Loschmidt echo for local perturbations: non-monotonous cross-over from the Fermi-golden-rule to the escape-rate regime

TL;DR

This paper investigates how the Loschmidt echo decays in chaotic quantum systems under local perturbations, revealing a non-monotonous transition from Fermi-golden-rule to escape-rate regimes supported by semiclassical and numerical analyses.

Contribution

It provides a comprehensive semiclassical theory describing the full decay behavior of the Loschmidt echo for local perturbations, including a non-monotonous transition between regimes.

Findings

FGR decay regime for weak perturbations

Escape-rate decay for strong perturbations

Non-monotonous oscillations in decay rate as a function of perturbation strength

Abstract

We address the sensitivity of quantum mechanical time evolution by considering the time decay of the Loschmidt echo (LE) (or fidelity) for local perturbations of the Hamiltonian. Within a semiclassical approach we derive analytical expressions for the LE decay for chaotic systems for the whole range from weak to strong local perturbations and identify different decay regimes which complement those known for the case of global perturbations. For weak perturbations a Fermi-golden-rule (FGR) type behavior is recovered. For strong perturbations the escape-rate regime is reached, where the LE decays exponentially with a rate independent of the perturbation strength. The transition between the FGR regime and the escape-rate regime is non-monotonic, i.e. the rate of the exponential time-decay of the LE oscillates as a function of the perturbation strength. We further perform extensive quantum mechanical calculations of the LE based on numerical wave packet evolution which strongly support our semiclassical theory. Finally, we discuss in some detail possible experimental realizations for observing the predicted behavior of the LE.