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 $\propto e^{-\gamma t},$ 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 $U>\mathrm{}\sqrt{\lambda \Delta }$ for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].