The Ground State And Thermodynamics Of One-Dimensional Quantum Gas

Recent years have witnessed great development of laser cooling and optical trapping technology,with remarkable achievements in several aspects, e.g.manipulation of the interaction strength between particles according to Feshbach resonance technique,the Bose-Einstein Condensation of quasi-one-dimensional Bose gas trapped in magnetic and optical potential and atom chip,strongly interacting Tonks-Girardeau gases,and the Bose-Fermi mixture which rarely occur in nature.These progresses make ultra-cold atoms a popularly investigated platform for various many-body effects in strongly correlated one-dimensional systems.This dissertation will introduce several theories for one-dimensional systems,including the Mean-Field theory,the Tonks-Girardeau gases,and the exact solution with Bethe ansatz.We then devote to theoretical investigation of the ground state properties of an interacting few-boson system in a splited hard-wall potential and the thermodynamics of quasi-one-dimensional Bose-Fermi mixture at finite temperature.First of all,we carry out a detailed examination of the ground-state properties of a few-boson system in a one-dimensional hard-wall potential with aδsplit in the center.In the Tonks-Girardeau limit with infinite repulsion between particles,we use the Bose-Fermi mapping to construct the exact N-particle ground-state wave function,which allows us to study the correlation properties accurately.For the general case with finite interparticle interaction,the exact diagonalization method is exploited to study the ground-state density distribution,occupation number distribution,and momentum distribution for variable interaction strengths and barrier heights. The secondary peaks in the momentum distribution reveal the interference between particles on the two sides of the split,which is more prominent for large barrier strength and small interaction strength.Secondly,we investigate theoretically the behavior of one-dimensional interacting Bose-Fermi mixture at finite temperature in the scheme of thermodynamic Bethe ansatz.Combining the Yang-Yang thermodynamic formalism with local density approximation in a harmonic trap,we calculate the density distribution of bosons,which is mainly influenced by the number of Fermions and interacting strength between particles.It is shown that Bose-Fermi phase separation needs even lower temperature and very large interacting strength between particles,which will provide comparative materials for relative experiments.