Resonant generation of higher harmonics in nonlinear electrodynamics

We study the possibility of resonant amplification of higher harmonics of electromagnetic modes in a closed cavity (resonator) in nonlinear Euler-Heisenberg electrodynamics in two ways: by solving classical equations of effective field theory and by perturbative calculation of the corresponding quantum processes. Classical calculations show that for the cases of “one-dimensional” and three-dimensional rectangular resonator for an arbitrary set of modes, both the third harmonic and the mode of the combined frequency 2ω1+ω2 are not resonantly amplified, and the frequency mode 2ω1−ω2 is resonantly amplified at a certain aspect ratio of the resonator. In quantum theory, the cross sections of the fusion of three quanta of resonator modes into one (process 3→1) and scattering of two quanta of resonator modes (2→2) were calculated. We show that the amplitude of the fusion is always zero, and give an explanation based on the expansion of the resonator modes into plane waves. On the contrary, the scattering amplitude 2→2 is different from zero for a certain ratio of the sides of the resonator. This elementary process is the key to generating a quantum of the 2ω1−ω2 mode when two coherent modes interact.
Based on the materials of works: arXiv:2110.04490, arXiv:2304.10209.