Classical chaos vs. quantum dynamical localization
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The kicked rotor: H = p²/2 + K·cos(θ)·Σδ(t−n). Classically, above K≈0.97 (Chirikov criterion) the KAM tori break and chaos diffuses momentum: ⟨p²⟩ ~ DK·t, with DK ≈ K²/2. Quantum mechanics introduces dynamical localization: quantum interference suppresses diffusion after a break time t* ~ ℓ (localization length). The quantum momentum distribution becomes exponentially localized P(n) ~ exp(−2|n|/ℓ) — an exact analogy to Anderson localization in 1D disordered systems. Left: classical phase space (Poincaré section). Right: quantum probability |ψ(n)|².