Fidelity recovery in the Gaussian ensembles(1)Current topics of quantum chaos in nanosciences, Chaos and Nonlinear Dynamics in Quantum-Mechanical and Macroscopic Systems)

Abstract

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude f_ε(γ) = 1/NTr[exp(2πιHφγ)exp(-2πιH_0γ)] are obtained, where Hφ = H_0cosφ + H_1sinφ, and H_0 and H_1 are matrices of rank N, taken from the Gaussian unitary ensemble(GUE) or the Gaussian orthogonal ensemble(GOE), respectively. For small perturbation strengths e = 4Nφ^2 a Gaussian decay of the fidelity amplitude is observed, whereas for stronger perturbations a change to a single-exponential decay takes place. Close to the Heisenberg time γ = 1, however, a partial revival of the fidelity is found. It can be interpreted in terms of an spectral analogue of the Debye-Waller factor, describing in X-ray spectroscopy the decrease of von Laue reflexes with increasing temperature.