Symmetry and localization of quantum walk induced by extra link in cycles

TL;DR

This paper analyzes how adding a single extra link to a cycle affects continuous-time quantum walks, revealing changes in dynamics, localization effects, and spectral properties through analytical and numerical methods.

Contribution

It provides the first analytical calculation of the Laplacian spectrum for this system, demonstrating the impact of the extra link on quantum walk behavior and localization.

Findings

Extra link causes distinct dynamical behavior compared to regular cycle.

Localization occurs when the exciton starts at the link ends, influenced by the largest eigenvalue.

Long-time probability symmetry is proven using Chebyshev polynomial analysis.

Abstract

In this paper, we study the impact of single extra link on the coherent dynamics modeled by continuous-time quantum walks. For this purpose, we consider the continuous-time quantum walk on the cycle with an additional link. We find that the additional link in cycle indeed cause a very different dynamical behavior compared to the dynamical behavior on the cycle. We analytically treat this problem and calculate the Laplacian spectrum for the first time, and approximate the eigenvalues and eigenstates using the Chebyshev polynomial technique and perturbation theory. It is found that the probability evolution exhibits a similar behavior like the cycle if the exciton starts far away from the two ends of the added link. We explain this phenomenon by the eigenstate of the largest eigenvalue. We prove symmetry of the long-time averaged probabilities using the exact determinant equation for the eigenvalues expressed by Chebyshev polynomials. In addition, there is a significant localization when the exciton starts at one of the two ends of the extra link, we show that the localized probability is determined by the largest eigenvalue and there is a significant lower bound for it even in the limit of infinite system. Finally, we study the problem of trapping and show the survival probability also displays significant localization for some special values of network parameters, and we determine the conditions for the emergence of such localization. All our findings suggest that the different dynamics caused by the extra link in cycle is mainly determined by the largest eigenvalue and its corresponding eigenstate. We hope the Laplacian spectral analysis in this work provides a deeper understanding for the dynamics of quantum walks on networks.