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 proportional toexp(-constantxe(0)(2lambda)t) in the main part of the phase space. The coefficient lambda(0) is the self-averaging Lyapunov exponent. The average decay (M) over bar proportional toe(1)(-lambda)t is single exponential with a different coefficient lambda(1). The volume of phase space that contributes to (M) over bar vanishes in the classical limit (h) over bar -->0 for times less than the Ehrenfest time tau(E)=1/2 lambda(0)(-1)\ln h\. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.