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arXiv:2109.07762 · Physical review. B./Physical review. B

Scattering coefficients of superconducting microwave resonators. I. Transfer matrix approach

Qiming Chen, Meike Pfeiffer, Matti Partanen, Florian Fesquet, Kedar Honasoge, Fabian Kronowetter, Yuki Nojiri, Michael Renger, Kirill G. Fedorov, Achim Marx, Frank Deppe, Rudolf Groß

Published Dec 9, 2022• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.106.214505

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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.
We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries, and demonstrate its consistency with the classical approach [Q.-M. Chen et al., Phys. Rev. B 106, 214505 (2022)]. We also generalize the result to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal and can be used to design high-quality resonators. These results form a firm theoretical ground for analyzing the scattering coefficients of an arbitrary resonator network. They set a step forward to designing and characterizing superconducting microwave resonators in a complex superconducting quantum circuit.
Abstract We originally develop a network analysis method to study the non-ideal superconducting resonator circuits. We model the resonator device by a 3-block serial network, consisting of input feedline, resonator-coupler and output feedline. We show that the resonance curve will be modified if non-negligible discontinuity (reflection) exists at input or output side of the resonator. The extracted resonance parameters (such as coupling quality factor Q c ) from only fitting the transmission deviate from their true values as originally defined in matched impedance conditions, and the deviation depends on a single parameter which includes all the non-ideal effects from the circuits. We prove that the internal quality factor Q i can be accurately extracted from the modified transmission curve by using a very simple graphical method. Our methods and results are quantitatively verified by both simulations and experiments on superconducting microstrip resonators. Comparing to the usual circuit analysis, our network analysis method is more intuitive and easier to understand. It can serve as a general tool to study the non-ideal superconducting resonator circuits.
Superconducting microwave resonators are of interest for a wide range of applications, including for their use as microwave kinetic inductance detectors (MKIDs) for the detection of faint astrophysical signatures, as well as for quantum computing applications and materials characterization. In this paper, procedures are presented for the fabrication and characterization of thin-film superconducting microwave resonators. The fabrication methodology allows for the realization of superconducting transmission-line resonators with features on both sides of an atomically smooth single-crystal silicon dielectric. This work describes the procedure for the installation of resonator devices into a cryogenic microwave testbed and for cool-down below the superconducting transition temperature. The set-up of the cryogenic microwave testbed allows one to do careful measurements of the complex microwave transmission of these resonator devices, enabling the extraction of the properties of the superconducting lines and dielectric substrate (e.g., internal quality factors, loss and kinetic inductance fractions), which are important for device design and performance.
Superconducting microwave resonators are of interest for a wide range of applications, including for their use as microwave kinetic inductance detectors (MKIDs) for the detection of faint astrophysical signatures, as well as for quantum computing applications and materials characterization. In this paper, procedures are presented for the fabrication and characterization of thin-film superconducting microwave resonators. The fabrication methodology allows for the realization of superconducting transmission-line resonators with features on both sides of an atomically smooth single-crystal silicon dielectric. This work describes the procedure for the installation of resonator devices into a cryogenic microwave testbed and for cool-down below the superconducting transition temperature. The set-up of the cryogenic microwave testbed allows one to do careful measurements of the complex microwave transmission of these resonator devices, enabling the extraction of the properties of the superconducting lines and dielectric substrate (e.g., internal quality factors, loss and kinetic inductance fractions), which are important for device design and performance.
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.
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.