Floquet Fulde-Ferrell-Larkin-Ovchinnikov superfluids and Majorana fermions in a shaken fermionic optical lattice

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids, Cooper pairings with finite momentum, and Majorana fermions (MFs), quasiparticles with non-Abelian exchange statistics, are two topics under intensive investigation in the past several decades, but unambiguous experimental evidence for them has not been found yet. Here we show that the recent experimentally realized shaken optical lattice provides a pathway to realize FFLO superfluids and MFs. By tuning the shaking frequency and amplitude, various couplings between the s and p orbitals of the lattice (called the pseudospins) can be generated. We show that the combination of the s- and p-band dispersion inversion, the engineered pseudospin coupling, and the on-site attractive interaction naturally allows the observation of FFLO superfluids as well as MFs in different parameter regions.