Path integral approach to the quantum fidelity amplitude

TL;DR

This paper derives an exact path integral formula for quantum fidelity amplitude, leading to new semiclassical approximations that improve understanding of quantum irreversibility and time-reversal protocols.

Contribution

It introduces a rigorous path integral approach to quantum fidelity, deriving new semiclassical approximations including an improved dephasing representation.

Findings

Zeroth-order approximation is simple yet nontrivial.

First-order yields an alternative derivation of dephasing representation.

Second-order approximation addresses shortcomings of previous methods.

Abstract

The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet nontrivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called "dephasing representation", circumventing the use of semiclassical propagator as in the original derivation. We also obtain approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact.