Effective reflection mode measurement for hanger-coupled microwave resonators

TL;DR

This work tackles the problem of Fano asymmetry obscuring resonator parameters in hanger-coupled superconducting devices. By exploiting approximate T-junction symmetry, the authors derive an effective reflection mode (ERM) via a full scattering-matrix treatment, enabling a one-port-like interpretation of the system and removing the asymmetry in the common-mode response. They validate the theory with room-temperature measurements on a 3D aluminum cavity and extend it to a cryogenic multiplexed CPW resonator using perturbation theory to handle symmetry deviations, achieving quantitative agreement in $Q_i^{-1}$ and a significant reduction in measurement uncertainty. Practically, ERM-based measurements can boost throughput by up to 25× and enable extraction of parameters from data that are otherwise unfittable, facilitating faster studies of two-level system loss in superconducting quantum devices.

Abstract

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 $φ$ in the diameter correction method, results from the coupling schemes used to measure these devices, including the commonly used hanger method. $φ$ 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 $Q_i^{-1}$ 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, $-160$ 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.

Figures (10)

• Figure 1: An idealized reflection mode coupled resonator. In red, a circuit diagram of the resonator itself, which alone has a reflection coefficient of $\Gamma$. In blue, a series reactive coupling network which modifies the resonator's reflection coefficient to $M(\Gamma)$. This mapping is used to define a reflection mode.
• Figure 2: Schematic of a hanger coupled resonator and the associated circuit model. We find that when the coupling network adheres to a T-junction symmetry, the common mode is an effective reflection mode $S_{21}^R+S_{11}^R = M(\Gamma)$. (a) Mask for coplanar waveguide multiplexed hanger-coupled resonators WhitePaper. Across the center is a feedline and launch pads which connect to wirebonds. These constitute the coupling network, shown in blue. In red, a quarter wave resonator. (b) Circuit diagram to model one of the resonators on this device. In blue, inductors represent wirebonds while the feedline is modeled with transmission line segments. A capacitor models the reactive coupling between feedline and resonator. Both ports share a common ground with the resonator. In red, the resonator.
• Figure 3: Experimental demonstration of an ERM measurement for a hanger-coupled 3D aluminum cavity measured at room temperature with an ECal. Smith chart (a), scattering parameter magnitude $|S|$ ((b) and (d)), and scattering parameter phase $\angle S$ ((c) and (e)) are plotted for the ERM $S^R_{21}+S^R_{11}$ (purple), differential mode $S^R_{21}-S^R_{11}$ (black), transmission $S^R_{21}$ (blue), and reflection $S^R_{11}$ (red). The ERM displays characteristics of an ideal reflection mode: a $2\pi$ phase shift and no asymmetry, unlike the reflection and transmission data. The differential mode $S^R_{21}-S^R_{11}$ displays complete destructive interference with constant unit magnitude.
• Figure 4: Experimental demonstration of an ERM measurement at $10$ mK on a multiplexed hanger-coupled coplanar waveguide resonator with perturbed T-junction symmetry. (a) Comparison of inverse internal quality factor $Q_i^{-1}$ between hanger-mode $S_{\text{hanger}}=\tfrac{1}{2}(S^R_{21}+S^R_{21})$ and ERM $S_{\text{CM}} = \tfrac{1}{2}(S^R_{21}+S^R_{21}) + \tfrac{1}{2}(S^R_{11}+S_{22}^R)$ measurements. All values agree over a large range of powers delivered to the device, expressed in terms of average photon number $\langle n \rangle$. $S_{\text{hanger}}$ values have been slightly offset for readability. Uncertainty is reported as 95% confidence intervals. (b) Smith chart showing the predicted splitting in reflection-type scattering parameters (red and orange). Eigenmodes $S_{\text{CM}}$ and $S_{\text{DM}} = \tfrac{1}{2}(S^R_{21}+S^R_{21}) - \tfrac{1}{2}(S^R_{11}+S_{22}^R)$ shown in purple and black, recovered by averaging the reflection at the two ports as prescribed by the perturbation theory calculations. Data are shown at $\langle n \rangle = 8.2\times 10^6$. (c) Magnitude of junction asymmetry $|\mu|$ obtained from scattering parameter measurements as a function of measurement frequency $f$ where $f_{\text{offset}} = 4.7076$ GHz.
• Figure 5: Illustration of a waveguide T-junction. The two collinear arms end at ports 1 and 2. These two ports are interchangeable. The coupling arm, which points to the right, ends at port 3.