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Journal of Applied Physics

Efficient methods for extracting superconducting resonator loss in the single-photon regime

Cliff Chen, David Perello, Shahriar Aghaeimeibodi, Guillaume Marcaud, Ignace Jarrige, Hanho Lee, Warren Fon, Matthew H. Matheny, Jiansong Gao

Published Jan 28, 2025• 4 citations
Publisher

Abstract

Superconducting micro-resonators have application in sensors and quantum computing. Measurement of the resonator internal loss in the single-photon regime is a common tool to study the origins of dissipation, noise, and decoherence in quantum circuits, as well as characterization of materials used for quantum devices. However, such measurements are challenging and time-consuming with large uncertainties due to the poor signal-to-noise ratio when measured at single-photon power levels. We developed methods to extract a subset of the resonator fitting parameters at high power and fix them in the analysis of low power data, which reduce the parameter space in the regression analysis. By comparing the analysis with and without partial extraction, we show that these methods reduce the uncertainty of the results, while improving the robustness and efficiency of the loss measurement.

DOI
10.1063/5.0242201

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Superconducting micro-resonators have application in sensors and quantum computing. Measurement of the resonator internal loss in the single-photon regime is a common tool to study the origins of dissipation, noise, and decoherence in quantum circuits, as well as characterization of materials used for quantum devices. However, such measurements are challenging and time-consuming with large uncertainties due to the poor signal-to-noise ratio when measured at single-photon power levels. We developed methods to extract a subset of the resonator fitting parameters at high power and fix them in the analysis of low power data, which reduce the parameter space in the regression analysis. By comparing the analysis with and without partial extraction, we show that these methods reduce the uncertainty of the results, while improving the robustness and efficiency of the loss measurement.
The performance of superconducting circuits for quantum computing is limited by materials losses. In particular, coherence times are typically bounded by two-level system (TLS) losses at single photon powers and millikelvin temperatures. The identification of low loss fabrication techniques, materials, and thin film dielectrics is critical to achieving scalable architectures for superconducting quantum computing. Superconducting microwave resonators provide a convenient qubit proxy for assessing performance and studying TLS loss and other mechanisms relevant to superconducting circuits such as non-equilibrium quasiparticles and magnetic flux vortices. In this review article, we provide an overview of considerations for designing accurate resonator experiments to characterize loss, including applicable types of losses, cryogenic setup, device design, and methods for extracting material and interface losses, summarizing techniques that have been evolving for over two decades. Results from measurements of a wide variety of materials and processes are also summarized. Finally, we present recommendations for the reporting of loss data from superconducting microwave resonators to facilitate materials comparisons across the field.
Superconducting microwave resonators are reliable circuits widely used for detection and as test devices for material research. A reliable determination of their external and internal quality factors is crucial for many modern applications, which either require fast measurements or operate in the single photon regime with small signal to noise ratios. Here, we use the circle fit technique with diameter correction and provide a step by step guide for implementing an algorithm for robust fitting and calibration of complex resonator scattering data in the presence of noise. The speedup and robustness of the analysis are achieved by employing an algebraic rather than an iterative fit technique for the resonance circle.
Superconducting qubits, though promising for both near-term problem-solving as well as the development of large-scale quantum computing systems, are limited in performance largely by materials-induced decoherence channels which are maximized at millikelvin temperatures, single photon powers, and microwave frequencies. Superconducting microwave resonators are ideal tools for measuring these materials losses due to their simplicity, well-understood behavior, and sensitivity to these same losses. In order to better understand and mitigate dominant qubit decoherence channels, accurate and precise characterization of superconducting microwave resonator performance is critical. In this paper, current measurement techniques for superconducting microwave resonators are discussed, including experimental set-up, error analysis, and cryogenic calibration of a vector network analyzer.
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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.