Universal Reconstruction of Complex Magnetic Profiles with Minimum Prior Assumptions

TL;DR

The paper tackles the inverse problem of reconstructing 2D magnetization maps from measured vector magnetic fields under noise and experimental uncertainty. It introduces a GPU-accelerated inversion with a forward dipole-moment model where the measured field obeys the linear relation $B = A m$, and a physics-informed loss that combines data misfit, smoothness, magnitude regularization, and a topological term to steer toward physically meaningful solutions. It is validated on simulated spin textures (skyrmions, merons, multi-domain ferromagnets) and on experimental NV-based measurements of a lunar basalt and a twisted CrI3 moiré, showing accurate reconstruction and parameter inference for experimental geometry. The approach offers a versatile, fast tool for quantum sensing and magnetic imaging, enabling universal reconstruction of complex magnetization profiles across materials and geological samples.

Abstract

Understanding intricate magnetic structures in materials is essential for advancing materials science, spintronics, and geology. Recent developments of quantum-enabled magnetometers, such as nitrogen-vacancy (NV) centers in diamond, have enabled direct imaging of magnetic field distributions across a wide range of magnetic profiles. However, reconstructing the magnetization from an experimentally measured magnetic field map is a complex inverse problem, further complicated by measurement noise, finite spatial resolution, and variations in sample-to-sensor distance. In this work, we present a novel and efficient GPU-accelerated method for reconstructing spatially varying magnetization density from measured magnetic fields with minimal prior assumptions. We validate our method by simulating diverse magnetic structures under realistic experimental conditions, including multi-domain ferromagnetism and magnetic spin textures such as skyrmion, anti-skyrmion, and meron. Experimentally, we reconstruct the magnetization of a micrometer-scale Apollo lunar mare basalt (sample 10003,184) and a nanometer-scale twisted double-trilayer CrI3. The basalt exhibits soft ferromagnetic domains consistent with previous paleomagnetic studies, whereas the CrI3 system reveals a well-defined hexagonal magnetic Moire superlattice. Our approach provides a versatile and universal tool for investigating complex magnetization profiles, paving the way for future quantum sensing experiments.

Figures (7)

• Figure 1: (a) Schematics of calculating magnetic fields on grid $(m,n)$generated by spins in allocated region. We discretize the magnetization map into a 2D grid with additional rearrangement of the source region into an array of dipole moment vectors $\vec{\mathbf{m}}_p$ with a single index$\mathbf{p} = (p_1, p_2, \ldots, p_N)$, and the magnetic field is obtained by summing the contributions from each element of $\mathbf{p}$. The green vector denotes the relative position of the origin of the magnetization map shifted from magnetic field map. The two maps are assumed to be parallel. (b) A demonstration of the complete reconstruction process is as follows: the green sections represent the experimental procedures, the red sections represent our customizable loss function, and the blue sections represent the computational procedures. Any quantum sensor capable of measuring 2D magnetic field maps would be suitable for the design. The illustration on the left side of the figure shows a quantum diamond magnetometer as an example. The spin textures displayed on the right side of the figure illustrate a Néel-type anti-skyrmion at the top and a skyrmion at the bottom.
• Figure 2: Case study of Néel-type skyrmion. (a) Schematic of the computational experiment setup, where $d$ is the sample-to-sensor distance, $r$ the skyrmion radius, and $w$ the full width at half maximum (FWHM) of the 2D Gaussian convolution modeling the experimental resolution. All parameters are expressed in dimensionless values normalized to pixel length. The skyrmion radius is fixed at $r = 8$ throughout this study. (b) Comparison of simulated and reconstructed magnetic field maps at small and large sample-to-sensor distances $d$. All maps use the same color scale and dimensions ($144 \times 144$ pixels). (c) Demonstration of the error calculation. Here, the reconstructed magnetization is obtained with parameters $d = 5$, $w = 2$, and SNR $= 5$. (d) Color map showing reconstruction error across different combinations of spatial resolution ($w$) and sample-to-sensor distance ($d$). (e) Visualization of parameters optimization with the shift vector $\vec{s}$, where each point represents a single optimization run, and the star denotes the target shift vector.
• Figure 3: Comparison between target and reconstructed magnetization for various spin configurations. The magnetic field was measured at $d=5$ with an optical resolution $w=2$ and an SNR of $5$ and shown below the magnetization. The arrow length is used to represent the magnitude of the dipole moment. (a) Ferromagnetic regions, each with distinct randomly assigned uniform magnetization, separated by a distance of approximately 10. (b$\sim$d) Topological charge reconstruction for samples with a radius of 8. The color bar highlights the z component, providing a clearer representation of the topological charge structures. (b) Néel-type skyrmion with topological charge $Q=1$. (c) Meron with $Q=1/2$. (d) Anti-skyrmion with $Q=-1$.
• Figure 4: (a) Optical microscope image of the moon rock sample 10003,184, with the yellow frame highlighting the region selected for subsequent SEM imaging. (b) Backscatterd SEM image showing the precise locations of potential magnetic rocks, visible as bright phases. (c) Infrared (IR) image acquired using the wide-field NV setup, covering the area where magnetic field measurements were performed. The IR image is overlaid on the SEM image, with the circled spots corresponding to the white spots in the SEM image, indicating the locations of magnetic sources (assigned magnetization region). (d) Comparison of experimental magnetic field data with reconstructed magnetic field components, displaying $B_x$, $B_y$, and $B_z$ from left to right. Measurements were conducted at room temperature with an external magnetic field applied in the direction of $(\text{-}1.07, \text{-}0.74, 0.65)$ mT. (e) $\cos{\varphi}$ map showing the alignment between the reconstructed magnetization vector and the expected direction (aligned with the applied field $B_\text{ext}$). As shown in the inset, $\varphi$ is defined as the angle between reconstructed magnetization and the applied external magnetic field at each pixel.
• Figure 5: Magnetization reconstruction of 2D van der Waals material ($\mathrm{CrI}_{3}$). (a) Magnetic field map along NV axis obtained using scanning NV magnetometer song2021moire. Three-dimensional map is generated using method described in Appendix A lima2009obtaining. (b) Reconstructed surface magnetization. (c) Three-dimensional auto-correlation map of the magnetization with lattice sites highlighted in black dots. (d) Illustration of two $\mathrm{CrI}_{3}$ layers twisted at a small angle generate a periodic stacking.