Recovery of the fidelity amplitude for the Gaussian ensembles

TL;DR

This paper derives analytical expressions for the average fidelity amplitude in Gaussian ensembles using supersymmetry, revealing decay behaviors and a partial revival near the Heisenberg time, supported by simulations.

Contribution

It provides the first analytical derivation of the fidelity amplitude for Gaussian ensembles, including a novel observation of partial revival near the Heisenberg time.

Findings

Gaussian decay for small perturbations

Single-exponential decay for stronger perturbations

Partial revival of fidelity near Heisenberg time

Abstract

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude f_epsilon(tau)=< psi(0)| exp(2 pi i H_epsilon tau) exp(-2 pi i H_0 tau)| psi(0) > are obtained, where H_epsilon=H_0+(sqrt{epsilon}/(2 pi) )*V, and H_0 and H_epsilon are taken from the Gaussian unitary ensemble (GUE) or the Gaussian orthogonal ensemble (GOE), respectively. As long as the perturbation strength is small compared to the mean level spacing, a Gaussian decay of the fidelity amplitude is observed, whereas for stronger perturbations a change to a single-exponential decay takes place, in accordance with results from literature. Close to the Heisenberg time tau=1, however, a partial revival of the fidelity is found, which hitherto remained unnoticed. Random matrix simulations have been performed for the three Gaussian ensembles. For the case of the GOE and the GUE they are in perfect agreement with the analytical results.