Mesoscopic fluctuations of the Loschmidt echo

We investigate the time-dependent variance of the fidelity with which an initial narrow wave packet is reconstructed after its dynamics is time reversed with a perturbed Hamiltonian. In the semiclassical regime of perturbation, we show that the variance first rises algebraically up to a critical time t(c), after which it decays. To leading order in the effective Planck's constant h(eff), this decay is given by the sum of a classical term similar or equal to exp[-2 lambda t], a quantum term similar or equal to 2h(eff) exp[-Gamma t], and a mixed term similar or equal to 2 exp[-(Gamma+lambda)t]. Compared to the behavior of the average fidelity, this allows for the extraction of the classical Lyapunov exponent lambda in a larger parameter range. Our results are confirmed by numerical simulations.