Superfluidity of Fermion Atoms in Optical Lattices

In this dissertation, we investigate two major areas of study concerning 2D optical lattices: mass-imbalance and its associated superfluid phases and collective excitations, and the emerging physics of Spin Orbit Coupling (SOC) in optical lattices. With the former, we find that the phase diagram presents with a unique topology that exhibits a Fulde-Ferrell (FF) phase where phase separation is typically found, and we also find that the collective excitations exhibit anomalous dispersion more closely related to Bose versus Fermi superfluidity. In the latter, we have found the Bardeen Cooper Schrieffer (BCS) - Bose Einstein Condensation (BEC) crossover for a two dimensional lattice. We show that SOC, and SOC with a Zeeman field, substantially change the nature of the crossover not only by reducing the interaction strength at which it occurs, but we have identified three different 'crossovers': (1) The emergence of gapped Dirac cones when chemical potential becomes zero, (2) The emergence of a Dirac point with positive chemical potential, such that a condensate of Rashbons may exist in the BCS regime, analogous to the occurrence of the Rashbon bound state on the BCS side of the Feshbach resonance in 3D systems, and (3) a crossover that occurs without the Dirac point or cones when chemical potential drops below zero.