Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System

Details Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System

2002-12-03

arxiv.org/abs/cond-mat/0212065v1

Phys. Rev. E 67, 056217 (2003)

Physics

Condensed Matter

Disordered Systems and Neural Networks

9 pages, 7 figures

Scientific paper

10.1103/PhysRevE.67.056217

We investigate the sensitivity of a disordered system with diffractive scatterers to a weak external perturbation. Specifically, we calculate the fidelity M(t) (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(t) decays exponentially with a rate governed by the classical Lyapunov exponent.

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