Propagation and spectral properties of quantum walks in electric fields

TL;DR

This paper investigates how one-dimensional quantum walks behave under a homogeneous electric field, revealing that their long-term propagation depends critically on the rationality of the field's phase, with implications for phenomena like localization and revivals.

Contribution

It provides a detailed analysis of the spectral and propagation properties of quantum walks in electric fields, linking these behaviors to the continued fraction expansion of the field phase.

Findings

Propagation properties depend on whether the phase ratio is rational or irrational.

Finite accuracy in the phase allows predictions of behavior over finite time scales.

The study connects spectral properties to continued fraction expansions of the electric field.

Abstract

We study one-dimensional quantum walks in a homogeneous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion and Anderson localization, depend very sensitively on the value of the electric field $\Phi$, e.g., on whether $\Phi/(2\pi)$ is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about the behavior on finite time scales.