  The notion of Loschmidt echo (also called "quantum fidelity") has been
introduced in order to study the (in)-stability of the quantum dynamics under
perturbations of the Hamiltonian. It has been extensively studied in the past
few years in the physics literature, in connection with the problems of
"quantum chaos", quantum computation and decoherence. In this paper, we study
this quantity semiclassically (as $\hbar \to 0$), taking as reference quantum
states the usual coherent states. The latter are known to be well adapted to a
semiclassical analysis, in particular with respect to semiclassical estimates
of their time evolution. For times not larger than the so-called "Ehrenfest
time" $C | \log \hbar |$, we are able to estimate semiclassically the Loschmidt
Echo as a function of $t$ (time), $\hbar$ (Planck constant), and $\delta$ (the
size of the perturbation). The way two classical trajectories merging from the
same point in classical phase-space, fly apart or come close together along the
evolutions governed by the perturbed and unperturbed Hamiltonians play a major
role in this estimate. We also give estimates of the "return probability"
(again on reference states being the coherent states) by the same method, as a
function of $t$ and $\hbar$.
