Mapping a 50-spin-qubit network through correlated sensing

Optically interfaced spin qubits associated to defects in solids provide a versatile platform for quantum simulation, quantum networks, and quantum sensing, with systems including diamond, silicon carbide, silicon, hexagonal boron nitride, and rare-earth ions. The defect electron spin offers high-fidelity control, optical initialization and readout, and a photonic interface, and can sense and control multiple surrounding nuclear spins. This ancillary nuclear-spin network enables quantum information processing and nanoscale magnetic resonance imaging, with applications ranging from many-body quantum simulations to quantum networks where nuclear spins act as quantum memories, for entanglement distillation, and for error correction. State-of-the-art experiments have mapped up to 27 nuclear spins. Mapping larger networks could enable currently intractable simulations, improve understanding of noise in spin-qubit registers, and advance nanoscale imaging of external spin systems. A key challenge is spectral crowding: finite linewidths cause overlapping nuclear-spin frequencies, creating ambiguity when assigning signals to individual spins and their interactions. Here, the authors develop correlated sensing sequences that measure both network connectivity and characteristic spin frequencies with high spectral resolution, and apply these to map a 50-nuclear-spin network comprising 1225 pairwise interactions around a single NV center in diamond.

Prior work established NV centers and other defect spins as platforms for quantum sensing, simulation, and networks, with demonstrated optical initialization/readout and control of nearby nuclear spins. Experiments have imaged nuclear-spin networks up to 27 spins using methods such as spin-echo double resonance (SEDOR), which isolates pairwise nuclear-nuclear couplings by applying simultaneous echo pulses at selected nuclear frequencies and mapping nuclear polarization to the electron spin. However, when nuclear-spin frequencies overlap due to finite T2-limited linewidths, pairwise SEDOR can become ambiguous, leading to overlapping signals that cannot be uniquely assigned to individual spins or specific interactions without additional assumptions. Spectral crowding increases for spins farther from the NV as hyperfine shifts decrease with distance (∼r^−3), increasing spectral density. Previous approaches were limited in accessing couplings within these crowded regions. The present work builds on these foundations by introducing correlated, concatenated double-resonance sequences that directly reveal multi-spin connectivity (spin chains) and a correlated electron–nuclear double resonance for high-resolution hyperfine shift measurements, thereby overcoming ambiguities from spectral overlap.

Overview: The study develops correlated double-resonance protocols that (1) sense and concatenate multi-spin chains via nuclear–nuclear interactions and (2) perform high-resolution electron–nuclear double resonance to measure individual nuclear hyperfine shifts (Δi) with T2-limited resolution. These methods are combined to map connectivity in spectrally crowded regions and reconstruct a 50-spin network around a single NV center in diamond at cryogenic temperature. Experimental system: A naturally occurring NV center in diamond (natural 13C abundance 1.1%) at 3.7–4 K is used. The NV electron spin serves as a quantum sensor and optical interface. A magnetic field Bz = 403.553 G is aligned along the NV axis. The electron spin is initialized and read out optically with high-fidelity single-shot readout. Nuclear spins (13C) exhibit T2* ≈ 5–10 ms and Hahn-echo T2 up to 0.77 s; for spins with Δ near the nuclear Larmor frequency, T2 can decrease due to instantaneous diffusion. Nuclear spins are polarized using dynamical nuclear polarization (PulsePol), with 500–10,000 repetitions depending on the experiment. Control: Microwave transitions ms=0↔−1 and ms=0↔+1 are driven at 1.746666 and 4.008650 GHz with Hermite-shaped pulses. RF pulses (error-function envelope) in the 400–500 kHz range address nuclear transitions. Dynamical decoupling (DD, DDRF) and SEDOR-type sequences are employed, using ‘xx’ or ‘yx’ rotation-axis conventions for π/2 pulses; π pulses are applied sequentially. Nuclear signals are read out via the electron spin with phase-sensitive DD/DDRF blocks; SWAP gates can reinitialize the readout nucleus to boost signal. Spin-network Hamiltonian: In an external field along z, the nuclear-spin network is described by H = Σi Δi Izi + Σi<j Cij Izi Izj, valid because hyperfine shifts typically dominate over nuclear–nuclear couplings (freezing flip-flops). Here Δi = ωL + Ai are the nuclear precession frequencies (ωL Larmor frequency, Ai hyperfine shift). Spin-chain sensing (concatenated nuclear–nuclear double resonance): The core protocol concatenates SEDOR blocks to traverse chains of coupled nuclei. Starting from a nuclear spin at frequency Δ1 identified and polarized via the electron, a SEDOR block with free-evolution time t12 (e.g., 50 ms) is used while sweeping an RF frequency RF2 to find a coupled partner at Δ2, yielding coupling C12 from t12 sweeps. To extend the chain, the phase of the first π/2 pulse (e.g., ‘yx’) is set and t12 tuned to 1/(2C12) to maximize state transfer from Spin 2 back through Spin 1 to the electron. An additional double-resonance block at RF2=Δ2 and RF3 explores couplings from Spin 2 (and so on), enabling chains up to N=5 spins in this work. The concatenation correlates a list {Δ1, C12, Δ2, C23, Δ3, ...}, directly capturing connectivity even when some Δi overlap spectrally. This approach also accesses spins in crowded regions by using intermediate strongly coupled links as local sensors. Importantly, the sequences are sensitive to both magnitude and sign of Cij. High-resolution electron–nuclear double resonance (correlated Δi spectroscopy): To overcome Δi resolution limited by nuclear T2* (~5 ms), the authors recouple the nuclear hyperfine shift with the electron while decoupling from quasi-static noise. Microwave pulses synchronously transfer the electron population between |−1⟩ and |+1⟩ during a nuclear echo-like block so that the nuclear spin accumulates phase from Δi while environmental dephasing is suppressed, extending coherence and yielding T2-limited Δi. Example: For a spin at Δ1, the method measures Δ1 = 14549.91(5) Hz with T2 = 0.36(2) s and linewidth ≈1.8 Hz, about 75× narrower than standard SEDOR spectroscopy in that band. 2D correlated spectroscopy in crowded bands: The electron–nuclear block can be embedded within a spin-chain sequence to perform high-resolution Δi labeling in chains containing multiple spectrally overlapping spins. A 2D experiment concatenates an electron–nuclear evolution t1 (to encode Ai) with a nuclear–nuclear SEDOR evolution t2 (to encode Cij), sweeping both t1 and t2. In a demonstration with two overlapping spins (2 and 3) both coupled to spins at Δ1 and Δ4, the 1D t1 sweep resolves Δ2 and Δ3; adding the t2 block yields a 2D PSD revealing correlations between Ai and specific couplings, allowing extraction of C24 and C34 despite overlap and an internal splitting due to C23. Network mapping and reconstruction: The overall mapping is formulated as a graph-search problem rooted at the NV electron. The protocol performs breadth-first-like exploration using chained SEDOR to discover vertices at new frequencies with single resolvable couplings, and merges chains that share overlapping sections. A CheckVertex routine uses additional chained and electron–nuclear measurements to disambiguate duplicates when Δi overlap. In total, 249 interactions (pairwise and chained) are measured to hypothesize the 50-spin connectivity. Spatial reconstruction then uses a positioning algorithm constrained by dipolar nuclear–nuclear couplings, measured coupling signs, and measured hyperfine shifts Ai to find a consistent 3D configuration, predicting the remaining unmeasured couplings. Practical details: Signal strength decreases with chain length due to finite polarization and decoherence of all spins in the chain; chain lengths are thus limited in practice. Electron-spin flipping between ms=±1 slightly shortens the observed nuclear T2 due to changes in nuclear quantization axes and residual coupling terms; multiple refocusing pulses and higher magnetic fields can further improve resolution for weakly coupled spins. Data processing includes undersampling and PSD estimation; fits use shot-noise-limited error propagation.

• Developed correlated, concatenated double-resonance sensing that directly measures multi-spin chains and high-resolution hyperfine shifts, overcoming spectral-crowding ambiguities.
• Mapped a 50-nuclear-spin 13C network around a single NV center in diamond, comprising 1225 pairwise nuclear–nuclear couplings. A total of 249 interactions were measured (pairwise and chained) to reconstruct connectivity; 976 additional weak couplings were predicted by spatial reconstruction.
• Demonstrated spin-chain sensing up to N=5 spins, enabling identification of connectivity and measurement of individual couplings (including signs) in spectrally crowded regions that are otherwise inaccessible from the electron.
• High-resolution electron–nuclear double resonance measured Δ1 = 14549.91(5) Hz with T2 = 0.36(2) s and linewidth ≈1.8 Hz, providing ~75× narrower spectral features than standard nuclear SEDOR spectroscopy in that band.
• 2D correlated spectroscopy resolved two overlapping spins at A2 and A3 with Δ2 = 8019.5(2) Hz, Δ3 = 7695.2(1) Hz, split by an internal coupling C23 ≈ 7.6(1) Hz, and determined their distinct couplings to a third spin: C24 = −11.8(2) Hz and C34 = −0.2(5) Hz.
• Combined connectivity (including coupling signs) and high-resolution Δi labeling enabled reliable merging of separately measured chains and validation via a dipolar 3D spatial reconstruction consistent with measured data.

The work addresses the core challenge of mapping large, interacting spin networks under spectral crowding, where pairwise SEDOR alone yields ambiguous assignments. By correlating multiple frequencies and couplings in single measurements through concatenated spin chains, the method directly reveals network connectivity, even when some spins are spectrally degenerate. The addition of electron–nuclear double resonance upgrades the frequency resolution from ~1/T2* to ~1/T2, allowing distinct labeling of spins within crowded bands. Together, these techniques enable access to interactions deeper in the network by using already identified spins as local sensors. The successful reconstruction of a 50-spin network, including validation through dipolar spatial modeling and prediction of many weak couplings, demonstrates the approach’s power and generality. These capabilities expand the number of usable qubits in solid-state registers, improve microscopic understanding of noise environments, and open avenues for high-resolution nano-MRI and complex-spin-system imaging beyond the host crystal.

Correlated double-resonance sensing—combining concatenated nuclear–nuclear spin-chain measurements with high-resolution electron–nuclear spectroscopy—enables unambiguous mapping of large, spectrally crowded spin networks. Applied to an NV center in diamond, the method reconstructed a 50-spin 13C network with 1225 couplings, resolved overlapping spins with T2-limited precision, and accessed interactions previously unreachable from the electron sensor alone. The approach is platform-agnostic and applicable to other spin systems, including electron–electron networks. Future directions include scaling to larger networks with machine-learning-enhanced protocols and adaptive sampling to reduce acquisition time, integrating with advanced control fields for universal control and readout, and leveraging the detailed network characterization for optimized gate design, tests of many-body and open-quantum-system models, and high-resolution nano-MRI of external samples.

• Chain length is limited by cumulative signal loss: finite nuclear polarization and decoherence across all spins in the chain reduce signal with each added link.
• Spectral resolution and T2 typically degrade for spins farther from the NV (increased crowding, instantaneous diffusion near the Larmor frequency), eventually limiting the number of uniquely identifiable spins.
• Electron spin flips between ms=±1 alter nuclear quantization axes and residual interactions, slightly reducing nuclear T2 in the high-resolution protocol; higher fields and additional refocusing may be required for weaker couplings.
• Some measurements rely on assumptions (e.g., dipolar coupling model, small perpendicular hyperfine components) for spatial reconstruction and prediction of unmeasured couplings.
• 2D spectra showed small discrepancies in extracted splittings across measurement modalities; limited resolution or parameter drifts can introduce uncertainties.
• The approach requires appreciable nuclear polarization (DNP or selective initialization), which adds experimental overhead and may not be equally effective for all spins.