Re-localization due to finite response times in a nonlinear Anderson chain

TL;DR

This paper investigates how a finite response time in a disordered nonlinear Schrödinger equation causes energy drift towards the band edge, leading to re-localization of initially localized excitations, contrasting with subdiffusive spreading observed when response time is zero.

Contribution

It provides a numerical explanation for the suppression of spreading in a nonlinear Anderson chain with finite response time, linking energy drift to re-localization.

Findings

Energy drifts towards the band edge with finite response time

Re-localization occurs due to reduced population of localized modes

Spreading is suppressed compared to the zero response time case

Abstract

We study a disordered nonlinear Schr\"odinger equation with an additional relaxation process having a finite response time $\tau$. Without the relaxation term, $\tau=0$, this model has been widely studied in the past and numerical simulations showed subdiffusive spreading of initially localized excitations. However, recently Caetano et al.\ (EPJ. B \textbf{80}, 2011) found that by introducing a response time $\tau > 0$, spreading is suppressed and any initially localized excitation will remain localized. Here, we explain the lack of subdiffusive spreading for $\tau>0$ by numerically analyzing the energy evolution. We find that in the presence of a relaxation process the energy drifts towards the band edge, which enforces the population of fewer and fewer localized modes and hence leads to re-localization. The explanation presented here is based on previous findings by the authors et al.\ (PRE \textbf{80}, 2009) on the energy dependence of thermalized states.