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Limits of absolute vector magnetometry with nitrogen-vacancy centers in diamond

Dennis Lönard, Isabel Cardoso Barbosa, Stefan Johansson, Jonas Gutsche, and Artur Widera*

  • *Contact author: widera@rptu.de
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Phys. Rev. Research 8, 013054 – Published 20 January, 2026

DOI: https://doi.org/10.1103/28t7-7gcn

Abstract

The nitrogen-vacancy (NV) center in diamond has become a widely used platform for quantum sensing. The four NV axes in monocrystalline diamond specifically allow for vector magnetometry, with magnetic-field sensitivities reaching down to fT/Hz. The current literature primarily focuses on improving the precision of NV-based magnetometers. Here, we study the experimental accuracy of determining the magnetic field from measured spin-resonance frequencies via solving the NV Hamiltonian. We derive exact, analytical, and fast-to-compute formulas for calculating resonance frequencies from a known magnetic-field vector, and vice versa (formulas for calculating the magnetic-field vector from measured resonance frequencies). Additionally, the accuracy of often-used approximations is assessed. Finally, we promote using the Voigt profile as a fit model to determine the linewidth of measured resonances accurately. An open-source Python package accompanies our analysis.

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References (65)

  1. J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, Sensitivity optimization for NV-diamond magnetometry, Rev. Mod. Phys. 92, 015004 (2020).
  2. N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, Quantum sensors for biomedical applications, Nat. Rev. Phys. 5, 157 (2023).
  3. T. F. Segawa and R. Igarashi, Nanoscale quantum sensing with nitrogen-vacancy centers in nanodiamonds—A magnetic resonance perspective, Prog. Nucl. Magn. Reson. Spectrosc. 134–135, 20 (2023).
  4. S. C. Scholten, A. J. Healey, I. O. Robertson, G. J. Abrahams, D. A. Broadway, and J.-P. Tetienne, Widefield quantum microscopy with nitrogen-vacancy centers in diamond: Strengths, limitations, and prospects, J. Appl. Phys. 130, 150902 (2021).
  5. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  6. A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. V. Borczyskowski, Scanning confocal optical microscopy and magnetic resonance on single defect centers, Science 276, 2012 (1997).
  7. C. L. Degen, Scanning magnetic field microscope with a diamond single-spin sensor, Appl. Phys. Lett. 92, 243111 (2008).
  8. M. W. Doherty, N. B. Manson, P. Delaney, and L. C. L. Hollenberg, The negatively charged nitrogen-vacancy centre in diamond: The electronic solution, New J. Phys. 13, 025019 (2011).
  9. L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, Magnetometry with nitrogen-vacancy defects in diamond, Rep. Prog. Phys. 77, 056503 (2014).
  10. J. H. N. Loubser and J. A. V. Wyk, Electron spin resonance in the study of diamond, Rep. Prog. Phys. 41, 1201 (1978).
  11. J. M. Schloss, J. F. Barry, M. J. Turner, and R. L. Walsworth, Simultaneous broadband vector magnetometry using solid-state spins, Phys. Rev. Appl. 10, 034044 (2018).
  12. J. F. Barry, M. H. Steinecker, S. T. Alsid, J. Majumder, L. M. Pham, M. F. O’Keeffe, and D. A. Braje, Sensitive ac and dc magnetometry with nitrogen-vacancy-center ensembles in diamond, Phys. Rev. Appl. 22, 044069 (2024).
  13. T. Wolf, P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, and J. Wrachtrup, Subpicotesla diamond magnetometry, Phys. Rev. X 5, 041001 (2015).
  14. Y. Xie, H. Yu, Y. Zhu, X. Qin, X. Rong, C.-K. Duan, and J. Du, A hybrid magnetometer towards femtotesla sensitivity under ambient conditions, Sci. Bull. 66, 127 (2021).
  15. Z. Wang, F. Kong, P. Zhao, Z. Huang, P. Yu, Y. Wang, F. Shi, and J. Du, Picotesla magnetometry of microwave fields with diamond sensors, Sci. Adv. 8, eabq8158 (2022).
  16. Z.-G. Hu, Y.-M. Gao, J.-F. Liu, H. Yang, M. Wang, Y. Lei, X. Zhou, J. Li, X. Cao, J. Liang, C.-Q. Hu, Z. Li, Y.-C. Lau, J.-W. Cai, and B.-B. Li, Picotesla-sensitivity microcavity optomechanical magnetometry, Light Sci. Appl. 13, 279 (2024).
  17. T. Yamaguchi, Y. Matsuzaki, S. Saito, S. Saijo, H. Watanabe, N. Mizuochi, and J. Ishi-Hayase, Bandwidth analysis of AC magnetic field sensing based on electronic spin double-resonance of nitrogen-vacancy centers in diamond, Jpn. J. Appl. Phys. 58, 100901 (2019).
  18. M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, and L. C. L. Hollenberg, Theory of the ground-state spin of the NV- center in diamond, Phys. Rev. B 85, 205203 (2012).
  19. N. B. Manson, J. P. Harrison, and M. J. Sellars, Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics, Phys. Rev. B 74, 104303 (2006).
  20. I. Harari and U. Albocher, Computation of eigenvalues of a real, symmetric 3 × 3 matrix with particular reference to the pernicious case of two nearly equal eigenvalues, Int. J. Numer. Methods Eng. 124, 1089 (2023).
  21. G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, Nanoscale imaging magnetometry with diamond spins under ambient conditions, Nature (London) 455, 648 (2008).
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge University Press, Cambridge, 2007).
  23. F. Viète, Opera Mathematica in Unum Volumen Congesta (Bonaventure & Abraham Elzevier, Leiden, 1646).
  24. D. J. Wolters, Practical algorithm for solving the cubic equation (2025), https://quarticequations.com/Cubic.pdf.
  25. R. W. D. Nickalls, Viète, Descartes and the cubic equation, Math. Gaz. 90, 203 (2006).
  26. S.-Y. Lee, M. Niethammer, and J. Wrachtrup, Vector magnetometry based on S = 3/2 electronic spins, Phys. Rev. B 92, 115201 (2015).
  27. J.-F. Ye, Z. Jiao, K. Ma, Z.-Y. Huang, H.-J. Lv, and F.-J. Jiang, Reconstruction of vector static magnetic field by different axial NV centers using continuous wave optically detected magnetic resonance in diamond, Chin. Phys. B 28, 047601 (2019).
  28. D. G. Stone, B. Whitefield, M. Kianinia, and C. Bradac, Fast characterization of optically detected magnetic resonance spectra via data clustering, J. Phys. Chem. C 128, 13147 (2024).
  29. S. Lourette, A. Jarmola, V. M. Acosta, A. G. Birdwell, D. Budker, M. W. Doherty, T. Ivanov, and V. S. Malinovsky, Temperature sensitivity of N14−V and N15−V ground-state manifolds, Phys. Rev. Appl. 19, 064084 (2023).
  30. T. Weggler, C. Ganslmayer, F. Frank, T. Eilert, F. Jelezko, and J. Michaelis, Determination of the three-dimensional magnetic field vector orientation with nitrogen vacancy centers in diamond, Nano Lett. 20, 2980 (2020).
  31. A. Aitken, IV. On least squares and linear combination of observations, Proc. R. Soc. Edinburgh 55, 42 (1936).
  32. N. M. Beaver, N. Voce, P. Meisenheimer, R. Ramesh, and P. Stevenson, Optimizing off-axis fields for two-axis magnetometry with point defects, Appl. Phys. Lett. 124, 254001 (2024).
  33. Y. Silani, J. Smits, I. Fescenko, M. W. Malone, A. F. McDowell, A. Jarmola, P. Kehayias, B. A. Richards, N. Mosavian, N. Ristoff, and V. M. Acosta, Nuclear quadrupole resonance spectroscopy with a femtotesla diamond magnetometer, Sci. Adv. 9, eadh3189 (2023).
  34. D. Simin, F. Fuchs, H. Kraus, A. Sperlich, P. Baranov, G. Astakhov, and V. Dyakonov, High-precision angle-resolved magnetometry with uniaxial quantum centers in silicon carbide, Phys. Rev. Appl. 4, 014009 (2015).
  35. B. J. Maertz, A. P. Wijnheijmer, G. D. Fuchs, M. E. Nowakowski, and D. D. Awschalom, Vector magnetic field microscopy using nitrogen vacancy centers in diamond, Appl. Phys. Lett. 96, 092504 (2010).
  36. S. Steinert, F. Dolde, P. Neumann, A. Aird, B. Naydenov, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, High sensitivity magnetic imaging using an array of spins in diamond, Rev. Sci. Instrum. 81, 043705 (2010).
  37. M. Garsi, R. Stöhr, A. Denisenko, F. Shagieva, N. Trautmann, U. Vogl, B. Sene, F. Kaiser, A. Zappe, R. Reuter, and J. Wrachtrup, Three-dimensional imaging of integrated-circuit activity using quantum defects in diamond, Phys. Rev. Appl. 21, 014055 (2024).
  38. M. Tsukamoto, S. Ito, K. Ogawa, Y. Ashida, K. Sasaki, and K. Kobayashi, Accurate magnetic field imaging using nanodiamond quantum sensors enhanced by machine learning, Sci. Rep. 12, 13942 (2022).
  39. J. Homrighausen, L. Horsthemke, J. Pogorzelski, S. Trinschek, P. Glösekötter, and M. Gregor, Edge-machine-learning-assisted robust magnetometer based on randomly oriented NV-ensembles in diamond, Sensors 23, 1119 (2023).
  40. E. Zhang, W. Zhang, X. Jiang, and X. Qin, Deep-neural-network-based magnetic-field acquisition method for quantum sensing with nitrogen-vacancy centers, Phys. Rev. A 110, 052417 (2024).
  41. E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys. 93, 025010 (2021).
  42. X.-F. He, N. B. Manson, and P. T. H. Fisk, Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the A3 ground state, Phys. Rev. B 47, 8809 (1993).
  43. S. Felton, A. M. Edmonds, M. E. Newton, P. M. Martineau, D. Fisher, D. J. Twitchen, and J. M. Baker, Hyperfine interaction in the ground state of the negatively charged nitrogen vacancy center in diamond, Phys. Rev. B 79, 075203 (2009).
  44. J. Homrighausen, F. Hoffmann, J. Pogorzelski, P. Glösekötter, and M. Gregor, Microscale fiber-integrated vector magnetometer with on-tip field biasing using N-V ensembles in diamond microcrystals, Phys. Rev. Appl. 22, 034029 (2024).
  45. V. M. Acosta, E. Bauch, M. P. Ledbetter, A. Waxman, L.-S. Bouchard, and D. Budker, Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond, Phys. Rev. Lett. 104, 070801 (2010).
  46. M. W. Doherty, V. V. Struzhkin, D. A. Simpson, L. P. McGuinness, Y. Meng, A. Stacey, T. J. Karle, R. J. Hemley, N. B. Manson, L. C. Hollenberg, and S. Prawer, Electronic properties and metrology applications of the diamond NV-center under pressure, Phys. Rev. Lett. 112, 047601 (2014).
  47. F. Dolde, H. Fedder, M. W. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, Electric-field sensing using single diamond spins, Nat. Phys. 7, 459 (2011).
  48. J. H. Shim, S.-J. Lee, S. Ghimire, J. I. Hwang, K.-G. Lee, K. Kim, M. J. Turner, C. A. Hart, R. L. Walsworth, and S. Oh, Multiplexed sensing of magnetic field and temperature in real time using a nitrogen-vacancy ensemble in diamond, Phys. Rev. Appl. 17, 014009 (2022).
  49. A. Dréau, M. Lesik, L. Rondin, P. Spinicelli, O. Arcizet, J.-F. Roch, and V. Jacques, Avoiding power broadening in optically detected magnetic resonance of single NV defects for enhanced dc magnetic field sensitivity, Phys. Rev. B 84, 195204 (2011).
  50. M. L. Citron, H. R. Gray, C. W. Gabel, and C. R. Stroud, Experimental study of power broadening in a two-level atom, Phys. Rev. A 16, 1507 (1977).
  51. N. Vitanov, B. Shore, L. Yatsenko, K. Böhmer, T. Halfmann, T. Rickes, and K. Bergmann, Power broadening revisited: Theory and experiment, Opt. Commun. 199, 117 (2001).
  52. Z. Shippony and W. Read, A highly accurate Voigt function algorithm, J. Quant. Spectrosc. Radiat. Transfer 50, 635 (1993).
  53. Z. Shippony and W. Read, A correction to a highly accurate Voigt function algorithm, J. Quant. Spectrosc. Radiat. Transfer 78, 255 (2003).
  54. J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31, 1497 (1994).
  55. The SciPy Community, Scipy Faddeeva function (2025), https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.wofz.html.
  56. S. G. Johnson, The Faddeeva package (2025), http://ab-initio.mit.edu/faddeeva/.
  57. S. G. Johnson, Faddeeva package: Complex error functions (2025), https://de.mathworks.com/matlabcentral/fileexchange/38787-faddeeva-package-complex-error-functions.
  58. J. Olivero and R. Longbothum, Empirical fits to the Voigt line width: A brief review, J. Quant. Spectrosc. Radiat. Transfer 17, 233 (1977).
  59. E. Whiting, An empirical approximation to the Voigt profile, J. Quant. Spectrosc. Radiat. Transfer 8, 1379 (1968).
  60. J. F. Kielkopf, New approximation to the Voigt function with applications to spectral-line profile analysis, J. Opt. Soc. Am. 63, 987 (1973).
  61. D. Lönard, I. C. Barbosa, S. Johansson, J. Gutsche, and A. Widera, Data for: “Limits of absolute vector magnetometry with NV centers in diamond”, Zenodo (2025), https://doi.org/10.5281/zenodo.15631435.
  62. D. Lönard, I. C. Barbosa, S. Johansson, A. Erlenbach, J. Gutsche, and A. Widera, PyPI nvtools (2025), https://pypi.org/project/nvtools/.
  63. D. Lönard, I. C. Barbosa, S. Johansson, A. Erlenbach, J. Gutsche, and A. Widera, Microscope experiment control application (2025), https://gitlab.rlp.net/agwidera/public/waveguide-control-python.
  64. S. Johansson, D. Lönard, I. Cardoso Barbosa, J. Gutsche, J. Witzenrath, and A. Widera, Miniaturized magnetic-field sensor based on nitrogen-vacancy centers, Phys. Rev. Appl. 24, 054078 (2025).
  65. NumPy Developers, Numpy.Linalg.Eigvalsh (2025), https://numpy.org/doc/stable/reference/generated/numpy.linalg.eigvalsh.html.

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Phys. Rev. Research 8, 013054– Published 20 January, 2026
  • Received 26 June 2025
  • Accepted 5 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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