Localization and Recurrence of Quantum Walk in Periodic Potential on a Line

TL;DR

This paper numerically investigates quantum walks on a line with periodic potentials, revealing that four out of six configurations exhibit localization and recurrence phenomena, depending on the potential pattern.

Contribution

It introduces a simple model of quantum walk in periodic potential with two coins and analyzes six cases, highlighting localization effects in four of them.

Findings

Four cases show significant localization.

Localization leads to recurrence of the walker's position.

Potential pattern influences the walker's confinement.

Abstract

We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness, we assume the walker's coin state is unaffected at sites without potential, and is rotated in an unbiased way according to Hadamard matrix at sites with potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks were studied numerically. It is found that of the six cases, four cases display significant localization effect, where the walker is confined in the neighborhood of the origin for sufficiently long times. Associated with such localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.