| Condensate in ultracold atomic gases is one of the most exciting research areas in condensed matter and atomic physics in recent years. One can tune the interaction between atoms from very weak to very strong through Feshbach resonances. This makes it possible to observe superfluid or condensate in quantum degenerate Fermi gases directly over the entire range of the BCS-BEC crossover. Furthermore, this has created a strong hope that study of these systems may help us understand high T_c superconductivity. First we give an introduction of theory of BCS-BEC crossover and related experimental advance. Basic theoretical Model of BCS-BEC crossover is also given, we find this model is applicable by comparison with BCS and BEC limit. One channel model of asymmetric two component Fermi gas is considered, and we obtain an effective action by Bogoliubov transformation. Two equations is derived by saddle point approximation: Order parameter equation and number equation, which is important to investigate BCS-BEC crossover. By numerical calculation we get some very interesting physics and give the relation between Polarization and transition temperature. Based on previous theory we consider the asymmetric Fermi gas trapped in external magnetic field. By local density approximation (LDA) we find an area where chemical difference is approximately proportional to polarization and give related phase diagram. At last we discuss relation between local density and temperature or polarization, the PS (phase separated) phase is obtained.
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