The Loschmidt echo M(t) = |⟨ψ₀|e^{iH't}e^{-iHt}|ψ₀⟩|² measures how well a quantum state returns to its origin under time-reversal with a perturbed Hamiltonian. Explore Fermi-golden-rule, Lyapunov, and revival regimes.
Loschmidt echo regimes: (1) Perturbative/FGR: M(t) ~ e^{-Γt} where Γ = 2πε²·ρ(E) (Fermi golden rule — ε-dependent). (2) Lyapunov regime (strong pert): M(t) ~ e^{-λ_L·t} — independent of ε! (OTOC / quantum butterfly effect). (3) Quantum Zeno for very short times. Connected to: out-of-time-order correlators (OTOCs), quantum chaos diagnostics, thermalization, quantum computing fidelity decay. Jalabert-Pastawski 2001 discovered Lyapunov regime.