Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo

The overlap of two wave packets evolving in time with slightly different Hamiltonians decays exponentially (proportional to)e(-gammat), for perturbation strengths U greater than the level spacing Delta. We present numerical evidence for a dynamical system that the decay rate gamma is given by the smallest of the Lyapunov exponent lambda of the classical chaotic dynamics and the level broadening U-2/Delta that follows from the golden rule of quantum mechanics. This implies the range of validity U>root lambda Delta for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].