| The fermion spins are of half-odd, whereas the boson spins are of integer. As a result, Fermions and bosons have completely different quantum statistical properties. Fermions obey Fermi-Dirac statistics while bosons obey Bose-Einstein statistics. Therefore, when the temperature falls below a critical value, There are macroscopic magnitude of bosons start to occupy the zero-momentum quantum state, which is called Bose-Einstein condensation (BEC). However, for the fermion system, due to the Pauli exclusion principle, fermions occupy Fermi sphere with radius kF,as their ground state. After46years of the discovery of superconductivity, in1957, Bardeen, Cooper and Schrieffer proposed a microscopic theory of superconductivity, the BCS theory. In this theory, electronics around Fermi surface with attractive interaction (mediated by phonons) from Cooper pairs. Superconductivity is actually the Bose condensation of Cooper pairs. BCS theory is a weak coupling theory. In ultracold atomic Fermi gases, one can tune the coupling strength (or scattering length) between two atoms, using the Feshbach Resonance, to realize crossover from BCS superfluid to BEC.This thesis is organized as follows:In chapter1, we give brief introductions to BCS theory, Bose-Einstein Condensation, and the BCS-BEC crossover, respectively. In chapter2, we discuss some related experimental methods, which are used for implementing the mixed-dimensional system. Including, the Feshbach resonance and the optical lattices. And also, we will talk about the significance of the manuscript. In chapter3, we will study the superfluidity in the mixed-dimensional system, throught the BCS-BEC crossover. Firstly,we discuss the behavior of the superfluid transition temperatures Tc for the entire BCS-BEC crossover with different optical lattice parameters, the amplitude, the intra-species mass ratio and polarization. We find that the increasing of the lattice spacing and the amplitude will decrease the mismatch between the Fermi surfaces of the two species, and thus, enhance the transition temperature. In addition, the increasing of the mass ratio of the two species are also propitious to improve the transition temperature. On the other hand, the transition temperature will be suppressed by the polarization. We find that there is the competitive relationship between the mass ratio and the polarization. Secondly, we give the behavior of chemical potential in various situations. Similarly, the lattice spacing, the amplitude have effects on it by changing the mismatch between the Fermi surfaces of the two species. However, the effects of the mass and number of particles on chemical potential are contrary. Finally, we study the behavior of the excitation gap, it is found that the gap will have the abnormal behavior at low temperatures in the presence of the polarization. Some qualitative explanations are proposed. |
