Theoretical Study Of Bose-Einstein Condensation And Structure-preserving Computation

This thesis concentrates on the numerically study of Bose-Einstein Condensation (BEC) in harmonically trapped, weekly-interacting dilute gases. The motivation for this study is the experimentally realization of BEC in trapped alkali gases (Rubidium, Sodium, and Lithium) since 1995. The week inter-particle interactions and the diluteness of the gases allow for a mean-field description of the system of BEC, and consequently granted the widely use of the mathematical model, the Gross-Pitaevskii equation (GP), which is the mathematic model in our study. What is unique in our computation is that the symplectic algorithm is employed since the GP equation can be discretized into a Hamiltonian system which has symplectic structure. Symplectic algorithm is the difference method that preserves the symplectic structure, and is a better method in the calculation of long-time many-step and preserving the structure of system. The fundamental theorem of Hamiltonian mechanics says that the time-evolution of Hamiltonian system is the evolution of symplectic transformation. In this sense, the Hamiltonian system has a symplectic structure. Therefore, the symplectic algorithm is appropriate for solving the Hamiltonian system, and for solving the Hamiltonian mechanics. This thesis mainly consists of three specific topics in the field of numerically study of BEC: (i) Fundamental study of the ground state wavefunction of BEC in spherically harmonic trap by time-independent method. (ii) Dynamic study of the numerical solution of BEC by the time-dependent method. (iii) Numerical study of the interference of two condensates. The main parts of the thesis are summarized here: (i) We solved the time-independent GP equation which describes the harmonically-trapped and weekly-interacting gases numerically by symplectic shooting method. Being a nonlinear equation, the GP equation is not very easy going in numerical computation when the strictly normalized wavefunctions were required. In this case, we exercise the symplectic shooting method with two parameters. The eigenvalues of the ground state and the first excited state as well as the corresponding wavefunctions are obtained, and our results manifest that the numerical method we used is efficient. (ii) Dynamic properties of the BEC in spherical harmonic trap are examined by solving the time-dependent GP equation. With the procedure proposed by Ruprecht et al, we tackled the equation by Euler-center algorithm which is also a symplectic algorithm. The eigenvalues of the ground state as well as the corresponding wavefunctions are obtained, and the results are consistent with those obtained by the time-independent method. We multiply the external potential term by 2 (or 1/2) to a stable wavefunction, and the evolution...