Collective oscillation modes of a superfluid Bose-Fermi mixture

In this work, we present a theoretical study for the transverse and monopole modes of a mixture of Bose and Fermi superfluids in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) of molecules in harmonic trapping potentials with cylindrical symmetry of experimental interest. To this end, we start from the coupled superfluid hydrodynamic equations for the dynamics of Bose-Fermi superfluid mixtures and use the scaling theory that has been developed for a coupled system. The collective oscillation modes of Bose-Fermi superfluid mixtures are found to crucially depend on the overlap integrals of the spatial derivations of density profiles of the Bose and Fermi superfluids at equilibrium. By applying a perturbative analysis for the equilibrium density profiles, we present the explicit expressions for the overlap density integrals, as well as the frequencies of the collective modes provided that the effective Bose-Fermi coupling is weak. Subsequently, the valid regimes of the analytical approximations are demonstrated by numerical calculations in realistic experimental conditions. In the presence of a repulsive Bose-Fermi interaction, we find that the frequencies of the collective modes of the Bose and Fermi superfluids are all upshifted, and the frequency shifts can characterize the different groundstate phases of the Bose-Fermi superfluid mixtures in the BCS-BEC crossover for different trap geometry. Our results may provide a further understanding for experiments on the collective oscillation modes of a mixture of Bose and Fermi superfluids.