Superfluidity and collective modes in a uniform gas of Fermi atoms with a Feshbach resonance

We investigate strong-coupling superfluidity in a uniform two-component gas of ultracold Fermi atoms attractively interacting via quasimolecular bosons associated with a Feshbach resonance. This interaction is tunable by the threshold energy $2\ensuremath{\nu}$ of the Feshbach resonance, becoming large as $2\ensuremath{\nu}$ is decreased (relative to $2{\ensuremath{\varepsilon}}_{\mathrm{F}},$ where ${\ensuremath{\varepsilon}}_{\mathrm{F}}$ is the Fermi energy of one component). In recent work, we showed that the enhancement of this tunable pairing interaction naturally leads to the BCS-BEC (Bose-Einstein condensation) crossover, where the character of the superfluid phase transition changes from the BCS type to a BEC of composite bosons consisting of preformed Cooper-pairs and Feshbach-induced molecules. In this paper, we extend our previous work and study both the quasiparticles and the collective dynamics of the superfluid phase below the phase-transition temperature ${T}_{\mathrm{c}},$ limiting ourselves to a uniform gas. We show how the superfluid order parameter changes from the Cooper-pair amplitude $\ensuremath{\Delta}$ to the square root of the number of condensed molecules $({\ensuremath{\varphi}}_{\mathrm{m}})$ associated with the Feshbach resonance, as the threshold energy $2\ensuremath{\nu}$ is lowered. In the intermediate coupling regime, the superfluidity is shown to be characterized by an order parameter consisting of a superposition of $\ensuremath{\Delta}$ and ${\ensuremath{\varphi}}_{\mathrm{m}}.$ We also discuss the Goldstone mode associated with superfluidity, and show how its character smoothly changes from the Anderson-Bogoliubov phonon in the BCS regime to the Bogoliubov phonon in the BEC regime in the BCS-BEC crossover. The velocity of this Goldstone phonon mode is shown to strongly depend on the value of $2\ensuremath{\nu}.$ We also show that this Goldstone mode appears as a resonance in the spectrum of the density-density correlation function, which is experimentally accessible.