Decoherence and the Loschmidt echo

Abstract

Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $\varsigma =\mathrm{Tr}{\rho }^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $\bar{M}(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $\varsigma (t)$ and $\bar{M}(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of $\varsigma$ and $\bar{M}$ has a regime where it is dominated by the classical Lyapunov exponents