Numerical Study Of The Bose-Einstein Condensates Dynamics

Bose-Einstein condensation (BEC) opens a new area of research in physics. It is a new form of matter which can show quantum property in large scale, and has wide prospect for application. A large number of Bose gases without interaction at certain low temperatures, a part of the atoms can all reside in the same lowest energy single atom quantum state which is called Bose-Einstein condensation. After 70 years of efforts, BEC was realized in nearly ideal alkali gases Rubidium, Sodium, and Lithium in 1995, and then the research of BEC began prosperous year after year.In the theoretical research of BEC, many of the numerical research are based on the Gross-Pitaevskii (GP) equation. In the static property of BEC, the ground state solution of the condensate, the eigenvalue problem, and the transition temperature etc can be studied by GP equation. In the dynamic property of BEC, vortex, breather, and oscillation etc can be examined by GP equation. In the coherence study of BEC, the interference effect can be studied by GP equation. GP equation utilizes mean field approximation, and has a form of nonlinear Schrodinger equation, which is an important model in physics.Since some dynamic property and the interference of BEC deserve more researches in depth, this paper concentrates on the numerical study of the Bose-Einstein condensates dynamics. Based on the GP equation describing the one dimensional neutral BEC in harmonic potential, apply the symplectic method in studying the dynamics of the cubic and quintic Schrodinger equation, propose an improved shooting method for solving the time independent GP equation, and present the symplectic method for solving the time dependent GP equation; Study the interference of two and three condensates and make comparison, discuss the dynamics of two and three condensates in a global potential; Take tunneling effect and interference effect as consequence process, study the dynamics of condensate in the harmonic trap and the Gauss energy barrier, and study the possible interference effect.This paper does some research on how to solve the GP equation, because a good ground state wavefunction is the basis for solving physics quantities and numerical simulating. GP equation is a mathematic model for describing BEC, and it has a form similar with the nonlinear Schrodinger equation. It is difficult to find the analytical solution of it, and numerical computation is more efficient in solving it. Nonlinear Schrodinger equation is a Hamiltonian system of infinite dimensions, and the time evolution of the system is the evolution of symplectic transformation, that is, the Hamiltonian system has symplectic structure. Based on these, Ruth and Feng Kang derive out the symplectic method for solving the Hamiltonian system. Symplectic method is a difference method which can preserve the symplectic structure of the Hamiltonian system. It is superior to many other methods on long-time, many-step computing, and on preserving the general structure of the system. Solve the cubic and quintic nonlinear Schrodinger equation by symplectic method and studies its dynamic property. Give out the N soliton solution for the cubic nonlinear Schrodinger equation; The solution of the cubic nonlinear Schrodinger equation changes from the quasiperiodic solution, the chaotic solution to the periodic solution with the increasing of the cubic nonlinear parameter; With the increasing of the quintic nonlinear parameter, the breather solution of the cubic and quintic nonlinear Schrodinger equation collapses into the fundamental soliton solution. Solve the GP equation by symplectic method, propose an improved shooting method with two parameters, and apply it to solve the one dimension time independent GP equation, and give out the ground state eigenvalue and the corresponding wavefunction of the condensate which can be tested to be stable; Give out the symplectic method for solving the one dimension time dependent GP equation, and adopt the procedure of gradually increasing the nonlinear parameter in our calculation. This method not only can solve out the ground state wavefunction, but also can be used for dynamical calculation; Introduce other numerical methods such as the spectrum method and the imaginary time evolution method. The results reached are consist with those obtained by other methods and are tested to be correct and efficient, and our methods can preserve the norm of the wavefunction simultaneously during calculation, and are superior to other methods in long-time many-step computation.Study the interference phenomenon of the condensates. Today in laboratory a dipole barrier can be erected in the middle of the trapping potential which can repulse the atoms, thus produce two atom clouds, and the technique of optical tweezers also can be used to displace the atom cloud, etc, all these lay the experimental ground for further studying the dynamics of the condensate. Mainly discuss the interference phenomenon of the condensates numerically, and study the interference of two condensates of the same particles and the same phases, further study the interference of three condensates, comparison is made between them and the interference patterns are presented. It shows that interference happen when the two condensates overlap, and the spacing between the interference fringes widens with time evolution. In order to prevent the reflection at the boundary from influencing the computation, adopt zero boundary condition at large enough distance, and it shows that the probability density at the reflection point of the interference pattern evolutes smoothly with time. The interference of three condensates has two stages, firstly the two neighbor condensates interfere, and it is the same as the former case, then the three condensates interfere together, in this stage the interference pattern shows oscillation. This phenomenon manifested itself by the oscillation of the probability density at the reflection point of the interference pattern, and the oscillation changes with the phase difference of the condensates, and it is believed that the increasing nonlinear interaction plays an important role in the oscillation phenomenon. This paper also studies the dynamics of two and three condensates in a global trapping potential. The condensates present periodic evolution, just like the harmonic oscillation, and the condensates evolute as the inter-passing of solitons. Due to the existence of the trapping, the interference is much stronger when the condensates overlap. The dynamic behavior of two and especially of three condensates brings much physics information, and the discussion of its dynamic phenomenon is of much interest.Considering the possibility of realizing in laboratory, propose to study the dynamics of the condensate in a harmonic trapping potential and a Gauss energy barrier, discuss the tunneling effect and the possible interference effect. Firstly propose to displace the condensate in the harmonic trapping potential with a distance from its equilibrium, and the condensate is expected to make periodic evolution like the harmonic oscillation. Secondly shut off the potential after some time of evolution and let free the condensates, at the same time erect a Gauss energy barrier, and the tunneling effect is discussed. It shows that when the released condensate has relatively large kinetic energy, part of the condensate tunnels through the barrier and forms a small wave packet; the other part reflected back to interfere with the incident wave and a small wave packet forms after the interference. In order to control the moving of the two small wave packets, and study the interference of two moving condensates, propose to retain the existing of the trapping potential when the barrier is erected. Study shows that tunneling effect could happen and interference effect happens between the incident and reflected wave. and the interference becomes stronger with the increasing of the height of the barrier.The tunneling effect is inhibited when the height of the barrier is increased to a certain value. Two moving condensates are obtained after the tunneling process, and interference could happen when they overlap after being released under certain conditions. It not only proves the existence of the global phase, but also provides convenience for experimental probing.This paper improves the shooting method and applies it for solving GP equation, and it shows to be correct and efficient. This method can be expected to apply for solving other similar nonlinear Schrodinger equation. This paper considers the tunneling effect and the interference effect together and studies the dynamics of condensates in a harmonic trapping potential and a Gauss energy barrier, it is a very interesting work and provides a new method for interference study experimentally. Theoretical study of BEC has reached wider area, and the numerical discussion of it is also very widely, and the symplectic method has broad application in the study of BEC.