Effective gaps in continuous Floquet Hamiltonians

We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$i\psi_t (t,x) = H^\varepsilon(t)\psi=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)\psi,\quad H^0=-\Delta +V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) by an effective time-periodic Dirac equation with a gap in its quasi-energy spectrum. This quasi-energy gap is typical of many reduced models of time-periodic (Floquet) materials and plays a role in conclusions drawn about the full system: conduction vs. insulation, topological vs. non-topological bands. Much is unknown about nature of the quasi-energy spectrum of original time-periodic Schroedinger equation, and it is believed that no such quasi-energy gap occurs. In this paper, we explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schroedinger dynamics. We introduce the notion of an {\it effective quasi-energy gap}, and establish its existence in the Schroedinger model. In the current setting, an effective quasi-energy gap is an interval of quasi-energies which does not support modes with large spectral projection onto band-limited Dirac wave-packets. The notion of effective quasi-energy gap is a physically relevant relaxation of the strict notion of quasi-energy spectral gap; if a system is tuned to drive or measure at momenta and energies near the Dirac point of $H^0$, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.