Ergodic Theorems for Continuous-Time Quantum Walks on Crystal Lattices and the Torus

Anne Boutet de Monvel, Mostafa Sabri

Research output: Contribution to journalArticlepeer-review

Abstract

We give several quantum dynamical analogs of the classical Kronecker–Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk exp(-itΔ)ψ starting from a localized initial state ψ. Then, the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.

Original languageEnglish (US)
JournalAnnales Henri Poincare
DOIs
StateAccepted/In press - 2024

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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