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The response of electrons under linearly polarized light in Dirac materialsas borophene or graphene is analyzed in a continuous wave regime for anarbitrary intense field. Using a rotation and a time-dependent phasetransformation, the wave function evolution is shown to be governed by aspinor-component decoupled Whittaker-Hill equation. The numerical solution ofthese equations enables to find the quasienergy spectrum. For borophene itreveals a strong anisotropic response. By applying an extra unitarytransformation, the wave functions are proven to follow an Ince equation. Theevolution of the real and imaginary parts of the wave function is interpretedas the trajectory of a classical charged particle under oscillating electricand magnetic field. The topological properties of this forced quantum systemare studied using this analogy. In particular, in the adiabatic driving regime,the system is described with an effective Matthieu equation while in thenon-adiabatic regime the full Whittaker-Hill equation is needed. From there, itis possible to separate the dynamical and Berry phase contributions to obtainthe topological phase diagram due to the driving. Therefore, a different pathto perturbation theory is developed to obtain time-driven topological phases.

Dirac materials under linear polarized light: quantum wave function  evolution and topological Berry phases as classical charged particles  trajectories under electromagnetic fields