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

Nonautonomous solitons in nonlinear optics and Bose-Einstein condensates

V. N. Serkin, Akira Hasegawa, Tatyana Leonidovna Belyaeva

Published Dec 22, 2008• 0 citations
Publisher

Abstract

We consider the generalized Gross-Pitaevskii models with varying nonlinearities from the point of view of their exact integrability and obtain the general solutions for these models both for confining and expulsive external potentials. We show that exact soliton solutions exist only under certain conditions and that varying in time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability conditions. The main features of nonautonomous matter-wave solitons near the Feshbach resonance with continuously tuned scattering length are investigated. We focus on the most physically important situations where the applied magnetic field is varying in time linearly and periodically. It is proved that near a Feshbach resonance the matter wave solitons can be stabilized even without a trapping potential. In the case of periodically varying interaction strength among atoms, variations of the external harmonic potential are found to be sign-reversal. The main difference from the canonical soliton case is that the matter-wave solitons management concept must be consistent with variations of confining potential.

DOI
10.1117/12.822526

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We consider the generalized Gross-Pitaevskii models with varying nonlinearities from the point of view of their exact integrability and obtain the general solutions for these models both for confining and expulsive external potentials. We show that exact soliton solutions exist only under certain conditions and that varying in time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability conditions. The main features of nonautonomous matter-wave solitons near the Feshbach resonance with continuously tuned scattering length are investigated. We focus on the most physically important situations where the applied magnetic field is varying in time linearly and periodically. It is proved that near a Feshbach resonance the matter wave solitons can be stabilized even without a trapping potential. In the case of periodically varying interaction strength among atoms, variations of the external harmonic potential are found to be sign-reversal. The main difference from the canonical soliton case is that the matter-wave solitons management concept must be consistent with variations of confining potential.
By means of analytical and numerical methods, we reveal the main features of nonautonomous matter-wave solitons near the Feshbach resonance in a one-dimensional Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. Based on the generalized nonautonomous Gross-Pitaevskii model, we show that solitons in nonautonomous physical systems exist only under certain conditions so that varying-in-time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability scenarios and complement each other. We focus on the most physically important examples where the applied magnetic field is either a linearly or a periodically varying-in-time function. In the case of periodically varying scattering length, variations of confining harmonic potential are found to be sign-reversible (periodic attractive and repulsive) to support the soliton-management regime. We investigate the losses of validity of one-dimensional (1D) approximation in the cases when, by the joint action of varying-in-time nonlinearity and confining potential, the atom cloud can be compressed from an initially elongated quasi-1D cigar-shaped geometry to a final ball-shaped three-dimensional geometry and the induced soliton collapse may occur.
We review our recent theoretical advances in the dynamics of Bose-Einstein condensates with tunable interactions using Feshbach resonance and external potential. A set of analytic and numerical methods for Gross-Pitaevskii equations are developed to study the nonlinear dynamics of Bose-Einstein condensates. Analytically, we present the integrable conditions for the Gross-Pitaevskii equations with tunable interactions and external potential, and obtain a family of exact analytical solutions for one- and two-component Bose-Einstein condensates in one and two-dimensional cases. Then we apply these models to investigate the dynamics of solitons and collisions between two solitons. Numerically, the stability of the analytic exact solutions are checked and the phenomena, such as the dynamics and modulation of the ring dark soliton and vector-soliton, soliton conversion via Feshbach resonance, quantized soliton and vortex in quasi-two-dimensional are also investigated. Both the exact and numerical solutions show that the dynamics of Bose-Einstein condensates can be effectively controlled by the Feshbach resonance and external potential, which offer a good opportunity for manipulation of atomic matter waves and nonlinear excitations in Bose-Einstein condensates.
We study exact single-soliton solutions of an attractive Bose—Einstein condensate governed by a one-dimensional nonautonomous Gross—Pitaevskii system. For several different forms of time-dependent atom—atom interaction and external parabolic potential which satisfy the exact integrability scenario, we construct a set of new analytical nonautonomous deformed-soliton solutions, including the macroscopic wave function and the position of soliton's center of mass. The soliton characteristics are modulated by the external field parameters and deformation factors related to the number of the condensed atoms and the initial conditions. The results suggest a simple and effective method for experimentally generating matter-wave deformed solitons and manipulating their motions.
Mathematical similarities and parallels between two different physical objects, optical solitons and matter-wave solitons, both described by similar mathematical models: the nonlinear Schrodinger equation (NLSE) and the Gross-Pitaevskii equation (GPE) model, open the possibility to study both systems in parallel and because of the obvious complexity of experiments with matter-wave solitons, offer outstanding possibilities in studies of BEC system by performing experiments in the nonlinear optical system and vise versa. In this report we briefly overview recent theoretical studies of the existence and stability of 3D solitons. With contributions from major groups who have pioneered research in this field, the report describes the historical development of the subject, provides a background to the associated nonlinear optical processes, the generation mechanisms of soliton bullets. The main features of nonautonomous matter-wave solitons near the Feshbach resonance with continuously tuned scattering length are investigated. We focus on the most physically important situations where the applied magnetic field is varying in time linearly and periodically. In nonlinear optical applications, this kind of periodic graded-index nonlinear structure with alternating waveguiding and antiwaveguiding segments can be used to simulate different and complicated processes in the total scenario of matterwave soliton bullets generation.
We present three families of exact matter-wave soliton solutions for an effective one-dimension two-component Bose—Einstein condensates (BECs) with tunable interactions, harmonic potential and gain or loss term. We investigate the dynamics of bright-bright solitons, bright-dark solitons and dark-dark solitons for the time-dependent expulsive harmonic trap potential, periodically modulated harmonic trap potential, and kinklike modulated harmonic trap potential. Through the Feshbach resonance, these dynamics can be realized in experiments by suitable control of time-dependent trap parameters, atomic interactions, and interaction with thermal cloud.
Exact solitonic solutions of the Gross–Pitaevskii equation with time-modulated nonlinearity of a(t) = a0 / (t + t0) are obtained. With help of these solutions, we analyze the properties of Feshbach-managed solitons in Bose–Einstein condensates in details. Our results show that the parameters of atomic matter waves can be manipulated by proper variation of the scattering length. In particular, an exact two-soliton solution is given, from which, it is shown that the separation between the neighboring solitons can be effectively maintained by allowing the solitons to have unequal initial amplitudes.