  We study theoretically AB-stacked honeycomb bilayers driven by light in
resonance with an infrared phonon within a tight-binding description. We
characterize the phonon properties of honeycomb bilayers with group theory and
construct an electronic time-dependent tight-binding model for the system
following photo-excitation in resonance with an infrared phonon. We adopt an
"atomically adiabatic" approximation, introduced by Mohantya and Heller PNAS
116, 18316 (2019) to describe classically vibrating nuclei, but obtain the
Floquet quasienergy spectrum associated with the time-dependent model exactly.
We introduce a general scheme to disentangle the complex low-frequency Floquet
spectrum to elucidate the relevant Floquet bands. As a prototypical example, we
consider bilayer graphene. We find that light in the low-frequency regime can
induce a bandgap in the quasienergy spectrum in the vicinity of the K points
even if it is linearly polarized, in contrast with the expectations within the
Born-Oppenheimer approximation and the high-frequency regime. Finally, we
analyze the diabaticity of the driven electron and driven phonon processes and
found contrasting effects on the autocorrelation functions at the same driving
frequency: driven phonons preserve the character of the initial state while
driven electrons exhibit strong deviations within a few drive cycles. The
procedure outlined here can be applied to other materials to describe the
combined effects of low-frequency light on phonons and electrons.
