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

Qi-Ming Chen, Meike Pfeiffer, Matti Partanen, Florian Fesquet, Kedar E. Honasoge, Fabian Kronowetter, Yuki Nojiri, Michael Renger, Kirill G. Fedorov, Achim Marx, Frank Deppe, and Rudolf Gross

Phys. Rev. B 106, 214505 – Published 9 December 2022

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.

Received 22 February 2022
Revised 22 November 2022
Accepted 28 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.214505

Physics Subject Headings (PhySH)

Light propagation, transmission & absorption Waveguides

Superconducting devices

Cavity resonators Light scattering Microwave techniques

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Qi-Ming Chen ^1,2,*, Meike Pfeiffer², Matti Partanen¹, Florian Fesquet^1,2, Kedar E. Honasoge^1,2, Fabian Kronowetter^1,2, Yuki Nojiri^1,2, Michael Renger^1,2, Kirill G. Fedorov^1,2, Achim Marx¹, Frank Deppe^1,2,3,†, and Rudolf Gross^1,2,3,‡

¹Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
²Physik-Department, Technische Universität München, 85748 Garching, Germany
³Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 Munich, Germany

^*qiming.chen@wmi.badw.de
^†frank.deppe@wmi.badw.de
^‡rudolf.gross@wmi.badw.de

Scattering coefficients of superconducting microwave resonators. II. System-bath approach

Qi-Ming Chen, Matti Partanen, Florian Fesquet, Kedar E. Honasoge, Fabian Kronowetter, Yuki Nojiri, Michael Renger, Kirill G. Fedorov, Achim Marx, Frank Deppe, and Rudolf Gross

Phys. Rev. B 106, 214506 (2022)

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 106, Iss. 21 — 1 December 2022

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematics of several different microwave resonators. (a) In lumped-element circuits, the series (left) and parallel (right) RLC resonant circuits are the two fundamental types of microwave resonators. (b) In distributed-element circuits, there are three types of transmission-line resonators: the short-circuited $λ / 4$ (left), the short-circuited $λ / 2$ (middle), and the open-circuited $λ / 2$ (right) resonators. (c) Microwave resonators can be also made by bulky 3D devices, which can be modeled by a short-circuited $λ / 2$ (left) or a short-circuited $λ / 4$ (right) transmission-line resonator. In all the panels, red and green vectors depict the spatial modes of the resonator, which are determined by the boundary conditions.
Reuse & Permissions
Figure 2
Schematics of three typical resonator designs and the corresponding circuit diagrams. They are (a) the hanger-type $λ / 4$ resonator, (b) the necklace-type $λ / 2$ resonator, and (c) the cross-type $λ / 2$ resonator. In all the panels, we denote $l_{1}$ and $l_{2}$ as the lengths of the feedlines that are coupled to the resonator, $C_{1}$ and $C_{2}$ the coupling capacitors, and $Δ Z_{1}$ and $Δ Z_{2}$ the circuit asymmetries. Without loss of generality, we label the left and right ports by 1 and 2.
Reuse & Permissions
Figure 3
Simulated scattering coefficients of the three types of microwave resonators shown in Fig. 2. They are (a) the hanger-type $λ / 4$ resonator, (b) the necklace-type $λ / 2$ resonator, and (c) the cross-type $λ / 2$ resonator. In all the panels, the black dashed curves represent the analytical results. They fit closely to the red and green curves, which are the numerical results calculated by using the transfer matrices. The shift between the dashed and solid curves is around $2 π \times 1 MHz$ , which is due to the approximation error of the resonant frequency $ω_{r}$ . This shift can be removed by calculating $ω_{r}$ in an iterative manner. In these examples, one extra iteration decreases the shift to $2 π \times 4 kHz$ .
Reuse & Permissions
Figure 4
(a) Sketch of the experimental setup. Here, a necklace-type transmission-line resonator is coupled to the feedlines with two mechanically tunable coupling capacitors. (b)–(g) The step-by-step correction procedure for the measured reflection coefficient $S_{11}$ for the third harmonic mode. In specifics, (b) and (c) show the raw data. The blue dots represent the whole data set, from which the green dots are selected for characterization. The latter is chosen within $\pm 2$ times of the FWHM around the resonant frequency. (d), (f), (g) Show the data after removing the frequency-dependent phase shift, the global constant phase shift and attenuation, and the local constant phase shift, respectively. (e) Shows the procedure to identify the resonance point, which is shown as the open red dot in (d)–(g). Detail of the correction procedure can be found in Appendix pp6.
Reuse & Permissions
Figure 5
The step-by-step procedure for removing distortions and characterizing a necklace-type microwave resonator (shown are $S_{11}$ only). (a) Rough phase correction with linear fit and data selection with Lorentzian fit, where the green dots are within $\pm 2$ times of FWHM around the estimated resonant frequency. (b) The raw data (blue) and the selected data (green) shown in the complex plane. (c) Fine phase correction with circle fit. (d) Resonant frequency determination with phase versus frequency fit. (e) Correction of cable attenuation and frequency-independent phase. (f) Correction of asymmetry. In all the plots, the fitting results are colored in red. The open and filled red dots represent the scattering coefficients at $| ω - ω_{r} | \to 0$ or $| ω - ω_{r} | \to \infty$ , respectively. The small red dot represents the center of the fitted circle.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics