  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 evidences for them have not been
found yet in any physical system. Here we show that the recent experimentally
realized cold atom shaken optical lattice provides a new pathway to realize
FFLO superfluids and MFs. By tuning shaken lattice parameters (shaking
frequency and amplitude), various coupling between the s- and p-orbitals of the
lattice (denoted as the pseudo-spins) can be generated. We show that the
combination of the inverted s- and p-band dispersions, the engineered
pseudo-spin coupling, and the attractive on-site atom interaction, naturally
allows the observation of FFLO superfluids as well as MFs in different
parameter regions. While without interaction the system is a topological
insulator (TI) with edge states, the MFs in the superfluid may be found to be
in the conduction or valence band, distinguished from previous TI-based schemes
that utilize edge states inside the band gap.
