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  • Open Access

Equilibrium thermometry in the multilevel quantum Rabi model

Tabitha Doicin1,*, Luis A. Correa2,3, Jonas Glatthard4, Andrew D. Armour4,5, and Gerardo Adesso1,5

  • *Contact author: tabitha.doicin@nottingham.ac.uk
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Phys. Rev. A 113, 062449 – Published 22 June, 2026

DOI: https://doi.org/10.1103/mb4g-5npp

Abstract

The temperature sensitivity of a probe in equilibrium can be gauged by its thermal quantum Fisher information (QFI). It is known that probes exhibiting degeneracy in their energy-level structure can achieve larger sensitivities, while probes with a more uniform spectrum may remain sensitive over a broader temperature range. Here, we study the thermometric performance of a multilevel quantum Rabi model in which two well-separated atomic manifolds of near-degenerate levels couple to a single cavity mode. We generalize the standard quantum Rabi treatment in the adiabatic regime to find an approximate closed-form expression for the thermal QFI. We then characterize two complementary limits. On the one hand, a large dark-state manifold (dark-manifold saturation) produces a robust peak in thermal sensitivity due to bright-dark population transfer. Such increase in sensitivity is further maximized at an intermediate light-matter coupling strength. Maximizing instead the number of bright states (bright-manifold saturation) generates a broadband thermal response that becomes increasingly stable under random light-matter couplings as the number of levels is increased. The rich spectral structure of our cavity-QED model thus makes it a versatile and sensitive equilibrium thermometer over a broad range of temperatures.

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References (48)

  1. M. Mehboudi, A. Sanpera, and L. A. Correa, Thermometry in the quantum regime: Recent theoretical progress, J. Phys. A: Math. Theor. 52, 303001 (2019).
  2. A. De Pasquale and T. M. Stace, Quantum thermometry, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018), p. 503.
  3. S. Campbell, I. d'Amico, M. A. Ciampini, J. Anders, N. Ares, S. Artini, A. Auffèves, L. B. Oftelie, L. P. Bettmann, M. V. Bonança, et al., Roadmap on quantum thermodynamics, Quantum Sci. Technol. 11, 012501 (2026).
  4. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  5. C. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A 25, 101 (1967).
  6. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  7. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A: Math. Theor. 53, 023001 (2020).
  8. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 07, 125 (2009).
  9. T. Jahnke, S. Lanéry, and G. Mahler, Operational approach to fluctuations of thermodynamic variables in finite quantum systems, Phys. Rev. E 83, 011109 (2011).
  10. L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Individual quantum probes for optimal thermometry, Phys. Rev. Lett. 114, 220405 (2015).
  11. S. Campbell, M. G. Genoni, and S. Deffner, Precision thermometry and the quantum speed limit, Quantum Sci. Technol. 3, 025002 (2018).
  12. M. Płodzień, R. Demkowicz-Dobrzański, and T. Sowiński, Few-fermion thermometry, Phys. Rev. A 97, 063619 (2018).
  13. W.-K. Mok, K. Bharti, L.-C. Kwek, and A. Bayat, Optimal probes for global quantum thermometry, Commun. Phys. 4, 62 (2021).
  14. P. Abiuso, P. Andrea Erdman, M. Ronen, F. Noé, G. Haack, and M. Perarnau-Llobet, Optimal thermometers with spin networks, Quantum Sci. Technol. 9, 035008 (2024).
  15. A. K. Srivastava, U. Bhattacharya, M. Lewenstein, and M. Płodzień, Topological quantum thermometry, Phys. Rev. A 111, 052216 (2025).
  16. V. Mukherjee, A. Zwick, A. Ghosh, X. Chen, and G. Kurizki, Enhanced precision bound of low-temperature quantum thermometry via dynamical control, Commun. Phys. 2, 162 (2019).
  17. J. Glatthard and L. A. Correa, Bending the rules of low-temperature thermometry with periodic driving, Quantum 6, 705 (2022).
  18. D. Reeb and M. M. Wolf, Tight bound on relative entropy by entropy difference, IEEE Trans. Inf. Theory 61, 1458 (2015).
  19. A. Ullah, O. E. Müstecaplıoğlu, and M. G. A. Paris, Configuration-dependent precision in magnetometry and thermometry using multiqubit quantum sensors, Phys. Rev. A 113, 012408 (2026).
  20. A. Ullah, M. T. Naseem, and O. E. Müstecaplıoğlu, Low-temperature quantum thermometry boosted by coherence generation, Phys. Rev. Res. 5, 043184 (2023).
  21. A. Ullah, M. Tahir Naseem, and O. E. Müstecaplıoğlu, Mixing thermal coherent states for precision and range enhancement in quantum thermometry, Quantum Sci. Technol. 10, 015044 (2025).
  22. E. T. Jaynes and F. W. Cummings, Comparison of quantum and semi-classical radiation theories with application to beam maser, Proc. IEEE 51, 89 (1963).
  23. Q. Xie, H. Zhong, M. T. Batchelor, and C. Lee, The quantum Rabi model: Solution and dynamics, J. Phys. A: Math. Theor. 50, 113001 (2017).
  24. D. Braak, Q.-H. Chen, M. T. Batchelor, and E. Solano, Semi-classical and quantum Rabi models: In celebration of 80 years, J. Phys. A: Math. Theor. 49, 300301 (2016).
  25. T. Tufarelli, D. Friedrich, H. Groß, J. Hamm, O. Hess, and B. Hecht, Single quantum emitter Dicke enhancement, Phys. Rev. Res. 3, 033103 (2021).
  26. T. Doicin, A. D. Armour, and T. Tufarelli, Multilevel quantum Rabi models, J. Phys. A: Math. Theor. 58, 315203 (2025).
  27. P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
  28. J. Keller, G. Scalari, F. Appugliese, S. Rajabali, M. Beck, J. Haase, C. A. Lehner, W. Wegscheider, M. Failla, M. Myronov, D. R. Leadley, J. Lloyd-Hughes, P. Nataf, and J. Faist, Landau polaritons in highly nonparabolic two-dimensional gases in the ultrastrong coupling regime, Phys. Rev. B 101, 075301 (2020).
  29. V. M. Muravev, I. V. Andreev, I. V. Kukushkin, S. Schmult, and W. Dietsche, Observation of hybrid plasmon-photon modes in microwave transmission of coplanar microresonators, Phys. Rev. B 83, 075309 (2011).
  30. C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, Ultrastrong coupling in the near field of complementary split-ring resonators, Phys. Rev. B 90, 205309 (2014).
  31. L. V. Abdurakhimov, R. Yamashiro, A. O. Badrutdinov, and D. Konstantinov, Strong coupling of the cyclotron motion of surface electrons on liquid helium to a microwave cavity, Phys. Rev. Lett. 117, 056803 (2016).
  32. F. W. Wise, Lead salt quantum dots: The limit of strong quantum confinement, Acc. Chem. Res. 33, 773 (2000).
  33. H. Groß, J. M. Hamm, T. Tufarelli, O. Hess, and B. Hecht, Near-field strong coupling of single quantum dots, Sci. Adv. 4, eaar4906 (2018).
  34. H. Liu and P. Guyot-Sionnest, Photoluminescence lifetime of lead selenide colloidal quantum dots, J. Phys. Chem. C 114, 14860 (2010).
  35. Z. Hens and I. Moreels, Light absorption by colloidal semiconductor quantum dots, J. Mater. Chem. 22, 10406 (2012).
  36. D. Z. Rossatto, C. J. Villas-Bôas, M. Sanz, and E. Solano, Spectral classification of coupling regimes in the quantum Rabi model, Phys. Rev. A 96, 013849 (2017).
  37. E. K. Twyeffort Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Dynamics of a two-level system strongly coupled to a high-frequency quantum oscillator, Phys. Rev. B 72, 195410 (2005).
  38. S. Schweber, On the application of Bargmann Hilbert spaces to dynamical problems, Ann. Phys. (NY) 41, 205 (1967).
  39. M. D. Crisp, Application of the displaced oscillator basis in quantum optics, Phys. Rev. A 46, 4138 (1992).
  40. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Bužek, Properties of displaced number states, Phys. Rev. A 41, 2645 (1990).
  41. M. Chiani, Distribution of the largest eigenvalue for real wishart and gaussian random matrices and a simple approximation for the tracy–widom distribution, J. Multivar. Anal. 129, 69 (2014).
  42. J. Kolodynski, Precision bounds in noisy quantum metrology, arXiv:1409.0535.
  43. P. Blanchard, D. J. Higham, and N. J. Higham, Accurate computation of the log-sum-exp and softmax functions, arXiv:1909.03469.
  44. E. S. R. Gopal, Specific Heats at Low Temperatures (Springer, Berlin, 1966).
  45. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002).
  46. H. Dette, Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials, J. Approx. Theory 118, 290 (2002).
  47. F. Marcellan, A. Martinez-Finkelshtein, and P. Martinez-Gonzalez, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207, 258 (2007).
  48. T. Doicin, TabithaDoicin/EquilibriumThermometryMQRM, Version 1.0.0 [Software], Zenodo, 2026, https://doi.org/10.5281/zenodo.18626760.

Outline

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Phys. Rev. A 113, 062449– Published 22 June, 2026
  • Received 17 February 2026
  • Accepted 3 June 2026
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