Quantum theory of the third-order nonlinear electrodynamic effects in graphene

The linear energy dispersion of graphene electrons leads to a strongly nonlinear electromagnetic response of this material. We develop a general quantum theory of the third-order nonlinear local dynamic conductivity of graphene $\sigma_{\alpha\beta\gamma\delta}(\omega_1,\omega_2,\omega_3)$, which describes its nonlinear response to a uniform electromagnetic radiation. The derived analytical formulas describe a large number of different nonlinear phenomena such as the third harmonic generation, the four wave mixing, the saturable absorption, the second harmonic generation stimulated by a dc electric current, etc., which may be used in different terahertz and optoelectronic devices.