- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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
  • Editors' Suggestion
  • Open Access

Universal quantum melting of quasiperiodic attractors in driven-dissipative cavities

Caroline Nowoczyn1,2, Ludwig Mathey1,2, and Kilian Seibold3

PDF

Phys. Rev. A 113, 052208 – Published 8 May, 2026

DOI: https://doi.org/10.1103/pdnh-1yxs

Abstract

Nonlinear classical mechanics has established rich phenomena. These include limit tori defined by toroidal attractors supporting quasiperiodic motion with incommensurate frequencies. We study the fate of such structures in open quantum systems using two coupled driven-dissipative Kerr cavities modeled via the Lindblad master equation. Combining Liouvillian spectral theory with the truncated Wigner approximation, we characterize the quantum-to-classical crossover. In the classical limit, two pairs of purely imaginary Liouvillian eigenvalues signal persistent quasiperiodic modes. Quantum fluctuations induce small negative real parts to these eigenvalues, giving rise to finite lifetimes and leading to the quantum melting of the torus. The associated Liouvillian gaps vanish algebraically in the classical limit, indicating a dynamical critical crossover with spontaneous breaking of time-translational symmetry. Quantum trajectory analysis reveals that this melting is driven by fluctuation-induced dephasing. Using a circular-variance-based order parameter, we uncover universal scaling in system size and time. These results establish quantum melting of limit tori as a distinct and robust nonequilibrium critical phenomenon, with clear experimental signatures in trapped ions and superconducting circuits.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (109)

  1. P. D. Drummond and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A: Math. Gen. 13, 725 (1980).
  2. S. Aldana, C. Bruder, and A. Nunnenkamp, Equivalence between an optomechanical system and a Kerr medium, Phys. Rev. A 88, 043826 (2013).
  3. N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions, Phys. Rev. A 94, 033841 (2016).
  4. H. Landa, M. Schiró, and G. Misguich, Multistability of driven-dissipative quantum spins, Phys. Rev. Lett. 124, 043601 (2020).
  5. A. M. Lyapunov, The general problem of the stability of motion, Int. J. Control 55, 531 (1992).
  6. V. Sharma, Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: Perspectives on a new frontier, Open Cardiovasc Med. J. 3, 110 (2009).
  7. T. Laffargue, J. Tailleur, and F. van Wijland, Lyapunov exponents of stochastic systems—From micro to macro, J. Stat. Mech. (2016) 034001.
  8. H. Goto and T. Kanao, Chaos in coupled Kerr-nonlinear parametric oscillators, Phys. Rev. Res. 3, 043196 (2021).
  9. G. Datseris and U. Parlitz, Nonlinear Dynamics: A Concise Introduction Interlaced with Code (Springer International, Cham, 2022).
  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. A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, Measures of quantum synchronization in continuous variable systems, Phys. Rev. Lett. 111, 103605 (2013).
  12. S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of two van der Pol oscillators, Ann. Phys. 527, 131 (2014).
  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. C. Davis-Tilley, C. K. Teoh, and A. D. Armour, Dynamics of many-body quantum synchronisation, New J. Phys. 20, 113002 (2018).
  15. Y. Kato, N. Yamamoto, and H. Nakao, Semiclassical phase reduction theory for quantum synchronization, Phys. Rev. Res. 1, 033012 (2019).
  16. N. Thomas, S. Karthiga, and M. Senthilvelan, High-order synchronization in a system of nonlinearly coupled Stuart-Landau oscillators, Eur. Phys. J. Plus 136, 1222 (2021).
  17. N. Thomas and M. Senthilvelan, Quantum synchronization in quadratically coupled quantum van der Pol oscillators, Phys. Rev. A 106, 012422 (2022).
  18. C. Lu, M. Kim, Y. Yang, Y. S. Gui, and C.-M. Hu, Synchronization of dissipatively coupled oscillators, J. Appl. Phys. 134, 221101 (2023).
  19. C. W. Wächtler and G. Platero, Topological synchronization of quantum van der Pol oscillators, Phys. Rev. Res. 5, 023021 (2023).
  20. J. N. Moreno, C. W. Wächtler, and A. Eisfeld, Synchronized states in a ring of dissipatively coupled harmonic oscillators, Phys. Rev. E 109, 014308 (2024).
  21. D. Luo, X. Wang, D. Zhu, and M. Han, Bifurcation Theory and Methods of Dynamical Systems (World Scientific, Singapore, 1997).
  22. H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to Partial Differential Equations (Springer, New York, 2012).
  23. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. 12, 47 (1883).
  24. J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).
  25. S. Strogatz, Nonlinear Dynamics and Chaos, 3rd ed. (CRC Press, Boca Raton, FL, 2024).
  26. A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR 98, 527 (1954).
  27. J. Möser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, II, 1 (1962).
  28. A. B. Givental, B. A. Khesin, J. E. Marsden, A. N. Varchenko, V. A. Vassiliev, O. Y. Viro, and V. M. Zakalyukin, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, in Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965 (Springer, Berlin, Heidelberg, 2009), pp. 267–294.
  29. I. I. Yusipov and M. V. Ivanchenko, Quantum Neimark-Sacker bifurcation, Sci. Rep. 9, 17932 (2019).
  30. J. G. Cosme, P. Kongkhambut, A. Bölian, R. J. L. Tuquero, J. Skulte, L. Mathey, A. Hemmerich, and H. Keßler, Torus bifurcation of a dissipative time crystal, Phys. Rev. Lett. 134, 223601 (2025).
  31. A. Garfinkel, P. S. Chen, D. O. Walter, H. S. Karagueuzian, B. Kogan, S. J. Evans, M. Karpoukhin, C. Hwang, T. Uchida, M. Gotoh, O. Nwasokwa, P. Sager, and J. N. Weiss, Quasiperiodicity and chaos in cardiac fibrillation, J. Clin. Invest. 99, 305 (1997).
  32. Y. Suzuki, M. Lu, E. Ben-Jacob, and J. N. Onuchic, Periodic, quasi-periodic and chaotic dynamics in simple gene elements with time delays, Sci. Rep. 6, 21037 (2016).
  33. C. Bick, M. Goodfellow, C. R. Laing, and E. A. Martens, Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: A review, J. Math. Neurosci. 10, 9 (2020).
  34. D. N. Voskresensky, Evolution of quasiperiodic structures in a non-ideal hydrodynamic description of phase transitions, Universe 6, 42 (2020).
  35. Y. J. F. Kpomahou, A. Adomou, J. A. Adéchinan, A. E. Yamadjako, and I. V. Madogni, Chaotic behaviors and coexisting attractors in a new nonlinear dissipative parametric chemical oscillator, Complexity 2022, 9350516 (2022).
  36. D. Zou, R. Liu, H. Liu, J. Chen, H. Dang, J. Li, A. Zhang, Y. Song, P. P. Shum, and M. Hu, Quasi-period dynamics of soliton molecules: Route to chaos and intrinsic frequency entrainment, Ultrafast Sci. 4, 0061 (2024).
  37. F. Ferrari, L. Gravina, D. Eeltink, P. Scarlino, V. Savona, and F. Minganti, Dissipative quantum chaos unveiled by stochastic quantum trajectories, Phys. Rev. Res. 7, 013276 (2025).
  38. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  39. F. Wilczek, Quantum time crystals, Phys. Rev. Lett. 109, 160401 (2012).
  40. D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016).
  41. H. Keßler, J. G. Cosme, M. Hemmerling, L. Mathey, and A. Hemmerich, Emergent limit cycles and time crystal dynamics in an atom-cavity system, Phys. Rev. A 99, 053605 (2019).
  42. K. Seibold, R. Rota, and V. Savona, Dissipative time crystal in an asymmetric nonlinear photonic dimer, Phys. Rev. A 101, 033839 (2020).
  43. P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continuous time crystal, Science 377, 670 (2022).
  44. L. Ben Arosh, M. C. Cross, and R. Lifshitz, Quantum limit cycles and the Rayleigh and van der Pol oscillators, Phys. Rev. Res. 3, 013130 (2021).
  45. S. Dutta, S. Zhang, and M. Haque, Quantum origin of limit cycles, fixed points, and critical slowing down, Phys. Rev. Lett. 134, 050407 (2025).
  46. J. Skulte, P. Kongkhambut, H. Keßler, A. Hemmerich, L. Mathey, and J. G. Cosme, Realizing limit cycles in dissipative bosonic systems, Phys. Rev. A 109, 063317 (2024).
  47. P. Zanardi and N. Anand, Information scrambling and chaos in open quantum systems, Phys. Rev. A 103, 062214 (2021).
  48. T. Yoshimura and L. Sá, Robustness of quantum chaos and anomalous relaxation in open quantum circuits, Nat. Commun. 15, 9808 (2024).
  49. J. Richter, L. Sá, and M. Haque, Integrability versus chaos in the steady state of many-body open quantum systems, Phys. Rev. E 111, 064103 (2025).
  50. P. Solanki and F. Minganti, Chaos in time: A dissipative continuous quasi time crystals, Phys. Rev. B 112, 134311 (2025).
  51. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  52. F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
  53. K. Macieszczak, M. Guţă, I. Lesanovsky, and J. P. Garrahan, Towards a theory of metastability in open quantum dynamics, Phys. Rev. Lett. 116, 240404 (2016).
  54. K. Vogel and H. Risken, Quasiprobability distributions in dispersive optical bistability, Phys. Rev. A 39, 4675 (1989).
  55. A. Polkovnikov, Phase space representation of quantum dynamics, Ann. Phys. 325, 1790 (2010).
  56. C. Gardiner and P. Zoller, The Quantum World of Ultra-cold Atoms and Light Book I: Foundations of Quantum Optics (Imperial College Press, London, 2014).
  57. P. D. Drummond and B. Opanchuk, Truncated Wigner dynamics and conservation laws, Phys. Rev. A 96, 043616 (2017).
  58. T. Yoneya, K. Fujimoto, and Y. Kawaguchi, Path-integral formulation of truncated Wigner approximation for bosonic Markovian open quantum systems, Ann. Phys. 479, 170072 (2025).
  59. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
  60. C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed., Springer Series in Synergetics Vol. 56 (Springer, Berlin, 2000), pp. 419–425.
  61. H. Spohn, Approach to equilibrium for completely positive dynamical semigroups of N-level systems, Rep. Math. Phys. 10, 189 (1976).
  62. H. Spohn, An algebraic condition for the approach to equilibrium of an open N-level system, Lett. Math. Phys. 2, 33 (1977).
  63. D. Nigro, On the uniqueness of the steady-state solution of the Lindblad–Gorini–Kossakowski–Sudarshan equation, J. Stat. Mech. (2019) 043202.
  64. M.-J. Hwang and M. B. Plenio, Quantum phase transition in the finite Jaynes-Cummings lattice systems, Phys. Rev. Lett. 117, 123602 (2016).
  65. W. Casteels and C. Ciuti, Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative Bose-Hubbard dimer, Phys. Rev. A 95, 013812 (2017).
  66. W. Casteels, R. Fazio, and C. Ciuti, Critical dynamical properties of a first-order dissipative phase transition, Phys. Rev. A 95, 012128 (2017).
  67. R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, Probing the dynamics of a superradiant quantum phase transition with a single trapped ion, Phys. Rev. Lett. 118, 073001 (2017).
  68. M.-J. Hwang, P. Rabl, and M. B. Plenio, Dissipative phase transition in the open quantum Rabi model, Phys. Rev. A 97, 013825 (2018).
  69. C. Lledó, T. K. Mavrogordatos, and M. H. Szymańska, Driven Bose-Hubbard dimer under nonlocal dissipation: A bistable time crystal, Phys. Rev. B 100, 054303 (2019).
  70. G. Dagvadorj, J. Fellows, S. Matyjaśkiewicz, F. Marchetti, I. Carusotto, and M. Szymańska, Nonequilibrium phase transition in a two-dimensional driven open quantum system, Phys. Rev. X 5, 041028 (2015).
  71. P. Deuar, A. Ferrier, M. Matuszewski, G. Orso, and M. H. Szymańska, Fully quantum scalable description of driven-dissipative lattice models, PRX Quantum 2, 010319 (2021).
  72. J. Dujardin, A. Argüelles, and P. Schlagheck, Elastic and inelastic transmission in guided atom lasers: A truncated Wigner approach, Phys. Rev. A 91, 033614 (2015).
  73. G. Orso, J. Zakrzewski, and P. Deuar, Self-organized cavity bosons beyond the adiabatic elimination approximation, Phys. Rev. Lett. 134, 183405 (2025).
  74. F. Vicentini, F. Minganti, R. Rota, G. Orso, and C. Ciuti, Critical slowing down in driven-dissipative Bose-Hubbard lattices, Phys. Rev. A 97, 013853 (2018).
  75. F. Vicentini, F. Minganti, A. Biella, G. Orso, and C. Ciuti, Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices, Phys. Rev. A 99, 032115 (2019).
  76. See Supplemental Material at http://link.aps.org/supplemental/10.1103/pdnh-1yxs for additional details on the derivation of the Gross-Pitaevskii equation, Lyapunov phase diagrams, frequency spectra of the classical dynamics, the truncated Wigner approximation, additional visualizations of the time evolution of Wigner functions, numerical extraction of Liouvillian gaps, universal scaling analysis, robustness against single-photon loss and thermal noise, and implementation schemes in trapped-ion systems, which includes Refs. [55, 60, 104, 105, 106, 107, 108, 109].
  77. M. I. Dykman, Critical exponents in metastable decay via quantum activation, Phys. Rev. E 75, 011101 (2007).
  78. H. B. Chan and C. Stambaugh, Activation barrier scaling and crossover for noise-induced switching in micromechanical parametric oscillators, Phys. Rev. Lett. 99, 060601 (2007).
  79. The time-resolved spectral analysis enables a high-precision extraction of the individual frequency components and their lifetimes, directly corresponding to the imaginary and real parts of Liouvillian eigenvalues, respectively. It provides a robust alternative to linewidth-based estimates, particularly in the presence of noise and non-stationarity.
  80. A. L. Schawlow and C. H. Townes, Infrared and optical masers, Phys. Rev. 112, 1940 (1958).
  81. A. Chia, M. Hajdušek, R. Fazio, L.-C. Kwek, and V. Vedral, Phase diffusion and the small-noise approximation in linear amplifiers: Limitations and beyond, Quantum 3, 200 (2019).
  82. F. Ferrari, F. Minganti, C. Aron, and V. Savona, Chaotic and quantum dynamics in driven-dissipative bosonic chains, Commun. Phys. 8, 407 (2025).
  83. L. Kruglikov, F. Ferrari, and V. Savona, Chaos and quantum regimes in n-photon-driven dissipative bosonic chains, Phys. Rev. A 112, 052216 (2025).
  84. We note that the same scaling analysis can equivalently be performed using the time-resolved spectrum Eq. (9), which resolves both temporal decay at fixed ℵ and its complementary dependence on ℵ at fixed time.
  85. Y. Hidaka and Y. Minami, Spontaneous symmetry breaking and Nambu–Goldstone modes in open classical and quantum systems, Prog. Theor. Exp. Phys. 2020, 033A01 (2020).
  86. S. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed. (CRC Press, Boca Raton, FL, 2019), pp. 470–482.
  87. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (CRC Press, Boca Raton, FL, 2018).
  88. H. Alaeian, G. Giedke, I. Carusotto, R. Löw, and T. Pfau, Limit cycle phase and Goldstone mode in driven dissipative systems, Phys. Rev. A 103, 013712 (2021).
  89. K. Ptaszyński and M. Esposito, Dynamical signatures of discontinuous phase transitions: How phase coexistence determines exponential versus power-law scaling, Phys. Rev. E 110, 044134 (2024).
  90. G. Beaulieu, F. Minganti, S. Frasca, V. Savona, S. Felicetti, R. Di Candia, and P. Scarlino, Observation of first- and second-order dissipative phase transitions in a two-photon driven Kerr resonator, Nat. Commun. 16, 1954 (2025).
  91. J. Steinbach, J. Twamley, and P. L. Knight, Engineering two-mode interactions in ion traps, Phys. Rev. A 56, 4815 (1997).
  92. M. Kajita, Ion Traps, 1st ed. (Institute of Physics Publishing, Bristol, 2022).
  93. G. S. Agarwal and J. Banerji, Fractional revivals in systems with two timescales, Phys. Rev. A 57, 3880 (1998).
  94. J. Walz, I. Siemers, M. Schubert, W. Neuhauser, R. Blatt, and E. Teloy, Ion storage in the rf octupole trap, Phys. Rev. A 50, 4122 (1994).
  95. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, Laser cooling of trapped ions in a standing wave, Phys. Rev. A 46, 2668 (1992).
  96. F. Reiter and A. S. Sørensen, Effective operator formalism for open quantum systems, Phys. Rev. A 85, 032111 (2012).
  97. T. Hillmann and F. Quijandría, Designing Kerr interactions for quantum information processing via counterrotating terms of asymmetric Josephson-junction loops, Phys. Rev. Appl. 17, 064018 (2022).
  98. R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys. 16, 509 (2020).
  99. A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller, S. O. Mundhada, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Robust concurrent remote entanglement between two superconducting qubits, Phys. Rev. X 6, 031036 (2016).
  100. S. Ding, G. Maslennikov, R. Hablützel, H. Loh, and D. Matsukevich, Quantum parametric oscillator with trapped ions, Phys. Rev. Lett. 119, 150404 (2017).
  101. L. G. Lutterbach and L. Davidovich, Method for direct measurement of the Wigner function in cavity QED and ion traps, Phys. Rev. Lett. 78, 2547 (1997).
  102. M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O'Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, Nature (London) 459, 546 (2009).
  103. K. He, M. Yuan, Y. Wong, S. Chakram, A. Seif, L. Jiang, and D. I. Schuster, Efficient multimode Wigner tomography, Nat. Commun. 15, 4138 (2024).
  104. L. Escot Mangas and J. Sandubete Galán, A brief methodological note of chaos theory based on recent application on computing resources, Energeia VII, 53 (2020).
  105. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica 15, 9 (1980).
  106. G. Datseris, DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics, J. Open Source Softw. 3, 598 (2018).
  107. G. Baier and M. Klein, A Chaotic Hierarchy (World Scientific, Singapore, 1991).
  108. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  109. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).

Outline

Information

Phys. Rev. A 113, 052208– Published 8 May, 2026
  • Received 4 July 2025
  • Accepted 10 April 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