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Quantum spin van der Pol oscillator: Spin-based limit-cycle oscillator exhibiting quantum synchronization

Yuzuru Kato1,* and Hiroya Nakao2

  • *Contact author: katoyuzu@fun.ac.jp
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Phys. Rev. Research 7, 043133 – Published 5 November, 2025

DOI: https://doi.org/10.1103/mfvy-bbzm

Abstract

We introduce a quantum spin van der Pol (vdP) oscillator as a prototypical model of quantum spin-based limit-cycle oscillators, which coincides with the quantum optical vdP oscillator in the high-spin limit. The system is described as a noisy limit-cycle oscillator in the semiclassical regime at large spin numbers, exhibiting frequency entrainment to a periodic drive. Even in the smallest spin-1 case, mutual synchronization, Arnold tongues, and entanglement tongues in two dissipatively coupled oscillators, and collective synchronization in all-to-all coupled oscillators are clearly observed. The proposed quantum spin vdP oscillator will provide a useful platform for analyzing quantum spin synchronization.

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References (64)

  1. A. T. Winfree, The Geometry of Biological Time (Springer, New York, 2001).
  2. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  3. G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience (Springer, New York, 2010).
  4. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  5. L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, NJ, 1988).
  6. S. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, Cambridge, MA, 1994).
  7. A. W. Laskar, P. Adhikary, S. Mondal, P. Katiyar, S. Vinjanampathy, and S. Ghosh, Observation of quantum phase synchronization in spin-1 atoms, Phys. Rev. Lett. 125, 013601 (2020).
  8. V. R. Krithika, P. Solanki, S. Vinjanampathy, and T. S. Mahesh, Observation of quantum phase synchronization in a nuclear-spin system, Phys. Rev. A 105, 062206 (2022).
  9. M. Koppenhöfer, C. Bruder, and A. Roulet, Quantum synchronization on the IBM Q system, Phys. Rev. Res. 2, 023026 (2020).
  10. T. E. Lee and H. R. Sadeghpour, Quantum synchronization of quantum van der Pol oscillators with trapped ions, Phys. Rev. Lett. 111, 234101 (2013).
  11. S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of a driven self-sustained oscillator, Phys. Rev. Lett. 112, 094102 (2014).
  12. S. Sonar, M. Hajdušek, M. Mukherjee, R. Fazio, V. Vedral, S. Vinjanampathy, and L.-C. Kwek, Squeezing enhances quantum synchronization, Phys. Rev. Lett. 120, 163601 (2018).
  13. N. Lörch, E. Amitai, A. Nunnenkamp, and C. Bruder, Genuine quantum signatures in synchronization of anharmonic self-oscillators, Phys. Rev. Lett. 117, 073601 (2016).
  14. T. E. Lee, C.-K. Chan, and S. Wang, Entanglement tongue and quantum synchronization of disordered oscillators, Phys. Rev. E 89, 022913 (2014).
  15. S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of two van der Pol oscillators, Ann. Phys. 527, 131 (2015).
  16. N. E. Sani, G. Manzano, R. Zambrini, and R. Fazio, Synchronization along quantum trajectories, Phys. Rev. Res. 2, 023101 (2020).
  17. V. M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, and T. Brandes, Quantum signatures of chimera states, Phys. Rev. E 92, 062924 (2015).
  18. Y. Kato, N. Yamamoto, and H. Nakao, Semiclassical phase reduction theory for quantum synchronization, Phys. Rev. Res. 1, 033012 (2019).
  19. W.-K. Mok, L.-C. Kwek, and H. Heimonen, Synchronization boost with single-photon dissipation in the deep quantum regime, Phys. Rev. Res. 2, 033422 (2020).
  20. A. Roulet and C. Bruder, Synchronizing the smallest possible system, Phys. Rev. Lett. 121, 053601 (2018).
  21. A. Roulet and C. Bruder, Quantum synchronization and entanglement generation, Phys. Rev. Lett. 121, 063601 (2018).
  22. M. Koppenhöfer and A. Roulet, Optimal synchronization deep in the quantum regime: Resource and fundamental limit, Phys. Rev. A 99, 043804 (2019).
  23. M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson, and M. J. Holland, Synchronization of two ensembles of atoms, Phys. Rev. Lett. 113, 154101 (2014).
  24. M. Xu and M. J. Holland, Conditional Ramsey spectroscopy with synchronized atoms, Phys. Rev. Lett. 114, 103601 (2015).
  25. Y. Kato and H. Nakao, A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint, Chaos 32, 063133 (2022).
  26. Y. Kato and H. Nakao, Quantum asymptotic phases reveal signatures of quantum synchronization, New J. Phys. 25, 023012 (2023).
  27. P. Solanki, F. M. Mehdi, M. Hajdušek, and S. Vinjanampathy, Symmetries and synchronization blockade, Phys. Rev. A 108, 022216 (2023).
  28. P. Solanki, N. Jaseem, M. Hajdušek, and S. Vinjanampathy, Role of coherence and degeneracies in quantum synchronization, Phys. Rev. A 105, L020401 (2022).
  29. A. Chia, L. C. Kwek, and C. Noh, Relaxation oscillations and frequency entrainment in quantum mechanics, Phys. Rev. E 102, 042213 (2020).
  30. L. B. Arosh, M. C. Cross, and Ron Lifshitz, Quantum limit cycles and the Rayleigh and van der Pol oscillators, Phys. Rev. Res. 3, 013130 (2021).
  31. F. Galve, G. Luca Giorgi, and R. Zambrini, in Lectures on General Quantum Correlations and their Applications, edited by F. F. Fanchini, D. d. O. Soares Pinto, and G. Adesso (Springer International Publishing, Cham, 2017), p. 393.
  32. B. Buča, C. Booker, and D. Jaksch, Algebraic theory of quantum synchronization and limit cycles under dissipation, SciPost Phys. 12, 097 (2022).
  33. F. Schmolke and E. Lutz, Noise-induced quantum synchronization, Phys. Rev. Lett. 129, 250601 (2022).
  34. R. Tan, C. Bruder, and M. Koppenhöfer, Half-integer vs. integer effects in quantum synchronization of spin systems, Quantum, 6, 885 (2022).
  35. H. Eneriz, D. Z. Rossatto, F. A. Cárdenas-López, E. Solano, and M. Sanz, Degree of quantumness in quantum synchronization, Sci. Rep. 9, 19933 (2019).
  36. N. Jaseem, M. Hajdušek, P. Solanki, L.-C. Kwek, R. Fazio, and S. Vinjanampathy, Generalized measure of quantum synchronization, Phys. Rev. Res. 2, 043287 (2020).
  37. J. Tindall, C. S. Munoz, B. Buča, and D. Jaksch, Quantum synchronisation enabled by dynamical symmetries and dissipation, New J. Phys. 22, 013026 (2020).
  38. A. Cabot, G. L. Giorgi, F. Galve, and R. Zambrini, Quantum synchronization in dimer atomic lattices, Phys. Rev. Lett. 123, 023604 (2019).
  39. A. Cabot, G. L. Giorgi, and R. Zambrini, Metastable quantum entrainment, New J. Phys. 23, 103017 (2021).
  40. T. Nadolny and C. Bruder, Macroscopic quantum synchronization effects, Phys. Rev. Lett. 131, 190402 (2023).
  41. B. van der Pol, VII. Forced oscillations in a circuit with non-linear resistance (reception with reactive triode), London Edinburgh Dublin Philos. Mag. J. Sci. 3, 65 (1927).
  42. O. V. Zhirov and D. L. Shepelyansky, Synchronization and bistability of a qubit coupled to a driven dissipative oscillator, Phys. Rev. Lett. 100, 014101 (2008).
  43. Á. Parra-López and J. Bergli, Synchronization in two-level quantum systems, Phys. Rev. A 101, 062104 (2020).
  44. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Atomic coherent states in quantum optics, Phys. Rev. A 6, 2211 (1972).
  45. J. M. Radcliffe, Some properties of coherent spin states, J. Phys. A: Gen. Phys. 4, 313 (1971).
  46. C. W Gardiner, Quantum Noise (Springer, New York, 1991).
  47. H. J. Carmichael, Statistical Methods in Quantum Optics (Springer, New York, 2007), Vols. 1 and 2.
  48. R. Gilmore, C. M. Bowden, and L. M. Narducci, Classical-quantum correspondence for multilevel systems, Phys. Rev. A 12, 1019 (1975).
  49. T. Weiss, A. Kronwald, and F. Marquardt, Noise-induced transitions in optomechanical synchronization, New J. Phys. 18, 013043 (2016).
  50. If ρss=|u〉〈u|, we obtain S1=2Ju/(1+|u|2)2J|u|2/(1+|u|2)=u|u|21+|u|2. In the high-spin limit, denoting α=reiϕ of the quantum optical coherent state, it converge to α|α|2=eiϕ.
  51. K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Volume of the set of separable states, Phys. Rev. A 58, 883 (1998).
  52. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  53. J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum reservoir engineering with laser cooled trapped ions, Phys. Rev. Lett. 77, 4728 (1996).
  54. N. Lütkenhaus, J. I. Cirac, and P. Zoller, Mimicking a squeezed-bath interaction: Quantum-reservoir engineering with atoms, Phys. Rev. A 57, 548 (1998).
  55. Y. Li et al., Experimental realization and synchronization of a quantum van der Pol oscillator, Sci. Adv. 11, eady5649 (2025).
  56. J. R. Johansson, P. D. Nation, and F. Nori, Qutip: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  57. J. R. Johansson, P. D. Nation, and F. Nori, Qutip 2: A Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  58. L. M. Narducci, C. M. Bowden, V. Bluemel, and G. P. Carrazana, Phase-space description of the thermal relaxation of a (2j+1)-level system, Phys. Rev. A 11, 280 (1975).
  59. S. Dutta, S. Zhang, and M. Haque, Quantum origin of limit cycles, fixed points, and critical slowing down, Phys. Rev. Lett. 134, 050407 (2025).
  60. E. Inonu and E. P. Wigner, On the contraction of groups and their representations, Proc. Natl. Acad. Sci. USA 39, 510 (1953).
  61. I. E. Segal et al., A class of operator algebras which are determined by groups, Duke Math. J. 18, 221 (1951).
  62. T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58, 1098 (1940).
  63. G. Schaller, C. W. Wachtler, V. M. Bastidas, and W. J. Munro, Dissipative nonequilibrium synchronization of topological edge states via self-oscillation, Phys. Rev. B 102, 014309 (2020).
  64. H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Mod. Phys. 52, 569 (1980).

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Phys. Rev. Research 7, 043133– Published 5 November, 2025
  • Received 13 September 2024
  • Accepted 25 September 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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