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The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, a superconducting statewith non-zero total momentum Cooper pairs in a large magnetic field, was firstpredicted about 50 years ago, and since then became an important concept inmany branches of physics. Despite intensive search in various materials,unambiguous experimental evidence for the FFLO phase is still lacking inexperiments. In this paper, we show that both FF (uniform order parameter withplane-wave phase) and LO phase (spatially varying order parameter amplitude)can be observed using fermionic cold atoms in spin-orbit coupled opticallattices. The increasing spin-orbit coupling enhances the FF phase over the LOphase. The coexistence of superfluid and magnetic orders is also found in thenormal BCS phase. The pairing mechanism for different phases is understood byvisualizing superfluid pairing densities in different spin-orbit bands. Thepossibility of observing similar physics using spin-orbit coupledsuperconducting ultra-thin films is also discussed.

Competing superfluid orders in spin-orbit coupled fermionic cold atom optical lattices