Abstract
Nuclear magnetic resonance imaging (MRI) at the atomic scale offers exciting prospects for determining the structure and function of individual molecules and proteins. Quantum defects in diamond have recently emerged as a promising platform towards reaching this goal, and allowed for the detection and localization of single nuclear spins under ambient conditions. Here, we present an efficient strategy for extending imaging to large nuclear spin clusters, fulfilling an important requirement towards a singlemolecule MRI technique. Our method combines the concepts of weak quantum measurements, phase encoding and simulated annealing to detect threedimensional positions from many nuclei in parallel. Detection is spatially selective, allowing us to probe nuclei at a chosen target radius while avoiding interference from stronglycoupled proximal nuclei. We demonstrate our strategy by imaging clusters containing more than 20 carbon13 nuclear spins within a radius of 2.4 nm from single, nearsurface nitrogen–vacancy centers at room temperature. The radius extrapolates to 5–6 nm for ^{1}H. Beside taking an important step in nanoscale MRI, our experiment also provides an efficient tool for the characterization of large nuclear spin registers in the context of quantum simulators and quantum network nodes.
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Introduction
Nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI) are powerful tools for molecular analysis and medical diagnostics. While conventional NMR operates on millimetersized samples containing large ensembles of molecules, much effort has been directed toward improving the resolution to the nanometer scale^{1,2} where the atomic structure could be analyzed at the level of individual molecules^{3}. Such a “singlemolecule MRI” technique would enable direct imaging of molecular structures with threedimensional resolution and elemental specificity^{1} and the monitoring of chemical reactions and binding^{4,5}. This capability could lead to many applications in biology, chemistry, and nanosciences, especially because MRI avoids important limitations of other structural techniques (like Xray diffraction or electron microscopy) such as radiation damage and the need for ensemble averaging^{6,7}.
Quantum sensors based on nitrogenvacancy (NV) centers in diamond have recently generated exciting progress in micronscale^{8,9} and nanoscale^{10,11,12} NMR spectroscopy. Early experiments have demonstrated the detection of single nuclear spins within the diamond crystal^{13,14,15} as well as of nanoscale films deposited on diamond surfaces^{10,11,12,16}. A recent refinement of protocols has led to tremendous advances in sensitivity and spectral resolution^{8,17,18}, allowing for the threedimensional localization of individual nuclear spins^{19,20,21,22}, spin pairs^{23,24,25}, and the chemical fingerprinting of molecular ensembles with high spectral resolution^{8,26}. Most recently, Abobeih et al.^{27} reported the milestone achievement of a complete mapping of a 27nuclearspin cluster at cryogenic temperatures.
To extend experiments to the imaging of single molecules, approaches are needed that are compatible with nearsurface NV centers (≲5 nm) and preferably an ambient environment. Moreover, methods are required that can efficiently detect and precisely localize a large number of distant nuclear spins in parallel. While advanced strategies have been developed to solve the latter challenge of nuclear spin detection and localization^{27,28,29,30,31,32}, many of these strategies require very long coherence times or a singleshot readout of the quantum sensor to reach adequate sensitivity and spectral resolution^{27,33}. These conditions are difficult to realize with shallow defect centers at room temperature^{34,35}.
In this work, we demonstrate a powerful method for the sensitive detection and spatial mapping of individual nuclei in large nuclear spins clusters. Our approach combines the concepts of weak quantum measurements^{36,37}, phase encoding^{21,22}, and simulated annealing^{38,39} to detect signals and extract precise threedimensional distances from many nuclei in parallel. We further show that our detection is spatially selective, allowing us to probe nuclei at a chosen target radius while avoiding interference from stronglycoupled proximal nuclei. We demonstrate our strategy by mapping the ^{13}C environment of two NV centers containing 20 and 29 nuclei, respectively. Because our experiments are performed on nearsurface spin defects and at room temperature, they are compatible with the demanding environment of prospective singlemolecule MRI investigations. Besides taking an important step in developing a singlemolecule MRI platform, our experiment also provides an efficient tool for the characterization of large qubit registers in the context of quantum simulators^{40}, quantum network nodes^{33,41,42}, and multiqubit quantum processors^{43,44}.
Results
Imaging concept
Our concept and the experimental situation is sketched in Fig. 1a. We consider a central electronic spin surrounded by a group of nuclear spins whose threedimensional locations we aim to determine. Here, both the electronic and nuclear spins are embedded in the solid matrix of a diamond crystal, but our concept is applicable to a general situation of a localized electronic spin^{45,46,47} and a nearby nuclear ensemble, including surface molecules^{10,11,48} or crystalline layers^{49}. The electronic spin plays a dual role in our arrangement^{30}: first, it acts as a local sensor for the weak magnetic fields produced by the nearby nuclei. Second, it generates a strong magnetic dipole field that we exploit for spatial imaging. In a reference frame where z is the common quantization axis (Fig. 1b), the dipole field is given by:
where a is the hyperfine vector (see Fig. 1b), r = (r, ϑ, ϕ) are the polar coordinates of the nuclear spin relative to the electron spin situated at the origin, e_{z} is a unit vector along z, m_{S} is the magnetic quantum number of the electronic spin (\({m}_{S}\in \left\{1,0,1\right\}\) for the NV center). Because we will be detecting distant ^{13}C spins, the Fermi contact interaction can be safely neglected^{20,50}. Further, μ_{0} is the vacuum permeability, ℏ the reduced Planck constant, and γ_{e} and γ_{n} are the electronic and nuclear gyromagnetic ratios, respectively. Thus, by measuring the three components of the hyperfine vector a, the distance vector r can be directly inferred (up to an inversion symmetry at the origin), revealing a spin’s threedimensional spatial location.
Parallel signal acquisition
While this scheme of threedimensional localization has been demonstrated on individual nuclear spins^{19,20,21,22}, the principal challenge lies in extending these experiments to large numbers of nuclei. We address this challenge by exploiting the principle of weak quantum measurements^{36,51}, which closely resembles the detection of a free induction decay (FID) signal in canonical Fourier NMR spectroscopy. Figure 2 introduces our experimental protocol, consisting of a polarization, excitation, and readout step. We begin by hyperpolarizing nuclear spins through a polarization transfer from the opticallyaligned electronic spin (Fig. 2a). This initial step, when applied repetitively and for a sufficiently long time, leads to a volume of nearfully polarized nuclei around the central electronic spin^{52}. We then excite all nuclei simultaneously using a broadband π/2 pulse and detect the free nuclear precession signal by sampling the transverse nuclear magnetization using weak measurements^{36}. The procedure yields an FID signal of the form:
where n is the number of nuclear spins. Further, A(β_{i}) is the probability amplitude^{17}, Γ(β_{i}) the dephasing rate, ω_{i} the precession frequency, and ϕ_{i} the initial phase of the signal belonging to the i’th nucleus. The parameter:
is a “measurement gain” parameter that is proportional to the hyperfine coupling constant a_{⊥,i} multiplied by the interaction time t_{β} of the ac detection (Fig. 2b). The parameter β_{i}, discussed in Section V, plays an important role in singlespin FID detection as it governs the balance between signal gain and quantum backaction^{36,37,53}. We sample the FID at instances t = kt_{s}, where t_{s} is the sampling time, k = 1 \(\ldots\) K, and where K is the number of points in the FID trace (Fig. 2).
Hyperfine parameters
We next show that an FID trace described by Eq. (2) contains all the information needed to reconstruct the hyperfine vectors a_{i}, and hence, the threedimensional locations r_{i} of the nuclear spins. First, the parallel components a_{∣∣,i} (see Fig. 1b) can be determined from the spectral positions of the nuclear resonances, given by the free precession frequencies:
where the approximation is for small a_{∣∣,i}, a_{⊥,i} ≪ ω_{0} (fulfilled in our experiments but not required for the analysis) and where ω_{0} = −γ_{n}B_{0}, with B_{0}∣∣e_{z} being the external magnetic bias field^{54}.
The amplitudes A(β_{i}) and decay rates Γ(β_{i}) encode information about the perpendicular components a_{⊥,i} (See Supplementary Materials accompanying this manuscript):
where p_{0,i} is the initial polarization of the i’th nuclear spin. Note that the p_{0,i} also contain any pulse errors and other imperfections of the pulse sequence, and therefore rather reflect prescaling factors and a lower bound for the nuclear polarization. The dephasing rate Γ(β_{i}) is influenced by three effects: a measurementinduced dephasing proportional to \({a}_{\perp ,i}^{2}{t}_{\beta }^{2}\) due to quantum backaction^{36}. (ii) An additional decay rate proportional to \({a}_{  ,i}^{2}{t}_{\ell }^{2}\) that is specific to the stochastic optical readout process of the NV center with effective duration t_{ℓ}^{36}. (iii) An intrinsic \({T}_{2,i}^{* }\) decay that accounts for all dephasing mechanisms not associated with the readout process, such as spinlattice effects or unresolved nuclearnuclear couplings.
Finally, the azimuth ϕ_{i} is encoded in the complex phase of the nuclear FID signal. Because we initiate the FID by applying a π/2 pulse with an external RF coil, all nuclei are rotated around a common laboratoryframe axis and start precession with the same phase. By contrast, the ac detection of the FID is phasesensitive with respect to each nucleus’ individual hyperfine field. As a consequence, the phase ϕ_{i} is equal to the spatial angle between the coil and hyperfine axes in the laboratory frame (Fig. 1b)^{21,22}. Analysis of the complex FID signal therefore directly reveals the desired azimuth ϕ_{i}.
In the following, we will make three important assumptions that are necessary for keeping the maximum likelihood fitting of spectra tractable. First, we assume that all nuclei carry approximately the same polarization, p_{0,i} ≈ p_{0}. Because we repeat the polarization transfer process for typically >10^{3} cycles (i.e., longer than the FID duration), we expect that all nuclei within the sensitive radius become close to fully polarized. The polarization level may be slightly reduced for spin pairs^{55} or due to residual spin diffusion, however, these effects are small for our dilute ^{13}C concentration. Our assumption of nearly full polarization is consistent with the observation that spectra show little change in peak intensities once the number of cycles is increased beyond ≳10^{3} ^{36}. Similar saturation behavior is suggested by ref. ^{52}. Second, we treat \({T}_{2,i}^{* }\approx {T}_{2}^{* }\) as a global parameter. Although this assumption is likely wrong in general, the role of \({T}_{2}^{* }\) here is that of an upper bound in the FID decay. Because the FID is for most spins dominated by the measurementinduced dephasing (i) and (ii), our method is not very sensitive to variations in \({T}_{2}^{* }\). We find that our fit results for \({T}_{2}^{* }\) are similar to those expected from the drift in the bias field, suggesting that our linewidths are limited by external field stability. Third, we will neglect nuclearnuclear couplings, discussed further below.
Sensitive slice
The magnitude of the FID signal strongly depends on a spin’s threedimensional position r, because of the position dependence of the hyperfine interaction. We can capture the spatial dependence by calculating a sensitivity function \({{{{{{{\mathcal{S}}}}}}}}({{{{{{{\bf{r}}}}}}}})\) that quantifies the signal contribution as a function of spin location r. The sensitivity function is expressed as a signaltonoise ratio and given by:
where β = a_{⊥}t_{β}/π (Eq. (3)) encodes the spatial position (via the hyperfine parameter a_{⊥}), and where K, t_{s}, t_{β}, and t_{pol} are experimental parameters defined in Fig. 2.
Figure 3 plots \({{{{{{{\mathcal{S}}}}}}}}({{{{{{{\bf{r}}}}}}}})\) as a function of vertical and radial distance to the central electronic spin. Interestingly, the sensitivity does not monotonically decay with distance, as might be expected from the a_{⊥} ∝ r^{−3} scaling of the hyperfine interaction. Rather, \({{{{{{{\mathcal{S}}}}}}}}\) is initially low, and increases with r until it reaches a maximum at a characteristic radius r_{slice} before showing the expected r^{−3} decay. The suppression of signal from close spins is a consequence of quantum backaction^{36}: Because these spins are strongly coupled, their measurement strength parameter β is large, leading to a rapid signal decay 1/Γ(β) ∝ β^{−2} → 0 (Eq. (5b)). Conversely, distant spins with small β generate weak signals because A(β) → 0 (Eq. (5a)). Maximum sensitivity results at an intermediate value where the two effects are balanced,
The optimum point of sensitivity is approximately reached when intrinsic and induced decay rates are commensurate, \({({T}_{2,{{{{{{{\rm{n}}}}}}}}}^{* })}^{1}={\beta }^{2}/(4{t}_{{{{{{{{\rm{s}}}}}}}}})\) and when the FID record length is matched to the decay rate, Kt_{s} = 1/Γ.
As shown in Fig. 3, the points of maximum sensitivity are located along a contour of constant a_{⊥} = πβ^{(opt)}/t_{β}. We denote this contour as the “sensitive slice” associated with the interaction time t_{β}. By varying t_{β}, we can vary the radius of the sensitive slice r_{slice} and tune detection from close to distant nuclear spins (Fig. 3, left to right). Because a_{⊥} ∝ r^{−3}, the radius of the sensitive slice scales as \({r}_{{{{{{{{\rm{slice}}}}}}}}}\propto {t}_{\beta }^{1/6}\). The spatial selectivity is an important feature of our method, since it allows us to selectively probe nuclear spins at a defined (far) distance from the central electronic spin while avoiding interference from stronglycoupled nuclei in close proximity. Further, by sweeping t_{β}, we can collect FID traces from several sensitive slices and cover an extended spatial volume in the sample.
Maximum likelihood estimation by simulated annealing
Armed with a protocol for measuring the signals and coupling constants from many nuclei in parallel, we develop a maximum likelihood protocol to extract the hyperfine parameters and position vectors from an FID trace (Eq. 2).
We begin by setting up a likelihood model. Assuming n spins are contributing to the signal, our model contains M = 3n + 3 unknown parameters, including the three hyperfine parameters a_{∣∣,i}, a_{⊥,i}, and ϕ_{i} for each spin i plus three additional, global parameters accounting for an initial polarization p_{0} and dephasing times \({T}_{2,{{{{{{{\rm{n}}}}}}}}}^{* }\) and t_{ℓ} [see Eq. (5)]. Note that because the number of spins n is a priori unknown, M is itself a free parameter. To proceed, we collect the unknown parameters in the parameter vector \({{{{{{{\boldsymbol{\theta }}}}}}}}=\left\{{\theta }_{m}\right\}\), where m = 1…M. Our goal is to balance goodness of the fit and model complexity by minimizing a cost function of the form:
where G(θ, x) is a measure of the goodness of the fit, \({{{{{{{\bf{x}}}}}}}}=\left\{{x}_{k}\right\}\), where k = 1…K, is the set of measured data points, and P(K, M) is a penalty term to prevent overfitting^{56}. Eq. (8) is the generic form of a socalled information criterion (IC). In our likelihood framework, G(θ, x) can be expressed in terms of a negative likelihood function^{57}:
where the argument of the logarithm is the residual sum of squares. The function \({\tilde{x}}_{k}({{{{{{{\boldsymbol{\theta }}}}}}}})\) represents the estimated data points calculated from Eq. (2) using the parameter vector θ. For the penalty term P(K, M), we choose the socalled weightedaverage information criterion (WIC)^{58} that is a weighted average of the Akaike and Bayesian information criteria (AIC^{59} and BIC^{60}, respectively, see (See Supplementary Materials accompanying this manuscript for definition). The advantage of the WIC is that it performs well regardless of sample size K.
Minimization of Eq. (8) is highly nontrivial, as the number of fit parameters is large and M is itself an unknown. In this work, we implement generalized simulated annealing (GSA)^{38,39,61} algorithm to address this challenge. GSA is wellposed in our case because it works well for the global optimization of complicated, multidimensional systems with large numbers of local minima. To further improve the GSA, we run the minimization over a large number of random starting configurations for θ. Finally, once a bestestimate (lowest IC) set of parameters has been found, we compute the threedimensional locations r_{i} of nuclei from the hyperfine vector a_{i} by inverting Eq. (1) (see Eqs. (10, 11) in the Methods).
Experimental demonstration
We experimentally demonstrate our threedimensional nuclear localization strategy by imaging the ^{13}C environment of shallow NV centers in diamond. We focus on two NV centers in this work, labeled NV1 and NV2, out of five recorded datasets. The two NV centers are selected for favorable optical contrast and electron spin coherence times, but not for their ^{13}C environment. Their shallow depth (~10 nm) is not important for this study except for demonstrating that our method is compatible with nearsurface NV centers. We probe the NV centers at room temperature using nonresonant optical excitation and a singlephoton counting module. Electronic and nuclear spins are manipulated via two arbitrary waveform generators connected to a separate microwave transmission line and RF microcoil circuits, respectively^{21,62}. Experiments use a bias field B_{0} ~ 200 mT aligned to within 1^{∘} of the NV symmetry axis (Fig. 1b). A description of diamond samples and the experimental setup is provided in the Methods section.
Figure 4a shows an example of an FID time trace from NV2 for t_{β} = 4.944 μs, and Fig. 4b, c shows the complete dataset of ^{13}C Fourier spectra obtained for both NV centers. For each NV center, we record four spectra with different values of the interaction time t_{β} to sample different radii of the sensitive slice and to add redundancy. For each dataset, we plot the power spectrum, the real and imaginary parts of the complex Fourier spectrum, as well as the fit residues. Clearly, the spectra show a rich peak structure, indicating that we are detecting a large number of ^{13}C resonances.
To fit the spectra, we add the likelihood functions [Eq. 9] from all four spectra and minimize the total cost function [Eq. 8] using a single set of hyperfine parameters. We begin by randomly initializing each parameter, and then minimize the residues between the experimental and computed spectra using a GSA algorithm on a highperformance computer cluster^{63}. To improve robustness, we fit the spectra, rather than the FID traces. We compute separate residues for real and imaginary parts of the complex spectrum as well as for their magnitude squared and minimize the sum of all residues. Additionally, we penalize configurations where the distance between any two ^{13}C is less than one bond length. To accelerate the search for a global minimum, we repeat the procedure for a large number (~10^{2}) of starting values randomly chosen from predefined parameter intervals. Finally, to determine the number of spins we run the minimization routine for different n and select the configuration with the smallest global IC value (Fig. 4d). Once the minimization has terminated and n has been determined, we perform bootstrapping on the final fit residues^{64} to obtain an estimate for the fit uncertainties for all parameters.
The calculated spectra for the most likely nuclear configurations are displayed as blue solid lines in Fig. 4b, c. For the two datasets, we find n = 20 for NV1 and n = 29 for NV2 (Fig. 4d). The large IC >10 of the nextbest configurations n ± 1 indicates that our n are welldefined. (An IC >10 is equivalent to an evidence ratio of e^{0.5IC} > 10^{2}, meaning that our estimated n is >10^{2} more likely than neighboring n ± 1.) This statistical finding is supported by the good agreement between fit and experimental data (gray dots) and the small residues (green), which are of the same order as the measurement noise. We have verified the calculated spectra by performing a full density matrix simulation using the final parameter set (Fig. S2). All fit results are collected in Tables S3 and S4.
Figure 5 shows visualizations of the threedimensional locations of nuclei. We find that nuclear positions are clustered between ca. r = 0.7–2.4 nm and ca. ϑ = 30–75^{∘}. This clustering is a consequence of the spatial selectivity of our method: although the ^{13}C nuclei are distributed randomly over the diamond lattice, only spins falling within the sensitive slice are picked up by the weakmeasurement detection sequence. Proximal spins (r < 0.7 nm) could be detected by further reducing t_{β} or by employing techniques developed for stronglycoupled nuclei^{19,20}. More distant spins (r > 2.4 nm) can be addressed by increasing t_{β}, up to a limit set by the electronic coherence time T_{2,e}.
We find that the combined sensitive slice for all measurements (colorcoded in Fig. 5a, d), taking the t_{β} values of the four spectra, t_{ℓ} and \({T}_{2,{{{{{{{\rm{n}}}}}}}}}^{* }\) dephasing into account, agrees well with the extracted ^{13}C positions. (The t_{ℓ} dephasing suppresses signal from nuclei with large a_{∣∣}, leading to low sensitivity for spins near ϑ = 0^{∘} and 90^{∘}). Figure 5 also clearly shows that the spatial precision of our method is highest and well below 1 Å for ^{13}C’s that are located near the maximum of the sensitive slice, while the precision can be poor for spins located at the fringe of the slice.
The solution presented in Fig. 5 is a maximum likelihood estimate based on the assumptions of global values for the polarization and intrinsic dephasing as well as an absence of nuclearnuclear couplings (see Section “Hyperfine parameters”). Our solution represents the threedimensional nuclear spin configuration that has the highest likelihood, but there is no guarantee that it represents the “true” configuration of ^{13}C nuclei on the diamond carbon lattice. This probabilistic aspect is a standard feature of largescale structure determination^{65}. For example, because our method does not account for nuclear couplings, it is possible that spin pairs or clusters of ^{13}C are erroneously assigned a singlespin position. In addition, because of the inversion symmetry of the dipolar interaction [Eq. (1)], our method does not discriminate between spins lying in the upper and lower hemisphere. Looking forward, this ambiguity will be naturally lifted for outside molecules. Further, both issues can be alleviated by introducing additional spatial constraints, especially by measuring nuclear spinspin distances using twodimensional NMR spectroscopy^{27} (see Discussion).
To verify the threedimensional nuclear configuration, we perform a set of basic statistical tests on the density, spatial distribution, and fit uncertainty of ^{13}C positions. Comparing the volume uncertainties δV of the ^{13}C (indicated by orange shading in Fig. 5c, e and tabulated in Supplementary Materials accompanying this manuscript.) with the volume per carbon atom in the diamond lattice (V = 5.67 Å^{3} ^{66}), we find that 13 out of 20 spins (NV1) and 21 out of 29 spins (NV2) have δV < V and therefore likely represent single nuclei. Next, defining the volume of the sensitive region by the volume in space contributing 50% to the total signal^{10}, we find sensitive volumes of V = 9.3 nm^{3} and 14.8 nm^{3} for NV1 and NV2, respectively. Considering an average density ρ(^{13}C) = 1.94 nm^{−3} for ^{13}C nuclei in diamond at natural isotope abundance (1.1%), the average number of ^{13}C in the sensitive slices are 18.0 and 28.6, respectively, in good agreement with our experimental result. Further, a χ^{2} test for the angular distributions of the azimuth angles yields pvalues well above the 5% level (54 and 70%, respectively), as expected for a random ^{13}C distribution. Within these statistics, our experimental results are fully consistent with a stochastic distribution of ^{13}C atoms around the NV centers. Finally, the r > 0.7 nm in retrospect justifies neglecting the Fermi contact interaction in our model^{20,50}.
Discussion
We conclude with a roadmap for extending the present experiment to samples outside of diamond, including individual molecules and complex spin structures (Fig. 6). Reaching this ambitious goal requires overcoming four central challenges:

(i)
To image outside nuclear spins, nearsurface NV centers are needed. Although our scheme is compatible with nearsurface NV centers, their properties are known to degrade. The degradation manifests itself both in a reduced coherence time and reduced stability of the negative NV charge state. Recently, there has been remarkable progress in stabilizing very shallow NV centers. In particular, ref. ^{67} reports coherence times of T_{2,e} ~ 40 μs for 5nmdeep NV centers. In our experiments, we find similar values (T_{2,e} ~ 50 μs for a 3.5nmdeep NV center). Extrapolating the r ~ 2.4 nm for ^{13}C (#12) in Fig. 5 using the \(r\propto {\gamma }_{{{{{{{{\rm{n}}}}}}}}}^{1/3}{t}_{\beta }^{1/6}\) scaling of the sensitive radius, the above T_{2,e} implies a maximum radius of ca. 5–6 nm for single őr ^{19}F nuclei (see Suppl. Fig. S5). This radius is compatible with the above shallow NV centers.

(ii)
Single molecules must be immobilized and isolated on the diamond surface. This step can be achieved using surface functionalization^{5,48,68}. To further protect the molecules and inhibit diffusion, molecules may be embedded in a spinfree matrix layer (Fig. 6)^{69}.

(iii)
Nuclear spins in molecules will behave differently from internal ^{13}C nuclei. In particular, nuclear spin interactions will likely dominate over the hyperfine interaction. Nuclear spin interactions can be mitigated by homo and heteronuclear decoupling techniques^{70} and by isotope dilution^{71}; however, they also are an important resource for structural information and a central element in our molecular imaging strategy, see next point.

(iv)
Advanced spectroscopy and imaging techniques are needed that can efficiently retrieve the desired structure. A promising strategy is to combine the global distance information available by our scheme with local distance constraints obtained by solidstate NMR methods^{27} (dashed and dotted lines in Fig. 6). For example, twodimensional spectroscopy^{9,25,72} and doubleecho techniques^{27,73} can be conveniently integrated with our protocol, because of its similarity to conventional FID detection. Additionally, efficient computational methods will be required to speed up data analysis and optimally extract the desired spatial information, including, for example, sparse sampling^{29,74}, gradient Monte Carlo^{75}, and machine learning techniques^{76,77,78}.
By demonstrating threedimensional mapping of nuclear spins in ambient conditions, our work takes an important step forward towards the ambitious goal of singlemolecule MRI. Looking beyond the milestone of structural imaging, the technique could be extended to study chemical binding and chemical surface reactions at the singlemolecule level. For example, in a biological context, surface NMR may allow investigation of a priori unknown substrate binding sites in enzymes^{79}, as well as conformational changes in biomolecules associated with allosteric regulation^{80}, protein aggregation^{81}, or aptamertarget recognition^{82}. These applications are particularly appealing because NMR can provide complementary information to other surfacesensitive techniques, such as surfaceenhanced Raman spectroscopy (SERS), photoelectron spectroscopies, and fluorescence microscopy.
Besides magnetic imaging of single spins, our work also provides interesting perspectives for the characterization of large qubit registers in quantum applications. For example, our method can be applied to efficiently map out the coupling network of quantum nodes built from a central electronic spin backed by a nuclear spin register^{83,84}. Such quantum nodes are central elements in emerging optical^{85} or electronic^{86} quantum interconnects. Another application is a nuclear quantum simulation using an electronic qubit for initialization and readout^{40}. Finally, our parallel measurement protocol could provide a rapid means for calibrating crosstalk in superconducting qubit architectures^{43,44}.
Methods
Diamond samples
Two singlecrystal diamond plates were used for experiments. Both sample A (NV1) and sample B (NV2) were electronicgrade, natural abundance (1.1% ^{13}C) diamond membranes. NV centers were created by ^{15}N^{+} ion implantation at an energy of 5 keV and doses of 5 × 10^{11} cm^{−2} and 4 × 10^{10} cm^{−2} for samples A and B, respectively. Samples were subsequently annealed at 850 °C to form NV centers. We chose the ^{15}N species to discriminate implanted NV centers from native (^{14}N) NV centers. Both samples were cleaned in a 1:1:1 mixture of H_{2}SO_{4}:HNO_{3}:HClO_{4} and baked at 465 °C in the air before mounting them in the setup. Whenever organic contamination was spotted, samples were cleaned in a 2:1 mixture of H_{2}SO_{4}:H_{2}O_{2} (Piranha). We etched nanopillars into the membrane surfaces to increase the photon collection efficiency. The continuous wave (CW) photon count rate was 250–500 kC/s.
Experimental setup
Experiments were performed using a custombuilt confocal microscope equipped with a green λ = 532 nm frequencydoubled Nd:YAG excitation laser (CNI Laser MSLFN532nm) and a 630−800 nm detection path using a singlephoton avalanche photodiode (APD, Pelkin Elmer SPCMAQR Series). Optical pulses were generated by an acoustooptic modulator (AOM, Crystal Technology 3200144) in a doublepass configuration, and gating of arriving photons was realized by timetagging (NIPCIe6363) and software binning of photon counts. Typical laser excitation powers were on the order of 100 μW.
We synthesized microwave pulses for manipulating the electronic spin using an arbitrary waveform generator (AWG, Tektronix AWG5012C) and upconverted them to ~2.5 GHz using a local oscillator (Hittite HMCT2100) and a quadrature mixer (Marki microwave IQ1545). Pulses were subsequently amplified (Gigatronics GT1000A) prior to delivery to the NV center using a coplanar waveguide (CPW) photolithographically defined on a quartz cover slip. The transmission line was terminated on an external 50 Ω load (Meca 4902). We synthesized radiofrequency (RF) pulses for nuclear spin manipulation using an AWG (National Instruments PCI5421) and subsequently amplified them (MiniCircuits LZY22+). The pulses were transmitted using a planar microcoil connected in series with a 50 Ω termination (Meca 697301). The measured microcoil inductance was L = 0.77 μH. The 50 Ω termination increased the rfcircuit bandwidth (Q = L/R) at the expense of efficiency (most power was dissipated in the load). The microcoil circuit had a 3dBbandwidth of ~19 MHz. ^{13}C Rabi frequencies were typically around 25 kHz. A layout of the microcoil arrangement is given in ref. ^{62}.
We used a cylindrical samariumcobalt permanent magnet (TCSmCo, reversible temperature coefficient 0.001%/^{°}C) to create a bias field B_{0} ~ 190 mT at the NV center location. To align B_{0} with the NV symmetry axis, we adjusted the relative location of the permanent magnet by fitting it to a set of electron paramagnetic resonance (EPR) lines recorded at different magnet locations and subsequently by maximizing the CW photon count rate.
Tracking of magnetic field drifts
The net magnetic bias field drifted by typically a few μT, leading to variations in the EPR frequency of order ±100 kHz and variations in the ^{13}C Larmor frequency of order ±50 Hz. This limited the observed ^{13}C linewidths to ~100 Hz. Field drifts were the dominating source of line broadening for the ^{13}C NMR. We continuously tracked and logged magnetic field drifts by measuring the EPR resonance of the NV center and periodically readjusted the microwave excitation frequency during the course of the experiment.
Hyperfine vector a and position r
The radius r and polar angle ϑ are computed from the parallel and transverse hyperfine parameters a_{∣∣} and a_{⊥} as follows (see ref. ^{54}, Eq. S28 and S29):
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The code that was used to perform the GSA minimization is available from the corresponding author upon request.
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Acknowledgements
We thank M. L. Palm for helpful discussions and acknowledge technical support from the machine shop of the Department of Physics at ETH Zurich. This work has been supported by Swiss National Science Foundation (SNSF) Project Grant No. 200020_175600, the National Center of Competence in Research in Quantum Science and Technology (NCCR QSIT), and the Advancing Science and TEchnology thRough dIamond Quantum Sensing (ASTERQIS) program, Grant No. 820394, of the European Commission.
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C.L.D. initiated the project. K.S.C. performed the experiments on a setup built by K.S.C. and K.H. K.S.C. developed the GSA minimization and analyzed the data. K.S.C. and C.L.D. cowrote the paper with input from K.H., J.Z., and J.M.A. All authors discussed the results.
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Cujia, K.S., Herb, K., Zopes, J. et al. Parallel detection and spatial mapping of large nuclear spin clusters. Nat Commun 13, 1260 (2022). https://doi.org/10.1038/s4146702228935z
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DOI: https://doi.org/10.1038/s4146702228935z
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