Letter | Published:

Simultaneous tracking of spin angle and amplitude beyond classical limits

Nature volume 543, pages 525528 (23 March 2017) | Download Citation

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.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

    , , & A subfemtotesla multichannel atomic magnetometer. Nature 422, 596–599 (2003)

  2. 2.

    , , , & Optical atomic clocks. Rev. Mod. Phys. 87, 637–701 (2015)

  3. 3.

    , & Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer. Appl. Phys. Lett. 97, 151110 (2010)

  4. 4.

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

  5. 5.

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

  6. 6.

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

  7. 7.

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

  8. 8.

    , , & Planar quantum squeezing and atom interferometry. Phys. Rev. A 84, 022107 (2011)

  9. 9.

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

  10. 10.

    , & Uncertainty relations for angular momentum. New J. Phys. 17, 093046 (2015)

  11. 11.

    , & Quantum non-demolition measurements in optics. Nature 396, 537–542 (1998)

  12. 12.

    , , & Quantum nondemolition measurement of large-spin ensembles by dynamical decoupling. Phys. Rev. Lett. 105, 093602 (2010)

  13. 13.

    , , , & Certified quantum non-demolition measurement of a macroscopic material system. Nat. Photon. 7, 517–520 (2013)

  14. 14.

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

  15. 15.

    , & Nondestructive measurement of the transition probability in a Sr optical lattice clock. Phys. Rev. A 79, 061401 (2009)

  16. 16.

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

  17. 17.

    , , & Subfemtotesla scalar atomic magnetometry using multipass cells. Phys. Rev. Lett. 110, 160802 (2013)

  18. 18.

    , , & Measurement noise 100 times lower than the quantum-projection limit using entangled atoms. Nature 529, 505–508 (2016)

  19. 19.

    , , & Sensing of fluctuating nanoscale magnetic fields using nitrogen-vacancy centers in diamond. Phys. Rev. Lett. 103, 220802 (2009)

  20. 20.

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

  21. 21.

    & Optical magnetometry. Nat. Phys. 3, 227–234 (2007)

  22. 22.

    , , , & Atom interferometry. Phys. Scr. 74, C15–C23 (2006)

  23. 23.

    , & SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033–4054 (1986)

  24. 24.

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

  25. 25.

    , & Quantum nondemolition measurements. Science 209, 547–557 (1980)

  26. 26.

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

  27. 27.

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

  28. 28.

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

  29. 29.

    , , , & Continuous weak measurement and nonlinear dynamics in a cold spin ensemble. Phys. Rev. Lett. 93, 163602 (2004)

  30. 30.

    , , & Sub-projection-noise sensitivity in broadband atomic magnetometry. Phys. Rev. Lett. 104, 093602 (2010)

  31. 31.

    , , & Real-time shot-noise-limited differential photodetection for atomic quantum control. Opt. Lett. 41, 2946–2949 (2016)

  32. 32.

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

  33. 33.

    & Quantum control and measurement of atomic spins in polarization spectroscopy. Opt. Commun. 283, 681–694 (2010)

  34. 34.

    , & Generation of spin squeezing via continuous quantum nondemolition measurement. Phys. Rev. Lett. 85, 1594–1597 (2000)

  35. 35.

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

  36. 36.

    & Entanglement detection. Phys. Rep. 474, 1–6 (2009)

  37. 37.

    & Entanglement and extreme spin squeezing. Phys. Rev. Lett. 86, 4431–4434 (2001)

  38. 38.

    , , & Sub-poissonian loading of single atoms in a microscopic dipole trap. Nature 411, 1024–1027 (2001)

  39. 39.

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

  40. 40.

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

  41. 41.

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

  42. 42.

    , , & Spin squeezing of a cold atomic ensemble with the nuclear spin of one-half. Phys. Rev. Lett. 102, 033601 (2009)

  43. 43.

    , & States of an ensemble of two-level atoms with reduced quantum uncertainty. Phys. Rev. Lett. 104, 073604 (2010)

  44. 44.

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

  45. 45.

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

  46. 46.

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

  47. 47.

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

  48. 48.

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

Download references

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.

Author information

Affiliations

  1. ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain

    • Giorgio Colangelo
    • , Ferran Martin Ciurana
    • , Lorena C. Bianchet
    • , Robert J. Sewell
    •  & Morgan W. Mitchell
  2. ICREA—Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain

    • Morgan W. Mitchell

Authors

  1. Search for Giorgio Colangelo in:

  2. Search for Ferran Martin Ciurana in:

  3. Search for Lorena C. Bianchet in:

  4. Search for Robert J. Sewell in:

  5. Search for Morgan W. Mitchell in:

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.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Giorgio Colangelo or Morgan W. Mitchell.

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

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/nature21434

Further reading

Comments

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.