1202.4208 (Xin-Ping Xu et al.)

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

Xin-Ping Xu, Yusuke Ide, Norio Konno

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.

View original: http://arxiv.org/abs/1202.4208