Simultaneous tracking of spin angle and amplitude beyond classical limits

Abstract

Measurement of spin precession is central to extreme sensing in physics1,2, geophysics3, chemistry4, nanotechnology5 and neuroscience6, and underlies magnetic resonance spectroscopy7. Because there is no spin-angle operator, any measurement of spin precession is necessarily indirect, for example, it may be inferred from spin projectors at different times. Such projectors do not commute, and so quantum measurement back-action—the random change in a quantum state due to measurement—necessarily enters the spin measurement record, introducing errors and limiting sensitivity. Here we show that this disturbance in the spin projector can be reduced below N1/2—the classical limit for N spins—by directing the quantum measurement back-action almost entirely into an unmeasured spin component. This generates a planar squeezed state8 that, because spins obey non-Heisenberg uncertainty relations9,10, enables simultaneous precise knowledge of spin angle and spin amplitude. We use high-dynamic-range optical quantum non-demolition measurements11,12,13 applied to a precessing magnetic spin ensemble to demonstrate spin tracking with steady-state angular sensitivity 2.9 decibels below the standard quantum limit, simultaneously with amplitude sensitivity 7.0 decibels below the Poissonian variance14. The standard quantum limit and Poissonian variance indicate the best possible sensitivity with independent particles. Our method surpasses these limits in non-commuting observables, enabling orders-of-magnitude improvements in sensitivity for state-of-the-art sensing15,16,17,18 and spectroscopy19,20.

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Figure 1: Simultaneous, precise tracking of spin angle and amplitude.
Figure 2: Experimental results.

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Acknowledgements

We thank G. Vitagliano, M. D. Reid, P. D. Drummond, G. Tóth, N. Behbood, M. Napolitano, S. Palacios, X. Menino and the ICFO mechanical workshop, and J.-C. Cifuentes and the ICFO electronic workshop. We also thank D. T. Campbell and M. M. Fría. Work supported by MINECO/FEDER, MINECO projects MAQRO (reference FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa grant SEV-2015-0522, Catalan 2014-SGR-1295, by the European Union Project QUIC (grant agreement 641122), European Research Council project AQUMET (grant agreement 280169) and ERIDIAN (grant agreement 713682), and by Fundació Privada CELLEX. L.C.B. was supported by the International Fellowship Programme ‘La Caixa’ - Severo Ochoa, awarded by the ‘La Caixa’ Foundation.

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Authors

Contributions

M.W.M. and G.C. conceived the project, experimental protocols were designed by G.C., F.M., R.J.S. and M.W.M., and the experiment was performed by G.C. and F.M. with help from L.C.B.; G.C. analysed the results with the help of R.J.S.; M.W.M. developed the theoretical model; and G.C., F.M., R.J.S. and M.W.M. wrote the manuscript with feedback from L.C.B.

Corresponding authors

Correspondence to Giorgio Colangelo or Morgan W. Mitchell.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks F. Wilhelm-Mauch and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Optical pumping efficiency.

We prepare an input atomic state with via stroboscopic optical pumping in the presence of a small magnetic field along the x axis. Data are fitted with the exponentially growing curve a(1 − et/τ) (solid line); we obtain a = 0.979 ± 0.004 and τ = 0.26 ± 0.02. The orange dashed line shows the optical pumping efficiency of 98%. Error bars (±1 s.e.m.) are smaller than the points for most of the data. Source data

Extended Data Figure 2 Calibration of average Faraday rotation signal.

We calibrate the rotation angle φ against input atom number N, measured via absorption imaging. Solid line, the fit curve φ = a0 + μ1N, with which we obtain μ1 = (7.07 ± 0.04) × 10−8 and a0 = (3.9 ± 0.3) × 10−3. Error bars, ±1 s.e.m. Source data

Extended Data Figure 3 Calibration of quantum-noise-limited Faraday rotation probing of atomic spins.

We plot the measured variance var(φ) as a function of the number of atoms N in an input coherent spin state with 〈F〉 = {0, N, 0}. Solid curve, a fit using the polynomial var(φ) = a0 + a1N + a2N2. The linear term a1 = αμ2N/2 corresponds to the atomic quantum noise from atoms in the input coherent spin state. We estimate a0 = (11.7 ± 0.7) × 10−10, a1 = (6.5 ± 0.8) × 10−15 and a2 = (2.8 ± 12) × 10−22, consistent with negligible technical noise in the atomic state preparation. Dashed line, var(φ) = a0 + a1N. Error bars indicate ±1 standard error in the variance for 206 repetitions. Source data

Extended Data Figure 4 Fit gain.

We compare the estimated Fz and Fy from a fit using equation (2): first, with the classical parameters g, ωL, T2 and φ0 fixed (labelled ) for measurements n = 1 and 2; and second, with these parameters free to vary as independent parameters (labelled ). Blue (green), Fz (Fy) of the first measurement; red (orange), Fz (Fy) of the second measurement. A linear fit γx + δ to points in plots a–d gives γa = 0.9981(8), γb = 1.0026(8), γc = 0.9923(4) and γd = 1.0007(5), and δa = 0.003(1), δb = 0.0001(9), δc = 0.0004(3) and δd = −0.0023(3). A grey line y = x is plotted for comparison in each panel. Source data

Extended Data Figure 5 Tracking precision as function of Δt.

An optimum is found at Δt = 270 μs. Error bars indicate ±1 standard error in the variance for 453 repetitions. Source data

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Colangelo, G., Ciurana, F., Bianchet, L. et al. Simultaneous tracking of spin angle and amplitude beyond classical limits. Nature 543, 525–528 (2017). https://doi.org/10.1038/nature21434

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