Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

We reexamine the problem of the ``Loschmidt echo,'' that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared $M(t)$ of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay ∝exp(-constant×e^2λ₀t) in the main part of the phase space. The coefficient λ₀ is the self-averaging Lyapunov exponent. The average decay M̄∝e^-λ₁t is single exponential with a different coefficient λ₁. The volume of phase space that contributes to M̄ vanishes in the classical limit →0 for times less than the Ehrenfest time τE=1/2λ₀^-1|ln|. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.