Properties Of Quantum Phase In One Dimensional Ultracold Fermi Gas With Spin-Orbit Coupling

Ultracold atoms are atoms that are maintained at temperatures close to 0 kelvin(absolute zero).At these temperatures,the fluctuation characteristic of the atoms become remarkable.Therefore,the atom's quantum-mechanical properties become very important.Ultracold atoms can be achieved by loading alkali atoms in optical lattices and using evaporative cooling of atoms.Feshbach resonance can be easily used to tune the strength of inter-particle interactions.So ultracold atoms can be used to simulate strong correlation systems with different intensities of interactions conveniently.Ultracold atoms offer a ideal platform to investigate quantum properties.Such as superfluid,quantum magnetism and underlying mechanisms of superconductive in condensed matters.In recent years,artificial spin-orbit coupling(SOC)in low-dimensional optical lattice is realized in experiment,and it has caused widely attention.People predict that ultracold atoms will have some new properties after introduced SOC.In this thesis.We considerate attract Fermi gas in one-dimentional optical lattice.And using Fermi-Hubbard model to investigate the quantum properties with SOC.In mean field method,energy gap of each lattice and energy spectrum of quasi-particles are obtained.These results are used for analyzing quantum phase properties of the system.With magnetic field and SOC coexist,we verify the exist of these two quantum phase,which are Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)phase with nonzero center-of-mass momentum cooper paring and Topological superfluid(TSF)phase.In each specific phase,we discussion about the properties of energy gap and energy spectrum.Through the contrastive analysis of phase diagram and the trend of energy gap,the influence of SOC on FFLO and TSF phase is investigated.The characteristics of the chain under different lattices scale are also concerned.In the Lanczos iteration method,we study a few lattices chain under the condition of half-filling.The ground wave function and ground state energy are calculated.According to paring correlation and the trend of magnetic susceptibility,the characteristics of quantum phase can be conducted.First,SOC or magnetic field with the impact on superconducting pairing is investigated.And we find that FFLO phase can't exist without SOC.Then,we analyse the chain in the case of magnetic field and SOC coexist.The relationship between FFLO phase,magnetic susceptibility and occupation probability of spin are studied.In particular,for moderate magnetic fields and SOC strengths,the FFLO and BCS paring can coexist,we call this specific phase as BCS-FFLO mixed phase.At last,the relationship between the specific phase diagram and the sudden change of magnetic susceptibility have been discussed.It is proved that the properties of Fermi chain are tightly related to the lattice scales.