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.

  • Article
  • Published:

Quantum-enhanced sensing on optical transitions through finite-range interactions


The control over quantum states in atomic systems has led to the most precise optical atomic clocks so far1,2,3. Their sensitivity is bounded at present by the standard quantum limit, a fundamental floor set by quantum mechanics for uncorrelated particles, which can—nevertheless—be overcome when operated with entangled particles. Yet demonstrating a quantum advantage in real-world sensors is extremely challenging. Here we illustrate a pathway for harnessing large-scale entanglement in an optical transition using 1D chains of up to 51 ions with interactions that decay as a power-law function of the ion separation. We show that our sensor can emulate many features of the one-axis-twisting (OAT) model, an iconic, fully connected model known to generate scalable squeezing4 and Greenberger–Horne–Zeilinger-like states5,6,7,8. The collective nature of the state manifests itself in the preservation of the total transverse magnetization, the reduced growth of the structure factor, that is, spin-wave excitations (SWE), at finite momenta, the generation of spin squeezing comparable with OAT (a Wineland parameter9,10 of −3.9 ± 0.3 dB for only N = 12 ions) and the development of non-Gaussian states in the form of multi-headed cat states in the Q-distribution. We demonstrate the metrological utility of the states in a Ramsey-type interferometer, in which we reduce the measurement uncertainty by −3.2 ± 0.5 dB below the standard quantum limit for N = 51 ions.

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

Access options

Buy this article

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

Fig. 1: Realization of squeezing by non-collective interactions.
Fig. 2: Structure factor.
Fig. 3: Husimi Q-distributions.
Fig. 4: Phase estimation with a CSS and a SSS of 51 qubits.

Similar content being viewed by others

Data availability

The experimental data generated and analysed during this study are available in the Zenodo repository,

Code availability

The code used for simulations in this study is available from the corresponding author on reasonable request.


  1. Bothwell, T. et al. Resolving the gravitational redshift across a millimetre-scale atomic sample. Nature 602, 420–424 (2022).

    CAS  PubMed  ADS  Google Scholar 

  2. Oelker, E. et al. Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks. Nat. Photon. 13, 714–719 (2019).

  3. McGrew, W. F. et al. Atomic clock performance enabling geodesy below the centimetre level. Nature 564, 87–90 (2018).

    CAS  PubMed  ADS  Google Scholar 

  4. Kitagawa, M. & Ueda, M. Squeezed spin states. Phys. Rev. A 47, 5138 (1993).

    CAS  PubMed  ADS  Google Scholar 

  5. Agarwal, G., Puri, R. & Singh, R. Atomic Schrödinger cat states. Phys. Rev. A 56, 2249 (1997).

    CAS  ADS  Google Scholar 

  6. Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835 (1999).

    ADS  Google Scholar 

  7. Song, C. et al. Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits. Science 365, 574–577 (2019).

    MathSciNet  CAS  PubMed  ADS  Google Scholar 

  8. Comparin, T., Mezzacapo, F. & Roscilde, T. Multipartite entangled states in dipolar quantum simulators. Phys. Rev. Lett. 129, 150503 (2022).

    CAS  PubMed  ADS  Google Scholar 

  9. Wineland, D. J., Bollinger, J. J., Itano, W. M., Moore, F. L. & Heinzen, D. J. Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev, A 46, R6797 (1992).

    CAS  PubMed  ADS  Google Scholar 

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

    CAS  PubMed  ADS  Google Scholar 

  11. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).

  12. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    MathSciNet  ADS  Google Scholar 

  13. 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).

    MathSciNet  ADS  Google Scholar 

  14. Ludlow, A. D., Boyd, M. M., Ye, J., Peik, E. & Schmidt, P. O. Optical atomic clocks. Rev. Mod. Phys. 87, 637 (2015).

    CAS  ADS  Google Scholar 

  15. Norcia, M. A. et al. Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser. Science 361, 259–262 (2018).

    CAS  PubMed  ADS  Google Scholar 

  16. Ritsch, H., Domokos, P., Brennecke, F. & Esslinger, T. Cold atoms in cavity-generated dynamical optical potentials. Rev. Mod. Phys. 85, 553 (2013).

    CAS  ADS  Google Scholar 

  17. Leroux, I. D., Schleier-Smith, M. H. & Vuletić, V. Implementation of cavity squeezing of a collective atomic spin. Phys. Rev. Lett. 104, 073602 (2010).

    PubMed  ADS  Google Scholar 

  18. Hosten, O., Engelsen, N. J., Krishnakumar, R. & Kasevich, M. A. Measurement noise 100 times lower than the quantum-projection limit using entangled atoms. Nature 529, 505–508 (2016).

    CAS  PubMed  MATH  ADS  Google Scholar 

  19. Cox, K. C., Greve, G. P., Weiner, J. M. & Thompson, J. K. Deterministic squeezed states with collective measurements and feedback. Phys. Rev. Lett. 116, 093602 (2016).

    PubMed  ADS  Google Scholar 

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

    PubMed  ADS  Google Scholar 

  21. Bohn, J. L., Rey, A. M. & Ye, J. Cold molecules: progress in quantum engineering of chemistry and quantum matter. Science 357, 1002–1010 (2017).

    MathSciNet  CAS  PubMed  MATH  ADS  Google Scholar 

  22. Schine, N., Young, A. W., Eckner, W. J., Martin, M. J. & Kaufman, A. M. Long-lived Bell states in an array of optical clock qubits. Nat. Phys. 18, 1067–1073 (2022).

    CAS  Google Scholar 

  23. Britton, J. W. et al. Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins. Nature 484, 489–492 (2012).

    CAS  PubMed  ADS  Google Scholar 

  24. Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016).

    MathSciNet  CAS  PubMed  MATH  ADS  Google Scholar 

  25. Perlin, M. A., Qu, C. & Rey, A. M. Spin squeezing with short-range spin-exchange interactions. Phys. Rev. Lett. 125, 223401 (2020).

    CAS  PubMed  ADS  Google Scholar 

  26. Bilitewski, T. et al. Dynamical generation of spin squeezing in ultracold dipolar molecules. Phys. Rev. Lett. 126, 113401 (2021).

    CAS  PubMed  ADS  Google Scholar 

  27. Comparin, T., Mezzacapo, F. & Roscilde, T. Robust spin squeezing from the tower of states of U(1)-symmetric spin Hamiltonians. Phys. Rev. A 105, 022625 (2022).

    MathSciNet  CAS  ADS  Google Scholar 

  28. Young, J. T., Muleady, S. R., Perlin, M. A., Kaufman, A. M. & Rey, A. M. Enhancing spin squeezing using soft-core interactions. Phys. Rev. Res. 5, L012033 (2023).

    CAS  Google Scholar 

  29. Block, M. et al. A universal theory of spin squeezing. Preprint at (2023).

  30. Pezzé, L. & Smerzi, A. Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett. 102, 100401 (2009).

    MathSciNet  PubMed  ADS  Google Scholar 

  31. Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    CAS  Google Scholar 

  32. Bruzewicz, C. D., Chiaverini, J., McConnell, R. & Sage, J. M. Trapped-ion quantum computing: progress and challenges. Appl. Phys. Rev. 6, 021314 (2019).

    ADS  Google Scholar 

  33. Tscherbul, T. V., Ye, J. & Rey, A. M. Robust nuclear spin entanglement via dipolar interactions in polar molecules. Phys. Rev. Lett. 130, 143002 (2023).

    CAS  PubMed  ADS  Google Scholar 

  34. Gorshkov, A. V. et al. Tunable superfluidity and quantum magnetism with ultracold polar molecules. Phys. Rev. Lett. 107, 115301 (2011).

    PubMed  ADS  Google Scholar 

  35. Foss-Feig, M., Gong, Z.-X., Gorshkov, A. V., and Clark, C. W. Entanglement and spin-squeezing without infinite-range interactions. Preprint at (2016).

  36. Rey, A. M., Jiang, L., Fleischhauer, M., Demler, E. & Lukin, M. D. Many-body protected entanglement generation in interacting spin systems. Phys. Rev. A 77, 052305 (2008).

    ADS  Google Scholar 

  37. Kranzl, F. et al. Controlling long ion strings for quantum simulation and precision measurements. Phys. Rev. A 105, 052426 (2022).

    CAS  ADS  Google Scholar 

  38. Greenberger, D. M., Horne, M. A. & Zeilinger, A. in Bell’s Theorem, Quantum Theory and Conceptions of the Universe (ed. Kafatos, M.) 69–72 (Springer, 1989).

  39. Qiao, M. et al. Observing frustrated quantum magnetism in two-dimensional ion crystals. Preprint at (2022).

  40. Kiesenhofer, D. et al. Controlling two-dimensional Coulomb crystals of more than 100 ions in a monolithic radio-frequency trap. PRX Quantum 4, 020317 (2023).

    ADS  Google Scholar 

  41. Itano, W. M. et al. Bragg diffraction from crystallized ion plasmas. Science 279, 686–689 (1998).

    CAS  PubMed  ADS  Google Scholar 

  42. Barredo, D., Lienhard, V., de Léséleuc, S., Lahaye, T. & Browaeys, A. Synthetic three-dimensional atomic structures assembled atom by atom. Nature 561, 79–82 (2018).

    CAS  PubMed  ADS  Google Scholar 

  43. Bornet, G. et al. Scalable spin squeezing in a dipolar Rydberg atom array. Preprint at (2023)

  44. Eckner, W. J. et al. Realizing spin squeezing with Rydberg interactions in a programmable optical clock. Preprint at (2023).

  45. Campbell, S. L. et al. A Fermi-degenerate three-dimensional optical lattice clock. Science 358, 90–94 (2017).

    CAS  PubMed  ADS  Google Scholar 

  46. Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett. 116, 053601 (2016).

    PubMed  ADS  Google Scholar 

  47. Liu, Y. C., Xu, Z. F., Jin, G. R. & You, L. Spin squeezing: transforming one-axis twisting into two-axis twisting. Phys. Rev. Lett. 107, 013601 (2011).

    CAS  PubMed  ADS  Google Scholar 

  48. Marciniak, C. D. et al. Optimal metrology with programmable quantum sensors. Nature 603, 604–609 (2022).

    CAS  PubMed  ADS  Google Scholar 

  49. 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).

    CAS  Google Scholar 

  50. Hines, J. A. et al. Spin squeezing by Rydberg dressing in an array of atomic ensembles. Preprint at (2023).

  51. Joshi, M. K. et al. Observing emergent hydrodynamics in a long-range quantum magnet. Science 376, 720–724 (2022).

    CAS  PubMed  ADS  Google Scholar 

  52. Zhu, S.-L., Monroe, C. & Duan, L.-M. Trapped ion quantum computation with transverse phonon modes. Phys. Rev. Lett. 97, 050505 (2006).

    PubMed  ADS  Google Scholar 

  53. Wu, C. F. J. Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Stat. 14, 1261–1295 (1986).

    MathSciNet  MATH  Google Scholar 

  54. Foss-Feig, M., Hazzard, K. R. A., Bollinger, J. J. & Rey, A. M. Nonequilibrium dynamics of arbitrary-range Ising models with decoherence: an exact analytic solution. Phys. Rev. A 87, 042101 (2013).

    ADS  Google Scholar 

  55. Schachenmayer, J., Pikovski, A. & Rey, A. M. Many-body quantum spin dynamics with Monte Carlo trajectories on a discrete phase space. Phys. Rev. X 5, 011022 (2015).

    Google Scholar 

  56. Zhu, B. H., Rey, A. M. & Schachenmayer, J. A generalized phase space approach for solving quantum spin dynamics. New J. Phys. 21, 082001 (2019).

    MathSciNet  CAS  ADS  Google Scholar 

  57. Huber, J., Rey, A. M. & Rabl, P. Realistic simulations of spin squeezing and cooperative coupling effects in large ensembles of interacting two-level systems. Phys. Rev. A 105, 013716 (2022).

    MathSciNet  CAS  ADS  Google Scholar 

  58. Muleady, S. R., Yang, M., White, S. R. & Rey, A. M. Validating phase-space methods with tensor networks in two-dimensional spin models with power-law interactions. Preprint at (2023).

  59. Gardiner, C. W. Stochastic Methods: A Handbook for the Natural and Social Sciences 4th edn (ed. Haken, H.) (Springer, 2009).

  60. Roscilde, T., Comparin, T. & Mezzacapo, F. Entangling dynamics from effective rotor/spin-wave separation in U(1)-symmetric quantum spin models. Preprint at (2023).

  61. Roscilde, T., Comparin, T. & Mezzacapo, F. Rotor/spin-wave theory for quantum spin models with U(1) symmetry. Preprint at (2023).

Download references


We acknowledge stimulating discussions with members of the LASCEM collaboration about realizing spin squeezing in trapped ions with short-range interactions that initiated the project, as well as valuable feedback on the manuscript by W. Eckner and A. Kaufman. We also acknowledge support by the Austrian Science Fund through the SFB BeyondC (F7110) and funding by the Institut für Quanteninformation GmbH, by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation, and by LASCEM through AFOSR no. 64896-PH-QC. Support is also acknowledged from the AFOSR grants FA9550-18-1-0319 and FA9550-19-1-0275, by the NSF JILA-PFC PHY-1734006, QLCI OMA-2016244, NSF grant PHY-1820885, by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator and by NIST.

Author information

Authors and Affiliations



The research was devised by S.R.M., M.K.J., R.K., A.M.R. and C.F.R. S.R.M. and A.M.R. developed the theoretical protocols. J.F., M.K.J., F.K., R.B. and C.F.R. contributed to the experimental setup. J.F., M.K.J. and F.K. performed the experiments. M.K.J., J.F. and R.K. analysed the data and S.R.M. carried out numerical simulations. S.R.M., M.K.J., J.F., R.K., A.M.R. and C.F.R. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript.

Corresponding authors

Correspondence to Ana Maria Rey or Christian F. Roos.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Sebastian Carrasco, Luming Duan, Michael Goerz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

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

Extended data figures and tables

Extended Data Fig. 1 Assessment of the experimentally prepared spin state.

a, The outcome of the entangling interaction on the spins pointing along the x-axis is depicted as a variance ellipse. ξ2 is evaluated from two sets of measurements that are obtained after a rotation \(\widehat{R}(\widetilde{\theta },\widetilde{\phi })\) has been applied to the state. b, Applying \(\widehat{R}(\pi /2,\widetilde{\phi })\) for various values of \(\widetilde{\phi }\) allows us to measure the spin projection \(\widehat{S}\)π/2,ϕ in any direction along the equator. From the sinusoidal fits (solid lines), we obtain the Bloch vector length and orientation and the angle ϕ0 between the x-axis and the mean spin orientation. c, Applying \(\widehat{R}(\widetilde{\theta },{\phi }_{0})\) for various values of θ allows us to measure the variance \(\langle {(\Delta {\widehat{S}}_{\theta ,{\phi }_{0}})}^{2}\rangle \) in any direction orthogonal to the mean spin direction. From the fitted data, we extract the minimal orthogonal variance \(\langle {(\Delta {\widehat{S}}_{{{\bf{n}}}_{\perp }})}^{2}\rangle \) and the angle θ0 for which the minimal variance is aligned with the z-axis.

Extended Data Fig. 2 Simulated effect of global dephasing on spin-squeezing preparation.

a, Dependence of ξ2 on system size, for varying levels of the global dephasing as obtained from analytic calculations of the OAT model with \(\chi =\overline{J}\). The opacity of the markers increases with the associated T2 coherence time, for which we show results for coherence times of 69 ms (lightest), 2 × 69 ms, 3 × 69 ms and 4 × 69 ms, as well as for infinite coherence time (darkest). b, Analogous results for numerical calculations of the power-law XY model. c, Analytical calculations for the power-law Ising model. We use couplings Ji,j as characterized in our platform for each system size (the dotted lines are a guide to the eye).

Extended Data Fig. 3 Simulated effect of interaction range on spin-squeezing preparation.

a, Dependence of ξ2 on the interaction range, for varying values of the power-law exponent α obtained from numerical calculations of the XY model in the absence of decoherence. The opacity of the markers increases with the interaction range, for which we show results for α = 1.5 (lightest), 1.2, 1.0, 0.8 and 0.5 (darkest). Compared with our analysis elsewhere in the text, in which we use the experimentally characterized Ji,j approximating a power-law potential, here we directly use the ideal power-law interaction Ji,j = J0|i − j|α. We also use the results of simulations through DDTWA without decoherence for all system sizes. The solid black curve indicates the corresponding spin squeezing for an OAT model. b, Analogous results for analytical calculations of the power-law Ising model.

Supplementary information

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

Franke, J., Muleady, S.R., Kaubruegger, R. et al. Quantum-enhanced sensing on optical transitions through finite-range interactions. Nature 621, 740–745 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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