  Based on density matrix renormalization group method, we investigate the
spin-orbit coupled Fermi gas with attractive interactions in one-dimensional
optical lattice and present a complete phase diagram for a quarter-filling
system with intermediate strong interactions. We unveil the exotic pairings
induced by spin-orbit couplings and the phase transitions between different
pairing states, and further demonstrate the existence of the long-sought
topological superfluid state at moderate magnetic field and spin-orbit coupling
strength. For the particle conserved system, this topological superfluid state
can be fixed by its gapless single particle excitation, Luttinger parameter and
non-trivial boundary effect. Our study removes out the widespread doubt on the
existence of topological superfluid in one dimension and paves the way for
simulating topological superfluid states in cold atom settings.
