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arXiv:cond-mat/0703779 · Physical Review A

Ansatz from nonlinear optics applied to trapped Bose-Einstein condensates

Murat Keçeli, Fatih Ömer Ilday, Mehmet Özgür Oktel

Published Mar 13, 2007• 5 citations
arXivHTMLPDF

Abstract

A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser (a laser with self-similar propagation of ultrashort pulses), is used as a variational wave function to solve the Gross-Pitaevskii equation for a wide range of interaction parameters. The variational form interpolates between the noninteracting density profile and the strongly interacting Thomas-Fermi profile smoothly. The simple form of the ansatz is modified for both cylindrically symmetric and completely anisotropic harmonic traps. The resulting ground-state density profile and energy are in very good agreement with both the analytical solutions in the limiting cases of interaction and the numerical solutions in the intermediate regime.

DOI
10.1103/physreva.75.035601

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A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser (a laser with self-similar propagation of ultrashort pulses), is used as a variational wave function to solve the Gross-Pitaevskii equation for a wide range of interaction parameters. The variational form interpolates between the noninteracting density profile and the strongly interacting Thomas-Fermi profile smoothly. The simple form of the ansatz is modified for both cylindrically symmetric and completely anisotropic harmonic traps. The resulting ground-state density profile and energy are in very good agreement with both the analytical solutions in the limiting cases of interaction and the numerical solutions in the intermediate regime.
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.
We consider the ground state of an attractively interacting atomic Bose-Einstein condensate in a prolate, cylindrically symmetric harmonic trap. If a true quasi-one-dimensional limit is realized, then for sufficiently weak axial trapping this ground state takes the form of a bright soliton solution of the nonlinear Schr\"odinger equation. Using analytic variational and highly accurate numerical solutions of the Gross-Pitaevskii equation, we systematically and quantitatively assess how solitonlike this ground state is, over a wide range of trap and interaction strengths. Our analysis reveals that the regime in which the ground state is highly solitonlike is significantly restricted and occurs only for experimentally challenging trap anisotropies. This result and our broader identification of regimes in which the ground state is well approximated by our simple analytic variational solution are relevant to a range of potential experiments involving attractively interacting Bose-Einstein condensates.
In this paper we present an analytical method for solving the Gross-Pitaevskii equation for the Bose-Einstein condensation in the dilute atomic alkali gases. Using a variational ansatz, we are able to obtain an analytical solution for the order parameter and for the chemical potential as a function of a unique universal parameter: λ/(ħ ω ℓ) (ℓ is the oscillator length, ω the trap frequency and λ an effective coupling constant). We also perform a comparative analysis between our method of solution and two other methods with defined range of validity: the perturbation theory and the Thomas-Fermi Approximation (TFA). The presented method is valid for negative and for positive self interactions as well. In the positive coupling constant region it is consistent with the TFA, extending its validity beyond the limits of validity of this well known approximation, towards small and negative values of the self-interaction coupling constant. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
It has been shown previously that the modified Thomas–Fermi (MTF) equation can be solved analytically by using a higher order iteration method (MTF(p)) for a spherically trapped atomic Bose–Einstein condensate (BEC). The ground-state properties, e.g. chemical potential, peak gas parameter and the ground state density of atoms (except in the surface region), thus obtained can successfully reproduce the results obtained by solving the modified Gross–Pitaevskii (MGP) equation numerically. In this paper, we have extended this analytical method for obtaining the ground-state and dynamical properties of a cylindrically trapped atomic BEC. In this analytical approach, the ground-state density of atoms at the surface of the trap fails to reproduce the correct behaviour due to the neglect of kinetic energy in the MTF equation. We have proposed a model function which when fitted with the analytical wavefunction (from the MTF equation) near the boundaries of the trap can reproduce the correct behaviour of the ground-state density of atoms at the surface region. The kinetic energy and the other energy components thus obtained using the modified analytical method (MTF(p)+model function fitting) can successfully reproduce the results obtained by solving the MGP equation numerically for the 85Rb BEC containing 104 atoms both for spherical and cylindrical traps. The excitation frequency thus obtained for the compressional mode of both spherically and axially symmetric traps is in good agreement with the numerical results. For these calculations, the virial relation is satisfied within the limit ≤10−4.
Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross–Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 (1996) 2477] This new method is simple compared to other methods used to solve numerically the Gross–Pitaevskii equation, and one can obtain analytic and reliable results.