Loschmidt Echo & Quantum Fidelity

Sensitivity of quantum evolution to perturbation
M(t) = |⟨ψ₀|e^{iH't} e^{-iHt}|ψ₀⟩|²
Perturbation ε 0.050
Chaos λ 1.00
N modes 8
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The Loschmidt Echo measures quantum irreversibility: evolve forward under H, then backwards under a perturbed H' = H + εV. For integrable systems: M(t) decays as a Gaussian ~ exp(-ε²t²/ℏ²). For chaotic systems: M(t) ~ exp(-λt) where λ is the classical Lyapunov exponent — perturbation-independent! (Jalabert-Pastawski regime). This reveals how quantum chaos erases memory of the initial state, and connects to quantum information scrambling.