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Communications in Theoretical Physics

Elliptic Function Waves of Spinor Bose–Einstein Condensates in an Optical Lattice

Xie Yuan-Dong

Published Mar 1, 2009• 10 citations
Publisher

Abstract

An improved nonlinear Schrodinger equation different from usual one of spinor Bose–Einstein condensates (BECs) in an optical lattice are obtained by taking into account a nonlinear term in the equation of motion for probability amplitude of spins carefully. The elliptic function wave solutions of the model are found under specific boundary condition, for example, the two ends of the atomic chain are fixed. In the case of limit the elliptic function wave solutions are reduced into spin-wave-like or solitons.

DOI
10.1088/0253-6102/51/3/12

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An improved nonlinear Schrodinger equation different from usual one of spinor Bose–Einstein condensates (BECs) in an optical lattice are obtained by taking into account a nonlinear term in the equation of motion for probability amplitude of spins carefully. The elliptic function wave solutions of the model are found under specific boundary condition, for example, the two ends of the atomic chain are fixed. In the case of limit the elliptic function wave solutions are reduced into spin-wave-like or solitons.
The present work is devoted to the theoretical investigation of the dynamics of Bose –Einstein condensates (BEC) in optical lattice with the long range magnetic dipole-dipole interaction between atoms. The system is described by a nonlocal nonlinear Schrodinger equation (NLSE). We consider the case when the lattice depth is sufficiently large (tight binding approximation). In this case nonlocal NLSE can be reduced to discrete NLSE equation. We show that discrete NLSE equation for some range of parameters can be reduced further to integrable Ablowitz-Ladik equation which supports moving solitons. Taking the one-soliton solution of Ablowitz-Ladik equation as an initial condition we have performed detailed numerical study of a soliton dynamics in both nonlocal NLSE and discrete NLSE. The conditions of existence and stability of moving solitons in dipolar BEC in an optical lattice are numerically revealed. Also the applicability limits for the tight binding approximation for the experimentally achievable range of parameters are investigated.
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).