Floquet Theory — Periodically Driven Two-Level System

Floquet's theorem: for a periodically driven Hamiltonian H(t+T)=H(t), solutions take the form |ψ(t)⟩ = e^{−iε_αt}|u_α(t)⟩ where ε_α are quasienergies (analogous to crystal momentum) and |u_α(t)⟩ are T-periodic Floquet modes. Avoided crossings in the quasienergy spectrum reveal resonance and chaos.

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Floquet Hamiltonian
H(t) = (Δ/2)σ_z + A·cos(ωt)·σ_x
Quasienergy: ε ∈ [−ω/2, +ω/2]
Resonance: ω ≈ nΔ (n-photon)
At resonance: Rabi splitting 2A
Heating: many avoided crossings