The Solution Of Bethe-Ansatz Equations For One Dimensional Quantum Gas

The development of laser cooling and magnetic (or optical) trapping in experiments as well as the remarkable achievements in the Feshbach resonance technique leads to the observation of quasi-one-dimensional Bose-Einstein condensation of alkali atomic gases, and the researches have been expanded to Bose-Fermi mixture and two component Bose gas. Lower dimensional systems, such as one-dimensional (1D) many-body quantum system, attract more and more attention in the strongly interacting regime with anomalous, quantum fluctuations and correlations.The advantage of Bethe-Ansatz method is that it gives the exact solution of arbitrary coupling interactions. In the thermodynamic limit, the result of thermodynamic Bethe-Ansatz equations is to proven to be consistent with the experiments. Therefore, the Bethe-Ansatz method plays an important role in the theoretical research of 1D system and gives a lot of meaningful conclusions. The exact solutions of 1D solvable models are readily applicable to the research of 1D quantum gases.In this paper, We reviewed the Bethe-Ansatz methods and thermodynamic Bethe-Ansatz equations for single component bosons and two component mixture under periodic and open boundary conditions. We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equations in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the Tonks-Girardeau gas and super Tonks-Girardeau gas show the same profiles at the strongly interacting point-1/γ=0, and in the weakly interacting limitγ=0, the ground states TG and bound state join to each other smoothly.Based on the Lieb-Liniger model, we study the ground state of the 1D Bose-Fermi mixture. In the strongly interaction condition, we iterated the thermodynamic Bethe-Ansatz equations and expanded the ground state energy up to the orders of 1/γ3, which is compared with the numerical solutions.