Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

Faure, Frédéric

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2525-2599.

On considère une application $M$, Anosov non linéaire qui conserve l’aire sur le tore $T^2$. C’est un des exemples les plus simples d’une dynamique chaotique. On s’intéresse à la dynamique quantique pour les temps longs, générée par un opérateur unitaire $\widehat{M}$. La formule des traces semi-classique habituelle exprime $Tr\left({\widehat{M}}^t\right)$ pour $t$ fini, dans la limite $\hslash \to 0$, en termes d’orbites périodiques de $M$ de période $t$. Des travaux récents atteignent des temps $t\ll t_E/6$ où $t_E=log(1/\hslash )/\lambda$ est le temps d’Ehrenfest, et $\lambda$ est le coefficient de Lyapounov. En utilisant une description uniforme de la dynamique au moyen d’une forme normale semi-classique, nous montrons comment étendre la formule des traces pour des temps plus longs, de la forme $t=C.t_E$, où $C$ est une constante arbitraire, et avec une erreur arbitrairement petite.

We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\widehat{M}$. The usual semi-classical Trace formula expresses $Tr\left({\widehat{M}}^t\right)$ for finite time $t$, in the limit $\hslash \to 0$, in terms of periodic orbits of $M$ of period $t$. Recent work reach time $t\ll t_E/6$ where $t_E=log(1/\hslash )/\lambda$ is the Ehrenfest time, and $\lambda$ is the Lyapounov coefficient. Using a semi-classical normal form description of the dynamics uniformly over phase space, we show how to extend the trace formula for longer time of the form $t=C.t_E$ where $C$ is any constant, with an arbitrary small error.

Classification : 81Q50, 37D20
Keywords: Quantum chaos, hyperbolic map, semiclassical trace formula, Ehrenfest time
Mot clés : chaos quantique, application hyperbolique, formule des traces semi-classique, temps d’Ehrenfest

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