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arXiv:2507.14394 · Physical Review A

Effective reflection mode measurement for hanger-coupled microwave resonators

John R. Pitten, Nicholas Materise, Wei-Ren Syong, Jorge L Ramirez Ortiz, D. A. Bennett, Corey Rae McRae

Published Oct 24, 2025• 2 citations
arXivHTMLPDF

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 $\ensuremath{\phi}$ 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.

DOI
10.1103/71lw-rsrj

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•Apr 9, 2015•1 citation
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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ in the diameter correction method, results from the coupling schemes used to measure these devices, including the commonly used hanger method. \ensuremathϕ\ensuremath{\phi}\ensuremathϕ 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 Qi\ensuremath−1{Q}_{i}^{\ensuremath{-}1}Qi\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.
Abstract Superconducting circuit testing and materials loss characterization requires robust and reliable methods for the extraction of internal and coupling quality factors of microwave resonators. A common method, imposed by limitations on the device design or experimental configuration, is the single-port reflection geometry, i.e. reflection-mode. However, impedance mismatches in cryogenic systems must be accounted for through calibration of the measurement chain while it is at low temperatures. In this paper, we demonstrate a data-based, single-port calibration using commercial microwave standards and a vector network analyzer with samples at millikelvin temperature in a dilution refrigerator, making this method useful for measurements of quantum phenomena. Finally, we cross reference our data-based, single-port calibration and reflection measurement with over-coupled 2D- and 3D-resonators against well established two-port techniques corroborating the validity of our method.
We examine the transmission through nonideal microwave resonant circuits. The general analytical resonance line shape is derived for both inductive and capacitive coupling with mismatched input and output transmission impedances, and it is found that, for certain non-ideal conditions, the line shape is asymmetric. We describe an analysis method for extracting an accurate internal quality factor (Qi), the diameter correction method (DCM), and compare it to the conventional method used for millikelvin resonator measurements, the φ rotation method (φRM). We analytically find that the φRM deterministically overestimates Qi when the asymmetry of the resonance line shape is high, and that this error is eliminated with the DCM. A consistent discrepancy between the two methods is observed when they are used to analyze both simulations from a numerical linear solver and data from asymmetric coplanar superconducting thin-film resonators.
We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experiment and theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct their influences in typical experiments. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or cross-type resonator also contains a so-called reference point that can be used to characterize the internal quality factor of the resonator. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.
A method is proposed and experimentally explored for in-situ calibration of complex transmission data for superconducting microwave resonators. This cryogenic calibration method accounts for the instrumental transmission response between the vector network analyzer reference plane and the device calibration plane. Once calibrated, the observed resonator response is analyzed in detail by two approaches. The first, a phenomenological model based on physically realizable rational functions, enables the extraction of multiple resonance frequencies and widths for coupled resonators without explicit specification of the circuit network. In the second, an ABCD-matrix representation for the distributed transmission line circuit is used to model the observed response from the characteristic impedance and propagation constant. When used in conjunction with electromagnetic simulations, the kinetic inductance fraction can be determined with this method with an accuracy of 2%. Datasets for superconducting microstrip and coplanar-waveguide resonator devices were investigated and a recovery within 1% of the observed complex transmission amplitude was achieved with both analysis approaches. The experimental configuration used in microwave characterization of the devices and self-consistent constraints for the electromagnetic constitutive relations for parameter extraction are also presented.
Superconducting resonators provide a convenient way to measure loss tangents of various dielectrics at low temperature. For the purpose of examining the microscopic loss mechanisms in dielectrics, precise measurements of the internal quality factor at different values of energy stored in the resonators are required. Here, we present a consistent method to analyze a LC superconducting resonator coupled to a transmission line. We first derive an approximate expression for the transmission S-parameter S21(ω), with ω the excitation frequency, based on a complete circuit model. In the weak coupling limit, we show that the internal quality factor is reliably determined by fitting the approximate form of S21(ω). Since the voltage V of the capacitor of the LC circuit is required to determine the energy stored in the resonator, we next calculate the relation between V and the forward propagating wave voltage Vin+, with the latter being the parameter controlled in experiments. Due to the dependence of the quality factor on voltage, V is not simply proportional to Vin+. We find a self-consistent way to determine the relation between V and Vin+, which employs only the fitting parameters for S21(ω) and a linear scaling factor. We then examine the resonator transmission in the cases of port reflection and impedance mismatch. We find that resonator transmission asymmetry is primarily due to the reflection from discontinuity in transmission lines. We show that our analysis method to extract the internal quality factor is robust in the non-ideal cases above. Finally, we show that the analysis method used for the LC resonator can be generalized to arbitrary weakly coupled lumped and distributed resonators. The generalization uses a systematic approximation on the response function based on the pole and zero which are closest to the resonance frequency. This Closest Pole and Zero Method is a valuable tool for analyzing physical measurements of high-Q resonators.
Superconducting quantum devices have gained increasing interest over the past 15 years because of their role as promising candidates for implementing a large-scale quantum computer and as ultra-sensitive detectors. Superconducting resonators are one type of superconducting device which are used in a variety of applications including quantum computing and radioastronomy. They can achieve large quality factors which are important for these applications and a variety of others. Specific examples of these applications include the readout and coupling of superconducting qubits and also microwave kinetic inductance detector arrays for sub-mm wavelength astronomical measurements. High quality factors also enable the study of very subtle features in the electromagnetic response of the resonator. This feature in turn offers provides a method to characterize loss mechanisms which affect the performance of superconducting qubits, by measuring the quality factor of a resonator. Dielectric loss is an example of such a loss mechanism, and is considered to be one of, if not \\it the \\rm major obstacle to overcome in order to construct scalable quantum technologies based on superconducting devices. This type of loss arises from the coupling of the electric field in the device to spurious degrees of freedom which arise naturally the dielectric environment of the device. The fast-growing interest in the physics and applications of superconducting resonators has renewed interest in an improving our fundamental understanding of dielectric loss. This interest involves the study of a much bigger and older question in condensed matter physics; namely, what is the microscopic physics which governs the behaviour and of an amorphous material at low-temperatures? \n \nIn this thesis, we present a systematic study of dielectric loss in amorphous aluminum oxide using superconducting microwave resonators. Measurements of the intrinsic loss tangent over a large temperature range of 40~mK to 1~K are presented and analyzed in detail. Some aspects of the data are well understood in terms of the standard tunnelling two-level system model, while others require a deeper analysis of the electrodynamics of the resonator and insights from more advanced theories. We employ a consistent approach using numerical simulations of the resonator to understand loss mechanisms arising from both an inhomogeneous electric field and multiple dielectric layers. We measure saturation effects which provide insight into coherent properties of two-level system ensembles and the mutual interaction between them. Excellent agreement with the two-level system model is found for intrinsic losses, however the strong dependence on temperature which we observe for saturation effects is not predicted in the standard form of the model. We find that this dependence can be understood in terms of the theory of spectral diffusion which is consistent with a strong mutual interaction between two-level systems contributes to broadening of their spectral linewidths.