Recovery of the fidelity amplitude for the Gaussian ensembles

Abstract. Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude fǫ(τ) = 〈ψ(0)| exp(2πıHǫτ) exp(−2πıH0τ)|ψ(0)〉 are obtained, where the H0 and Hǫ are taken from the Gaussian unitary ensemble (GUE) or the Gaussian orthogonal ensemble (GOE), and ǫ is a parameter, characterizing the strength of a perturbation. 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 τ = 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.