An initial value representation for the Loschmidt echo

TL;DR

This paper introduces an improved initial value representation for the quantum Loschmidt echo based on semiclassical Wigner function evolution, enhancing accuracy especially for quadratic Hamiltonians.

Contribution

It develops a new initial value representation that incorporates mean Hamiltonian trajectories, improving upon the dephasing approximation for quantum Loschmidt echo calculations.

Findings

Works well for quadratic Hamiltonians, where semiclassical evolution is exact.

Extends the applicability of the dephasing representation beyond quantum chaos.

Provides a more accurate phase correction via second derivative of the action.

Abstract

We obtain an initial value representation for the quantum Loschmidt echo from the semiclassical theory of Wigner function evolution, together with classical first-order perturbation theory. In the limit of small actions, the amplitude of each trajectory reduces to unity, just as in the dephasing representation introduced by Vanicek, but these trajectories are here generated by the mean Hamiltonian for both the forward and the backward motion. This slight change of action may substantially alter the phase. The amplitude correction depends on the second derivative of the action. This improved dephasing approximation is verified to work even for quadratic Hamiltonians, for which the semiclassical evolution is exact, thus extending the range of application beyond its original scope in quantum chaos.