Near invariance of quasi-energy spectrum of Floquet Hamiltonians

Abstract

The spectral analysis of the unitary monodromy operator, associated with a time-periodically (parametrically) forced Schrödinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form

where $H^0$ is an unperturbed autonomous Hamiltonian, $a\ge 1$, and $W(T,\cdot )$ has a period of $T_{\mathrm{per}}>0$. In particular, in the small $\epsilon >0$ regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the $H^{\epsilon }(t)$ dynamics, and that of the autonomous (unforced) flow, $\exp [-i{H}^0{T}_{\mathrm{per}}{\epsilon }^{-a}]$. We consider $H^0$ which is spatially periodic on ${\mathbb{R}}^n$ with respect to a lattice. Using the decomposition of $H^0$ and $H^{\epsilon }(t)$ 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 $\epsilon$-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, $H^0$, of interest in the study of photonic and quantum materials.