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Exponential Suppression of Transport in Electric Quantum Walks

Houssam Abdul-Rahman, Christopher Cedzich, Günter Stolz, Albert H. Werner

TL;DR

The paper analyzes transport in one-dimensional quantum walks under periodic (electric) fields, establishing an exact exponential suppression of the maximal group velocity with the field period $m$ via $v(W_\Phi)=|a|^m$ for rational fields $\Phi=2\pi n/m$. The authors develop a Fourier-based approach combined with a sieving technique that decomposes the square of a shift-coin walk into a direct sum of two split-step walks on even/odd sublattices, enabling precise velocity and spectral analyses. They prove revival relations and show the spectrum of the electrified walk is absolutely continuous with a $2m$-band structure, described by explicit dispersion relations, linking the dynamics to CMV matrix formalisms. These results extend previous bounds to exact equalities for all $|a|\in[0,1]$ and provide a rigorous bridge to generalized CMV/GECMV matrices, potentially informing quantum-control strategies for transport in periodically driven quantum systems.

Abstract

We establish exact scalings for the maximal group velocity of translation-invariant quantum walks in periodic electric fields. Our main result shows that the maximal group velocity decays exponentially with the period of the field in the whole parameter range, thus affirming a conjecture of arXiv:2302.01869 and at the same time augmenting it to an exact equality. We further demonstrate explicit revival relations and characterize the absolutely continuous spectrum in these models. Our results apply directly also to generalized CMV matrices.

Exponential Suppression of Transport in Electric Quantum Walks

TL;DR

The paper analyzes transport in one-dimensional quantum walks under periodic (electric) fields, establishing an exact exponential suppression of the maximal group velocity with the field period via for rational fields . The authors develop a Fourier-based approach combined with a sieving technique that decomposes the square of a shift-coin walk into a direct sum of two split-step walks on even/odd sublattices, enabling precise velocity and spectral analyses. They prove revival relations and show the spectrum of the electrified walk is absolutely continuous with a -band structure, described by explicit dispersion relations, linking the dynamics to CMV matrix formalisms. These results extend previous bounds to exact equalities for all and provide a rigorous bridge to generalized CMV/GECMV matrices, potentially informing quantum-control strategies for transport in periodically driven quantum systems.

Abstract

We establish exact scalings for the maximal group velocity of translation-invariant quantum walks in periodic electric fields. Our main result shows that the maximal group velocity decays exponentially with the period of the field in the whole parameter range, thus affirming a conjecture of arXiv:2302.01869 and at the same time augmenting it to an exact equality. We further demonstrate explicit revival relations and characterize the absolutely continuous spectrum in these models. Our results apply directly also to generalized CMV matrices.

Paper Structure

This paper contains 11 sections, 9 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Quantum walks
  4. Electric fields
  5. The speed limit of electric quantum walks
  6. Proof of Theorem \ref{['thm:speed_limit_W']}
  7. The maximal velocity of $U_\Phi$
  8. Sieving split-step walks
  9. Proof of Theorem \ref{['thm:sieving']}
  10. Speed limit of direct sums and powers of quantum walks
  11. Connection to generalized extended CMV matrices

Key Result

Theorem 2.1

Let $m,n\in\mathbb{N}$ be coprime and let $\Phi=2\pi n/m$. Consider the electric split-step walk $W_\Phi=\tilde{F}_{\Phi} S_+ C S_- C$ with translation-invariant coin $C$. Then, for $|a|\in[0,1]$ as in def:C:

Figures (3)

  • Figure 1: The standard deviation of the dynamics under $U_\Phi$ and $W_\Phi$ with the Hadamard coin $C=C_H$ for $\Phi/(2\pi)=1/5$ (left) and $\Phi/(2\pi)=21/106=[0,5,21]$ (right) with initial state $\psi=|0\rangle\otimes[1,i]^\top/\sqrt{2}$. As the inset shows, the dynamics is initially very similar until about the order of $t=20$ where the errors committed in each revival accumulated enough such that the next term in the continued fraction expansion kicks in, see ewalks for a thorough explanation of the interplay between the continued fraction expansion of $\Phi$ and the revivals.
  • Figure 2: The spectra resp. the dispersion relations $\omega_\pm(\theta,m)$ of $U_\Phi$ for $m=1,3,5$ (left to right). The colors distinguish the different signs in \ref{['def:omega_intro']}.
  • Figure 3: The spectra resp. the dispersion relations $\omega_\pm(\theta,m)$ of $U_\Phi$ for $m=2,4,6$ (left to right). The colors distinguish the different signs in \ref{['def:omega_intro']}. Note that opposed to the case of $m$ odd, here the choices for $\omega_\pm$ are not analytic at the degenerate points. However, such an analytic choice is always possible in the present setting ahlbrechtAsymptoticEvolutionQuantum2011ahlbrechtAsymptoticBehaviorDecoherent2013.

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:d-omega-bound']}
  • Remark 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • ...and 8 more