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Kluwer Academic Publishers eBooks

Two-Component Bose-Einstein Condensates in Optical Lattices

Nikolay A. Kostov, Vladimir Stefanov Gerdjikov, V. Z. Enolski, Mario Salerno, V. V. Konotop

Published Mar 1, 2006• 1 citation
Publisher

Abstract

Coupled nonlinear Schrödinger equations (CNLS) with an external elliptic function potential model a quasi one-dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New stationary solutions of the CNLS with a periodic potential are derived and interpreted as exact Bloch states at the edge of the Brillouin zone. The modulationally unstable solutions lead to formation of localized ground states of the coupled BEC system.

DOI
10.1007/1-4020-2190-9_22

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Coupled nonlinear Schrödinger equations (CNLS) with an external elliptic function potential model a quasi one-dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New stationary solutions of the CNLS with a periodic potential are derived and interpreted as exact Bloch states at the edge of the Brillouin zone. The modulationally unstable solutions lead to formation of localized ground states of the coupled BEC system.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.
Coupled nonlinear Schrödinger (CNLS) equations with an external elliptic function potential model with high accuracy a quasi-one-dimensional interacting two-component Bose-Einstein condensate (BEC) trapped in a standing wave generated by a few laser beams. The construction of stationary solutions of the two-component CNLS equation with a periodic potential is detailed and their stability properties are studied by direct numerical simulations. Some of these solutions allow reduction to the Manakov system. From a physical point of view the trivial phase solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against weak modulations of long wavelength. By numerical simulations it is shown that the modulationally unstable solutions lead to the formation of localized ground states of the coupled BEC system.
The cubic vector nonlinear Schrödinger equation with an external trigonometric potential models a quasi-one-dimensional multi-component Bose–Einstein condensate trapped in a standing light wave. We construct families of exact stationary solutions for the more general case of an elliptic function potential. Some of these solutions degenerate to zero as the effect of the external potential disappears, whereas others limit to solutions of the free vector nonlinear Schrödinger equation. The stability of these solutions is examined both analytically and numerically. The stability results depend on the nature of the atomic interactions not only within the components but also between components. As in the scalar case (one component) with repulsive interaction, all linearly stable solutions are deformations of the ground state of the linear Schrödinger equation. Unlike the scalar case with attractive interaction, no solutions are found to be stable if there is any attractive interaction present.
We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.
We study the nonlinear dynamics of two-component Bose–Einstein condensates in one-dimensional periodic optical lattice potentials. The stationary state perturbation solutions of the coupled two-component nonlinear Schrödinger/Gross–Pitaevskii equations are constructed by using the direct perturbation method. Theoretical analysis revels that the perturbation solution is the chaotic one, which indicates the existence of chaos and chaotic region in parameter space. The corresponding numerical calculation results agree well with the analytical results. By applying the chaotic perturbation solution, we demonstrate the atomic spatial population and the energy distribution of the system are chaotic generally.
The cubic–quintic nonlinear Schrödinger equation with a periodic potential, which is expressed in terms of the Jacobian function dn, is used to model a Bose–Einstein condensate trapped in an optical lattice (in most cases, the dn potential is very close to the ordinary sinusoidal one). The quintic term in the equation represents either effects of three-body collisions (which is relevant in the case when the ordinary two-body collisions are very weak), or an effective deviation of the trapped condensate from the one-dimensionality. The cases of both repulsive and attractive two-body interactions are considered. The same model governs the planar propagation of light in waveguides with the cubic–quintic optical nonlinearity, the periodic potential being induced by transverse modulation of the refractive index. In the case of the attractive quintic term (which is always the case if it accounts for the residual non-one-dimensionality of the tightly trapped condensate), a new family of exact periodic solutions is constructed in terms of Jacobian elliptic functions. Using semi-analytical and numerical methods, a stability region for these periodic states is identified. The effect of boundaries is shown to be very weak. E Kengne dedicates this work to Taa Bah Deffo Takotue Innocent, the Chairman of the Bandjoun Community in Douala (Cameroon).