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Phases of Bosons in Optical Lattices and Coupled to Artifical Gauge Fields Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2014 Phases of bosons in optical lattices and coupled to artificial gauge fields Qinqin Lu Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Digital Par t of the Physical Sciences and Mathematics Commons Commons Network Recommended Citation Logo Lu, Qinqin, "Phases of bosons in optical lattices and coupled to artificial gauge fields" (2014). LSU Doctoral Dissertations. 80. https://digitalcommons.lsu.edu/gradschool_dissertations/80 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. PHASES OF BOSONS IN OPTICAL LATTICES AND COUPLED TO ARTIFICIAL GAUGE FIELDS A Dissertation Submied to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Qinqin Lu B.S., University of Science and Technology of China, 2008 May 2014 The degree candidate's name is 吕骎骎 in Chinese. Due to variations of transliter- ating Chinese characters into the Latin alphabet, the candidate's name can be spelt as any combination of Qinqin, QinQin, Qin-Qin (given name), and Lu, Lü, Lyu, Lv, Lui (surname). ii 为⾃⼰的⽇⼦ 在⾃⼰的脸上留下伤⼜ 因为没有别的⼀切为我们作证 —— 海⼦《我，以及其他的证⼈》 iii ACKNOWLEDGMENTS During my Ph.D. study, there have been many people who I am grateful to. I would like to thank my advisor Dr. Daniel Sheehy for his insightful directions and kind guidance in my daily research. I would like to thank Dr. Kelly Paon, a former postdoctoral member in our group, for the useful discussions with him. I would like to thank my commiee members for their time devoted. I would like to thank the professors who have instructed or helped me, especially Dr. A. Ravi. P. Rau, for their enthusiasm on students. I would like to thank my friends, local and international, for their support which is indispensable to my career and my life. I would like to express my special grati- tude to my old friends from my hometown or from my undergraduate university, for our long and steadfast friendship. We will take a cup of kindness, for auld lang syne. I would like to thank my parents and my extended family for their unswerv- ing love to which I am forever indebted. 爸爸妈妈，谢谢你们。 I would like to thank the world for its massive diversity and thank all human civilizations for their breathtaking creativity, as seeking truth, beauty and virtue in them makes a meaningful life. Lastly, to quote Bertrand Russell, “Of all forms of caution, caution in love is perhaps the most fatal to true happiness.” iv TABLE OF CONTENTS ACKNOWLEDGMENTS . iv LIST OF FIGURES . vii ABSTRACT . x 1 INTRODUCTION . 1 1.1 A Brief History of Ultracold Bosons . 2 1.2 Structure of This Thesis . 3 2 NON-INTERACTING AND INTERACTING BEC . 6 2.1 Thermodynamics of Non-Interacting BEC . 6 2.2 Interacting Systems . 8 2.3 Interacting Systems: Mean Field Theory . 9 2.3.1 Gross-Pitaevskii Equation . 9 2.3.2 Bogoliubov Approximation . 11 2.3.3 Hartree Fock Approximation . 13 2.4 Interacting Systems: Bose-Hubbard Model . 17 2.5 Interacting Spinor BEC . 20 3 BOSONS IN AN OPTICAL LATTICE . 24 3.1 Optical Laice . 25 3.2 Superfluid Transition of Bosons in an Optical Laice . 26 3.3 BEC Transition Results . 31 3.3.1 Unit Filling f = 1 ........................... 31 3.3.2 Multiple Filling f > 1 ......................... 32 3.3.3 In Comparison: Mathieu Approach and Approximating with Effective Mass . 33 3.3.4 Effect of Higher Bands . 35 3.4 Free Expansion . 36 3.5 Non-Interacting Density Profile after Free Expansion Results . 40 3.6 Interacting Bosons in Optical Laice: Hartree Fock Approximation . 46 4 INTERACTING BOSONS IN AN OPTICAL LATTICE: HARTREE FOCK SELF-CONSISTENT SCHEME . 49 4.1 Model . 50 4.2 Results . 56 4.2.1 Transition Temperature . 57 4.2.2 Condensate Fraction . 59 4.2.3 Boson Density . 60 4.2.4 Density Profile after Free Expansion . 62 v 5 SETUP FOR BOSONS WITH SPIN-ORBIT COUPLING . 68 5.1 Background . 68 5.2 Raman Scheme . 70 6 Phases in Bosons with Spin-Orbit Coupling . 76 6.1 Model . 76 6.1.1 Band Structure of the Bare Hamiltonian . 76 6.1.2 Low-Energy Effective Hamiltonian . 78 6.2 Phase Diagram . 81 6.2.1 Fixed Density Phase Diagram . 81 6.2.2 Fixed Chemical Potential Phase Diagram . 83 6.3 Trapped Bosons with SOC . 95 6.3.1 Model with Local Density Approximation . 96 6.3.2 Results . 98 6.4 Sound Mode . 108 6.5 Concluding Remarks . 112 7 CONCLUSION . 113 BIBLIOGRAPHY . 116 APPENDIX A BOGOLIUBOV HAMILTONIAN . 120 APPENDIX B GREEN'S FUNCTION . 123 APPENDIX C BLOCH FUNCTION AND WANNIER FUNCTION . 127 C.1 Bloch Function . 127 C.2 Wannier Function . 128 APPENDIX D MATHIEU EQUATION . 131 APPENDIX E BOUNDARY CONDITIONS AND FINITE SIZE EFFECT . 138 E.1 Vanishing Boundary Condition . 138 E.2 Periodic Boundary Condition . 139 APPENDIX F PRECISION TEST . 143 APPENDIX G FREE EXPANSION . 148 APPENDIX H RABI OSCILLATION . 155 APPENDIX I EFFECTIVE LOW-ENERGY HAMILTONIAN . 159 VITA . 165 vi LIST OF FIGURES 1-1 Velocity distribution of atomic cloud at 400, 200, 50 nK . 2 2-1 Superfluid-Mo insulator phase diagram . 19 2-2 Superfluid-Mo insulator phase transition . 20 3-1 Optical laice . 26 3-2 Mathieu characteristic functions at different laice depths V0 .... 29 3-3 Effective mass for dispersion in optical laice . 30 3-4 BEC transition temperature for unit filling . 32 3-5 BEC transition temperature at multiple filling . 33 3-6 Comparing Mathieu approach to the effective mass approximation . 34 3-7 Mathieu approach and the effective mass: near unit filling . 35 3-8 Mathieu approach and effective mass: small filling . 36 3-9 Characteristic function vs. free particle dispersion . 37 3-10 Characteristic function vs. free particle dispersion: high energy . 37 3-11 Multiple fillings with multiple bands . 38 3-12 Large fillings with high bands . 39 3-13 Imaging of free expansion . 42 3-14 Calculated density distributions after free expansion . 43 3-15 1D total density after free expansion . 45 3-16 1D m = 0 component density after free expansion . 46 3-17 1D ground state density after free expansion . 47 3-18 Single band Hartree Fock calculation in effective mass approximation 48 4-1 Transition temperature vs. laice depth, Hartree Fock, f = 1 .... 59 4-2 Transition temperature vs. laice depth, Hartree Fock, f = 5 .... 60 vii 4-3 Transition temperature vs. laice depth, Hartree Fock, f = 5 .... 61 4-4 Condensate fraction vs. laice depth, Hartree Fock, f = 1 ...... 62 4-5 Condensate fraction vs. temperature, Hartree Fock, f = 5 ...... 63 4-6 Boson density at the center of unit cell . 64 4-7 Boson density near the edge of unit cell . 64 4-8 Boson density contour plots . 65 4-9 Calculated densities after expansion for interacting bosons . 66 4-10 1D calculated density distribution after free expansion . 67 5-1 Energy level structure used in the NIST scheme of SOC . 71 5-2 NIST SOC scheme setup geometry . 72 6-1 Dispersion of the bare SOC hamiltonian . 78 6-2 Fixed density phase diagram for bosons with SOC . 84 6-3 Comparison of the free energy solutions for fixed m .......... 88 6-4 Fixed chemical potential densities with varying interaction: c"#0<0 . 89 6-5 Fixed chemical potential densities with varying interaction: c"#0>0 . 90 6-6 Phase diagram for fixed chemical potential . 91 6-7 Fixed chemical potential phase diagram, more chemical potentials . 93 6-8 The combined phase diagram for the experimental parameter regime 94 6-9 Phase diagram when one chemical potential is negative . 96 6-10 Cases for trapped BEC with SOC . 99 6-11 Schematic graph showing phases for parameter region III . 100 6-12 Schematic graph showing phases for parameter region I . 102 6-13 Atom density profiles for a BEC with SOC . 104 6-14 Spin-# density profiles . 105 6-15 Spin-" density profiles . 106 6-16 1D Atom density profiles: total density and magnetization . 107 viii 6-17 Atom density profiles: total density and magnetization . 108 6-18 Bogoliubov sound velocity in the mixed BEC phase . 111 D-1 Even Mathieu function Example 1 . 131 D-2 Even Mathieu function Example 2 . 132 D-3 Odd Mathieu function Example 1 . 132 D-4 Odd Mathieu function Example 2 . 133 D-5 Mathieu characteristic function for the even Mathieu function . 133 D-6 Mathieu characteristic functions at different laice depths . 134 D-7 Minima of Mathieu characteristic function . 134 D-8 Symmetry of characteristic functions and discontinuity at integer n . 135 D-9 Symmetry of characteristic functions about q .............. 135 D-10 Jumps of characeristic functions at integer n's . 136 D-11 n and periodicity of Mathieu functions . 136 D-12 Complex Mathieu functions at non-characterisitc a .......... 137 E-1 Mathieu functions of the lowest four states . 139 F-1 Testing band number included: no laice . ..
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