Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interférences pour les chats quantiques

Jean-Michel Pipeau

TL;DR

The paper addresses the long-time quantum dynamics of quantized linear toral maps (quantum cat maps) beyond the Ehrenfest time. It shows that the residual interference after Ehrenfest time can be described by sums of Birkhoff for a parabolic fibred system on the torus, connecting quantum propagators to a concrete dynamical sum. The Marklof method is developed to bound these Birkhoff sums under an arithmetic (Diophantine) condition, yielding uniform bounds on Husimi measures for generic quantizations $h=1/N$ and clarifying the role of special arithmetic sequences. The work further provides a unified perspective that links quantum cat maps to Schrödinger dynamics on negatively curved surfaces and outlines extensions to broader geometric settings, with implications for understanding quantum chaos and eigenfunction statistics. Overall, the paper combines microlocal analysis, metaplectic representation, and arithmetical dynamics to describe and control quantum interference phenomena in chaotic toral maps and suggests dynamical interpretations of arithmetical phenomena in quantum chaos.

Abstract

In this text I study quantum dynamics of quantized linear automorphisms of the torus after the Ehrenfest time. I show that, in the wave packet basis, the 'matrix' of the associated propagator is well approximated by Birkhoff sums of nilrotations on the torus. In the second part, I conduct a thorough study of these sums and relate the equidistribution of evolved wave packets to a problem of Diophantine approximation.

Interférences pour les chats quantiques

TL;DR

The paper addresses the long-time quantum dynamics of quantized linear toral maps (quantum cat maps) beyond the Ehrenfest time. It shows that the residual interference after Ehrenfest time can be described by sums of Birkhoff for a parabolic fibred system on the torus, connecting quantum propagators to a concrete dynamical sum. The Marklof method is developed to bound these Birkhoff sums under an arithmetic (Diophantine) condition, yielding uniform bounds on Husimi measures for generic quantizations and clarifying the role of special arithmetic sequences. The work further provides a unified perspective that links quantum cat maps to Schrödinger dynamics on negatively curved surfaces and outlines extensions to broader geometric settings, with implications for understanding quantum chaos and eigenfunction statistics. Overall, the paper combines microlocal analysis, metaplectic representation, and arithmetical dynamics to describe and control quantum interference phenomena in chaotic toral maps and suggests dynamical interpretations of arithmetical phenomena in quantum chaos.

Abstract

In this text I study quantum dynamics of quantized linear automorphisms of the torus after the Ehrenfest time. I show that, in the wave packet basis, the 'matrix' of the associated propagator is well approximated by Birkhoff sums of nilrotations on the torus. In the second part, I conduct a thorough study of these sums and relate the equidistribution of evolved wave packets to a problem of Diophantine approximation.
Paper Structure (79 sections, 21 theorems, 356 equations, 4 figures)

This paper contains 79 sections, 21 theorems, 356 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Chaos quantique
  3. Propagation et états propres
  4. Temps d'Ehrenfest et interférences
  5. Chats quantiques et interférences constructives
  6. La genèse de l'article.
  7. Une bonne remarque
  8. Une tentative d'explication heuristique
  9. Fonction lagrangiennes et droites de $\mathbb{R}^2$
  10. Premières interférences le long d'un réseau
  11. Retour au bercail
  12. Contenu du texte
  13. Propagation après le temps d'Ehrenfest
  14. Quantification de hamiltoniens quadratiques et d'applications linéaires de $\mathbb{R}^2$
  15. L' (les) application(s) du chat
  16. ...and 64 more sections

Key Result

Proposition 2.3

On a l'expression suivante pour $\widehat{M}^h_t$ : où on a noté

Figures (4)

  • Figure 1: Le temps (nombre d'itérations du chat quantique) s'écoule d'abord de gauche à droite et ensuite de bas en haut. La première image, au temps $t=0$, correspond à la mesure associée au paquet d'ondes centrée en $0$. Le comportement reste classique sur les deux premières images, à la différence notable du fait que les variétés stables ne sont pas contractées (c'est le principe d'incertitude). La 4$^{\text{ème}}$ image est probablement la plus intéressante: on a dépassé le temps d'interférences et la fonction d'onde interfère non trivialement. On voit nettement des zones plus ou moins chargées, sans lien apparent avec la direction instable. Ce jeu d'interférences est non trivial; en itérant encore un peu l'application du chat quantique on parvient à reconstruire le paquet d'ondes original, sur la sixième image. (Source : Frédéric Faure
  • Figure 2: Intersections homoclines (des variétés stable et instable du point $(0,0)$)
  • Figure 3: Les branches lagrangiennes au premier temps d'interférence
  • Figure 4: Le graphe de la fonction $\varphi_h$ pour $h \sim 0.1$

Theorems & Definitions (37)

  • Proposition 2.3
  • Proposition 2.11
  • proof
  • Proposition 2.13
  • proof
  • Proposition 2.15
  • Proposition 2.16
  • proof
  • Proposition 2.17
  • proof
  • ...and 27 more