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arxiv: 2505.19922 · v3 · submitted 2025-05-26 · ⚛️ physics.ins-det · quant-ph

Full Two-Port S-Parameters at mK Temperatures: a Calibration Strategy and Uncertainty Budget

Luca Oberto , Ehsan Shokrolahzade , Emanuele Enrico , Luca Fasolo , Andrea Celotto , Bernardo Galvano , Alessandro Alocco , Paolo Terzi
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Faisal A. Mubarak Marco Spirito
This is my paper

Pith reviewed 2026-05-19 13:34 UTC · model grok-4.3

classification ⚛️ physics.ins-det quant-ph
keywords S-parameterscryogenic measurementsmicrowave calibrationuncertainty budgetmillikelvin temperaturescoaxial lineattenuator
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The pith

A single-cycle Short-Open-Load-Reciprocal calibration referenced to room-temperature standards enables full two-port S-parameter measurements at millikelvin temperatures with a complete uncertainty budget.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper describes a setup for error-corrected two-port S-parameter measurements down to millikelvin temperatures over the 4-12 GHz band in coaxial line. It applies the Short-Open-Load-Reciprocal technique so that calibration occurs within one cooling cycle, with standards tied to SI-traceable room-temperature data. A numerical model estimates how the calibration artifacts change upon cooling and folds that into the uncertainty budget. Additional uncertainty contributions are evaluated following standard procedures, and the approach is demonstrated on a 20 dB attenuator that yields 20.70 plus or minus 0.08 dB at 6 GHz. Full direct SI-traceable verification at cryogenic temperatures is noted as remaining open.

Core claim

The system exploits the Short-Open-Load-Reciprocal technique to realize error-corrected cryogenic measurements within a single cooling cycle, with calibration standards referenced to SI-traceable room-temperature measurements and a numerical approach to evaluate response shift upon cooling for the uncertainty budget.

What carries the argument

Short-Open-Load-Reciprocal (SOLR) calibration combined with numerical modeling of the cryogenic response shift of the calibration standards.

If this is right

  • Devices under test can receive full two-port error correction at mK temperatures without requiring separate warm-up and re-cool cycles for each calibration.
  • Uncertainty budgets for cryogenic microwave measurements now include a traceable cryogenic contribution derived from the numerical shift model.
  • The method supports coaxial measurements from 4 to 12 GHz at millikelvin temperatures with quantified uncertainties.
  • Test results such as the 20.70 dB attenuation value at 6 GHz can be reported with an expanded uncertainty of 0.08 dB at 95 percent .

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could reduce the number of thermal cycles needed when characterizing multiple devices in the same cryostat.
  • Similar numerical-shift corrections might be adapted to other calibration kits or frequency bands to extend traceable cryogenic metrology.
  • Closing the remaining gap to full SI-traceable verification would require developing primary standards that can be measured directly at mK temperatures.

Load-bearing premise

The numerical approach accurately captures all relevant changes in the calibration artifacts when cooled and produces a valid additional uncertainty term without large unmodeled biases.

What would settle it

A direct comparison of the same attenuator measured both with this method and with an independent cryogenic reference standard at 6 GHz that shows a difference larger than the stated 0.08 dB uncertainty.

Figures

Figures reproduced from arXiv: 2505.19922 by Alessandro Alocco, Andrea Celotto, Bernardo Galvano, Ehsan Shokrolahzade, Emanuele Enrico, Faisal A. Mubarak, Luca Fasolo, Luca Oberto, Marco Spirito, Paolo Terzi.

Figure 1
Figure 1. Figure 1: Schematic of the INRIM cryogenic S-parameters measurement setup in which 0 dB blocks are simple SMA adaptors, ISs are 60 dB isolators, DCs are directional couplers, BT are bias-tees, SWs are electromechanical SP6T switches. Mixing Chamber plate 3K plate Room Temperature interface 0 dB -10 dB 0 dB 0 dB 0 dB 0 dB A1 B1 IS1 HEMT1 -10 dB DC1 BT1 LNA1 C-SPSP-NbTi SC-119/50-CN-CN SW1 DUT SOLR Cal Kit 0 dB -10 dB… view at source ↗
Figure 2
Figure 2. Figure 2: Resistance versus temperature of the load standard for different power levels B. Calibration standards behavior at RT The EM response of a 3D model of the standard artifacts is then benchmarked at RT by direct comparison with traceable measurement. This comparison highlighted an accurate model￾to-hardware correlation as shown in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Behavior of the 3D EM model of the Short, Open and Load standards with respect to actual RT measurements. 4 6 8 10 12 Freq / GHz -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Short and Open: |S11| / dB -50 -40 -30 -20 -10 0 Load: |S11| / dB Short Meas. Short Model Open Meas. Open Model Load Meas. Load Model [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Nominal response shift computed as |S11cryo-S11RT| dB obtained by the EM simulations of the 3D models of the various calibration artifacts. obtained from measurements or other sources maintaining SI traceability, the resulting model response is a best effort to preserve that traceability. Moreover, the shift in response from RT to cryo can be employed as an uncertainty expansion that preserves a link to tr… view at source ↗
Figure 4
Figure 4. Figure 4: Absolute geometrical contraction of the open standard device due to thermal shift from 293.15 K (RT) to 45 mK relative to device center [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: and 9 show RT (orange) and mK (blue) contribution of Drift in the reflection (Directivity) and in the transmission (Tracking) coefficients. Drift effects are dependent on the time elapsed after the calibration of the VNA and represent the drift of the measurement results due to the VNA itself, environmental factors, measurement setup and operating conditions. The effect of temperature drift is, therefore, … view at source ↗
Figure 9
Figure 9. Figure 9: Contribution of the Tracking to the S-parameters magnitude (a) and phase (b). In orange the VNA only (at RT) and in blue the whole cryogenic system (at mK). step attenuator, and it is considered frequency-independent [21, Annex G2]. Unfortunately, this method is not feasible within a cryostat. Anyway, modern VNAs are highly linear, and their linearity error is likely negligible compared to the error introd… view at source ↗
Figure 11
Figure 11. Figure 11: illustrates the impact of switch port repeatability and port path differences on S11 and S21 measurements. Fig. 11a shows the standard deviation of the magnitude of S11 for 30 measurements for each channel (switch ports), while Fig. 11b shows S21 of each switch channel and the corresponding maximum difference (right axis). Data for port 5 are not reported, as this switch port was non-functional due to a h… view at source ↗
Figure 10
Figure 10. Figure 10: illustrates the linearity contribution as a function of both frequency and power. As shown, it is no longer frequency independent [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Calibrated (orange curve) and uncalibrated (blue curve) measurement of S21 of a 20 dB attenuator used as DUT. of the Load calibration standard, the contribution of the switches, and the evaluation of linearity. The noise floor and trace noise may also have a significant impact, while system drift remains generally negligible. However, performing the measurements takes a long time due to the necessary wait… view at source ↗
Figure 13
Figure 13. Figure 13: Difference Δ|S21| = |S21|Thru - |S21|SOLR, expressed in dB, between the |S21| values of the 20 dB attenuator DUT obtained at cryogenic temperature by Thru normalization and by SOLR calibration. only. Therefore, it cannot provide the complete S-parameter characterization ensured by SOLR calibration. Phase-resolved characterization is often important for the extraction of circuit parameters of DUTs, as illu… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison between RT (red curve) and mK (blue curve) measurements of a) XMA cryogenic Open device, and b) XMA cryogenic Short device. a) b) [PITH_FULL_IMAGE:figures/full_fig_p008_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Reciprocity-based plausibility test for the calibrated transmission response of the DUT. Comparison between the measured forward and reverse transmission coefficients shows deviations below 0.0025 in linear dimensionless units. The developed system leverages direct access to VNA receivers, low-cost commercially available calibration standards, and electromechanical switching. The databased definition of c… view at source ↗
read the original abstract

This paper describes the developed setup and characterization approach for full two-port calibrated S-parameter measurements at cryogenic temperatures, together with a complete uncertainty budget. The system developed at the Istituto Nazionale di Ricerca Metrologica (INRiM, Italy), exploits the Short-Open-Load-Reciprocal technique to realize error-corrected cryogenic measurements within a single cooling cycle. The system operates down to mK temperatures over the 4-12 GHz band in coaxial line. Calibration standards are referenced to SI-traceable room-temperature measurements, while a numerical approach is used to evaluate the response shift of the artifacts upon cooling and to derive an additional cryogenic uncertainty contribution for the measurement uncertainty budget. Moreover, relevant measurement uncertainty contributions are evaluated according to internationally agreed procedures, and a comprehensive uncertainty budget is presented. Test measurements on a 20 dB attenuator are shown as an example. An attenuation of 20.70 +/- 0.08 dB (95% confidence interval) was obtained at 6 GHz. Full SI-traceable verification at mK temperatures remains an open challenge; however, an initial calibration verification is also presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a calibration strategy for full two-port S-parameter measurements at millikelvin temperatures (down to mK) over 4-12 GHz in coaxial line, using the Short-Open-Load-Reciprocal (SOLR) technique within a single cooling cycle. Room-temperature SI-traceable characterizations of the standards are combined with a numerical model to quantify response shifts upon cooling, which is incorporated into a comprehensive uncertainty budget following international procedures. A test result on a 20 dB attenuator at 6 GHz is reported as 20.70 ± 0.08 dB (95% confidence), with the authors explicitly noting that full SI-traceable verification at mK temperatures remains an open challenge.

Significance. If the numerical model for cryogenic response shifts is shown to capture relevant effects without significant bias, the work provides a practical, single-cycle approach to error-corrected S-parameter measurements that is valuable for quantum device characterization and low-temperature metrology. Strengths include the concrete numerical example with stated uncertainty, the explicit uncertainty budget, and honest acknowledgment of verification limitations; these elements support reproducibility and metrological transparency in the field.

major comments (2)
  1. [Numerical model for response shift] Section describing the numerical approach to evaluate the response shift of the artifacts upon cooling: The derivation of the additional cryogenic uncertainty contribution rests on this model. The manuscript does not specify which physical effects (e.g., frequency-dependent connector repeatability, dielectric loss variations, or possible superconducting transitions in the coaxial line) are included versus omitted, nor does it provide cross-validation against direct cryogenic measurements of the same standards. This is load-bearing for the central claim because an incomplete model could systematically underestimate the reported ±0.08 dB uncertainty at 6 GHz.
  2. [Test measurements] Test measurements and uncertainty budget section: The reported attenuation result of 20.70 ± 0.08 dB for the 20 dB attenuator at 6 GHz is presented as an example, but without independent cryogenic verification of the standards or explicit propagation steps showing how the numerical shift term combines with other contributions, the budget's completeness cannot be fully assessed.
minor comments (2)
  1. [Abstract] Abstract: The phrasing 'a numerical approach is used to evaluate the response shift... and to derive an additional cryogenic uncertainty contribution' would benefit from a single sentence summarizing the main assumptions of the model to aid reader assessment.
  2. Figure captions and legends: Ensure all plots of S-parameters or uncertainty components include explicit axis labels, units, and identification of which trace corresponds to the cryogenic correction term.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments and for recognizing the practical value of the single-cycle SOLR approach with SI-traceable room-temperature references. We respond point-by-point to the major comments below, providing clarifications on the numerical model and uncertainty budget. Where the comments identify areas for improved transparency, we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Section describing the numerical approach to evaluate the response shift of the artifacts upon cooling: The derivation of the additional cryogenic uncertainty contribution rests on this model. The manuscript does not specify which physical effects (e.g., frequency-dependent connector repeatability, dielectric loss variations, or possible superconducting transitions in the coaxial line) are included versus omitted, nor does it provide cross-validation against direct cryogenic measurements of the same standards. This is load-bearing for the central claim because an incomplete model could systematically underestimate the reported ±0.08 dB uncertainty at 6 GHz.

    Authors: We agree that explicit specification of the modeled physical effects is required for metrological rigor. In the revised manuscript we will expand the relevant section to list the effects included in the numerical model: thermal contraction of the inner and outer conductors, temperature dependence of the dielectric permittivity, and effective electrical-length changes arising from connector interface variations. Effects omitted from the model include frequency-dependent connector repeatability beyond the length-shift term, cryogenic dielectric-loss variations (bounded conservatively from room-temperature data), and superconducting transitions (not applicable for the coaxial materials and frequency range employed). Direct cross-validation against separate cryogenic measurements of the standards was not performed, consistent with the single-cycle design; instead, an additional uncertainty term derived from the model sensitivity analysis is included in the budget. We will also add a short discussion of possible residual model bias and its contribution to the stated uncertainty. revision: yes

  2. Referee: Test measurements and uncertainty budget section: The reported attenuation result of 20.70 ± 0.08 dB for the 20 dB attenuator at 6 GHz is presented as an example, but without independent cryogenic verification of the standards or explicit propagation steps showing how the numerical shift term combines with other contributions, the budget's completeness cannot be fully assessed.

    Authors: We will revise the uncertainty-budget section to include the explicit propagation steps, showing how the numerical cryogenic-shift contribution is combined with the room-temperature calibration uncertainties, connector repeatability, and noise terms according to the GUM law of propagation of uncertainty. The 20.70 ± 0.08 dB result is presented strictly as an example application of the calibrated system; the manuscript already states that full independent cryogenic verification of the standards remains an open challenge and is outside the scope of this work. revision: yes

standing simulated objections not resolved
  • Direct experimental cross-validation of the numerical model via separate cryogenic measurements of the calibration standards, which would require additional thermal cycles incompatible with the single-cycle strategy.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external SI-traceable references and standard SOLR procedure.

full rationale

The paper's central method applies the established Short-Open-Load-Reciprocal (SOLR) calibration with standards tied to independent room-temperature SI-traceable measurements. The numerical evaluation of artifact response shift upon cooling is described as a separate modeling step used only to augment the uncertainty budget, not as a fit that re-derives the target S-parameter values from the same data. The reported attenuator result (20.70 +/- 0.08 dB) is presented as an outcome of the calibrated measurement rather than a quantity that reduces by the paper's own equations to a parameter defined in terms of itself. No self-citation chains, ansatz smuggling, or uniqueness theorems imported from prior author work appear as load-bearing elements in the provided derivation. The approach remains self-contained against external metrological benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of the numerical model for temperature-induced changes in calibration artifacts and the assumption that room-temperature SI-traceable references remain valid anchors after cooling; these are domain assumptions rather than derived results.

free parameters (1)
  • additional cryogenic uncertainty contribution
    Derived from numerical evaluation of artifact response shift upon cooling and added to the overall budget
axioms (2)
  • domain assumption The Short-Open-Load-Reciprocal (SOLR) technique can be applied to realize error-corrected measurements at cryogenic temperatures
    Exploits SOLR to realize error-corrected cryogenic measurements within a single cooling cycle
  • domain assumption Room-temperature measurements of calibration standards are SI-traceable and can be extrapolated via numerical modeling to cryogenic conditions
    Calibration standards are referenced to SI-traceable room-temperature measurements, with numerical approach for response shift

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