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

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 [Schwiete and Finkel'stein, 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 indicate different mechanisms through which the nonlinearity may influence localization of the expanding wave packet.

Figure 1

The retarded ($G^R$) and advanced ($G^A$) Green's functions. The time arrow runs from left to right.

Figure 2

In a Keldysh many-body approach to interacting bosons, two classes of vertices appear: the classical vertices shown on the left and the quantum vertices shown on the right. In the MSR-type approach used in this paper, only the classical vertices are present.

Figure 3

Upon averaging with respect to the fields $\psi$ and $\eta$, the (classical) interaction vertices in our approach give rise to the two subdiagrams shown above.

Figure 4

General structure of the perturbation theory: The density evolution is represented by the infinite sum of all diagrams of the type displayed in this figure.

Figure 5

Diagrammatic representation of the diffusion process in the absence of the nonlinearity.

Figure 6

On a diagrammatic level, the solution of the kinetic equation corresponds to the sum of all diagrams of the type shown in this figure.

Figure 7

The two box diagrams: ${\mathcal{B}}_R$ to the left and ${\mathcal{B}}_A$ to the right. The interaction line carries frequency ${\omega }_1$ and momentum ${\mathbf{q}}_1$ (incoming). For the individual diagrams, a constant term remains in the limit of vanishing external frequencies and momenta. This constant cancels, however, between the two diagrams. The cancellation is related to number conservation as is discussed in the main text.

Figure 8

For dimension $d\ne 2$, the density of states is not constant and the $\vartheta$ dependence of the Green's function entering the SCBA becomes important. In this case, the box diagrams ${\mathcal{B}}_R$ and ${\mathcal{B}}_A$ should be generalized as displayed.

Figure 9

Corrections to ${\Sigma }^R$ according to Eq. (5.2).

Figure 10

Corrections to ${\Sigma }^S$ according to Eq. (5.1).