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Journal of Physics Conference Series

Partially coherent matter wave soliton solutions: Multimode approach

Jun Chen, Qiang Lin

Published Dec 17, 2012• 0 citations
Publisher

Abstract

We introduce a multimode approach to describe the partially coherent matter wave(PCMW) and construct a set of localized soliton solutions of the one-dimensional multimode nonlinear Schrödinger equation with a periodic lattice potential and a periodic interaction potential. Based on our analytical expression of the multimode PCMW, we investigate the density distribution and coherence property of a two-mode case.

DOI
10.1088/1742-6596/400/1/012003

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We introduce a multimode approach to describe the partially coherent matter wave(PCMW) and construct a set of localized soliton solutions of the one-dimensional multimode nonlinear Schrödinger equation with a periodic lattice potential and a periodic interaction potential. Based on our analytical expression of the multimode PCMW, we investigate the density distribution and coherence property of a two-mode case.
As a coherent matter wave,the Bose-Einstein condensate(BEC) provides a playground for the investigation of the macroscopic quantum phenomenon.The nonlinear optical lattice,which is consisted of two counter-propagating laser beams,is a powerful and widely used tool for the controlling of the dynamics of the coherent matter wave.In view of the potential applications of BEC,two or more independently grown condensates provide extra controllable degrees of freedom and can be used in many areas,such as quantum metrology and quantum computation.In this paper,a multi-mode description is introduced to characterize the multi-component coherent matter wave.To determine the evolution property of the multi-mode matter wave in a nonlinear optical lattice,we solve the multi-mode coupled nonlinear equations.A set of analytical bright soliton solutions are constructed with a similarity transformation.
We theoretically study the role of the mode structure of a multicomponent Bose–Einstein condensate (BEC) in the potential created by a nonlinear optical lattice. We describe a multisoliton complex (MSC) as a superposition of different fundamental soliton modes in the matter-wave system. Using a similarity transformation, we solve the nonlinear evolution equation of the multimode coupled matter-wave field and construct a set of analytical bright soliton solutions. A perturbation method is used to examine the linear stability of the constructed solitons. Based on these particular solutions, we numerically analyze the mode structure of a MSC. The results show that the periodicity causes a Bloch modulation in the envelopes of the density distribution. When different fundamental modes collide with each other in the nonlinear lattice, the collision-induced shifts, and the space-dependent modulation of external potentials change the density profile of the multimode soliton complex. Therefore, the mode structure, which is absent in a one-mode BEC, provides the possible multiscale modeling of the matter-wave field with extra degrees of freedom.
The Wigner function, a kind of wavelet transform is studied for one- and two-soliton solutions to the nonlinear Schrödinger equation. The exact expression of the Wigner transform for the one-soliton solution is obtained. The features of soliton-soliton interaction of different two-soliton states, the states of coherence analogous to that of two-localized wave packets in quantum mechanics, are displayed. Suitability of this transform for analyzing “ soliton ” in open systems is also discussed.
Dynamics and stability of solitons in two-dimensional (2D) Bose-Einstein condensates (BEC), with one-dimensional (1D) conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media (with attractive atomic systems), the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media (repulsive BEC case) with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.
We overview our recent theoretical studies on nonlinear atom optics of the Bose-Einstein condensates (BECs) loaded into optical lattices. In particular, we describe the band-gap spectrum and nonlinear localization of BECs in one- and two-dimensional optical lattices. We discuss the structure and stability properties of spatially localized states (matter-wave solitons) in 1D lattices, as well as trivial and vortex-like bound states of 2D gap solitons. To highlight similarities between the behavior of coherent light and matter waves in periodic potentials, we draw useful parallels with the physics of coherent light waves in nonlinear photonic crystals and optically-induced photonic lattices.
The dynamics of matter waves in linear and nonlinear optical lattices subject to a spatially uniform linear force is studied both analytically and numerically. It is shown that by properly designing the spatial dependence of the scattering length it is possible to induce long-living Bloch oscillations of gap-soliton matter waves in optical lattices. This occurs when the effective nonlinearity and the effective mass of the soliton have opposite signs for all values of the crystal momentum in the Brillouin zone. The results apply to all systems modeled by the periodic nonlinear Schrödinger equation, including propagation of light in photonic and photorefractive crystals with tilted band structures.