Localization of wave packets in one-dimensional random potentials

TL;DR

This paper investigates how wave packets localize in one-dimensional random potentials, providing an analytical framework that matches numerical results for the localized density profile, including both tails and center.

Contribution

It extends Berezinskii's diagrammatic method to wave packets and derives a comprehensive analytical description of Anderson localization in 1D random potentials.

Findings

Analytical expression for Lyapunov exponent across energies

Self-consistent Born approximation for energy distribution

Good agreement between theory and numerical simulations

Abstract

We study the expansion of an initially strongly confined wave packet in a one-dimensional weak random potential with short correlation length. At long times, the expansion of the wave packet comes to a halt due to destructive interferences leading to Anderson localization. We develop an analytical description for the disorder-averaged localized density profile. For this purpose, we employ the diagrammatic method of Berezinskii which we extend to the case of wave packets, present an analytical expression of the Lyapunov exponent which is valid for small as well as for high energies and, finally, develop a self-consistent Born approximation in order to analytically calculate the energy distribution of our wave packet. By comparison with numerical simulations, we show that our theory describes well the complete localized density profile, not only in the tails, but also in the center.