PROPERTIES OF DIRTY BOSONS IN DISORDERED OPTICAL LATTICES BY USHNISH RAY DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2015 Urbana, Illinois Doctoral Committee: Professor Anthony Leggett, Chair Professor David Ceperley, Director of Research Professor Brian DeMarco Professor Paul Kwiat Abstract The study of the disordered Bose-Hubbard model is key to understanding the interplay of disorder and interactions. Despite many studies with uniform diagonal disorder, few have inquired into experimental realizations with an additional correlated off-diagonal disorder. The presence of a trap and finite temperature effects in experiments lead to multiple do- mains of the Superfluid, Mott-Insulator/normal and the Bose-Glass phase. Previous studies using approximate theories produced results that are not in accordance with experiments. Stochastic Series Expansion is a finite temperature technique that can solve Bosonic lattice Hamiltonians exactly for large systems. Here, studies are performed for an extensive range of parameters using disorder distributions that are similar to experiments. Insights are first acquired by studying trap-free situations. Constant density calculations show that, although the qualitative features of the phase diagram remain robust between speckle disorder and uniform box disorder, there are quantitative differences. Studies of the Bose-glass phase ex- plicitly show that it is composed of superfluid puddles that are stable to finite temperature effects for large temperature ranges. Finite temperature behavior of a strongly correlated sys- tem reveals that at unit filling, the transition temperature of the superfluid is increased due to the addition of disorder. Inquires are then extended to discern the properties of trapped systems. Extensive calculations show that domain-like structures that develop can be rig- orously demarcated using the single-particle eigenstates extracted from the single-particle density matrix. Observables are calculated for the system at the single-site and global scales, showing that intermediate length scales provide the correct description of the physics of the domains in these systems. These techniques are used to conclusively show the possibility of the re-entrant superfluid that should be accessible to experiments. The temperature de- pendence of the re-entrant domain is explicitly calculated to be within experimental limits provided interactions are not too large. Comparisons with the local density approximation show reasonable agreement at low disorder strengths. At large disorder strengths there can be quantitative errors and can also result in qualitative errors. The phase diagram due to speckle disorder is presented for a range of values that are readily accessible to experiments. It is quantitatively shown that the effects of off-diagonal disorder are minimal. The superfluid remains unaffected despite large disorder in the tunneling term. Full scale ab initio calcu- lations of the largest trapped disordered systems to date are performed in order to identify the superfluid-Bose-glass phase boundary in collaboration with experiments. Results show remarkable agreement, but there are open questions with regards to the possibility of glassy dynamics. ii Acknowledgements This doctoral work would not have been possible without the help of my advisor, Professor David Ceperley. I would like to thank him for the generous support he gave me through- out my tenure as a graduate student. My work with David also made it possible to meet outstanding collaborators who contributed a great deal to my development. Among these scholars, I would especially like to thank Professor Brian DeMarco for the opportunity to work with him and his group. Brian was exceptionally generous to me with his time and guidance. I thoroughly enjoyed the numerous hours we spent discussing physics, academia, and life. I have had the distinct pleasure of learning from these pioneers of physics, both of whom played important roles in showing me that it was indeed possible to be a good scientist and a good human being. Numerous other individuals contributed to my intellectual growth and graduate career. I would like to thank my collaborators and colleagues, Carolyn Meldgin, Stefan Natu, David McKay, Philip Russ, Norm Tubman, Jeremy McMinis, Hitesh Changlani, Fei Lin, Sheng- Quan Zhou, Stanomir Kondov, Matthew Pasienski, Matthew White, Mingwu Lu, Seo-Ho Youn, Ethan Brown, Raymond Clay, Ilka Kylanpaa, and Victor Chua. I would especially like to thank ChangMo Yang for encouraging me to talk to David Ceperley for a possible position in his group (in 2010). Also, special thanks to Professors Anthony Leggett, Paul Kwiat, Vito Scarola, Bryan Clark, Lucas Wagner, Bryce Gadway, Nigel Goldenfeld, and Paul Goldbart for taking the time to listen to my (hopefully only) occasional ramblings, dis- sertation and defense preparations, and coursework. Also thank you to Professor Benjamin Lev for kickstarting my graduate career at the University of Illinois, Urbana-Champaign. Physics research is one part of graduate school, the other part involves social aspects without which my graduate career would have been impossible. My father has been a role model and exemplar of benevolence, wisdom, and fortitude in my life. I would like to thank him for his constant guidance and support in pursuing my interests. A big thank you also to my mother. My earnest gratitude and regards to Mr. Sudipta Sinha (uncle, as I call him) for helping me start my journey to the US for higher education, and also for being an indomitable force and advocate of my interests. Mrs. Durga Sinha (aunty) was also a guiding light in this regard; my sincere gratitude to her, as well. Throughout the highs and lows of graduate school, one companion was a constant source of love, amusement and support: my dog Zero. I must admit that I am perpetually amazed iii at his ability to perceive my moods and react in ways that always makes me \lighter" and not take life too seriously. Friends were of great help in this regard as well, and I am lucky to have so many that I couldn't possibly list them all here { but a big thank you to them all. Special thanks to Zsuzs´annaMagd´o,Brendan Pavlow, Brandon Lansche, David Coonan, Giovanni Fiorre, and Frank Filippi who took the brunt of my personality on a regular basis. Finally, my warmest regards and gratitude to the staff at Bacaro for keeping my spirits up with the excellent food, wine, and atmosphere of congeniality. iv Contents I Introduction ................................... 1 1 Overview ........................................ 2 1.1 Synthetic Materials with Ultra-Cold Atomic Gases . 2 1.2 Bosons in Lattices . 3 1.3 The Dirty Boson Problem . 4 1.4 Methodology . 5 1.5 Applications . 7 II Methods ...................................... 11 2 Monte Carlo Techniques .............................. 12 2.1 Random Variables and Probability Distributions . 12 2.2 Monte Carlo Quadrature: Estimators . 13 2.3 Sampling Distributions via Transformation of Random Variables . 14 2.4 Rejection Techniques . 16 2.5 The M(RT )2 Algorithm . 18 3 Stochastic Series Expansion ............................ 20 3.1 Diagonal Update . 23 3.2 Loop Update . 25 3.3 Observables . 28 3.3.1 Z-sector Observables . 29 3.3.2 G-sector Observables . 32 4 Calculating Single-Particle Modes ........................ 40 4.1 Non-linear Transformations of the Single-Particle Density Matrix: Important Aspects of Diagonalization . 41 4.1.1 Variational Estimate of The Dominant Eigenvalue . 44 4.1.2 Ultra-Cold Atomic Gases in Traps . 45 v 4.2 Diagonalization Strategies for Large Systems . 48 4.2.1 Krylov Subspace . 49 4.2.2 Rayleigh-Ritz Approximation . 50 4.2.3 Arnoldi Decomposition . 51 5 Coarse-Graining Single-Particle Hamiltonians ................ 59 5.1 Indirect Coarse-Graining . 60 5.2 Bloch's Theorem and Periodic Potentials . 61 5.3 Coarse-Graining with Wannier Functions . 64 5.4 Low Energy Hamiltonian for Optical Lattice Experiments . 67 5.5 Generalized Coarse-Graining Procedure . 71 5.6 Coarse-Graining Disordered Potentials from Speckle Fields . 75 5.6.1 Coarse-Graining Strategies for Large Systems . 77 III Applications .................................. 83 6 Bose-Hubbard Model ................................ 84 6.1 Mean Field Approximations . 87 6.1.1 Weak-Coupling Approximation . 88 6.1.2 Strong Coupling Approximation . 95 6.2 Inhomogeneous Systems . 96 6.2.1 Local Density Approximation . 98 6.3 Experiments . 100 6.3.1 Setup . 100 6.3.2 Measurements . 101 6.4 Quantum Monte Carlo Measurements . 105 6.4.1 Finite Time-of-Flight Effects . 105 6.4.2 Entropy Estimation . 110 6.5 Comparisons between MFT and QMC . 111 6.6 Metastability of the Condensate . 114 6.7 Theory for Metastability of Condensate: Landau Damping . 118 6.8 Conclusions and Outlook . 122 7 The Disordered Bose-Hubbard Model: Preliminaries ............ 124 7.1 General Effects of Disorder on Continuous Phase Transitions . 126 7.2 Theorem of Inclusions . 129 7.3 The Percolation Picture . 131 vi 7.3.1 Local Hamiltonian Approach . 132 7.3.2 Mean Field approach and Scaling based on Classical Percolation . 135 7.4 Features of the Phase Diagram . 137 7.4.1 Commensurate Filling . 138 7.4.2 Incommensurate Filling . 141 7.5 Equilibrium Properties of the Bose-Glass . 142 7.5.1 Commensurate Filling . 142 7.5.2 Incommensurate Filling . 146 7.5.3 Stability of the Bose-Glass State to Finite Temperature Effects . 148 7.6 Finite Temperature Effects . 149 7.7 Summary of Results . 151 8 Properties of Trapped Systems with Speckle Disorder ........... 153 8.1 General Experimental Setup . 154 8.2 Measurement of Transport Properties . 156 8.3 QMC Results . 159 8.3.1 Comparison with Experiments: A Preview . 162 8.4 Properties of Trapped Systems . 164 8.4.1 Measurement of Global Properties . 172 8.4.2 Considerations in Domain Identification in Trapped Systems . 174 8.4.3 Measurements of Local Observables for Domains . 175 8.4.4 Relationship of Trapped System Domains with the Local Density Ap- proximation .