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

Frédéric Faure[1]

  • [1] Institut Fourier 100 rue des Maths, BP74 38402 St Martin d’Heres (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2525-2599
  • ISSN: 0373-0956

Abstract

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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 M ^ . The usual semi-classical Trace formula expresses T r M ^ t for finite time t , in the limit ℏ → 0 , in terms of periodic orbits of M of period t . Recent work reach time t ≪ t E / 6 where t E = log ( 1 / ℏ ) / λ is the Ehrenfest time, and λ 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.

How to cite

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Faure, Frédéric. "Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula." Annales de l’institut Fourier 57.7 (2007): 2525-2599. <http://eudml.org/doc/10305>.

@article{Faure2007,
abstract = {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 $\hat\{M\}$. The usual semi-classical Trace formula expresses $\mbox \{Tr\}\left(\hat\{M\}^\{t\}\right)$ for finite time $t$, in the limit $\hbar\rightarrow 0$, in terms of periodic orbits of $M$ of period $ t$. Recent work reach time $t \ll t_\{E\}/6$ where $t_\{E\} = \log (1/\hbar)/\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.},
affiliation = {Institut Fourier 100 rue des Maths, BP74 38402 St Martin d’Heres (France)},
author = {Faure, Frédéric},
journal = {Annales de l’institut Fourier},
keywords = {Quantum chaos; hyperbolic map; semiclassical trace formula; Ehrenfest time; quantum chaos},
language = {eng},
number = {7},
pages = {2525-2599},
publisher = {Association des Annales de l’institut Fourier},
title = {Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula},
url = {http://eudml.org/doc/10305},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Faure, Frédéric
TI - Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2525
EP - 2599
AB - 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 $\hat{M}$. The usual semi-classical Trace formula expresses $\mbox {Tr}\left(\hat{M}^{t}\right)$ for finite time $t$, in the limit $\hbar\rightarrow 0$, in terms of periodic orbits of $M$ of period $ t$. Recent work reach time $t \ll t_{E}/6$ where $t_{E} = \log (1/\hbar)/\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.
LA - eng
KW - Quantum chaos; hyperbolic map; semiclassical trace formula; Ehrenfest time; quantum chaos
UR - http://eudml.org/doc/10305
ER -

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