Ultracold Quantum Matter In Lower Dimensions

Rapid progress in the field of ultracold atoms allows the study of many new and old models of quantum many-body physics. In this doctoral dissertation we theoretically explore exotic phases of ultracold quantum gases, with a special focus spin-imbalanced attractive Fermi gases in lower dimensional situations. Chapter 2 reviews the mean-field theory approach to pairing in twocomponent Fermi gases. Applications of this theory are illustrated in Chapter 3, where we discuss mostly well-known results of mean-field theory applied to imbalanced Fermi gases. Adapted from the author's prior publications, Chapters 4, 5 use the theory developed in Chapters 2, 3. In Chapter 6 we discuss the physics of Fermi gases, squeezed into one spatial dimension. In this and Chapter 7, we go beyond mean-field theory, approaching the problem through the Bethe ansatz, exact solutions to few-body problems and Fermi-Bose mappings ("fermionization"). We also show results from a joint effort with the experimental group of Randy Hulet at Rice University to experimentally realize and probe a strongly interacting one dimensional paired Fermi gas. In Chapter 8, after a brief introduction to rapidly rotating two dimensional Bose gases, we introduce a new protocol to create few atom fractional quantum Hall states. Finally, in Chapter 9 we study the effects of two-body losses on lattice Bose gases with hardcore interactions in one and two spatial dimensions