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 (\(3 \rightarrow 1\) process) as well as the scattering of two cavity mode quanta (\(2 \rightarrow 2\) 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 \(2 \rightarrow 2\) 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 (Kopchinskii and Satunin in Phys Rev A 105:013508, 2022).