A phase-space study of the quantum Loschmidt echo in the semiclassical limit

The notion of Loschmidt echo (also called quantum. delity) 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 (h) over bar --> 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 (h) over bar|, we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (h) over bar (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 (h) over bar.