M(t) = |⟨ψ|e^{iHt}e^{-i(H+εV)t}|ψ⟩|² · chaos · decoherence regimes
The Loschmidt echo M(t) measures how well a quantum state returns to itself after evolving forward under a perturbed Hamiltonian H+εV and backward under H. It quantifies how sensitive quantum dynamics is to small perturbations:
M(t) = |⟨ψ|e^{iHt} e^{-i(H+εV)t}|ψ⟩|²
Three decay regimes are known: (1) Gaussian for weak perturbations ε → short-time expansion; (2) Exponential ~ e^{−Γt} for intermediate ε (Lyapunov regime, decay rate = classical Lyapunov exponent λ); (3) Power law for very strong perturbations (perturbative FGR regime). The transition reveals quantum signatures of classical chaos.