Collective modes and the effect of single-particle excitations in the BCS-BEC crossover region of a trapped Fermi superfluid

We investigate the collective oscillations in the BCS-BEC crossover region of a trapped Fermi superfluid at T=0. We explicitly include the effect of a Feshbach resonance, which leads to a tunable pairing interaction adjusted by the threshold energy of the resonance, as well as the associated mol ecules. In a previous paper,we obtained solutions of the Bogoliubov-de Gennes coupled equations, describing a trapped Fermi superfluid ga s. Using these Bogoliubov quasi-particle excitations, we calculate the density correlation function in the Hartree-Fock-Bogoliubov generalized random phase approximation (HFB-GRPA). Previous discussions of collective modes were based on solving the T=0 equation of motion for the superfluid in a trap, ignoring the effect of single-particle excitations. We determine the frequencies of the quadrupole and the monopole modes. In the crossover region, the frequencies of these two modes are shown to be stro ngly depressed by the threshold energy of the two-particle continuum, originating from Andreev single-particle bound states (or in-gap states) localized at the edge of the trap. This effect becomes less important as one enters the BEC regime, due to the absence of these low energy fermion bound states. This suppression of collective mode frequencies by the two-particle continuum may be a useful experimental signature of the single-particle energy gap in the crossover region of trapped Fermi superfluids.