Resonant generation of electromagnetic modes in nonlinear electrodynamics: Quantum perturbative approach

The paper studies resonant generation of higher-order harmonics in a closed cavity in Euler-Heisenberg electrodynamics from the point of view of pure quantum field theory. We consider quantum states of the electromagnetic field in a rectangular cavity with conducting boundary conditions, and calculate the cross-section for the merging of three quanta of cavity modes into a single one ($3 \to 1$ process) as well as the scattering of two cavity mode quanta ($2 \to 2$ process). We show that the amplitude of the merging process vanishes for a cavity with an arbitrary aspect ratio, and provide an explanation based on plane wave decomposition for cavity modes. Contrary, the scattering amplitude is nonzero for specific cavity aspect ratio. This $2 \to 2$ scattering is a crucial elementary process for the generation of a quantum of a high-order harmonics with frequency $2\omega_1 - \omega_2$ in an interaction of two coherent states of cavity modes with frequencies $\omega_1$ and $\omega_2$. For this process we calculate the mean number of quanta in the final state in a model with dissipation, which supports the previous result of resonant higher-order harmonics generation in an effective field theory approach.