Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

Abstract

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 $\propto \exp (-\mathrm{constant}\times e^{2{\lambda }_{0}t})$ 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 $\hslash \to 0$ for times less than the Ehrenfest time ${\tau }_E=\frac{1}{2}{\lambda }_0^{-1}|\ln \hslash |$. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.

4 pages, 4 figures, [2017: fixed broken postscript figures]

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• Uniform Semiclassical Approach to Fidelity Decay: from Weak to Strong Perturbation
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• WKB Propagation of Gaussian Wavepackets
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• Chaos Beyond Linearized Stability Analysis: Folding of the Phase Space and Distribution of Lyapunov Exponents