Power Dissipation in an Asymmetric Superconducting Resonator

Resonator asymmetry is a common feature in superconducting resonators, often arising from parasitic effects or non-ideal bias conditions in the microwave system. While the effects of asymmetric line shapes on extracting resonator quality factors has been well investigated, the impact of asymmetry on the power dissipation as well as the resonator response to perturbation of resonator properties remains poorly understood. In this paper, we derive the full scattering matrix of an arbitrarily complex reciprocal passive network with a single embedded shunt resonator. We identify a minimal set of approximations that accurately extract the resonator behavior while preserving the relationship between the continuum and resonator modes of the network. The full scattering matrix can be decomposed into the unperturbed continuum, a perturbation to the continuum and the resonant response. The continuum is unperturbed when the resonator is symmetric. Our analysis therefore reveals that resonator asymmetry can significantly affects continuum transmission through the network - an effect that becomes particularly relevant in large resonator arrays. Additionally, using the generalized relationship between S11,S21, and S22, we demonstrate that power dissipation in the resonator is independent of the asymmetry angle. However, the power dissipation is sensitive to the symmetry (even or odd) of the coupling between the resonator and the continuum modes. This approach also provides a framework for modeling dynamic changes in resonator coupling, baseline, and asymmetry in response to variations induced by temperature changes, magnetic fields, applied bias currents and/or optical loading.