The Studies On One-dimensional Ultracold Atoms With Spin-orbit Coupling

In recent years, a new state of matter named topological superfluid has at-tracted intense attentions. Being similar to topological insulators, they are fully gapped in the bulk and yet possess gapless exotic excitations on the edges called the Majorana fermions, which can be used for topological quantum computation since they are robust against local perturbations. As a result, the experimental realization of topological superfluid and associated Majorana fermions has be-come a desirable task and feasible proposals from the side of the theorists are greatly needed. In this paper, we investigate one-dimensional three-component spin-orbit coupled Fermi gases in the presence of Zeeman field. Furthermore, we discuss the possibility of realizing topological superfluid.In Chapter1, we give a brief introduction of topological superfluid and relative research progress in ultracold atom system.In Chapter2, firstly, we introduce single-particle spin-orbit coupling, which is originated from the Dirac equation in relativistic quantum mechanics. Sec-ondly, we illustrate the basic properties of single particle with spin-orbit cou-pling (Rashba type and Dresselhaus type), including the dispersion relation, wavefunction, and spin polarization etc. Thirdly, we discuss the properties of Rashba-Drcsselhaus spin-orbit coupling by introducing a pseudo-spin rotation.In Chapter3, we discuss the one-dimensional two-component topological superfluid system. Firstly, we introduce the experimental realization on the syn-thetic gauge field and the spin-orbit coupling, which is accomplished by the spiel-man’s group. Then we give the model Hamiltonian, by solving the Bogoliubov-de-Gennes (BdG) equation. From that, we know the main characteristics of two-component superfluid, such as the quasiparticle dispersion relation, topological invariant, zero-energy modes and localized wave functions.In Chapter4, we discuss the one-dimensional three-component topological superfluid system. For simplicity, we mainly discuss the homogenous situation. Being similar to the discussion in Chapter3, we solve the BdG equation and ob-tain the dispersion relation. The Berry phase is defined to describe the topology of the system. It turns out that, the Berry phase changes its sign only when the system undergoes a topological transition. Then we describe Majorana zero modes and localized wave functions of the system. Finally, we present the μ-h phase diagram at given order parameter△, where μ is the chemical potential of the system and h is the magnetic field. According to the phase diagram, we find the novel topological transition behaviours of the three-component sys-tem, distincting from the conventional two-component case, due to the three order parameters of system can not keep gauge invariant simultaneously. The inhomogenous situations are also discussed by considering the effect of Zeeman energy and inhomogenous order parameters.Conclusion and outlook are given in Chapter5.In conclusion, the three-component topological superfluid system has new topological transition behaviours due to the hidden new physics, and in certain parameter range the new system is more optimizing for experimental realization due to the smaller magnetic field needed.