Dynamics Of Two Particles In A Harmonic Trap With The Propagator Method

In recent years, the study of one-dimensional quantum gases has got great progress in experiments and theoreties. The various quantum phases can be achieved in experiments by adjusting the interaction strength among particles with Feshbash resonance including the Tonks-Girardeau (TG) gas, Cluster state and Super-Tonks-Girardeau (STG) gas. In this paper, analytical methods are applied to the system of two interacting particles. Firstly, we get the exactly analytical solutions for two heteronuclear atoms in one-dimensional ring trap with the Bethe-Ansatz method. We found that large mass ratio reduces the system energy, and strong interaction tends to fermionize the particles. Secondly, we study the dynamics of two bosons in one-dimensional harmonic trap with the propagator method, with the two particles prepared initially in the three typical quantum phases. The free particle propagator result indicates that the density distribution spreads to whole space as time evolves, suggesting strong interaction tends to fermionize the particles. However, for weak attractive interaction the time evolution of single particle density is similar to the diffuse process of free particle Gaussian wave packet. The harmonic oscillator propagator result reveals the periodicity of the time evolution of the single particle density.