Topological Superfluid in one-dimensional Ultracold Atomic System with Spin-Orbit Coupling

We propose a one-dimensional Hamiltonian $H_{1D}$ which supports Majorana fermions when $d_{x^{2}-y^{2}}$-wave superfluid appears in the ultracold atomic system and obtain the phase-separation diagrams both for the time-reversal-invariant case and time-reversal-symmetry-breaking case. From the phase-separation diagrams, we find that the single Majorana fermions exist in the topological superfluid region, and we can reach this region by tuning the chemical potential $μ$ and spin-orbit coupling $α_{R}$. Importantly, the spin-orbit coupling has realized in ultracold atoms by the recent experimental achievement of synthetic gauge field, therefore, our one-dimensional ultra-cold atomic system described by $H_{1D}$ is a promising platform to find the mysterious Majorana fermions.

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