Fourier magnetic imaging with nanoscale resolution and compressed sensing speed-up using electronic spins in diamond

Journal name:
Nature Nanotechnology
Year published:
Published online

Optically detected magnetic resonance using nitrogen–vacancy (NV) colour centres in diamond is a leading modality for nanoscale magnetic field imaging1, 2, 3, as it provides single electron spin sensitivity4, three-dimensional resolution better than 1 nm (ref. 5) and applicability to a wide range of physical6, 7, 8, 9, 10, 11, 12, 13 and biological14, 15 samples under ambient conditions. To date, however, NV-diamond magnetic imaging has been performed using ‘real-space’ techniques, which are either limited by optical diffraction to ∼250 nm resolution16 or require slow, point-by-point scanning for nanoscale resolution, for example, using an atomic force microscope17, magnetic tip5, or super-resolution optical imaging18, 19. Here, we introduce an alternative technique of Fourier magnetic imaging using NV-diamond. In analogy with conventional magnetic resonance imaging (MRI), we employ pulsed magnetic field gradients to phase-encode spatial information on NV electronic spins in wavenumber or ‘k-space’20 followed by a fast Fourier transform to yield real-space images with nanoscale resolution, wide field of view and compressed sensing speed-up.

At a glance


  1. Fourier magnetic imaging experiment.
    Figure 1: Fourier magnetic imaging experiment.

    a, Schematic of the Fourier magnetic imaging microscope. NV-centre magnetic sensors are located near the surface of a diamond chip (for example, as represented by spheres with arrows). NV spin states are initialized and read out with a green (532 nm) laser and coherently manipulated with resonant pulses using a microwave loop. Controlled magnetic field gradients for NV spin phase encoding are generated by sending currents through pairs of gold wires (gradient microcoils) separated by 100 μm and connected in an anti-Helmholtz configuration. An external-field wire is used to create a non-uniform d.c. or a.c. magnetic field for demonstrations of nanoscale Fourier magnetic imaging. The NV quantization axis, represented by ζ, is offset from the surface normal (z-axis) of the [100]-cut diamond sample and aligned with a static, uniform magnetic field of ∼30 G created by a permanent magnet (not shown). b, Top-view schematic of the Fourier magnetic imaging microscope, as well as a simulation (using COMSOL Multiphysics) of the magnetic field gradient √ when a current of 1 A is sent through both microcoil pairs, with current directions indicated by white arrows. Scale bar, 20 μm. c, Energy-level diagram of the NV centre (see Methods for details). d, Fourier magnetic imaging experimental sequence. Spins are polarized into the |0〉 state with a green laser pulse. A microwave pulse sequence with 2n π pulses dynamically decouples NV centres from magnetic field noise from the environment. A pulsed magnetic field gradient of alternating direction is applied during each precession interval. Spins at different locations accumulate phase at different rates. A final π/2 pulse projects the spins into the |0〉–|1〉 manifold, and state populations are read out optically via spin-dependent fluorescence. An a.c. magnetic field Bext produced by current in the external-field wire can be sensed using the procedure described in the main text.

  2. Fourier imaging of NV centres with nanoscale resolution.
    Figure 2: Fourier imaging of NV centres with nanoscale resolution.

    a, One-dimensional k-space image of a single NV centre in diamond (sample A) using a four-pulse CPMG sequence (n = 2). As the gradient strength dBζ/dx is incrementally stepped by varying the current through the microcoil, the NV fluorescence shows sinusoidal oscillations. Here, we show NV fluorescence normalized to a reference measurement of |0〉 state fluorescence and with a constant background level subtracted. b, One-dimensional real-space image data (black) obtained from the absolute value of the Fourier transform of the k-space data. For comparison, the diffraction-limited, real-space point spread function of the microscope is shown (pink shaded area, full-width at half-maximum of 300 nm). c, Scanning electron micrograph of a 400-nm-diameter NV-containing diamond nanopillar fabricated on sample B. d, Scanning confocal fluorescence image of the same nanopillar (full-width at half-maxima of corresponding x and y profiles of ∼400 nm). e, Two-dimensional k-space image of two proximal NV centres inside this same nanopillar using a Hahn-echo sequence (n = 1/2). f, Bottom: Fourier-transformed, two-dimensional real-space image (absolute value) with a threshold set at 5σ above the noise level, where σ is the standard deviation of observed optical noise. Top: cross-section of the two peaks along the direction indicated by the white dashed line in the two-dimensional image. Two NV centres separated by 121(9) nm are clearly resolved. Scale bars: 200 nm (c,d,f) and 0.0066 nm−1 (e).

  3. Fourier magnetic gradient sensing below the optical diffraction limit.
    Figure 3: Fourier magnetic gradient sensing below the optical diffraction limit.

    One-dimensional k-space magnetic images for the two NV centres shown in Fig. 2f are acquired by incrementally stepping the dBζ/dx gradient strength for fixed values of a.c. current sent through the external-field wire. For each current value, the corresponding k-space image is Fourier-transformed and thresholded at 5σ to obtain a one-dimensional real-space image. a, Absolute value of the real-space image shows peaks corresponding to the two NV centres separated by 121(9) nm. The vertical axis is real-space position along the x direction and the horizontal axis corresponds to different ranges of a.c. currents at a frequency of 50 kHz, increasing from left to right: 0–0.51, 2.56–3.08 and 5.13–5.64 mA. b, Cosine of the argument of the real-space images of the two NV centres shown in a, for the same ranges of a.c. currents. c, Overlay of measured values for cos θ (symbols) and corresponding fits (solids curves) as a function of a.c. current amplitude for the two NV centres, where θ = γBextτ. The observed differential phase shift between the data for the two NV centres shows that these spatially separated NV centres measure a magnetic field difference ΔBext arising from a gradient in the external a.c. magnetic field magnitude Bext.

  4. Fourier magnetic imaging with wide FOV and nanometre-scale resolution.
    Figure 4: Fourier magnetic imaging with wide FOV and nanometre-scale resolution.

    The a.c. magnetic field produced by passing a 50 kHz, 5.13 mA electric current through the external-field wire (indicated by a thick yellow line at the top-left corner) is imaged using a hybrid real + k-space technique over a wide FOV spanning 167 diamond nanopillars. Imaged magnetic field amplitudes are indicated with a colour scale, with numerical values (and associated uncertainties) given for some example nanopillars. A low-resolution real-space magnetic image is acquired over the full FOV by scanning the microscope across all nanopillars (see Methods). The spatial resolution is limited by optical diffraction and NV centres within individual nanopillars are not resolved. Fourier (k-space) magnetic imaging is then performed on individual nanopillars to determine NV centre positions and local a.c. magnetic field amplitudes with ∼30 nm resolution (right boxes of inset panels). To check for consistency, the measured long-range magnetic field gradient provided by the low-resolution real-space image, together with the NV positions determined via Fourier imaging, are used to estimate the variation in a.c. magnetic field amplitude within each nanopillar (left boxes of inset panels). Good agreement is found between the measured and estimated values for the magnetic field difference between NV centres within each nanopillar (Supplementary Section IV). Scale bars: 2 µm (main figure) and 100 nm (insets).

  5. Compressed sensing speed-up of NV Fourier magnetic imaging.
    Figure 5: Compressed sensing speed-up of NV Fourier magnetic imaging.

    a, For two NV centres in sample B: fully sampled one-dimensional k-space signal with N = 2,048 data points (blue line) and randomly under-sampled k-space signal with M = 128 data points (red symbols). The under-sampling (speed-up) factor is N/M = 16. b, Blue trace: absolute value of the Fourier transform of fully sampled k-space data, indicating a real-space NV separation of 116 nm along the x axis. Red traces (offset for clarity): real-space signals reconstructed from under-sampled k-space data sets via compressed sensing techniques, in good agreement with fully sampled k-space data for M ≥ 128 (see text for details). c, Inverse Fourier transform of data reconstructed via compressed sensing for M = 128 with (black trace) and without (red trace) an a.c. current (50 kHz, 10.26 mA) sent through an external-field wire. The observed phase shift between the data sets provides a measure of the magnetic field difference between the positions of the two NV centres, in good agreement with the results from fully sampled k-space data, thereby showing that compressed sensing reconstruction retains reliable information about imaged magnetic fields.


  1. Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale resolution. Nature Phys. 4, 810816 (2008).
  2. Maze, J. R. et al. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature 455, 644647 (2008).
  3. Balasubramanian, G. et al. Nanoscale imaging magnetometry with diamond spins under ambient conditions. Nature 455, 648651 (2008).
  4. Grinolds, M. S. et al. Nanoscale magnetic imaging of a single electron spin under ambient conditions. Nature Phys. 9, 215219 (2013).
  5. Grinolds, M. S. et al. Sub-nanometer resolution in three-dimensional magnetic resonance imaging of individual dark spins. Nature Nanotech. 9, 279284 (2014).
  6. Rondin, L. et al. Stray-field imaging of magnetic vortices with a single diamond spin. Nature Commun. 4, 2279 (2013).
  7. Mamin, H. J. et al. Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor. Science 339, 557560 (2013).
  8. Staudacher, T. et al. Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume. Science 339, 561563 (2013).
  9. Sushkov, A. O. et al. Magnetic resonance detection of individual proton spins using quantum reporters. Phys. Rev. Lett. 113, 197601 (2014).
  10. Fu, R. R. et al. Solar nebula magnetic fields recorded in the Semarkona meteorite. Science 346, 10891092 (2014).
  11. Sushkov, A. O. et al. All-optical sensing of a single-molecule electron spin. Nano Lett. 14, 64436448 (2014).
  12. Luan, L. et al. Decoherence imaging of spin ensembles using a scanning single-electron spin in diamond. Sci. Rep. 5, 8119 (2015).
  13. van der Sar, T. et al. Nanometre-scale probing of spin waves using single-electron spins. Nature Commun. 6, 7886 (2015).
  14. Le Sage, D. et al. Optical magnetic imaging of living cells. Nature 496, 486489 (2013).
  15. Rahn-Lee, L. et al. A genetic strategy for probing the functional diversity of magnetosome formation. PLoS Genet. 11, e1004811 (2015).
  16. Pham, L. M. et al. Magnetic field imaging with nitrogen-vacancy ensembles. New J. Phys. 13, 045021 (2011).
  17. Maletinsky, P. et al. A robust scanning diamond sensor for nanoscale imaging with single nitrogen-vacancy centres. Nature Nanotech. 7, 320324 (2012).
  18. Maurer, P. C. et al. Far-field optical imaging and manipulation of individual spins with nanoscale resolution. Nature Phys. 6, 912918 (2010).
  19. Wildanger, D. et al. Solid immersion facilitates fluorescence microscopy with nanometer resolution and sub-ångström emitter localization. Adv. Mater. 24, OP309OP313 (2012).
  20. Sodickson, A. & Cory, D. G. A generalized k-space formalism for treating the spatial aspects of a variety of NMR experiments. Prog. Nucl. Magn. Reson. Spectrosc. 33, 77108 (1998).
  21. Ernst, R. R. & Anderson, W. A. Application of Fourier transform spectroscopy to magnetic resonance. Rev. Sci. Instrum. 37, 93102 (1966).
  22. Nichol, J. M. et al. Nanoscale Fourier-transform magnetic resonance imaging. Phys. Rev. X 3, 031016 (2013).
  23. Candès, E., Romberg, J. & Tao, T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489509 (2006).
  24. Lustig, M., Donoho, D. L., Santos, J. M. & Pauly, J. M. Compressed sensing MRI. Signal Process. Mag. IEEE 25, 7282 (2008).
  25. De Lange, G., Wang, Z. H., Ristè, D., Dobrovitski, V. V. & Hanson, R. Universal dynamical decoupling of a single solid-state spin from a spin bath. Science 330, 6063 (2010).
  26. Pham, L. M. et al. Enhanced solid-state multispin metrology using dynamical decoupling. Phys. Rev. B 86, 045214 (2012).
  27. Glenn, D. R. et al. Single-cell magnetic imaging using a quantum diamond microscope. Nature Methods 12, 736738 (2015).
  28. DeVience, S. J. et al. Nanoscale NMR spectroscopy and imaging of multiple nuclear species. Nature Nanotech. 10, 129134 (2015).
  29. Pham, L. M. et al. Enhanced metrology using preferential orientation of nitrogen-vacancy centers in diamond. Phys. Rev. B 86, 121202(R) (2012).
  30. Le Sage, D. et al. Efficient photon detection from color centers in a diamond optical waveguide. Phys. Rev. B 85, 121202(R) (2012).
  31. Bar-Gill, N. et al. Suppression of spin-bath dynamics for improved coherence of multi-spin-qubit systems. Nature Commun. 3, 858 (2012).
  32. Bar-Gill, N., Pham, L. M., Jarmola, A., Budker, D. & Walsworth, R. L. Solid-state electronic spin coherence time approaching one second. Nature Commun. 4, 1743 (2013).
  33. Dolde, F. et al. Electric-field sensing using single diamond spins. Nature Phys. 7, 459463 (2011).
  34. Kucsko, G. et al. Nanometre-scale thermometry in a living cell. Nature 500, 5458 (2013).
  35. Koehl, W. F., Buckley, B. B., Heremans, F. J., Calusine, G. & Awschalom, D. D. Room temperature coherent control of defect spin qubits in silicon carbide. Nature 479, 8487 (2011).
  36. Stanwix, P. L. et al. Coherence of nitrogen-vacancy electronic spin ensembles in diamond. Phys. Rev. B 82, 201201(R) (2010).

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Author information

  1. Present address: Department of Applied Physics and Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

    • N. Bar-Gill
  2. These authors contributed equally to this work

    • K. Arai,
    • C. Belthangady &
    • H. Zhang


  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • K. Arai
  2. Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA

    • C. Belthangady,
    • H. Zhang,
    • N. Bar-Gill &
    • R. L. Walsworth
  3. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • C. Belthangady,
    • H. Zhang,
    • N. Bar-Gill,
    • A. Yacoby &
    • R. L. Walsworth
  4. Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA

    • S. J. DeVience
  5. Nuclear Science and Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • P. Cappellaro
  6. Center for Brain Science, Harvard University, Cambridge, Massachusetts 02138, USA

    • R. L. Walsworth


K.A., C.B. and H.Z. contributed equally to this work. R.L.W. conceived the idea of NV Fourier magnetic imaging and supervised the project. K.A., C.B. and H.Z. developed measurement protocols, hardware and software for NV Fourier magnetic imaging, performed the measurements and analysed the data. C.B. and N.B.-G. developed the NV-diamond confocal microscope used in the study. N.B.-G. also aided the development of data acquisition software. S.J.D. and A.Y. advised on Fourier imaging techniques and applications. P.C. advised on compressed sensing techniques and applications. All authors discussed the results and participated in writing the manuscript.

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