Floquet-Bloch Driven Lattice Micromotion

Floquet theorem: for periodic drive H(t+T)=H(t), solutions are Ψ(t)=e^−iεt/ℏu(t) where u(t+T)=u(t) (Floquet modes). Quasienergies ε lie in the first Brillouin zone [−ℏω/2, ℏω/2]. Micromotion = fast oscillation within each period atop slow quasiparticle drift. The effective Hamiltonian H_eff = J₀(A/ω)t₀ shows hopping renormalized by Bessel function — at zeros of J₀, dynamic localization occurs.