Abstract
Spin-squeezed states provide a seminal example of how the structure of quantum mechanical correlations can be controlled to produce metrologically useful entanglement1,2,3,4,5,6,7. These squeezed states have been demonstrated in a wide variety of quantum systems ranging from atoms in optical cavities to trapped ion crystals8,9,10,11,12,13,14,15,16. By contrast, despite their numerous advantages as practical sensors, spin ensembles in solid-state materials have yet to be controlled with sufficient precision to generate targeted entanglement such as spin squeezing. Here we report the experimental demonstration of spin squeezing in a solid-state spin system. Our experiments are performed on a strongly interacting ensemble of nitrogen–vacancy colour centres in diamond at room temperature, and squeezing (−0.50 ± 0.13 dB) below the noise of uncorrelated spins is generated by the native magnetic dipole–dipole interaction between nitrogen–vacancy centres. To generate and detect squeezing in a solid-state spin system, we overcome several challenges. First, we develop an approach, using interaction-enabled noise spectroscopy, to characterize the quantum projection noise in our system without directly resolving the spin probability distribution. Second, noting that the random positioning of spin defects severely limits the generation of spin squeezing, we implement a pair of strategies aimed at isolating the dynamics of a relatively ordered sub-ensemble of nitrogen–vacancy centres. Our results open the door to entanglement-enhanced metrology using macroscopic ensembles of optically active spins in solids.
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Source data are available at Harvard Dataverse (https://doi.org/10.7910/DVN/LVPLRI)62.
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Acknowledgements
We thank M. Aidelsburger, S. Chern, P. Crowley, H. Gao, B. Kobrin, N. Leitao, F. Machado, M. Schleier-Smith, T. Schuster and B. Zhu for their insights and discussions. This work was supported by the US Department of Energy through the Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator and the BES grant no. DE-SC0019241, as well as the Army Research Office (grant no. W911NF-24-1-0079) and the MURI program (grant no. W911NF-20-1-0136). We acknowledge the use of shared facilities of the UCSB Quantum Foundry through Q-AMASE-i program (NSF DMR-1906325), the UCSB MRSEC (NSF DMR 1720256) and the Quantum Structures Facility within the UCSB California NanoSystems Institute. A.C.B.J. acknowledges support from the NSF QLCI program (grant no. OMA-2016245) and the Gordon and Betty Moore Foundation’s EPiQS Initiative via grant GBMF10279. L.B.H. acknowledges support from the NSF Graduate Research Fellowship Program (DGE 2139319) and the UCSB Quantum Foundry. D.K. acknowledges support from Generation-Q AWS and HQI fellowships. T.O. acknowledges support from the Ezoe Memorial Recruit Foundation. S.A.M. acknowledges support from the UCSB Quantum Foundry (NSF DMR-1906325) and support from the Canada NSERC (grant no. AID 516704-2018).
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W.W., E.J.D. and Z.W. performed experiments with the help of T.O., C.L. and H.Y.; L.B.H., S.A.M. and A.C.B.J. synthesized the diamond sample. W.W. and E.J.D. developed the experimental protocols and performed data analysis. E.J.D., B.Y. and M.B. developed the theoretical models and methodology. W.W., D.K., B.Y. and Z.W. performed the numerical simulation. A.C.B.J. and N.Y.Y. supervised the project. W.W., E.J.D., B.Y., Z.W. and N.Y.Y. wrote the paper with input from all authors.
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Harvard University (co-inventors W.W., E.J.D., L.B.H., B.Y., Z.W., D.K., A.C.B.J. and N.Y.Y.) filed for a provisional patent that relates to reducing the amount of positional disorder in a spin ensemble using lattice engineering and for measuring the quantum variance by the decay of spin polarization.
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Extended data figures and tables
Extended Data Fig. 1 Sample characterization.
(a) Secondary ion mass spectroscopy (SIMS) measurements of the sample showing the densities of 15N (i), 14N (ii), and 13C atoms (iii) as a function of distance from the surface of the diamond. (b) A typical confocal scan of the irradiated spot used in our mesaurements. (c) Normalized quench decay profile Sx(t)/Sx(0) measured by XY-8 dynamical decoupling sequence with different interpulse spacings τp. We fit each curve to a stretched exponential \({e}^{-{(t/{T}_{2})}^{2/3}}\) to extract T2. We compare the optimal XY-8 timescale (11.5μs) with the DROID decay timescale (91μs), which demonstrates that the NV ensemble dynamics (during an XY-8 sequence) are limited by internal dipole-dipole interactions. The DROID sequence is performed with an optimal interpulse spacing τp = 100 ns, which balances between pulse error accumulation and decoupling fast noise. (d) We observe a saturation of T2 as the interpulse spacing τp is decreased, indicating that NV-P1 interactions are sufficiently decoupled. (e) Nuclear spin polarization as measured via ODMR. At 393 Gauss, the nuclear spin exhibits a hyperpolarization of 89(1)% from a double-Lorentzian fit to the blue data. Red data shows the ODMR spectrum after shelving the less populated mI = − 1/2 nuclear spin subgroup in the \(| {m}_{s}=1\rangle \) state. (f, g) Measurement of NV density. We compare the XY-8 decoherence profile on both a linear (f) and log (g) scale with cluster DTWA numerics for NV densities ranging from 5.0 to 11.0 ppm ⋅ nm, and extract an NV density of 8 ppm ⋅ nm.
Extended Data Fig. 2 Full experiment sequence.
(a) The experiment consists of three parts: lattice engineering, squeezing generation, and squeezing readout. (b) In the lattice engineering step, we prepare an effective lattice from the disordered system by removing NV dimers with either the shelving method (i) or the depolarization method (ii). (c) To generate squeezing, we evolve the system under its native dipolar interactions HXXZ for a time tg. (d) In the readout step, we first apply a pulse with variable rotation angle θ along the x axis (yellow) to map the spin operator Sθ into Sz. Then we probe the variance Var(Sθ) by measuring the quench dynamics of the rotated squeezed state for a time tr. The initial and final π/2 pulses that initialize and readout the state on + x axis are shown in black. Throughout the time evolution, we apply XY-8 pulses to decouple the Ising interaction between NV and P1 centers. We conclude the experiment by a fluorescence readout while shining the 532 nm laser (not shown).
Extended Data Fig. 3 Stretched power as a measure of spatial geometry.
Krylov subspace numerics averaged over 40 realizations of positional disorder for N = 15 spins. Data shows quench dynamics of spins evolving under HXXZ with minimum spin spacing \({r}_{\min }=0.1,4,8,16\,{\rm{nm}}\) on linear (left) and log (right) scales. The shaded area indicates the standard deviation of the positional disorder averaging. The deviation from p = 2/3 at long times can be explained by the crossover from the early-time ballistic regime into the late-time random-walk regime (p = 1/3)56.
Extended Data Fig. 4 Minimum spin spacing after lattice engineering.
(a, b).i. The shelving probability pshelve (a) and the dimer depolarization probability pdepol (b) as a function of nearest-neighbor distance rNN. (a, b).ii-iii We measure the XY-8 decoherence profile after the shelving (a) and depolarization (b) protocols, and compare the experimental result with the cluster DTWA numerical simulation on linear (ii) and log (iii) scales. The curves are the decoherence profiles assuming different cut radii rshelve or rdepol. The NV density is fixed at 8 ppm ⋅ nm. We extract a dimer removal radius rshelve = 7 nm for the shelving protocol and rdepol = 14 nm for the adiabatic depolarization protocol.
Extended Data Fig. 5 Data analysis.
(a) Sample decay curves ⟨Sx(tr)⟩ comparing experimental and numerical data. Here, shelving has been used to prepare the initial state which is then squeezed for a time tg = 1.6μs. (b) The same data plotted in (a) after shifting each curve by to(tg, θ). (c) Comparison of the offset time calculated using the OAT model (purple) with direct numerical extraction of the offset time for these data (blue). (d, e) Comparison between the fitted decay timescales for experimental (d) and numerical (e) data. Here we also show the change in extracted timescale if the upper bound on the fitting window \({t}_{\max }\) is adjusted between 12 and 16 μs. (f, g) We plot the extracted T2 decay timescale for the numerical data (f) in the full parameter space {tg, θ}. Maximum and minimum T2 are indicated with the dashed blue and red lines, respectively. The same plot after mapping T2 to Var(Sθ) is shown in (g). (h) Red and blue curves show the maximum and minimum variances Var(Sθ) as a function of tg, corresponding to the dashed lines in (g). (i) Final mapping obtained by matching the timescale data plotted in (f) to the variance data plotted in (g). We observe that the dictionaries at different squeezing generation times tg (solid to transparent green) overlap, as expected. (j) Experimentally measured decay timescale, T2, as a function of preparation time tg and probing angle θ for the shelving method. For tg = 0, the measured decay timescale is independent of θ. As the NV ensemble evolves under the XXZ Hamiltonian, for a large range of global rotation angles, the decay timescale becomes longer; we note that such behavior is not expected for any external noise sources, but is naturally expected for spin squeezing dynamics.
Extended Data Fig. 6 Probing spin projection noise via quench dynamics under one-axis twisting dynamics.
From an initial spin-polarized state (black at t = 0), a squeezed state is generated after time tg. Then, a rotation Xθ is applied and the readout quench begins, during which ⟨Sx⟩ is measured as a function of readout time tr. For two angles θ, the fiducial state is prepared at tr = 0; the first rotates the antisqueezed quadrature to z (red), and the second rotates the squeezed quadrature to z (blue). For these two curves, the fit \({S}_{x}\propto {e}^{-{({t}_{{\rm{r}}}/{T}_{2})}^{2}}\) begins at tr = 0. For all other rotation angles θ, the fiducial state occurs at some other time tr = to, here indicated by the circular marker. For example, fitting of the black curve should start at tr = to = − tg indicated by the black marker, not at tr = 0; similarly, fitting of the purple curve should start at tr = to indicated by the purple marker, not at tr = 0.
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Wu, W., Davis, E.J., Hughes, L.B. et al. Spin squeezing in an ensemble of nitrogen–vacancy centres in diamond. Nature 646, 74–80 (2025). https://doi.org/10.1038/s41586-025-09524-8
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DOI: https://doi.org/10.1038/s41586-025-09524-8
