Nonlinear optical conductivity and subharmonic instabilities of graphene in a strong electromagnetic field

We study theoretically the second-order nonlinear optical conductivity $\sigma^{(2)}$ of graphene as a function of frequency and momentum. We distinguish two regimes. At frequencies $\omega$ higher than the temperature-dependent electron-electron collision rate $\gamma^{-1}_{ee}$, the conductivity $\sigma^{(2)}$ can be derived from the semiclassical kinetic equation. The calculation requires taking into account the photon drag (Lorentz force) due to the ac magnetic field. In the low-frequency hydrodynamic regime $\omega \ll \gamma^{-1}_{ee}$, the nonlinear conductivity has a different form and the photon drag effect is suppressed. As a consequence of the nonlinearity, a strong enough photoexcitation can cause spontaneous generation of collective modes in a graphene strip: plasmons in the high-frequency regime and energy waves (demons) in the hydrodynamic one. The dominant instability occurs at frequency $\omega / 2$.