The Dynamic Property Of Bose-Einstein Condensation In A Double-well Potential

Bose-Einstein condensate (BEC) is a new state of matter, and research activities around it is very active since 1995, when Bose-Einstein condensation was first carried out. Today, many research laboratories of the world has been dedicated to ultra-cold atoms, and has made great progress in a series of major experiments, in particular, such as atom optics, phonons, superfluid, vortex, and optical lattice. The two-mode approximation is widely used in a variety of dynamical properties of BEC, and has obtained a number of very interesting phenomena, such as tunneling, self-trapping and chaos.Theoretical research of Bose-Einstein condensate is extensive. In the theoretical study of BEC there is a very good mathematical model, namely, Gross-Pitaevskii equation (GP equation). The GP equation of weakly interacting Bose gas is a nonlinear Schrodinger equation. The GP equation is widely used, so accurate solution of the GP equation is very important and a variety of approximate numerical methods are developed. Approximate methods for solving a GP equation is very much, such as ODA law, Imaginary time evolution method, steepest descent method, Runge-Kutta method, step by step Crank-Nicolson method, and symplectic method.This article lists the specific formula of the one-dimensional double-well potential. It is a harmonic potential with a Gaussian barrier. By changing the parameters of formula, we have simulated the one-dimensional double-well potential, and find that changing the parameters can control the height and width of the double-well potential.With applying the standard two-mode model or improved two-mode model, we can change the time-dependent GP equation of a single BEC into the first order differential equations which are composed of the population imbalance and the relative phase. We solve the fixed points of the equations, and discuss dynamic properties around the fixed points by thirteen-step seventh-order Runge-Kutta method. By studying the population imbalance and the relative phase with changing interaction parameters and initial variables, we find that BEC can change from Josephson oscillations to the self-trapping state in the double-well potential.GP equation of a mixture of two BECs is two coupled Schrodinger equations. With applying the standard two-mode model or improved two-mode model, we can change the equations into the first order differential equations which are composed of the population imbalance and the relative phase.In this paper, we study dynamic properties of a mixture of the two BECs in a symmetric double-well potential by thirteen-step seventh-order Runge-Kutta method. When all the initial variables of the relative phase are zero, increasing the interaction parameters can change the condensates from the Josephson oscillations to the self-trapping state and two BECs are in the different potential wells; When all the initial variables of the relative phase areπ, increasing the interaction parameters can also change the condensates from the Josephson oscillations to the self-trapping state and two BECs are in the same potential well.Two-mode model is an ideal platform of the study of dynamic properties of BEC. It's simple, but has rich physical meaning.