Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System

TL;DR

This paper studies how a quantum disordered system's fidelity decays over time, revealing different regimes where decay is governed by diffusive dynamics or classical chaos, characterized by the Lyapunov exponent.

Contribution

It provides a detailed analysis of Loschmidt echo decay in quantum disordered systems, connecting quantum fidelity decay to classical Lyapunov exponents.

Findings

Fidelity decays exponentially initially due to the golden rule.

A power-law decay follows in the diffusive regime.

In systems with long-range disorder, decay is governed by the Lyapunov exponent.

Abstract

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.