The dynamics of quantum systems under the influence of time-periodic modulations has recently attracted growing attention for the possibility to realize unconventional phases of matter, including topological phases.
When a time-periodic field is applied to electrons in a periodic lattice the Bloch theorem can be applied twice, both in space and in time. Bloch theorem in the time domain is the essence of the Floquet approach that provides an exact representation of the time dependent states as a superposition of Floquet modes.
The solution of the Floquet Hamiltonian provides information on the true observables of the system: the k-dispersion of the zero-mode Floquet eigenvalue describes itself the time average of the true time-dependent k-resolved energies but much more detailed information on physical observables can be obtained by considering expectation values over the exact states written as a superposition of Floquet eigenvectors. In particular the time-dependent energies can be calculated as the expectation value of the exact Hamiltonian over these exact states. The same is true of charge and current density oscillation.
Within this scheme we have studied different regimes of intensity, frequency and polarization of laser fields on graphene and graphene nanoribbons. We show that for specific values of frequency and intensity the resulting oscillating current density is responsible of a non-linear optical response in 2D graphene and in particular of saturable absorption and High Harmonic Generation where high harmonic components may have an intensity comparable or even higher than the fundamental one.
Extending the analysis to 1D nanoribbons under circularly polarized fileds we have explored the dynamics of electrons localized on the edges which are responsible of charge oscillations across the ribbon and exhibit a robust dynamics in the presence of localized atomic vacancies. This is a direct signature of the non-trivial topological character induced by the external laser field. Moreover, performing the evolution of these states in the presence of a potential step, we show that it is possible to control them modifying the step height, in order to switch from a perfect transmission condition, to a perfect reflection on the opposite edge.This is very promising in view of future applications of topologically protected Floquet edge states for the realization of quantum computing architectures.