Uniform semiclassical approach to fidelity decay: from weak to strong perturbation

We study fidelity decay by a uniform semiclassical approach, in the three perturbation regimes: namely, the perturbative regime, the Fermi golden rule (FGR) regime, and the Lyapunov regime. A semiclassical expression is derived for the fidelity of initial Gaussian wave packets with width of the order sqare root h (h being the effective Planck constant). The short-time decay of the fidelity of initial Gaussian wave packets is also studied with respect to two time scales introduced in the semiclassical approach. In the perturbative regime, it is confirmed numerically that fidelity has FGR-type decay before Gaussian decay sets in. An explanation is suggested for a non-FGR decay in the FGR regime of a system with weak chaos in the classical limit by using the Levy distribution as an approximation for the distribution of the action difference. In the Lyapunov regime, it is shown that the average of the logarithm of fidelity may have roughly Lyapunov decay within some time interval in systems possessing large fluctuations in the finite-time Lyapunov exponent in the classical limit.