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Optics and Photonics News

Solutions of the Nonlinear Schrödinger Equation for Bose-Einstein Condensates of Alkali Atoms

R. J. Dodd, Mark Edwards, Keith R. Burnett, Charles W. Clark

Published Jan 1, 1996• 0 citations
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Stability of the solution of the nonlinear Schrödinger equation for Bose-Einstein condensation

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Spatially periodic structures of an atomic Bose-Einstein condensate

Nikolai N Rozanov

The conditions providing the formation of periodic vortex lattices of an interference nature in an atomic Bose-Einstein condensate (i.e., in the absence of rotation of the condensate) are determined. Spatially periodic exact solutions of the nonlocal nonlinear Schrödinger equation (the generalized Gross-Pitaevskii equation) that describes the Bose-Einstein condensate of a dilute gas of alkali metal atoms with due regard for the nonlocality of interatomic interactions are obtained in the form of a set of two or three plane waves. It is shown that periodic vortex lattices can be produced in interference experiments with a Bose-Einstein condensate of a dilute gas of alkali metal atoms.

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Stability of exact solutions of the defocusing nonlinear Schrödinger equation with periodic potential in two dimensions

Bernard Deconinck, Béla A. Frigyik, J. Nathan Kutz

The cubic nonlinear Schrodinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose--Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our results imply that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Key words: Bose-Einstein condensates, two-dimensional nonlinear Schrodinger equation, periodic potential, elliptic functions PACS: 65.65 1 Introduction The cubic nonlinear Schrodinger equation with attractive or repulsive nonlinearity and a potential is used as a mean-field model for the dynamics of a dilute-gas Bose Einstein condensate (BEC) [1,2]. In this case, the equation is often referred to as the Gross-Pitaevskii equation. These BECs are of interest in both the theoretical and experim...

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Theoretical study of Bose-Einstein condensation in alkali atom gases

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Resonant reflection of a Bose–Einstein condensate by a double barrier within the Gross–Pitaevskii equation

V. P. Kraynov, H. A. Ishkhanyan

Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrodinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.

Physica Scripta•Sep 1, 2010•2 citations

Higher Dimensional Nonlinear Schrödinger Equations of Bose–Einstein Condensates

Wu‐Ming Liu, Emmanuel Kengne

Springer eBooks•Jan 1, 2019•0 citations
The conditions providing the formation of periodic vortex lattices of an interference nature in an atomic Bose-Einstein condensate (i.e., in the absence of rotation of the condensate) are determined. Spatially periodic exact solutions of the nonlocal nonlinear Schrödinger equation (the generalized Gross-Pitaevskii equation) that describes the Bose-Einstein condensate of a dilute gas of alkali metal atoms with due regard for the nonlocality of interatomic interactions are obtained in the form of a set of two or three plane waves. It is shown that periodic vortex lattices can be produced in interference experiments with a Bose-Einstein condensate of a dilute gas of alkali metal atoms.
The cubic nonlinear Schrodinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose--Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our results imply that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Key words: Bose-Einstein condensates, two-dimensional nonlinear Schrodinger equation, periodic potential, elliptic functions PACS: 65.65 1 Introduction The cubic nonlinear Schrodinger equation with attractive or repulsive nonlinearity and a potential is used as a mean-field model for the dynamics of a dilute-gas Bose Einstein condensate (BEC) [1,2]. In this case, the equation is often referred to as the Gross-Pitaevskii equation. These BECs are of interest in both the theoretical and experim...
Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrodinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.