Friday, July 12, 2013

1307.3186 (C. -I. Chou et al.)

Localization and Recurrence of Quantum Walk in Periodic Potential on a

C. -I. Chou, C. -L. Ho
We present 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. Thus we suppose the coin state is changed according to a coin operator $C_0$ when there is no potential, and $C_p$ when the field is present. For simplicity and definiteness, we choose in this work $C_0=I$ and $C_p=H$. This means that 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. In these four cases, the number of points with Hadamard coin is "sufficiently" larger than the number of points with identity coin. This implies the existence of critical values of period $N$ for these cases.
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