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  • Letter
  • Open Access

Timekeeping precision enhancements at constant power

K. J. H. Peters1, B. Braeckeveldt2, B. Maes2, and S. R. K. Rodriguez1,*

  • 1Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
  • 2Micro- and Nanophotonic Materials Group, Research Institute for Materials Science and Engineering, University of Mons, 20 Place du Parc, Mons B-7000, Belgium

  • *Contact author: s.rodriguez@amolf.nl
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Phys. Rev. Research 8, L012009 – Published 12 January, 2026

DOI: https://doi.org/10.1103/pym4-3csj

Abstract

We study the precision of a noisy clock comprising laser-driven coupled optical cavities sustaining limit cycles. We quantify the timekeeping precision of this system via the standard deviation of the limit-cycle period and demonstrate how it changes when varying the cavity length. We find timekeeping precision enhancements at constant power and regardless of the operation frequency. Through a phase space analysis of the limit-cycle fluctuations, we reveal how the proximity of different bifurcations determines the timekeeping precision of our system regardless of the input power and oscillation frequency. We expect our results to assist in the design of clocks that must operate in the presence of strong fluctuations, such as small clocks influenced by thermal noise. While fluctuations inevitably limit the maximum precision that can be attained, our results elucidate how that limited precision can be substantially enhanced while maintaining the energy efficiency and operation frequency of the clock.

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References (54)

  1. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  2. A. C. Barato and U. Seifert, Cost and precision of Brownian clocks, Phys. Rev. X 6, 041053 (2016).
  3. P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Autonomous quantum clocks: Does thermodynamics limit our ability to measure time? Phys. Rev. X 7, 031022 (2017).
  4. M. Monti, D. K. Lubensky, and P. R. ten Wolde, Robustness of clocks to input noise, Phys. Rev. Lett. 121, 078101 (2018).
  5. G. J. Milburn, The thermodynamics of clocks, Contemp. Phys. 61, 69 (2020).
  6. A. N. Pearson, Y. Guryanova, P. Erker, E. A. Laird, G. A. D. Briggs, M. Huber, and N. Ares, Measuring the thermodynamic cost of timekeeping, Phys. Rev. X 11, 021029 (2021).
  7. D. Ziemkiewicz, Entropy of timekeeping in a mechanical clock, Phys. Rev. E 105, 055001 (2022).
  8. P. Pietzonka, Classical pendulum clocks break the thermodynamic uncertainty relation, Phys. Rev. Lett. 128, 130606 (2022).
  9. A. Gopal, M. Esposito, and N. Freitas, Thermodynamic cost of precise timekeeping in an electronic underdamped clock, Phys. Rev. B 109, 085421 (2024).
  10. S. H. Strogatz, Nonlinear Dynamics and Chaos (Perseus Books Publishing, Reading, MA, 1994).
  11. A. Jenkins, Self-oscillation, Phys. Rep. 525, 167 (2013).
  12. J. del Pino, J. Košata, and O. Zilberberg, Limit cycles as stationary states of an extended harmonic balance ansatz, Phys. Rev. Res. 6, 033180 (2024).
  13. C. Troein, J. C. Locke, M. S. Turner, and A. J. Millar, Weather and seasons together demand complex biological clocks, Curr. Biol. 19, 1961 (2009).
  14. W. Pittayakanchit, Z. Lu, J. Chew, M. J. Rust, and A. Murugan, Biophysical clocks face a trade-off between internal and external noise resistance, eLife 7, e37624 (2018).
  15. V. A. Makarov, V. I. Nekorkin, and M. G. Velarde, Spiking behavior in a noise-driven system combining oscillatory and excitatory properties, Phys. Rev. Lett. 86, 3431 (2001).
  16. V. Buendía, S. di Santo, P. Villegas, R. Burioni, and M. A. Muñoz, Self-organized bistability and its possible relevance for brain dynamics, Phys. Rev. Res. 2, 013318 (2020).
  17. B. Braeckeveldt, K. J. H. Peters, B. Verdonschot, B. Maes, and S. R. K. Rodriguez, Memory-induced excitability in optical cavities, Phys. Rev. Res. 6, 023008 (2024).
  18. L. A. Orozco, A. T. Rosenberger, and H. J. Kimble, Intrinsic dynamical instability in optical bistability with two-level atoms, Phys. Rev. Lett. 53, 2547 (1984).
  19. M. Marconi, F. Raineri, A. Levenson, A. M. Yacomotti, J. Javaloyes, S. H. Pan, A. El Amili, and Y. Fainman, Mesoscopic limit cycles in coupled nanolasers, Phys. Rev. Lett. 124, 213602 (2020).
  20. S. Kim, Y. G. Rubo, Timothy C. H. Liew, S. Brodbeck, C. Schneider, S. Höfling, and H. Deng, Emergence of microfrequency comb via limit cycles in dissipatively coupled condensates, Phys. Rev. B 101, 085302 (2020).
  21. N. C. Zambon, S. R. K. Rodriguez, A. Lemaître, A. Harouri, L. Le Gratiet, I. Sagnes, P. St-Jean, S. Ravets, A. Amo, and J. Bloch, Parametric instability in coupled nonlinear microcavities, Phys. Rev. A 102, 023526 (2020).
  22. J. Abad-Arredondo, Z. Geng, G. Keijsers, F. Bijloo, F. J. García-Vidal, A. I. Fernández-Domínguez, and S. R. K. Rodriguez, Spontaneous symmetry breaking in diffraction, Phys. Rev. Lett. 133, 133803 (2024).
  23. K. J. H. Peters and S. R. K. Rodriguez, Limit cycles and chaos induced by a nonlinearity with memory, Eur. Phys. J. Spec. Top. 231, 247 (2022).
  24. E. Zheng, M. Brandenbourger, L. Robinet, P. Schall, E. Lerner, and C. Coulais, Self-oscillation and synchronization transitions in elastoactive structures, Phys. Rev. Lett. 130, 178202 (2023).
  25. C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt, Self-induced oscillations in an optomechanical system driven by bolometric backaction, Phys. Rev. Lett. 101, 133903 (2008).
  26. M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, Photonic cavity synchronization of nanomechanical oscillators, Phys. Rev. Lett. 111, 213902 (2013).
  27. P. Piergentili, W. Li, R. Natali, D. Vitali, and G. Di Giuseppe, Absolute determination of the single-photon optomechanical coupling rate via a Hopf bifurcation, Phys. Rev. Appl. 15, 034012 (2021).
  28. V. Liška, T. Zemánková, P. Jákl, M. Šiler, S. H. Simpson, P. Zemánek, and O. Brzobohatý, PT-like phase transition and limit cycle oscillations in non-reciprocally coupled optomechanical oscillators levitated in vacuum, Nat. Phys. 20 1622 (2024).
  29. L. G. Villanueva, E. Kenig, R. B. Karabalin, M. H. Matheny, R. Lifshitz, M. C. Cross, and M. L. Roukes, Surpassing fundamental limits of oscillators using nonlinear resonators, Phys. Rev. Lett. 110, 177208 (2013).
  30. A. Shapere and F. Wilczek, Classical time crystals, Phys. Rev. Lett. 109, 160402 (2012).
  31. J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, et al., Observation of a discrete time crystal, Nature (London) 543, 217 (2017).
  32. J. Smits, L. Liao, H. T. C. Stoof, and P. van der Straten, Observation of a space-time crystal in a superfluid quantum gas, Phys. Rev. Lett. 121, 185301 (2018).
  33. T. L. Heugel, M. Oscity, A. Eichler, O. Zilberberg, and R. Chitra, Classical many-body time crystals, Phys. Rev. Lett. 123, 124301 (2019).
  34. J. Randall, C. E. Bradley, F. V. van der Gronden, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau, Many-body–localized discrete time crystal with a programmable spin-based quantum simulator, Science 374, 1474 (2021).
  35. H. Keßler, P. Kongkhambut, C. Georges, L. Mathey, J. G. Cosme, and A. Hemmerich, Observation of a dissipative time crystal, Phys. Rev. Lett. 127, 043602 (2021).
  36. L. Oberreiter, U. Seifert, and A. C. Barato, Stochastic discrete time crystals: Entropy production and subharmonic synchronization, Phys. Rev. Lett. 126, 020603 (2021).
  37. P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continuous time crystal, Science 377, 670 (2022).
  38. M. P. Zaletel, M. Lukin, C. Monroe, C. Nayak, F. Wilczek, and N. Y. Yao, Colloquium: Quantum and classical discrete time crystals, Rev. Mod. Phys. 95, 031001 (2023).
  39. I. Carraro-Haddad, D. L. Chafatinos, A. S. Kuznetsov, I. A. Papuccio-Fernández, A. A. Reynoso, A. Bruchhausen, K. Biermann, P. V. Santos, G. Usaj, and A. Fainstein, Solid-state continuous time crystal in a polariton condensate with a built-in mechanical clock, Science 384, 995 (2024).
  40. R. Daviet, C. P. Zelle, A. Asadollahi, and S. Diehl, Kardar-Parisi-Zhang scaling in time-crystalline matter, Phys. Rev. Lett. 135, 047101 (2025).
  41. T. L. Heugel, A. Eichler, R. Chitra, and O. Zilberberg, The role of fluctuations in quantum and classical time crystals, SciPost Phys. Core 6, 053 (2023).
  42. Z. Geng, K. J. H. Peters, A. A. P. Trichet, K. Malmir, R. Kolkowski, J. M. Smith, and S. R. K. Rodriguez, Universal scaling in the dynamic hysteresis, and non-Markovian dynamics, of a tunable optical cavity, Phys. Rev. Lett. 124, 153603 (2020).
  43. K. J. H. Peters, Z. Geng, K. Malmir, J. M. Smith, and S. R. K. Rodriguez, Extremely broadband stochastic resonance of light and enhanced energy harvesting enabled by memory effects in the nonlinear response, Phys. Rev. Lett. 126, 213901 (2021).
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/pym4-3csj for details about the experimental setup, validation of our model across a wide parameter range, experimental signatures of predicted bifurcations, changes in limit-cycle amplitude with detuning, and the precision in relation to intracavity intensity, oscillation amplitude, and phase space velocity.
  45. A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, matcont: A matlab package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Softw. 29, 141 (2003).
  46. M. Dykman and M. Krivoglaz, Fluctuations in nonlinear systems near bifurcations corresponding to the appearance of new stable states, Physica A 104, 480 (1980).
  47. E. Knobloch and K. Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach, J. Stat. Phys. 33, 611 (1983).
  48. C. Meunier and A. Verga, Noise and bifurcations, J. Stat. Phys. 50, 345 (1988).
  49. R. Marsland, III, W. Cui, and J. M. Horowitz, The thermodynamic uncertainty relation in biochemical oscillations, J. R. Soc. Interface 16, 20190098 (2019).
  50. https://zenodo.org/records/17963773.
  51. A. A. P. Trichet, P. R. Dolan, D. M. Coles, G. M. Hughes, and J. M. Smith, Topographic control of open-access microcavities at the nanometer scale, Opt. Express 23, 17205 (2015).
  52. S. Dufferwiel, F. Li, A. Trichet, L. Giriunas, P. Walker, I. Farrer, D. Ritchie, J. Smith, M. Skolnick, and D. Krizhanovskii, Tunable polaritonic molecules in an open microcavity system, Appl. Phys. Lett. 107, 201106 (2015).
  53. L. Flatten, A. Trichet, and J. Smith, Spectral engineering of coupled open-access microcavities, Laser Photonics Rev. 10, 257 (2016).
  54. E. Stein, F. Vewinger, and A. Pelster, Collective modes of a photon Bose-Einstein condensate with thermo-optic interaction, New J. Phys. 21, 103044 (2019).

Outline

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Phys. Rev. Research 8, L012009– Published 12 January, 2026
  • Received 13 August 2025
  • Accepted 25 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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