The SI second is expected to be redefined within the next decade, and one candidate ('option 2') fixes a defining constant NN equal to the weighted geometric mean of several optical clock transition frequencies. Fang, Wang, Zuo, and Dai (arXiv:2603.14858, 2026) derived how a single laboratory can realize NN from asynchronous optical clocks referenced to a hydrogen-maser flywheel (a continuously running, stable-but-drifting oscillator that bridges intervals when no optical clock is being compared), using geometric- or arithmetic-mean reconstruction routes and a time-segmented coefficient/covariance-matrix combination, and their conclusions identify three realization-level directions left open for future work: (1) the covariance formalism does not incorporate correlations between different reference standards (e.g., flywheel oscillators sharing calibration chains or common references); (2) the combination weights are fixed rather than adapted to measured campaign performance, nonstationary noise, and evolving recommended frequencies; (3) the methodology is single-laboratory and has not been extended to multi-laboratory optical clock networks. This record states the resulting open problem: a realization of NN whose total uncertainty is kept as low as practicable and is not underestimated under real-world operational constraints, including inter-laboratory operation.

Background

The SI second is currently defined by the hyperfine transition of the cesium-133 atom in the microwave domain, and the best cesium fountain realizations (clocks in which laser-cooled cesium atoms are launched repeatedly through a microwave cavity) reach fractional frequency uncertainties of order 10−1610^{-16}. Optical atomic clocks, based on ultra-narrow transitions in the visible/near-infrared, now reach uncertainties of 10−1810^{-18} and below, and the 27th General Conference on Weights and Measures (CGPM, the body that decides SI revisions) has resolved to replace the cesium-based definition, with a redefinition expected around 2030 (Dimarcq et al. 2024). Because many atomic species are serious contenders, one proposed definition ('option 2', Lodewyck 2019) avoids choosing a single transition: it fixes, as a defining constant with no uncertainty, the number N=∏i=1nviwiN=\prod_{i=1}^{n} v_i^{w_i}, the weighted geometric mean of the frequencies viv_i of nn selected clock transitions, with normalized weights wiw_i (∑iwi=1\sum_i w_i=1) published by the International Committee for Weights and Measures (CIPM), either as fixed values or updated according to the latest frequency-ratio measurements. Individual transitions keep associated uncertainties through their recommended values, which the CIPM derives from a global least-squares adjustment of all published absolute-frequency and frequency-ratio measurements (Margolis et al. 2024); a frequency ratio is the dimensionless quotient of two transition frequencies, measurable directly between two clocks without cesium. A transition used in this way at the realization level parallels the established category of secondary representations of the second (SRS), i.e., transitions with recommended values that may serve to realize the second besides the primary cesium definition.

Realizing NN in practice is nontrivial: a laboratory typically cannot measure all defining transitions; its clocks differ in Type A (statistical, averaging-time-dependent) and Type B (systematic, evaluation-limited) uncertainties; operations are asynchronous with dead time (intervals when a clock is not running), commonly bridged by a hydrogen maser acting as a flywheel — a continuously running, stable-but-drifting oscillator to which the optical clocks are compared; and measurements share systematic effects, so their uncertainties are correlated. Fang et al. (2026) derived consistent uncertainty expressions for two reconstruction routes (forming a weighted geometric mean of measured frequencies supplemented where necessary by recommended values or ratios, versus a weighted arithmetic mean of per-clock estimates of NN), identified in analytic three-transition case studies when each route gives lower total uncertainty, and introduced a time-segmented, time-weighted combination based on coefficient and covariance matrices (matrices that encode, per measurement interval, which clocks operated and how interval results correlate) that accounts for dead time and maser instability. Related single-institution demonstrations of optical-clock timescales with maser flywheels include Grebing et al. (2016) and Yao et al. (2019). The framework of Fang et al., however, is restricted to a single laboratory, uses fixed weights, and omits correlations between distinct reference standards. At the definition level, Lodewyck and Ido (2025) reviewed properties of the multi-transition definition and introduced a quantitative method to set the weights, but did not treat realization-level adaptive weighting, reference-standard correlations, or network operation.

Problem Statement

Extend the single-laboratory realization methodology of Fang et al. (2026) for the option-2 defining constant N=∏i=1nviwiN=\prod_{i=1}^{n} v_i^{w_i} (weighted geometric mean of clock transition frequencies) into a realization that is valid and uncertainty-honest under real-world operational constraints. The common objective is a multi-clock — ultimately multi-laboratory — realization of NN whose total uncertainty is kept as low as practicable and is not underestimated, suitable for calibrating International Atomic Time (TAI, the continuous weighted-average atomic timescale computed by the International Bureau of Weights and Measures from worldwide clock data) and steering local time scales. Three directions for future work identified in the source paper's conclusions:

  1. Correlated reference standards. Incorporate correlations between different reference standards — e.g., flywheel oscillators (hydrogen masers, i.e., continuously running reference oscillators bridging optical-clock dead time) calibrated against the same primary frequency standard, or shared references across laboratories — into the coefficient/covariance-matrix formalism, and derive consistent uncertainty-propagation rules for recommended frequencies and frequency ratios when such correlations are present, so that total realization uncertainty is not underestimated.

  2. Adaptive weighting. Develop data-driven or adaptive weighting schemes for the realization-level combination weights (e.g., the arithmetic-combination weights and the interval weights of the time-segmented combination) in response to measured campaign performance, nonstationary noise, and evolving recommended-frequency/ratio information, keeping the realization optimal over time rather than tied to fixed initial combination weights. Because NN and the defining weights wiw_i are defining constants fixed by the CIPM — which may itself update them as frequency-ratio measurements evolve — the adaptive weighting concerns how measurements are combined to realize a given defined NN, not a laboratory redefining NN locally.

  3. Multi-laboratory extension. Extend the time-segmented, time-weighted reconstruction methodology to multi-laboratory optical-clock networks — clocks at different sites with their own reference standards, differing performance, uptime, correlation structures, and time-transfer (fiber/satellite link) uncertainties — so that the methodology applies to inter-laboratory realizations of NN.

These three directions together define the open realization-level problem: an uncertainty-honest, ultimately distributed realization of the geometric-mean second. A complete realization requires all three, yet each direction is independently attackable and has its own minimal bar: for direction 1, a covariance treatment that keeps the realization uncertainty consistent — in particular not underestimated — when at least two reference standards are correlated; for direction 2, an adaptive weighting rule shown to match or beat the fixed weights under nonstationary noise; for direction 3, a reconstruction in which cross-site correlations and time-transfer uncertainties are propagated through to the realization uncertainty. A self-contained result that meets the bar for one direction is a valid contribution to the joint objective.

References

  1. Fang Fang, Chaowei Wang, Yani Zuo, Shaoyang Dai, 'Realization of the SI Second Defined by Geometric Mean of Multiple Clock Transitions', arXiv:2603.14858 (2026), doi:10.48550/arXiv.2603.14858, https://arxiv.org/abs/2603.14858
  2. Jérôme Lodewyck, 'On a definition of the SI second with a set of optical clock transitions', Metrologia 56, 055009 (2019), doi:10.1088/1681-7575/ab3a82, https://doi.org/10.1088/1681-7575/ab3a82
  3. Jérôme Lodewyck, Tetsuya Ido, 'Properties of a definition of the SI second based on several optical transitions', Metrologia 62, 065003 (2025), doi:10.1088/1681-7575/ae033f, https://doi.org/10.1088/1681-7575/ae033f
  4. N Dimarcq, M Gertsvolf, G Mileti, S Bize, C W Oates, E Peik, D Calonico, T Ido, P Tavella, F Meynadier, G Petit, G Panfilo, J Bartholomew, P Defraigne, E A Donley, P O Hedekvist, I Sesia, M Wouters, P Dubé, F Fang, F Levi, J Lodewyck, H S Margolis, D Newell, S Slyusarev, S Weyers, J-P Uzan, M Yasuda, D-H Yu, C Rieck, H Schnatz, Y Hanado, M Fujieda, P-E Pottie, J Hanssen, A Malimon, N Ashby, 'Roadmap towards the redefinition of the second', Metrologia 61, 012001 (2024), doi:10.1088/1681-7575/ad17d2, https://doi.org/10.1088/1681-7575/ad17d2
  5. H S Margolis, G Panfilo, G Petit, C Oates, T Ido, S Bize, 'The CIPM list "Recommended values of standard frequencies": 2021 update', Metrologia 61, 035005 (2024), doi:10.1088/1681-7575/ad3afc, https://doi.org/10.1088/1681-7575/ad3afc
  6. Thomas Lindvall, Marco Pizzocaro, Rachel M. Godun, Michel Abgrall, Daisuke Akamatsu, Anne Amy-Klein, Erik Benkler, Nishant M. Bhatt, Davide Calonico, Etienne Cantin, Elena Cantoni, Giancarlo Cerretto, Christian Chardonnet, Miguel Angel Cifuentes Marin, Cecilia Clivati, Stefano Condio, E. Anne Curtis, Heiner Denker, Simone Donadello, Sören Dörscher, Chen-Hao Feng, Melina Filzinger, Thomas Fordell, Irene Goti, Kalle Hanhijärvi, H. Nimrod Hausser, Ian R. Hill, Kazumoto Hosaka, Nils Huntemann, Matthew Y. H. Johnson, Jonas Keller, Joshua Klose, Takumi Kobayashi, Sebastian Koke, Alexander Kuhl, Rodolphe Le Targat, Thomas Legero, Filippo Levi, Burghard Lipphardt, Christian Lisdat, Hongli Liu, Jérôme Lodewyck, Olivier Lopez, Maxime Mazouth-Laurol, Tanja E. Mehlstäubler, Alberto Mura, Akiko Nishiyama, Tabea Nordmann, Adam O. Parsons, Gérard Petit, Benjamin Pointard, Paul-Eric Pottie, Matias Risaro, Billy I. Robertson, Marco Schioppo, Haosen Shang, Kilian Stahl, Martin Steinel, Uwe Sterr, Alexandra Tofful, Mads Tønnes, Dang-Bao-An Tran, Jacob Tunesi, Anders E. Wallin, Helen S. Margolis, 'Coordinated international comparisons between optical clocks connected via fiber and satellite links', Optica 12, 843 (2025), doi:10.1364/optica.561754, https://doi.org/10.1364/optica.561754
  7. Marco Pizzocaro, Clara Zyskind, Anne Amy-Klein, Erik Benkler, Sebastien Bize, Davide Calonico, Etienne Cantin, Christian Chardonnet, Cecilia Clivati, Stefano Condio, E. Anne Curtis, Simone Donadello, Sören Dörscher, Chen-Hao Feng, Melina Filzinger, Jacques-Olivier Gaudron, Rachel M. Godun, Irene Goti, Ian R. Hill, Wei Huang, Nils Huntemann, Matthew Johnson, Joshua Klose, Jochen Kronjäger, Alexander Kuhl, Rodolphe Le Targat, Filippo Levi, Burghard Lipphardt, Christian Lisdat, Jerome Lodewyck, Olivier Lopez, Helen S. Margolis, Maxime Mazouth-Laurol, Alberto Mura, Benjamin Pointard, Paul-Eric Pottie, Matias Risaro, Billy I. Robertson, Marco Schioppo, Kilian Stahl, Martin Steinel, Alexandra Tofful, Mads Tønnes, Jacob Tunesi, 'International Optical Clock Comparison Using the European Optical Fiber Network', arXiv:2604.27963 (2026), doi:10.48550/arXiv.2604.27963, https://arxiv.org/abs/2604.27963
  8. Alexander Aeppli, Willa J. Arthur-Dworschack, Kyle Beloy, Caitlin M. Berry, Tobias Bothwell, Angela Folz, Tara M. Fortier, Tanner Grogan, Youssef S. Hassan, Zoey Z. Hu, David B. Hume, Benjamin D. Hunt, Kyungtae Kim, Amanda Koepke, Dahyeon Lee, David R. Leibrandt, Ben Lewis, Andrew D. Ludlow, Mason C. Marshall, Nicholas V. Nardelli, Harikesh Ranganath, Daniel A. Rodriguez Castillo, Jeffrey A. Sherman, Jacob L. Siegel, Suzanne Thornton, William Warfield, Jun Ye, 'Atomic clock frequency ratios with fractional uncertainty ≤3.2×10−18\leq 3.2\times 10^{-18}', Physical Review Letters 137, 033201 (2026), doi:10.1103/g865-9mk1, https://doi.org/10.1103/g865-9mk1
  9. Christian Grebing, Ali Al-Masoudi, Sören Dörscher, Sebastian Häfner, Vladislav Gerginov, Stefan Weyers, Burghard Lipphardt, Fritz Riehle, Uwe Sterr, Christian Lisdat, 'Realization of a timescale with an accurate optical lattice clock', Optica 3, 563 (2016), doi:10.1364/OPTICA.3.000563, https://doi.org/10.1364/OPTICA.3.000563
  10. Jian Yao, Jeff A. Sherman, Tara Fortier, Holly Leopardi, Thomas Parker, William McGrew, Xiaogang Zhang, Daniele Nicolodi, Robert Fasano, Stefan Schäffer, Kyle Beloy, Joshua Savory, Stefania Romisch, Chris Oates, Scott Diddams, Andrew Ludlow, Judah Levine, 'Optical-clock-based time scale', Physical Review Applied 12, 044069 (2019), doi:10.1103/PhysRevApplied.12.044069, https://doi.org/10.1103/PhysRevApplied.12.044069
  11. Nils Nemitz, Claudio E. Calosso, 'An Accessible Formulation for an SI Second Based on Multiple Atomic Transitions', URSI AP-RASC 2025 (2025), doi:10.46620/ursiaprasc25/dqlf3252, https://doi.org/10.46620/ursiaprasc25/dqlf3252