Difference frequency generation of surface plasmon-polaritons in Landau quantized graphene

We develop a rigorous quantum-mechanical theory of the nonlinear optical process of difference frequency generation of surface plasmon-polaritons in Landau-quantized graphene. Although forbidden in the electric-dipole approximation, the second-order susceptibility is surprisingly high, equivalent to the bulk magnitude above ${10}^{-3}$ m/V. We consider the graphene monolayer as a nonlinear optical component of a monolithic photonic chip with integrated pump fields. The nonlinear power conversion efficiency of the order of tens $\mu$W/W${}^2$ is predicted from structures of $10-100$ $\mu$m size. We investigate a variety of waveguide configurations to identify the optimal geometry for maximum efficiency.