Simultaneous tracking of spin angle and amplitude beyond classical limits


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.


  1. 1

    Kominis, I. K., Kornack, T. W., Allred, J. C. & Romalis, M. V. A subfemtotesla multichannel atomic magnetometer. Nature 422, 596–599 (2003)

    CAS  Article  ADS  Google Scholar 

  2. 2

    Ludlow, A. D., Boyd, M. M., Ye, J., Peik, E. & Schmidt, P. O. Optical atomic clocks. Rev. Mod. Phys. 87, 637–701 (2015)

    CAS  Article  ADS  Google Scholar 

  3. 3

    Dang, H. B., Maloof, A. C. & Romalis, M. V. Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer. Appl. Phys. Lett. 97, 151110 (2010)

    Article  ADS  CAS  Google Scholar 

  4. 4

    Gomes, M. D. et al. 129Xe NMR relaxation-based macromolecular sensing. J. Am. Chem. Soc. 138, 9747–9750 (2016)

    CAS  PubMed  Article  Google Scholar 

  5. 5

    Wolf, T. et al. Subpicotesla diamond magnetometry. Phys. Rev. X 5, 041001 (2015)

    Google Scholar 

  6. 6

    Jensen, K. et al. Non-invasive detection of animal nerve impulses with an atomic magnetometer operating near quantum limited sensitivity. Sci. Rep. 6, 29638 (2016)

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  7. 7

    Plewes, D. B. & Kucharczyk, W. Physics of MRI: a primer. J. Magn. Reson. Imaging 35, 1038–1054 (2012)

    PubMed  Article  Google Scholar 

  8. 8

    He, Q. Y., Peng, S.-G., Drummond, P. D. & Reid, M. D. Planar quantum squeezing and atom interferometry. Phys. Rev. A 84, 022107 (2011)

    Article  ADS  CAS  Google Scholar 

  9. 9

    Robertson, H. P. The uncertainty principle. Phys. Rev. 34, 163–164 (1929)

    Article  ADS  Google Scholar 

  10. 10

    Dammeier, L., Schwonnek, R. & Werner, R. F. Uncertainty relations for angular momentum. New J. Phys. 17, 093046 (2015)

    Article  ADS  Google Scholar 

  11. 11

    Grangier, P., Levenson, J. A. & Poizat, J.-P. Quantum non-demolition measurements in optics. Nature 396, 537–542 (1998)

    CAS  Article  ADS  Google Scholar 

  12. 12

    Koschorreck, M., Napolitano, M., Dubost, B. & Mitchell, M. W. Quantum nondemolition measurement of large-spin ensembles by dynamical decoupling. Phys. Rev. Lett. 105, 093602 (2010)

    CAS  PubMed  Article  ADS  Google Scholar 

  13. 13

    Sewell, R. J., Napolitano, M., Behbood, N., Colangelo, G. & Mitchell, M. W. Certified quantum non-demolition measurement of a macroscopic material system. Nat. Photon. 7, 517–520 (2013)

    CAS  Article  ADS  Google Scholar 

  14. 14

    Béguin, J.-B. et al. Generation and detection of a sub-Poissonian atom number distribution in a one-dimensional optical lattice. Phys. Rev. Lett. 113, 263603 (2014)

    PubMed  Article  ADS  CAS  Google Scholar 

  15. 15

    Lodewyck, J., Westergaard, P. G. & Lemonde, P. Nondestructive measurement of the transition probability in a Sr optical lattice clock. Phys. Rev. A 79, 061401 (2009)

    Article  ADS  CAS  Google Scholar 

  16. 16

    Sander, T. H. et al. Magnetoencephalography with a chip-scale atomic magnetometer. Biomed. Opt. Express 3, 981–990 (2012)

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  17. 17

    Sheng, D., Li, S., Dural, N. & Romalis, M. V. Subfemtotesla scalar atomic magnetometry using multipass cells. Phys. Rev. Lett. 110, 160802 (2013)

    CAS  PubMed  Article  ADS  Google Scholar 

  18. 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  Article  ADS  Google Scholar 

  19. 19

    Hall, L. T., Cole, J. H., Hill, C. D. & Hollenberg, L. C. L. Sensing of fluctuating nanoscale magnetic fields using nitrogen-vacancy centers in diamond. Phys. Rev. Lett. 103, 220802 (2009)

    CAS  PubMed  Article  ADS  Google Scholar 

  20. 20

    Bienfait, A. et al. Reaching the quantum limit of sensitivity in electron spin resonance. Nat. Nanotechnol. 11, 253–257 (2016)

    CAS  PubMed  Article  ADS  Google Scholar 

  21. 21

    Budker, D. & Romalis, M. Optical magnetometry. Nat. Phys. 3, 227–234 (2007)

    CAS  Article  Google Scholar 

  22. 22

    Miffre, A., Jacquey, M., Buchner, M., Trenec, G. & Vigue, J. Atom interferometry. Phys. Scr. 74, C15–C23 (2006)

    CAS  Article  Google Scholar 

  23. 23

    Yurke, B., McCall, S. L. & Klauder, J. R. SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033–4054 (1986)

    CAS  Article  ADS  Google Scholar 

  24. 24

    Colangelo, G. et al. Quantum atom–light interfaces in the Gaussian description for spin-1 systems. New J. Phys. 15, 103007 (2013)

    Article  ADS  CAS  Google Scholar 

  25. 25

    Braginsky, V. B., Vorontsov, Y. I. & Thorne, K. S. Quantum nondemolition measurements. Science 209, 547–557 (1980)

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  26. 26

    Tsang, M. & Caves, C. M. Evading quantum mechanics: engineering a classical subsystem within a quantum environment. Phys. Rev. X 2, 031016 (2012)

    Google Scholar 

  27. 27

    Polzik, E. S. & Hammerer, K. Trajectories without quantum uncertainties. Ann. Phys. 527, A15–A20 (2015)

    MathSciNet  Article  Google Scholar 

  28. 28

    Møller, C. B. et al. Back action evading quantum measurement of motion in a negative mass reference frame. Preprint at (2016)

  29. 29

    Smith, G. A., Chaudhury, S., Silberfarb, A., Deutsch, I. H. & Jessen, P. S. Continuous weak measurement and nonlinear dynamics in a cold spin ensemble. Phys. Rev. Lett. 93, 163602 (2004)

    PubMed  Article  ADS  CAS  Google Scholar 

  30. 30

    Koschorreck, M., Napolitano, M., Dubost, B. & Mitchell, M. W. Sub-projection-noise sensitivity in broadband atomic magnetometry. Phys. Rev. Lett. 104, 093602 (2010)

    CAS  PubMed  Article  ADS  Google Scholar 

  31. 31

    Ciurana, F. M., Colangelo, G., Sewell, R. J. & Mitchell, M. W. Real-time shot-noise-limited differential photodetection for atomic quantum control. Opt. Lett. 41, 2946–2949 (2016)

    Article  ADS  Google Scholar 

  32. 32

    Kubasik, M. et al. Polarization-based light-atom quantum interface with an all-optical trap. Phys. Rev. A 79, 043815 (2009)

    Article  ADS  CAS  Google Scholar 

  33. 33

    Deutsch, I. H. & Jessen, P. S. Quantum control and measurement of atomic spins in polarization spectroscopy. Opt. Commun. 283, 681–694 (2010)

    CAS  Article  ADS  Google Scholar 

  34. 34

    Kuzmich, A., Mandel, L. & Bigelow, N. P. Generation of spin squeezing via continuous quantum nondemolition measurement. Phys. Rev. Lett. 85, 1594–1597 (2000)

    CAS  PubMed  Article  ADS  Google Scholar 

  35. 35

    Appel, J. et al. Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit. Proc. Natl Acad. Sci. USA 106, 10960–10965 (2009)

    CAS  PubMed  Article  ADS  Google Scholar 

  36. 36

    Gühne, O. & Tóth, G. Entanglement detection. Phys. Rep. 474, 1–6 (2009)

    MathSciNet  Article  ADS  CAS  Google Scholar 

  37. 37

    Sørensen, A. S. & Mølmer, K. Entanglement and extreme spin squeezing. Phys. Rev. Lett. 86, 4431–4434 (2001)

    PubMed  Article  ADS  CAS  Google Scholar 

  38. 38

    Schlosser, N., Reymond, G., Protsenko, I. & Grangier, P. Sub-poissonian loading of single atoms in a microscopic dipole trap. Nature 411, 1024–1027 (2001)

    CAS  PubMed  Article  ADS  Google Scholar 

  39. 39

    Hofmann, C. S. et al. Sub-Poissonian statistics of Rydberg-interacting dark-state polaritons. Phys. Rev. Lett. 110, 203601 (2013)

    CAS  PubMed  Article  ADS  Google Scholar 

  40. 40

    Gajdacz, M. et al. Preparation of ultracold atom clouds at the shot noise level. Phys. Rev. Lett. 117, 073604 (2016)

    CAS  PubMed  Article  ADS  Google Scholar 

  41. 41

    Stockton, J. K. Continuous Quantum Measurement of Cold Alkali-atom Spins. PhD thesis, California Institute of Technology (2007)

  42. 42

    Takano, T., Fuyama, M., Namiki, R. & Takahashi, Y. Spin squeezing of a cold atomic ensemble with the nuclear spin of one-half. Phys. Rev. Lett. 102, 033601 (2009)

    CAS  PubMed  Article  ADS  Google Scholar 

  43. 43

    Schleier-Smith, M. H., Leroux, I. D. & Vuleticć, V. States of an ensemble of two-level atoms with reduced quantum uncertainty. Phys. Rev. Lett. 104, 073604 (2010)

    PubMed  Article  ADS  CAS  Google Scholar 

  44. 44

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

    CAS  PubMed  Article  ADS  Google Scholar 

  45. 45

    Bohnet, J. G. et al. Reduced spin measurement back-action for a phase sensitivity ten times beyond the standard quantum limit. Nat. Photon. 8, 731–736 (2014)

    CAS  Article  ADS  Google Scholar 

  46. 46

    Madsen, L. B. & Mølmer, K. Spin squeezing and precision probing with light and samples of atoms in the Gaussian description. Phys. Rev. A 70, 052324 (2004)

    Article  ADS  CAS  Google Scholar 

  47. 47

    Behbood, N. et al. Generation of macroscopic singlet states in a cold atomic ensemble. Phys. Rev. Lett. 113, 093601 (2014)

    CAS  PubMed  Article  ADS  Google Scholar 

  48. 48

    Kendall, M. & Stuart, A. The Advanced Theory of Statistics Vol. 1, Ch. 9 (Griffin, 1979)

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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.

Author information




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.

Additional information

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).

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