The Loschmidt echo M(t)=|⟨ψ|e^{iH't}e^{-iHt}|ψ⟩|² measures how well a quantum state returns to its origin after evolving under a perturbed Hamiltonian H'=H+εV. Chaotic systems show exponential decay M∝e^{-λt}.
Perturbation ε=0.05
Chaos κ=2.0
Left: Husimi phase-space distribution. Right: Fidelity M(t) vs time — exponential decay rate ≈ Lyapunov exponent for strong perturbation.