BEC And BCS-BEC Crossover In Ultracold Atomic Gases

From the microscopic level,bosons and fermions possess completely different quantum statistical characters:in terms of bosons,quantum statistical effect is associated with the phenomenon of Bose-Einstein-Condensation(BEC),while for noninteracting fermionic system,its evolution is the smooth crossover between classical and quantum behavior as the temperature is dropping;Contrary to the case that repulsive interaction stabilizes BEC,the occurrence of a superfluid phase in a fermionic gas can only be due to the presence of attractive interaction.In order to look into these behaviors caused by different quantum statistics,we must get so low temperature that bosonic or fermionic system becomes degenerate.Fortunately in many kinds of ultracold atomic gas,irrespective of their constituents,we have successfully observed BEC and Bardeen-Cooper-Schrieffer (BCS) superfiuidity.What is more important is that fermionic system evolves continuously from BCS superfiuidity to BEC when attractive interaction between fermions is tuned from weak to strong coupling by Feshbach resonance(This is the so-called BCS-BEC crossover).Hence it is possible to probe into two different kinds of quantum statistical effect in a single fermionic system.In this doctoral thesis,we firstly review some fundamental theories about BEC,BCS superconductor,BCS-BEC crossover,Feshbach resonance and corresponding experimental evidences,then we mainly discuss following questions in ultracold atomic gases:(1) The collective modes in a uniform fermionic atomic gas with Feshbach resonance are investigated with coherent state path integral method in the frame of boson-fermion model.We mainly concentrate on the long-wavelength and low-frequency limits at T = OK and get an analytical expression for the collective modes across the whole BCS-BEC crossover.We completely recover the Anderson-Bogoliubov modes in the BCS limit and the Bogoliubov modes of the bosonic systems in the BEC limit.The numerical results show that there exists a continuous interpolation for sound velocity between BCS and BEC limits.In this way we can also solve the equation of state of the system,which makes an explicit difference from the hydrodynamics.(2) First and second sound modes in a uniform fermionic atom gas with Feshbach resonance are investigated in the frame of two-fluid model with a microscopic calculation of thermodynamical variables at finite temperature.The analytical results for thermodynamic quantities and sound velocities in BCS and BEC limits are obtained.The numerical results show that there exists a continuous interpolation for sound velocities of first and second sound modes between BCS and BEC limits,so we testify the validity of extrapolation for sound velocities made in some early work.(3) Based on single-band approximation for Hubbard model and only by analyzing some physical constraints,breached-pair(BP) phase in imbalance fermionic atom gases placed in a three-dimensional optical lattice is studied theoretically. According to the distribution of unpaired particles in the first Brillouin zone, the breached-pair phase is classified into two classes(â… ) normal-superfluid phase and(â…¡) superfluid-normal-superfluid phase which are stable in strong and weak coupling limits respectively at nonzero temperature.Moreover it is possible to predict a classical phase transition between these two different BP phase although further determination still depends on numerical solution.(4) Taking advantage of gauge transformation and dipolar approximation, we qualitatively derive rotating Bose-Hubbard model.Then finite temperature phase boundary between superfluid phase and normal state in rotating Bose-Hubbard model is analytically derived by studying the stability of normal state in rotating bosonic optical lattice.We also analytically prove that as the function of rotation angular velocity the oscillation of critical hopping matrix directly follows the upper boundary of Hofstadter butterfly.(5) Using Bogoliubov theory and forward-scattering approximation,we derive a reduced Hamiltonian describing an impurity in BEC.By using Lee-Low-Pines (LLP) theory,we obtain the effective mass of impurity,the phonon number carried by impurity and the energy related to the existence of impurity.Moreover according to Landau superfluidity criterion,we also analyze the validity of forward-scattering approximation.