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In a system with a Dirac-like linear dispersion there are always states thatfulfill the resonance condition for electromagnetic radiation of arbitraryfrequency $\Omega$. When a flat band is present two kinds of resonanttransitions are found. Considering the $\alpha-\mathcal{T}_3$ graphene model asa minimal model with a flat band and Dirac cones, and describing the dynamicsusing the interaction picture, we study the band transitions induced by anexternal electromagnetic field. We found that transitions depend upon therelative angle between the electron momentum and the electromagnetic field wavevector. For parallel incidence, the transitions are found using Floquet theorywhile for other angles perturbation theory is used. In all cases, thetransition probabilities and the frequencies are found. For some special valuesof the parameter $\alpha$ or by charge doping, the system behaves as a threelevel or a two-level Rabi system. All these previous results were compared withnumerical simulations. A good agreement was found between both. The obtainedresults are useful to provide a quantum control of the system.

physical review. b./physical review. b

Electron transitions for Dirac Hamiltonians with flat-bands under  electromagnetic radiation and its application to the $\alpha-\mathcal{T}_3$  graphene model