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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm

Maria Luisa Chiofalo, Sauro Succi A, M. P. Tosi

Published Nov 1, 2000• 388 citations
Publisher

Abstract

We show that an explicit time-marching method previously developed for the numerical study of the dynamics of Bose-Einstein condensates can be profitably adapted to the numerical determination of their ground state. After reduction to a one-dimensional model, we first reproduce and test known results on condensates in harmonic traps and then determine the ground state of a condensate in a harmonically bound optical lattice in the range of parameters which are relevant to existing experiments.

DOI
10.1103/physreve.62.7438

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We show that an explicit time-marching method previously developed for the numerical study of the dynamics of Bose-Einstein condensates can be profitably adapted to the numerical determination of their ground state. After reduction to a one-dimensional model, we first reproduce and test known results on condensates in harmonic traps and then determine the ground state of a condensate in a harmonically bound optical lattice in the range of parameters which are relevant to existing experiments.
For Bose–Einstein condensates trapped in a harmonic potential well, we present numerical results from solving the time-dependent nonlinear Schrodinger equation based on the Crank–Nicolson method. With this method we are able to find the ground state wave function and energy by evolving the trial initial wave function in real and imaginary time spaces, respectively. In real time space, the results are in agreement with [Phys. Rev. A 51 (1995) 4704], but the trial wave function is restricted in a very small range. On the contrary, in imaginary time space, the trial wave function can be chosen widely, moreover, the results are stable.
We present the application of a fast, explicit time-marching scheme for the solution of the Gross-Pitaevskii equation in cylindrical geometry. The scheme is validated on simple analytical tests and demonstrated for two situations of physical interest in experiments on the Bose-Einstein condensation (BEC) of trapped alkali-metal vapors. It is tested by reproducing known results on the free expansion of a BEC after removing a cylindrical trap, and it is then used to address the formation of matter-wave pulses that result from gravity-induced transport of a condensate in an optical potential.
We derive closed analytical expressions for the order parameter $\ensuremath{\Phi}(x)$ and for the chemical potential $\ensuremath{\mu}$ of a Bose-Einstein condensate loaded into a harmonically confined, one-dimensional optical lattice for sufficiently weak, repulsive, or attractive interaction and not too strong laser intensities. Our results are compared to exact numerical calculations in order to map out the range of validity of the perturbative analytical approach. We identify parameter values where the optical lattice compensates the interaction-induced nonlinearity, such that the condensate ground state coincides with a simple, single-particle harmonic-oscillator wave function.
We present numerical results from solving the time-dependent nonlinear Schr\"odinger equation (NLSE) that describes an inhomogeneous, weakly interacting Bose-Einstein condensate in a small harmonic trap potential at zero temperature. With this method we are able to find solutions for the NLSE for ground state condensate wave functions in one dimension or in three dimensions with spherical symmetry. These solutions corroborate previous ground state results obtained from the solution of the time-independent NLSE. Furthrmore, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering lengths in one-dimensional (1D) and 3D systems for a fixed number of atoms. In both the 1D and 3D cases, these negative scattering length solutions have solitonlike properties. In 3D, however, these solutions are only stable for a modest range of nonlinearities. We analyze the prospects for diagnosing Bose-Einstein condensation in a trap using several experiments that exploit the time-dependent behavior of the condensate.
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.