Two-band superfluidity from the BCS to the BEC limit

We analyze the evolution of two-band superfluidity from the weak coupling Bardeen-Cooper-Schrieffer (BCS) to the strong coupling Bose-Einstein condensation (BEC) limit. When the interband interaction is tuned from negative to positive values, a quantum phase transition occurs from a 0-phase to a $\pi$-phase state, depending on the relative phase of two order parameters. Furthermore, population imbalances between the two bands can be created by tuning the intraband or interband interactions. We also find two undamped low energy collective excitations corresponding to in-phase and out-of-phase modes. Lastly, we derive the coupled Ginzburg-Landau equations, and show that they reduce to coupled Gross-Pitaevskii equations for two types of bosons in the BEC limit.