Fidelity recovery in chaotic systems and the Debye-Waller factor

Using supersymmetry calculations and random matrix simulations, we studied the decay of 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)>, where H_epsilon differs from H_0 by a slight perturbation characterized by the parameter epsilon. For strong perturbations a recovery of f_epsilon(tau) at the Heisenberg time tau=1 is found. It is most pronounced for the Gaussian symplectic ensemble, and least for the Gaussian orthogonal one. Using Dyson's Brownian motion model for an eigenvalue crystal, the recovery is interpreted in terms of a spectral analogue of the Debye-Waller factor known from solid state physics, describing the decrease of X-ray and neutron diffraction peaks with temperature due to lattice vibrations.