Fermion-mediated BCS-BEC crossover in ultracold K40 gases: Mean-field theory and collective modes

Microscopic theories of Feshbach resonance phenomena in fermionic alkali-metal gases have, in the main, relied upon the intuition afforded by a "Fermi-Bose" theory which presents the Feshbach molecule as a featureless pointlike Bose particle. While this model may provide a suitable platform to explore the condensation phenomena in the Li6 system, it has been argued recently that its application to the K40 system, where the hyperfine structure is inverted, is inappropriate. Here, the formation of a Feshbach resonance involves the formation of a Feshbach molecule which shares a spin state with the open channel. In such an arrangement, the mechanism of condensate formation cannot escape the effects of Pauli exclusion. Building on a model three-state fermion Hamiltonian recently introduced by Parish to describe the K40 system, we elucidate the mean-field structure of the model, investigate the internal structure of the condensate wave function in the crossover region, and develop a field theory of the transition.