Inverse problem for stationary state of a Bose-Einstein condensate

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.