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New Journal of Physics

Controllable exact self-similar evolution of the Bose–Einstein condensate

Lei Wu, Lu Li, Gang Chen, Qing Tian, Jie-Fang Zhang

Published Feb 15, 2008• 5 citations
Publisher

Abstract

We show analytically that the profile of the Bose–Einstein condensate can evolve exact self-similarly when the time-dependent interatomic interaction and harmonic potential satisfy a certain condition for one-, two- and three-dimensional geometries, respectively. Based on the exact scaling laws for the amplitude and width of the condensate profile, we propose experimentally feasible ways to improve the accuracy in measuring the stationary-state profile of the condensate when its spatial size is so small that the direct measurement is not available, and to squeeze the condensate to high local particle density to check the validity of the Gross–Pitaevskii equation via the controllable exact self-similar evolution of the condensate profile.

DOI
10.1088/1367-2630/10/2/023021

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We show analytically that the profile of the Bose–Einstein condensate can evolve exact self-similarly when the time-dependent interatomic interaction and harmonic potential satisfy a certain condition for one-, two- and three-dimensional geometries, respectively. Based on the exact scaling laws for the amplitude and width of the condensate profile, we propose experimentally feasible ways to improve the accuracy in measuring the stationary-state profile of the condensate when its spatial size is so small that the direct measurement is not available, and to squeeze the condensate to high local particle density to check the validity of the Gross–Pitaevskii equation via the controllable exact self-similar evolution of the condensate profile.
We compare the exact evolution of an expanding three-dimensional Bose-Einstein condensate with that obtained from the effective scaling approach introduced in D. Guéry-Odelin [Phys. Rev. A 66, 033613 (2002)].This approach, which consists in looking for self-similar solutions to be satisfied on average, is tested here in different geometries and configurations.We find that, in case of almost isotropic traps, the effective scaling reproduces with high accuracy the exact evolution dictated by the Gross-Pitaevskii equation for arbitrary values of the interactions, in agreement with the proof-of-concept of M. Modugno, G. Pagnini, and M. A. Valle-Basagoiti [Phys. Rev. A 97, 043604 (2018)].Conversely, it is shown that the hypothesis of universal self-similarity breaks down in case of strong anisotropies and trapped geometries.
We discuss a simplified description of the dynamics of inhomogeneous dilute Bose-Einstein condensates (BECs), based on the assumption that the time-evolving BEC preserves its shape (i.e. the density profile remains similar to the initial profile). While the validity of the self-similarity assumption is not a priori obvious, we show that this description becomes exact in the opposite limits of large and small condensates contained by time-dependent spherically symmetric harmonic oscillator trap potentials. We discuss applications of this simple description and we show how the insights derived from it suggest that phase-coherent BECs expanding from optical lattice sites can be used for the purpose of imaging.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
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.
Abstract By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.