Transport of localized and extended excitations in a nonlinear Anderson model

Abstract

We study the propagation of electrons (or excitations) through a one-dimensional tight-binding chain in the simultaneous presence of nonlinearity and diagonal disorder. The evolution of the system is given by a disordered version of the discrete nonlinear Schr\"odinger equation. For an initially localized excitation we examine its mean square displacement $〈{n}^{2}(t)〉$ for relatively long times $\mathrm{Vt}\ensuremath{\sim}{10}^{4},$ for different degrees of nonlinearity. We found that the presence of nonlinearity produces a subdiffusive propagation $〈{n}^{2}(t)〉\ensuremath{\sim}{t}^{\ensuremath{\alpha}},$ with $\ensuremath{\alpha}\ensuremath{\sim}0.27$ and depending weakly on nonlinearity strength. This nonlinearity effect seems to persist for a long time before the system converges to the usual Anderson model. We also compute the transmission of plane waves through the system. We found an average transmissivity that decays exponentially with system size $〈T〉\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}L),$ where \ensuremath{\beta} increases with nonlinearity. We conclude that the presence of nonlinearity favors (inhibits) the propagation of localized (extended) excitations.