  The recent experimental realization of spin-orbit coupling (SOC) for
ultra-cold atoms opens a completely new avenue for exploring new quantum
matter. In experiments, the SOC is implemented simultaneously with a Zeeman
field. Such spin-orbit coupled Fermi gases are predicted to support Majorana
fermions with non-Abelian exchange statistics in one dimension (1D). However,
as shown in recent theory and experiments for 1D spin-imbalanced Fermi gases,
the Zeeman field can lead to the long-sought Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) superfluids with non-zero momentum Cooper pairings, in contrast to the
zero momentum pairing in Majorana superfluids. Therefore a natural question to
ask is which phase, FFLO or Majorana superfluids, will survive in spin-orbit
coupled Fermi gases in the presence of a large Zeeman field. In this paper, we
address this question by studying the mean field quantum phases of 1D
(quasi-1D) spin-orbit coupled fermionic cold atom optical lattices.
