Dynamics of localized waves in 1D random potentials: statistical theory of the coherent forward scattering peak

TL;DR

This paper investigates the long-time behavior of the coherent forward scattering peak in 1D disordered systems, linking it to eigenfunction statistics and spectral properties, and confirms the dynamics with numerical and theoretical methods.

Contribution

It provides a detailed statistical theory of the CFS peak dynamics in 1D random potentials, extending understanding beyond perturbative regimes.

Findings

CFS peak dynamics governed by logarithmic level repulsion

Peak width inversely proportional to localization length

Peak height is twice the background, indicating Poisson eigenfunction statistics

Abstract

As recently discovered [PRL ${\bf 109}$ 190601(2012)], Anderson localization in a bulk disordered system triggers the emergence of a coherent forward scattering (CFS) peak in momentum space, which twins the well-known coherent backscattering (CBS) peak observed in weak localization experiments. Going beyond the perturbative regime, we address here the long-time dynamics of the CFS peak in a 1D random system and we relate this novel interference effect to the statistical properties of the eigenfunctions and eigenspectrum of the corresponding random Hamiltonian. Our numerical results show that the dynamics of the CFS peak is governed by the logarithmic level repulsion between localized states, with a time scale that is, with good accuracy, twice the Heisenberg time. This is in perfect agreement with recent findings based on the nonlinear $\sigma$-model. In the stationary regime, the width of the CFS peak in momentum space is inversely proportional to the localization length, reflecting the exponential decay of the eigenfunctions in real space, while its height is exactly twice the background, reflecting the Poisson statistical properties of the eigenfunctions. Our results should be easily extended to higher dimensional systems and other symmetry classes.