Ultra-cold Fermi gases with tunable attractive interactions: Population imbalance and optical lattices

This thesis summarizes our studies on two topical issues on Fermi gases with tunable attractive interactions. When the attraction is weak, fermions form loosely bound pairs describable by Bardeen-Cooper-Schrieffer (BCS) theory. When the attraction is strong, fermions form tightly bound pairs which behave like composite bosons. At low temperature these tightly bound pairs form a Bose-Einstein condensation (BEC) similar to that in bosonic systems. The subject of the intermediate state between these two limits is referred to as the BEC-BCS crossover. Past work on theories of BEC-BCS crossover is reviewed. Pairing fluctuation effects, which are important at finite temperature, are included in a manner consistent with the standard ground state. One important feature is that the superfluid transition temperature Tc, is to be distinguished from the pairing onset temperature T*;Our first topic is on attractive Fermi gases with population imbalance in the two spin species. We provide a unified description of possible phases such as the Sarma phase (a polarized superfluid), the phase separation structure, and the Larkin-Ovchinnikov-Fulde-Ferrell phase (pairing with finite momentum). Their properties and stabilities are investigated systematically for a broad range of interaction strength and temperature. We extend these to a trap and find that particle density profiles and central densities obtained from our theory agree well with experimental results. We also present phase diagrams of polarized Fermi gases in the BEC-BCS crossover and make predictions. Relevant topics such as radio frequency measurement and open questions on highly polarized Fermi gases are discussed. Further extensions of our studies may be applied to color superconductivity of quarks, neutron stars, etc.;The second topic in this thesis is on unpolarized Fermi gases in optical lattices, which has become a very exciting research area. We found that BEC-BCS crossover in lattices is only achieved at low filling. When the attraction and the filling are sufficiently large, the crossover in lattices is interrupted and pairing fluctuations destroy superfluidity. In this situation another ground state with local pairs may become a better ground state in the strongly attractive regime while the BCS-type ground state breaks down. We also study d-wave pairing in lattices, which is relevant to the high-temperature superconductivity (HTSC) of the cuprates. The phase diagram of d-wave pairing shows that BEC-BCS crossover in this case is interrupted at almost all filling. Implications for simulations of attractive Hubbard models using ultra-cold fermions and their relation to the high-temperature superconductivity are presented.;We emphasize that in future simulations of the HTSC and other strongly correlated phenomena using ultra-cold fermions, one has as much to learn from the attractive Hubbard model and BEC-BCS crossover in lattices as from the repulsive Hubbard model or other models.