Spectral Theory for Non-linear Superconducting Microwave Systems: Extracting Relaxation Rates and Mode Hybridization

The accurate modeling of mode hybridization and calculation of radiative relaxation rates have been crucial to the design and optimization of superconducting quantum devices. In this work, we introduce a spectral theory for the electrohydrodynamics of superconductors that enables the extraction of the relaxation rates of excitations in a general three-dimensional distribution of superconducting bodies. Our approach addresses the long-standing problem of formulating a modal description of open systems that is both efficient and allows for second quantization of the radiative hybridized fields. This is achieved through the implementation of finite but transparent boundaries through which radiation can propagate into and out of the computational domain. The resulting spectral problem is defined within a coarse-grained formulation of the electrohydrodynamical equations that is suitable for the analysis of the non-equilibrium dynamics of multiscale superconducting quantum systems.