Studies of cold fermionic systems near the Feshbach resonance

Cold Fermi gas with a tunable interaction is experimentally realized in trapped cold alkali atoms. Through evaporative cooling these atoms can be brought well into the degenerate regime. In addition, with magnetic field induced Feshbach resonances the underlying interaction between atoms prepared in two different spin states can be manipulated via the Zeeman effect. Cold Fermi atoms can be made to undergo a BCS-BEC crossover, where the ground state wavefunction evolves smoothly from the BCS-type to the BEC-type. The different interaction regimes in the crossover can be characterized by the dimensionless parameter 1/askF defined from the s-wave scattering as and the Fermi momentum kF. At Feshbach resonance, where the scattering length essentially diverges up to +/-infinity, 1/ask F = 0 resulting in a special scenario for a low-density Fermi gas termed the 'unitary limit' by many authors. At such limit cold Fermi gas should exhibit universal behavior in the sense that physical properties becomes independent of the two-body interaction and determined only by k F. In particular, the total energy at zero-temperature E0 is expected to depend on kF through the simple relation E0 = xEfree0 where Efree0 is the corresponding quantity in a non-interacting gas. The proportionality constant xi should be an universal constant being the same for any type of underlying particles. In this dissertation we present several studies on cold Fermi gas at and near the unitary limit. Our primary focus is the calculation of xi with neutron matter. As is well-known, the 1 S0 channel of neutron matter has a fairly large scattering length (as = --18.97fm), therefore ordinary neutron matter is already close to the unitary limit. To obtain xi accurately we need 'modified' neutron matter much closer to the unitary limit than the ordinary one. By slightly tuning the meson-exchange CD-Bonn potential, neutron-neutron potentials with various 1S0 scattering lengths such as as = --12070fm and +21fm are constructed. Such potentials are renormalized with rigorous procedures to give the corresponding as-equivalent low-momentum potentials Vlow--k, with which the low-momentum particle-particle hole-hole ring diagrams are summed up to all orders, giving the ground state energy E0 of neutron matter for various scattering lengths. At the limit of as → +/-infinity, our calculated ratio of E0 to that of the non-interacting case is found remarkably close to a constant of 0.44 over a wide range of Fermi momenta. This result reveals an universality that is well consistent with the recent experimental and Monte-Carlo computational study on low-density cold Fermi gas at the unitary limit. Apart from ground state properties, low-lying excitations also offer lots of insights into the rich physics underlying the BCS-BEC crossover process. We have calculated the quadrupole excitations of cold Fermi gas near the unitary limit using a simple model where atoms are confined in a harmonic oscillator potential. By summing up exactly the ladder diagrams between a pair of interacting atoms to all orders, we first obtained a renormalized atomic interaction which has well defined and identical limits as the scattering length tends to +/-infinity. Employing both the Tamm-Dancoff and random phase approximations we obtained the excitation frequency and decay width. The experimentally observed abrupt rise in frequency and an associated large decay width in the radial compression mode and radial quadrupole mode are satisfactorily reproduced by our calculation.