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Efficient and robust analysis of complex scattering data under noise in microwave resonators
S. Probst, F. B. Song, P. A. Bushev, A. V. Ustinov, M. Weides
TL;DR
Determining resonator quality factors from noisy complex scattering data requires a robust analysis that handles measurement-environment effects and low signal-to-noise conditions. The paper combines circle fitting with diameter correction, algebraic fitting, and automatic calibration of environmental prefactors. The resulting procedure determines resonator and mismatch parameters efficiently and remains robust at SNR < 20, while calibration becomes unreliable below SNR 200 and internal-quality-factor estimation requires at least 200 points at SNR 65.
Problem
Reliable extraction of internal and external quality factors is needed for fast and single-photon-regime resonator measurements, but noise and uncalibrated environmental effects complicate complex scattering-data analysis.
Method
The paper uses a constrained algebraic resonance-circle fit with diameter correction and fits measurement-environment factors, including amplitude, phase, and cable delay, during calibration.
Results
The method determines internal and loaded quality factors, resonance frequency, impedance mismatch, and measurement-circuit contributions automatically, remaining robust even at SNR < 20.
Takeaways & Limitations
At sufficient signal-to-noise ratio, complex VNA data can be analyzed without calibration, while the algebraic fit provides a fast and robust analysis of noisy resonator data.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Superconducting microwave resonators support sensitive detection and material research, making precise extraction of resonance and quality-factor parameters important. The introduction motivates notch-type transmission measurements because pure S21 data lack the baseline needed to determine internal and coupling losses and impedance mismatch.
- Motivation: Superconducting resonators provide low internal loss and are used for detection, material studies, and circuit quantum electrodynamics down to the single-photon regime.Applications include dispersive readout and coupling to qubits or spin systems.
- Resonator parameters: A resonator’s internal and coupling losses are characterized by Qi and Qc, while the loaded quality factor combines their reciprocal contributions.The coupling quality factor may be complex when impedance mismatch is included.
- Measurement geometry: Notch-type geometry couples a resonator to a transmission line and supports frequency-division multiplexed readout of multiple resonators.The measured S21 response appears as a transmission dip with amplitude and phase varying across frequency.
- Model choice: The introduction adopts a complex Qc = |Qc| exp(−iφ) to model impedance mismatch, while noting that resonance-dip asymmetry may also be attributed to port reflections.The internal quality factor is emphasized because it quantifies internal losses and can indicate coupling to spin ensembles.
- Measurement problem: Pure S21 data cannot determine internal and coupling quality factors or impedance mismatch because an arbitrary measurement amplitude masks the resonance-circle diameter and the reference baseline is missing.The uncalibrated diameter becomes a · Ql/|Qc|, while the off-resonant point cannot distinguish α from φ0.
II. ALGEBRAIC FIT OF THE RESONANCE CIRCLE
The method fits the resonance circle algebraically rather than iteratively, using a constrained general circle parametrization and an eigenvalue problem. This yields a fast, reliable estimate of the circle diameter and its center and radius, without start parameters or iterative fitting.
- Fitting objective: The core fitting task is to determine the resonance-circle diameter d = Ql/|Qc| from complex scattering data.The diameter links the fitted circle to the loaded and coupling quality factors.
- Robustness and speed: The algebraic fit needs neither start parameters nor iterations, making its runtime independent of signal-to-noise ratio and providing a fast, reliable result under heavy noise.The method is presented as especially advantageous over iterative approaches in noisy data.
- Circle parametrization: The circle is represented by A(x^2 + y^2) + Bx + Cy + D = 0 with an additional constraint to remove over-parametrization.This parametrization also represents lines when A = 0 and circles when A > 0.
- Constrained fit: The fitting objective minimizes the squared algebraic residuals of n measured points that approximately lie on the circle.In matrix form, the objective is F = A^TMA for coefficient vector A = (A, B, C, D)^T, subject to A^TBA = 1.
- Eigenvalue solution: The constrained minimization reduces to a generalized eigenvalue problem, whose smallest non-negative eigenvalue η* gives the minimizing solution.The characteristic polynomial is decreasing and concave up between zero and η*, so Newton’s method starting at zero converges to η*.
- Implementation: The practical implementation computes moments, solves the characteristic polynomial from η = 0, finds the corresponding eigenvector, and extracts the circle center and radius.The radius and center are obtained from the eigenvector associated with η*.
III. INFLUENCE OF THE ENVIRONMENT
The method removes environmental distortions from complex resonator data through sequential cable-delay correction, circle fitting, phase fitting, and canonical transformation. These steps recover the environmental prefactors and the transformed off-resonant point needed for calibration.
- Environmental effects: Cable damping changes amplitude, cable delay rotates and deforms the resonance circle, and the initial phase may differ from zero.The method fits these environmental quantities rather than relying solely on VNA calibration.
- Cable-delay correction: A rough linear phase fit estimates the cable delay, after which nonlinear least-squares fitting minimizes deviation from an ideal circular shape.The robustness of the algebraic circle fit makes this correction procedure possible.
- Circle fitting: After delay correction, an algebraic circle fit determines the circle center and radius, while translation to the origin enables estimation of attenuation and phase prefactors.The remaining affine transformation is represented by a e^iα.
- Phase fitting: The translated-circle phase fit yields the resonance frequency, loaded quality factor Q_l, and offset phase θ0.The phase result determines the angular position β of the transformed off-resonant point.
- Canonical calibration: The transformed off-resonant point P′ is obtained geometrically, with |P′| giving amplitude scaling a and arg(P′) giving phase offset α.The final transformation places P′ at the canonical value 1.
IV. IMPLEMENTATION
The implementation uses a high-resolution, high-SNR resonance scan followed by fitting and covariance-based uncertainty estimation. The authors also provide an implementation of the algorithm online.
- Measurement: A high-resolution S21(f) scan should use high power and low IF bandwidth to achieve SNR larger than 100.Points should be evenly spaced around resonance across approximately four 3 dB bandwidths.
- Measurement: The fitting scan should contain sufficient evenly spaced points around the resonance over a span of approximately four 3 dB bandwidths.The 3 dB bandwidth is defined using the loaded quality factor.
- Availability: An implementation of the algorithm is available on the authors’ website.
V. TEST OF THE ALGORITHM
The algorithm remains robust under strong noise for calibrated data, while automatic calibration requires substantially higher SNR and enough recorded points for reliable quality-factor extraction.
- Calibrated fit: Above SNR=40, fitted Qi, Qc, and Ql are very stable, while Qi remains robust down to SNR=20 for properly calibrated data.The test uses typical superconducting-resonator parameters and assumes the measurement is properly calibrated.
- Noise model: The generated-data tests add Gaussian radial noise with width σ = r0/SNR to artificial S21 resonance-circle points.Here, r0 is the circle radius, and the fitted center and radius introduce uncertainty when estimating SNR from real data.
- Automatic calibration: Below a SNR of 200, automatic calibration becomes inaccurate, making resonator-parameter determination unreliable; for the given parameters, the procedure works only above SNR=300.The raw data include arbitrary cable delay, start phase, and amplitude that must be corrected automatically.
- Automatic calibration: Calibration is typically performed only once at large SNR for a given experiment, so its high-SNR requirement does not prevent later measurements at lower SNR.This separates the calibration constraint from the robustness of the subsequent calibrated fit.
- Number of points: At fixed SNR of 65, at least 200 recorded points are necessary for reliable determination of the internal quality factor.Above 200 points, the fit appears limited primarily by the SNR in this test.
VI. CONCLUSION
The presented method analyzes noisy complex microwave-resonator scattering data quickly and accurately using automated fitting and calibration. It remains robust at low SNR and uses a non-iterative circle fit that converges without start values.
- Conclusion: The method provides fast and accurate analysis of noisy complex scattering data from microwave resonators.It is designed for applications requiring real-time analysis of calibrated resonator data.
- Conclusion: At sufficient SNR, complex VNA data can determine six generalized resonator parameters automatically without calibration.These include internal and loaded quality factors, resonance frequency, impedance mismatch, arbitrary amplitude and phase factors, and cable delay.
- Conclusion: The calibrated resonator-fit algorithm remains robust under strong noise, including SNR < 20.Its circle parameters are obtained algebraically rather than through an iterative fit.
- Conclusion: The non-iterative circle fit converges immediately without supplied start values, improving robustness and speed relative to existing methods.The method therefore supports real-time analysis of calibrated resonator data.