TY - JOUR
T1 - Ergodic Theorems for Continuous-Time Quantum Walks on Crystal Lattices and the Torus
AU - Boutet de Monvel, Anne
AU - Sabri, Mostafa
N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
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U2 - 10.1007/s00023-024-01470-x
DO - 10.1007/s00023-024-01470-x
M3 - Article
AN - SCOPUS:85197741710
SN - 1424-0637
JO - Annales Henri Poincare
JF - Annales Henri Poincare
ER -