- 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

Analytical phase boundary of a quantum driven-dissipative Kerr oscillator from classical stochastic instantons

Théo Sépulcre*,†

  • *Contact author: theo.sepulcre@riken.jp
  • †Present address: Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan.
PDF

Phys. Rev. Research 8, L012058 – Published 9 March, 2026

DOI: https://doi.org/10.1103/528d-l76d

Abstract

The framework of Keldysh path integral concisely describes quantum systems driven away from thermal equilibrium, such as the two-photon driven Kerr oscillator. Within the thermodynamic limit of diverging photon number, we map it to a Martin-Siggia-Rose-Janssen-de Dominicis path integral and obtain a purely classical, stochastic equivalent where photon self-interaction plays the role of temperature. This perspective sheds light on the difficulties encountered in the search for an effective thermodynamic potential to describe the bistability of the model. It allows us to estimate the bistable tunneling rates using a real-time instanton technique leading to an analytical expression of the phase boundary, the first to our knowledge. It opens the way to powerful semianalytical techniques to be applied to various quantum optics models displaying bistability.

View figure in article

Physics Subject Headings (PhySH)

See Also

Nonperturbative Switching Rates in Bistable Open Quantum Systems: From Driven Kerr Oscillators to Dissipative Cat Qubits

Léon Carde, Ronan Gautier, Nicolas Didier, Alexandru Petrescu, Joachim Cohen, and Alexander McDonald
Phys. Rev. Lett. 136, 100402 (2026)

Article Text

References (49)

  1. J. Clerk-Maxwell, On the dynamical evidence of the molecular constitution of bodies, Nature (London) 11, 357 (1875).
  2. H. Poincaré, Science et méthode, Bibliothèque de Philosophie Scientifique (Flammarion, Paris, 1908).
  3. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011).
  4. E. G. Dalla Torre, E. Demler, T. Giamarchi, and E. Altman, Quantum critical states and phase transitions in the presence of non-equilibrium noise, Nat. Phys. 6, 806 (2010).
  5. L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Dynamical critical phenomena in driven-dissipative systems, Phys. Rev. Lett. 110, 195301 (2013).
  6. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016).
  7. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, New York, 2002).
  8. P. D. Drummond and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A: Math. Gen. 13, 725 (1980).
  9. 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).
  10. C. Castillo-Moreno, T. Sépulcre, T. Hillmann, K. R. Amin, M. Kervinen, and S. Gasparinetti, Experimental observation of multimode quantum phase transitions in a superconducting Bose-Hubbard simulator, arXiv:2508.20116.
  11. W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A 93, 033824 (2016).
  12. S. R. K. Rodriguez, W. Casteels, F. Storme, N. Carlon Zambon, I. Sagnes, L. Le Gratiet, E. Galopin, A. Lemaître, A. Amo, C. Ciuti, and J. Bloch, Probing a dissipative phase transition via dynamical optical hysteresis, Phys. Rev. Lett. 118, 247402 (2017).
  13. K. Petrovnin, J. Wang, M. Perelshtein, P. Hakonen, and G. S. Paraoanu, Microwave photon detection at parametric criticality, PRX Quantum 5, 020342 (2024).
  14. 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).
  15. P. R. Muppalla, O. Gargiulo, S. I. Mirzaei, B. P. Venkatesh, M. L. Juan, L. Grünhaupt, I. M. Pop, and G. Kirchmair, Bistability in a mesoscopic Josephson junction array resonator, Phys. Rev. B 97, 024518 (2018).
  16. C. K. Andersen, A. Kamal, N. A. Masluk, I. M. Pop, A. Blais, and M. H. Devoret, Quantum versus classical switching dynamics of driven dissipative Kerr resonators, Phys. Rev. Appl. 13, 044017 (2020).
  17. G. Yu. Kryuchkyan and K. V. Kheruntsyan, Exact quantum theory of a parametrically driven dissipative anharmonic oscillator, Opt. Commun. 127, 230 (1996).
  18. K. V. Kheruntsyan, Wigner function for a driven anharmonic oscillator, J. Opt. B: Quantum Semiclassical Opt. 1, 225 (1999).
  19. F. Minganti, N. Bartolo, J. Lolli, W. Casteels, and C. Ciuti, Exact results for Schrödinger cats in driven-dissipative systems and their feedback control, Sci. Rep. 6, 26987 (2016).
  20. D. Roberts and A. A. Clerk, Driven-dissipative quantum Kerr resonators: New exact solutions, photon blockade and quantum bistability, Phys. Rev. X 10, 021022 (2020).
  21. X. H. H. Zhang and H. U. Baranger, Driven-dissipative phase transition in a Kerr oscillator: From semiclassical PT symmetry to quantum fluctuations, Phys. Rev. A 103, 033711 (2021).
  22. K. J. H. Peters, J. Busink, P. Ackermans, K. G. Cognée, and S. R. K. Rodriguez, Scalar potentials for light in a cavity, Phys. Rev. Res. 5, 013154 (2023).
  23. V. Yu. Mylnikov, S. O. Potashin, G. S. Sokolovskii, and N. S. Averkiev, Emergent equilibrium and quantum criticality in a two-photon dissipative oscillator, Phys. Rev. Res. 7, 013061 (2025).
  24. K. Petrovnin, J. Wang, and G. S. Paraoanu, Numerical simulation methods for quantum sensing at parametric criticality, Commun. Phys. 8, 378 (2025).
  25. M. F. Maghrebi and A. V. Gorshkov, Nonequilibrium many-body steady states via Keldysh formalism, Phys. Rev. B 93, 014307 (2016).
  26. F. Thompson and A. Kamenev, Field theory of many-body Lindbladian dynamics, Ann. Phys. 455, 169385 (2023).
  27. M. Titov and D. B. Gutman, Korshunov instantons out of equilibrium, Phys. Rev. B 93, 155428 (2016).
  28. F. Thompson and A. Kamenev, Qubit decoherence and symmetry restoration through real-time instantons, Phys. Rev. Res. 4, 023020 (2022).
  29. C.-W. Lee, P. Brookes, K.-S. Park, M. H. Szymańska, and E. Ginossar, Real-time instanton approach to quantum activation, Phys. Rev. A 112, 012216 (2025).
  30. L. Carde, R. Gautier, N. Didier, A. Petrescu, J. Cohen, and A. McDonald, companion paper, Nonperturbative switching rates in bistable open quantum systems: From driven Kerr oscillators to dissipative cat qubits, Phys. Rev. Lett. 136, 100402 (2026).
  31. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011).
  32. U. C. Täuber, Field-theory approaches to nonequilibrium dynamics, in Ageing and the Glass Transition, edited by M. Henkel, M. Pleimling, and R. Sanctuary (Springer, Berlin, 2007), pp. 295–348.
  33. J. A. Hertz, Y. Roudi, and P. Sollich, Path integral methods for the dynamics of stochastic and disordered systems, J. Phys. A: Math. Theory 50, 033001 (2017).
  34. R. Zakine and E. Vanden-Eijnden, Minimum-action method for nonequilibrium phase transitions, Phys. Rev. X 13, 041044 (2023).
  35. A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2010).
  36. M. Foss-Feig, P. Niroula, J. T. Young, M. Hafezi, A. V. Gorshkov, R. M. Wilson, and M. F. Maghrebi, Emergent equilibrium in many-body optical bistability, Phys. Rev. A 95, 043826 (2017).
  37. The photon occupation number also diverges when U/γ→0, which can be interpreted as the large number of degrees of freedom usually expected at the thermodynamic limit.
  38. R. Bonifacio, M. Gronchi, and L. A. Lugiato, Photon statistics of a bistable absorber, Phys. Rev. A 18, 2266 (1978).
  39. K. Vogel and H. Risken, Quasiprobability distributions in dispersive optical bistability, Phys. Rev. A 39, 4675 (1989).
  40. I. Carusotto and C. Ciuti, Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies, Phys. Rev. B 72, 125335 (2005).
  41. I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).
  42. V. P. Singh and H. Weimer, Driven-dissipative criticality within the discrete truncated Wigner approximation, Phys. Rev. Lett. 128, 200602 (2022).
  43. C. D. Mink, A. Pelster, J. Benary, H. Ott, and M. Fleischhauer, Variational truncated Wigner approximation for weakly interacting Bose fields: Dynamics of coupled condensates, SciPost Phys. 12, 051 (2022).
  44. C. Rackauckas and Q. Nie, Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory, Discrete Contin. Dyn. Syst. B 22, 2731 (2017).
  45. P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990).
  46. G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36, 823 (1930).
  47. The full tunneling trajectory will continue via a trapping, actionless path toward the next stable fixed point, and is ignored in our analysis.
  48. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, 2007).
  49. G. Beaulieu, F. Minganti, S. Frasca, M. Scigliuzzo, S. Felicetti, R. Di Candia, and P. Scarlino, Criticality-enhanced quantum sensing with a parametric superconducting resonator, PRX Quantum 6, 020301 (2025).

Outline

Information

Phys. Rev. Research 8, L012058– Published 9 March, 2026
  • Received 12 August 2025
  • Accepted 20 January 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