Theoretical Study On 1D Many-Body Bose System

The experimental realization of Bose-Einstein condensates (BECs) of trapped alkali atomic clouds has stimulated active studies in many new regimes. From then on, BECs have become a popularly investigated platform for various effects of quantum many-body interaction in strongly correlated systems. After giving a brief review for the one dimensional (1D) Bose gases, this dissertation is devoted to theoretical investigation of several interesting problems.Firstly, the low-energy-level macroscopic wave functions of the BECs trapped in a symmetric double-well and a periodic potential are obtained by solving the Gross-Pitaevskii equation numerically. The ground state tunnel splitting is evaluated in terms of the even and odd wave functions corresponding to the global ground and excited states respectively. We show that the numerical result is in good agreement with the analytic level splitting obtained by means of the periodical instanton method.Secondly, the density distributions of a spin-1 bosonic gases in its ground state are evaluated numerically within a modified Gross-Pitaevskii theory, which is obtained by the combination of the exact solution of the corresponding integrable model with a local density approximation. The population of atoms in different components depends on the overall magnetization and the (anti-)ferromagnetism of the Bose gases in both weakly and strongly interacting regimes. When the system is in the Tonks regime, the density profiles show obvious Fermi-like distribution, however,for strong enough spin-spin interaction, apparent deviation of the density distribution from the Fermi-like distribution. The present work also investigate the effect of the anisotropic spin-spin interaction on the ground state density distribution. It is found that for ferromagnetic spinor gas the phase separation appears.Then, we investigate the ground state of the system of N bosons enclosed in a hard-wall trap interacting via a repulsive or attractive S -function potential. Based on the Bethe ansatz method, the explicit ground state is got for the full physical regime from the Tonks limit to the strong attractive limit. In the Tonks limit the density profiles display the Fermi-like behavior, while in the strong attractive limit the Bosons form a bound state of N atoms corresponding to the N -string solution. The density profiles show the continuous crossover behavior in the entire regime. Further the correlation function indicates that the Bose atoms bunch closer as the interaction constant decreases.Finally, based on the exact solution of the time-dependent Schrodinger equation for two-species BECs consisting of two hyperfine states of the atoms coupled by a tuned adiabatic and time-varying Raman coupling, we obtain analytically the entanglement dynamics of the system with various initial states, particularly the SU(2) coherent state, for both of cases with and without the nonlinear interactions. It is shown that the effect of nonlinear interaction on the entanglement appears only in a longer time period depending on the BEC parameters.