Effective theory for the propagation of a wave-packet in a disordered and nonlinear medium

Abstract

The propagation of a wave-packet in a nonlinear disordered medium exhibits interesting dynamics. Here, we present an analysis based on the nonlinear Schrödinger equation (Gross-Pitaevskii equation). This problem is directly connected to experiments on expanding Bose gases and to studies of transverse localization in nonlinear optical media. In a nonlinear medium the energy of the wave-packet is stored both in the kinetic and potential parts, and details of its propagation are to a large extent determined by the transfer from one form of energy to the other. A theory describing the evolution of the wave-packet has been formulated in [G. Schwiete and A. Finkelstein, Phys. Rev. Lett. 104, 103904 (2010)] in terms of a nonlinear kinetic equation. In this paper, we present details of the derivation of the kinetic equation and of its analysis. As an important new ingredient we study interparticle-collisions induced by the nonlinearity and derive the corresponding collision integral. We restrict ourselves to the weakly nonlinear limit, for which disorder scattering is the dominant scattering mechanism. We find that in the special case of a white noise impurity potential the mean squared radius in a two-dimensional system scales linearly with t. This result has previously been obtained in the collisionless limit, but it also holds in the presence of collisions. Finally, we mention different mechanisms through which the nonlinearity may influence localization of the expanding wave-packet.

21 pages, 10 figures

References in corpus (5)

• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Quantum corrections to the dynamics of interacting bosons: beyond the truncated Wigner approximation
• Quantum versus classical statistical dynamics of an ultracold Bose gas
• Expansion of a Bose-Einstein Condensate in the Presence of Disorder
• Anisotropic 2D diffusive expansion of ultra-cold atoms in a disordered potential

Cited by in corpus (7)

• Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos
• Thermalization of matter waves in speckle potentials
• Keldysh approach to the renormalization group analysis of the disordered electron liquid
• Nonlinear quantum transport of light in a cold atomic cloud
• Quench dynamics of a weakly interacting disordered Bose gas in momentum space
• A self-consistent theory of localization in nonlinear random media
• Kinetics of the disordered Bose gas with collisions