Three-dimensional magnetic resonance tomography with sub-10 nanometer resolution

The study addresses the challenge of extending nanoscale spin detection to true three-dimensional imaging with nanometer resolution. While MRFM and NV centers have enabled detection of small ensembles and even single spins, translating this capability into scalable 3D imaging at the nanoscale has remained elusive. Applying magnetic field gradients can encode spatial information, and Fourier-accelerated methods promise efficient acquisition of large volumetric datasets, akin to clinical MRI. However, prior nanoscale demonstrations were limited to 1D/2D or used static or intrinsic gradients that could not be switched within spectroscopy sequences, restricting scalability and accessible volumes. The purpose of this work is to realize Fourier-accelerated 3D magnetic resonance tomography at the nanoscale using switchable, linearly independent gradients to image individual NV centers in diamond, achieving sub-10 nm resolution and enabling practical paths toward 3D structure determination of spin systems.

Prior work established nanoscale spin detection via MRFM and NV centers, including detection of individual electron and nuclear spins. Imaging efforts included 3D localization using intrinsic dipolar fields of color centers, which confine imaging to a few nanometers around a defect; 1D and 2D imaging with engineered gradients; and 3D imaging of intrinsic spins with static gradients where switching within a spectroscopy sequence was not possible. Fourier-accelerated imaging with switchable gradients was shown conceptually and in lower-dimensional or lower-resolution settings, but nanoscale 3D Fourier imaging remained unachieved. Alternative gradient sources (e.g., hard-drive wire heads) enabled 2D demonstrations. Compressed sensing has been explored to reduce acquisition load in nanoscale Fourier magnetic imaging by exploiting signal sparsity. This work advances beyond these by demonstrating fully 3D, Fourier-accelerated nanoscale tomography with sub-10 nm resolution and introduces an aliasing-based compressed sensing approach that requires no iterative reconstruction.

Device and gradients: Three microfabricated gold wires arranged in a U-shaped structure are lithographically patterned (lift-off photolithography) on a diamond substrate hosting a densely doped ensemble of NV centers (Element Six General Grade CVD diamond, NV density ≈1.3 ppb). The U-structure consists of a ~200 nm gold film. When driven with currents, the three wires generate linearly independent magnetic field gradients in a region a few micrometers beneath the structure, enabling 3D spatial encoding despite the 2D wire layout. Imaging is performed in the lateral center of the U and at depths 2–6 μm below the diamond surface. Pulse sequences and k-space encoding: A Hahn echo sequence is used with a trailing π/2 pulse for readout. Rectangular current pulses on each of the three wires are applied sequentially within the echo sequence to impart position-dependent Larmor phase shifts. The time-domain signal constitutes k-space; a 3D inverse Fourier transform maps it to real-space, yielding a 3D image of NV positions. Only NVs within the optical confocal volume contribute to the signal. Electronics and pulse linearity: To approximate ideal rectangular gradient pulses, a stable voltage source (Keithley 2230-30-5) is rapidly switched using fast switches (ic-Haus HGP), providing high pulse stability and fast edges (targeting ≈100 MHz bandwidth). Residual nonlinearities and shot-to-shot fluctuations are corrected by measuring each pulse (hardware integration) on a fast A/D converter (M441-45) and applying online post-processing to compute an effective pulse and correct the time-domain phase evolution, suppressing drift and improving coherence. Signal model and reconstruction: For an ensemble of NVs, the measured spin projection Sz(t) is a linear superposition of oscillatory components whose frequencies encode spatial positions relative to the gradient fields. In 1D, inverse FFT yields line images; in 3D, a 3D inverse FFT of the acquired k-space yields a 3D point distribution. Gradient non-orthogonality introduces a known distortion that can be corrected numerically by inverting the gradient mapping if desired; even without correction, the raw FFT provides a valid 3D image in gradient coordinates. Compressed sensing via aliasing (Fourier zooming): To reduce the large sampling burden in 3D k-space (e.g., up to ~10^6 points), the experiment implements equidistant undersampling to induce controlled aliasing. When the true signal occupies a narrow frequency band (region of interest), choosing an undersampled Nyquist frequency just large enough to cover the bandwidth shifts the spectrum (by aliasing) near zero frequency without distortion, effectively zooming into the ROI and reducing acquisition time. This approach enables direct visual interpretation via FFT without iterative optimization (e.g., L1 minimization), while still allowing comparison to conventional sparse recovery if desired. Resolution considerations: Spatial resolution is set by the gradient magnitude and spectral (frequency) resolution of the time-domain acquisition, the latter given by the inverse of the acquisition window length. The experiment examines decoherence and stability limits, employs pulse-by-pulse integration and correction to suppress current noise, and discusses potential spatial drift of current paths due to heating in the conductors.

- Demonstrated 3D Fourier-accelerated magnetic resonance tomography of NV centers with spatial resolution down to 5.9 ± 0.1 nm. - Realized three linearly independent, switchable magnetic field gradients using a microfabricated U-shaped, three-wire gold structure on diamond, enabling true 3D imaging from a 2D layout. - Produced 3D images of individual NV centers within the confocal volume via a 3D inverse Fourier transform of time-domain (k-space) data; for the given NV density, ~5–15 NVs are expected within the imaged volume. - Implemented an aliasing-based compressed sensing scheme (Fourier zooming) that directly yields interpretable images via FFT and reduced acquisition time by approximately a factor of 10 in measured 2D demonstrations, consistent with simulations showing controlled aliasing without distortion when the bandwidth is within the undersampled Nyquist limit. - Managed large k-space datasets (up to ~10^6 points) and showed that undersampling can target narrow regions of interest effectively without iterative reconstruction. - Characterized coherence and resolution limits: example analysis shows a time-domain decay with timescale ~10 μs and a fitted timescale T2,1 = 8.64 ± 0.1 μs; identified current stability and spatial drift of current paths (likely due to local heating) as dominant resolution-limiters, varying between wires, rather than purely electrical fluctuations. - The achieved resolution compares favorably with leading super-resolution optical methods, approaching the positioning accuracy of site-directed spin labeling and enabling rigid distance constraints beyond 80 Å for electron-spin-labeled proteins.

The results answer the central question of whether nanoscale, switchable-gradient, Fourier-accelerated 3D magnetic resonance imaging is feasible: the demonstrated 5.9 nm resolution and clear 3D localization of individual NVs confirm it is. Using a compact microfabricated U-wire device to create three independent gradients enables 3D encoding beneath a 2D layout, solving a key hardware bottleneck. The aliasing-based compressed sensing approach further addresses the data-volume challenge by exploiting prior knowledge of a limited field of view to reduce acquisition without sacrificing interpretability. The findings are significant for several fields. In quantum information with solid-state spins, super-resolved 3D addressing allows selective manipulation and readout within dense NV ensembles, supporting construction of quantum registers and advanced sensing modalities. For materials and particle detection concepts, 3D mapping of defect distributions at the nanoscale becomes practical. In bioimaging, applying similar gradient-based tomography to electron-spin-labeled proteins could provide direct real-space 3D constraints over >80 Å distances, complementing and extending electron spin resonance techniques. Compared to optical super-resolution, the achieved resolution outperforms common localization microscopy methods (PALM, STORM) and approaches the domain where only specialized techniques (e.g., MINFLUX, COLD) claim superior performance, though those methods have limited adoption and potential pitfalls. The ability to generalize the technique to ‘dark’ spins outside diamond, with a single NV acting solely as a detector while the U-wire device performs the imaging, broadens its relevance.

This work establishes Fourier-accelerated 3D magnetic resonance tomography at the nanoscale using switchable, microfabricated gradient sources, achieving sub-10 nm (5.9 ± 0.1 nm) resolution and direct 3D imaging of individual NV centers. It introduces a simple, non-iterative compressed sensing strategy based on controlled aliasing that effectively zooms into regions of interest and reduces acquisition time. Together, these advances provide a practical route to 3D super-resolution imaging of spin qubits and lay groundwork for applications ranging from quantum registers and particle detection to prospective single-molecule MRI of spin-labeled proteins. Future directions include shrinking the gradient structure by an order of magnitude to target angstrom-scale resolution, extending the method to dark spins external to diamond using a single NV as a sensor, and identifying fluorescent molecules with optically readable electron spins to apply the technique directly in broader systems.

- Gradient axes are linearly independent but not fully orthogonal, causing geometric distortion in the raw 3D FFT image; accurate correction requires calculating and inverting the gradient mapping. - Resolution is limited by decoherence and stability: shot-to-shot current fluctuations and especially spatial drift of current paths (likely from local heating) degrade effective T2* and differ between wires. - The approach requires strict pulse rectangularity and fast, stable switching; residual electronic noise and waveform nonidealities can blur frequency-domain images. - Acquisition of dense 3D k-space can be memory- and time-intensive (up to ~10^6 samples); while aliasing-based undersampling mitigates this, it relies on prior knowledge that the signal bandwidth fits within the undersampled Nyquist limit. - Imaging is confined to NVs within the optical confocal volume; thus, field of view is limited by optics and NV distribution. - Thermal/mechanical drift (e.g., due to heating) in the device or sample can affect repeatability and image sharpness.