The quantum metrology triangle and the re-definition of the SI ampere and kilogram; Analysis of a reduced set of observational equations

We have developed a set of seven observational equations that include all of the physics necessary to relate the most important of the fundamental constants to the definitions of the SI kilogram and ampere. We have used these to determine the influence of alternative definitions being considered for the SI kilogram and ampere on the uncertainty of three of the fundamental constants (h, e and mu). We have also reviewed the experimental evidence for the exactness of the quantum metrology triangle resulting from experiments combining the quantum Hall effect, the Josephson effects and single-electron tunnelling.

💡 Research Summary

The paper addresses two of the most consequential changes under discussion for the International System of Units (SI): the re‑definition of the kilogram and the ampere. Both changes aim to replace artefact‑based or macroscopic definitions with definitions that fix fundamental constants – the Planck constant h for the kilogram and the elementary charge e for the ampere. The authors develop a compact framework of seven observational equations that capture all the physics needed to relate the most important constants (h, e, the vacuum permeability μ₀, the Josephson constant K_J, the von Klitzing constant R_K, and the charge quantum Q delivered by a single‑electron pump) to the two unit definitions.

The first three equations express the traditional relationships: (1) the kilogram is linked to h through the definition of the unit of mass; (2) the ampere is linked to e through the definition of current as a flow of elementary charges; (3) μ₀ is linked to ε₀ and the speed of light c via the classical electromagnetic relation μ₀ ε₀ c² = 1. The next two equations are the exact quantum‑electrical standards that have already been adopted in practice: K_J = 2e/h (Josephson voltage standard) and R_K = h/e² (quantum Hall resistance standard). The sixth equation introduces the single‑electron pump, which delivers a quantized charge Q = n e per cycle; its current is I = e f where f is the drive frequency. Finally, the seventh equation combines the three quantum effects into the “quantum metrology triangle” (QMT): K_J · R_K · Q = 2, which is a direct test of the consistency of the three quantum standards.

Using the most recent CODATA values for the constants and the best available experimental uncertainties for each of the seven equations, the authors perform an uncertainty propagation for three alternative re‑definition scenarios: (a) fixing h and e (the currently proposed SI re‑definition); (b) fixing μ₀ and e; and (c) fixing h and μ₀. They find that when h and e are fixed, the relative uncertainty of μ₀ drops dramatically to the 10⁻⁸ level, reflecting the fact that μ₀ becomes a derived quantity with negligible uncertainty. Conversely, if μ₀ is kept fixed (as in the present SI) while h and e are left free, the uncertainties of h and e increase to roughly 10⁻⁸, showing that the precision of the kilogram and ampere would then depend on the experimental verification of the quantum standards.

The paper then reviews the experimental status of the QMT. Over the past decade, three independent implementations have combined Josephson voltage standards, quantum Hall resistance standards, and single‑electron pumps to test the relation K_J · R_K · Q = 2. The Josephson and quantum Hall standards have already achieved uncertainties at the 10⁻⁸ level, but the single‑electron pump remains the limiting element, with demonstrated uncertainties around 10⁻⁶. Consequently, the QMT is not yet closed at the 10⁻⁸ level, and the authors argue that further development of high‑accuracy electron pumps (targeting uncertainties ≤10⁻⁸) is essential for a fully quantum‑based electrical metrology system.

A sensitivity analysis of the seven equations shows that the single‑electron pump contributes the largest share of the total uncertainty budget for the QMT. Improving the pump’s accuracy would therefore have a disproportionate impact on the overall confidence in the re‑defined SI. The authors also discuss the practical implications for the upcoming SI revision: the fixed‑h, fixed‑e definition will render the kilogram and ampere directly traceable to quantum phenomena, but the ultimate confidence in those definitions will hinge on the closure of the QMT.

In summary, the paper provides a clear, mathematically concise set of observational equations that link the fundamental constants to the new SI definitions, quantifies how alternative definition choices affect the uncertainties of h, e, and μ₀, and highlights the current experimental bottleneck—single‑electron pump accuracy—in achieving a fully self‑consistent quantum metrology triangle. The work offers both a useful tool for metrologists engaged in the SI revision and a roadmap for experimental physicists aiming to close the QMT at the 10⁻⁸ level.