BCS-BEC Crossover In Ultracold Fermi Gas

Fermions and Bosons obey different statistical properties as for the different sym-metry of the spin. The Fermi systems obey the Fermi-Dirac statistics as for the Fermionsobeying the Pauli principal. The Boson systems obey the Boson-Einstein statistics. Atultra low temperature, due to the Boson not restrict by the Pauli principal, there willbe a phenomena that Bosons condense at zero momentum state which is called Boson-Einstein Condense (BEC). It is different for Fermi system. The condensation phenom-ena occur in the Fermi system with attractive interaction. Two Fermions will forminto a Cooper pair firstly, and then the Cooper pairs condense. This is the Bardeen-Cooper-Schrieffer(BCS) theory. The temperature should be reduced to lower than thedegenerate temperature of the Boson (Fermi) system to observe the condensation phe-nomena in the Boson (Fermi) system. At 1995 year, the groups of Cornell,Wieman,Hulet and Ketterle had observed the BEC in ultracold alkali metal Boson gas. Fewyears later, the condensation phenomena had been observed in Fermi system. The in-teraction between Fermions can be modulated by the Feshbach Resonance. Therefor,the Fermi system can undergo from BCS to BEC. The cold atoms be heated by thecontinuously development of the experiment. Very recently, the gauge field is realizedin the cold atomic gas. Therefor, a effective spin-orbit coupling is achieved. The spin-orbit coupling significantly change the Fermi surface. Such that the ground state of theFermi system is significant different from the case without SOC. Many new physicswill be show up like topological superfluid, topological phase separation and so on.In this thesis, we will review some fundamental theories and experiments (BCS,BEC, BCS-BEC crossover Feshbach resonance and optical lattice) firstly, and then wemainly discuss the following parts in Fermi system:(1) The transition temperature of the 2D-3D mixed Fermi system was investi-gated in the mean field framework including the effect of fluctuation by G0G schemewhich is improved on the NSR scheme. The gap, pseudogap and the number equationswere derived. The transition temperature was achieved by selfconsistently solving the equations. The transition temperature smoothly vary as the system undergo BCS-BECcrossover. The effect on the transition temperature by the imbalance of mass and pop-ulation also has been considered. We find that the transition temperature reduces whilethe imbalance of mass is increased or lattice spacing is reduced. In spin imbalancecase, the stability of superfuid is sharply destroyed by increasing the polarization.(2) The trapped two-dimensional Fermi gas with spin-orbit coupling was investi-gated. In the mean field framework and under local-density approximation(LDA), wequalitative analyzed the ground state of the trapped system by calculating the topolog-ical properties. We found that there exist topological phase separation in the trappedregion. The topological phase separation is a ubiquitous phenomena in the trapped two-dimensional polarized Fermi system and very different from the topologically trivialphase separation. There is Majorana edge states at the boundaries between the super-fluid phases with different topological properties. This Majorana edge states are robustand protected by the bulk topological properties for bulk-edge correspondence. For thetopologically trivial phase separation system, there have not this robust Majorana edgestates.(3) For the two-dimensional spin-orbit coupled Fermi system, we determined theground state by minimizing the thermodynamic potential taking the stability of thephase separation in to account in the framework of mean field. Then, we mapped outthe phase diagram. As a result of that the spin-orbit coupling topologically change theFermi surface, the helicity is a good quantum number instead of the spin. Thereby,the ground also been topologically changed. The spin-orbit coupling enrich the groundstate of the system. Due to the topologically changing of Fermi surface, the phaseseparation is very different from the case without spin-orbit coupling. There are twotopologically distinct phase separation(NPS and TPS). The two topologically distinctphase separation can not be connected with each other without a topological phasetransition.(4) We determined the collective mode of the three dimensional spin-orbit cou-pled Fermi system. In the mean field framework, we considered the gauss fluctuationbeyond the saddle point approximation to derive the behavior of the collective modein the entire BCS-BEC crossover. We found that the spin-orbit coupling suppress the collective mode only in the BCS side.