Loschmidt Echo

TL;DR

This review surveys the Loschmidt echo, a quantitative measure of quantum irreversibility and perturbation sensitivity defined by $M(t) = \left| \langle \psi_0 | e^{i H_2 t / \hbar} e^{-i H_1 t / \hbar} | \psi_0 \rangle \right|^2$, and its connections to fidelity and decoherence. It integrates experimental progress across NMR, microwave billiards, elastic waves, cold atoms, and time-reversal mirrors with theoretical frameworks—semiclassics, random-matrix theory, and numerical simulations—to classify decay regimes in chaotic and regular dynamics. Central results include the parabolic, Gaussian/Fermi-Golden-Rule, Lyapunov, and saturation regimes, with global and local perturbations yielding distinct behaviors and crossover scales such as $t_E$, $t_H$, and $\Gamma$. The paper highlights fluctuations, disorder, and many-body extensions as frontier areas, emphasizing theLoschmidt echo as a bridge between quantum chaos, decoherence, and information processing, and outlines open questions and future directions for experimental observation of the Lyapunov regime and robust semiclassical formalisms.

Abstract

In this article we review the past, present, and future of the Loschmidt Echo.

Figures (17)

• Figure 1: Schematic flow of the time-evolution for (a) the Loschmidt echo and (b) the fidelity.
• Figure 2: Wave-packet evolution in a Lorentz gas. (a) Initial state at time $t=0$ with momentum pointing to the left. (b) State evolved with the Hamiltonian $H_1$ in the interval $(0,t)$. (c) State evolved with the Hamiltonian $H_2$ in the same time interval. (d) State evolved from that depicted in panel (b) with the $-H_2$ for the time interval $(t,2t)$. In panels (b) and (c) the classical trajectories corresponding to three initial positions within the original wave-packet are shown for reference. The square of the overlap between the states (a) and (d), the Loschmidt Echo, is $M(t)=0.09$, the same as that between the states of panels (b) and (c). Adapted from Ref. CPW02Decoherence. Copyright (2002), American Physical Society.
• Figure 3: Decay regimes of the Loschmidt echo in quantum systems with chaotic classical limit. Characteristic crossover times are $t_\mathrm{c} \simeq \hbar / \eta$, signaling the end of the initial parabolic decay, and $t_\mathrm{s}$ ($\simeq \Gamma^{-1} \ln N$ for the non-perturbative/exponential regime) indicating the onset of the saturation.
• Figure 4: Typical regimes of the Loschmidt echo decay in chaotic systems.
• Figure 5: Exponential decay rate $\Gamma$ as a function of the perturbation strength $\kappa$ in the case of a global Hamiltonian perturbation.