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

TL;DR

This paper develops a detailed nonlinear kinetic theory for wave-packet propagation in disordered, nonlinear media, analyzing energy transfer, collisions, and localization effects, with implications for Bose gases and optical systems.

Contribution

It derives a kinetic equation including collision effects for wave propagation in nonlinear disordered media, extending previous models to include interparticle collisions and their impact.

Findings

Mean squared radius scales linearly with time in 2D with white noise disorder

Collision effects do not alter the linear time scaling in this regime

Nonlinearity influences localization mechanisms of the wave-packet

Abstract

The propagation of a wave-packet in a nonlinear disordered medium exhibits interesting dynamics. Here, we present an analysis based on the nonlinear Schr\"odinger 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.