| Ultracold atomic gas experiments have emerged as a new testing ground for finding elusive, exotic states of matter. One such state that has eluded detection is the Larkin-Ovchinnikov (LO) phase predicted to exist in a system with unequal populations of up and down fermions. This phase is characterized by periodic domain walls across which the order parameter changes sign and the excess polarization is localized Despite fifty years of theoretical and experimental work, there has so far been no unambiguous observation of an LO phase. In this thesis, we propose an experiment in which two fermion clouds, prepared with unequal population imbalances, are allowed to expand and interfere. We show that a pattern of staggered fringes in the interference is unequivocal evidence of LO physics.;Finally, we study the superconductor-insulator quantum phase transition. Both superconductivity and localization stand on the shoulders of giants -- the BCS theory of superconductivity and the Anderson theory of localization. Yet, when their combined effects are considered, both paradigms break down, even for s-wave superconductors. In this work, we calculate the dynamical quantities that help guide present and future experiments. Specifically, we calculate the conductivity sigma(o) and the bosonic (pair) spectral function P(o) from quantum Monte Carlo simulations across clean and disorder-driven superconductor-insulator transitions (SIT). Using these quantities, we identify characteristic energy scales in both the superconducting and insulating phases that vanish at the transition due to enhanced quantum fluctuations, despite the persistence of a robust fermionic gap across the SIT. The clean and disordered transition are compared throughout, and we find that disorder leads to enhanced absorption in sigma(o) at low frequencies and a change in the universality class, although the underlying T = 0quantum critical point remains in both transitions. |
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