to the P3/2 state, and due to the hyperfine splitting of the excited state into four levels, spin selection rules allow certain transitions to be used as cycling transitions4. 4The small excited state splittings in lithium (both 6Li [152] and 7Li [153]) and potassium [154] (39K, 40K and 41K) are likely to make the cycling transition for the respective atoms less effective due to off-resonant excitation and consequently optical pumping (discussed in section 4.5.c). One could use polarization selection (using circularly polarized light, analogous to sodium, discussed in 72 F=1 F=23S 1/2 3P3/2 F'=3 589 nm 1.7 GHz F'=2 F'=1 F'=0 | i > | u >?u i | o > (b)(a) Figure 4.1: (a) General PTAI scheme: PTAI is implemented by coherently transferring a small fraction of the cloud from |g? to |i? and then imaging on the cycling transition |i? to |c?. For the scheme to work, the frequency difference ?ig should be much larger than the natural linewidth, ?c, of the cycling transition. (b) Sodium D2 hyperfine structure : Our implementation of PTAI with 23Na uses the F=1 (|g?), F=2(|i?) and F?=3(|c?) states. For sodium, ?ig ? ?c. Optical pumping due to off-resonant excitation to other 3 P3/2 states followed by decay to F=1 state is low enough to allow the technique to be used. For sodium (figure 4.1b), we implement PTAI by keeping the atoms in the initial S1/2 F=1 state |g?, then transferring a small fraction to the S1/2 F=2 state |i? using a single-photon microwave process, and then imaging the transferred fraction on the S1/2 F=2 to P3/2 F?=3 (|c?) cycling transition5. For this transition, ?ig = (2?)?1.77 GHz ? ? = (2?)?9.8 MHz. 4.4 Practical examples using PTAI In order to illustrate the usefulness of the technique, I will describe three specific case examples of experiments in our setup, which used the PTAI technique, each for a separate reason. the next footnote), to enhance the cycling transition for implementing PTAI. 5If one were to use a suitable polarization of light to excite to the mF = ?3 magnetic sublevels of the F?=3 state, spin selection rules guarantee that the atom can return only to the F=2, mF = ?2 states, which enhances the effect of the cycling transition. 73 4.4.a In situ imaging of an optically thick BEC PTAI allows for the use of absorption imaging without any change in the optical setup. This is advantageous in situations where one may need to image a high OD cloud in situ and a low OD cloud in time-of-flight [52, 54, 155], possibly even within the same experimental run. In our critical velocity experiment (discussed in chapter 7), condensates were created in a ring-shaped trap with a repulsive, permeable barrier cutting across one part of it. The cloud had an on-resonance OD of up to 20 making it unsuitable for absorption imaging in situ (shown in figure 4.2d), which obscured the azimuthal density variations. By imaging the cloud using PTAI (figure 4.2b), where we transferred between 15-40% of the atoms depending on the cloud OD, we are able to observe the density variations due to inhomogeneities in the azimuthal potential (figure 4.2c). Plotting the azimuthal profile (figure 4.2e), one can see that PTAI (black) clearly shows the nearly 50% density variation, while absorption imaging (gray) does not. 4.4.b Eliminating trap drift For calibration purposes of the critical velocity experiment, it was necessary to measure the full profile of the barrier beam. Since the diameter of the repulsive barrier was larger than the annular width of the condensate, we looked at the effect of the beam on a BEC in the sheet optical dipole trap (described in section 3.6). For technical reasons, the sheet trap was found to drift up to 30 ?m between shots. Due to this shot-to-shot variation in the position of the dipole trap, we could not use sequential runs to compare images of the BEC with and without the barrier. Instead, by utilizing PTAI, an image with the barrier beam and an image without the barrier beam could be taken in quick succession without destroying the BEC (see figure 4.3) to minimize the relative drift of the BEC and get an accurate reference. 74 ?pi0 1 pi0?pi2 pi2 Col um n D ens ity Angle (e) 4 0 (a) (b) Angle ?pi pi pi/2 ?pi/2 0 (c) (d) 1 0 Tra nsm iss ion OD Figure 4.2: Imaging an optically thick cloud: (a) Standard absorption image of an optically thick cloud (grayscale: probe transmission) (b) The corresponding PTAI image of an almost identical cloud using a 16% transfer fraction for the image. From this image we determine the maximum OD to be ?19. (c),(d) Corresponding column density profiles (colorbar: measured OD). (e) Normalized azimuthal density profiles of the PTAI (black) and absorption (gray) images (angles shown in (d)). Due to the severe attenuation of the probe (seen in (a)), the absorption image fails to show the full optical depth of the cloud and consequently, spatial features such as the azimuthal density variation (plotted in (e)) are unclear and affected by shot noise (discussed in section 4.5). In contrast, the PTAI image shows clear spatial features, particularly the density variations due to azimuthal inhomogeneities of the toroidal potential. 4.4.c Accounting for number fluctuations In ultracold gas experiments the initial conditions may vary from run to run. This is due to the fact that the process of producing these samples involves several steps of cooling and evaporation, each of which causes fluctuations in the number of atoms and consequently other properties such as the condensate fraction. Although it is possible to average over several experimental runs to overcome the initial fluc- tuations, one can also, more easily, make a minimally-destructive measurement of the cloud before the experimental procedure to know the initial atom number. 75 0 70 40 Column D ensit y ( at/ ?m 2 ) Figure 4.3: Eliminating noise due to shot-to-shot fluctuations in trap position : In one example of an application of PTAI, the effect of drift on measurements of a barrier beam is minimized. In order to obtain the profile of a barrier beam, we take two images, in quick succession, of the BEC in a drifting optical dipole trap, without (left) and with (right) the barrier beam. By keeping the time between the 2 images short (300 ms in this case), we can minimize trap drift to get an accurate comparison of the profile of the atoms, with and without the barrier. Both images are taken with PTAI, transferring 18% and 32% of the atoms respectively. PTAI images have been corrected for saturation and optical pumping using an expression derived in section 4.6 A simple example of this is measuring trap loss of atoms in a sample, which is commonly used to study Feshbach resonances [141?143] and Efimov states [144, 145]. Here, we demonstrate the measurement of the lifetime of a BEC in a trap. The primary loss mechanism is background gas collisions. For every measurement, we take two images, first a reference PTAI image at time t = 0, transferring ? 15% of the cloud and then a second PTAI image at 300 ms 20, typical of some of the largest BECs, PCI gives a lower uncertainty for a given perturbation, while at low column densities (? < 1), PTAI gives a lower uncertainty as shown in figure 4.6. 84 Forintermediate ODs(1 ?r). 107 There are 3 simplified ways to treat the ground state BEC in the ring trap. 1. Harmonic oscillator ground state regime : In this case, we neglect the inter- action energy. The condensate assumes the ground state of the 2D harmonic oscillator. The density distribution n3D is: n3D(r,z,?) = n3D,0e?z2/z2we?(r?rM)2/r2w, (5.22) where zw and rw are the harmonic oscillator lengths given by radicalbigplanckover2pi1/m?z and radicalbigplanckover2pi1/m? r respectively and n3D,0 is the maximum (peak) 3D density. 2. Radial Thomas-Fermi and vertical harmonic oscillator : In this case, the con- densate expands due to the mean field in the radial (about the annulus) di- rection, but is still confined to the ground state in the vertical5: n3D(r,z,?) = e?z2/z2wg parenleftbigg ?0 ? 12m?2r(r?rM)2 parenrightbigg for |r?rM| 5 units) of angular momentum, necessary to observe effects such as vortex lattices [47], or large vortex aggregates [218], or to attain critical rotation where the rotation velocity is close to the radial trapping frequency [221]. Phase engineering is important in situations where one needs to transfer a fixed amount of rotation, usually one or two units. Transferring large amounts of circulation by the repeated use of phase engineering is difficult because of the perturbation tothe cloud by each transfer process (discussed in section 6.5). Transfer of rotation by phase engineering has been used for observing persistent currents [52] and for sculpting a vortex state in a spinor BEC [237]. 129 For our experiments, where we observed persistent currents and studied the decay of superflow (chapter 7), we used phase engineering to transfer circulation. We implemented phase engineering using a two-photon Raman2 transfer with an LG beam [52, 117, 126], which transferred exactly one unit of circulation. 6.3 Transferring orbital angular momentum using a LG beam Having already discussed the connection between circulation and phase wind- ing, we will now talk about transferring OAM using LG beams. Light can carry two kinds of angular momentum: internal or spin angular momentum associated with its polarization and external or OAM associated with its spatial mode [240]. A light beam with a phase singularity, such as an LG beam, has a well-defined OAM along its propagation axis [241]. As with circulating atoms, LG beams have a phase winding around the central phase singularity, sometimes termed an optical vortex. Coherent transfer of atoms using such a beam will cause the atomic cloud to acquire the optical phase winding. In order to perform a coherent transfer, one needs a stable final state. Such a state cannot be in the 3P3/2 (D2 transition) of sodium, as the excited state has a very short lifetime and the spontaneous emission is likely to destroy the condensate. Hence, two-photon transfers leading back into the 3S1/2 state are needed. In a two- photon coherent transfer, the difference in the phase winding of the two beams is imprinted onto the atoms. The two-photon transfer process works identically to the single photon Rabi flopping, which we discussed in section 3.1.a. The two laser beams used for the transfer have different frequencies, ?1 and ?2. The resonance condition is given by ?1 ??2 = ?if, (6.4) where ?if is the energy separation between the initial and final states (shown in 2A Raman transition refers to a transition which involves the absorption of a photon by an atom (or molecule) and the emission of a photon of a slightly different color [238]. For a history of the Raman effect, see Krishnan and Shankar [239]. 130 figure 6.2) of the atom3. The two-photon Rabi frequency ?R depends on the Rabi frequencies of the individual beams, ?1 and ?2 as well as their mutual detuning from the single-photon resonance (normalized by the transition half-linewidth ?/2), ?? (see figure 6.2), and is given by [80] ?R = ?1?2??? . (6.5) For efficient coherent transfer and to ensure that four-photon or other multi-photon processes do not occur, the final state has to be energetically separated from the initial state4. The circulation by itself does notcreate asufficient difference in energy. Previous experiments in our group [52, 117] used counter-propagating beams for the two-photon transfer, thereby giving a net momentum kick to the transferred cloud (see figure 6.1). Owing to the spatial separation of the transferred and remnant clouds, one could choose to either image one of the two clouds (shown in figure 6.1) or selectively remove the non-circulating component [52]. In our scheme, we make an internal-state transfer between two hyperfine states of the 3S1/2 manifold (see figure 6.2) using co-propagating Gaussian and Laguerre- Gaussian beams (similar to Wright et al. [126]), exploiting the fact that optical dipole traps can trap all spin states. The transferred and remnant clouds are in the same momentum state (at rest), but in different internal states (F = 1 mF=0 and F = 1 mF=-1 respectively, as shown in figure 6.2). To selectively remove the remnant, we need to have the two clouds in different hyperfine manifolds (remnant in F = 2 and transferred component in F = 1), so that we can shine light resonant with only the remnant cloud. Choosing a final state that is already in a different hyperfine manifold would allow this selective removal. For situations where that is not the case (figure 6.2), we separate the clouds by choosing a suitable microwave transition (|1,?1? to |2,?2?). 3?2 ??1 = ?if works equally well. In such a situation, the imprinted phase winding and hence the rotation would be in the opposite direction. 4In principle, a suitable change in spin state would be sufficient to ensure that no multi-photon processes occur (say transfer from mF=-1 to mF=+1 of the F = 1 hyperfine manifold). 131 Figure 6.1: OAM transfer via counterpropagating beams [117]: (a) Counterpropagating LG10 and Gaussian laser beams, with the same linear polarization and a frequency dif- ference (?1 ??2 = ?if), were applied to a BEC. (b) The atoms that had undergone the Raman transition (right cloud) would separate from those that had not (left cloud). A spatially localized pump beam was used for independent imaging of each cloud by the absorption of a probe beam propagating along the direction of linear momentum trans- fer. (c) Diagram illustrating energy and linear momentum conservation of the 2-photon Raman process for one and two consecutive pulses (?? here has not been normalized by ?/2). For a single pulse planckover2pi1?if = 4Er, where Er is the recoil energy of the atom. The recoil energy is calculated from Er = planckover2pi12k2/2m, where k ??1/c??2/c is the momentum of the photon. (taken from Andersen et al. [117]) 6.4 Changes from the previous experiment Since the goal of our current experiment was to realize persistent currents, similar to Ryu et al. [52], it is useful to compare the similarities and differences between the two experiments. The two experiments were broadly the same. Both experiments observed persistent flow by looking at the decay of a single quantum of flow around a toroidal geometry. Both experiments used a two-photon Raman trans- fer with an LG beam for transferring OAM. Both experiments detected the survival of flow by looking for a density minimum in the time-of-flight (TOF) expansion of the cloud (discussed in section 6.6). However, despite the general similarity, there 132 Figure 6.2: OAM transfer scheme : (left) We use co-propagating Gaussian and LG beams for the transfer of OAM, and hence do not transfer any linear momentum to the BEC. The beams travel along the same beam path as the trapping infrared LG beam. (right) The transfer occurs between the mF=-1 and mF=0 magnetic sublevels of the 3S1/2 F = 1 hyperfine manifold. 70 Hz 50 Hz Plug 6?m)? 35 Hz (? ? (e) Figure 6.3: Plugged TOP trap : (a) Toroidal trap from the combined potentials of the TOP magnetic trap and Gaussian laser ?plug? beam. (b) In-situ image of a BEC in the toroidal trap. (c) TOF image of a non-circulating BEC released from the toroidal trap. (d) TOF image of a circulating BEC, released after transfer of one unit of orbital angular momentum. (e) 3D rendering of the spheroidal trap and the optical plug. (from Ryu et al. [52]) 133 were a few key differences: ? In the plugged TOP trap, the inner diameter of the BEC was around 10 ?m (see figure 6.3). In our all-optical trap, it was around 20 ?m. Since the atoms on the inner edge have the highest velocity, depending on the inner radius, this difference is important. ? In transferring circulation to the atoms via a two-photon transfer, the previous experiments used a change of momentum state to energetically separate the initial and final states. In the new experiments, we used a change of internal state (figure 6.2). ? In detecting circulation, the previous experiments did a simple 13 ms TOF. As a consequence of our relatively larger inner radius, we employed a slightly different scheme where we lowered the toroidal confinement before doing a 6 ms TOF. ? In the previous experiment, the persistent current lifetime was limited to 10 seconds by the relative drift between the magnetic trap and the optical plug. In the current experiment, we observed flow lasting up to 40 seconds (figure 6.8), limited by the lifetime of the BEC (see section 6.7). Now that we understand the differences between the current and the previous persistent current experiments, we can move on to discuss some of the more practical considerations. 6.5 Practical considerations for the transfer of circulation Before we arrived at our current scheme of transferring circulation to the atoms, we tried out several different initial and final state combinations. The current scheme was arrived at after optimizing the OAM transfer process to have the most efficient transfer to the final state, with minimum heating of the BEC and minimum perturbation to the cloud. I shall now go over some of the technical issues associated 134 with the transfer process, and describe the changes implemented to overcome or minimize them. 6.5.a Single photon scattering Although detuned far from resonance, the Raman beams still cause some single-photon off-resonant excitation. The number of atoms lost can be expressed as the number of photons absorbed, i.e. Nl = ?Mabs. Applying the far-detuned limit, i.e. ?? ? 1, to the photon absorption ?Mabs by a cloud as described in equation (4.11), we obtain Nl = ?M? ??2 = ?MN?0 A??2 , (6.6) where ?M is the incident number of photons and ? is the on-resonant optical depth of the cloud, N is the number of atoms in the region of interest or area A. For a Rabi ?-pulse, necessary to transfer the cloud to the circulating state, the pulse area (intensity ? time) is given by ?R? = ?, (6.7) where ? is the pulse duration. Using equation (6.5) and expressing the Rabi fre- quency in terms of Isat (see section 3.1.c), we get ???I1I2 2Isat ?? = ?, (6.8) where I1 and I2 are the intensities of the two Raman beams. To express ?M in terms of intensity, we use ?M = I?A/planckover2pi1?, (6.9) and assume that the two beams have roughly equal intensities, I1 ? I2 and so, substituting I = I1,I2 from equation (6.8) into (6.9), we obtain ?M = 2?Isat?A??/planckover2pi1?. (6.10) The frequency difference has been neglected for this calculation (? = ?1 ? ?2). Finally, substituting ?M into equation (6.6), and using ?0 = ?planckover2pi1?/2Isat from equation (4.2), we obtain Nl = ???N. (6.11) 135 For simplicity, we have assumed a 3-level system. The initial and final states couple via an intermediate excited state, which in the process also causes single photon scattering. In reality, there are multiple intermediate states, each with a different coupling strength. Not all intermediate states contribute to the two-photon process. Since the intermediate states have different energies, the detuning ?? also varies between the states. We are justified in the assumption of a 3-level system in that the detuning is typically much larger than the spacing of the intermediate states, and so they behave like an effective state, although the losses are typically higher than predicted, because of the many states only contributing to loss, but not two-photon coupling. The single photon loss scales simply as a?/?? fraction of the number of atoms. Hence, it is advantageous to detune as far from resonance as possible5, so long as the laser power is still sufficient to drive the two-photon transition. We initially used a detuning of around (2?)?500 MHz (?? ? 100), which gives a (lower bound) calculated loss of 3%. However, the actual losses were much more, ?20%, for reasons mentioned above. We later used a detuning of (2?)?2.3 GHz (?? ? 50), which gives a calculated loss of 0.6%. The actual loss was around 5%. 6.5.b Dipole force Apart from the single-photon scattering, the Raman beams individually also cause an energy shift. The spatial inhomogeneity of the beams translates to a gradient dipole force on the atoms. The dipole force can cause the BEC to be excited by breathing and sloshing oscillations (section 5.7), which leads to heating and loss of atoms. As discussed in section 3.1.b, a far off-resonant beam causes an energy shift given by ?? = planckover2pi1?2/2???, which gives a total energy shift due to the two Raman 5In detuning very far from resonance, one must keep in mind possible coupling via other excited states, say the D1 line, that may affect the overall coupling strength. 136 beams ?? = ? 2 1 + ? 2 2 2??? (6.12) which, like the expression for the two-photon Rabi frequency has a 1/?? dependence. For ?1 ? ?2, we get ?? = ? 2 2??? = ?R 2 . Since the energy shift scales with the Rabi frequency, changing the intensity or detuning does not make any difference to the magnitude of the light shift. If we were to consider the case of two uniform beams, there would be a uniform light shift which would result in no dipole force since there would not be any field gradient. For that reason, we made the Gaussian beam large compared to the size of the BEC to make the light shift as uniform as possible. However, the LG beam necessarily has an intensity gradient with a length scale the size of the BEC. To minimize the dipole force due to the LG beam, we lowered the power of the LG beam, while increasing the power of the Gaussian beam to keep the two-photon Rabi frequency constant until the dipole force was smaller than other perturbations from the two-photon Raman process. 6.5.c Cleaning up the remnant In our two-photon Raman transfer scheme, the transferred cloud and the rem- nant cloud, which are in different magnetic sub-levels, are both at rest and occupy the same area in physical space. In such a situation, interactions between atoms in different magnetic sub-levels play a role. In sodium, the inter-species scatter- ing length (scattering between atoms in different magnetic sub-levels) is usually higher than the intra-species scattering length (scattering between atoms in the same magnetic sub-level) and hence any hyperfine mixture tends to spatially phase- separate [242]. This phase separation affects the condensate dynamics of both clouds and could cause the circulation to decay6. 6We observed that phase separation of the cloud due to a significant remnant tended to break the continuity of the circulating cloud around the ring, causing the flow to decay. 137 While microwave or rf transfer processes can be made to transfer 100%, the transfer of circulation cannot be 100% because of the non-uniform radial intensity profile of the LG beam, and hence there will always be a remnant that must be removed from the trap. The removal of atoms using resonant light occurs as follows. The atom absorbs a photon, acquiring a momentum planckover2pi1k from the photon, and correspondingly a recoil energy of Er = planckover2pi12k2/2m. The atom then spontaneously decays back to the ground state (not necessarily the state it started in) by emitting a photon in a random direction, acquiring another momentum kick planckover2pi1k. From the two momentum kicks, the atom, on average, acquires translational kinetic energy of 2Er [87]. At first sight, it might seem like as long as the transferred fraction and the remnant are in different hyperfine manifolds, we can simply shine light resonant with the remnant and eject those atoms. However, it is always better to have the remnant in the F = 2 state (and the transferred fraction in the F = 1), since the F = 2 can be removed with light resonant with the cycling transition, scattering many photons per atom, acquiring many recoil energies (? nEr, where n > 50 for sodium) ensuring that the atom leaves the trap. If the remnant was in the F = 1 state and F = 1 to F? = 2 imaging light was shone, a good fraction of atoms would undergo only a single recoil event before decaying to the F = 2 state acquiring only a single recoil energy (0 to 4 Er). Such atoms may not have sufficient energy to leave the trap and may collide with other atoms causing heating. To give an estimate, Er = h?25 kHz (calculated) for sodium. The total trap depth for our trap (from table 5.1) is 25 kHz. An atom absorbing a single recoil may not leave the trap, whereas an atom absorbing n? 20 recoils is almost certain to leave the trap. Hence, it is important to always remove the remnant using the F = 2 to F? = 3 cycling transition. 6.5.d Our optimized scheme After several changes and improvements, we arrived at our optimized scheme, through which we could transfer 90% of the atoms to the circulating state. In 138 our scheme (shown in figure 6.4), we make an internal-state transfer between two magnetic sublevels (mF=-1 to mF=0) of the 3S1/2 F = 1 hyperfine manifold. We then selectively perform a microwave transfer, transferring the remnant to the 3S1/2 F = 2 (mF = ?2), which can then be ejected using resonant light. Our Raman beams were detuned 2.3 GHz from single-photon resonance, limited by the various laser beam frequencies we had available7. We used a shallow trap (trap depth ? 25 kHz) so that single-photon scattered atoms could more easily leave the trap without causing heating. At low magnetic fields, the energy difference between the mF=-1 and mF=0 states is equal to the difference between the mF=0 to mF=+1 states. It is not possible toensure transfer onlytothe singlycirculatingmF=0state if the degeneracy is not broken. We break the degeneracy using the quadratic Zeeman effect (shown in figure 3.7) by applying a higher magnetic bias field (? 5 G). The bias field strength was limited by the ? 1 A current that we were able to pass through the bias field producing current coils. At 4.9 G, the linear Zeeman shift caused a splitting of around 3.4 MHz, while the quadratic Zeeman shift caused the energy differences to be split by around 20 kHz, for which we used a 100 ?s Raman pulse to minimize coupling to the mF=+1 state. We obtained a transfer efficiency of around 90% (see figure 6.5) to the final rotating state. There were some small oscillations created during the transfer of circulation due to the dipole force of the Raman beams and the sudden ? 10% loss of atoms due to the 90% transfer efficiency. These oscillations decayed in several 100 ms (similar to figure 5.11, when we deliberately excited them), and caused no problems to the circulation or its detection. However, as the energy of oscillations ultimately leads to heating, we typically waited a few seconds for the cloud to equilibriate in the trap before performing any further experiments. 7In the lab, the various laser beams used for the MOT, the Zeeman slower, absorption imaging, the Raman beams, etc. are all derived from two lasers, appropriately frequency shifted using acousto-optic modulators [114]. Acousto-optic modulators are limited in how much they can shift the frequency, thereby limiting how far from single-photon resonance the Raman beams can be detuned. 139 (c) Microwave Remnant | 1, 0 > | 2, -2> LG G (b) Raman transfer | 1, -1> | 1, 0 > (d) Clean up Remnant | 1, 0 > (e) Circulating Cloud | 1, 0 >| 1, -1> (a) Initial BEC Figure 6.4: The OAM transfer with co-propagating Gaussian and LG beams is shown : We start with an initial |F = 1,mF = ?1 > BEC, which undergoes a 2-photon Raman transfer (b) to the |1,0 > state. We then transfer the remnant |1,?1 > cloud to the |2,?2 > state via a microwave process (c) and then remove the remnant |2,?2 > atoms usingF = 2 resonant light (d). TheF = 2 resonant light, which is also used for absorption imaging of the final cloud, is used to obtain an image of the remnant (shown under (d)) to get information about the efficiency of the transfer process. Finally we obtain a circulating cloud in the |1,0 > state (e). The circulating cloud does not look very different from the stationary cloud in situ. 6.6 Detecting the circulation Vortices in a BEC are usually detected by doing a simple TOF (for example see Madison et al. [45]). The vortex core, with a characteristic size given by the condensate healing length, is too small to be resolved by an in situ image. In TOF, the vortex core expands and one can see a clear central hole. Since circulation was detected by a simple 13 ms TOF in the previous per- sistent currents experiment [52], we initially tried to detect circulation by a simple similar TOF. However, we overlooked a crucial difference between our current exper- iment and the previous persistent current experiment, which delayed our detection 140 5 0 1 0 Tra nsm iss ion OD | 1, 0 >| 1, 1 >| 2,-2 > | 1, -1 > Figure 6.5: OAM transfer efficiency : The figure shows the distribution of atoms in the various spin states after the transfer of OAM. The atoms are transferred from the initial non-circulating |F = 1,mF = ?1? state to the circulating |1,0? state via a two-photon Raman transfer process. In the process, some atoms (< 4% of initial number) undergo a four-photon transfer to the |1,1? state with two units of circulation. The |1,?1? remnant (< 5%) is then transferred to the |2,?2? state via a microwave pulse. The different spin states are then split by applying a magnetic field gradient and releasing the cloud in TOF. of circulation and persistent flow by more than four months! In performing a simple TOF for the circulating cloud (see figure 6.6 top), we found that the central hole did not persist. It appeared to close in TOF (due to mean-field driven ballistic ex- pansion), qualitatively indistiguishable from the behavior of a non-circulating cloud (figure 5.6). The reason for vortices to be visible in a simply-connected BEC in TOF is as follows: The velocity of flow around a vortex has a 1/r dependence (r is the distance from the center of circulation)8. Due to the mean-field of the cloud in the trap, atoms can climb the 1/r2 (comes from the v2s scaling of the kinetic energy) centrifugal pseudo-potential of the vortex, acquiring a kinetic energy comparable 8Here, I am assuming a lengthscale small compared to the size of the condensate or any other perturbation. 141 0 ms 2 ms 12 ms10 ms6 ms4 ms3 ms Figure 6.6: TOF sequence after relaxing radial confinement : The figure shows TOF sequences of a circulating cloud with no relaxation (top), after relaxing the strength of the radial confinement to 28% of the unrelaxed value (middle) and relaxing to 7% (bottom) respectively. With no relaxation (top), the TOF is indistinguishable from that of a non- rotating cloud. The hole due to circulation becomes larger and more resolvable with increasing relaxation. The fringe-like structure on the top left and bottom right of some of the images is due to inhomogeneities in the sheet beam potential, and does not affect the detection of circulation. The sheet beam propagation axis is at 45 degrees, from bottom left to top right, and hence there are no structures along that direction. to the mean-field9. In TOF, for the bulk of the cloud, the mean-field energy is distributed amongst the three axis in the form of kinetic energy, causing expansion. For the atoms close to the vortex core, there is a strong outward expansion due to 9In fact, the radius r? at which the kinetic energy of the vortex pseudo-potential equals the BEC mean-field, planckover2pi12/2mr2? = ? ? r? = radicalbigplanckover2pi12/2?m = (8pi?a)?1/2, is the healing length ?. Hence, for estimates of the energy of a vortex, the vortex core size can be taken to be the condensate healing length. 142 the tangential velocity of circulation. This expansion is primarily responsible for the visibility of a vortex. In our ring trap, since the inner radius of the cloud is 10 ?m, the tangential velocity of such atoms is ? 0.3 mm/s for one unit of circulation. Given a mean-field of about 1 kHz, our in-plane expansion velocities (directed radially for a ring shaped cloud) are nearly an order of magnitude higher. Expansion is radially symmetric in TOF and atoms acquire little additional tangential velocity as the cloud expands towards the center. Since the radial velocity is much higher than the tangential velocity, atoms get very close to the center of circulation. While the singularity persists in TOF, atoms nevertheless get very close to it preventing us from optically resolving the hole due to the singularity. The tangential velocities due to circulation are too small to have any impact on the TOF expansion for the durations probed. To overcome this issue, we adiabatically relax the radial confinement over 100 ms to less than 10% of its initial value before doing a release into TOF. This does three things. It allows atoms to move closer to the center gaining a higher tangential velocity. It lowers the mean-field and hence the radial velocity in TOF. Finally, by lowering the annular confinement (?r), it causes more mean-field energy to be released along the vertical direction than the radial direction (discussed in section 5.5.b), further lowering the radial velocity. This allowed us to clearly resolve the hole due to circulation (see figure 6.6). To give an estimate of numbers for the above argument, consider a typical trap, ?z = 2??550 Hz, ?r = 2??110 Hz, N = 2?105, rM =20 ?m. The chemical potential ?0 = 1 kHz (see table 5.2). The initial TF radius rTF=9 ?m. This gives a circulating velocity of 0.25mm/s for atoms on the inner of the condensate. The velocity of radial expansion of the condensate (see section 5.5.b) is ? 1.2mm/s, a factor of 5 more than the velocity of circulation. On relaxing the radial confinement to 10% of its initial value, ?r = 2??35 Hz, for the same number of atoms, ?0 ? 580 Hz. We get rTF=20 ?m, which is equal to the radius of the ring, rM, implying that the inner edge reaches the center and hence the local healing length (? ?0.6 ?m). The circulating velocity becomes 2.7 mm/s. 143 Figure 6.7: Distinguishing between circulation and no circulation : After our standardized sequence of adiabatically relaxing the radial confinement to 5% of its initial value followed by a 6 ms TOF,we could clearly distinguish a circulating cloud (left) from a non-circulating cloud (right) by its central hole. The velocity of radial expansion for this configuration is 0.3 mm/s which is a factor of 10 smaller than the circulation velocity. Hence, relaxing the radial confinement lets the circulating velocity to play a greater role in the BEC expansion, allowing us to more easily detect the presence of flow. 6.7 Lifetime of persistent currents We settled upon a a sequence where we relax the radial confinement to around 5% of its initial value, followed by a 6 ms TOF. We could clearly distinguish the case of a circulating cloud vs a non-circulating cloud (see figure 6.7). Once we realized how to detect the flow, we had no difficulty in observing the persistent currents. We found the current to last 40 seconds (figure 6.8). The lifetime of the current was limited by the lifetime of the BEC, which was ? 30 s for the data shown in figure 6.8. In our apparatus, the lifetime of the BEC was limited by background gas scattering due to imperfect vacuum. On days when the vacuum limited lifetime of the BEC was shorter, the lifetime of the persistent current was shorter. 144 Hold time (s) Survi va l Proba bil ity Figure 6.8: 40 second persistent current : We find that the persistent current lasted 40 seconds, limited only by atom loss from the BEC due to background gas scattering. The vacuum limited trap lifetime of the atoms was ? 30 s for the above measurements. The number of points for each time, from shortest to longest hold times are 4 (for 3 s), 5, 2, 2, 5 and 1 (for 60 s) respectively. The above paragraph may imply that the persistent current survives as long as there is some BEC. This is not the case. The flow decays when the chemical potential reaches the level of bumps in the potential, breaking the continuity of the annulus. The reader may be curious to know the exact decay mechanism, whether the flow decays just before continuity is broken or just after, etc. The next chapter investigates precisely this point, and so a discussion here, with the limited data that we have, would not give a complete picture. 6.8 Multiply-charged circulation Having successfully demonstrated transferring and detecting circulation, one of the next things to try was transferring multiple units of circulation. We did so by repeating the transfer process multiple times. After each transfer, we moved atoms back to the mF=-1 state using an rf pulse (figure 6.9). In this manner, we successfully transferred up to 3 units of circulation. While transferring multiple units of circulation was straightforward, detection 145 Charge 1 Charge 2 Charge 3 | 1, -1 > | 1, 0 > LG G RF pulse Figure 6.9: Multiply-charged circulation : By repeating the process of transfer of circula- tion (using an rf pulse to get the atoms back to the |1,?1? state), we were able to observe transfer of up to 3 units of circulation. The larger rotational kinetic energy due to the higher circulation causes a larger central hole in TOF, when the cloud is controlled to have about the same number of atoms. The BEC peripheral ?lobes? on the top left and bottom right of the main ?lobe? that contains the hole, are due to inhomogeneities in the sheet potential. They do not affect the measurement of circulation. was more tricky. In performing our standard relaxation and TOF, the difference between no circulation and one unit of circulation is simply the appearance of a hole in TOF (figure 6.7). The difference between one unit and say two units of circulation is the size of the hole, as the cloud with two units of circulation has four times the kinetic energy. The size of the hole gets larger with additional units of circulation, as shown in figure 6.7. However, the size of the hole also depends on the radial relaxation. Accurately distinguishing the different levels of circulation requires more care and calibration, which is one of the future directions of the experiment. Another possibility for detecting multiply-charged circulation is to remove the radial confinement and allow the BEC to become simply connected10. In a simply- connected geometry, it is energetically favorable for a multiply-charged vortex to break up into individual vortices [52]. In some of our preliminary attempts, we 10Even in the absence of toroidalconfinement, the sheet beam confines the BEC in the horizontal. See section 3.6.a. 146 were not able to find a suitable relaxation time after removing the annular con- finement, when the multiply-charged vortex split into individual vortices without any individual vortices leaving the system11. In our attempts, we found that either the multiply-charged vortex had not split up, or one of the individual vortices had already left the system. However, with a better understanding of the vortex dynam- ics, it may be possible to find a suitable relaxation sequence that would allow us to determine the presence of multiply-charged circulation. 6.9 Conclusion We were able to create and observe persistent currents lasting 40 seconds, limited only by the vacuum lifetime of the BEC. In a toroidal BEC, circulation is quantized and hence, on transferring only one unit of circulation, the survival of flow indicates that there is no dissipation (not counting the loss of atoms). This is unambiguous evidence of superflow and hence superfluidity. From apractical stand point, we found the persistent currents to be remarkably robust. We found them to survive even in bumpy traps (discussed quantitatively in chapter 7) and sub-optimal conditions such as misaligned beams (the LG Raman beam misaligned by half a radius from the center), imperfect detuning of the Raman beams (even for transfer fractions of around 40%) and lower mean-field (so long as there was continuity in the BEC around the ring). It usually took something severely wrong for persistent currents not to work. This is also a testament to superfluidity and dissipationless flow. We look forward to the day when, just like how making a BEC is so reliable, making a persistent current also becomes a routine part of the process. While making a clean ring trap merely set the stage for creating persistent currents, obtaining a reliable persistent current was an important milestone and opened the door to several experiments. The most obvious experiment is studying the decay of flow in the presence of a barrier, which constitutes the next chapter. 11Vortices in a simply-connected trapped BEC are by nature unstable excitations. 147 However, there are other possibilities, two of which I shall briefly mention. Although the kinetic energy of a single vortex is much lower than the mean- field and has no impact on the radial spatial distribution of the BEC, going to high units of circulation can significantly increase the kinetic energy. It would be interesting to know how much circulation the ring could sustain, and if there was any effect of the high kinetic energy on the radial spatial profile of the BEC [47, 218]. Primordial circulation is also interesting [151, 208]. Primordial circulation refers to circulation that was entrapped during the BEC condensation process. As with persistent currents, any such circulation is expected to be stable and last the lifetime of the BEC. During the initial experiments of persistent currents, we noticed some false positives (around 1 in 10)in vortex formation, which we later realized were due to primordial circulation from the condensation process. We could eliminate such circulation (to prevent it from interfering with our measurement) by putting a barrier in the flow path of the initial BEC to break the continuity of the ring. Nevertheless, primordial circulation is interesting and has implications on critical fluctuations. There are also interesting physics in the ring trap that can be studied using the spinor degree of freedom, which are yet to be explored. However, rather than describe other possible experiments, it suffices to say that having realized persistent currents, there are lots of interesting possibilities. We shall now move on to studies of the critical velocity of flow. 148 Chapter 7 Superfluid flow in the presence of a barrier Having successfully observed superflow and persistent currents, we went on to make a more quantitative test of superfluidity by introducing a barrier in the flow path of the ring. Given the superfluid nature of the BEC, for a weak barrier in the flow path, the superfluid property should ensure that there is no dissipation. In the opposite limit, that of a high barrier, the ring is no longer continuous and hence cannot sustain flow. Somewhere in between, there is a critical point at which the flow decays. The breakdown of superfluidity can be understood in terms of the Landau criterion (discussed in section 2.4), which states that dissipation occurs at a critical flow velocity when it is energetically favorable to create excitations. Naturally, such excitations are expected to be from the low-energy excitation spectrum, which could be phonons or vortex-like excitations. For low-level phononic excitations, the critical velocity is the speed of sound in the condensate1. For vortex-like excitations, one can use Feynman?s approximate expression for the critical velocity. Before we go on to the details of the experiment and how the various criti- cal velocities could play a role, let us take a look at previous related work. The breakdown of superfluidity in BECs has been experimentally probed in different ways. Experiments at MIT [48, 49] scanned a blue detuned laser beam back and forth through a BEC and found that excitations were created when the scan speed crossed a certain critical velocity. Engels and Atherton [50] swept a penetrable 1Often, this velocity is termed as the Landau critical velocity, which I feel is incorrect for two reasons. Firstly, it takes away credit from Landau, who stated the general expression for any excitation and then took the specific case of liquid helium considering phonons and rotons [13]. Secondly, until Bogoliubov?s theory of weakly-interacting gases [16], the expression for the sound velocity in a BEC as well as the possibility of pure phononic (without rotons) breakdown did not exist. 149 barrier along a cigar-shaped BEC and observed that excitations in the wake of the beam were created when sweep speed crossed a certain critical velocity. Experiments at the University of Arizona [51] forced a Gaussian obstacle through a superfluid and found that at a critical velocity, a vortex-antivortex pair was nucleated. In all experiments, the inhomogeneity and finite size of the sample were such that the breakdown of superfluidity could not be simply explained in terms of a phononic or vortex-like critical velocity for a uniform system. Our current work has three key differences from previous work: ? The superfluid is made to flow, while the barrier is stationary. While in the frame of reference of the atoms, this should not matter, our experiment is closer to the original superfluidity experiments [9, 10]. ? In our experiment, the superfluid is in a metastable circulating state, which is different from the stationary ground state. Previous work had the BEC in the ground state. ? In contrast to the previous work, which looked for signs of excitations in the BEC, the current work detects the survival of flow. In that sense, the detection does not depend on whether we can see the excitations or heating, and with the additional feature of a binary detection (flow survives or does not), our detection of dissipation is more direct and sensitive. Given the binary detection of the breakdown of flow, our data does not reveal the decay mechanisms, and hence, we need a good understanding of our system. In the following sections, I will first discuss the various aspects of superflow in a toroidal geometry, the stability of the persistent current and the concept of phase slips. I will then describe the dimensionality aspects of the problem, particularly with respect to quasi-2D physics, and in that context, describe some of our previous experiments on 2D Berezinskii-Kosterlitz-Thouless physics. Following that, I will go into the experimental details, our observations, the analysis of the data and subsequently make comparisons to the phononic and vortex-like critical velocities, before concluding. 150 7.1 Superfluid flow in a toroidal condensate In the previous chapter, we discussed the velocity profile of a uniform toroidal condensate with one unit of OAM (equation (6.3)). The flow profile, J, defined as the number of atoms moving through a unit cross-section in unit time, is given by J(r,?,z) = planckover2pi1mr|?(r,?,z)|2??. (7.1) As mentioned previously, the 1/r dependence comes from the fact that as one goes further from the center of circulation, the phase winding is distributed over a larger circumference leading to a lower gradient and hence lower flow. The expression holds for any system with cylindrical symmetry. The case of a ring with a barrier (non-uniform ring) is more complicated, since one cannot simply use the symmetry of the problem. Before we solve the problem of flow in a ring with a barrier, let us first look at the effect of a barrier in the ring. 7.1.a Effect of a barrier The barrier, in effect locally changes the potential of the trap. The effect can be easily seen in an in situ image (figure 7.1a). For a BEC, particularly if it is radially TF, this causes a decrease in the local mean field of the condensate (figure 7.1b). The decrease in mean field thereby causes fewer atoms to be present at the location of the barrier, which is accentuated by the transverse TF profile (radially integrated number of atoms, n1D ? ?2 from table (5.2), where ? is the mean field)), as can be seen in the azimuthal atom distribution of figure 7.1c. A second consequence of lowering the mean field of the (TF) condensate, is a slight constriction in the transverse directions (radial and vertical) of the BEC at the location of the barrier. This may seem counter-intuitive at first since we are not actively squeezing the condensate at the location of the barrier. However, because of the harmonic condition, (nearly) uniformly raising the potential at the location of the barrier causes some regions to be raised beyond the chemical potential. As the condensate is TF, particularly in the transverse direction, this leads to the 151 ?3 ?2 ?1 0 1 2 3 0 500 1000 1500 2000 Theta (radians) At oms (pe r 0 .05 ra dia n) ?3 ?2 ?1 0 1 2 30 200 400 600 800 1000 1200 1400 Theta (radians) Lo ca l Me an Fi eld (H z) (b) (c) Angle ?pi pi pi/2 ?pi/2 0 0 95 190 (a) ? Figure 7.1: Effect of a barrier on a toroidal BEC: (a) PTAI image of the BEC (colorbar shows column density in atoms/?m2). The barrier has been inserted at an azimuthal position of pi/2. (b) Azimuthal plot of the condensate mean field (see appendix C.7 for details). The condensate was broken down into 128 angular sections of 0.1 radians each, and a fully ring TF profile (column density distribution given by equation (5.35)) was fitted to each section to obtain a peak height, which was related to the mean field using equation (5.36). The barrier causes a strong reduction in the local mean field. (c) Azimuthal plot of the distribution of the number of atoms. The effect of the barrier is more pronounced because of the transverse TF profile and causes a sharper dip in the region of the barrier (for a TF BEC, radially integrated number of atoms, n1D ??2, see table (5.2)). condensate having a smaller transverse extent at the location of the barrier2. The change in shape and density of the condensate at the location of the barrier breaks the azimuthal symmetry of the problem and prevents us from using equation (7.1) to obtain the flow pattern. However, all is not lost. Some general assumptions are still valid: ? Steady state condition: If one assumes an steady state configuration where the density distribution does not change with time, flow has to be conserved for any path along a direction of flow3. For a given flow path l(s), where s is 2In principle, the displacementof atomsfromthe barrierregionshouldcausethe overallchemical potential to increase. However, in practice, for our system, this effect was small (< 5%) and we chose to ignore it in light of our nearly 10% calibration uncertainty. 3This assumes a well defined flow velocity vector at all points in the BEC. Starting at any point, one can follow the flow velocity vector around the ring, which for a steady state should form a path around the ring that closes upon itself. 152 a parameter defining the location on the flow path: J(s)squaresmallsolid ?dl = J0 = const (7.2) v(s)squaresmallsolid ?dl = J0|?(l(s))|2 (7.3) integraldisplay v squaresmallsolid ?dl = planckover2pi1m2? 2? phase winding, (7.4) where ?dl refers to a unit vector along dl. ? No flow across boundary: This is a corollary to the steady state condition. The flow being parallel to the boundary implies that there is no phase gradient normal to the boundary. Lines of constant phase are normal to the boundaries. One can use v(l(s)) ? 1|?(l(s))|2, (7.5) to obtain the flow velocity at any given point. While the exact flow pattern for a ring with a barrier cannot be simply solved, we can still learn some things. The most obvious, from equation (7.5), is that the flow velocity at the location of the barrier is higher than that of the rest of the ring due to the local depletion of the density. This effect can also be understood in terms of conservation of flow from classical fluid dynamics. The purpose of this section is to give the reader the gist of what happens when a barrier is introduced, and now we will move on to other aspects of flow in a toroidal potential. We will return to this topic when we actually try to obtain a flow velocity. 7.1.b Stability of flow in a ring So far, we have talked about superfluidity breaking down by the dissipation of flow into low-energy excitations. An implicit assumption in this picture is that the flow is somehow fixed. This assumption is valid if there is an external driver of flow, as is the case for the original superfluidity experiments [9, 10], where gravity (or a pressure difference) was used to make helium flow from one container to another via a narrow path. In such situations, a dissipation event leads to a loss of energy 153 from the flow, but will not stop the flow itself. In considering the local flow in the constrained narrow path, the flow is not expected to dissipate into other excitations if it is below the critical velocity. We could use the above picture for our toroidal BEC, and state that superflow is expected to dissipate if the local velocity at some points exceeds a critical velocity. However, unlike the case of flowing helium, the flow is not fixed and any dissipation in one part of the ring not only causes local excitations, but also causes the stopping of flow everywhere in the ring. Hence, any dissipation is a global effect, not just a local effect. The question that arises is: Is there any validity to a local critical velocity for the onset of dissipation? Taking a global picture, the stationary non-circulating BEC is the ground state. On transferring one unit of circulation to the ring, the BEC is no longer in the ground state. If one were to apply an energetics argument, the BEC is expected to decay to the ground state, which has lower energy. Then, how is it that we see stable flow? The answer to the question is that the flow is not stable, but metastable. The reason for metastability becomes more obvious if one looks at a phase plot of a circulating condensate (see figure 7.2b). The 2? phase winding of a circulating condensate gives it topological stability and does not allow any simple continuous relaxation to cause decay to the ground state (figure 7.2a). Hence, even though the circulating BEC is not in the ground state, the topology yields a metastability that allows for persistent currents. 7.1.c Phase slips Having arrived at the metastability of flow from a global picture, we shall now look at phase slips, which are necessary to cause decay of the circulating state to the non-circulating ground state. A phase slip event results in the phase of a particular location shifting by 2?, and for a ring geometry, it results in a change of the phase winding around the ring. As can be seen in figure 7.2, a phase slip event causes a change in the phase winding, changing the phase pattern in (b) to that of (d). The phase pattern in (d) has no phase winding, and can subsequently continuously relax 154 0 pi 2pi(a) (b) (c) (d) Figure 7.2: (Note: Plot is in color) Flow stability and phase slips: (a) Phase plot of a stationary BEC in a ring with a barrier. (b) Phase plot of a BEC with one unit of circulation. There is no simple, continuous deformation of the phase from the circulating state that would lead to the non-circulating state. This topological stability leads to persistent currents in the metastable circulating state. (c) and (d) show an example of a phase slip event via a single vortex moving from the inside to the outside of the annulus. In (c), the vortex, shown by a hole in the ring, is at the location of the barrier. For a path around the annulus that passes on the inside of the vortex (does not enclose the vortex), the phase winding is zero. The phase is always belowpi(see colorbar). For a path that passes on the outside of the vortex, the phase winding is still 2pi (passes through all colors). After the vortex moves to the outside of the annulus, the phase winding is zero as seen in (d). The phase can subsequently relax to a uniform phase (stationary BEC) as in (a). (generated using Scilab) 155 to (a). Phase slip events are known to occur in superconductors [1] and have also been seen in superfluids [33, 34]. Tinkham [1] describes a phase slip as an event where the coherence is mo- mentarily broken at a location in a superconductor (often at a Josephson junction), during which time the local phase changes by a multiple of 2?. Phase slips are quan- tum in nature [243], and in superconductors, phase slips have been found to interfere across junctions [244]. In superfluids, phase slips can have spatial structure. Experi- ments studying helium superflow through small apertures by Avenel and Varoquaux [33] and Amar et al. [34] have shown evidence of a phase slip mechanisms similar to dissipation via vortex rings. Coming back to our system, a phase slip is most likely to occur at the loca- tion of the barrier since the density is low, and hence the phase coherence can be more easily broken. In principle, it is possible for the phase slip to occur via the momentary formation of a plane of zero density (line when projected to a 2D plane as shown in the plots of figure 7.2) across a part of the torus, during which time the phase changes by 2?. However, an energy cost for such a mechanism is high given the width of the condensate even at the barrier. The energetics of a phase slip via the vortex movement through the barrier region are more favorable. Figure 7.2(c) illustrates a phase slip occurring via a single vortex moving through the barrier from the inside to the outside of the ring. As the vortex moves through the barrier, paths around the annulus that do not enclose the vortex (or equivalently, paths that the vortex has cut across) have zero phase winding, while the paths that do enclose the vortex (paths which the vortex has not cut across) still have a 2? phase winding. Once the vortex reaches the outside of the BEC, all paths around the annulus have a zero phase winding as seem in figure 7.2(d). This way, a vortex moving through the BEC causes the circulation to decay4. We will discuss the role of vortices for flow decay in the context of quasi-2D physics (section 7.2.e). 4A vortex passing through the BEC can be seen as a mechanism for the continuous decay of flow. 156 7.2 Dimensionality and quasi-2D physics Until now, through all the discussions of the ring trap and persistent currents, we have implicitly assumed our BEC to be a three-dimensional (3D) system. In section 5.1.b, we briefly discussed the possibility of studying quasi-1D physics as motivation for the ring trap. As mentioned before, highly anistropic 3D condensates can exhibit certain 1D-like behavior [184, 186]. In that context, it is worth exploring the dimensionality aspect of the system and how it plays a role. For the persistent current experiments we used, the dimensions of the ring were approximately ? Circumference : 120 ?m ? Radial TF full width : 18 ?m ? Vertical TF full width : 3 ?m While the above conditions were typical, we could increase the radial confine- ment (?r) by up to a factor of 2 and the vertical (?z) by up to a factor of 3. The radial confinement could also be decreased by a factor of 3, as is done when we relax the radial confinement for detecting flow. In addition, by changing the focusing of the LG trapping beam, the ring diameter could be increased by a factor of two. The current aspect ratio, along with the extent we can vary different parameters implies the following: ? Excitations in different directions have different energy scales (energy scales approximately as 1/r2i, where ri is the length scale in a particular axis). For example, azimuthal excitations (? 1 Hz) have lower energy than the radial excitations (??r = 100 Hz), which have lower energy than vertical excitations (??z = 500 Hz). ? Depending on the temperature, it is possible to be in a regime where the condensate has thermal phase fluctuations in 1 or 2 dimensions [193]. 157 Finally, in inserting a barrier in the ring, the density depletion causes the local mean field to drop to well below the vertical trapping frequency, while still above the radial trapping frequency (see figure 7.1b). This localized region has dimensions of around 9?9?2 ?m, with the condensate being mainly in the vertical harmonic oscillator ground state, making it susceptible to quasi-2D physics. As this region is important in the decay of flow, we shall now spend some time on quasi-2D physics. While on the topic, I will also mention some of our previous experiments studying the phase transition of a quasi-2D gas [53], which were performed on the same setup, with just the sheet beam (no annular confinement). 7.2.a Superfluidity and the BKT transition The origins of quasi-2D (Bose gas) physics can be understood better if one started from a true 2D system. In this section, we will study the superfluid transition of a 2D gas, before moving on to the case of a quasi-2D gas in the next. While it was understood in the 1950s that a 2D Bose gas would not Bose- condense at finite temperature [245], Berezinskii [246] and Kosterlitz and Thou- less [247] showed that a 2D neutral superfluid would undergo a phase transition to a non-superfluid state at a temperature T = TBKT at a universal value ns?2T = 4, (7.6) where ns is the 2D superfluid density and ?T is the thermal de Broglie wavelength given by ?T = radicalBigg 2?planckover2pi12 mkBT. (7.7) The BKT transition is more general than just the 2D neutral superfluid and also occurs for defects in a 2D crystal and magnetism for spins in the 2D XY model [247]. For a neutral superfluid, the microscopic mechanism of the BKT transition is based on vortex excitations in a 2D system. The energy of a single vortex in a 2D 158 a) b) c) d) Figure 7.3: The phase profile of a 2D superfluid for a) single vortex, b) single vortex anti-vortex pair, c) a 2D Bose gas below the BKT transition (T TBKT). c) and d) were created for illustrative purposes only, with vortex pais and free vortices put in at random. Colorbar same as figure 7.2. Since the flow velocity is given by the gradient of phase, changes in color correspond to kinetic energy. For a single vortex (a), there is flow even at distances far from the center, while for a vortex pair (b), superfluid far away from it is not affected. Similarly, there is a clear qualitative difference between only vortex pair excitations (c) and free vortex excitations (d). (Generated using Scilab) superfluid is given by: Evortex = integraldisplay R rv nsplanckover2pi12 2mr22?rdr, Evortex = nsplanckover2pi1 2? m log R rv, (7.8) where rv is the size of the vortex core and R is the system size. The energy of a vortex diverges with system size. If one were to consider the energy of a vortex anti-vortex pair, one obtains: Epair = nsplanckover2pi1 22? m log rs rv, (7.9) where rs is the separation between the vortex and the anti-vortex. This energy is independent of the system size for a large enough system size (R ? rs,rv), which is what one expects since for large distances, the flow field of the vortex cancels the one from the antivortex. One can look at a pictorial representation of the two cases (see figure 7.3 (a) and (b)). For a 2D Bose gas at constant temperature, the free energy of a single vortex determines whether such an excitation is favored. The free energy of an excitation 159 is given by: F = E?TS, (7.10) where F is the free energy and S is the entropy. The entropy of an excitation can be expressed as the number of degenerate ways, ?, in which the excitation can occur S = kB log?. (7.11) For a vortex, whose size scales as r2v relative to the system size R2, the entropy is given by Svortex = 2kB log parenleftbiggR rv parenrightbigg . (7.12) As one can see from equations (7.8) and (7.12), both the energy and the entropy of a vortex scale similarly with system size. For the creation of a vortex to be favorable, the free energy has to be negative: Fvortex = nsplanckover2pi1 2? m log R rv ?T2kB log parenleftbiggR rv parenrightbigg < 0, ?T > nsplanckover2pi1 2? 2mkB. (7.13) Hence, free vortices can be spontaneously created at temperatures T >TBKT, given by TBKT = nsplanckover2pi1 2? 2mkB, (7.14) which is identical to equation (7.6). In contrast to a uniform, homogeneous BEC, where phase coherence does not decay at long distances, the BKT superfluid phase has an algebraic decay of coherence5. In the limit of T ? 0, the coherence length goes to infinity like a true BEC. With increasing temperature, the coherence length gets shorter, and as the gas goes through the BKT transition, the coherence function changes from algebraic to exponential. With further increasing temperature, the coherence length further decreases until the gas has a coherence length determined by the single 5For a system with algebraic decay of coherence, the correlation function ??(r)?(r+?r)? scales as (?r)??, where ?>0. In contrast, for a system with exponential decay of coherence, ??(r)?(r+ ?r)?? exp(??r/rc), where rc is correlation length. 160 particle de Broglie wavelength. The coherence function and its derivation from vortex excitations is discussed extensively in Kosterlitz and Thouless [247]. Equations (7.14) and (7.6) express the transition in terms of the superfluid density. In terms of the total density, n, the corresponding expression is [245, 248] n?2T = log(C/u), (7.15) where C is a constant and u is the 2D interaction strength. Although the transition point in terms of the superfluid density is universal, the transition point in terms of the total density depends on interactions. 7.2.b Case of a quasi-2D Bose gas Since we live in a 3D world, no gas can be truly confined to 2 dimensions, and we have to consider the more realistic quasi-2D case. A gas can be said to be quasi-2D if excitations along one of the dimensions are frozen out. A simple way to picture this is by having a 3D Bose gas harmonically confined in one dimension, having energy sufficiently low that all the atoms are in the ground state in the trapped dimension. In such a system, one needs to treat the gas in the 2D picture, with a dimensionless interaction (coupling) constant, g2D, given by Petrov et al. [249] g2D = ?8? aa ho , (7.16) where a is the 3D atom-atom s-wave scattering length, aho = radicalbigplanckover2pi1/m?0, and ?0 is the trapping frequency of the confining potential. Using the above expression, one can treat the quasi-2D case identical to the 2D case and hence a BKT transition is expected to occur. The BKT superfluid density still satisfies equation (7.6) and the total density satisfies an equation similar to equation (7.15): n?2T = log(C/g2D). (7.17) Using Monte-Carlo calculations, the constant C was calculated to be 380 [250]. While BKT theory deals with the phase transition from a state without free vortices to a state with, it does not talk about the high temperature limit. A quasi- 161 ?3 ?2 ?1 0 1 2 30 5 10 15 20 25 30  - C n 2 Thermal SuperfluidQuas i - Co nd en sate Figure 7.4: The phases of a quasi-2D gas: Calculated total density (green), quasi- condensate density (red) and superfluid density (blue). The chemical potential is nor- malized by g2D/?2T with the BKT transition occurring at ? = ?c, where the superfluid density jumps and is given by ns?2T = 4. (based on Prokof?ev and Svistunov [251]) 2D Bose gas at temperatures well above the BKT transition temperature is expected to behave like a classical gas. However, as one approaches the transition from above, there is an appearance of a non-superfluid quasi-condensate [251] phase, which has some coherence before a sudden transition to a superfluid quasi-condensate state. A quasi-condensate can be understood as a phase which behaves like a true condensate at short length scales, but does not have long range order. In figure 7.4, the phases of a quasi-2D gas are plotted. Until now, we have talked about the transition in terms of temperature. The BKT transition can also be discussed in terms of phase-space density (n?2T) as a function of the chemical potential. When the chemical potential is near the critical point, the conditions given by equations (7.6) and (7.17) are satisfied. As one can see, prior to the transition, there is the appearance of a significant (non-superfluid) quasi-condensate. The quasi-condensate component is indistinguishable from a condensate except that it has quasi-long range order (short coherence length) as opposed to the true long range order of a BEC. At a local level, it behaves like a condensate. The 162 Phase Appearance Coherence ? Thermal Classical gas Gaussian ?T Quasi-condensate QC component exponential >?T with fluctuations Superfluid SF and QC components algebraic longer Table 7.1: Brief summary of the different phases of a quasi-2D Bose gas. This table is only to give a general picture. Coherence refers to the functional form of the coherence decay. ? is the coherence length. QC = quasi-condensate. SF = superfluid. quasi-condensate region in figure 7.4 is characterized by large scale fluctuations. 7.2.c Previous experiments studying the BKT transition In one of the earliest experiments on BKT physics in neutral superfluids, Bishop and Reppy [252] studied the superfluid transition of a thin superfluid he- lium film adsorbed on an oscillating substrate. Analysis of the superfluid mass and dissipation supported a dynamic theory of an oscillating superfluid based on the BKT picture [253, 254]. Estimates of the change in superfluid density were in good agreement with predictions of the BKT theory. There have been several studies of the BKT transition for the 2D XY model using Josephson junction arrays. Resnick et al. [255] reported evidence for the BKT transition in triangular planar arrays of proximity coupled Pb-Sn junctions. Shaw et al. [256] saw a change in the characteristic of the magnetic flux noise in a Josephson junction array as one crossed the BKT transition. Leemann et al. [257] studied the ac response of large two-dimensional arrays of proximity-effect Joseph- son junctions to an oscillating driving field as a function of temperature, applied transverse magnetic field, and frequency. They observed that for an integer number of flux quanta per unit cell, a peak in dissipation and a drop in superfluid den- sity are observed near the superconducting transition of the array. They concluded 163 that these features and their frequency dependence provided clear evidence for the vortex-unbinding BKT transition. With ultracold gases, there were two path-breaking experiments in Paris study- ing the BKT transition in quasi-2D trapped Rubidium. In Hadzibabic et al. [258], using a matter wave heterodyning technique, the long-wavelength fluctuations of the quasi-condensate phase and free vortices were observed. Starting at low tem- peratures, where the gas is quasi-coherent on the length scale set by the system size, there was a loss of long-range coherence that coincided with the onset of the proliferation of free vortices as the temperature was increased, providing experimen- tal evidence for the microscopic mechanism underlying the BKT theory. In Kr?uger et al. [259], the critical atom number for condensation6 in an array of harmoni- cally trapped two-dimensional (2D) Bose gases of rubidium atoms was measured at different temperatures. The critical atom number being 5 times higher than that predicted by the semiclassical theory of Bose-Einstein condensation (BEC) in the ideal gas, the authors found that a simple heuristic model based on the Berezinskii- Kosterlitz-Thouless theory of 2D superfluidity and the local density approximation accounted well for the experimental results. While studying similar physics to the Paris experiments, our experiment dif- fered in key ways. We studied the quasi-2D Bose gas in a single plane, imaging perpendicular to the plane of interest. In addition, we used a homodyne technique to measure the coherence length in contrast to the heterodyne technique used in Paris. The reader should note that there have been recent experimental work on the quasi-2D Bose gas by the Paris group [260], studying the quasi-2D Bose gas with even tighter confinement. Also, Eric Cornell?s group at JILA [261] recently studied the in situ position and momentum distribution of a 2D Bose gas more carefully, to clearly observe non-mean-field physics in the non-superfluid quasi-condensate phase. 164 200 ms 500 ms 5 s Figure 7.5: Excitations in a 2D BEC: On transferring a BEC from our magnetic TOP trap to the optical sheet, there are excitations which damp out over time. Images are taken after a certain wait in the optical trap followed by a 5 ms TOF. The 2D BEC seems more susceptible to density fluctuations. 7.2.d NIST experiments in the quasi-2D trap Our trap was created by just the sheet beam (see section 3.6.a). Thez trapping frequency was? 1 kHz, and the horizontal trapping frequencies were? 20 Hz, giving an aspect ratio of 1:50. In preliminary experiments in the quasi-2D trap, we were able to get a con- densate in our trap, identified by the clear signature of a bimodal distribution (con- densed and thermal atoms) in TOF. Our first indications of quasi-2D physics were excitations in the BEC. In contrast to a 3D BEC where the thermal fraction can be taken as an indicator of heating or the presence of excitations, we noticed that excitations in the form of large density ripples (see figure 7.5) seen in 5 ms TOF. If we waited a few seconds, the density ripples were reduced. Furthermore, by deliberately exciting the quasi-2D BEC when there were few other excitations, we could see ?holes? (figure 7.6, image of 5 ms TOF) in the cloud in TOF. The BEC was excited by pulsing a strong magnetic field gradient to transfer impulse. On allowing the BEC to expand in 8 and 10 ms times-of-flight, 6It was determined by the appearance of a narrow feature, akin to a BEC, above the broad Gaussian thermal distribution of a gas in TOF. 165 5 ms 8 ms 10 ms Figure 7.6: Deliberate excitations in a 2D BEC: We deliberately cause excitations to the BEC by pulsing a strong magnetic field gradient, giving it a momentum impulse. There are ?hole?-like density ripples suggestive of vortices. The merging of these in longer times- of-flight (TOF indicated below figure) is similar to what one would expect if there were vortex-antivortex pairs. there were fewer, but larger holes, implying that some of the smaller holes ?merged? together, as one would expect of vortex-antivortex pairs. The nature of excitations was indicative of something different from 3D BECs, and suggestive of quasi-2D vortex excitations. We performed a set of measurements of the quasi-2D condensate varying the number of atoms (or equivalently, the chemical potential ?) close to the apparent transition point. From the absorption images obtained, we performed a bimodal fit to the cloud. The cloud fit well to a bimodal double Gaussian (wide and narrow Gaussian modes superposed). The wide Gaussian fit the non-condensed (or thermal) cloud, and stayed constant, both in width and amplitude7, across the range of the number of atoms probed for a given trap depth, indicating that the temperature was constant. The narrow Gaussian fit the quasi-condensed atoms. For a 5 ms TOF (fig- ure 7.7 (left)), we found that the width of the narrow Gaussian initially decreased 7Any change was immeasurably small. This is similar to a 3D BEC at equilibrium with a thermal cloud, where, on adding more atoms, the thermal cloud changes little while the BEC changes in height, and consequently in width and shape due to mean field repulsion. 166 as a function of ? or equivalently the peak 2D column density in TOF (presumably due to fewer phase fluctuations) and subsequently increased (due to mean field). The sharp change in the sign of the slope was indicative of a phase transition (el- bow point in fitted lines in figure 7.7 (left)). The obtained critical points agreed with predictions of the BKT theory (figure 7.7 right), although only after correcting for excitations in the tight confinement direction based on Holzmann et al. [262]. We also observed the expansion of the cloud for longer TOF. At 10 ms TOF, we observed a sudden appearance of a narrow peak (above the bimodal distribution), which was consistent with the sudden appearance of superfluid fraction at the BKT transition [53]. Such measurements were subsequently performed more carefully by the JILA group [261]. Phase coherence measurements also indicated an increase of coherence from the thermal de Broglie wavelength to the size of the condensate [53]. 0 10 20 15 5 Temperature (nK) Cri tica l D en sit y (a t/? m2 ) 10080 120 180140 16060 200 T = 177 nK T = 88 nK 0 5 10 15 20 25 300 10 20 30 40 50 60 Co nd en sa te wi dth (? m) Peak Density (at/?m2) critical point Figure 7.7: Critical density: (left) We obtain the critical density by finding the minimum width (elbow point) in a plot of the condensate width vs the peak density. The data is taken after 5 ms TOF for two different temperatures shown. (right) On comparing the critical density (translating the 5 ms TOF to in situ) to BKT theory, we find that our data is systematically higher than that predicted by BKT theory (dashed line: n?2T = log(380/g2D) for our conditions). However, if one were to correct the theory for excitations (occupation of higher modes) in the vertical direction based on Holzmann et al. [262], one gets better agreement (solid line). In effect, we identified three regions for a quasi-2D Bose gas near the critical 167 point. The gas underwent a phase transition at a point which agreed with the predictions of BKT theory. Our measurements were indirect in the sense that we could not directly observe free vortices or vortex-antivortex pairs or observe the change in functional form of the coherence, which some experts would consider as the hallmarks of the BKT transition. Also, our experiments (or any other atomic gas experiments) did not look at any superflow. Nevertheless, our measurements were in good quantitative and qualitative agreement with the quasi-2D BKT theory, and have been further confirmed by subsequent simulations by Bisset and Blakie [263]. Our experiments observed the quasi-2D gas near the critical point differently from the Paris experiments [258, 259]. Imaging perpendicular to the plane, we obtained a 2D image of the cloud, and could directly measure the 2D (column) density. We could therefore perform a 2D Gaussian fit to the cloud, and obtain the peak 2D density of the cloud, indicative of the chemical potential ?. Hence, our determination of the critical point was based on the minimum width of the narrow mode of the bimodal distribution as a function of the peak 2D density. In the Paris eperiments, it was based on the change in functional form of the correlation function [258] or the appearance of a bimodal distribution [259]. More importantly, our experiments were able to clearly identify the three phases; the thermal, the non-superfluid quasi-condensate and the superfluid quasi- condensate. The critical point separated the quasi-condensate and the superfluid phases. The quasi-condensate was distinguished from the thermal cloud by nature of its coherence length being longer than the thermal de Broglie wavelength ?T. 7.2.e Effects of BKT physics and quasi-2D dimensionality The BKT experiments [53] have shown that our vertical confinement by the sheet beam is sufficient to observe quasi-2D physics and even the BKT transition. As mentioned previously, the barrier region is particularly susceptible to quasi-2D excitations due to the reduced mean field interactions there. We will now look at what are the possible effects due to the quasi-2D nature of the gas in the barrier region. 168 As discussed earlier in the chapter, the circulating state can decay to the non- circulating state via a phase slip, which in our system is believed to be vortex motion through the region of the barrier. A vortex moving through the condensate has an energy barrier, which is the cost of putting a vortex at the center of the annulus. The energy of such a vortex is E = ?planckover2pi1 2n2D m log rt ?, (7.18) where rt is of the order of the width of the annulus at the location of the barrier, ? is the healing length of the condensate8 at the location of the vortex and n2D is the mean column density of the condensate in the region of the barrier. The exact energy depends on various parameters such as the exact shape and density profile of the condensate near the barrier and so equation (7.18) should be taken as the first approximation. As one can see, increasing the height of the barrier would cause the local density, n2D to decrease and the healing length, ? to increase, both of which lower the energy barrier for a vortex to leave. While one could have a vortex (of the same sign as the circulation) pass from the inside of the annulus to outside or an anti-vortex pass the other way, the mech- anism with the lowest energy barrier would be a vortex from the inside and an anti-vortex from the outside coming in and annihilating each other at the center. This is also consistent with the low energy excitation spectrum for a quasi-2D gas that is superfluid, which consists of vortex-antivortex pairs. Such a pair would have minimal effect on regions away from the barrier, which have high density. While the cost of putting a vortex and an anti-vortex in the middle of the condensate is higher than the cost of keeping them at the edge of the condensate, such a pair would have an attraction which would lower the overall energy barrier. The condensate flow causes an additional attractive force between the vortex and the antivortex9. The 8For simplicity, the vortex core sizerv can be taken as the healing length of the condensate [264]. The error in this approximation is small compared to that arising from other approximations we will make. 9From a classical perspective, this is the Magnus force [265]. The sign of the Magnus force 169 energy of a vortex-antivortex pair in the annulus can be expressed as [266, 267] E = 2?planckover2pi1 2n2D m log rs ? ?2?planckover2pi1n2Dvfrs, (7.19) where rs is the distance between the pair and vf is the velocity of flow10. We assume that the pair starts at a distance 2rt, which is the width of the annulus and attempts to move inward. The flow velocity causes an attractive force between pair, while the energy barrier is the cost of putting a vortex and an anti-vortex in the center of the annulus, where the density is maximum. Flow at the critical velocity causes the attractive force to overcome the energy barrier, which leads to the decay of circulation. If the superfluid already had thermally excited vortex-antivortex pairs, the flow could decay by causing a pair to split and leave in opposite directions11. In such a situation, the velocity term of equation (7.19) would oppose the vortex-antivortex interaction energy term. The critical velocity would depend on the temperature, which determines the initial separation of the pair. We have discussed some of the possible mechanisms for the decay of flow and how dimensionality may play a role. We shall now move on to the critical velocity experiments and will return to this discussion in the context of our experimental results. 7.3 Experimental setup and procedure Our experiment determined flow survival in the presence of a barrier. The barrier consisted of an elliptical blue-detuned 532 nm beam that intersected the annulus. The barrier beam was measured to be about 15 ?m by 4.3 ?m (1/e2 radii), with the long dimension extending from nearly the center of the ring across the atoms depends on the sign of the vortex, and hence, the force on the vortex is of the opposite sign to the force of the antivortex. This is an additional force to the vortex pair attraction; a mutual attraction or repulsion or shear force depending on the direction of flow. 10The sign of the velocity dependent energy term depends on the direction of flow. 11If the direction of flow were such that it initially caused attraction between a vortex pair, the vortex pair would reorient and the force would then become repulsive. 170 38 ?m 8.6 ?m 30 ?m (a) (b) Figure 7.8: Schematic of the barrier beam: (a) 3D rendering of the barrier beam and its effect on the BEC. (b) Relative orientation of the elliptical barrier beam with respect to the trapping LG beam. The barrier is sufficiently elongated in the transverse (radial) direction so that the BEC cannot flow around it. to well outside the annulus (figure 7.8) so that the atoms could not flow around it. The barrier was much larger than the condensate healing length (? < 0.5?m) in bulk and so we can model the depletion using TF formalism. Our experiment consisted of creating a persistent current, then turning on the barrier for 1-2 seconds, followed by turning it off and checking to see if the flow survived. By taking several data points at each of a range of different barrier heights and chemical potentials, we were able to obtain the critical point for the breakdown of flow. The procedure for taking data was the following (see figure 7.9). We created a BEC in the ring trap. After holding for about 3 seconds in the ring trap, we per- formed the Raman transition, obtaining a circulating cloud in the |1,0? state. After allowing the ring to relax, and excitations from the transfer process (discussed in section 6.5) to settle down, we then turned on the barrier and adiabatically ramped it to its full height, Vb in 100 ms. After holding the barrier at its maximum value for 2 seconds, we ramped down the barrier in 100 ms and turned it off completely. We then detected circulation (see section 6.6) by ramping down the ring and doing a 6 ms TOF. Rather than finding the critical barrier for a fixed BEC, we found it eas- 171 Im ag ing Trap O? Barrier Beam On: 1-2 seconds Seg425ling g415me for persistent current: (3-5 seconds)Ram an 6 ms TOF Figure 7.9: Experimental sequence studying critical flow. The sloped lines indicate inten- sity ramps of the barrier beam with adiabatic turn on/off in ?100 ms. The full sequence time is 30-40 seconds including around 25 seconds required to create the BEC. ier to vary the chemical potential ?0 for a fixed Vb and find the critical chemi- cal potential, ?c that was sufficient to overcome the barrier and allow the persis- tent current to survive. The chemical potential depends on the number of atoms, ?0 = radicalbig(1/2?2)Ng?z?r/mrM, (from equation (5.38) assuming a fully TF BEC), which we varied in a controlled way (in addition to uncontrolled shot-to-shot fluc- tuations) over a large range by doing one or more of the following: 1. Adding delays before or after the Raman process : Since the vacuum lifetime was about 15 seconds for many of the runs, a 3 second extra delay would lower the atom numbers by about 20%. By varying the delay, we could accurately control the range of the number of atoms. However, delays made the sequence time longer slowing down the rate of data and so for large changes, we needed other methods. 2. Doing a microwave cut : By removing a fixed fraction of atoms, we can deter- ministically lower the number of atoms. While in principle, we could have cut any fraction we wanted, large cuts (more than 25%) caused excitations which could lead to heating, and hence we limited cuts to 25%. After any cut, we waited at least 3 seconds for the excitations to damp out. 172 3. Lowering the Zeeman slower current : This works by lowering the number of atoms we load into our MOT and is thereby useful in significantly lowering the number of atoms. The only practical limit to this method was that we needed a sufficient number of atoms in the BEC to sustain a persistent current before the barrier was turned on. We interspersed our critical flow data with calibration data, necessary for calibrating ?0 and Vb. Calibration data was taken using an identical procedure, except that we imaged the cloud in situ while the barrier is still on. Since the cloud was optically thick, we used partial-transfer absorption imaging (see chapter 4), typically transferring 8-42% of the cloud. The calibrations are detailed in appendix C. 7.4 Observation of the breakdown of flow From each experimental run we obtain two pieces of information: a binary (survival or not) of flow and the number of atoms in the cloud (by integrating the atom column density over the area of the cloud), which correspondingly gives ?0. We find that for a given barrier height, the survival probability clearly depends on ?0 (figure 7.10). The flow survival dependence on the chemical potential can be divided into three regions. At high ?0, the flow always survives (above 1100 Hz for Vb/h = 780 Hz, shown in figure 7.10, number of atoms N > 225?103). Below that is a critical region, where the probability of survival increases with atom number (1000 to 1100 Hz for Vb/h = 780 Hz, N ? 185?225?103 atoms). At the low end, the flow always decays (below 1000 Hz for Vb/h = 780 Hz, N < 185?103). For a lower barrier height (figure 7.10 (top), for Vb/h = 650 Hz), the behavior is identical except that the critical region occurs at a lower point ?0 (800-950 Hz). In figure 7.11, we plot the critical chemical potential as a function of the barrier height. For each experimental run we calculate the chemical potential, ?0, assuming an annular Thomas-Fermi condensate profile, where N is the number of atoms, g is the interaction strength, rM is the radius of the annulus and ?z and ?r are the 173 P fl ow 800 900 1000 1100 1200 ?0 (Hz) 700 0 0 1 1 P fl ow Barrier 650 Hz ?c1 ?c2 Barrier 780 Hz Figure 7.10: Flow survival as a function of chemical potential, ?0, for two barrier heights: (Upper, blue)Vb/h = 650 Hz (barrier beam power = 75 ?W), and (lower, red) Vb/h = 780 Hz (barrier beam power = 91 ?W). Presence or absence of flow for a single condensate is shown by closed circles. Open circles are the average of data within the bins (vertical lines), representing the flow survival probability (Pflow) of each bin. A critical chemical potential ?c for stable flow is found from a sigmoidal fit (solid lines) to the data for each Vb/h. The width of the sigmoidal is believed to be primarily due to temporal drift of the trapping potential that causes the effective height of the barrier to vary (see appendix C.7). The vertical green lines indicate the height of the barrier (to compare against the chemical potential). vertical and radial trapping frequencies respectively. Here, we have neglected the azimuthal variations in ? and the volume of the barrier, which has a < 4% effect on the calculated chemical potential. We fit a sigmoidal [1+exp(?c??/?w)]?1 function to the data obtaining a critical chemical potential ?c and the 1/(1 +e2) half-width of the critical region, 2?w. We obtain the barrier height relative to the chemical potential by analyzing its effect on the BEC. We take a series of in situ images (figure 7.1a) of the BEC with varying barrier beam power. For each image, we divide the annulus into angular 174 Figure 7.11: Critical chemical potential vs barrier height (Vb): The chemical potential at the critical point (?c) is plotted against the barrier height. We find a linear scaling with a slope of 1.6 (continuous line). The chemical potential is calculated based on the number of atoms in the trap. For each barrier height, we fit a sigmoidal [1 + exp(?c ??/?w)]?1 to obtain the critical chemical potential ?c and critical region width, ?w. The vertical uncertainty is of size ?2?w (from sigmoidal fit), and the horizontal uncertainty is ?20 Hz, based on our calibration uncertainty. The images indicate regions of flow survival (top left) and regions of flow decay (bottom right) sections and measure the local peak mean field (gn3D,max for the section), ?l, as a function of the azimuthal coordinate, ? (see figure 7.1b). We use the fully Thomas- Fermi approximation to obtain ?l = parenleftbig(9/32)n22D,lm?2zg2parenrightbig1/3 (equation (5.35)) from the peak column density, n2D,l, of each section. For regions of the cloud close to the barrier, where the Thomas-Fermi approxi- mation is clearly not valid (?l < planckover2pi1?z), we assume a harmonic oscillator ground state profile along z giving ?l =radicalbig?planckover2pi1?z/m?n2D,l (equation (5.32)). To get the barrier height, we use Vb = ?0 ?min(?l), (7.20) 175 where min(?l) is the minimum ?l as a function of ?. We correct for our imaging resolution which is expected to cause our barrier to appear around 15% wider (see appendix C.7). We find that the critical chemical potential scales approximately linearly with the barrier height (figure 7.11) over the full range of barrier heights and chemical potentials. This supports that the breakdown of flow occurs due to a depletion of density in the barrier region, which can be compensated by raising the chemical potential. The linear dependence of ?c on the barrier height has slope of 1.6, which indicates that it is not simply a matter of the barrier cutting off the flow. The > 1 slope indicates some sort of flow velocity dependence, based on flow conservation, which leads to excitations and dissipation. Taking our analysis to the next step, we need to obtain the flow velocity so that we can compare the critical velocity to different criteria. 7.5 Flow velocity of a circulating ring We return to the problem of obtaining the flow pattern in a circulating ring. By imposing the conditions of phase winding, no flow across boundaries and steady- state flow, we could calculate the complete flow profile. However, since the purpose of the experiment is more to understand the physics rather than make a precision measurement, it is more useful to employ a simple model where the physics is easier to see. Also, since our calibrations of the barrier height and the chemical potential have significant uncertainties, there is little benefit from a precisely calculated flow velocity. We make the following simplification. We treat the ring as a 1D object by integrating out the transverse directions. That is, we neglect the r (and z) variation of the ring and assume it to be fixed at r = rM (and z = 0). The phase at any angle ? is fixed, and hence the flow and the flow velocity have only a ? dependence. We then use equations (7.3) and (7.4) to obtain the flow velocity. Due to the depleted 1D density at the location of the barrier, the velocity is maximum there. 176 ?3 ?2 ?1 0 1 2 3 Theta (radians) 0.00 0.10 0.20 0.30 0.40 Ve loci ty (mm/ s) Angle ?pi pi pi/2 ?pi/2 0 ? Figure 7.12: Sample flow velocity profile: We plot the azimuthal flow profile of a cloud computed after integrating the transverse profiles of the ring to create a 1D object. Con- servation of flow leads to the flow velocity being maximum at the location of the barrier, as seen by the prominent peak reaching nearly 3 times the velocity for the rest of the ring. The above plot does not correct for optical resolution. In the absence of the quantization condition, if flow was fixed (if it was ex- ternally driven for example), the (transverse integrated) 1D density at the barrier would be sufficient to determine the peak flow velocity (such a system is considered in Watanabe et al. [268]). However, because of the quantization condition, given in equation (7.4), the peak flow velocity depends on the complete density distribution. To illustrate the point, we start with our ring in the absence of a barrier, which is azimuthally more or less uniform. The flow velocity everywhere is ? planckover2pi1/mrM = 0.15 mm/s (ignoring the minor azimuthal non-uniformities). On adding a single barrier, we obtain the distribution shown in figure 7.12, which has a peak of ?0.4 mm/s and the flow at other regions to average < 0.15 mm/s. If another barrier was added somewhere else in the annulus, the overall average flow would further decrease. The peak flow velocity would also decrease, even though nothing changed in the vicinity of the barrier. If the entire ring had a potential as high as the peak of the barrier, the flow velocity would go back to case of a uniform ring and hence the velocity everywhere would be planckover2pi1/mrM. The counter-intuitive nature of the flow velocity and its dependence on the density distribution of the entire ring ties in with the global picture for dissipation 177 of flow. Both the condition for the dissipation of flow and the complete stoppage of flow are global in nature, and depend on and affect the flow in the entire ring. The flow velocity at the barrier follows directly from the density distribution (shown in figure 7.12). Using our in situ data and barrier calibration, we were able to obtain a density distribution for a given number of atoms (or ?0) and barrier height. This was then used to obtain the flow velocity at the barrier. As mentioned earlier in the context of the azimuthal smoothness of the ring, azimuthal variations in the density profile lower the accuracy to which we can determine the flow velocity. Hence, the smooth ring was key to a good measurement of the superfluid flow velocity. 7.6 The critical velocity While one may argue that the phononic critical velocity is irrelevant, as there needs to be a phase slip and hence vortex-like mechanisms are the likely cause of dissipation, I feel that it is still necessary to put the phononic critical velocity in the right context. As discussed earlier, the phononic critical velocity of a BEC is the speed of sound. The speed of sound in bulk has a physical meaning and is easy to calculate. Since the phononic critical velocity is nearly the same for all phononic modes (see section 2.2.c), flow at the speed of sound is likely to excite several modes and is almost certain to cause severe dissipation. Hence, the speed of sound serves as an upper bound on the critical velocity. For a BEC, the speed of sound c is given by [16, 269] c = radicalbigg? l m, (7.21) where ?l is the local mean field, and the size of the sample is large enough for c to be meaningful. In comparing our critical velocity to the sound speed, there is one detail: Our condensate has an inhomogeneous density distribution (given in equation (5.24)) and so the speed of sound is also spatially varying. Taking a cross-section of the ring, one finds that the local speed of sound is zero at the edges. Since the flow 178 velocity is non-zero across the cross-section, there is some point at which the flow velocity exceeds the critical velocity purely by virtue of an inhomogeneous profile. How do we resolve this inconsistency? The Landau criterion for the dissipation of flow is based upon the lowest energy excitations. If we were to take a step back and try to see what kinds of excitations are possible, we immediately notice that the transverse excitations (along r and z) have much higher energy than the longitudinal excitations (along ?). Hence, the longitudinal phononic excitations are the lowest energy phononic excitations and need to be considered separate from the other excitations. In our inhomogeneous system with a TF cross-section, the longitudinal effective speed of sound at the barrier is given by [215, 270, 271] ceff = c?2 = radicalbigg? l 2m, (7.22) where ?l is the (peak) local mean field at the location of the barrier. Plotting our obtained critical velocity vs ceff (see figure 7.13), we see that our data spans a range of sound speeds (1.1 to 1.6 mm/s) and critical flow velocities (0.7 to 1.0 mm/s). This was possible because of the azimuthal smoothness of the ring that allowed us to vary the chemical potential and therefore flow velocity by a large range. We find that the critical flow velocity is around 60% of ceff. This is consistent with it being lower than the sound speed. This is also to be compared to the measurement of ?15% for the Washington State University experiment [50], and ?25% (correcting for the cylindrical shape of the laser beam [48]) for the MIT experiments [49] and the University of Arizona experiments [51]. Experiments in superfluid helium have also found superflow to breakdown at velocities well below the critical velocity due to phonon and roton excitations (see Wilks [272] or Varoquaux [32]). Coming to the critical velocity of vortex-like excitations, we look at the energy cost of putting a vortex-antivortex pair with separation rs at the middle of the annulus in the barrier region. From equation (7.19), we can see that the energy is 179 Figure 7.13: Critical velocity comparison: The critical velocity vc normalized by the effective local sound speed,ceff is plotted. The flow breaks when the flow velocity is around 60% of theeffective sound speed. Thegray bandshowsthe predicted (2D) Feynman critical velocity. negative if12 vf ? planckover2pi1mr s log rs?, (7.23) where ? is the healing length. Again, we have the problem of the inhomogeneous density profile, and hence ? varies across the cross-section. For any vortex-pair recombination, the vortices have to pass through the center of the annulus, and hence the peak density and the corresponding ? are relevant. The log(rs)/rs dependence of vf makes it relatively insensitive to rs for the range of rs in our system (TF width at the barrier), and so the choice of rs is less critical. We set13 rs = 2rTF = 2radicalbig2?/m?2r and ? = radicalbigplanckover2pi12/2?m, where ?l/2 ? ? ? ?l is the transverse variation of the mean field at the location of the barrier. Plotted against our data (see figure 7.13), we can see that our data lies within the gray band 12This expression is very similar to Feynman?s [24] expression for the critical velocity for creating vortices in a channel (discussed in section 2.5.b). The difference is a factor of 2 (rs is the full width of the channel), which arises from the quasi-2D nature of our condensate near the barrier. 13While the condensate may not be TF along z, it is still TF along r. 180 indicative of the (2D Feynman) critical velocity for a vortex-pair excitation. The specified range of ? is a best guess to accommodate the inhomogeneous transverse profile, where the mean field varies from ?l at the center and 0 at the edge. While the match looks good, it must be kept in mind that both the expression for the Feynman critical velocity, and the calculation of the critical flow velocity are approximate, and a more careful calculation of the various parameters and the energetics are needed. Still, that all the points fall within the band (within error bars) is suggestive of the mechanism of flow decay by vortex production. Simulations of systems similar to ours [273], although with circulation of 8planckover2pi1, indicate decay of flow by a vortex entering the ring. This could be compoared to the annhilation of a vortex-antivortex pair at the center of the annulus. In their case, presumably due to the high circulation, the vortices were dragged along with the flow, which should not happen in our experiment since we have only one unit of circulation. Experiments studying helium superflow through small apertures by Avenel and Varoquaux [33] and Amar et al. [34] have also shown evidence of a phase slip mechanism, although the 3D nature of these systems makes vortex rings more likely. The critical velocities for such events have been in agreement with the Feynman critical velocities (discussed extensively in Varoquaux [32]). 7.7 Conclusion and future work In conclusion, we have been able to clearly identify the decay of flow due to the presence of a barrier. Our estimates of the critical velocity match well with that necessary for the decay via a vortex-antivortex pair. Our critical velocity experiment is a first step to studying several phenomenon. In the experiment, we worked with a cloud that was effectively at zero temperature. A possible experiment is studying the effect of temperature, which could lead to decay of flow via a different mechanism, the splitting of thermally excited vortex- antivortex pairs. Another possibility is studying the effect of dimensionality by varying the radial and vertical confinement. 181 Recently, we have had success in testing an acousto-optic deflector (AOD) to generate arbitrarilyshaped potentials. By engineering the barrier using the AOD, we could change the dimensionality of the BEC in the barrier region; that would allowus to explore the channel width dependence of the critical velocity. We could also create a definite constriction with nearly hard walls, which would more closely approach the superfluid helium experiments allowing more direct comparisons. Using the AOD, we could also create a moving barrier. Such a barrier could be used to deterministically stir the condensate, as described in Brand and Reinhardt [224]. Josephson effects have been observed in BECs separated by a weak link [274, 275]. The barrier in our setup serves as a weak link, and hence the weak link in a superfluid circuit constitutes a first step towards the atom analog of a superconduct- ing SQUID. A possible scheme for sensing rotation is to rotate the barrier around the ring close to the critical velocity for one unit of circulation. Small changes in the rotation speed would then translate to circulation being imparted to the BEC or not. Such changes could arise from the rotation of the apparatus as a whole or even the rotation of the earth, thereby creating a rotation sensor. As I conclude this thesis, I would say that our journey these last 4+ years has been quite long. From our first days of creating an all-optical toroidal potential, we now have very good control of the ring BEC, from how to create it to how to characterize it. We also now have a good understanding of what sustains a persistent current, and what it takes for the flow to breakdown. Along with being able to measure the critical velocity, we are well poised to perform experiments with moving barriers and atom SQUIDs. The future looks bright with possibilities of creating rotation sensors or other ?atomtronic? devices (atom analogs of electronic devices). With that, I would like to thank you for reading this thesis. 182 Appendices Appendix A Analyzing PTAI images A.1 Calculating the uncertainty of measurement using PTAI In this section, we calculate the uncertainty of measurement using PTAI. We start with the expression for the total noise (equation (4.18)), and use it to determine the uncertainty, ??m. The area of interest in the cloud has an OD of?. We transfer a fraction ?, giving a transferred OD of ?f = ??. By imaging the transferred fraction, we obtain an inferred optical depth, ??m, which can be expressed in terms of the photon counts on the detector ? ?Mt1, the latter being the measured quantity: ??m = ?f ? = ? 1 ? log ? ?Mt ? ?M (A.1) Correspondingly, the uncertainty in the measurement, ??m, can be expressed in terms of the statistical variation of the transmitted probe light. We assume that the reference signal, ? ?M, which comes from the image of the probe beam on CCD in the absence of atoms has been averaged over several realizations and so has no uncertainty associated with it: ??m = 1??(?Mt)? ?M t . (A.2) From the limit to the number of photons, eq. (4.17), we get: ? ?Mt = ??Nfe ??f 1?e??f (A.3) The variation in the detected photon counts comes from the noise, eq. (4.18). ?(?Mt) = radicaltpradicalvertex radicalvertexradicalbt?Nf?e??f 1?e??f parenleftBigg 1 + ??? 2 fe ??f 1?e??f parenrightBigg (A.4) 1The bar indicates an average, assuming an average over several realizations. The uncertainty is based on a single realization and hence in the notation, I do not use a bar. 184 Substituting equations (A.4) and (A.3) into equation (A.2), the uncertainty in the measurement can be calculated: ??m = 1? radicaltpradicalvertex radicalvertexradicalvertex radicalvertexradicalbt parenleftbigg 1+ ???2fe ??f 1?e??f parenrightbigg ??Nfe??f 1?e??f ??m = 1? radicalBigg 1?e??f +???2fe??f ??Nfe??f (A.5) where, the last step is only an algebraic simplification. Equation (A.5) expresses the uncertainty in measurement, ??m as a function of the OD of the cloud (?), the transferred fraction (?f) and the pixel resolution (one can express Nf in the denominator as ?fA/?0). The first two terms in the numerator under the square- root come from the photon shot noise, while the last term comes from the atom shot noise. Depending on which term is larger, the respective shot noise will dominate. A.2 Calculating the uncertainty in measurement using PCI In this section, we calculate the uncertainty in measurement of phase-contrast imaging as a function of the OD of the cloud and the perturbation to the sample. We choose a region of interest identical to the PTAI case, having an area, A, N atoms and an (on-resonant) OD ?. In phase-contrast imaging, the probe light is usually far detuned from resonance, (???0)/(?/2) = ? greaterorsimilar 10. Applying ?2 ? 1 to equations (4.10, 4.11 and 4.12), we obtain (see Lye et al. [140] for example): ?Mt = ?Me??/?2 (A.6) ?Mabs = ?M(1?e??/?2) (A.7) ?? = NA?0 12? = ?2? (A.8) To retain sensitivity at higher optical depths and not have the phase wrap around, we have to also ensure that ?? 2?? . (A.9) This condition ensures that e??/?2 ? 1, and so, the absorption of photons through the sample can be expressed as, ?Mabs = ?M ? ?2 (A.10) Since the absorption of photons through the sample is low, we can assume that there is very little change in the number of photons in the probe beam as it passes through the sample, i.e. Mt ?M. A.2.a Photon shot noise For simplicity of calculation, we will assume the following setup. The trans- mitted probe beam, ?M?, interferes with a reference local oscillator of ?ML photons on the detector. The calculation holds for the more typical phase-contrast imaging setups [136], if one uses a large enough probe beam and an appropriate interfer- ence scheme. Deviations from these ideal conditions causes some distortion, which is beyond the scope of this thesis. For our model interference setup, the detected number of photons is: ?Md = ?ML +? ?M + 2radicalbig ?ML? ?M cos(?0 +??) (A.11) where ?Md is the detected number of photons and?0 is the native phase shift between the beams. For maximum detectability, we set the relative phase ?0 = pi2. This simplifies equation (A.11): ?Md = ?ML +? ?M ?2radicalbig ?ML? ?M sin(??) (A.12) In the absence of atoms, the number of photons incident on the detector is: ?Md,0 = ?ML +? ?M, (A.13) 186 which is the reference signal. Substituting the phase shift ?? from equation (A.8), we obtain the detected number of photons, ?Md, as a function of the OD, ?: ?Md = ?ML +? ?M ?2radicalbig ?ML? ?M sin parenleftbigg ? 2? parenrightbigg . (A.14) The photon shot noise goes as the square root of the number of photons inci- dent on the detector, ?Md. The photon shot noise for phase-contrast imaging is: ?phot = radicalBigg ( ?ML +? ?M)?2 radicalbig ? ML? ?M sin parenleftbigg ? 2? parenrightbigg (A.15) A.2.b Uncertainty in the measured optical depth Analogous to the PTAI case, we start with an expression for the measured OD, ??m, and obtain an expression for the uncertainty ??m arising from the photon shot noise, ?phot. From equations (A.8),(A.12) and (A.13) we obtain an expression for the measured OD, ??m: ??m = 2?sin?1 parenleftBigg ? Md,0 ? ?Md 2 radicalbig ? ML? ?M parenrightBigg , (A.16) where ?ML and ? ?M are measured prior to taking the image. We assume that these quantities have been measured repeatedly and so have no shot noise associated with them. Again, we assume that the reference is averaged over several images and so has no shot noise associated with it (?Md,0 = 0). Differentiating equation (A.16) with respect to the signal, ?Md, we get: ??m = 2? ? ?1? parenleftBigg ? Md,0 ? ?Md 2 radicalbig ? ML? ?M parenrightBigg2? ? ?12 ?Md 2 radicalbig ? ML? ?M (A.17) Substituting the variation in photon number, ? ?Md from equation (A.15), the uncertainty in measurement is: ??m = ? radicalBig ( ?ML +? ?M)?2 radicalbig ? ML? ?M sinparenleftbig ?2?parenrightbig cosparenleftbig ?2?parenrightbig radicalbig ? ML? ?M (A.18) 187 In order to obtain the minimum uncertainty without increasing the amount of light passing through the atom cloud, we set ?ML >> ?M. This assumes a sufficient dynamic range of the detector, greaterorsimilar 2 ?ML. Equation (A.18) simplifies to ??m = ? 1cosparenleftbig ? 2? parenrightbig radicalBigg 1 ? ?M. (A.19) Although we detect only the phase-shift of the beam, there is some absorption of light (see equation (A.7)). Expressing the number of incident photons in terms of the absorption through the cloud, we find: ??m = 1cosparenleftbig ? 2? parenrightbig radicalBigg ? ? ?Mabs. (A.20) Each absorbed photon corresponds to an atom undergoing a recoil event. For ultracold gases, even a single recoil event transfers kinetic energy much higher than the temperature and other energy scales. We can set ?Mabs = Nd, where Nd is the number of atoms undergoing recoil events, which is a measure of the perturbation of the sample. We express the absorbed light in terms of the number of atoms undergoing recoil. We obtain: ??m = 1cosparenleftbig ? 2? parenrightbig radicalBigg ? ?Nd, (A.21) which expresses the uncertainty of measurement in terms of the OD and the number of atoms undergoing recoil. A.3 Change in transmitted intensity due to optical pumping In performing PTAI, there is some optical pumping due to off-resonant excita- tion. The optical pumping can be accounted for by solving for the change in optical depth as a function of time given an certain incident intensity. However, as seen in section 4.6.b, the differential equation is more easily cast in terms of the transmitted intensity It(?) than the optical depth ?(?). In this section, we will solve for It(?), given an incident probe intensity Ii. We start with the basic differential equation, 188 reproduced from equation (4.27): dIt(?) d? parenleftbigg 1 It(?) + 1 Isat parenrightbigg = ? ?bplanckover2pi1???2 parenleftbigg Ii ?It(?) + I 2 i ?I 2 t (?) 2Isat parenrightbigg . We will now make a suitable change of variables to make the algebra simple: x = It(?)I sat + 1, B = IiI sat + 1, and A = ?bIsatplanckover2pi1???2. Equation (4.27) then becomes: dx d? parenleftbigg 1 x?1 + 1 parenrightbigg = ?A parenleftbigg B?1?(x?1) + (B?1) 2 ?(x?1)2 2 parenrightbigg , (A.22) which then simplifies to dx d? parenleftbigg x x?1 parenrightbigg = ?A2 (B2 ?x2). (A.23) Equation (A.23) can be solved by first moving allxcontaining terms to the left-hand side, xdx (x?1)(B2 ?x2) = ? Ad? 2 , (A.24) then splitting the terms into partial fractions, dx 2(B2 ?1) parenleftbigg 2 x?1 + B?1 B +x + B + 1 B?x parenrightbigg = ?Ad?2 , (A.25) and finally integrating the equation to give dx 2(B2 ?1) log parenleftbigg(x?1)2(B +x)B?1 (B?x)B+1 parenrightbigg = ?A?2 +C, (A.26) where C is the constant of integration. Substituting for It(?), and using the ? = 0 condition for C, we finally obtain 1 ?Ii(?Ii + 2) log parenleftBigg It(?)2(Ii +It(?) + 2Isat)?Ii (Ii ?It(?))(?Ii+2) parenrightBigg = A? + 1 ?Ii(?Ii + 2) log parenleftBigg It(0)2(Ii +It(0) + 2Isat)?Ii (Ii ?It(0))(?Ii+2) parenrightBigg , (A.27) where ?Ii = Ii/Isat. The solution is not particularly insightful, and is presented here just for com- pleteness. 189 Appendix B Fully Thomas-Fermi condensate in the ring trap In this appendix, we will calculate the relationship between the number of atoms and the chemical potential for a fully Thomas-Fermi condensate. The poten- tial is given by V(r,z,?) = 12m?2zz2 + 12m?2r(r?rM)2. We start with the 3D density distribution: gn3D(r,z,?) = ?0 ? parenleftbigg1 2m? 2 zz 2 + 1 2m? 2 r(r?rM) 2 parenrightbigg for ?0 >V(r,z,?)(B.1) = 0 everywhere else For integrating along z, one needs to be aware that the limit of integration depends on the radial coordinate r. In the region of the condensate, we set up the integral with limits: gn2D(r,?) = integraldisplay zr ?zr dz parenleftbigg ?0 ? parenleftbigg1 2m? 2 zz 2 + 1 2m? 2 r(r?rM) 2 parenrightbiggparenrightbigg , (B.2) where zr = radicalBig ?0?12m?2r(r?rM)2 m?2z/2 is the distance up to which the condensate extends at a radial location r. gn2D(r,?) = parenleftbigg ?0 ? m? 2 r(r?rM) 2 2 parenrightbigg z|zr?zr ? 16m?2zz3|zr?zr = parenleftbigg ?0 ? m? 2 r(r?rM) 2 2 parenrightbigg 2zr ? 16m?2z2z3r Substituting zr, we get gn2D(r,?) = 2(?0 ? 1 2m? 2 r(r?rM) 2)3/2 radicalbigm?2 z/2 ?13m?2z parenleftbigg? 0 ? 12m?2r(r?rM)2 m?2z/2 parenrightbigg3/2 gn2D(r,?) = 4 ?2 3radicalbigm?2z (?0 ? m?2r(r?rM)2 2 ) 3/2 (B.3) 190 The 2D column density profile obtained scales as a 3/2 power of the expression for the radial TF case (see equation (5.31)), which makes it similar in shape except for a steeper drop-off near the TF width. I will call this functional form the 3D-TF profile. The peak 2D density is given by: n2D(r = rM,?) = 4 ?2?3/2 0 3radicalbigm?2zg. (B.4) We then set up the integral to integrate the 2D column density to get the number of atoms: N = integraldisplay rM+rTF rM?rTF rdr 4 ?2 3gradicalbigm?2z (?0 ? m?2r(r?rM)2 2 ) 3/2 integraldisplay d?, (B.5) where rTF = radicalBig 2?0 m?2r , as defined in chapter 5. Since the trap is radially symmetric, we can easily integrate over ? to obtain: N = integraldisplay rM+rTF rM?rTF rdr 8 ?2? 3gradicalbigm?2z (?0 ? m?2r(r?rM)2 2 ) 3/2. (B.6) In order to integrate over r, we first express ?0 in terms of rTF: N = integraldisplay rM+rTF rM?rTF rdr4?m? 3 r 3g?z (r 2 TF ?(r?rM) 2)3/2. (B.7) We will now make a change of variable. We then set q = r?rM. The change in integrand and limits of integration are given by: dq = dr, (B.8) for r = rM ?rTF ?q = ?rTF, and (B.9) for r = rM +rTF ?q = rTF. (B.10) Substituting the above expressions in equation (B.7), we get: N = integraldisplay rTF ?rTF rMdq4?m? 3 r 3g?z (r 2 TF ?q 2)3/2. (B.11) Since the integral is symmetric about q = 0, only rM, which is the even part of rM +q (comes from r in equation (B.7)) survives, and so the term containing q has not been shown in equation (B.11). 191 We then make a further change of variables and set ?q = qr TF . This gives us: N = integraldisplay 1 ?1 rMd?q4?m? 3 rr 4 TF 3g?z (1? ?q 2)3/2. (B.12) We can then use a trignometric integral to solve, setting sin? = ?q, which gives: cos? d? = d?q, (B.13) ?q = ?1 ?? = ??/2, and (B.14) cos? = radicalbig 1? ?q2. (B.15) Substituting these expressions for ?q, we get: N = integraldisplay pi/2 ?pi/2 rMd?4?m? 3 rr 4 TF 3g?z (cos?) 4. (B.16) This is a known trignometric integral giving us: N = rM 4?m? 3 rr 4 TF 3g?z ? 3? 8 . (B.17) Substituting for rTF, we obtain: N = 2?2 rM? 2 0 gm?z?r. (B.18) 192 Appendix C Calibrations for critical velocity This appendix explains the various calibration procedures used. C.1 Time calibration Our PulseBlaster board was tested against various (calibrated) oscilloscopes, and the timing was found to be accurate to the limit of the measuring device. This makes our knowledge of time (for Raman or microwave pulses and time of flight) the most accurate measurement, better than 10?4 (specified to be accurate to 10 ns). In the remainder of this section, I will neglect any inaccuracies in our knowledge of time, and treat time as perfectly known. C.2 Length calibration and pixel size Since we know the imaging system we have in place, we can determine the magnification and pixel size purely from the location of various optical elements. However, it is usually more accurate to measure a property of the atoms to determine the lengthscale. The simplest such method is to image a cloud falling under gravity. We first calibrate our horizontal imaging system to watch the change in position of a cloud falling in time of flight, and then use that to calibrate the vertical imaging system. C.2.a Calibrating the horizontal imaging system using gravity In data taken on October 29 2008, we took a series of measurements watching the cloud fall under gravity (see figure C.1). Our camera pixel size is given as 6.45 ?m. Since our imaging system consists of a 20 cm focal length lens followed by a 50 cm lens (both convex achromatic doublets), we estimate an initial pixel size (size 193 In - situ 4ms 12ms In-situ 1/2 x Time2 (ms)2 Di sta nce (? m) gravity slope = 9.8m/s2 initial estimate 1 pixel = 2.58 ?m best fit calibration 1 pixel = 2.30 ?m c) d) a) Figure C.1: Calibrating length scale using gravity. Absorption images (a), (b) and (c) show the cloud at different times of flight. We match the pixel size to acceleration due to gravity. As one can see, all points fall on the straight line. at the plane of the BEC corresponding to one pixel on the camera) of 2.58 ?m. Matching the location of the cloud on the camera with what is expected of a freely falling object (12gt2, where t is the time-of-flight and g = 9.8m/s2 is the acceleration due to gravity), we obtain a calibrated pixel size of 2.30 ?m (figure C.1(d)). Our pixel size calibration agrees with our initial estimate, the difference aris- ing from the first lens not being at the proper focal distance from the BEC. The distance between the two lenses were not the sum of focal lengths (analogous to 4F imaging), and so the magnification is sensitive to minor displacements of the lens. The difference between the predicted and actual pixel size correspond to the first lens being placed 1 cm closer to the plane of the BEC than the focal length. The discrepancy is quite plausible as the BEC sits inside vacuum in a 2 cm glass cell (see figure 3.8), and so its precise position cannot be determined. 194 C.2.b Calibratingthe verticalimaging system by matching cloud size The horizontal and vertical images share a common axis (which is along the Zeeman slower). If one were to take an image of the same cloud using both cameras and integrate out the non-common axis in the respective images, one should get an identical line profile from each camera. Taking a set of 3 images from each camera, we compare and match up line profiles to get the calibration of the vertical camera (figure C.2). We get a pixel size (at a 5x zoom lens setting) of 1.2 ?m. The uncertainty in the calibration is estimated to be less than 10%. The calibration matches what one would expect from an estimate of the magnification of the imaging set up. Green ? vertical cam @1.2 ?m/pixel Blue ? horizontal cam @ 2.3 ?m/pixel Axis along slower (?m) Line D ensit y (ar b units) Figure C.2: Matching the line profiles from the two cameras. By adjusting the x-axis scales of the vertical camera, we match up the line profiles of 6 ms time-of-flight images to get a calibration of the pixel size of the vertical camera from the known calibration of the horizontal camera. We use 3 images of each to get a fair sample size. While the regions of line density may not match up as the regions of high optical depth are not accurately measured, the slope at the edge of the cloud matches well. C.3 Measuring the trap frequencies We measured trap frequency by parametric heating. By summing a small AC signal to the amplitude control of the AOM driver for the ring or the sheet beam, 195 we could modulate the dipole trap. We sum a 20-50 mV signal (1-4%) for a time varying between 100 ms and 2 s. Modulating a beam at the corresponding trap frequency causes parametric heating. In our system, the trap depth is relatively low and only a weak diffuse thermal cloud can be contained. Therefore, it was easier to detect heating by looking at the loss of atoms from the trap (particularly the BEC) due to thermal excitation. We measured the number of atoms in situ immediately after the parametric heating. As we required only a relative measurement of the heating, this measurement sufficed for us to determine the trapping frequency. C.3.a Vertical trapping frequency The vertical confinement is due to the sheet beam1. Due to non-uniformities in the beam believed to be due to interference fringes from a secondary reflection as the beam passes through the glass cell, parametric heating occurs for a broad range of frequencies (more than 50 Hz 1/e2 full width in 540 Hz). However, when we performed the same experiment with the ring beam turned on, we were able to narrow the range. We measured the sheet trapping frequency, ?z on June 9 2010. ?z was found to be 545?5 Hz (uncertainty in mean) for a sheet beam power of 135 mW, which was the configuration we did our critical velocity experiments in. Measuring the trap frequency at different sheet beam intensities, we obtained values consistent with the relation ? ??P (?/2? is the trap frequency, and P is the beam power). We also measured heating at close to twice the fundamental frequency. C.3.b Radial trapping frequency The annular confinement is due to the LG ring beam. Performing a similar process for the LG beam (performed on June 11 2010), we found the radial trap- ping frequency, ?r to be 107?2 Hz for a ring beam power of 27 mW (our typical 1While it may seem that parametric heating should not occur in this direction at the trap frequency ?z because of the symmetry in the beam, a combination of the gravitational potential and the anharmonicity of the trap is believed to allow parametric heating to take place. 196 experimental condition). We verified the ? ? ?P scaling for the ring beam power also. C.4 Measuring imaging parameters In getting any quantitative measurement of atom column density out of ab- sorption imaging, one needs to know the scattering cross-section (?0), the incident and transmitted light intensities (Ii and It) and, to correct for the effects of satura- tion, the saturation intensity (Isat, see section 4.2). While it may seem more logical to get the scattering cross-section first, it is more complicated. Saturation intensity, on the other hand can be determined by comparing the uncorrected measured atom number for different the incident intensities of the light, without knowing anything else. 0 5000 10000 15000 0 50 100 150 200 250 Probe Intensity (counts) Number of Atoms (x1000) Figure C.3: Calibrating the saturation intensity: For a set of 6 images of nearly identical clouds, at varying probe intensity, we plot the number of atoms against the average probe count (green). We then make a correction (red), setting the saturation intensity to 10000 counts. The dotted lines are a guide to the eye. 197 C.4.a Measuring saturation intensity To determine the saturation intensity, we take a set of absorption images of nearly identical atom clouds (data taken on May 6 2010) and different probe intensities. The clouds are imaged in time-of-flight with full optical repump. To correct for saturation, we take an overall average correction factor as opposed to an individual pixel-by-pixel correction factor. This is because certain variations in the probe intensity are due to optics downstream of the atom cloud. We calculate the probe intensity by averaging the pixel counts of the reference probe image (in the absence of atoms) over the region of atoms, and using that average value ?Ii (equation (4.9)) as the incident intensity for all pixels. We correct using ?c(x,y) = ?u(x,y)+ ?Ii Isat(1?e ??u(x,y)), (C.1) where?u(x,y) = ?log(It(x,y)/Ii(x,y)) is the uncorrected optical depth and?c(x,y) is the corrected optical depth. We iteratively correct for Isat as shown in figure C.3. We obtain a saturation intensity of 10 000?1000 digital detector counts per pixel (figure C.3). This value is consistent with the estimated intensity (? 10 mW/cm2) of the probe beam at the location of the atoms based on the calibration of the camera measuring the intensity at the atoms. C.4.b Calibrating optical pumping for PTAI images For in situ images, imaging with partial-transfer absorption imaging (PTAI), we need to correct for optical pumping because we cannot use repumping light (see section 4.6). The reader should note that although off-resonant excitation is sufficiently large so as to cause optical pumping, it is nevertheless small compared to resonant excitation and so does not contribute to the scattering cross-section. We use the calibration for saturation from the time-of-flight, optically repumped images and then apply another correction for optical pumping. While an exact analytic expression for optical pumping is not possible, we use the approximation in equation (4.30), which acts as a simple overall scaling factor and which works well for low transfer optical depth. 198 0 1000 2000 3000 4000 5000 6000 7000 0 20 40 60 80 100 120 Probe Intensity (counts) Number of Atoms (x1000) Figure C.4: Calibrating the optical pumping: For a series of 17 partial transfer images taken of nearly identical clouds, we plot the saturation-corrected number of atoms as a function of probe intensity (blue and red). We are able to correct for optical pumping to obtain the atom number independent of the probe intensity (black and green). While the correction is not linear, for this limited range of probe intensities, it appears so. The dotted lines are a guide to the eye. The blue (and corresponding black) points were taken with a 23% microwave transfer, while the red (and green) were taken with a 34% transfer. As one can see, the lower transfer fraction gives a higher scatter in atom number. We took a series of measurements of an in situ cloud (data taken on May 28 2010) with varying probe power. We found optical pumping to be a factor of 3 more than the calculated value for the given intensity of light assuming isotropic light polarization. We do not have an explanation for this. The corrected and uncorrected plots are shown in figure C.4. We also test our calibration, which was only based on imaging in situ data at various probe intensities, against optically repumped images (figure C.5). On June 14 2010, we took several sets of images of identical clouds imaging certain clouds in situ by PTAI and others in time-of-flight using optical repump. By taking the sets in quick succession, we avoid long term drifts in atom number due to varying 199 conditions of oven temperature or transfer fractions. The atom numbers determined by the two methods were found to agree. The PTAI images had more shot to shot fluctuations. This is presumed to be due to atoms in situ giving a smaller signal. Checks taken closely in time agreed well. Those that were not taken closely in time did not agree as well, presumably due to other systematic drifts of experimental conditions during the time interval between the checks. Figure C.5: Verifying the optical pumping calibration : We take a series of in situ partial transfer images (red) and time-of-flight optically repumped images (green). We compare shots taken under similar conditions (indicated on graph). The measured number of atoms of images taken back to back by the two techniques agree very well, while those taken some duration apart agree, but not as well. C.4.c Measuring the scattering cross-section The natural next step in imaging calibration is determining the scattering cross-section. When imaging with a cycling transition, each atom absorbs several photons, undergoing spontaneous decay during the process. Given the polarization of light and the initial state of the atom, one can calculate the scattering cross-section from the transition dipole moment and the appropriate Clebsch-Gordon coefficients. However, when an atom undergoes spontaneous decay, one cannot predict which 200 state it will decay to and so cannot determine in which state an atom is starting from. Certain special cases can be calculated (see Steck [156]). ? |F = 2,mF = ?2? ? |F? = 3,mF = ?3? with ?? circularly polarized light, magnetic sub-level cycling transition : In this case, spin selection rules specify that the atom stays in the cycling transition, which keeps the initial state fixed allowing one to calculate the scattering cross-section. For sodium, this value is 1.657?10?13 m2 [156]. This transition uses the highest transition dipole matrix element and so is the upper bound for the scattering cross-section. ? F = 2 ? F? = 3 isotropic light polarization: In this case, the distribution is averaged over all magnetic sublevels, and the scattering cross-section is averaged over all the transition dipole matrix elements. For sodium, this value is 0.7734?10?13 m2 [156]. In our system, we use circularly polarized light. However, our bias field (?1 G) is perpendicular to the direction of propagation of light and so our system does not map on to either of the above mentioned cases. We expect our scattering cross- section to be in-between the above mentioned values. To determine the exact value, we use our observed atom-atom repulsion to independently determine the density of atoms. Knowing the density, one can calculate the scattering cross-section from the absorption of light going through the cloud. C.5 Calibrating the mean field At ultracold temperatures, the interaction between sodium atoms can be char- acterized by a single parameter, the s-wave scattering length (see Tiesinga et al. [101]): a = 2.75nm (C.2) Using mean field theory, in the TF regime, one can define a position dependent mean field energy for a BEC. For a condensate of density n3D(r), where r is the 201 position in space, the mean field energy, ?(r) is given by ?(r) = gn3D(r), (C.3) where g = 4?planckover2pi12a/m. To know the shape of our atom cloud, given the trapping potential and mean field interaction, we use the time-independent Gross-Pitaevskii (GP) equation (see Dalfovo et al. [63]): E?(r) = ? planckover2pi1 2 2m? 2?(r)+Vext(r)?(r) +g|?(r)|2?(r) (C.4) where ? is the condensate wavefunction, E is the energy of the eigenstate and Vext is the trapping potential. The GP equation is a non-linear Schrodinger equation. The first term on the right is the kinetic energy, the second term is the potential energy and the third term is the mean field energy. The lowest energy eigenstate of the GP equation is the ground state. The GP equation cannot in general be solved analytically. In the limit of low interaction (g|?|4 ? planckover2pi122m|??|2 ), one gets the linear Schrodinger equation, which can be solved exactly for a harmonic oscillator. In the limit of high interaction energy (Thomas-Fermi approximation) compared to the kinetic energy, (g|?|4 ? planckover2pi122m|??|2), one gets a simple analytic solution: n3D(r) = |?(r)|2 = 1g (?0 ?Vext(r)), (C.5) where ?0 is the energy per particle (or chemical potential of the system). The Thomas-Fermi (TF) approximation breaks down at the edge of the condensate (?0 = Vext(r)), due to the sharp discontinuity in the derivative of the condensate density. While this can be corrected for, its overall effect on the shape of the cloud away from the edge and on the peak cloud density is usually negligible. In this analysis, we use the Thomas-Fermi approximation. We perform various cross-checks and discuss possible corrections to account for deviations from the TF regime. Our trap can be approximated as (see chapter 5) V(r,z,?) = 12m?2zz2 + 12m?2r(r?r0)2 (C.6) 202 where ?z is the vertical trapping frequency and?r is the annular trapping frequency. We use cylindrical coordinates (r,z,?). Using equation (C.5), we obtain the con- densate density: gn3D(r,z,?) = ?0 ? parenleftbigg1 2m? 2 zz 2 + 1 2m? 2 r(r?r0) 2 parenrightbigg for ?0 >V(r,z,?) (C.7) = 0 everywhere else The column density for such a profile would be (see equation (5.35) in chapter 5) gn2D(r,?) = 4 ?2 3radicalbigm?2z (?0 ? m?2r(r?r0)2 2 ) 3/2 for (r?r0)