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