Scattering Expansion for Localization in One Dimension: from Disordered Wires to Quantum Walks

TL;DR

This paper introduces a perturbative method for analyzing localization in one-dimensional disordered systems, connecting phase disorder with localization length and providing analytical insights into various models including quantum walks.

Contribution

It develops a full-range phase disorder expansion that clarifies localization mechanisms and derives a simplified scaling theory for weak scattering regimes.

Findings

Localization length can depend non-monotonically on phase disorder.

The joint distribution of transmission and reflection is characterized by three parameters in weak scattering.

Analytical results are obtained for the Anderson model, periodic potentials, and quantum walks.

Abstract

We present a perturbative approach to disordered systems in one spatial dimension that accesses the full range of phase disorder and clarifies the connection between localization and phase information. We consider a long chain of identically disordered scatterers and expand in the reflection strength of any individual scatterer. We apply this expansion to several examples, including the Anderson model, a general class of periodic-on-average-random potentials, and a two-component discrete-time quantum walk, showing analytically in the latter case that the localization length can depend non-monotonically on the strength of phase disorder (whereas expanding in weak disorder yields monotonic decrease). More generally, we obtain to all orders in the expansion a particular non-separable form for the joint probability distribution of the transmission coefficient logarithm and reflection phase. Furthermore, we show that for weak local reflection strength, a version of the scaling theory of localization holds: the joint distribution is determined by just three parameters.