Abstract: The topological superfluid state is protected by the energy gap in the bulk, but it can accommodate the gapless Majorana fermions at the edge of the system. The Majorana fermions satisfy non-Abelian statistics and are protected by topology and have good stability, they can carry quantized information and can be used in the study of topological quantum computing. In recent years, theoretical work has predicted the possible topological superfluid states in various systems. Firstly, we introduce the topological superfluid in various optical lattice models. The ultracold atoms of optical lattice have good controllability and universality. It is an ideal model system to realize topological superfluid. Next, we introduce the topological superfluid under the control of spin orbit coupling. The spin orbit coupling effect is an important condition to induce the topological phase, and the artificial spin orbit coupling has been realized in the experiment. Which makes a breakthrough for the experimental observation of topological superfluid. With the improvement of experimental technology in recent years, the topological FFLO superfluid phase, which was difficult to observe in the experiment and ignored by people, has also become a research hotspot. Therefore, we next introduce the topological FFLO superfluid. In addition, we also introduce the progress in other aspects of topological superfluid, including topological superfluid induced by soliton, three-component topological superfluid, topological superfluid with large Chern number, and the high critical temperature of topological superfluid. In the experiment, how to detect and implement topological superfluid is the purpose and significance of our research. Therefore, we introduce the identification and implementation of topological superfluid at the end of the article.

Key words: Majorana fermion, Topological superfluid, Optical lattice, Spin orbit coupling; Topological FFLO superfluid