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Acta Physica Sinica

Study of the ground state wave function in optical lattice by using the genetic algorithm

XU Zhi-jun, Qingmiao Nie, Penghua Li, 浙江工业大学应用物理系，杭州 310032

Published Jan 1, 2009• 2 citations
Publisher

Abstract

Taking the gross-pitaevskii（G-P） mean-field energy functional of Bose condensed gas as the objective function, we study the properties of Bose condensed gas in 1D optical lattice system by using the genetic algorithm, and present a new method to find the ground-state wave function of the system. Through the optimized computation, the currently commonly used Thomas-Fermi approximation and Gaussian approximation model are revised and discussed, and the optimal ground state wave function is given.

DOI
10.7498/aps.58.2878

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Taking the gross-pitaevskii（G-P） mean-field energy functional of Bose condensed gas as the objective function, we study the properties of Bose condensed gas in 1D optical lattice system by using the genetic algorithm, and present a new method to find the ground-state wave function of the system. Through the optimized computation, the currently commonly used Thomas-Fermi approximation and Gaussian approximation model are revised and discussed, and the optimal ground state wave function is given.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
Based on the Gross-Pitaevskii theory, for a Bose-condensed gas confined in a harmonic trap and in a one-dimensional (1D) optical lattice, the relationship of the Gaussian width of the sub-condenses and the intensity of 1D-optical lattices potential is studied by using the G-P energy functional and the variational method. For the ideal non-interacting model, the Gaussian width of the sub-condenses is analyzed. Concerning interacting Bose-condensed atoms, the Gaussian approximate model is advanced and the analytic expression of the Gaussian width with the intensity of optical lattices potential is obtained. And then, those analytical results are compared with the straightforward numerical calculation. It indicates that the Gaussian approximate model is closer to the results of the numerical solution, and both are reaching unanimity with the intensity of ID-optical lattices potential increasing.
For the Bose-condensed gas in a combined potential consisting of an axially-symmetric harmonic magnetic trap and one-dimensional (1D) optical lattice, using the mean-field Gross-Pitaevskii (G-P) equation and the propagator method, we study the formation of matter-wave interference upon releasing the condensate from the combined potential. The analytical result of the wave function for matter wave interference at any time t is given, and the relation between the interference fringe period and the optical lattice period is obtained. The interference pattern of Bose-condensed gas in the optical lattices is analogous to the multiple slit interference pattern in optics, and coincides with the results of experiment performed by Andrews et al. and Peil et al. and the numerical solution of the G-P equation by Sadhan K. Adhikari.
Based on the Gross-Pitaevskii equation and by using the effective chemical poten tial,we study the distribution of the Bose-condensed gas in the combined potenti als (magnetic trap and three-dimensional optical lattices).The normalized ground-state wave function is given from the distribution of the Bose-condensed gas.The analytical expression of the density distribution of the Bose-condensed gas is given by using the propagator method when the combined potentials and only the optical lattices are switched off,respectively.When the combined potentials are switched off,the theoretical results of this work agree well with the experiment by Greiner et al.When only the optical lattices are switched off,it is show n that the interference pattern will oscillate in the magnetic trap.In addition,the evolution of the interference pattern is also investigated in the case of an isotropic magnetic trap.
Several models of a strongly interacting Bose gas in an optical lattice are studied within the functionalintegral approach. The one-dimensional Bose gas is briefly discussed. Then the Bose-Einstein condensate and the Mott insulator of a three-dimensional Bose gas are described in mean-field approximation, and the corresponding phase diagrams are evaluated. Other characteristic quantities, like the spectrum of quasiparticle excitations and the static structure factor, are obtained from Gaussian fluctuations around the mean-field solutions. We discuss the role of quantum and thermal fluctuations, and determine the behavior of physical quantities in terms of density and temperature of the Bose gas. In particular, we study the dilute limit, where the mean-field equation becomes the Gross-Pitaevskii equation. This allows us to extend the Gross-