Decay of the Loschmidt echo in a time-dependent environment

We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation $V(q,t)$ with typical strength $\hbar/\tau_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length $\xi_0$ and a correlation time $\tau_0$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi Golden Rule, $1/\tilde{\tau}= \tau_{c}/\tau_{V}^2$, where typically $\tau_{c} \sim \min[\tau_{0},\xi_0/v]$ with $v$ the particle velocity. Whenever the rate $1/\tilde{\tau}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence, one can protect the system against decoherence.