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Classical chaotic dynamics is characterized by the exponential sensitivity toinitial conditions. Quantum mechanics, however, does not show this feature. Weconsider instead the sensitivity of quantum evolution to perturbations in theHamiltonian. This is observed as an atenuation of the Loschmidt Echo, $M(t)$,i.e. the amount of the original state (wave packet of width $\sigma$) which isrecovered after a time reversed evolution, in presence of a classically weakperturbation. By considering a Lorentz gas of size $L$, which for large $L$ isa model for an {\it unbounded} classically chaotic system, we find numericalevidence that, if the perturbation is within a certain range, $M(t)$ decaysexponentially with a rate $1/\tau_{\phi}$ determined by the Lyapunov exponent$\lambda$ of the corresponding classical dynamics. This exponential decayextends much beyond the Eherenfest time $t_{E}$ and saturates at a time$t_{s}\simeq \lambda^{-1}\ln (\widetilde{N})$, where $\widetilde{N}\simeq(L/\sigma)^2$ 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.Comment: 3 ps figures, uses Revtex and epsfig. Major revision to the text, now  including discussion and references on averaging and Ehrenfest time. Figures  2 and 3 content and order change

Decoherence as decay of the Loschmidt echo in a Lorentz gas