Recovery of the fidelity amplitude for the Gaussian ensembles

Abstract

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude fϵ(τ) = ⟨ψ(0)|exp(2πiHϵτ) exp(−2πiH0τ)|ψ(0)⟩ are obtained, where , and H0 and Hϵ 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 the 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.