Abstract
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.
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Data availability
The experimental data generated and analysed during this study are available in the Zenodo repository, https://doi.org/10.5281/zenodo.8124375.
Code availability
The code used for simulations in this study is available from the corresponding author on reasonable request.
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Acknowledgements
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.
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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.
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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.
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Franke, J., Muleady, S.R., Kaubruegger, R. et al. Quantum-enhanced sensing on optical transitions through finite-range interactions. Nature 621, 740–745 (2023). https://doi.org/10.1038/s41586-023-06472-z
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DOI: https://doi.org/10.1038/s41586-023-06472-z
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