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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

On the soliton solutions of the spinor Bose-Einstein condensate

Nikolay A. Kostov, Victor Atanasov, Vladimir Stefanov Gerdjikov, Georgi Georgiev Grahovski

Published Mar 2, 2007• 10 citations
Publisher

Abstract

A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensates is studied. These types of χ3-interactions are integrable by the inverse scattering method. We analyze its Hamiltonian properties and outline an algebraic procedure to derive their three types of soliton solutions based on Zakharov-Shabat dressing method. Their applications to spinor model of Bose-Einstein condensates are discussed.

DOI
10.1117/12.727191

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A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensates is studied. These types of χ3-interactions are integrable by the inverse scattering method. We analyze its Hamiltonian properties and outline an algebraic procedure to derive their three types of soliton solutions based on Zakharov-Shabat dressing method. Their applications to spinor model of Bose-Einstein condensates are discussed.
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We propose an integrable model of a multicomponent spinor Bose-Einstein condensate in one dimension, which allows an exact description of the dynamics of bright solitons with spin degrees of freedom. We consider specifically an atomic condensate in the F=1 hyperfine state confined by an optical dipole trap. When the mean-field interaction is attractive (c(0)<0) and the spin-exchange interaction of a spinor condensate is ferromagnetic (c(2)<0), we prove that the system possesses a completely integrable point leading to the existence of multiple bright solitons. By applying results from the inverse scattering method, we analyze a collision law for two-soliton solutions and find that the dynamics can be explained in terms of the spin precession.
We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.
We present a general classification of one-soliton solutions as well as families of rogue-wave solutions for F=1 spinor Bose-Einstein condensates (BECs). These solutions are obtained from the inverse scattering transform for a focusing matrix nonlinear Schrödinger equation which models condensates in the case of attractive mean-field interactions and ferromagnetic spin-exchange interactions. In particular, we show that when no background is present, all one-soliton solutions are reducible via unitary transformations to a combination of oppositely polarized solitonic solutions of single-component BECs. On the other hand, we show that when a nonzero background is present, not all matrix one-soliton solutions are reducible to a simple combination of scalar solutions. Finally, by taking suitable limits of all the solutions on a nonzero background we also obtain three families of rogue-wave (i.e., rational) solutions.
Abstract Exact analytical solutions are good candidates for studying and explaining the dynamics of solitons in nonlinear systems. We further extend the region of existence of spin solitons in the nonlinearity coefficient space for the spin-1 Bose–Einstein condensate. Six types of spin soliton solutions can be obtained, and they exist in different regions. Stability analysis and numerical simulation results indicate that three types of spin solitons are stable against weak noise. The non-integrable properties of the model can induce shape oscillation and increase in speed after the collision between two spin solitons. These results further enrich the soliton family for non-integrable models and can provide theoretical references for experimental studies.