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Resonant tunneling effect for quantum walks on directed graphs

Kenta Higuchi

TL;DR

The work investigates how the scattering matrix of quantum walks on directed graphs with tails behaves in the small-perturbation limit when resonances approach the essential spectrum. By reducing the problem to a finite-dimensional matrix on the interior, the authors derive a resonance-expansion for the scattering matrix and prove that the discrepancy from the unperturbed diagonal case is governed by nearby resonances, attaining almost maximal norm near those resonances. A generalized resonant tunneling phenomenon is established, showing that transmission can be enhanced or suppressed depending on the symmetry of resonant states, with explicit formulas and corollaries. The study combines resolvent techniques, matrix perturbation theory, and concrete graph models (including matrix-Schrödinger-type graphs and cycle graphs) to illuminate how resonances control transport in quantum walks, with implications for quantum-network design and waveguide-like systems.

Abstract

Quantum walks that depend smoothly on a small parameter $\varepsilon\ge0$ are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as $\varepsilon\to+0$ is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for $\varepsilon=0$ 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.

Resonant tunneling effect for quantum walks on directed graphs

TL;DR

The work investigates how the scattering matrix of quantum walks on directed graphs with tails behaves in the small-perturbation limit when resonances approach the essential spectrum. By reducing the problem to a finite-dimensional matrix on the interior, the authors derive a resonance-expansion for the scattering matrix and prove that the discrepancy from the unperturbed diagonal case is governed by nearby resonances, attaining almost maximal norm near those resonances. A generalized resonant tunneling phenomenon is established, showing that transmission can be enhanced or suppressed depending on the symmetry of resonant states, with explicit formulas and corollaries. The study combines resolvent techniques, matrix perturbation theory, and concrete graph models (including matrix-Schrödinger-type graphs and cycle graphs) to illuminate how resonances control transport in quantum walks, with implications for quantum-network design and waveguide-like systems.

Abstract

Quantum walks that depend smoothly on a small parameter are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for 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.

Paper Structure

This paper contains 32 sections, 14 theorems, 234 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A question concerning the resonant tunneling effect
  3. Brief introduction of the setting and results
  4. Related works on quantum walks
  5. Scattering theory and the resonant tunneling effect in other context
  6. Semiclassical Schrödinger operators
  7. Semiclassical matrix Schrödinger operators
  8. Quantum graphs
  9. Waveguides
  10. Method and construction of this manuscript
  11. Setting and main results
  12. Definition of quantum walks on balanced directed graphs with tails
  13. Resonances and scattering matrix
  14. Asymptotic behavior of the scattering matrix in the semi-classical limit
  15. Comfortability
  16. ...and 17 more sections

Key Result

Theorem 1

Under Condition cond:pert, one has Moreover, for any fixed $z\in\mathbb{S}^1\setminus{\operatorname{Res}}(U(0))$ independent of $\varepsilon$, $M_{\lambda,\varepsilon}(z)={\mathcal{O}}(\varepsilon)$ holds for every $\lambda$, and One also has an estimate from below: This implies that there exists $C>0$ such that provided that $|\lambda_\varepsilon|<1$.

Figures (9)

  • Figure 1: An example of a balanced directed graph
  • Figure 2: An example of a balanced directed graph with tails
  • Figure 3: Construction of $(V,A)$ from a finite graph $(V_1,A_1)$ as in Remark \ref{['rem:make_tails']}
  • Figure 4: A model related to a matrix Schrödinger operator
  • Figure 5: Dynamics of the example
  • ...and 4 more figures

Theorems & Definitions (34)

  • Remark 2.1
  • Theorem 1
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2
  • Remark 3.1
  • Remark 3.2
  • Remark 5.1
  • Theorem 3
  • Lemma 5.2
  • ...and 24 more