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Higher symmetry breaking and nonreciprocity in a driven-dissipative Dicke model

Jacquelyn Ho1,2, Yue-Hui Lu1,2, Tai Xiang1,2, Tsai-Chen Lee1,2, Zhenjie Yan1,2, and Dan M. Stamper-Kurn1,2,3,*

  • *Contact author: dmsk@berkeley.edu
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Phys. Rev. Research 8, 013311 – Published 24 March, 2026

DOI: https://doi.org/10.1103/5hlh-9sfb

Abstract

Higher symmetries in interacting many-body systems often give rise to new phases and unexpected dynamical behavior. Here, we theoretically investigate a variant of the Dicke model with higher-order discrete symmetry, resulting from complex-valued coupling coefficients between quantum emitters and a bosonic mode. We propose a driven-dissipative realization of this model focusing on optomechanical response of a driven atom tweezer array comprised of n subensembles and placed within an optical cavity, with the phase of the driving field advancing stepwise between subensembles. Examining stationary points and their dynamical stability, we identify a phase diagram for n≥3 with three distinctive features: a Zn (Z2n) symmetry-breaking superradiant phase for even (odd) n, a normal unbroken-symmetry phase that is dynamically unstable due to nonreciprocal forces between emitters, and a first-order phase transition separating these phases. This n-phase Dicke model may be equivalently realized in a variety of optomechanical or optomagnonic settings, where it can serve as a test bed for studying high-order symmetry breaking and nonreciprocal interactions in open systems.

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References (49)

  1. H. Hinrichsen, Non-equilibrium phase transitions, Physica A 369, 1 (2006).
  2. M. H. Matheny, J. Emenheiser, W. Fon, A. Chapman, A. Salova, M. Rohden, J. Li, M. Hudoba De Badyn, M. Pósfai, L. Duenas-Osorio, M. Mesbahi, J. P. Crutchfield, M. C. Cross, R. M. D’Souza, and M. L. Roukes, Exotic states in a simple network of nanoelectromechanical oscillators, Science 363, eaav7932 (2019).
  3. K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Exploring symmetry breaking at the Dicke quantum phase transition, Phys. Rev. Lett. 107, 140402 (2011).
  4. J. T. Young, A. V. Gorshkov, M. Foss-Feig, and M. F. Maghrebi, Nonequilibrium fixed points of coupled Ising models, Phys. Rev. X 10, 011039 (2020).
  5. M. H. Szymańska, J. Keeling, and P. B. Littlewood, Nonequilibrium quantum condensation in an incoherently pumped dissipative system, Phys. Rev. Lett. 96, 230602 (2006).
  6. R. Belyansky, C. Weis, R. Hanai, P. B. Littlewood, and A. A. Clerk, Phase transitions in nonreciprocal driven-dissipative condensates, Phys. Rev. Lett. 135, 123401 (2025).
  7. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  8. K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. 76, 360 (1973).
  9. V. M. Schäfer, Z. Niu, J. R. K. Cline, D. J. Young, E. Y. Song, H. Ritsch, and J. K. Thompson, Continuous recoil-driven lasing and cavity frequency pinning with laser-cooled atoms, Nat. Phys. 21, 902 (2025).
  10. M. A. Norcia, M. N. Winchester, J. R. K. Cline, and J. K. Thompson, Superradiance on the millihertz linewidth strontium clock transition, Sci. Adv. 2, e1601231 (2016).
  11. D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, Prospects for a millihertz-linewidth laser, Phys. Rev. Lett. 102, 163601 (2009).
  12. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010).
  13. F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity QED with quantum gases: New paradigms in many-body physics, Adv. Phys. 70, 1 (2021).
  14. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013).
  15. A. Safavi-Naini, R. J. Lewis-Swan, J. G. Bohnet, M. Gärttner, K. A. Gilmore, J. E. Jordan, J. Cohn, J. K. Freericks, A. M. Rey, and J. J. Bollinger, Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition, Phys. Rev. Lett. 121, 040503 (2018).
  16. J. Ho, Y.-H. Lu, T. Xiang, C. C. Rusconi, S. J. Masson, A. Asenjo-Garcia, Z. Yan, and D. M. Stamper-Kurn, Optomechanical self-organization in a mesoscopic atom array, Nat. Phys. 21, 1071 (2025).
  17. Z. Zhiqiang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. S. Parkins, and M. D. Barrett, Nonequilibrium phase transition in a spin-1 Dicke model, Optica 4, 424 (2017).
  18. M. Nairn, L. Giannelli, G. Morigi, S. Slama, B. Olmos, and S. B. Jäger, Spin self-organization in an optical cavity facilitated by inhomogeneous broadening, Phys. Rev. Lett. 134, 083603 (2025).
  19. V. V. Temnov and U. Woggon, Superradiance and subradiance in an inhomogeneously broadened ensemble of two-level systems coupled to a low-Q cavity, Phys. Rev. Lett. 95, 243602 (2005).
  20. P. Das, S. Wüster, and A. Sharma, Dicke model with disordered spin-boson couplings, Phys. Rev. A 109, 013715 (2024).
  21. Z. Zhang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. L. Grimsmo, A. S. Parkins, and M. D. Barrett, Dicke-model simulation via cavity-assisted Raman transitions, Phys. Rev. A 97, 043858 (2018).
  22. R. M. Kroeze, B. P. Marsh, D. A. Schuller, H. S. Hunt, A. N. Bourzutschky, M. Winer, S. Gopalakrishnan, J. Keeling, and B. L. Lev, Directly observing replica symmetry breaking in a vector quantum-optical spin glass, Science 389, 1122 (2025).
  23. B. P. Marsh, R. M. Kroeze, S. Ganguli, S. Gopalakrishnan, J. Keeling, and B. L. Lev, Entanglement and replica symmetry breaking in a driven-dissipative quantum spin glass, Phys. Rev. X 14, 011026 (2024).
  24. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  25. S. Chatterjee, S. Puri, and R. Paul, Ordering kinetics in the q-state clock model: Scaling properties and growth laws, Phys. Rev. E 98, 032109 (2018).
  26. G. Wang, D. C. Spierings, M. L. Peters, M.-W. Chen, U. Delić, and V. Vuletić, Programmable few-atom Bragg scattering and ground-state cooling in a cavity, arXiv:2508.10748.
  27. T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58, 1098 (1940).
  28. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5hlh-9sfb for details on stability analyses, phase boundaries, and trajectories.
  29. D. G. Schaeffer and J. W. Cain, Ordinary Differential Equations: Basics and Beyond, Texts in Applied Mathematics (Springer, New York, NY, 2016).
  30. S. C. Endres, C. Sandrock, and W. W. Focke, A simplicial homology algorithm for Lipschitz optimisation, J. Global Optim. 72, 181 (2018).
  31. N. Dogra, M. Landini, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger, Dissipation-induced structural instability and chiral dynamics in a quantum gas, Science 366, 1496 (2019).
  32. E. I. R. Chiacchio, A. Nunnenkamp, and M. Brunelli, Nonreciprocal Dicke model, Phys. Rev. Lett. 131, 113602 (2023).
  33. J. Jachinowski and P. B. Littlewood, Spin-only dynamics of the multi-species nonreciprocal Dicke model, arXiv:2507.07960.
  34. G. Lyu and M.-J. Hwang, Nonreciprocal and geometric frustration in dissipative quantum spins, arXiv:2508.06444.
  35. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  36. A. Metelmann and A. A. Clerk, Nonreciprocal photon transmission and amplification via reservoir engineering, Phys. Rev. X 5, 021025 (2015).
  37. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering, Nat. Phys. 13, 465 (2017).
  38. H. Xu, L. Jiang, A. A. Clerk, and J. G. E. Harris, Nonreciprocal control and cooling of phonon modes in an optomechanical system, Nature (London) 568, 65 (2019).
  39. M. Reisenbauer, H. Rudolph, L. Egyed, K. Hornberger, A. V. Zasedatelev, M. Abuzarli, B. A. Stickler, and U. Delić, Non-Hermitian dynamics and non-reciprocity of optically coupled nanoparticles, Nat. Phys. 20, 1629 (2024).
  40. 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).
  41. H. Rudolph, U. Delić, K. Hornberger, and B. A. Stickler, Quantum optical binding of nanoscale particles, Phys. Rev. Lett. 133, 233603 (2024).
  42. K. Yokomizo and Y. Ashida, Non-Hermitian physics of levitated nanoparticle array, Phys. Rev. Res. 5, 033217 (2023).
  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. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Optomechanical crystals, Nature (London) 462, 78 (2009).
  45. J. Vijayan, J. Piotrowski, C. Gonzalez-Ballestero, K. Weber, O. Romero-Isart, and L. Novotny, Cavity-mediated long-range interactions in levitated optomechanics, Nat. Phys. 20, 859 (2024).
  46. Q. Xu, G. Zheng, Y.-X. Wang, P. Zoller, A. A. Clerk, and L. Jiang, Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits, npj Quantum Inf. 9, 78 (2023).
  47. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  48. G.-L. Zhu, C.-S. Hu, H. Wang, W. Qin, X.-Y. Lü, and F. Nori, Nonreciprocal superradiant phase transitions and multicriticality in a cavity QED system, Phys. Rev. Lett. 132, 193602 (2024).
  49. G.-Q. Zhang, S.-Y. Lin, W. Feng, Y.-H. Kang, and W. Xiong, Nonreciprocal superradiant quantum phase transition induced by the magnon Kerr effect, Phys. Rev. B 113, 094403 (2026).

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Phys. Rev. Research 8, 013311– Published 24 March, 2026
  • Received 7 October 2025
  • Accepted 23 February 2026
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