Loschmidt echo and Lyapunov exponent in a quantum disordered system

We investigate the sensitivity of a disordered system with diffractive scatterers to a weak external perturbation. Specifically, we calculate the fidelity $M\left(t\right)\mathrm{}$ (also called the Loschmidt echo) characterizing a return probability after a propagation for a time t followed by a backward propagation governed by a slightly perturbed Hamiltonian. For short-range scatterers, we perform a diagrammatic calculation showing that the fidelity decays first exponentially according to the golden rule, and then follows a power law governed by the diffusive dynamics. For long-range disorder (when the diffractive scattering is of small-angle character), an intermediate regime emerges where the diagrammatics is not applicable. Using the path-integral technique, we derive a kinetic equation and show that $M\left(t\right)\mathrm{}$ decays exponentially with a rate governed by the classical Lyapunov exponent.