  Classical chaotic dynamics is characterized by the 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 atenuation of the Loschmidt Echo, $M(t)$,
i.e. the amount of the original state (wave packet of width $\sigma$) which is
recovered after a time reversed evolution, in presence of a classically weak
perturbation. By considering a Lorentz gas of size $L$, which for large $L$ is
a model for an {\it unbounded} classically chaotic system, we find numerical
evidence that, if the perturbation is within a certain range, $M(t)$ 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
$t_{s}\simeq \lambda^{-1}\ln (\widetilde{N})$, 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.
