Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Optical magnetometry

Abstract

Some of the most sensitive methods of measuring magnetic fields use interactions of resonant light with atomic vapour. Recent developments in this vibrant field have led to improvements in sensitivity and other characteristics of atomic magnetometers, benefiting their traditional applications for measurements of geomagnetic anomalies and magnetic fields in space, and opening many new areas previously accessible only to magnetometers based on superconducting quantum interference devices. We review basic principles of modern optical magnetometers, discuss fundamental limitations on their performance, and describe recently explored applications for dynamical measurements of biomagnetic fields, detecting signals in NMR and MRI, inertial rotation sensing, magnetic microscopy with cold atoms, and tests of fundamental symmetries of nature.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Figure 1: A general schematic of an all-optical atomic magnetometer.
Figure 2: Comparison of Zeeman resonances for different modes of operation in a potassium vapour with density n = 7 × 1013 cm−3.
Figure 3: Examples of biological magnetic fields recorded with atomic magnetometers.
Figure 4: Detection of effective magnetic field by imaging of Larmor precession in a BEC of 87Rb (ref. 55).

Similar content being viewed by others

References

  1. Kastler, A. Some suggestions concerning the production and detection by optical means of inequalities in the populations of levels of spatial quantization in atoms. Application to the Stern and Gerlach and magnetic resonance experiments. J. Phys. Radium 11, 255–265 (1950).

    Article  Google Scholar 

  2. Dehmelt, H. Modulation of a light beam by precessing absorbing atoms. Phys. Rev. 105, 1924–1925 (1957).

    Article  ADS  Google Scholar 

  3. Bell, W. & Bloom, A. Optical detection of magnetic resonance in alkali metal vapor. Phys. Rev. 107, 1559–1565 (1957).

    Article  ADS  Google Scholar 

  4. Bell, W. & Bloom, A. Optically driven spin precession. Phys. Rev. Lett. 6, 280–281 (1961).

    Article  ADS  Google Scholar 

  5. Bloom, A. Principles of operation of the rubidium vapor magnetometer. Appl. Opt. 1, 61–68 (1962).

    Article  ADS  Google Scholar 

  6. Dupont-Roc, J., Haroche, S. & Cohen-Tannoudji, C. Detection of very weak magnetic fields (10−9 gauss) by 87Rb zero-field level crossing resonances. Phys. Lett. A 28, 638–639 (1969).

    Article  ADS  Google Scholar 

  7. Budker, D. et al. Resonant nonlinear magneto-optical effects in atoms. Rev. Mod. Phys. 74, 1153–1201 (2002).

    Article  ADS  Google Scholar 

  8. Alexandrov, E. B. et al. Dynamic effects in nonlinear magneto-optics of atoms and molecules: Review. J. Opt. Soc. Am. B 22, 7–20 (2005).

    Article  ADS  Google Scholar 

  9. Aleksandrov, E. B., Balabas, M. V., Vershovskii, A. K. & Pazgalev, A. S. Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer. Tech. Phys. 49, 779–783 (2004).

    Article  Google Scholar 

  10. Budker, D., Kimball, D. F., Rochester, S. M., Yashchuk, V. V. & Zolotorev, M. Sensitive magnetometry based on nonlinear magneto-optical rotation. Phys. Rev. A 62, 043403 (2000).

    Article  ADS  Google Scholar 

  11. Groeger, S., Bison, G., Schenker, J. L., Wynands, R. & Weis, A. A high-sensitivity laser-pumped Mx magnetometer. Eur. Phys. J. D 38, 239–247 (2006).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. Clarke, J. & Braginski, A. I. The SQUID Handbook (Wiley-VCH, Weinheim, 2004).

    Book  Google Scholar 

  14. Groeger, S., Pazgalev, A. S. & Weis, A. Comparison of discharge lamp and laser pumped cesium magnetometers. Appl. Phys. B. 80, 645–654 (2005).

    Article  ADS  Google Scholar 

  15. Vershovskii, A. K., Pazgalev, A. S. & Aleksandrov, E. B. The design of a λ-hfs magnetometer. Techn. Phys. 45, 88–93 (2000).

    Article  ADS  Google Scholar 

  16. Geremia, G. M., Stockton, J. K. & Mabuchi, H. Suppression of spin projection noise in broadband atomic magnetometry. Phys. Rev. Lett. 94, 203002 (2005).

    Article  ADS  Google Scholar 

  17. Auzinsh, M. et al. Can a quantum nondemolition measurement improve the sensitivity of an atomic magnetometer? Phys. Rev. Lett. 93, 173002 (2004).

    Article  ADS  Google Scholar 

  18. Savukov, I. M., Seltzer, S. J., Romalis, M. V. & Sauer, K. L. Tunable atomic magnetometer for detection of radio-frequency magnetic fields. Phys. Rev. Lett. 95, 063004 (2005).

    Article  ADS  Google Scholar 

  19. Happer, W. & Mathur, B. Effective operator formalism in optical pumping. Phys. Rev. 163, 12–25 (1967).

    Article  ADS  Google Scholar 

  20. Fleischhauer, M., Matsko, A. B. & Scully, M. O. Quantum limit of optical magnetometry in the presence of ac stark shifts. Phys. Rev. A 62, 013808 (2000).

    Article  ADS  Google Scholar 

  21. Novikova, I., Matsko, A. B., Velichansky, V. L., Scully, M. O. & Welch, G. R. Compensation of ac stark shifts in optical magnetometry. Phys. Rev. A 63, 063802 (2001).

    Article  ADS  Google Scholar 

  22. Robinson, H., Ensberg, E. & Dehmelt, H. Preservation of spin state in free atom-inert surface collisions. Bull. Am. Phys. Soc. 3, 9 (1958).

    Google Scholar 

  23. Bouchiat, M. A. & Brossel, J. Relaxation of optically pumped Rb atoms on paraffin-coated walls. Phys. Rev. 147, 41–54 (1966).

    Article  ADS  Google Scholar 

  24. Alexandrov, E. B. et al. Light-induced desorption of alkali-metal atoms from paraffin coating. Phys. Rev. A 66, 042903 (2002); Erratum. Phys. Rev. A 70, 049902(E) (2004).

    Article  ADS  Google Scholar 

  25. Happer, W. & Tang, H. Spin-exchange shift and narrowing of magnetic resonance lines in optically pumped alkali vapors. Phys. Rev. Lett. 31, 273–276 (1973).

    Article  ADS  Google Scholar 

  26. Savukov, I. M. & Romalis, M. V. Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields. Phys. Rev. A 71, 23405 (2005).

    Article  ADS  Google Scholar 

  27. Erickson, C. J. et al. Spin relaxation resonances due to the spin-axis interaction in dense rubidium and cesium vapor. Phys. Rev. Lett. 85, 4237–4240 (2000).

    Article  ADS  Google Scholar 

  28. Kadlecek, S., Anderson, L. W. & Walker, T. G. Field dependence of spin relaxation in a dense Rb vapor. Phys. Rev. Lett. 80, 5512–5515 (1998).

    Article  ADS  Google Scholar 

  29. Allred, J. C., Lyman, R. N., Kornack, T. W. & Romalis, M. V. A high-sensitivity atomic magnetometer unaffected by spin-exchange relaxation. Phys. Rev. Lett. 89, 130801 (2002).

    Article  ADS  Google Scholar 

  30. Seltzer, S. & Romalis, M. V. Unshielded three-axis vector operation of a spin-exchange-relaxation-free atomic magnetometer. Appl. Phys. Lett. 85, 4804–4806 (2004).

    Article  ADS  Google Scholar 

  31. Appelt, S., Ben-Amar Baranga, A., Young, A. R. & Happer, W. Light narrowing of rubidium magneticresonance lines in high-pressure optical-pumping cells. Phys. Rev. A 59, 2078–2084 (1999).

    Article  ADS  Google Scholar 

  32. Jau, Y. Y. et al. Intense, narrow atomic-clock resonances. Phys. Rev. Lett. 92, 110801 (2004).

    Article  ADS  Google Scholar 

  33. Smullin, S. J., Savukov, I. M., Vasilakis, G., Ghosh, R. K. & Romalis, M. V. A low-noise high-density alkali metal scalar magnetometer. Preprint at http://arxiv.org/abs/physics/0611085 (2006).

  34. Stahler, M., Knappe, S., Affolderbach, C., Kemp, W. & Wynands, R. Picotesla magnetometry with coherent dark states. Europhys. Lett. 54, 323–328 (2001).

    Article  ADS  Google Scholar 

  35. Andreeva, C. et al. Two-color coherent population trapping in a single Cs hyperfine transition, with application in magnetometry. Appl. Phys. B 76, 667–675 (2003).

    Article  ADS  Google Scholar 

  36. Alipieva, E. et al. Coherent population trapping for magnetic field measurements. Proc. SPIE 5830, 170–175 (2005).

    Article  ADS  Google Scholar 

  37. Acosta, V. et al. Nonlinear magneto-optical rotation with frequency-modulated light in the geophysical field range. Phys. Rev. A 73, 053404 (2006).

    Article  ADS  Google Scholar 

  38. Seltzer, S. J., Meares, P. J. & Romalis, M. V. Synchronous optical pumping of quantum revival beats for atomic magnetometry. Preprint at http://arxiv.org/abs/physics/0611014 (2006).

  39. Alexandrov, E. B., Pazgalev, A. S. & Rasson, J. L. Observation of four-quantum resonance in the Zeeman structure of the ground-state of 39K. Opt. Spectrosk. 82, 14–20 (1997).

    ADS  Google Scholar 

  40. Yashchuk, V. V. et al. Selective addressing of high-rank atomic polarization moments. Phys. Rev. Lett. 90, 253001 (2003).

    Article  ADS  Google Scholar 

  41. Pustelny, S. et al. Pump-probe nonlinear magnetooptical rotation with frequency-modulated light. Phys. Rev. A 73, 023817 (2006).

    Article  ADS  Google Scholar 

  42. Gravrand, O., Khokhlov, A., Mouël, J. L. L. & Léger, J. M. On the calibration of a vectorial 4He pumped magnetometer. Earth Planets Space 53, 949–958 (2001).

    Article  ADS  Google Scholar 

  43. Alexandrov, E. B. et al. Three-component variometer based on a scalar potassium sensor. Meas. Sci. Technol. 15, 918–922 (2004).

    Article  ADS  Google Scholar 

  44. Matsko, A. B., Strekalov, D. & Maleki, L. Magnetometer based on the opto-electronic microwave oscillator. Opt. Commun. 247, 141–148 (2005).

    Article  ADS  Google Scholar 

  45. Schwindt, P. D. D., Hollberg, L. & Kitching, J. Self-oscillating rubidium magnetometer using nonlinear magneto-optical rotation. Rev. Sci. Instrum. 76, 126103 (2005).

    Article  ADS  Google Scholar 

  46. Higbie, J., Corsini, E. & Budker, D. Robust, high-speed, all-optical atomic magnetometer. Rev. Sci. Instrum. 77, 113106 (2006).

    Article  ADS  Google Scholar 

  47. Bechhoefer, J. Feedback for physicists: A tutorial essay on control. Rev. Mod. Phys. 77, 783–836 (2005).

    Article  ADS  Google Scholar 

  48. Rife, D. C. & Boorstyn, R. R. Single-tone parameter estimation from discrete-time observations. IEEE Trans. Inform. Theory 20, 591–598 (1974).

    Article  MATH  Google Scholar 

  49. Weissman, M. B. 1/f noise and other slow, nonexponential kinetics in condensed matter. Rev. Mod. Phys. 60, 537571 (1988).

    Article  Google Scholar 

  50. Li, Z., Wakai, R. T. & Walker, T. G. Parametric modulation of an atomic magnetometer. Appl. Phys. Lett. 89, 134105 (2006).

    Article  ADS  Google Scholar 

  51. Balabas, M. V., Budker, D., Kitching, J., Schwindt, P. D. D. & Stalnaker, J. E. Magnetometry with millimeter-scale antirelaxation-coated alkali-metal vapor cells. J. Opt. Soc. Am. B 23, 1001–1006 (2006).

    Article  ADS  Google Scholar 

  52. Schwindt, P. D. D. et al. Chip-scale atomic magnetometer. Appl. Phys. Lett. 85, 6409–6411 (2004).

    Article  ADS  Google Scholar 

  53. Pustelny, S., Jackson Kimball, D. F., Rochester, S. M., Yashchuk, V. V. & Budker, D. Influence of magnetic-field inhomogeneity on nonlinear magneto-optical resonances. Phys. Rev. A 74, 063406 (2006).

    Article  ADS  Google Scholar 

  54. Wildermuth, S. et al. Sensing electric and magnetic fields with Bose-Einstein condensates. Appl. Phys. Lett. 88, 264103 (2006).

    Article  ADS  Google Scholar 

  55. Vengalattore, M. et al. High-resolution magnetometry with a spinor Bose-Einstein condensate. Preprint at http://arxiv.org/abs/cond-mat/0612685 (2006).

  56. Zhao, K. F. & Wu, Z. Evanescent wave magnetometer. Appl. Phys. Lett. 89, 261113 (2006).

    Article  ADS  Google Scholar 

  57. Fenici, R., Brisinda, D. & Meloni, A. M. Clinical application of magnetocardiography. Exp. Rev. Mol. Diagn. 5, 291–313 (2005).

    Article  Google Scholar 

  58. Hämäläinen, M., Hari, R., Ilmoniemi, R. J., Knuutila, J. & Lounasmaa, O. V. Magnetoencephalography theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65, 413–497 (1993).

    Article  ADS  Google Scholar 

  59. Papanicolaou, A. C., Castillo, E. M., Billingsley-Marshall, R., Pataraia, E. & Simos, P. G. A review of clinical applications of magnetoencephalography. Int. Rev. Neurobiol. 68, 223–247 (2005).

    Article  Google Scholar 

  60. Livanov, M. N. et al. Recording of human magnetic fields. Doklady Akademii Nauk SSSR 238, 253–256 (1977).

    Google Scholar 

  61. Bison, G., Wynands, R. & Weis, A. A laser-pumped magnetometer for the mapping of human cardiomagnetic fields. Appl. Phys. B. 76, 325–328 (2003).

    Article  ADS  Google Scholar 

  62. Xia, H., Baranga, A. B., Hoffman, D. & Romalis, M. V. Magnetoencephalography with an atomic magnetometer. Appl. Phys. Lett. 89, 211104 (2006).

    Article  ADS  Google Scholar 

  63. Murthy, S. A., Krause, J., D., Li, Z. L. & Hunter, L. R. New limits on the electron electric dipole moment from cesium. Phys. Rev. Lett. 63, 965–968 (1989).

    Article  ADS  Google Scholar 

  64. Berglund, C. J. et al. New limits on local Lorentz invariance from Hg and Cs magnetometers. Phys. Rev. Lett. 75, 1879–1882 (1995).

    Article  ADS  Google Scholar 

  65. Youdin, A. N., Krause, J., D., Jagannathan, K., Hunter, L. R. & Lamoreaux, S. K. Limits on spin-mass couplings within the axion window. Phys. Rev. Lett. 77, 2170–2173 (1996).

    Article  ADS  Google Scholar 

  66. Gilles, H., Monfort, Y. & Hamel, J. 3He maser for earth magnetic field measurement. Rev. Sci. Instrum. 74, 4515–4520 (2003).

    Article  ADS  Google Scholar 

  67. Romalis, M. V., Griffith, W. C., Jacobs, J. P. & Fortson, E. N. New limit on the permanent electric dipole moment of 199Hg. Phys. Rev. Lett. 86, 2505–2508 (2001).

    Article  ADS  Google Scholar 

  68. Baker, C. A. et al. Improved experimental limit on the electric dipole moment of the neutron. Phys. Rev. Lett. 97, 131801 (2006).

    Article  ADS  Google Scholar 

  69. Pospelov, M. & Ritz, A. Electric dipole moments as probes of new physics. Ann. Phys. 318, 119–169 (2005).

    Article  ADS  MATH  Google Scholar 

  70. Bear, D., Stoner, R. E., Walsworth, R. L., Kostelecký, V. A. & Lane, C. D. Limit on Lorentz and CPT violation of the neutron using a two-species noble-gas maser. Phys. Rev. Lett. 85, 5038–5041 (2000); Erratum. Phys. Rev. Lett. 89, 209902 (2002).

    Article  ADS  Google Scholar 

  71. Chin, C., Leiber, V., Vuletić, V., Kerman, A. J. & Chu, S. Measurement of an electron's electric dipole moment using Cs atoms trapped in optical lattices. Phys. Rev. A 63, 033401 (2001).

    Article  ADS  Google Scholar 

  72. Amini, J. M., Munger, C. T. Jr & Gould, H. Demonstration of a cold atom fountain electron electric dipole moment experiment. http://arxiv.org/physics/0602011 (2006).

  73. Lamoreaux, S. K. Solid-state systems for the electron electric dipole moment and other fundamental measurements. Phys. Rev. A 66, 022109 (2002).

    Article  ADS  Google Scholar 

  74. Budker, D., Lamoreaux, S. K., Sushkov, A. O. & Sushkov, O. P. On the sensitivity of condensed-matter P- and T-violation experiments. Phys. Rev. A 73, 022107 (2006).

    Article  ADS  Google Scholar 

  75. Acuna, M. H. in Encyclopedia of Planetary Sciences (eds Shirley, J. H. & Fairbridge, R. W.) 406–410 (Chapman & Hall, London, 1997).

    Book  Google Scholar 

  76. Balogh, A. in IEE Colloquium on Satellite Instrumentation Digest No. 12, 2/1–3 (IEE, London, 1988).

    Google Scholar 

  77. Dunlop, M. W., Dougherty, M. K., Kellock, S. & Southwood, D. J. Operation of the dual magnetometer on Cassini: science performance. Planet. Space Sci. (UK) 47, 1389–1405 (1999).

    Article  ADS  Google Scholar 

  78. Dougherty, M. K. et al. Cassini magnetometer observations during Saturn orbit insertion. Science 307, 1266–1270 (2005).

    Article  ADS  Google Scholar 

  79. Dougherty, M. K. et al. Identification of a dynamic atmosphere at Enceladus with the Cassini magnetometer. Science 311, 1406–1409 (2006).

    Article  ADS  Google Scholar 

  80. Slocum, R. E., Kuhlman, G., Ryan, L. & King, D. in IEEE Proc. Conf. Oceans 2002 Vol. 2, 945–951 (IEEE, London, 2002).

    Google Scholar 

  81. McGregor, D. D. High-sensitivity helium resonance magnetometers. Rev. Sci. Instrum. 58, 1067–1076 (1987).

    Article  ADS  Google Scholar 

  82. Burlaga, L. F. et al. A transition to fast flows and its effects on the magnetic fields and cosmic rays observed by Voyager 2 near 70 au. Astrophys. J. 618, 1074–1078 (2005).

    Article  ADS  Google Scholar 

  83. Greenberg, Y. S. Application of superconducting quantum interference devices to nuclear magnetic resonance. Rev. Mod. Phys. 70, 175–222 (1998).

    Article  ADS  Google Scholar 

  84. Cohen-Tannoudji, C., DuPont-Roc, J., Haroche, S. & Laloë, F. Detection of the static magnetic field produced by the oriented nuclei of optically pumped 3He gas. Phys. Rev. Lett. 22, 758–760 (1969).

    Article  ADS  Google Scholar 

  85. Yashchuk, V. V. et al. Hyperpolarized xenon nuclear spins detected by optical atomic magnetometry. Phys. Rev. Lett. 93, 160801 (2004).

    Article  ADS  Google Scholar 

  86. Savukov, I. M. & Romalis, M. V. NMR detection with an atomic magnetometer. Phys. Rev. Lett. 94, 123001 (2005).

    Article  ADS  Google Scholar 

  87. Moulé, A. J. et al. Amplification of xenon NMR and MRI by remote detection. Proc. Natl Acad. Sci. USA 100, 9122–9127 (2003).

    Article  ADS  Google Scholar 

  88. Xu, S., Rochester, S. M., Yashchuk, V. V., Donaldson, M. H. & Budker, D. Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation. Rev. Sci. Instrum. 77, 083106 (2006).

    Article  ADS  Google Scholar 

  89. Xu, S. et al. Magnetic resonance imaging with an optical atomic magnetometer. Proc. Natl Acad. Sci. USA 103, 12668–12671 (2006).

    Article  ADS  Google Scholar 

  90. Schaefer, S. R. et al. Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms. Phys. Rev. A 39, 5613–5623 (1989).

    Article  ADS  Google Scholar 

  91. Woodman, K. F., Franks, P. W. & Richards, M. D. The nuclear magnetic resonance gyroscope: a review. J. Navigation 40, 366–384 (1987).

    Article  ADS  Google Scholar 

  92. Kornack, T. W., Ghosh, R. K. & Romalis, M. V. Nuclear spin gyroscope based on an atomic comagnetometer. Phys. Rev. Lett. 95, 230801 (2005).

    Article  ADS  Google Scholar 

  93. Budker, D., Kimball, D. F., Rochester, S. M. & Urban, J. T. Alignment-to-orientation conversion and nuclear quadrupole resonance. Chem. Phys. Lett. 378, 440–448 (2003).

    Article  ADS  Google Scholar 

  94. Garroway, A. N. et al. Remote sensing by nuclear quadrupole resonance. IEEE Trans. Geosci. Remote Sens. (USA) 39, 1108–1118 (2001).

    Article  ADS  Google Scholar 

  95. Ledbetter, M. P. et al. Detection of radio frequency magnetic fields using nonlinear magneto-optical rotation. Phys. Rev. A 75, 023405 (2007).

    Article  ADS  Google Scholar 

  96. Lee, S.-K., Sauer, K. L., Seltzer, S. J., Alem, O. & Romalis, M. V. Subfemtotesla radio-frequency atomic magnetometer for detection of nuclear quadrupole resonance. Appl. Phys. Lett. 89, 214106 (2006).

    Article  ADS  Google Scholar 

  97. Savukov, I. M., Seltzer, S. J. & Romalis, M. V. Detection of NMR signals with a radio-frequency atomic magnetometer. J. Magn. Res. 185, 227–233 (2007).

    Article  Google Scholar 

  98. O'Hara, K. M. et al. Ultrastable CO2 laser trapping of lithium fermions. Phys. Rev. Lett. 82, 4204–4207 (1999).

    Article  ADS  Google Scholar 

  99. Moler, K. A., Kirtley, J. R., Hinks, D. G., Li, T. W. & Xu, M. Images of interlayer Josephson vortices in Tl2Ba2 CuO6+δ . Science 279, 1193–1196 (1998).

    Article  ADS  Google Scholar 

  100. Chatraphorn, S., Fleet, E. F., Wellstood, F. C., Knauss, L. A. & Eiles, T. M. Scanning SQUID microscopy of integrated circuits. Appl. Phys. Lett. 76, 2304–2306 (2000).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work is supported by DOD MURI grant No. N00014-05-1-0406. We are grateful to E. B. Alexandrov, M. V. Balabas, G. Bison, S. Bale, W. Gawlik, J. Higbie, M. Ledbetter, I. M. Savukov, D. Stamper-Kurn, A. Sushkov, M. Vengalattore and A. Weis for providing valuable input for this review.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Dmitry Budker or Michael Romalis.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Budker, D., Romalis, M. Optical magnetometry. Nature Phys 3, 227–234 (2007). https://doi.org/10.1038/nphys566

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys566

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing