A macroscopic mass from quantum mechanics in an integrated approach

Following the 2019 SI revision, electrical and mechanical quantities can be directly linked via defining constants (e and h), enabling mass to be realized electrically. Historically, the Kibble balance realized the kilogram by using quantum electrical standards indirectly through artifact resistors, requiring separate experiments to realize the von Klitzing constant. This study addresses that gap by integrating two quantum standards—the Josephson voltage standard and graphene quantum Hall array resistance standards—directly within a single Kibble balance experiment. The purpose is to directly relate a macroscopic mass to quantum standards by passing the weighing current through a quantum Hall array while comparing the resulting quantum Hall voltage to a Josephson voltage, thereby simplifying the traceability chain and reducing reliance on artifact resistors. The significance lies in consolidating mechanical and quantum electrical metrology, potentially democratizing access to quantum-traceable currents and enabling more direct closures of the metrology triangle.

Key developments enabling this work include: (1) the prediction and use of the Josephson effect (1962) and the discovery of the quantum Hall effect (1980), which provided highly stable voltage and resistance quantum standards; (2) the Kibble balance technique for mass realization from electrical power; (3) progress in single-electron devices for realizing the ampere, though presently limited to nanoampere currents, reducing practical applicability; and (4) advances in graphene-based quantum Hall resistance standards and quantum Hall array resistance standards (QHARS). Previous QHARS efforts were constrained by limited allowable current before breakdown. Epitaxial graphene on SiC has emerged as a robust platform with higher operating currents and relaxed cryogenic/magnetic requirements compared to GaAs devices. Past practice in Kibble balance experiments used separately realized resistance standards (wire or thin-film resistors) calibrated against GaAs QHR; the present work integrates graphene QHARS directly into the balance to serve as the current-to-voltage conversion standard, allowing a direct Josephson–QHE comparison during mass levitation.

Overview: The experiment integrates three NIST-built components: (i) a programmable Josephson voltage standard (PJVS) to realize voltage from a microwave frequency referenced to the Cs hyperfine transition; (ii) graphene quantum Hall array resistance standards (QHARS) to convert current to a quantized Hall voltage; and (iii) the NIST-4 Kibble balance to realize mass via the equality of virtual mechanical and electrical power. Programmable Josephson voltage standard (PJVS): A Josephson junction array converts a microwave frequency v to a dc voltage U = ku kv vCs with n junctions in series (ku = n), where KJ = 2e/h. The NIST PJVS used has 67,406 junctions in programmable segments following a ternary sequence, operated nominally at 18.2 GHz with ±10 MHz tuning, enabling up to 2.5 V output with nanovolt-level uncertainty. Graphene QHARS: Epitaxial graphene was grown on the Si face of 4H-SiC(0001) at 1900 °C in Ar using polymer-assisted sublimation growth. Device quality was assessed by confocal laser scanning microscopy. Fabrication included Pd/Au protection, Ar plasma etching into Hall bars, NbTiN superconducting contacts, Cr(CO)3 functionalization for homogeneous, tunable carrier density, and Delahaye multiple-series interconnections with split-contact geometry. Arrays were designed with i = 2 (filling factor) and j = 13 parallel Hall elements, giving R = RK/(i j) = RK/26 (~1 kΩ), facilitating 10:1 scaling by room-temperature direct current comparators. Four QHARS devices (QHARS1-D1, QHARS1-D2, QHARS2-D1, QHARS2-D2) were operated in a liquid He bath at T = 1.6 K and |B| = 9 T. Quantization verification used a cryogenic current comparator (CCC) bridge with multiple 100 Ω standards (periodically calibrated vs GaAs QHR) to measure RH vs source-drain current, plus magnetic field reversal tests to confirm absence of dissipation-related offsets. Kibble balance (NIST-4): The balance employs two modes. In force mode, mg = −N(∂Φ/∂z) I, equating weight to electromagnetic force. In velocity mode, a controlled coil motion through the same magnetic field induces U = −N(∂Φ/∂z) vz. Combining yields mg vz = U I, an equality of virtual mechanical and electrical power measured in separate phases. The weighing current I is determined by passing it through a resistor R and measuring U across it, so mg vz = U²/R. Here, R is the quantized Hall resistance of the QHARS, and U is realized by the PJVS, establishing a direct link to h via Josephson and QHE relations. The NIST-4 setup features a wheel-guided vertical coil motion in a radial Sm2Co17 magnet system, with velocity measured by heterodyne Michelson interferometers. A low-noise 30-bit programmable current source (battery powered, optically controlled) supplies current with <100 pA/√Hz at 1 Hz for 10 mA. Mass measurement protocol: Primary measurements were performed on 50 g and 100 g stainless steel masses. For 50 g and 100 g realizations, the balance required weighing currents of 0.35 mA and 0.7 mA, respectively, sourced through the QHARS. The experiment was conducted under high vacuum (~10⁻⁴ Pa). The electrical path alternated between (a) each of the four graphene QHARS and (b) artifact standard resistors (100 Ω and 1000 Ω, in air or oil baths) calibrated against a GaAs QHR, chosen for low temperature and power coefficients. Drift corrections (temporal drift, temperature, and power coefficients per Table 2) were applied to measurements with standard resistors. Quantization of QHARS was verified across operating currents, with field reversal measurements to bound systematic offsets. During the 100 g run, an emergency vacuum interruption induced additional drift, increasing scatter for certain datasets (notably QHARS1-D1 and 100 Ω in oil).

- Demonstration of a single, integrated experiment directly linking a macroscopic mass to two quantum electrical standards: the Josephson effect (voltage) and the quantum Hall effect (resistance), by passing the weighing current through graphene QHARS while comparing to a PJVS. - Robust quantization of graphene QHARS at T = 1.6 K and |B| = 9 T over operating currents used for mass realization (0.35 mA, 0.7 mA). Weighted deviations from RK/26 (k = 1): QHARS1-D1: −1.78 ± 0.29 nΩ/Ω; QHARS1-D2: −6.00 ± 0.51 nΩ/Ω; QHARS2-D1: +0.12 ± 0.28 nΩ/Ω; QHARS2-D2: −3.19 ± 1.00 nΩ/Ω. - Group average deviation when referenced via artifact standards: (−2.72 ± 0.31) × 10⁻⁹, attributed to resistor stability, cable insulation leakage, and CCC balance electronics. - Primary mass realizations for 50 g and 100 g achieved relative standard uncertainties (k = 1) on the order of 3 × 10⁻⁸. - Deviations between QHARS and traditional resistor-based realizations: for 100 g, (−21.5 ± 12.8) × 10⁻⁹; for 50 g, (−2.6 ± 20) × 10⁻⁹. After omitting data from a leaky 1000 Ω oil resistor, the 100 g deviation improves to (−3.7 ± 14.4) × 10⁻⁹. - Operational currents through QHARS of 0.35 mA and 0.7 mA validate higher-current capability of graphene arrays relative to prior QHARS, supporting practical quantum-traceable current sourcing. - The experiment advances closure prospects for a metrology triangle formed by the Kibble balance, Josephson effect, and quantum Hall effect (with noted scaling factors for the 100 g case: kR = 1/26, ku = 18,350, kv = 477,100).

The experiment validates the practical use of virtual quantum current sources at milliampere levels, directly traceable to the revised SI via the Josephson and quantum Hall effects. By integrating QHARS within the Kibble balance current path and directly comparing the quantum Hall voltage to a Josephson voltage, the work removes intermediate artifact resistors from the measurement chain, thereby simplifying traceability and potentially improving robustness. The demonstrated quantization at operating currents and the achieved relative uncertainties (~3 × 10⁻⁸) for 50 g and 100 g masses indicate that graphene QHARS can support precision mass metrology. In the broader context, the approach brings the quantum ampere closer to wide deployment by enabling larger, stable currents. The study also situates the results within a metrology triangle linking mass realization (Kibble balance) with the Josephson and quantum Hall standards. Although international dissemination currently relies on a consensus kilogram (with only minor correlation to NIST-4 data), the integrated method provides a pathway to triangle closure by comparing Kibble-balance-derived masses to independent realizations (e.g., XRCD) that do not involve the same quantum electrical standards. Practical limitations in traditional scaling (e.g., from Pt-Ir 1 kg artifacts to stainless steel masses) presently limit the uncertainty of such a comparison, but as dissemination shifts toward purely Planck-constant-based realizations, direct closure becomes more feasible.

By incorporating Josephson voltage and graphene quantum Hall resistance standards directly into the NIST-4 Kibble balance, the study demonstrates primary realizations of 50 g and 100 g masses with relative uncertainties on the order of 3 × 10⁻⁸, while reducing reliance on artifact resistors. The results highlight the capability of graphene QHARS to operate at higher currents with robust quantization and establish a direct connection between macroscopic mass and quantum electrical standards within a single experiment. This integrated approach simplifies the calibration chain and supports broader dissemination of quantum-traceable current and mass standards. Future work includes removing remaining artifact elements from verification steps, improving resistance leakage controls, and pursuing closure of the metrology triangle via independent mass realization methods such as XRCD.

- A 1000 Ω standard oil resistor exhibited resistance leakage discovered post-campaign, contributing to a larger deviation in the 100 g results; excluding this dataset improved agreement. - An emergency vacuum interruption during the 100 g measurement introduced additional drift, increasing data scatter for certain measurements (e.g., QHARS1-D1 and 100 Ω in oil). - Verification of QHARS quantization relied on artifact resistor comparisons (100 Ω standards), contributing a non-zero group average deviation (−2.72 ± 0.31) × 10⁻⁹ due to resistor stability, insulation leakage, and CCC electronics. - Full closure of the metrology triangle was not achieved in this work; current practical limitations include uncertainty introduced by traditional scaling from Pt-Ir 1 kg artifacts to stainless steel masses when comparing to the international consensus kilogram. - Quantization and current range, while improved with graphene, remain bounded by device breakdown constraints and the PJVS maximum voltage (2.5 V) for higher masses/currents.