  In a system with a Dirac-like linear dispersion there are always states that
fulfill the resonance condition for electromagnetic radiation of arbitrary
frequency $\Omega$. When a flat band is present two kinds of resonant
transitions are found. Considering the $\alpha-\mathcal{T}_3$ graphene model as
a minimal model with a flat band and Dirac cones, and describing the dynamics
using the interaction picture, we study the band transitions induced by an
external electromagnetic field. We found that transitions depend upon the
relative angle between the electron momentum and the electromagnetic field wave
vector. For parallel incidence, the transitions are found using Floquet theory
while for other angles perturbation theory is used. In all cases, the
transition probabilities and the frequencies are found. For some special values
of the parameter $\alpha$ or by charge doping, the system behaves as a three
level or a two-level Rabi system. All these previous results were compared with
numerical simulations. A good agreement was found between both. The obtained
results are useful to provide a quantum control of the system.
