Quantum chaos versus noise environment

Quantum dynamics of a classically chaotic 1D system in the presence of external noise is studied. Stability and reversibility properties of the motion (characterized by the Peres fidelity) as functions of the noise level $\sigma$ are considered. We calculate fidelity analytically in the cases of weak and very strong noise and find critical value, $\sigma_c(t)$, below which the effect of perturbation remains small. Decay of critical perturbation with time is found to be power-like after the Ehrenfest time $t_E$. An estimation of the decoherence time $t_d(\sigma)$ is presented after which the averaged density matrix becomes diagonal and its evolution turns into a Markovian process.