Stability of FFLO states in optical lattices with bilayer structure

We investigate the stability of the superfluid state in a bilayer fermionic optical lattice system with a confining potential, using the Bogoliubov de-Gennes equations. It is clarified that in the imbalanced case, the introduction of the interlayer hopping stabilizes the radial Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, while makes the angular FFLO state unstable. We also discuss the system size dependence of the superfluid ground state. It is clarified that in a certain ring region the A-FFLO state is indeed realized in a large system.