- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Dynamical slowing-down in the phase space of nanomechanical strings

Samer Houri1,2, Agnes Zinth3,4,5, and Menno Poot3,5,6,7,*

  • *Contact author: menno.poot@tum.de
PDF

Phys. Rev. Research 8, L042016 – Published 12 November, 2026

DOI: https://doi.org/10.1103/ynnb-2bv5

Abstract

The linear behavior of micro- and nanomechanical devices has been the topic of intense study over the past years. Their nonlinear behavior, on the other hand, continues to provide new and interesting dynamics. This work leverages a cavity optomechanical system whose very high sensitivity and low dissipation rate allow to capture the dynamics of fast transients that occur when strongly driven nonlinear resonators relax to their equilibrium state. We experimentally study the ringdown dynamics of a nanomechanical Duffing resonator in both the time domain and phase space. The resonator exhibits very different ringdown dynamics depending on whether it is subject to one- or two-tone driving. In the former case, the ringdown follows a regular exponential decay, whereas the latter case shows exotic dynamics such as libration, slowing-down while scattering in phase space, and ghost slowing-down. We demonstrate that this distinct difference can be attributed to the structure of the phase space itself. Surprisingly, we find that the energy in the resonator changes nonmonotonously and that long delay between turning off the drive and the decay of the motion amplitude results in dynamical metastates.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (42)

  1. A. Cleland, Foundations of Nanomechanics (Springer, Berlin, 2003).
  2. D. T. Nayfeh and A. H. Mook, Nonlinear Oscillations, 1st ed. (Wiley-VCH, New York, 1979).
  3. N. Minorsky, Introduction to Nonlinear Mechanics: Topological Methods (Literary Licensing, LLC, 2012).
  4. S. Houri, M. Asano, H. Okamoto, and H. Yamaguchi, Self-sustained libration regime in nonlinear microelectromechanical devices, Phys. Rev. Appl. 16, 064015 (2021).
  5. G. Villa, J. del Pino, V. Dumont, G. Rastelli, M. Michałek, A. Eichler, and O. Zilberberg, Topological classification of driven-dissipative nonlinear systems, Sci. Adv. 11, eadt9311 (2025).
  6. T. Mutschler, G. Villa, and O. Zilberberg, Flow topology classification of limit cycles, Phys. Rev. Res. 8, 033213 (2026).
  7. J. S. Aldridge and A. N. Cleland, Noise-enabled precision measurements of a Duffing nanomechanical resonator, Phys. Rev. Lett. 94, 156403 (2005).
  8. T. L. Heugel, R. Chitra, A. Eichler, and O. Zilberberg, Proliferation of unstable states and their impact on stochastic out-of-equilibrium dynamics in two coupled Kerr parametric oscillators, Phys. Rev. E 109, 064308 (2024).
  9. D. Koch, A. Nandan, G. Ramesan, I. Tyukin, A. Gorban, and A. Koseska, Ghost channels and ghost cycles guiding long transients in dynamical systems, Phys. Rev. Lett. 133, 047202 (2024).
  10. L. Catalini, J. del Pino, S. S. Kumar, V. Dumont, G. Margiani, O. Zilberberg, and A. Eichler, Slow and fast topological dynamical phase transitions in a Duffing resonator driven by two detuned tones, Phys. Rev. Res. 7, 033058 (2025).
  11. S. Houri, R. Ohta, M. Asano, Y. M. Blanter, and H. Yamaguchi, Pulse-width modulated oscillations in a nonlinear resonator under two-tone driving as a means for MEMS sensor readout, Jpn. J. Appl. Phys. 58, SBBI05 (2019).
  12. A. Chowdhury, M. G. Clerc, S. Barbay, I. Robert-Philip, and R. Braive, Weak signal enhancement by nonlinear resonance control in a forced nano-electromechanical resonator, Nat. Commun. 11, 2400 (2020).
  13. S. Houri, M. Asano, H. Yamaguchi, N. Yoshimura, Y. Koike, and L. Minati, Generic rotating-frame-based approach to chaos generation in nonlinear micro- and nanoelectromechanical system resonators, Phys. Rev. Lett. 125, 174301 (2020).
  14. J. S. Ochs, M. Seitner, M. I. Dykman, and E. M. Weig, Amplification and spectral evidence of squeezing in the response of a strongly driven nanoresonator to a probe field, Phys. Rev. A 103, 013506 (2021).
  15. J. S. Ochs, D. K. J. Boneß, G. Rastelli, M. Seitner, W. Belzig, M. I. Dykman, and E. M. Weig, Frequency comb from a single driven nonlinear nanomechanical mode, Phys. Rev. X 12, 041019 (2022).
  16. H. Choi, O. Shoshani, R. Kim, Y. Ryu, J. Jeong, J. Suh, S. W. Shaw, M. I. Dykman, and H. Choi, Nonexponential relaxation in the rotating frame of a driven nanomechanical mode, Phys. Rev. B 113, L060301 (2026).
  17. V. Dumont, M. Bestler, L. Catalini, G. Margiani, O. Zilberberg, and A. Eichler, Energy landscape and flow dynamics measurements of driven-dissipative systems, Phys. Rev. Res. 6, 043012 (2024).
  18. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Chapman and Hall/CRC, New York, 2024).
  19. The authors use the term metastate to indicate an unusually long-lived transient, which gives the appearance of a state, but in reality does not correspond to any fixed point, stable or otherwise.
  20. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  21. M. Poot and H. S. van der Zant, Mechanical systems in the quantum regime, Phys. Rep. 511, 273 (2012).
  22. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ynnb-2bv5 for details on the delayed ringdowns, simulations, and experimental techniques, including the device, which includes Refs. [1, 21, 23, 25, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42].
  23. T. Sommer, A. Zinth, Aditya, and M. Poot, Spatial mapping of intrinsic and readout nonlinearities in a strongly driven micromechanical membrane, Phys. Rev. Lett. 134, 203604 (2025).
  24. T. Antoni, K. Makles, R. Braive, T. Briant, P.-F. Cohadon, I. Sagnes, I. Robert-Philip, and A. Heidmann, Nonlinear mechanics with suspended nanomembranes, Europhys. Lett. 100, 68005 (2012).
  25. I. Kozinsky, H. W. Ch. Postma, O. Kogan, A. Husain, and M. L. Roukes, Basins of attraction of a nonlinear nanomechanical resonator, Phys. Rev. Lett. 99, 207201 (2007).
  26. M. Dykman, Fluctuating Nonlinear Oscillators: From Nanomechanics to Quantum Superconducting Circuits (Oxford University Press, Oxford, 2012).
  27. A. Bachtold, J. Moser, and M. I. Dykman, Mesoscopic physics of nanomechanical systems, Rev. Mod. Phys. 94, 045005 (2022).
  28. R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, High intermodulation gain in a micromechanical Duffing resonator, Appl. Phys. Lett. 88, 213509 (2006).
  29. A. Erbe, H. Krömmer, A. Kraus, R. Blick, G. Corso, and K. Richter, Mechanical mixing in nonlinear nanomechanical resonators, Appl. Phys. Lett. 77, 3102 (2000).
  30. M. Defoort, K. J. Lulla, C. Blanc, O. Bourgeois, E. Collin, and A. D. Armour, Modal “self-coupling” as a sensitive probe for nanomechanical detection, Appl. Phys. Lett. 103, 013104 (2013).
  31. S. S. Kumar, J. del Pino, L. Catalini, A. Eichler, and O. Zilberberg, Dynamical phase transitions across slow and fast regimes in a two-tone driven Duffing resonator, Phys. Rev. Res. 8, 023298 (2026).
  32. J. S. Huber, G. Rastelli, M. J. Seitner, J. Kölbl, W. Belzig, M. I. Dykman, and E. M. Weig, Spectral evidence of squeezing of a weakly damped driven nanomechanical mode, Phys. Rev. X 10, 021066 (2020).
  33. P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field (Springer, Berlin, 2013).
  34. M. Volvert and G. Kerschen, Resonant phase lags of a duffing oscillator, Int. J. Non-Linear Mech. 146, 104150 (2022).
  35. A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer Science & Business Media, 2013).
  36. W. H. Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).
  37. P. Zhang, Y. Jia, Z. Liu, X. Zhou, D. Xiao, Y. Chen, H. Jia, and R. Yang, Probing linear to nonlinear damping in 2d semiconductor nanoelectromechanical resonators toward a unified quality factor model, Nano Lett. 23, 9375 (2023).
  38. D. Hoch, X. Yao, and M. Poot, Geometric tuning of stress in predisplaced silicon nitride resonators, Nano Lett. 22, 4013 (2022).
  39. M. Poot, S. Etaki, H. Yamaguchi, and H. S. J. van der Zant, Discrete-time quadrature feedback cooling of a radio-frequency mechanical resonator, Appl. Phys. Lett. 99, 013113 (2011).
  40. D. Hoch, K.-J. Haas, L. Moller, T. Sommer, P. Soubelet, J. J. Finley, and M. Poot, Efficient optomechanical mode-shape mapping of micromechanical devices, Micromachines 12, 880 (2021).
  41. D. C. J. Hoch, Optomechanics with high-stress silicon nitride resonators, Ph.D. thesis, TU Munich, 2023.
  42. M. Poot and H. X. Tang, Broadband nanoelectromechanical phase shifting of light on a chip, Appl. Phys. Lett. 104, 061101 (2014).

Outline

Information

Phys. Rev. Research 8, L042016– Published 12 November, 2026
  • Received 23 April 2026
  • Accepted 16 October 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation