Destruction of Anderson Localization by a Weak Nonlinearity

Abstract

We study numerically a spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time ∝ t α , with the exponent α being in the range 0.3 − 0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case. [2,3,4,5,6]. An interesting new aspect in such systems is an importance of nonlinear effects since in a good approximation the evolution of BEC can be described by the nonlinear Gross-Pitaevskii equation (see e.g. [7]). An interplay of disorder, localization, and nonlinearity appears also in other physical systems like wave propagation in nonlinear disordered media (see e.g. [8,9]), chains of nonlinear oscillators (see e.g.[10]) with randomly distributed frequencies, and models of quantum chaos with a kicked soliton [11] and a kicked BEC [12,13,14].We focus here on the discrete Anderson nonlinear Schrödinger equation (DANSE)where β characterizes nonlinearity, V is hopping matrix element, on-site energies are randomly distributed in the range −W/2 < E n < W/2, and the total probability is normalized to unity n | ψ n | 2 = 1. For β = 0 and weak disorder all eigenstates are exponentially localized with the localization length l ≈ 96(V /W ) 2 at the center of the energy band [19]. Hereafter we set for convenience = V = 1, thus the energy coincides with the frequency. For nonlinear equation (1) we consider the following problem: how an initially localized field | ψ n (0) | 2 = δ n,0 is spreading? In the linear case the spreading saturates after excitation of all linear modes that have significant values at n = 0. The same process of "initial excitation" of modes happens in the nonlinear case as well, this initial stage of spreading has been analyzed recently in refs. [20,21] and is now well understood. However, a behavior at large time scales remains less clear. The results presented in [20] support the view of eventual exponential localization of the field. We demonstrate below that the spreading is unlimited, however it is rather slow -subdiffusive.The basic idea is to use the equivalence between the Anderson localization and the localization of quasienergy eigenstates in a kicked quantum rotator [15,16]. In the latter model the case of quantum chaos with nonlinearity has been considered analytically and numerically in [17,18] and it has been shown that above a certain nonlinearity level, nonlinear phase shifts lead to a complete delocalization with a subdiffusive spreading over all states [17]. Furthermore it has been argued that the same situation should appear for the DANSE (1).We first apply the theoretical arguments of paper [17] to model (1), and then perform large scale numerical simulations of a wave packet spreading on a time scale which is by 5-6 orders of magnitude larger compared to that i...

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