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We consider the quasi-one dimensional system realized by an array of weaklycoupled parallel one-dimensional "tubes" in a two-dimensional lattice whichpermits free motion of atoms in an axial direction in the presence of a Zeemanfield, Rashba type spin orbit coupling (SOC), and an s-wave attractiveinteraction, while the radial motion is tightly confined. We solve thezero-temperature (T=0) Bogoliubov-de Gennes (BdG) equations for the quasi-1DFermi gas with the dispersion modified by tunneling between the tubes, and showthat the T=0 phase diagram hosts the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)phase with non-zero center of mass momentum Cooper pairs for small values ofthe SOC while for larger values of the SOC and high Zeeman fields the uniformsuperfluid phase with zero center of mass momentum Cooper pairs has aninstability towards the topological uniform superfluid phase with Majoranafermions at the tube ends. Also, we show that tuning the two-dimensionaloptical lattice strength in this model allows one to explore the crossoverbehaviors of the phases during the transition between the 3D and the 1D systemand in general the FFLO (for small SOC) and the topological uniform superfluidphase (for large SOC) are favored as the system becomes more one-dimensional.We also find evidence of the existence of a Zeeman tuned topological quantumphase transition (TQPT) within the FFLO phase itself and for large values ofthe Zeeman field and small SOC the TQPT gives rise to a topologically distinctFFLO phase.

Topological Uniform Superfluid and FFLO Phases in 3D to 1D crossover of spin-orbit coupled Fermi gases