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Brazilian Journal of Physics

An analytical solution for the critical number of particles for stable bose-einstein condensation under the influence of an anisotropic potential

Alvaro S. de Sousa, Vanderlei Salvador Bagnato, Albérico B. F. da Silva

Published Mar 1, 2008• 6 citations
Publisher

Abstract

We have considered a Bose gas in an anisotropic potential. Applying the the Gross-Pitaevskii Equation (GPE) for a confined dilute atomic gas, we have used the methods of optimized perturbation theory and self-similar root approximants, to obtain an analytical formula for the critical number of particles as a function of the anisotropy parameter for the potential. The spectrum of the GPE is also discussed.

DOI
10.1590/s0103-97332008000100020

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We have considered a Bose gas in an anisotropic potential. Applying the the Gross-Pitaevskii Equation (GPE) for a confined dilute atomic gas, we have used the methods of optimized perturbation theory and self-similar root approximants, to obtain an analytical formula for the critical number of particles as a function of the anisotropy parameter for the potential. The spectrum of the GPE is also discussed.
We apply controlled perturbation theory to calculate the spectrum of the Gross-Pitaevskii equation for a system composed of attractive bosons confined in an anisotropic harmonic trap. The energy spectrum is calculated as a function of the coupling parameters for traps going from cigar to pancake shapes. The critical number of particles that ensures real values for the energy spectrum is obtained as a function of the potential anisotropic parameter, showing strong dependence of the critical number on the anisotropy of the trap. For a number of particles above the critical value the metastability of the system is characterized through the calculation of the condensate lifetime, using the imaginary part of the energy values. The obtained results are relevant for experiments where highly anisotropic traps are considered.
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic trap, modeled by an anisotropic harmonic potential. We evaluate the wave function and the energy of the Bose-Einstein condensate as a function of the particle number, both for positive and negative scattering length. The results for the transverse and the vertical size of the cloud of atoms, as well as for the kinetic and potential energy per particle, are compared with the predictions of approximated models. We also compare the aspect ratio of the velocity distribution with experimental estimates available for 87Rb^{87}\mathrm{Rb}87Rb. Vortex states are considered and the critical angular velocity for production of vortices is calculated. We show that the presence of vortices significantly increases the stability of the condensate in the case of attractive interactions. \textcopyright{} 1996 The American Physical Society.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
It is well known that bosonic particles with attractive interaction in a uniform gas do not form a condensate. Here we investigate a dilute Bose gas and study stationary solutions of the Gross–Pitaevskii equation with attractive interaction. We have also used a higher order stabilising term in the presence of a harmonic confining potential. We show that there are three possible types of stationary solutions corresponding to stable, metastable and unstable phases. These results are discussed in relation to a 7Li condensate.
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)
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.