From percolation transition to Anderson localization in one-dimensional speckle potentials

TL;DR

This paper investigates the transition from classical percolation to quantum Anderson localization in a one-dimensional speckle potential, revealing a continuous crossover influenced by potential correlations and non-Gaussian features.

Contribution

It provides a comprehensive semi-classical analysis of the crossover between percolation and Anderson localization in 1D speckle potentials, highlighting the breakdown of standard models and predicting bimodal transmission distributions.

Findings

Algebraic divergence of percolation correlation length connects smoothly to localization length.

Correlations and non-Gaussian features cause deviations from standard DMPK predictions.

Bimodal transmission distribution emerges in the crossover regime.

Abstract

Classical particles in random potentials typically experience a percolation phase transition, being trapped in clusters of mean size $\chi$ that diverges algebraically at a percolation threshold. In contrast, quantum transport in random potentials is controlled by the Anderson localization length, which shows no distinct feature at this classical critical point. Here, we present a comprehensive theoretical analysis of the semi-classical crossover between these two regimes by studying particle propagation in a one-dimensional, red speckle potential, which hosts a percolation transition at its upper bound. As the system deviates from the classical limit, we find that the algebraic divergence of $\chi$ continuously connects to a smooth yet non-analytic increase of the localization length. We characterize this behavior both numerically and theoretically using a semi-classical approach. In this crossover regime, the correlated and non-Gaussian nature of the speckle potential becomes essential, causing the standard Dorokhov-Mello-Pereyra-Kumar (DPMK) description for uncorrelated disorder to break down. Instead, we predict the emergence of a bimodal transmission distribution, a behavior normally absent in one dimension, which we capture within our semi-classical analysis. Deep in the quantum regime, the DMPK framework is recovered and the universal features of Anderson localization reappear.