The Study of the BCS-BEC Crossover in Optical Lattices and of Quenches in Paired Superfluids

This dissertation presents the theoretical study of two-component fermionic gases. It firstly studies the BCS-BEC crossover for the system of the two-component fermionic gas in optical lattices. If the system were loaded into deep three-dimensional optical lattices, a tight-binding model with the lowest band is applicable. If the band is more than half-filled and by increasing the attractive interactions, then there exists a novel crossover, from a paired BCS superfluid to a BEC of holes, back to the BCS superfluid, and finally to a conventional BEC regime of diatomic molecules. If the band is less than half-filled, the crossover is similar to that without optical lattices. If the band is fully filled, a quantum phase transition from band insulator to BCS-BEC superfluid occurs. Then, this dissertation studies a model describing the one-dimensional Feshbach resonance. Because this model satisfies the Yang-Baxter equations, it was assumed that the Bethe Ansatz could solve this model. However, this dissertation demonstrates that it is not integrable by calculating the chemical potential from the Bethe Ansatz, and by studying the scattering between the bosons. To the researcher\u27s knowledge, this model is the first continuum model that satisfies the Yang-Baxter equations, but is not integrable. After that, this dissertation studies the quenches in the system of paired fermionic superfluids. If the system is slightly driven from its equilibrium state, the asymptotic behaviors of this small oscillation are calculated with both s- and p-wave interactions, and in both the BCS and the BEC regimes, under the mean field collisionless approximation