Atomic-scale imaging of a 27-nuclear-spin cluster using a single-spin quantum sensor

TL;DR

This paper demonstrates atomic-scale imaging of a 27-nuclear-spin cluster using a single NV centre sensor, achieving high spectral resolution and sub-angstrom spatial reconstruction, advancing magnetic imaging of complex molecules.

Contribution

It introduces a multidimensional spectroscopy method for high-resolution imaging of large spin clusters with sub-angstrom precision.

Findings

Successfully imaged a 27-nuclear-spin cluster.

Achieved spectral resolution below 80 mHz.

Reconstructed the 3D structure with 0.1 Å accuracy.

Abstract

Nuclear magnetic resonance (NMR) is a powerful method for determining the structure of molecules and proteins. While conventional NMR requires averaging over large ensembles, recent progress with single-spin quantum sensors has created the prospect of magnetic imaging of individual molecules. As an initial step towards this goal, isolated nuclear spins and spin pairs have been mapped. However, large clusters of interacting spins - such as found in molecules - result in highly complex spectra. Imaging these complex systems is an outstanding challenge due to the required high spectral resolution and efficient spatial reconstruction with sub-angstrom precision. Here we develop such atomic-scale imaging using a single nitrogen-vacancy (NV) centre as a quantum sensor, and demonstrate it on a model system of $27$ coupled $^{13}$C nuclear spins in a diamond. We present a new multidimensional spectroscopy method that isolates individual nuclear-nuclear spin interactions with high spectral resolution ($< 80\,$mHz) and high accuracy ($2$ mHz). We show that these interactions encode the composition and inter-connectivity of the cluster, and develop methods to extract the 3D structure of the cluster with sub-angstrom resolution. Our results demonstrate a key capability towards magnetic imaging of individual molecules and other complex spin systems.

Equations

$$\begin{array}{rl}{f}_{\text{DE}}& (m_s=\pm 1)\approx \frac{1}{4\pi }|C+\\ & +\Delta {\lambda }_1(m_s)+\Delta {\lambda }_2(m_s)+\Delta {\lambda }_3(m_s)|,\end{array}$$

$$\Delta {\lambda }_2^{(0)}=\frac{(A_{zx}^{(1)}+A_{zx}^{(2)})C_{zx}+(A_{zy}^{(1)}+A_{zy}^{(2)})C_{zy}}{{\gamma }_c{B}_z},$$

$$C_{ij}=\frac{{\alpha }_{ij}}{\Delta r_{ij}^3}\left(\frac{3(z_j-z_i{)}^2}{\Delta r_{ij}^2}-1\right),$$

Full text

Atomic-scale imaging of a 27-nuclear-spin cluster

using a single-spin quantum sensor

M. H. Abobeih1,2

  

J. Randall1,2

  

C. E. Bradley1,2

  

H. P. Bartling1,2

  

M. A. Bakker1,2

  

M. J. Degen1,2

  

M. Markham3

  

D. J. Twitchen3

  

T. H. Taminiau1,2

T.H.Taminiau@TUDelft.nl

1QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands

2Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands

3Element Six, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire, OX11 0QR, United Kingdom

Nuclear magnetic resonance (NMR) is a powerful method for determining the structure of molecules and proteins rule2006fundamentals . While conventional NMR requires averaging over large ensembles, recent progress with single-spin quantum sensors mamin2013nanoscale ; staudacher2013nuclear ; shi2015single ; Lovchinsky_Science2016 ; Aslam_Science2017 ; glenn2018high ; smits2019two ; lovchinsky2017magnetic has created the prospect of magnetic imaging of individual molecules ajoy2015atomic ; kost2015resolving ; perunicic2016quantum ; wang2016positioning . As an initial step towards this goal, isolated nuclear spins and spin pairs have been mapped sushkov2014magnetic ; Muller_NatComm2014 ; Shi_NatPhys2014 ; Zopes_NatComm2018 ; Zopes_PRL2018 ; sasaki_prb2018 ; Abobeih_NatComm2018 ; Yang_arXiv2019 . However, large clusters of interacting spins — such as found in molecules — result in highly complex spectra. Imaging these complex systems is an outstanding challenge due to the required high spectral resolution and efficient spatial reconstruction with sub-angstrom precision. Here we develop such atomic-scale imaging using a single nitrogen-vacancy (NV) centre as a quantum sensor, and demonstrate it on a model system of $27$ coupled 13C nuclear spins in a diamond. We present a new multidimensional spectroscopy method that isolates individual nuclear-nuclear spin interactions with high spectral resolution ($<80\,$mHz) and high accuracy ($2$ mHz). We show that these interactions encode the composition and inter-connectivity of the cluster, and develop methods to extract the 3D structure of the cluster with sub-angstrom resolution. Our results demonstrate a key capability towards magnetic imaging of individual molecules and other complex spin systems ajoy2015atomic ; kost2015resolving ; perunicic2016quantum ; wang2016positioning ; lovchinsky2017magnetic .

The nitrogen-vacancy (NV) centre in diamond has emerged as a powerful quantum sensor mamin2013nanoscale ; staudacher2013nuclear ; Lovchinsky_Science2016 ; Aslam_Science2017 ; glenn2018high ; lovchinsky2017magnetic ; shi2015single ; smits2019two ; ajoy2015atomic ; kost2015resolving ; perunicic2016quantum ; wang2016positioning ; rosenfeld2018sensing ; knowles2016demonstration . The NV electron spin provides long coherence times Lovchinsky_Science2016 ; Aslam_Science2017 ; Abobeih_NatComm2018 and high-contrast optical readout Lovchinsky_Science2016 ; Cramer_NatureComm2016 ; pfender2019high , enabling high sensitivity over a large range of temperatures Lovchinsky_Science2016 ; Aslam_Science2017 ; Abobeih_NatComm2018 ; pfender2019high ; Cujia_arXiv2018 . Pioneering experiments with near-surface NV centers have demonstrated spectroscopy of small ensembles of nuclear spins in nano-scale volumes mamin2013nanoscale ; staudacher2013nuclear ; smits2019two ; Lovchinsky_Science2016 ; Aslam_Science2017 ; glenn2018high , and electron-spin labelled proteins shi2015single . Furthermore, single nuclear spin sensitivity has been demonstrated and isolated individual nuclear spins and spin pairs have been mapped Muller_NatComm2014 ; sushkov2014magnetic ; Zopes_NatComm2018 ; Zopes_PRL2018 ; sasaki_prb2018 ; Shi_NatPhys2014 ; Abobeih_NatComm2018 ; Yang_arXiv2019 . Together, these results have established the NV center as a promising platform for magnetic imaging of complex spin systems and single molecules ajoy2015atomic ; kost2015resolving ; perunicic2016quantum ; wang2016positioning .

In this work, we realise a key ability towards that goal: the 3D imaging of large nuclear-spin structures with atomic resolution. The main idea of our method is to obtain structural information by accessing the couplings between individual nuclear spins. The key open challenges are: (1) to realize high spectral resolution so that small couplings can be accessed, (2) to isolate such couplings from complex spectra, and (3) to transform the revealed connectivity into the 3D spatial structure with sub-angstrom precision.

The basic elements of our experiment are illustrated in Fig. 1a. We consider a cluster of 13C nuclear spins in the vicinity of a single NV centre in diamond at $4$ K. This cluster provides a model system for the magnetic imaging of single molecules and spin structures external to the diamond. Each 13C spin precesses at a shifted frequency due to the hyperfine interaction with the electron spin, resembling a chemical shift in traditional NMR Slichter_1996 ; rule2006fundamentals . These shifts enable different nuclear spins in the cluster to be distinguished.

We use the NV electron spin as a sensor to probe the nuclear-nuclear interactions (Fig. 1b). Inspired by NMR spectroscopy Slichter_1996 ; rule2006fundamentals , we develop sequences that employ spin-echo double-resonance (SEDOR) techniques to isolate and measure individual couplings with high spectral resolution. First, we polarise a nuclear “probe” spin (frequency $RF1$) using newly developed quantum sensing sequences that can detect spins in any direction from the NV, enabling access to a large number of spins (see methods) Bradley_arXiv2019 . Second, we let this probe spin evolve for a time $t$ and apply $N$ echo pulses that decouple it from the other spins and environmental noise. Simultaneously, pulses on a “target” spin in the cluster (frequency $RF2$) re-couple it to the probe spin, selecting the interaction between these two spins. Finally, a second sensing sequence detects the resulting polarisation of the probe spin through a high-contrast readout of the electron spin (see methods) Cramer_NatureComm2016 , which enables fast data collection. This double-resonance sequence provides a high spectral resolution through a long nuclear phase accumulation time. Importantly, the resolution is not limited by the relatively short coherence time of the electron spin sensor (see methods) Cramer_NatureComm2016 ; laraoui2013high .

It is instructive to first consider the case without echo pulses ($N=0$), for which all couplings act simultaneously. This results in complex spectra that indicate many nuclear-nuclear spin interactions and/or simultaneous signals from multiple spins (Fig. 1c). The underlying structure of individual spins and couplings is obscured by the many frequencies ($2^{j}$ for coupling to $j$ spins) and the low spectral resolution of $\sim\,$30 Hz FWHM, set by the nuclear spin linewidth.

The echo sequence enables couplings between specific spins to be isolated and measured with high resolution. We first scan the target frequency $RF2$ for a fixed probe frequency $RF1$ (Fig. 1d). This reveals the spectral positions of nuclear spins coupled to the probe spin. We then sweep the evolution time $t$ and Fourier transform the signal to quantify the coupling strengths (Fig. 1e). For a single echo pulse ($N=1$), the nuclear spin coherence time is $0.56(2)\,$s, yielding a spectral resolution of $1.807(7)\,$Hz and a centre frequency accuracy of $10$ mHz. The spectral resolution can be further enhanced by applying more echo pulses. For $N=256$, a resolution of $78(1)$ mHz and an accuracy of $2$ mHz are obtained for the spin in Fig. 1f, making it possible to detect sub-Hertz interactions. The obtained resolution is an order of magnitude higher than in previous experiments on spin signals maurer2012room ; pfender2019high ; smits2019two ; Pfender_NatComm2017 ; glenn2018high ; Aslam_Science2017 .

To characterise the complete cluster, we perform 3D spectroscopy by varying the probe frequency $RF1$, the target frequency $RF2$, and the evolution time $t$. The combinations of $RF1$ and $RF2$ reveal the spectral positions of the spins in the cluster. The coupling between spins is retrieved from the Fourier transform of the time dimension $t$. This yields a 3D data set that in principle encodes the composition and connectivity of the spin cluster (Fig. 2).

To identify the spins and their couplings from the 3D spectra, we need to resolve ambiguities due to overlapping signals from multiple spins at near-identical frequencies. The first challenge is to retrieve individual couplings when multiple couplings are probed at the same time. Figure 3a shows an example where the couplings between one probe spin and three target spins are measured simultaneously. Whilst the spectrum becomes more complex, the high spectral resolution of our method enables retrieval of the underlying couplings. The second challenge is to determine the number of spins in the cluster and to assign the measured couplings to them. While the observation of a coupling at frequencies $RF1$ and $RF2$ is by itself not enough to assign it to particular spins, our method extracts multiple couplings that together constrain the problem. In particular, each spin couples predominantly to spins in its vicinity, so that spins can be uniquely identified from their connections to the rest of the cluster (see Fig. 3b for an example).

Transforming the 3D spectra into a spatial structure requires a precise relation between the measured couplings and the relative positions of the spins. A complication is that the presence of electronic spins can modify the nuclear couplings dutt2007quantum , causing the measured value to deviate from a basic dipole-dipole coupling. We use perturbation theory to derive a set of many-body corrections that depend on the electron-nuclear and nuclear-nuclear couplings, and the magnetic field direction (see methods). For the type of cluster considered here, the corrections could be significant. However, the signs of the leading terms depend on the electron spin state. By averaging the measured couplings for the $m_{s}=+1$ and $m_{s}=-1$ states, the deviations are strongly reduced. Together with a novel method to align the magnetic field to within $0.07$ degrees (see methods), this enables us to approximate the nuclear-nuclear couplings as dipolar.

Finally, we determine the structure of the spin cluster. Figure 4a summarises all extracted couplings. We identify $M=27$ nuclear spins and retrieve a total of $171$ pairwise couplings, out of the total of $M(M-1)/2=351$ couplings. The structure of the cluster is completely described by $3M-4=77$ spatial coordinates (see methods), so that the problem is overdetermined. However, due to the large number of parameters and local minima, a direct least-squares minimisation ajoy2015atomic is challenging. Instead, we sequentially build the structure by progressively adding spins, while keeping track of all possible structures that match the measured couplings within a certain tolerance.

We use two different methods. The first method constrains the spin coordinates to the diamond lattice. The second method discretises space in a general cubic lattice, with voxel spacing down to $5\times 10^{-3}\,$nm ($\sim 1/70$th of the lattice constant, see methods). While this second method is more computationally intensive, it uses minimum a priori knowledge and can be applied on arbitrary spin systems. We run these analyses in parallel with the measurements, so that sets of the most promising spin assignments and structures are regularly created. These yield predictions for which unmeasured couplings (combinations of $RF1$ and $RF2$) are required to decide between different assignments and structures, which we use to guide the experiments and reduce the total measurement time (see methods).

Figure 4b shows the structure obtained for the $27$ spins using the diamond-lattice. The blue connections show the strongest couplings ($>3$ Hz) and visualise the inter-connectivity of the cluster. The cubic-lattice method yields a nearly identical structure (see methods); the average distance between the spin positions for the two solutions is $0.58\,$Å, a fraction of the bond length of $\sim 1.54\,$Å. As a final step, we use these structures as inputs for least-squares minimisation, where the $x,y,z$ coordinates are allowed to relax to any value. The solution obtained lies close to the initial guess with an average distance of $0.46$ Å. The uncertainties for the spatial coordinates ($\delta x$, $\delta y$, $\delta z$) are below a diamond bond length for all 27 spins (Fig. 4c,d), indicating atomic-scale imaging of the complete $27$-spin cluster.

Additionally, we determine the position of the NV sensor relative to the cluster. Although not required to reconstruct the cluster, this provides a control experiment. We measure the coupling of the 14N nuclear spin to $12$ of the 13C spins (Supplementary Fig. 9). This unambiguously determines the location of both the 14N atom and the vacancy (fit uncertainties $<0.3\,$Å). We can now compare the electron-13C hyperfine couplings to previous density functional theory (DFT) calculations for $5$ of our spins Nizovtsev_2018 . All $5$ couplings agree with the DFT calculations (Supplementary Fig. 9), providing an independent corroboration of the extracted structure, as well as a direct test of the DFT calculations. Looking beyond quantum sensing, this precise microscopic characterisation of the NV environment provides new opportunities for improved control of quantum bits for quantum information Bradley_arXiv2019 ; Abobeih_NatComm2018 ; Cramer_NatureComm2016 ; maurer2012room ; dutt2007quantum , and for investigating many-body physics in coupled spin systems.

In conclusion, we have developed and demonstrated 3D atomic-scale imaging of large clusters of nuclear spins using a single-spin quantum sensor. Our approach is compatible with room temperature operation maurer2012room ; Pfender_NatComm2017 ; Cujia_arXiv2018 ; pfender2019high and can be extended to larger structures, as the number of required measurements scales linearly with the number of spins. Future improvements in the data acquisition and the computation of 3D structures can further reduce time requirements. In particular, recent methods to polarise and measure nuclear spins are expected to improve sensitivity Cujia_arXiv2018 ; pfender2019high , especially for samples with weak couplings to the NV sensor. Optimised sampling of the measurements Yang_arXiv2019 and adaptive algorithms based on a real-time structure analysis can further reduce the total number of required measurements. Therefore, when combined with recent progress in nanoscale NMR with near-surface NV centres mamin2013nanoscale ; staudacher2013nuclear ; Lovchinsky_Science2016 ; Aslam_Science2017 ; glenn2018high ; shi2015single ; smits2019two , our results provide a path towards the magnetic imaging of individual molecules and complex spin structures external to diamond ajoy2015atomic ; kost2015resolving ; perunicic2016quantum ; wang2016positioning .

METHODS

Sample and NV centre sensor. We use a naturally occurring NV centre in a homoepitaxially chemical-vapor-deposition (CVD) grown diamond with a $1.1\%$ natural abundance of 13C and a $\langle 111\rangle$ crystal orientation (Element Six). The NV is placed in a solid-immersion lens to enhance photon collection efficiency Robledo_Nature2011 . The NV centre has been selected for the absence of 13C spins with hyperfine couplings $>$ 500 kHz. The NV electron spin coherence times are $T_{2}^{*}=4.9(2)\,\mu$s and $T_{2}=1.182(5)\,$ms.

The NV sensor spin is used to create and detect polarisation (Fig. 1b). The two key requirements for the sensor spin are (1) a high-contrast readout to keep measurement times manageable, and (2) that it does not limit the spectral resolution by disturbing the phase evolution of the nuclear spins through relaxation maurer2012room ; Pfender_NatComm2017 . We work at 4 Kelvin, so that the electron relaxation is negligible ($T_{1}=3.6(3)\times 10^{3}\,$s Abobeih_NatComm2018 ), and use high-fidelity readout through resonant optical excitation (average $F=94.5\%$) Robledo_Nature2011 . Note that recent experiments have satisfied both these requirements at room temperature Lovchinsky_Science2016 ; maurer2012room ; Pfender_NatComm2017 ; pfender2019high ; Cujia_arXiv2018 , so that the spectroscopy and imaging methods developed here can be applied at ambient conditions.

Magnetic field alignment. A magnetic field of $\sim 403\,$G is applied using a permanent magnet. We align the magnetic field along the NV axis to avoid electron-mediated shifts that cause the measured couplings to deviate from nuclear-nuclear dipolar coupling (see Supplementary Information section III). We use a “thermal” echo sequence — previously introduced to measure temperature Toyli8417 — to decouple the electron spin from magnetic noise along the NV axis, while retaining the sensitivity to the magnetic field in the $x,y$ directions (see Supplementary Fig. 4). This extends the sensing time from $T_{2}^{*}\approx 5\,\mu$s to $T_{2}\approx 1$ ms, resulting in an uncertainty in the alignment of $0.07$ degrees.

Quantum sensing sequences. We employ two different sensing sequences. Sequence A consists of dynamical decoupling sequences of $N$ equally spaced $\pi$-pulses on the electron spin of the form ($\tau-\pi-\tau)^{N}$ Taminiau_PRL2012 ; Kolkowitz_PRL2012 ; Zhao_NatureNano2012 . This sequence is only sensitive to nuclear spins with a significant electron-nuclear hyperfine component perpendicular to the applied magnetic field Taminiau_PRL2012 . The inter-pulse spacing 2$\tau$ determines the spin frequency that is being probed.

Sequence B is a newly developed method, described in more detail in Bradley et al. Bradley_arXiv2019 , that interleaves the dynamical decoupling sequence with RF pulses. This method enables the detection of spins with a weak or negligible perpendicular hyperfine component Bradley_arXiv2019 ; Pfender_NatComm2017 , so that spins in any direction from the NV can be detected. In this work, this enables us to access a greater number of spins in the cluster. For this sequence, the frequency of the RF pulse sets the targeted spin frequency, while $\tau$ can be freely chosen Bradley_arXiv2019 .

Electron-nuclear spectroscopy. As a starting point, we use the electron spin as a sensor to roughly characterise some of the nuclear spins in the cluster. We perform spectroscopy by sweeping the interpulse delay $\tau$ in sequence A (see for example Abobeih et al. Abobeih_NatComm2018 ) and the RF frequency for sequence B Bradley_arXiv2019 . This identifies the frequency range at which spins are present in the cluster and provides the parameters to polarise and detect several spins Cramer_NatureComm2016 .

Nuclear-nuclear double-resonance spectroscopy. The sequence for the double resonance experiments is given in Fig. 1b. To polarise and detect the probe spin, we either use sequence A (without the RF1 pulses in the dashed box) or sequence B (with the RF1 pulses), depending if the perpendicular hyperfine coupling to the electron spin is significant or not. For sequence A, we set the interpulse delay as $\tau=(2k-1)\pi/(\omega_{0}+RF1)$, with $k$ an integer and $\omega_{0}$ the 13C Larmor frequency for the electron $m_{s}=0$ state, and calibrate the number of pulses $N$ to maximise the signal Taminiau_PRL2012 . For sequence B we calibrate the RF power to maximise the signal.

We create nuclear polarisation by projective measurements Cramer_NatureComm2016 . First the electron is prepared in a superposition state through resonant excitation Robledo_Nature2011 and a $\pi/2$ pulse. Second, the sensing sequence correlates the phase of the electron with the nuclear spin state. Finally, the electron is read out so that the nuclear spin is projected into a polarised state Cramer_NatureComm2016 . To enhance the signal-to-noise ratio and to ensure that the electron measurement does not disturb the nuclear spin evolution, we only perform the double resonance sequence if a photon was detected during the electron readout Cramer_NatureComm2016 . The resulting signal contrast for different spins varies from $20\%$ to $96\%$.

For the double resonance sequence, the phases of the RF1 echo pulses are calibrated so that the phase difference with respect to the polarisation axis is [math] or $\pi/2$. For the target spins, the phase of the RF2 pulse does not affect the signal and is arbitrarily set.

To mitigate pulse errors we alternate the phases of the pulses following the XY8 scheme XY8 , both for the electron and nuclear spins. For the electron spin, we use Hermite pulse envelopes warren1984effects with Rabi frequency $\sim 14\,$MHz to obtain effective microwave pulses without initialisation of the intrinsic 14N nuclear spin. The nuclear-spin Rabi frequencies are in the range $0.3-0.7\,$kHz.

Data analysis. We extract the spin-spin couplings $f$ and their uncertainties from fitting the time-domain double resonance signals (e.g. Fig. 1e-f, top) to $S=a+A\cdot e^{-(t/T_{2})^{2}}\cos{(2\pi{f}t+\phi)}$, where $T_{2}$ is the coherence time (also a fit parameter). The PSD is obtained from a Fourier transform of the time domain signal with zero filling rule2006fundamentals and the D.C. component filtered out (e.g. Fig. 1e-f, bottom). The spectral resolution (FWHM) is obtained from a Gaussian fit of the PSD, and its uncertainty is obtained from the fit of the time domain signal. Alternatively we can define the spectral resolution (FWHM) directly from the time domain signal as $\frac{2\sqrt{ln2}}{\pi T_{2}}$, which yields $0.945(7)$ Hz for Fig. 1e and $50(1)$ mHz for Fig. 1f.

Electron-mediated interactions. We calculate corrections to the nuclear-nuclear couplings using perturbation theory up to second order. In contrast to previous results for strong electron-nuclear couplings dutt2007quantum ; Zhao_NatureNat2011 , here many-body interactions due to the non-secular nuclear-nuclear couplings must be taken into account. The resulting frequency in a double resonance experiment is of the form (see Supplementary Information section IV)

[TABLE]

where $C$ is the parallel ($zz$) component of the dipole-dipole interaction between the nuclear spins and $\Delta\lambda_{i}$ are correction terms due to the presence of the electron spin. See Supplementary Information for the full analysis of all terms.

The dominant correction for our parameter regime is $\Delta\lambda_{2}$, which depends on both the electron-nuclear and nuclear-nuclear interactions. We make a Taylor expansion up to first order in $A_{zz}^{(j)}/\gamma_{c}B_{z}$, where $A_{zz}^{(j)}$ is the parallel electron-nuclear hyperfine coupling for spin $j$, $\gamma_{c}$ is the nuclear gyromagnetic ratio and $B_{z}$ is the component of the magnetic field along the NV axis. This yields an expression of the form $\Delta\lambda_{2}(m_{s})\approx m_{s}\Delta\lambda_{2}^{(0)}+\Delta\lambda_{2}^{(1)}$, where the leading, zeroth-order, correction $m_{s}\Delta\lambda_{2}^{(0)}$ is given by

[TABLE]

where $A_{zx}^{(j)}$ ($C_{zx}$) and and $A_{zy}^{(j)}$ ($C_{zy}$) are the perpendicular electron-nuclear (nuclear-nuclear) coupling components. We cancel this term by averaging the double resonance frequencies measured for the $m_{s}=\pm 1$ electron spin projections.

The remaining electron-mediated corrections depend on the angles of the electron-nuclear hyperfine interactions. Because these angles are unknown, we estimate the maximum possible shift for each spin-spin interaction by maximising over all angles. For our cluster (Fig. 4), most of these maximum possible shifts are small (their average value is $\sim 0.03\,$Hz). In rare cases, the maximum possible correction runs up to $0.6\,$Hz (see Supplementary Information section IV), but as the locations of the involved spins are already precisely fixed through strong ($>20\,$Hz) interactions with several other spins, this would have a negligible effect on the obtained structure. Therefore, we can base the structural analysis on dipole-dipole interactions.

3D structure analysis. The 3D structure of the nuclear spins is obtained using the dipole-dipole coupling formula, which relates the $zz$ couplings $C_{ij}$ to the spatial $x,y,z$ coordinates of spins $i$ and $j$ as

[TABLE]

where $\Delta r_{ij}=\sqrt{(x_{j}-x_{i})^{2}+(y_{j}-y_{i})^{2}+(z_{j}-z_{i})^{2}}$, $\alpha_{ij}=\mu_{0}\gamma_{i}\gamma_{j}\hbar/4\pi$, $\mu_{0}$ is the permeability of free space, $\gamma_{i}$ is the gyromagnetic ratio of nuclear spin $i$ and $\hbar$ is the reduced Planck constant.

The goal is to minimise the sum of squares $\xi=\sum_{i<j}|\Delta f_{ij}|^{2}$, where $\Delta f_{ij}=f_{ij}-|C_{ij}|/4\pi$ are the residuals and $f_{ij}$ are the measured coupling frequencies. For $M=27$ spins, there are $3M-4=77$ free coordinates and $M(M-1)/2=351$ pairwise couplings, of which $171$ were determined in this work. $\xi$ can in principle be minimised using standard fitting methods, however tests with randomly generated spin clusters indicate that the initial guess for the coordinates should be within $\sim 0.5\,$Å in order for the fit to converge to the correct solution. For 27 spins, this corresponds to an intractable $\sim 10^{100}$ possible initial guesses. Instead we sequentially build the structure by adding spins one-by-one.

For the diamond lattice positioning method, we first use the strongest measured coupling to any spin that is already positioned to reduce the position of a new spin to a number of possible lattice coordinates. For each possible coordinate, we then check if the predicted couplings to all other spins satisfy $\Delta f_{ij}<\mathcal{T}$, where $\mathcal{T}=1.1\,$Hz is a tolerance that is chosen to ensure that all promising configurations are included while maintaining reasonable computation time. Configurations are discarded if they do not satisfy this requirement for one or more of the pairwise couplings. If more than $X_{\text{cutoff}}=5000$ possible configurations are identified, only the best $X_{\text{cutoff}}$ solutions are kept, according to their $\xi$ values.

For the cubic lattice positioning method, the same procedure is followed, with the key difference being that the lattice is adaptively generated depending on the strongest coupling to an already positioned spin in the cluster (see Supplementary Information section V). This ensures that in each case the lattice spacing is fine enough to appropriately sample the volume associated with the dipole-dipole coupling between the nuclear spins.

Data availability. The data that support the findings of this study are available from the corresponding author upon request.

Acknowledgements. We thank V.V. Dobrovitski, W. Hahn, M. Scheer and R. Zia for valuable discussions. We thank R. F. L. Vermeulen and R. N. Schouten for assistance with the RF electronics, and M. Eschen for assistance with the experimental setup. We acknowledge support from the Netherlands Organisation for Scientific Research (NWO) through a Vidi grant.

Author contributions. MHA and THT devised the experiments. MHA performed the experiments. JR developed the 3D structure analysis. MHA, JR and THT analyzed the data. MHA, JR, MJD and CEB prepared the experimental apparatus. CEB and JR developed the RF electronics. HPB and MAB performed preliminary experiments. MHA, MJD, JR, CEB and THT developed the magnetic field alignment procedure and the 14N echo spectroscopy. MM and DJT grew the diamond sample. MHA, JR and THT wrote the manuscript with input from all authors. THT supervised the project.