Quantum theory of third-harmonic generation in graphene

A quantum theory of third-harmonic generation in graphene is presented. An analytical formula for the nonlinear conductivity tensor σ_αβγδ⁽³⁾(ω,ω,ω) is derived. Resonant maxima of the third harmonic are shown to exist at low frequencies ωEF/, as well as around the frequency ω=2EF/, where EF is the Fermi energy in graphene. At an input power of a CO₂ laser (λ≈10 μm) of about 10.28em0exMW/cm², the output power of the third harmonic (λ≈3.3 μm) is expected to be 50 W/cm².