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Journal of Hebei Normal University of Science & Technology

Density Distribution of Bose-Einstein Condensates in a One-Dimensional Optical Lattice

Yan Li-hua

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

This paper studied a one-dimensional nonlinear Schrodinger equation and described a Bose-Einstein condensation with G-P function and variation in a three-dimensional axisymmetrical harmonic trap and a one-dimensional optical lattice.The connection of density distribution of Bose-Einstein condensation and the depth of optical lattice was investigated with numerical method.The influence of magnetic trap on density distribution of Bose-Einstein condensates was also analyzed in this paper.

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This paper studied a one-dimensional nonlinear Schrodinger equation and described a Bose-Einstein condensation with G-P function and variation in a three-dimensional axisymmetrical harmonic trap and a one-dimensional optical lattice.The connection of density distribution of Bose-Einstein condensation and the depth of optical lattice was investigated with numerical method.The influence of magnetic trap on density distribution of Bose-Einstein condensates was also analyzed in this paper.
We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.
We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.
We investigate the effect of solitons and vortices, with a range of different topologies, on the dynamics of Bose-Einstein condensates in a one-dimensional optical lattice and a three-dimensional harmonic trap. The solitons and vortices are imposed on the initial state of the atom cloud, which is set in oscillatory motion by a sudden displacement of the harmonic trap. We study this motion using both quantum-mechanical calculations, based on the nonlinear Schr\"odinger equation, and a semiclassical model for transport through the lowest energy band of the optical lattice. We demonstrate that the geometry of the vortices and solitons in the initial state plays a crucial role in the evolution of the atom cloud. In particular, we find that the center-of-mass motion of a condensate containing a vortex is only weakly damped if the vortex core lies parallel to the optical lattice direction, but is strongly damped when the core is orientated perpendicular to the lattice direction. Severe damping is observed for a condensate containing a soliton whose nodal plane is perpendicular to the optical lattice.
We investigate the stability of dark solitons (DSs) in an effectively one-dimensional Bose-Einstein condensate in the presence of the magnetic parabolic trap and an optical lattice (OL). The analysis is based on both the full Gross-Pitaevskii equation and its tight-binding approximation counterpart (discrete nonlinear Schr\"odinger equation). We find that DSs are subject to weak instabilities with an onset of instability mainly governed by the period and amplitude of the OL. The instability, if present, sets in at large times and it is characterized by quasiperiodic oscillations of the DS about the minimum of the parabolic trap.
Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.
The stability of an attractive Bose–Einstein condensate on a joint one-dimensional optical lattice and an axially symmetrical harmonic trap is studied using the numerical solution of the time-dependent mean-field Gross–Pitaevskii equation and the critical number of atoms for a stable condensate is calculated. We also calculate this critical number of atoms in a double-well potential which is always greater than that in an axially symmetrical harmonic trap. The critical number of atoms in an optical trap can be made smaller or larger than the corresponding number in the absence of the optical trap by moving a node of the optical lattice potential in the axial direction of the harmonic trap. This variation of the critical number of atoms can be observed experimentally and compared with the present calculations.