Decoherence as decay of the Loschmidt echo in a Lorentz gas

Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo $M\left(t\right),$ i.e., the amount of the original state (wave packet of width $\sigma )$ which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, $M\left(t\right)\mathrm{}$ decays exponentially with a rate $1/{\tau }_{\phi }$ determined by the Lyapunov exponent $\lambda$ of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time $t_E$ and saturates at a time $t_s\simeq {\lambda }^{-1}\ln \left[Ñ\right],$ where $Ñ\simeq (L/\sigma {)}^2$ is the effective dimensionality of the Hilbert space. Since ${\tau }_{\phi }$ quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.