Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

Concurrent spin squeezing and field tracking with machine learning

Nature Physics volume 21, pages 909–915 (2025)Cite this article

Subjects

Abstract

Squeezing and entanglement play crucial roles in approaches for quantum metrology. Yet, demonstrating quantum enhancement in continuous signal tracking remains a challenging endeavour because simultaneous entanglement generation and signal perturbations are often incompatible. We demonstrate that concurrent steady-state spin squeezing and sensing are possible using continuous quantum non-demolition measurements under constant optical pumping. We achieve a sustained spin-squeezed state with a large ensemble of hot atoms using metrologically relevant steady-state squeezing. We further employ the system to track different types of continuous time-fluctuating magnetic fields, and we demonstrate the use of deep learning models to infer the time-varying fields from an optical measurement. The quantum enhancement due to spin squeezing was verified by a degraded performance in test experiments where the spin squeezing was deliberately prevented. These results represent an advance in continuous quantum-enhanced metrology with entangled atoms, including the training and application of a deep neural network to infer complex time-dependent perturbations.

Access through your institution
Buy or subscribe

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

Access options

Access through your institution

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Experiment schematics and working principle.
Fig. 2: Steady spin squeezing and pulse magnetometry.
Fig. 3: Magnetic field tracking results.
Fig. 4: Verification of quantum enhancement in continuous field tracking.

Similar content being viewed by others

Data availability

The data supporting the findings of this study are included in the paper and its Supplementary Information. The DL training data used in this study are available via Figshare at https://doi.org/10.6084/m9.figshare.25623525.v1 (ref. 62).

Code availability

The DL code for this study is available via Figshare at https://doi.org/10.6084/m9.figshare.25623525.v1 (ref. 62).

References

  1. 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  ADS  MathSciNet  Google Scholar 

  2. Pedrozo-Peñafiel, E. et al. Entanglement on an optical atomic-clock transition. Nature 588, 414–418 (2020).

    Article  ADS  Google Scholar 

  3. Robinson, J. M. et al. Direct comparison of two spin-squeezed optical clock ensembles at the 10−17 level. Nat. Phys. 20, 208–213 (2024).

    Article  Google Scholar 

  4. Greve, G. P., Luo, C., Wu, B. & Thompson, J. K. Entanglement-enhanced matter-wave interferometry in a high-finesse cavity. Nature 610, 472–477 (2022).

    Article  ADS  Google Scholar 

  5. Malia, B. K., Wu, Y., Martínez-Rincón, J. & Kasevich, M. A. Distributed quantum sensing with mode-entangled spin-squeezed atomic states. Nature 612, 661–665 (2022).

    Article  ADS  Google Scholar 

  6. Sewell, R. J. et al. Magnetic sensitivity beyond the projection noise limit by spin squeezing. Phys. Rev. Lett. 109, 253605 (2012).

    Article  ADS  Google Scholar 

  7. Bao, H. et al. Spin squeezing of 1011 atoms by prediction and retrodiction measurements. Nature 581, 159–163 (2020).

    Article  ADS  Google Scholar 

  8. Zheng, W., Wang, H., Schmieg, R., Oesterle, A. & Polzik, E. S. Entanglement-enhanced magnetic induction tomography. Phys. Rev. Lett. 130, 203602 (2023).

    Article  ADS  Google Scholar 

  9. Haine, S. A. & Hope, J. J. Machine-designed sensor to make optimal use of entanglement-generating dynamics for quantum sensing. Phys. Rev. Lett. 124, 060402 (2020).

    Article  ADS  Google Scholar 

  10. Lin, Y. et al. Dissipative production of a maximally entangled steady state of two quantum bits. Nature 504, 415–418 (2013).

    Article  ADS  Google Scholar 

  11. Shankar, S. et al. Autonomously stabilized entanglement between two superconducting quantum bits. Nature 504, 419–422 (2013).

    Article  ADS  Google Scholar 

  12. Krauter, H. et al. Entanglement generated by dissipation and steady state entanglement of two macroscopic objects. Phys. Rev. Lett. 107, 080503 (2011).

    Article  ADS  Google Scholar 

  13. Ockeloen-Korppi, C. et al. Stabilized entanglement of massive mechanical oscillators. Nature 556, 478–482 (2018).

    Article  ADS  Google Scholar 

  14. Mercier de Lépinay, L., Ockeloen-Korppi, C. F., Woolley, M. J. & Sillanpää, M. A. Quantum mechanics free subsystem with mechanical oscillators. Science 372, 625–629 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  15. Aasi, J. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photonics 7, 613–619 (2013).

    Article  ADS  Google Scholar 

  16. Yonezawa, H. et al. Quantum-enhanced optical-phase tracking. Science 337, 1514–1517 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  17. Casacio, C. A. et al. Quantum-enhanced nonlinear microscopy. Nature 594, 201–206 (2021).

    Article  ADS  Google Scholar 

  18. Wolfgramm, F. et al. Squeezed-light optical magnetometry. Phys. Rev. Lett. 105, 053601 (2010).

    Article  ADS  Google Scholar 

  19. Li, B.-B. et al. Quantum enhanced optomechanical magnetometry. Optica 5, 850–856 (2018).

    Article  ADS  Google Scholar 

  20. Gammelmark, S. & Mølmer, K. Bayesian parameter inference from continuously monitored quantum systems. Phys. Rev. A 87, 032115 (2013).

    Article  ADS  Google Scholar 

  21. Genoni, M. G. Cramér-Rao bound for time-continuous measurements in linear Gaussian quantum systems. Phys. Rev. A 95, 012116 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  22. Orenes, D. B., Sewell, R. J., Lodewyck, J. & Mitchell, M. W. Improving short-term stability in optical lattice clocks by quantum nondemolition measurement. Phys. Rev. Lett. 128, 153201 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  23. Bouten, L., Van Handel, R. & James, M. R. An introduction to quantum filtering. SIAM J. Control Optimiz. 46, 2199–2241 (2007).

    Article  MathSciNet  Google Scholar 

  24. Belavkin, V. in Quantum Communications and Measurement (eds Hirota, O. et al.) 381–391 (Springer, 1995).

  25. Zhang, C. & Mølmer, K. Estimating a fluctuating magnetic field with a continuously monitored atomic ensemble. Phys. Rev. A 102, 063716 (2020).

    Article  ADS  Google Scholar 

  26. Warszawski, P., Wiseman, H. & Mabuchi, H. Quantum trajectories for realistic detection. Phys. Rev. A 65, 023802 (2002).

    Article  ADS  Google Scholar 

  27. Khanahmadi, M. & Mølmer, K. Time-dependent atomic magnetometry with a recurrent neural network. Phys. Rev. A 103, 032406 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  28. Schmidhuber, J. Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015).

    Article  Google Scholar 

  29. Sivak, V. et al. Real-time quantum error correction beyond break-even. Nature 616, 50–55 (2023).

    Article  ADS  Google Scholar 

  30. Orazbayev, B. & Fleury, R. Far-field subwavelength acoustic imaging by deep learning. Phys. Rev. X 10, 031029 (2020).

    Google Scholar 

  31. Ness, G., Vainbaum, A., Shkedrov, C., Florshaim, Y. & Sagi, Y. Single-exposure absorption imaging of ultracold atoms using deep learning. Phys. Rev. Appl. 14, 014011 (2020).

    Article  ADS  Google Scholar 

  32. Li, T. et al. Photonic-dispersion neural networks for inverse scattering problems. Light: Sci. Appl. 10, 154 (2021).

    Article  ADS  Google Scholar 

  33. Tranter, A. D. et al. Multiparameter optimisation of a magneto-optical trap using deep learning. Nat. Commun. 9, 4360 (2018).

    Article  ADS  Google Scholar 

  34. Vendeiro, Z. et al. Machine-learning-accelerated Bose–Einstein condensation. Phys. Rev. Res. 4, 043216 (2022).

    Article  Google Scholar 

  35. Chen, Y. et al. A neural network assisted 171Yb+ quantum magnetometer. npj Quantum Inf. 8, 152 (2022).

    Article  ADS  Google Scholar 

  36. Liu, Z.-K. et al. Deep learning enhanced Rydberg multifrequency microwave recognition. Nat. Commun. 13, 1997 (2022).

    Article  ADS  Google Scholar 

  37. Meng, X. et al. Machine learning assisted vector atomic magnetometry. Nat. Commun. 14, 6105 (2023).

    Article  ADS  Google Scholar 

  38. Vasilakis, G. et al. Generation of a squeezed state of an oscillator by stroboscopic back-action-evading measurement. Nat. Phys. 11, 389–392 (2015).

    Article  Google Scholar 

  39. Gammelmark, S., Julsgaard, B. & Mølmer, K. Past quantum states of a monitored system. Phys. Rev. Lett. 111, 160401 (2013).

    Article  ADS  Google Scholar 

  40. Zhang, J. & Mølmer, K. Prediction and retrodiction with continuously monitored Gaussian states. Phys. Rev. A 96, 062131 (2017).

    Article  ADS  Google Scholar 

  41. Budker, D. & Kimball, D. F. J. Optical Magnetometry (Cambridge Univ. Press, 2013).

  42. Smith, G. A., Silberfarb, A., Deutsch, I. H. & Jessen, P. S. Efficient quantum-state estimation by continuous weak measurement and dynamical control. Phys. Rev. Lett. 97, 180403 (2006).

    Article  ADS  Google Scholar 

  43. LeCun, Y., Bengio, Y. & Hinton, G. Deep learning. Nature 521, 436–444 (2015).

    Article  ADS  Google Scholar 

  44. Hochreiter, S. & Schmidhuber, J. Long short-term memory. Neural Comput. 9, 1735–1780 (1997).

    Article  Google Scholar 

  45. Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at arxiv.org/abs/1412.6980 (2014).

  46. Wineland, D. J., Bollinger, J. J., Itano, W. M. & Heinzen, D. Squeezed atomic states and projection noise in spectroscopy. Phys. Rev. A 50, 67 (1994).

    Article  ADS  Google Scholar 

  47. Wasilewski, W. et al. Quantum noise limited and entanglement-assisted magnetometry. Phys. Rev. Lett. 104, 133601 (2010).

    Article  ADS  Google Scholar 

  48. Gardiner, C. W. et al. Handbook of Stochastic Methods, vol. 3 (Springer, 1985).

  49. Amorós-Binefa, J. & Kołodyński, J. Noisy atomic magnetometry in real time. New J. Phys. 23, 123030 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  50. Amorós-Binefa, J. & Kołodyński, J. Noisy atomic magnetometry with Kalman filtering and measurement-based feedback. Preprint at arxiv.org/abs/2403.14764 (2024).

  51. Shankar, A., Greve, G. P., Wu, B., Thompson, J. K. & Holland, M. Continuous real-time tracking of a quantum phase below the standard quantum limit. Phys. Rev. Lett. 122, 233602 (2019).

    Article  ADS  Google Scholar 

  52. Kaubruegger, R., Vasilyev, D. V., Schulte, M., Hammerer, K. & Zoller, P. Quantum variational optimization of Ramsey interferometry and atomic clocks. Phys. Rev. X 11, 041045 (2021).

    Google Scholar 

  53. Kaubruegger, R. et al. Variational spin-squeezing algorithms on programmable quantum sensors. Phys. Rev. Lett. 123, 260505 (2019).

    Article  ADS  Google Scholar 

  54. Katz, O., Shaham, R., Polzik, E. S. & Firstenberg, O. Long-lived entanglement generation of nuclear spins using coherent light. Phys. Rev. Lett. 124, 043602 (2020).

    Article  ADS  Google Scholar 

  55. Serafin, A., Fadel, M., Treutlein, P. & Sinatra, A. Nuclear spin squeezing in helium-3 by continuous quantum nondemolition measurement. Phys. Rev. Lett. 127, 013601 (2021).

    Article  ADS  Google Scholar 

  56. Santagati, R. et al. Magnetic-field learning using a single electronic spin in diamond with one-photon readout at room temperature. Phys. Rev. X 9, 021019 (2019).

    Google Scholar 

  57. Turner, E., Wu, S.-H., Li, X. & Wang, H. Real-time magnetometry with coherent population trapping in a nitrogen-vacancy center. Phys. Rev. A 105, L010601 (2022).

    Article  ADS  Google Scholar 

  58. Rossi, M., Mason, D., Chen, J. & Schliesser, A. Observing and verifying the quantum trajectory of a mechanical resonator. Phys. Rev. Lett. 123, 163601 (2019).

    Article  ADS  Google Scholar 

  59. Meng, C., Brawley, G. A., Bennett, J. S., Vanner, M. R. & Bowen, W. P. Mechanical squeezing via fast continuous measurement. Phys. Rev. Lett. 125, 043604 (2020).

    Article  ADS  Google Scholar 

  60. Julsgaard, B., Sherson, J., Sørensen, J. & Polzik, E. S. Characterizing the spin state of an atomic ensemble using the magneto-optical resonance method. J. Opt. B: Quantum Semiclass. Opt. 6, 5 (2003).

    Article  ADS  Google Scholar 

  61. Han, B. et al. Retrodiction beyond the Heisenberg uncertainty relation. Nat. Commun. 11, 5658 (2020).

    Article  ADS  Google Scholar 

  62. Concurrent spin squeezing and field tracking with machine learning. Figshare https://doi.org/10.6084/m9.figshare.25623525.v1 (2025).

Download references

Acknowledgements

This work is supported by Innovation Program for Quantum Science and Technology (Grant No. 2023ZD0300900), the National Natural Science Foundation of China (Grant Nos. 12027806 and 12161141018), the Fund for the Shanxi 1331 Project and in part by NSF Grant No. PHY-2309135 to the Kavli Institute for Theoretical Physics. K.M. acknowledges support from the Carlsberg Foundation through the Semper Ardens project QcooL and Innovation Fund Denmark through the European QuantERA project C’MON-QSENS! (Grant No. 9085-00002).

Author information

Authors and Affiliations

  1. Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai, China

    Junlei Duan, Zhiwei Hu, Xingda Lu & Yanhong Xiao

  2. State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, China

    Liantuan Xiao, Suotang Jia & Yanhong Xiao

  3. Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, China

    Liantuan Xiao, Suotang Jia & Yanhong Xiao

  4. Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    Klaus Mølmer

Authors
  1. Junlei Duan
    View author publications

    Search author on:PubMed Google Scholar

  2. Zhiwei Hu
    View author publications

    Search author on:PubMed Google Scholar

  3. Xingda Lu
    View author publications

    Search author on:PubMed Google Scholar

  4. Liantuan Xiao
    View author publications

    Search author on:PubMed Google Scholar

  5. Suotang Jia
    View author publications

    Search author on:PubMed Google Scholar

  6. Klaus Mølmer
    View author publications

    Search author on:PubMed Google Scholar

  7. Yanhong Xiao
    View author publications

    Search author on:PubMed Google Scholar

Contributions

J.D., K.M. and Y.X. conceived the idea. J.D., Z.H., X.L. and Y.X. designed the experiments, performed the measurements and analysed the data together with all other authors. Z.H. and J.D. built the DL model. J.D. carried out the numerical simulation and theoretical analysis under K.M. and Y.X.’s guidance. J.D., K.M. and Y.X. wrote the manuscript with contributions from all other authors. K.M. and Y.X. supervised the project.

Corresponding authors

Correspondence to Junlei Duan, Klaus Mølmer or Yanhong Xiao.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Alice Sinatra and the other, anonymous, reviewer(s) 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.

Extended data

Extended Data Table 1 Robustness of the deep learning
Full size table

Supplementary information

Supplementary Information

Supplementary Sections 1–13, Figs. 1–12, Tables 1–3 and references.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, J., Hu, Z., Lu, X. et al. Concurrent spin squeezing and field tracking with machine learning. Nat. Phys. 21, 909–915 (2025). https://doi.org/10.1038/s41567-025-02855-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-025-02855-3

This article is cited by

Search

Advanced search

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