Detecting $\pi$-phase superfluids with $p$-wave symmetry in a quasi-1D optical lattice*

We propose an experimental protocol to create a $p$-wave superfluid in a spin-polarized cold Fermi gas tuned by an $s$-wave Feshbach resonance. A crucial ingredient is to add an anisotropic 3D optical lattice and tune the fillings of two spins to the $s$ and $p$ band, respectively. The pairing order parameter is confirmed to inherit $p$-wave symmetry in its center-of-mass motion. We find that it can further develop into a state of unexpected $\pi$-phase modulation in a broad parameter regime. Experimental signatures are predicted in the momentum distributions, density of states and spatial densities for a realistic experimental setup. The $\pi$-phase $p$-wave superfluid is reminiscent of the $\pi$-state in superconductor-ferromagnet heterostructures but differs in symmetry and physical origin. The spatially-varying phases of the superfluid gap provide a novel approach to synthetic magnetic fields for neutral atoms. It would represent another example of $p$-wave pairing, first discovered in He-3 liquids.