  The nonlinear magneto-optic responses are investigated for gapped graphene
and doped graphene in a perpendicular magnetic field. The electronic states are
described by Landau levels, and the electron dynamics in an optical field is
obtained by solving the density matrix in the equation of motion. In the linear
dispersion approximation around the Dirac points, both linear conductivity and
third order nonlinear conductivities are numerically evaluated for infrared
frequencies. The nonlinear phenomena, including third harmonic generation, Kerr
effects and two photon absorption, and four wave mixing, are studied. All
optical conductivities show strong dependence on the magnetic field. At weak
magnetic fields, our results for doped graphene agree with those in the
literature. We also present the spectra of the conductivities of gapped
graphene. At strong magnetic fields, the third order conductivities show peaks
with varying the magnetic field and the photon energy. These peaks are induced
by the resonant transitions between different Landau levels. The resonant
channels, the positions, and the divergences of peaks are analyzed. The
conductivities can be greatly modified, up to orders of magnitude. The
dependence of the conductivities on the gap parameter and the chemical
potential is studied.
