Abstract
Spins associated to optically accessible solidstate defects have emerged as a versatile platform for exploring quantum simulation, quantum sensing and quantum communication. Pioneering experiments have shown the sensing, imaging, and control of multiple nuclear spins surrounding a single electron spin defect. However, the accessible size of these spin networks has been constrained by the spectral resolution of current methods. Here, we map a network of 50 coupled spins through highresolution correlated sensing schemes, using a single nitrogenvacancy center in diamond. We develop concatenated doubleresonance sequences that identify spinchains through the network. These chains reveal the characteristic spin frequencies and their interconnections with high spectral resolution, and can be fused together to map out the network. Our results provide new opportunities for quantum simulations by increasing the number of available spin qubits. Additionally, our methods might find applications in nanoscale imaging of complex spin systems external to the host crystal.
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Introduction
Optically interfaced spin qubits associated to defects in solids provide a versatile platform for quantum simulation^{1}, quantum networks^{2,3} and quantum sensing^{4,5,6}. Various systems are being explored^{7}, including defects in diamond^{1,2,3,8,9}, silicon carbide^{10,11}, silicon^{12,13}, hexagonal boron nitride (hBN)^{14}, and rareearth ions^{15}. The defect electron spin provides a qubit with highfidelity control, optical initialization and readout, and a (longrange) photonic quantum network interface^{2,3}. Additionally, the electron spin can be used to sense and control multiple nuclear spins surrounding the defect^{15,16,17}. This additional network of coupled spins provides a qubit register for quantum information processing, as well as a test bed for nanoscale magnetic resonance imaging^{18,19,20,21,22,23}. Examples of emerging applications are quantum simulations of manybody physics^{1,24,25,26,27}, as well as quantum networks^{2,3}, where the nuclear spins provide qubits for quantum memory^{28}, entanglement distillation^{29}, and error correction^{30,31,32}.
Stateoftheart experiments have demonstrated the imaging of spin networks containing up to 27 nuclear spins^{18,19,33,34,35}. The ability to map larger spin networks can be a precursor for quantum simulations that are currently intractable, would provide a precise understanding of the noise environment of spinqubit registers^{32,36,37}, and might contribute towards efforts to image complex spin systems outside of the host material^{20,21,22,23,38,39}. A key open challenge for mapping larger networks is spectral crowding, which causes overlapping signals and introduces ambiguity in the assignment of signals to individual spins and the interactions between them.
Here, we develop correlated sensing sequences that measure both the network connectivity as well as the characteristic spin frequencies with high spectral resolution. We apply these sequences to map a 50nuclearspin network comprised of 1225 spinspin interactions in the vicinity of a nitrogenvacancy (NV) center in diamond. The key concept of our method is to concatenate doubleresonance sequences to measure chains of coupled spins through the network. The mapping of spin chains removes ambiguity about how the spins are connected and enables the sensing of spins that are farther away from the electron spinsensor in spectrally crowded regions. These results significantly increase the size and complexity of the accessible spin network. Additionally, our methods are applicable to a wide variety of systems and might inspire future methods to magnetically image complex samples such as individual molecules or proteins^{22,33,39}.
Results
Spinnetwork mapping
We consider a network of Ncoupled nuclear spins in the vicinity of a single electron spin that acts as a quantum sensor^{18,19}. The effective dynamics of the nuclearspin network, with an external magnetic field along the zaxis, are described by the Hamiltonian (see Supplementary Note 1):
where \(\, {\hat{\! I}\, }_{{{{{{{{\rm{z}}}}}}}}}^{(i)}\) denotes the nuclear Pauli spin\(\frac{1}{2}\) operator for spin i, A_{i} are the precession frequencies associated with each spin, and C_{ij} denotes the nuclearnuclear coupling between spin i and j. The frequencies A_{i} might differ due to differences in species (gyromagnetic ratio), the local magnetic field and spin environment, and due to coupling to the sensor electron spin. Our goal is to extract the characteristic spin frequencies A_{i} and spinspin couplings C_{ij} that capture the structure of the network.
Figure 1 shows an example network, with colored disks denoting frequency regions, and numbered dots inside signifying spins at these frequencies. Although in principle all spins are coupled to all spins, we draw edges only for strong, resolvable, spinspin couplings, defined by: C_{ij} ≳ 1/T_{2}, where T_{2} is the nuclear Hahnecho coherence time (∼0.5 s)^{16}. The network connectivity constitutes the presence (or absence) of such resolvable couplings. In general, the number of frequency disks is smaller than the number of spins, as multiple spins might occupy the same frequency region (i.e., overlap in frequency).
Stateoftheart spin network mapping relies on isolating individual nuclearnuclear interactions through spinecho double resonance (SEDOR)^{18}. Applying simultaneous echo pulses at frequencies A_{i} and A_{j} preserves the interaction C_{ij} between spins at A_{i} and A_{j}, while decoupling them from (quasistatic) environmental noise and the rest of the network so that the coupling C_{ij} is encoded in the nuclearspin polarization with high spectral resolution (set by the nuclear T_{2}time rather than \({T}_{2}^{\star }\)time). The signal is acquired by mapping the resulting nuclearspin polarization, for example at frequency A_{i}, on the NV electron spin and reading it out optically^{16}. Such a measurement yields a correlated list of three frequencies {A_{i}, C_{ij}, A_{j}} (Fig. 1a). If all spins are spectrally isolated so that the A_{i} do not overlap, these pairwise measurements completely characterize the network.
However, due to their finite spectral line widths (set by \(1/{T}_{2}^{\star }\)), multiple spin frequencies A_{i} may overlap (indicated by multiple spins occupying a disk). This introduces ambiguity when assigning measured couplings to specific spins in the network, and causes complex overlapping signals, which are difficult to resolve and interpret^{18,19}. Figure 1b shows an example where pairwise measurements break down; spins 2 and 5 overlap in frequency (A_{2} ≈ A_{5}). Applying pairwise SEDOR between frequencies A_{1}, A_{3}, A_{4} and a frequency that overlaps with A_{2} and A_{5} returns three independent pairwise correlations: {A_{1}, C_{12}, A_{2}}, {A_{3}, C_{23}, A_{2}} and {A_{4}, C_{45}, A_{5}}. Crucially, however, such measurements cannot distinguish this uncoupled 2spin and 3spin chain (Fig. 1b) from a single 4spin network (with a single central spin at A_{2}), nor from a network of 3 uncoupled 2spin chains (three spectrally overlapping spins). Without introducing additional apriori knowledge or assumptions about the system, pairwise measurements cannot be assigned to specific spins and are thus insufficient to reconstruct the network^{18}.
Our approach is to measure connected chains through the network and combine these with highresolution spinfrequency measurements. First, spinchain sensing (detailed in the “Spinchain sensing” section) correlates multiple frequencies and spinspin couplings, directly accessing the underlying network connectivity, and thus reducing ambiguity due to (potential) spectral overlap. Consider the previous example: by probing the correlation between the three frequencies A_{1}, A_{2}, and A_{3} in a single measurement, we directly reveal that Spin 1 and Spin 3 are connected to the same spin at A_{2} (Spin 2). Such a spinchain measurement yields a correlated list of 5 frequencies: {A_{1}, C_{12}, A_{2}, C_{23}, A_{3}}, characterizing the 3spin chain. Applying the same method but now with spin 4 (A_{3} ← A_{4}) reveals that it is not connected to Spin 2, but couples to another spin (spin 5) that overlaps in frequency with Spin 2.
Second, spinchain sensing enables measuring couplings that are otherwise challenging to access, enabling exploration further into the network. Consider the case where starting from some spin (e.g. Spin 1 in Fig. 1c) it is challenging to probe a part of the network, either because the couplings to Spin 1 are too weak to be observed or spectral crowding causes signals to overlap. The desired interactions (e.g. those belonging to Spin 4 in Fig. 1c) can be reached by constructing a spin chain, in which each link is formed by a strong and resolvable spinspin interaction. The chain iteratively unlocks new spins that can be used as sensors of their own local spatial environment.
Finally, we combine the spinchain measurements with a correlated doubleecho spectroscopy scheme that increases the resolution with which different A_{i} are distinguished from \(\sim 1/{T}_{2}^{\star }\) to ∼1/T_{2} (Fig. 1d). This directly reduces spectral overlap of spin frequencies, further removing ambiguity.
In principle, the entire network can be mapped by expanding and looping a single chain. In practice, measuring limitedsize chains is sufficient. An Nspinchain measurement yields a correlated list of N spin frequencies A, alongside N − 1 coupling frequencies C, which quickly becomes uniquely identifiable, even when some spin frequencies in the network are degenerate. This allows for the merging of chains that share a common section to reconstruct the network (see “Methods” section).
Experimental system
We demonstrate these methods on a network of 50 ^{13}C spins surrounding a single NV center in diamond at 4 K. The NV electron spin is initialized and measured optically and is used as the sensor spin^{18}. We employ dynamical decoupling sequences to sense nuclear spins at selected frequency bands, using sequences with and without radiofrequency driving (DDRF) of the nuclear spins to ensure sensitivity in all directions from the NV (see “Methods” section)^{16,18}. The nuclear spins are polarized via the electron spin, using global dynamicalnuclearpolarization techniques (PulsePol sequence^{1,40}), or by selective projective measurements or SWAP gates^{16,18}.
The ^{13}C nuclearspin frequencies are given by A_{i} = ω_{L} + m_{s}Δ_{i}, with ω_{L} the global Larmor frequency and Δ_{i} a local shift due to the hyperfine interaction with the NV center (see for example ref. ^{41} and Supplementary Note 1). Here, we neglected corrections due to the anisotropy of the hyperfine interaction, which are treated in Supplementary Note 4. The experiments are performed with the electronic spin in the m_{s} = ±1 states. Because, for the spins considered, Δ_{i} is typically two to three orders of magnitude larger than the nuclearnuclear couplings C_{ij}, nuclearspin flipflop interactions are largely frozen, and Eq. (1) applies (Supplementary Note 1).
In the NVnuclear system, spectral crowding forms a natural challenge for determining the spin network structure. The spin frequencies are broadened by the inhomogeneous linewidth \(\sim 1/{T}_{2}^{\star }\), which is mainly set by the coupling to all other nuclear spins. A limited number of nuclear spins close to the NV center are spectrally isolated (defined as: \( {A}_{i}{A}_{j} \, > \, 1/{T}_{2}^{\star }\,\,\forall \,\,j\,\)), making them directly accessible with electronnuclear gates^{16,18}, and making pairwise measurements sufficient to map the interactions. However, the hyperfine interaction, and thus Δ_{i}, decreases with distance (∼r^{−3}), resulting in an increasing spectral density for lower Δ_{i} (larger distance). Interestingly, there exists a spectrally crowded region (\( {A}_{i}{A}_{j} \, < \, 1/{T}_{2}^{\star }\)) for which nuclear spins still do not couple strongly to other spins in the same spectral region (C_{ij} ≲ 1/T_{2} ∀ j ), for example when they are on opposite sides of the NV center. Contrary to previous work^{18}, the methods outlined in the “Spinnetwork mapping” section allow us to measure interactions between spins in the spectrally crowded region (see Supplementary Note 2), unlocking a part of the network that was previously not accessible.
Spinchain sensing
We experimentally demonstrate the correlated sensing of spin chains up to five nuclear spins (Fig. 2), by sweeping a multidimensional parameter space (set by 5 spin frequencies and 4 spinspin couplings). We start by polarizing the spin network^{1,40} and use the electron spin to sense a nuclear spin (Spin 1) at frequency A_{1}, which marks the start of the chain.
First, we perform a doubleresonance sensing sequence (Fig. 2b) consisting of a spinecho sequence at frequency A_{1} and an additional πpulse at frequency RF_{2}. The free evolution time t_{12} is set to 50 ms, to optimize sensitivity to nuclearnuclear couplings (typically ∼10 Hz). By sweeping RF_{2}, strong connections (C_{1j} ≫ 1/T_{2}) are revealed through dips in the coherence signal of Spin 1 (Fig. 2d, left). We select a connection to a spin at RF_{2} = A_{2} (Spin 2) and determine (C_{12}) by sweeping t_{12} (Fig. 2d, right).
Next, we extend the chain. To map the state of Spin 2 back to the electron sensor through Spin 1, we change the phase of the first \(\frac{\pi }{2}\)pulse (labeled ‘yx’) and set t_{12} = 1/(2C_{12}) to maximize signal transfer (see Supplementary Note 3). We then insert a doubleresonance block for frequencies RF_{2} = A_{2} and RF_{3} in front of the sequence (Fig. 2c, e, left) to explore the couplings of Spin 2 to the network. This concatenating procedure can be continued to extend the chain, with up to 5 nuclear spins shown in Fig. 2. In general, the signal strength decreases with increasing chain length, as it is set by a combination of the degree of polarization and decoherence (T_{2} relative to C_{ij}) of all spins in the chain (See Supplementary Note 3). This limits the chain lengths that can be effectively used.
By mapping back the signal through the spin chain, the five spin frequencies and the 4 coupling frequencies are directly correlated: they are found to originate from the same branch of the network. As spins are now characterized by their connection to the chains, rather than by their individual, generally degenerate, frequencies (Fig. 1b), they can be uniquely identified. Additionally, the chains enable measuring individual spinspin couplings in spectrally crowded regions (Fig. 1c). As an example, the expected density of spins at frequency A_{4} is around 30 spins per kHz (Supplementary Fig. 6), making Spin 4 challenging to access directly from the electron spin. However, because Spin 3 probes only a small part of space, Spin 4 can be accessed through the chain, as demonstrated by the singlefrequency oscillation in Fig. 2f. Another advantage over previous methods^{18} is that our sequences are sensitive to both the magnitude and the sign of the couplings, at the cost of requiring observable polarization of the spins in the chain. The sign of the couplings provides additional information for reconstructing the network (Fig. 2g).
Highresolution measurement of spin frequencies
While the sensing of spinchains unlocks new parts of the network and reduces ambiguity by directly mapping the network connections, the spectral resolution for the spin frequencies (A_{i}) remains limited by the nuclear inhomogeneous dephasing time \({T}_{2}^{\star } \sim 5\,{{{{{{{\rm{ms}}}}}}}}\)^{16}. Next, we demonstrate highresolution (T_{2}limited) measurements of the characteristic spinfrequency shifts Δ_{i}. These frequencies provide a way to label spins, and thus further reduce ambiguity regarding which spins participate in the measured chains, particularly when a spectral region in the chain contains multiple spins (see Fig. 1d).
We isolate the interaction of nuclear spins with the electron spin through an electronnuclear doubleresonance block acting at a selected nuclearspinfrequency region. The key idea is that the frequency shift imprinted by the electron spin sensor can be recoupled by controlling the electron spin state. We use microwave pulses that transfer the electron population from the \(\left\vert 1\right\rangle\) to the \(\left\vert+1\right\rangle\) state (Fig. 3b, see “Methods” section). The nuclear spin is decoupled from quasistatic noise and the rest of the spins, extending its coherence time, while the interaction of interest (Δ_{i}) is retained.
Figure 3 shows an example for a nuclear spin at A_{1}, for which we measure a hyperfine shift Δ_{1} = 14549.91(5) Hz and a spectral linewidth of 1.8 Hz (Fig. 3d and e). Besides a tool to distinguish individual spins in the network with high spectral resolution, this method has the potential for improved characterization of the hyperfine interaction in electronnuclearspin systems.
The observed coherence time T_{2} = 0.36(2) s is slightly shorter than the bare nuclearspinecho time T_{2,SE} = 0.62(5) s. This reduction is caused by a perturbative component of the hyperfine tensor in combination with the finite magnetic field strength (see Supplementary Note 4). Flipping the electron spin between m_{s} = ± 1 changes the quantization axes of the nuclear spins, which causes a change of the nuclearnuclear interactions^{18}, which is not decoupled by the spinecho sequence (see Supplementary Fig. 4). The effect is strongest for spins near the NV center. For larger fields or for spins with weak hyperfine couplings, we expect that further resolution enhancement is possible by applying multiple refocusing pulses (see Supplementary Note 4).
Finally, we combine spinchain sensing and electronnuclear double resonance to correlate highresolution spin frequencies (Δ_{i}) with specific spinspin couplings (C_{ij}), even when a chain contains multiple spins with overlapping frequencies. We illustrate this scheme on a chain of spins, where two spins (2 and 3) have a similar frequency (A_{2} ≈ A_{3}) and both couple to A_{1} and A_{4} (Fig. 4a). The goal is to extract Δ_{2}, Δ_{3} and the couplings to Spin 4 (C_{24}, C_{34}). As a reference, standard doubleresonance shows a quickly decaying timedomain signal, indicating couplings to multiple spins that are spectrally unresolved (Fig. 4b).
Figure 4 c shows how the electronnuclear doubleresonance sequence (mint green) is inserted in the spinchain sequence to perform highresolution spectroscopy of the A_{2}, A_{3} frequency region. Sweeping the interaction time t_{1} shows multiple frequencies (Fig. 4e), hinting at the existence of multiple spins with approximate frequency A_{2}. The result is consistent with two spins at frequencies Δ_{2} = 8019.5(2) Hz and Δ_{3} = 7695.2(1) Hz, split by an internal coupling of C_{23} = 7.6(1) Hz (Fig. 4a and Supplementary Fig. 2e, f).
Next, we add a nuclearnuclear block (pink block in Fig. 4d) and sweep both electronnuclear (t_{1}) and nuclearnuclear (t_{2}) doubleresonance times to correlate Δ_{2} and Δ_{3} with nuclearnuclear couplings C_{24} and C_{34}. After the t_{1} evolution, the hyperfine shifts Δ_{i} are imprinted in the zexpectation value of each spin, effectively modulating the nuclearnuclear couplings observed in t_{2}. The 2D power spectral density (PSD) shows signals in two distinct frequency regions along the f_{1}axis, corresponding to Δ_{2} and Δ_{3} (Fig. 4f). Analysing the nuclearnuclear (f_{2}) signal at these frequencies (Fig. 4g), we find C_{24} = −11.8(2) Hz and C_{34} = −0.2(5) Hz. We attribute the splitting to the coupling C_{23} between Spins 2 and 3 (see “Methods” section, Supplementary Fig. 2g, h). Varying RF_{4} enables the measurement of the interactions of spins 2 and 3 to other parts of the network (for example to determine C_{12}, C_{13}). Beyond the examples shown here, the electronnuclear block can be inserted at specific positions in the spinchain sequence (Fig. 2c) to extract Δ_{i} of all spins in the chain (Supplementary Fig. 9).
Reconstruction of a 50spin network
Finally, we apply these methods to map a 50spin network. The problem resembles a graph search (see “Methods” section)^{42}. By identifying a number of spin chains in the system, and fusing them together based on overlapping sections, we reconstruct the connectivity (Fig. 5). Limitedsized chains are sufficient because the couplings are highly nonuniform, so that a few overlapping vertices and edges enable fusing chains with high confidence. We use a total of 249 measured interactions through pairwise and chained measurements. Fusing these together provides a hypothesis for the network connectivity (Fig. 5b).
To validate our solution for the network we use the additional information that the nuclearnuclear couplings can be modeled as dipolar and attempt to reconstruct the spatial distribution of the spins. Compared to work based on pairwise measurements^{18}, our spinchain measurements provide additional information on the connectivity and coupling signs, reducing the complexity of the numerical reconstruction. Additionally, we constrain the position using the measured hyperfine shift Δ_{i} (see “Methods” section). Because the problem is highly overdetermined^{18}, the fact that a spatial solution is found that closely matches the measured frequencies and assignments validates the obtained network connectivity. Additionally, the reconstruction yields a spatial image of the spin network and predicts the remaining unmeasured 976 spinspin interactions, most of which are weak (<1 Hz). An overview of the complete 50spin cluster, characterized by 50spin frequencies and 1225 spinspin couplings can be found in Supplementary Table 1 and in Fig. 5b.
Discussion
In conclusion, we developed correlated doubleresonance sensing that can map the structure of large networks of coupled spins, with high spectral resolution. We applied these methods to reconstruct a 50spin network in the vicinity of an NV center in diamond. The methods can be applied to a variety of systems in different platforms, including electronelectron spin networks^{7,8,9,10,11,12,13,14,15,43}. Mapping larger spin systems might be in reach using machinelearningenhanced protocols and sparse or adaptive sampling techniques, which can further reduce acquisition times^{44,45}. Combined with control fields^{1,16,32}, the methods developed here provide a basis for universal quantum control and readout of the network, which has applications in quantum simulations of manybody physics^{1}. Furthermore, the precise characterization of a 50spin network provides new opportunities for optimizing quantum control gates in spinqubit registers^{16,32,36}, for testing theoretical predictions for defect spin systems^{46}, and for studying coherence of solidstate spins on the microscopic level, including quantitative tests of open quantum systems and approximations of the central spin model^{47}. Finally, these results might inspire highresolution nanoMRI of quantum materials and biologically relevant samples outside the host crystal.
Methods
Sample and setup
All experiments are performed on a naturally occurring NV center at a temperature of 3.7 K (Montana S50 Cryostation), using a homebuilt confocal microscopy setup. The diamond sample was homoepitaxially grown using chemical vapor deposition and cleaved along the 〈111〉 crystal direction (Element Six). The sample has a natural abundance of ^{13}C (1.1%). The NV center has been selected on the absence of couplings to ^{13}C stronger than ≈ 500 kHz. No selection was made on other properties of the ^{13}C nuclei distribution. A solid immersion lens (SIL) that enhances photon collection efficiency is fabricated around the NV center. A gold stripline is deposited close to the edge of the SIL for applying microwave (MW) and radiofrequency (RF) pulses. An external magnetic field of B_{z} = 403.553 G is applied along the symmetry axis of the NV center, using a (temperaturestabilized) permanent neodymium magnet mounted on a piezo stage outside the cryostat^{16}. The field is aligned to within 0.1 degrees using a thermal echo sequence^{18}.
Electron and nuclear spins
The sample was previously characterized in Abobeih et al.^{18} and the 27 nuclear spins imaged in that work are a subset of the 50 nuclearspin network presented here. The NV electron spin has a dephasing time of \({T}_{2}^{\star }=4.9(2)\, {{{{{{{\rm{\mu}}}{{\rm{s}}}}}}}}\), a Hahn spinecho time of T_{2} = 1.182(5) ms, and a relaxation time of T_{1} > 1 h^{18}. The spin state is initialized via spinpumping and read out in a single shot through spinselective resonant excitation, with fidelities F_{0} = 89.3(2) (F_{1} = 98.2(1)) for the m_{s} = 0 (m_{s} = −1) state, resulting in an average fidelity of F_{avg} = 0.938(2). The readout is corrected for these numbers to obtain a best estimate of the electronic spin state. The nuclear spins have typical dephasing times of \({T}_{2}^{\star }\) = 5–10 ms and Hahnecho T_{2}, up to 0.77(4) s^{16}. T_{2}times for spins with frequencies closer to the nuclear Larmor frequency (Δ_{i} ≲ 5 kHz) typically decrease to below 100 ms (see e.g. Fig. 2g, right panel), as the spinecho simultaneously drives other nuclear spins at these frequencies which are recoupled to the target (instantaneous diffusion).
Pulse sequences
We drive the electronic m_{s} = 0 ↔ m_{s} = −1 (m_{s} = 0 ↔ m_{s} = +1) spin transitions at 1.746666 (4.008650) GHz with Hermiteshaped pulses. For transferring the electron population from the m_{s} = −1 to the m_{s} = +1 state (Figs. 3 and 4), we apply two consecutive πpulses at the two MW transitions, spaced by a waiting time of 3 μs. For all experiments, we apply RF pulses with an errorfunction envelope in the frequency range 400–500 kHz. Details on the electronics to generate these pulses can be found in ref. ^{1}.
For most experiments described in this work, the measurable signal is dependent on the degree of nuclearspin polarization. We use a dynamicalnuclear polarization sequence, PulsePol, to transfer polarization from the electron spin to the nuclearspin bath^{1,40}. The number of repetitions of the sequence is dependent on the specific polarization dynamics of the spins being used in the given experiment but ranges from 500–10,000. The PulsePol sequence is indicated by the ‘Init’ block in the sequence schematics. All doubleresonance sequences follow the convention illustrated in the dotted boxes in Fig. 2b, c, and Fig. 3b, where the horizontal gray lines denote different RF frequencies and the top line the electronic MW frequency. The two letters in the doubleresonance blocks (‘xx’ or ‘yx’) denote the rotation axes of the first and final π/2pulses. The πpulses (along the xaxis) are applied sequentially (following ref. ^{18}). The lengths of all RF pulses are taken into account for calculating the total evolution time. Nuclear spins are read out via the electron by phasesensitive (‘yx’) dynamical decoupling; DD or DDRF sequences^{16}, indicated by the ‘RO’marked block in the sequence schematics. Typically, the spin that is read out with the electron is reinitialized via a SWAP gate before the final SEDOR block in order to maximize its polarization. However, all experiments presented here can be performed by using just the DNP initialization, albeit with a slightly lower signaltonoise ratio.
2D spectroscopy experiments
For the 2D measurement, we concatenate an electronnuclear double resonance with a nuclearnuclear SEDOR. For every t_{1}point, we acquire 20 t_{2} points, ranging from 10 to 260 ms. The final π/2pulse of the electron double resonance and the first of the SEDOR are not executed, as they can be compiled away. To correct for any slow magnetic field drifts that lead to miscalibration of the twoqubit gate used for readout, causing a small offset in the measured signal, we set our signal baseline to the mean of the final five points (≈200–260 ms), where we expect the signal to be mostly decayed. Note that these field drifts do not affect any of the doubleresonance blocks in which the quantities to be measured are encoded (due to the spinecho).
Both the 1D (Fig. 4e) and 2D (Fig. 4f) signals are undersampled to reduce the required bandwidth. To extract Δ_{2}, Δ_{3}, we fit a sum of cosines to the timedomain signal of Fig. 4e. To extract the frequencies along the f_{2}axis, which encode the nuclearnuclear couplings (C_{24}, C_{34}), we take an (extended) line cut at f_{1} = Δ_{2} and f_{1} = Δ_{3}. To increase the signal, we sum over the four bins indicated by the dotted lines. We fit two independent Gaussians to the f_{2}data to extract C_{24} and C_{34}. We find splittings of 7.8(2) Hz and 10.2(5) Hz, respectively, whose deviation with respect to measurements in Fig. 4e is unexplained. The skewed configuration of the two peaks (lower left, upper right) is a result of the correlation of the neighboring spin state between the t_{1} and t_{2} evolution times. The different ratio of signal amplitudes belonging to Spin 2 and Spin 3, between the 1D and 2D electronnuclear measurements are due to using different settings for the chained readout (evolution time, RF power). As we are only interested in extracting frequencies, we can tolerate such deviations.
Supplementary Fig. 2 shows numerical simulations of the experiments presented in Fig. 4. These are generated by evaluating the Hamiltonian in Supplementary Eq. 3, taking into account the two spins at A_{2}, A_{3}, the spin at A_{4,} and the electron spin.
Network reconstruction
Here, we outline a general procedure for mapping the network by performing specific spinchain and highresolution Δ_{i} measurements. The mappingproblem resembles a graph search, with the NV electron spin used as root^{42}. We base the protocol on a breadthfirstlike search, which yields a spanning tree as output, completely characterizing the network. The following pseudocode describes the protocol:
Algorithm 1
Input: physical spin network, initial vertex el
Output: breadthfirst tree T from root el
V_{0} = {el} ⊳ Make el the root of T, V_{i} denotes the set of vertices at distance i
i = 0
while\({V}_{i} \, \ne \, {{\emptyset}}\) do ⊳ Continue until network is exhausted
for each vertex v ∈ V_{i }do
for each frequency f do
C, singlecoupling = MeasureCoupling(v, f) ⊳ Returns coupling C between vertex v and frequency f
if singlecoupling then ⊳ Checks if MeasureCoupling returned a single, resolvable coupling
create vertex w
A_{w} = f
C_{vw} = C
unique, duplicate = CheckVertex(w, T) ⊳ Checks if w was already mapped in T
if unique then ⊳ w was not yet mapped
add w to V_{i+1} in T ⊳ w is added to T as a new vertex
end if
if not unique and duplicate == k then ⊳ w is the same vertex as k in T
add C_{vk} = C_{vw} in T ⊳ The measured coupling is assigned to k
delete w
end if
if not unique and duplicate == None then ⊳ Undecided if spin was mapped
delete w ⊳ w is not added to T
end if
end if
end for
end for
i = i + 1
end while
New vertices that are detected by chained measurements are iteratively added, once we verify that a vertex was not characterized before (i.e. has a duplicate in the spanning tree T). The function MeasureCoupling(v, f) performs a spinecho doubleresonance sequence between vertex v and a frequency f, (a spin chain of length i − 1 is used to access v) and checks whether a single, resolvable coupling is present (stored in the boolean variable ‘singlecoupling’). In the case that v is the electron spin (el) an electronnuclear DD(RF) sequence is performed^{16,41}. The function CheckVertex(w, T) instructs the experimenter to perform a number of spin chain and electronnuclear doubleresonance measurements, comparing the vertex w and its position in the network with that of the (possibly duplicate) vertex k (see Supplementary Note 5). If one of these measurements is not consistent with our knowledge of k, we conclude w is a unique vertex and add it to T. If all measurements coincide with our knowledge of k, we conclude it is the same vertex and merge w and k. If the CheckVertex(w, T) is inconclusive (e.g. due to limited measurement resolution), we do not add w to T. Note that the measurement resolution, determined by the nuclear T_{2}time, is expected to decrease for spins further away from the NV center (See Supplementary Note 2). This eventually limits the number of unique spins that can be identified and added to the network map.
The platformindependent procedure outlined above can be complemented by logic based on the 3D spatial structure of the system^{18}. For example, when the CheckVertex(w, T) function is inconclusive, one can sometimes still conclude that w must be unique (or vice versa equal to k), based on the restricted number of possible physical positions of these two spins in 3D space^{18}. In practice, we alternate the graph search procedure with calls to a positioning algorithm^{18}, which continuously checks whether the spanning tree T is physical and aides in the identification of possible duplicates.
3D spatial image
For the 3D reconstruction of the network, we use the positioning algorithm developed in ref. ^{18}. To limit the experimental time we reuse the data of ref. ^{18} and add the new measurements to it in an iterative way. We set the tolerance for the difference between measured and calculated couplings to 1 Hz. Although we only measure the new spinspin couplings and chains when the electron is in the m_{s} = −1 state, we can assume this is within tolerance to the average value of the coupling if the perpendicular hyperfine component is small (<10 kHz)^{18}. The spin positions are restricted by the diamond lattice. Spins that belong to the same chain are always added in the same iteration and up to 10000 possible configurations are kept. Chains starting from different parts of the known cluster can be positioned in a parallel fashion if they share no spins, reducing computational time. For spins that are relatively far away from the NV, we also make use of the interaction with the electron spin, approximating the hyperfine shift Δ_{i} to be of dipolar form within a tolerance of 1 kHz (neglecting the Fermi contact term^{46}). For those cases, we model the electron spin as a point dipole with origin at the center of mass, as computed by density functional theory^{46}. If multiple solutions are found, we report the standard deviation of the possible solutions as a measure of the spatial uncertainty (see Supplementary Table 1).
Error model and fitting
Confidence intervals assume the measurement of the electron state is limited by photon shot noise. The shotnoiselimited model is propagated in an absolute sense, meaning the uncertainty on fit parameters is not rescaled to match the sample variance of the residuals after the fit. For all quoted numbers, the number between brackets indicates one standard deviation or error indicated by the fitting procedure. We calculate the error on the PSD according to ref. ^{48}, assuming normally distributed errors.
Data availability
All data underlying the study are available on the open 4TU data server under accession code: https://doi.org/10.4121/aba1cc840aea4cdc93ca68b0db38bd81.v1.
Code availability
Code used to operate the experiments is available on request.
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Acknowledgements
We thank V. V. Dobrovitski for useful discussions, and H.P. Bartling and S.J.H. Loenen for experimental assistance. This work is part of the research program NWAORC (NWA.1160.18.208 and NWA.1292.19.194), (partly) financed by the Dutch Research Council (NWO). This work was supported by the Dutch National Growth Fund (NGF), as part of the Quantum Delta NL program. This work was supported by the Netherlands Organisation for Scientific Research (NWO/OCW) through a Vidi grant. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 852410). This work was supported by the Netherlands Organization for Scientific Research (NWO/OCW), as part of the Quantum Software Consortium Program under Project 024.003.037/3368. S.A.B. and L.C.B. acknowledge support from the National Science Foundation under grant ECCS1842655, and from the Institute of International Education Graduate International Research Experiences (IIEGIRE) Scholarship. S.A.B. acknowledges support from an IBM PhD Fellowship.
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GLvdS, DK, and THT devised the experiments. GLvdS performed the experiments and collected the data. CEB, JR, SAB, LCB, MHA, and THT performed and analyzed preliminary experiments. GLvdS, DK, CEB, and JR prepared the experimental apparatus. GLvdS, DK, and THT analyzed the data. MM and DJT grew the diamond sample. GLvdS and THT wrote the manuscript with input from all authors. THT supervised the project.
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T.H. Taminiau, G.L. van de Stolpe, C.E. Bradley, J. Randall, and D.P. Kwiatkowski declare competing interests in the form of a Dutch patent application. Patent applicant: Technische Universiteit Delft. Name of inventor(s): Taminiau, Tim Hugo; van de Stolpe, Guido Luuk; Bradley, Conor Eliot; Randall, Joe; Kwiatkowski, Damian Patryk. Application number: NL2035279. Status of application: filed, waiting for search report. Covered aspects: Full content of manuscript. M.H. Abobeih, S.A. Breitweiser, L.C. Basset, M. Markham, and D.J. Twitchen declare no competing interests.
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van de Stolpe, G.L., Kwiatkowski, D.P., Bradley, C.E. et al. Mapping a 50spinqubit network through correlated sensing. Nat Commun 15, 2006 (2024). https://doi.org/10.1038/s41467024460754
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DOI: https://doi.org/10.1038/s41467024460754
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