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We propose an experimental protocol to study $p$-wave superfluidity in aspin-polarized cold Fermi gas tuned by an $s$-wave Feshbach resonance. Acrucial ingredient is to add a quasi-1D optical lattice and tune the fillingsof two spins to the $s$ and $p$ band, respectively. The pairing order parameteris confirmed to inherit $p$-wave symmetry in its center-of-mass motion. We findthat it can further develop into a state of unexpected $\pi$-phase modulationin a broad parameter regime. Measurable quantities are calculated, includingtime-of-flight distributions, radio-frequency spectra, and in situphase-contrast imaging in an external trap. The $\pi$-phase $p$-wave superfluidis reminiscent of the $\pi$-state in superconductor-ferromagnetheterostructures but differs in symmetry and origin. If observed, it wouldrepresent another example of $p$-wave pairing, first discovered in He-3liquids.

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