The time-averaged limit measure of the Wojcik model

TL;DR

This paper studies the time-averaged limit measure of the Wojcik quantum walk model, revealing its relation to the stationary measure and demonstrating localization effects caused by a phase defect.

Contribution

It derives the time-averaged limit measure for the Wojcik model and clarifies its relationship with the previously obtained stationary measure.

Findings

The stationary measure is a special case of the time-averaged limit measure.

Localization occurs due to the phase defect at the origin.

The paper provides multiple methods to compute the time-averaged limit measure.

Abstract

We investigate "the Wojcik model" introduced and studied by Wojcik et al., which is a one-defect quantum walk (QW) having a single phase at the origin. They reported that giving a phase at one point causes an astonishing effect for localization. There are three types of measures having important roles in the study of QWs: time-averaged limit measure, weak limit measure, and stationary measure. The first two measures imply a coexistence of localized behavior and the ballistic spreading in the QW. As Konno et al. suggested, the time-averaged limit and stationary measures are closely related to each other for some models. In this paper, we focus on a relation between the two measures for the Wojcik model. The stationary measure was already obtained by our previous work. Here, we get the time-averaged limit measure by several methods. Our results show that the stationary measure is a special case of the time-averaged limit measure.