BCS-BCS crossover between atomic and molecular superfluids in a Bose-Fermi mixture

We examine theoretically a continuity between atomic and molecular Fermi superfluids in a Bose-Fermi mixture near the Feshbach resonance. Considering a two-channel model with Fermi atoms $f$, Bose atoms $b$, and the closed-channel molecules $F$, we construct a mean-field framework based on the perturbative expansion of the $b\text{\ensuremath{-}}f\text{\ensuremath{-}}F$ Feshbach coupling. The resulting effective Hamiltonian not only exhibits the continuity between atom-atom to molecule-molecule Cooper pairings but also becomes equivalent to the two-band-superconductor model with Suhl-Matthias-Walker--type pair-exchange coupling. We demonstrate how these atomic and molecular Fermi superfluids coexist within the two-band-like superfluid theory. The pair-exchange coupling $ff\ensuremath{\leftrightarrow}FF$ and resulting superfluid gaps $\ensuremath{\langle}ff\ensuremath{\rangle}$ and $\ensuremath{\langle}FF\ensuremath{\rangle}$ are found to be strongly enhanced near the Feshbach resonance due to the interplay between the infrared singularity of Bogoliubov phonons and their Landau damping arising from the coupling with fermions. The pair-exchange coupling can be probed via the observation of the intrinsic Josephson effect between atomic and molecular superfluids.