Resonant tunneling effect for quantum walks on directed graphs

Abstract

Quantum walks that depend smoothly on a small parameter $\epsilon \ge 0$ are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as $\epsilon \to +0$ is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for $\epsilon =0$ at points in the essential spectrum (the unit circle) that are asymptotically approached by a quantum resonance. Furthermore, a phenomenon resembling and extending the resonant tunneling effect is observed by analyzing this discrepancy through resonant states.

44 pages, 9 figures