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Quantum Electronics

Analytical trial functions for modelling a two-dimensional Bose condensate

Yu. V. Likhanova, С. Б. Медведев, М. П. Федорук, P. L. Chapovsky

Published May 9, 2017• 4 citations
Publisher

Abstract

Approximate analytical solutions to the time-independent two-dimensional Gross–Pitaevsky equation are obtained by a variational method. The solutions found are compared with the results of direct numerical calculations. The accuracy of analytical solutions is determined under various conditions of the problem, primarily the degree of nonlinear interaction of atoms in a Bose condensate.

DOI
10.1070/qel16348

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Approximate analytical solutions to the time-independent two-dimensional Gross–Pitaevsky equation are obtained by a variational method. The solutions found are compared with the results of direct numerical calculations. The accuracy of analytical solutions is determined under various conditions of the problem, primarily the degree of nonlinear interaction of atoms in a Bose condensate.
Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.
We try to find the analytical solutions to the time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in the simulation of Bose—Einstein condensates trapped in a harmonic potential. Both the homotopy analysis method and the Galerkin spectral method are applied. We investigate the one-dimensional case and obtain the approximate analytical solutions successfully. Comparison between the analytical solutions and the numerical solutions has been made. The results indicate that they agree very well with each other when the atomic interaction is not too strong.
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 Bose-Einstein condensate is generally described by the three-dimensional equations. However, the two-dimensional case is of great interest too. First, these results can be directly used to study the planar condensates. Second, it serves as a test version for numerical calculations and as a quasi-two-dimensional approximation for the complete three-dimensional problem. An additional advantage of the two-dimensional case is that with the help of variational approach one can obtain analytical results for the stationary state and for the expansion of the condensate. In this paper, we propose a new approach for investigation of the Bose condensate in harmonic potential based on the variational method. We consider the inverse problem of reconstructing coefficients of the two-dimensional Gross-Pitaevskii equation with harmonic potential under the initially given condensate parameters (particle number, interatomic interaction, cloud widths). In particular, we solve the stationary problem in the framework of three trial functions, i.e., the Gaussian, the super-Gaussian and the Thomas-Fermi functions. In addition to that, the virial relation valid for arbitrary trial functions is obtained. We introduce the concept of special nonlinearity parameter in order to differentiate various degrees of the interatomic interaction. We perform comparison the variational solutions for the problem with the results of numerical calculations at the various levels of nonlinearity. For example, the relative differences between the condensate widths (Q1 and Q2) obtained by the variational method and the results of the direct numerical simulation is presented. It can be seen that the Gaussian and the super-Gaussian approximate the widths of the condensate with the accuracy better than 5% at various levels of nonlinearity. We demonstrate that the suggested approach allows one to solve the inverse problem of reconstructing all the equation parameters under the initially given condensate parameters. The joint use of the variational approach to the problem together with the numerical calculations allows one efficiently link the characteristics of the Bose condensate with the parameters of the Gross-Pitaevskii equation. Moreover, the proposed analytical-numerical approach can be used for broader class of problems than we discuss in the paper, for example, for investigating the self-focusing electromagnetic waves.
Using the F-expansion method we present analytical matter-wave solutions to Bose—Einstein condensates with two- and three-body interactions through the generalized three-dimensional Gross—Pitaevskii equation with time-dependent coefficients, for the periodically time-varying interactions and quadratic potential strength. Such solutions exist under certain conditions, and impose constraints on the functions describing potential strength, nonlinearities, and gain (loss). Various shapes of analytical matter-wave solutions which have important applications of physical interest are studied in details.
It is shown using the Gross–Pitaevskii equation that resonance states of Bose–Einstein condensates with attractive interactions can be stabilized into true bound states. A semiclassical variational approximation and an independent quantum variational numerical method are used to calculate the energies (chemical potentials) and linewidths of resonances of the time-independent Gross–Pitaevskii equation; both methods produce similar results. Borders between the regimes of resonances, bound states and, in two and three dimensions, collapse are identified.