Near invariance of quasi-energy spectrum of Floquet Hamiltonians

The spectral analysis of the unitary monodromy operator, associated with a timeperiodically (parametrically) forced Schrödinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form [Formula Presented] where H⁰ is an unperturbed autonomous Hamiltonian, a ≥ 1, and W(T; ·) has a period of T_per > 0. In particular, in the small ɛ > 0 regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the H^ɛ(t) dynamics, and that of the autonomous (unforced) flow, exp[– iH⁰T_per ɛ^{– a} ]. We consider H⁰ which is spatially periodic on ℝⁿ with respect to a lattice. Using the decomposition of H⁰ and H^ɛ(t) into their actions on spaces (Floquet–Bloch fibers) of pseudo-periodic functions, we establish a spectral near-invariance property for the monodromy operator, when acting on data which are ɛ -localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by-emphband-limited (Floquet–Bloch) wavepackets; (iii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE; and (iii) an exact invariance property for band-limited Floquet– Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians, H⁰, of interest in the study of photonic and quantum materials.