Photon Frequency Conversion in High-$Q$ Superconducting Resonators: Axion Electrodynamics, QED & Nonlinear Meissner Radiation

High-Q superconducting resonators have been proposed and developed as detectors of light-by-light scattering mediated by the hypothesized axion or virtual electron-positron pairs in quantum electrodynamics - the Euler-Heisenberg (EH) interaction. Photon frequency and mode conversion is central to the scheme for detecting such rare events. Superconducting resonators are nonlinear devices. The Meissner screening currents that confine the electromagnetic fields to the vacuum region of a superconducting RF cavity are nonlinear functions of the EM field at the vacuum-superconducting interface, and as a result can generate source currents and frequency conversion of microwave photons in the cavity. In this report we consider photon frequency and mode conversion in superconducting resonators with high quality factors from Meissner currents in single and dual cavity setups proposed for axion and QED searches based on light-by-light scattering. In a single cavity with two pump modes photon frequency conversion by the Meissner screening current dominates photon generation by the EH interaction for cavities with $Q \lesssim 10^{12}$. The Meissner currents also generate background photons that limits the operation of the resonator for axion detection in three-mode, single cavity setups. We also consider the leakage of photons from pump modes into the signal mode for both axion and EH mediated light-by-light scattering. Photon frequency conversion by the EH interaction can compete with Meissner and leakage radiation in \emph{ultra-high-Q} cavities that are beyond current state of the art. Meissner radiation and leakage backgrounds can be suppressed in dual cavity setups with appropriate choices for pump and spectator modes, as well as the single-cavity setup proposed for heterodyne detection of galactic axion dark matter.