Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Enhanced metrology at the critical point of a many-body Rydberg atomic system


Interacting many-body systems display enhanced sensitivity close to critical transition points due to diverging quantum fluctuations. This criticality-based enhancement has been suggested as a potential resource for applications in precision metrology. Here we demonstrate many-body critical enhanced metrology for the sensing of external microwave electric fields in a non-equilibrium Rydberg atomic gas. We show that small variations in external driving lead to a large variation in the population of Rydberg states around criticality and to a notable change in the optical transmission signal. For continuous optical transmission at the critical point, we quantify the enhanced sensitivity extracting the Fisher information, which shows a three orders of magnitude increase due to many-body effects compared with single-particle systems. These results demonstrate that critical properties of many-body systems are promising resources for sensing and metrology applications.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Principle of single-body (top) (many-body (bottom)) Rydberg metrology.
Fig. 2: Optical transmission spectra with and without phase transition.
Fig. 3: Transmission spectra and associated FI.
Fig. 4: Change in transmission spectra by application of MW fields.
Fig. 5: Transmission under different amplitudes of MW field.
Fig. 6: Theoretical simulations of interacting two-level atoms.

Data availability

The data that support this study are available via GitHub56 at Source data are provided with this paper.


  1. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).

    Article  ADS  Google Scholar 

  2. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    Article  ADS  Google Scholar 

  3. Martin, M. J. et al. A quantum many-body spin system in an optical lattice clock. Science 341, 632–636 (2013).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Colombo, S., Pedrozo-Peñafiel, E., Adiyatullin, A.F. et al. Time-reversal-based quantum metrology with many-body entangled states. Nat. Phys. 18, 925–930. (2022).

  5. Lukin, M. D. et al. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001).

    Article  ADS  Google Scholar 

  6. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313 (2010).

    Article  ADS  Google Scholar 

  7. Carr, C., Ritter, R., Wade, C. G., Adams, C. S. & Weatherill, K. J. Nonequilibrium phase transition in a dilute Rydberg ensemble. Phys. Rev. Lett. 111, 113901 (2013).

    Article  ADS  Google Scholar 

  8. Ding, Dong-Sheng, Busche, H., Shi, Bao-Sen, Guo, Guang-Can & Adams, C. S. Phase diagram of non-equilibrium phase transition in a strongly-interacting Rydberg atom vapour. Phys. Rev. X 10, 021023 (2020).

    Google Scholar 

  9. Malossi, N. et al. Full counting statistics and phase diagram of a dissipative Rydberg gas. Phys. Rev. Lett. 113, 023006 (2014).

    Article  ADS  Google Scholar 

  10. de Melo, N. R. et al. Intrinsic optical bistability in a strongly driven Rydberg ensemble. Phys. Rev. A. 93, 063863 (2016).

    Article  ADS  Google Scholar 

  11. Šibalić, N., Wade, C. G., Adams, C. S., Weatherill, K. J. & Pohl, T. Driven-dissipative many-body systems with mixed power-law interactions: bistabilities and temperature-driven nonequilibrium phase transitions. Phys. Rev. A. 94, 011401 (2016).

    Article  ADS  Google Scholar 

  12. Wade, C. G. et al. A terahertz-driven non-equilibrium phase transition in a room temperature atomic vapour. Nat. Commun. 9, 3567 (2018).

    Article  ADS  Google Scholar 

  13. Wintermantel, T. M. et al. Epidemic growth and Griffiths effects on an emergent network of excited atoms. Nat. Commun. 12, 103 (2020).

    Article  ADS  Google Scholar 

  14. Ding, D.-S. et al. Epidemic spreading and herd immunity in a driven non-equilibrium system of strongly-interacting atoms. Preprint at (2021).

  15. Gibbs, H. M., McCall, S. L. & Venkatesan, T. N. C. Differential gain and bistability using a sodium-filled Fabry-Perot interferometer. Phys. Rev. Lett. 36, 1135 (1976).

    Article  ADS  Google Scholar 

  16. Wang, H., Goorskey, D. J. & Xiao, M. Bistability and instability of three-level atoms inside an optical cavity. Phys. Rev. A. 65, 011801 (2001).

    Article  ADS  Google Scholar 

  17. Wang, H., Goorskey, D. & Xiao, M. Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system. Phys. Rev. Lett. 87, 073601 (2001).

    Article  ADS  Google Scholar 

  18. Pickup, L. et al. Optical bistability under nonresonant excitation in spinor polariton condensates. Phys. Rev. Lett. 120, 225301 (2018).

    Article  ADS  Google Scholar 

  19. Hehlen, M. P. et al. Cooperative bistability in dense, excited atomic systems. Phys. Rev. Lett. 73, 1103 (1994).

    Article  ADS  Google Scholar 

  20. Lee, T. E., Haeffner, H. & Cross, M. C. Collective quantum jumps of Rydberg atoms. Phys. Rev. Lett. 108, 023602 (2012).

    Article  ADS  Google Scholar 

  21. Marcuzzi, M., Levi, E., Diehl, S., Garrahan, J. P. & Lesanovsky, I. Universal nonequilibrium properties of dissipative Rydberg gases. Phys. Rev. Lett. 113, 210401 (2014).

    Article  ADS  Google Scholar 

  22. Weimer, H. Variational principle for steady states of dissipative quantum many-body systems. Phys. Rev. Lett. 114, 040402 (2015).

    Article  ADS  Google Scholar 

  23. Levi, E., Gutiérrez, R. & Lesanovsky, I. Quantum non-equilibrium dynamics of Rydberg gases in the presence of dephasing noise of different strengths. J. Phys. B: At. Mol. Opt. Phys. 49, 184003 (2016).

    Article  ADS  Google Scholar 

  24. Fan, H. et al. Atom based RF electric field sensing. J. Phys. B: At. Mol. Opt. Phys. 48, 202001 (2015).

    Article  ADS  Google Scholar 

  25. Sedlacek, J. A. et al. Microwave electrometry with Rydberg atoms in a vapour cell using bright atomic resonances. Nat. Phys. 8, 819–824 (2012).

    Article  Google Scholar 

  26. Facon, A. et al. A sensitive electrometer based on a Rydberg atom in a Schrödinger-cat state. Nature 535, 262–265 (2016).

    Article  ADS  Google Scholar 

  27. Cox, K. C., Meyer, D. H., Fatemi, F. K. & Kunz, P. D. Quantum-limited atomic receiver in the electrically small regime. Phys. Rev. Lett. 121, 110502 (2018).

    Article  ADS  Google Scholar 

  28. Jing, M. et al. Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy. Nat. Phys. 16, 911–915 (2020).

    Article  Google Scholar 

  29. Liu, Zong-Kai et al. Deep learning enhanced Rydberg multifrequency microwave recognition. Nat. Commun. 13, 1997 (2022).

    Article  ADS  Google Scholar 

  30. Gammelmark, S. & Mølmer, K. Phase transitions and Heisenberg limited metrology in an Ising chain interacting with a single-mode cavity field. New J. Phys. 13, 053035 (2011).

    Article  ADS  Google Scholar 

  31. Macieszczak, K., Guţă, M., Lesanovsky, I. & Garrahan, J. P. Dynamical phase transitions as a resource for quantum enhanced metrology. Phys. Rev. A 93, 022103 (2016).

    Article  ADS  Google Scholar 

  32. Fernández-Lorenzo, S. & Porras, D. Quantum sensing close to a dissipative phase transition: symmetry breaking and criticality as metrological resources. Phys. Rev. A 96, 013817 (2017).

    Article  ADS  Google Scholar 

  33. Raghunandan, M., Wrachtrup, J. örg & Weimer, H. High-density quantum sensing with dissipative first order transitions. Phys. Rev. Lett. 120, 150501 (2018).

    Article  ADS  Google Scholar 

  34. Garbe, L., Bina, M., Keller, A., Paris, MatteoG. A. & Felicetti, S. Critical quantum metrology with a finite-component quantum phase transition. Phys. Rev. Lett. 124, 120504 (2020).

    Article  ADS  Google Scholar 

  35. Chu, Y., Zhang, S., Yu, B. & Cai, J. Dynamic framework for criticality-enhanced quantum sensing. Phys. Rev. Lett. 126, 010502 (2021).

    Article  MathSciNet  ADS  Google Scholar 

  36. Montenegro, V., Mishra, U. & Bayat, A. Global sensing and its impact for quantum many-body probes with criticality. Phys. Rev. Lett. 126, 200501 (2021).

    Article  MathSciNet  ADS  Google Scholar 

  37. Ilias, T., Yang, D., Huelga, S. F. & Plenio, M. B. Criticality enhanced quantum sensing via continuous measurement. PRX Quantum 3, 010354. 10.1103/PRXQuantum.3.010354 (2022).

  38. Garbe, L., Abah, O., Felicetti, S. & Puebla, R. Critical quantum metrology with fully-connected models: from Heisenberg to Kibble-Zurek scaling. Quantum Sci. Technol. 7 035010 (2022).

  39. Liu, R. et al. Experimental critical quantum metrology with the Heisenberg scaling. npj Quantum Inf. 7, 170 (2021).

    Article  ADS  Google Scholar 

  40. Zanardi, P., Paris, MatteoG. A. & Campos Venuti, L. Quantum criticality as a resource for quantum estimation. Phys. Rev. A 78, 042105 (2008).

    Article  ADS  Google Scholar 

  41. Rossini, D. & Vicari, E. Dynamic Kibble-Zurek scaling framework for open dissipative many-body systems crossing quantum transitions. Phys. Rev. Res. 2, 023211 (2020).

    Article  Google Scholar 

  42. Pelissetto, A., Rossini, D. & Vicari, E. Dynamic finite-size scaling after a quench at quantum transitions. Phys. Rev. E 97, 052148 (2018).

    Article  ADS  Google Scholar 

  43. Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. 90, 035005 (2018).

    Article  MathSciNet  ADS  Google Scholar 

  44. Braunstein, S. L., Caves, C. M. & Milburn, G. J. Generalized uncertainty relations: theory, examples, and Lorentz invariance. Ann. Phys. 247, 135–173 (1996).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  45. Zurek, W. H., Dorner, U. & Zoller, P. Dynamics of a quantum phase transition. Phys. Rev. Lett. 95, 105701 (2005).

    Article  ADS  Google Scholar 

  46. Clark, L. W., Feng, L. & Chin, C. Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition. Science 354, 606–610 (2016).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  47. Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature 568, 207–211 (2019).

    Article  ADS  Google Scholar 

  48. Trenkwalder, A. et al. Quantum phase transitions with parity-symmetry breaking and hysteresis. Nat. Phys. 12, 826–829 (2016).

    Article  Google Scholar 

  49. Negretti, A., Henkel, C. & Mølmer, K. Quantum-limited position measurements of a dark matter-wave soliton. Phys. Rev. A 77, 043606 (2008).

    Article  ADS  Google Scholar 

  50. Delaubert, V., Treps, N., Fabre, C., Bachor, H. A. & Réfrégier, P. Quantum limits in image processing. EPL 81, 44001 (2008).

    Article  ADS  Google Scholar 

  51. Šibalić, N., Pritchard, J. D., Adams, C. S. & Weatherill, K. J. ARC: an open-source library for calculating properties of alkali Rydberg atoms. Comput. Phys. Commun. 220, 319–331 (2017).

    Article  MATH  ADS  Google Scholar 

  52. Lehmann, E. L. & Casella, G. Theory of Point Estimation (Springer, 1998).

  53. Mardia, K. V. & Marshall, R. J. Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135–146 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  54. Miller, K. S. Complex Stochastic Processes: An Introduction to Theory and Application (Addison Wesley Publishing Company, 1974).

  55. Weller, D., Urvoy, A., Rico, A., Löw, R. & Kübler, H. Charge-induced optical bistability in thermal Rydberg vapor. Phys. Rev. A. 94, 063820 (2016).

    Article  ADS  Google Scholar 

  56. Zong-Kai, L. Original data for ‘Enhanced metrology at the critical point of a many-body Rydberg atomic system’. GitHub (2022).

Download references


D.-S.D. thanks the discussions with J. Ye (JILA). Z.-K.L. appreciates the instructive discussions with T.-Y. Xie. D.-S.D. acknowledges funding from the National Key Research and Development Program of China (2017YFA0304800), the National Natural Science Foundation of China (grant no. U20A20218), the Major Science and Technology Projects in Anhui Province (grant no. 202203a13010001) and the Youth Innovation Promotion Association of the Chinese Academy of Sciences under grant no. 2018490. B.-S.S. acknowledges funding from the National Natural Science Foundation of China (grant no. 11934013), the Innovation Program for Quantum Science and Technology (2021ZD0301100) and Anhui Initiative in Quantum Information Technologies (AHY020200). C.-S.A. acknowledges funding from the EPSRC through grant agreements EP/M014398/1 (‘Rydberg Soft Matter’), EP/R002061/1 (‘Atom-based Quantum Photonics’), EP/L023024/1 (‘Cooperative Quantum Optics in Dense Thermal Vapours’), EP/P012000/1 (‘Solid State Superatoms’), EP/R035482/1 (‘Optical Clock Arrays for Quantum Metrology’) and EP/S015973/1 (‘Microwave and Terahertz Field Sensing and Imaging using Rydberg Atoms’); the Danish National Research Foundation through the Center of Excellence for Complex Quantum Systems (grant agreement no. DNRF156); DSTL; and Durham University.

Author information

Authors and Affiliations



D.-S.D. conceived the idea and implemented the physical experiments with Z.-K.L. Z.-K.L., D.-S.D. and K.M. employed the FI. D.-S.D., Z.-K.L. and K.M. derived the equations, plotted the figures and wrote the manuscript. All the authors contributed to the discussions regarding the results and analysis contained in the manuscript. D.-S.D., B.-S.S., G.-C.G. and C.-S.A. sponsor this project.

Corresponding authors

Correspondence to Dong-Sheng Ding, Bao-Sen Shi, Klaus Mølmer or Charles S. Adams.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Shannon Whitlock and Abolfazl Bayat for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–4.

Supplementary Data 1

Data for Supplementary Fig. 1.

Supplementary Data 2

Data for Supplementary Fig. 2.

Supplementary Data 3

Data for Supplementary Fig. 3.

Source data

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Source Data Fig. 4

Statistical source data.

Source Data Fig. 5

Statistical source data.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ding, DS., Liu, ZK., Shi, BS. et al. Enhanced metrology at the critical point of a many-body Rydberg atomic system. Nat. Phys. 18, 1447–1452 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing