Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2002-06-30
Phys.Rev.E 67, 025204(R) (2003)
Nonlinear Sciences
Chaotic Dynamics
4 pages, 4 figures
Scientific paper
10.1103/PhysRevE.67.025204
We re-examine the problem of the ``Loschmidt echo'', which measures the sensitivity to perturbation of quantum chaotic dynamics. The overlap squared $M(t)$ of two wave packets evolving under slightly different Hamiltonians is shown to have the double-exponential initial decay $\propto \exp(-{\rm constant}\times e^{2\lambda_0 t})$ in the main part of phase space. The coefficient $\lambda_0$ is the self-averaging Lyapunov exponent. The average decay $\bar{M}\propto e^{-\lambda_1 t}$ is single exponential with a different coefficient $\lambda_1$. The volume of phase space that contributes to $\bar{M}$ vanishes in the classical limit $\hbar\to 0$ for times less than the Ehrenfest time $\tau_E=\fr{1}{2}\lambda_0^{-1}|\ln \hbar|$. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.
Beenakker C. W. J.
Silvestrov P. G.
Tworzydlo Jakub
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