Research spotlight: Grzegorz Szklarz

One-tenth of a kelvin reads like a rounding error in most laboratories. At 4 kelvin it is a 2.5 percent error in a material’s internal energy — roughly the equivalent of misreading room temperature by 3 degrees — and the authors of a new calibration study argue that a deviation of that size can push a quantum system out of coherence. That is the precision problem a team at the Institute of Low Temperature and Structure Research in Wrocław, Poland, set out to tame.

The group, led by Grzegorz Szklarz of the institute’s Temperature Standard Laboratory, has published an analytical framework for calibrating rhodium-iron (RhFe) and platinum-cobalt (PtCo) resistance thermometers below 25 kelvin. The work appears in the journal Measurement, with Ruslan Nikonkov and Aleksandra Kowal as co-authors.

The timing matters. Quantum computers and superconducting electronics, along with space and particle-physics detectors, all operate in this deep-cryogenic range. Since 2019 the kelvin has been defined through a fixed value of the Boltzmann constant, which has pushed national laboratories to prove that temperature measurements from 4 K to 300 K are genuinely traceable to thermodynamics. A sensor whose calibration curve wobbles quietly undermines that entire chain, no matter how precise the hardware around it.

A sensor that reports resistance, not temperature

A capsule resistance thermometer is a piece of alloy wire inside a sealed case, and its raw output is a number in ohms. Someone has to write down the rule that converts that resistance into temperature. Above about 14 K, standard platinum thermometers handle the job with a nearly linear response. Below that, their sensitivity fades, so metrologists turn to alloy sensors. Rhodium-iron thermometers, which typically contain about 0.5 atomic percent iron, have been in use since the 1970s. Platinum-cobalt is a newer alternative valued for how reproducibly it can be manufactured.

Below 25 K, that conversion rule bends sharply. Both alloys respond in a strongly non-linear way as the temperature drops toward the triple point of neon at 24.5561 K and beyond. Calibration has traditionally meant fitting a polynomial through the measured points. A high-degree polynomial can be forced through almost any data, but it tends to wobble between the points — an oscillation named after the mathematician Carl Runge — and it can chase noise rather than the sensor’s true behaviour, a failure known as overfitting. The paper notes that conventional RhFe calibration has required more than 30 measurement points, partly because choosing a model and removing outliers become tangled together.

Think of the fitting process as a suspension problem. A high-degree polynomial acts like a rigid axle that jolts over every bump in the data. A rational function, which is one polynomial divided by another, adds a denominator that behaves like a spring, absorbing the sharp changes in sensor sensitivity as the temperature nears absolute zero.

What the Python framework does

The code starts from the messiest stage of the process — raw resistance readings that arrive with noise and drift, along with the occasional outright bad point. It ingests those readings, detects statistical outliers, fits candidate models, and scores each candidate with several diagnostics rather than a single fit statistic. The authors describe the shift as moving from simple goodness-of-fit checks toward evidence that the model is physically coherent.

• Information criteria (AIC and BIC) that penalize a model for spending extra parameters just to track noise
• The Durbin-Watson test, which looks for autocorrelation — hidden structure — in the residuals left after fitting
• Iterative outlier detection that refines the model and the rejected points together
• Cross-validation against independent heating and cooling runs

Two of those tools deserve plain-language treatment. The Akaike information criterion and the Bayesian information criterion both ask whether a fancier curve earns its extra parameters; if it does not, the score tells you to prefer the simpler one. The Durbin-Watson test inspects the residuals, the differences between measured points and the fitted curve, to see whether they leave a pattern that the model failed to capture.

The design targets a circular problem the authors lay out in the paper’s introduction. You cannot reliably spot outliers without a reasonable preliminary model, yet the model choice is corrupted by the same outliers you are trying to find. A one-pass script cannot escape that loop, which is why the framework refines model and outlier set together, iteratively.

To verify the tool, the team ran it on measurements from 17 sensors spanning platinum, PtCo, and RhFe thermometers that had passed through multiple laboratories, all calibrated below 25 K. The paper, published in Measurement, is explicit that the contribution is the methodology itself, not the ranking of one manufacturer’s device against another.

Photo by Charly Álvarez on Unsplash

Why rational functions won

Across the 17 sensors, one family of curves kept coming out ahead. Rational functions described the RhFe and PtCo sensors better than the high-degree polynomials that remain common practice. They delivered higher numerical stability and smaller residuals while tracking the physical changes in sensor sensitivity more faithfully. They also sidestep Runge’s oscillation, which can make a polynomial look accurate on a plot while being wrong in the gaps between calibration points.

The authors say the result points toward something metrology laboratories have wanted for years: a single, universal calibration function for RhFe and PtCo thermometers that spans from the boiling point of helium to the triple point of water. Wide-range work today often means stitching several functions together, and each join is a place where error can hide.

In the discussion, Szklarz and his co-authors frame the stakes directly: modern quantum technologies require moving “beyond simplistic calibration routines toward evidence-based, automated procedures.” A version of the software, with documentation and example datasets, is archived on Zenodo as PrecisionThermometryFramework.

Where caution is warranted

The study is a methodology validation, not a declaration that one universal equation has been proven. The authors are explicit that they tested the framework on the pooled DireK-T dataset and are not presenting the absolute performance of any single sensor as the final word.

Rational functions carry a known caveat. “A rational function is an excellent interpolator, but its denominator can behave unpredictably outside the range where it was fitted,” said an independent metrologist not involved in the work. “The meaningful test is whether another laboratory, with its own cryostat and its own dataset, reproduces the result.”

That caution sits comfortably with the authors’ own framing. They present the framework as a standardized, statistically rigorous foundation and a step toward future machine-learning-driven optimization of calibration protocols, not as a finished universal model.

The European metrology backdrop

The work was conducted as part of European Partnership in Metrology project 22IEM02, “Dissemination of the redefined kelvin,” known as DireK-T. Coordinated through EURAMET and co-financed by the European Union’s Horizon Europe programme and participating states, the project runs from 2023 to 2026 and targets an uncertainty of 0.3 millikelvin for thermodynamic temperature dissemination between 4 K and 25 K.

Since the 2019 redefinition tied the kelvin to the Boltzmann constant, national metrology institutes have been working to turn thermodynamic temperature from a paper ideal into a practical, transferable measurement. INTiBS, the Wrocław institute where Szklarz works, calibrated the project’s capsule thermometers across the 4 K to 25 K range. The institute’s own summary of the work is available on its scientific news page, and the broader DireK-T design is described in an open overview paper.

Photo by sidi alexx on Unsplash

What the next decade could look like

If rational-function calibration catches on, the near-term prize is a single function that carries a RhFe or PtCo thermometer from helium’s boiling point to water’s triple point, eliminating the spliced curves that complicate wide-range instruments. The authors suggest their standardized evaluation metric also gives machine-learning-driven calibration a foundation to build on, which could turn the iterative model-plus-outlier loop into something a computer explores automatically.

For a quantum hardware startup or a national metrology institute, that means fewer hidden calibration errors and an easier defence of every millikelvin when results are audited. The harder test, cross-laboratory reproducibility, is now the one that matters most — and it is precisely what the DireK-T comparison network was built to provide.