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Quantum Theory of the Third-Harmonic Generation in Graphene

A quantum theory of the third-harmonic generation in graphene is presented. An analytical formula for the nonlinear conductivity tensor $σ^{(3)}_{αβγδ}(ω,ω,ω)$ is derived. Resonant maxima of the third harmonic are shown to exist at low frequencies $ω\ll E_F/\hbar$, as well as around the frequency $ω=2E_F/\hbar$, where $E_F$ is the Fermi energy in graphene. At the input power of a CO$_2$ laser ($λ\approx 10$ $μ$m) of about 1 MW/cm$^2$ the output power of the third-harmonic ($λ\approx 3.3$ $μ$m) is expected to be $\simeq 50$ W/cm$^2$.