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We study nonlinear optical response of Landau quantized graphene to an intense electromagnetic wave. In particular, we consider high harmonic generation process. It is shown that one can achieve efficient generation of high harmonics with strong radiation fields -when the work of the wave electric field on the magnetic length is larger than pump photon energy. At that high harmonics generation process takes place for a wide range of the pump wave frequencies and intensities even for significant broadening of Landau levels because of impurities in graphene.

Microscopic nonlinear quantum theory of interaction of coherent electromagnetic radiation with gapped bilayer graphene is developed. The Liouville-von Neumann equation for the density matrix is solved numerically at the multiphoton excitation regime. The developed theory of interaction of charged carriers with strong driving wave field is valid near the Dirac points of the Brillouin zone. We consider the harmonic generation process in the nonadiabatic regime of interaction when the Keldysh parameter is of the order of unity. On the basis of numerical solutions, we examine the rates of odd and even high-harmonics at the particle-hole annihilation in the field of a strong pump wave of arbitrary polarization. Obtained results show that the gapped bilayer graphene can serve as an effective medium for generation of even and odd high harmonics in the THz and far infrared domains of frequencies.

Microscopic quantum theory of nonlinear stimulated scattering of 2D massless Dirac particles in doped graphene on Coulomb field of impurity ions at the presence of an external strong coherent electromagnetic radiation is developed. We consider high Fermi energies and low frequencies (actually terahertz radiation) to exclude the valence electrons excitations. The Liouville-von Neumann equation for the density matrix is solved analytically, taking into account the interaction of electrons with the scattering potential in the Born approximation. With the help of this solution, the nonlinear inverse-bremsstrahlung absorption rate for a grand canonical ensemble of 2D Dirac fermions is calculated. It is shown that one can achieve the efficient absorption coefficient by this mechanism.

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