Near invariance of quasi-energy spectrum of Floquet Hamiltonians

Abstract: The spectral analysis of the unitary monodromy operator, associated with a time-periodically (paramatrically) forced Schrodinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form $$H^{{\varepsilon}(t)=H⁰} + \varepsilon^a W(\varepsilon^a t, -i\nabla)\, ,$$ where $H<sup>0$ is an unperturbed autonomous Hamiltonian, $a\geq 1$, and $W(T,\cdot)$ has a period of $T_{\rm per} &gt;0$. In particular, in the small $\varepsilon&gt;0$ regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the $H<sup>\varepsilon(t)$ dynamics, and that of the autonomous (unforced) flow, $$\exp[-iH⁰ T_{\rm per} \varepsilon <sup>{-a}]$$. We consider $H<sup>0$ which is spatially periodic on $\mathbb{R} <sup>n$ with respect to a lattice. Using the decomposition of $H<sup>0$ and $H<sup>\varepsilon(t)$ into their actions on spaces (Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near spectral-invariance property for the monodromy operator, when acting data which are $\varepsilon$-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by {\it band-limited (Floquet-Bloch) wavepackets}; (ii) 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, $H<sup>0$, of interest in the study of photonic and quantum materials.