Effective reflection mode measurement for hanger-coupled microwave resonators

Superconducting microwave resonators are used in many low-power applications, such as the study of two-level system loss in superconducting quantum devices. Fano asymmetry, characterized by a nonzero asymmetry angle $\ensuremath{\phi}$ in the diameter correction method, results from the coupling schemes used to measure these devices, including the commonly used hanger method. $\ensuremath{\phi}$ is an additional fitting parameter which contains no physically interesting information and can obscure device parameters of interest. The T-junction symmetry nominally present in these resonator devices provides an avenue for the elimination of Fano asymmetry using calibrated measurement. We show that the eigenvalue associated with the common mode excitation of the resonator is an effective reflection mode (ERM) which has no Fano asymmetry. Our analysis reveals the cause of Fano asymmetry as interference between common and differential modes. Practically, we obtain the ERM from a linear combination of calibrated reflection and transmission measurements. We utilize a three-dimensional aluminum cavity to experimentally demonstrate the validity and flexibility of this model. To extend the usefulness of this symmetry analysis, we apply perturbation theory to recover the ERM in a multiplexed coplanar waveguide resonator device and experimentally demonstrate quantitative agreement in the extracted ${Q}_{i}^{\ensuremath{-}1}$ between hanger mode and ERM measurements. We observe a fivefold reduction in uncertainty from the ERM compared to the standard hanger mode at the lowest measured power, \ensuremath{-}160 dBm delivered to the device. This method could facilitate an increase in throughput of low-power superconducting resonator measurements by up to a factor of 25, as well as allow the extraction of critical parameters from otherwise unfittable device data.