A Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements

Reconstructing complex magnetization textures from nitrogen-vacancy (NV) magnetometry stray-field measurements presents a challenging inverse problem. In this work, we introduce a physics-informed method that addresses this by incorporating the full micromagnetic energy directly into the variational formulation. Built on a PyTorch backend, our forward model integrates an auto-differentiable finite-differences micromagnetic framework with FFT-based stray-field calculations and Fourier-space upward continuation. This enables efficient gradient-based optimization via the adjoint method and allows the sensor-sample distance to be treated as an optimization parameter. By doing so, we eliminate the experimental uncertainty arising from unknown NV implantation depths and surface oxidation layers. Validation on synthetic data demonstrates high-fidelity reconstruction of spin textures and precise sensor height estimation. Furthermore, when applied to NV measurements of the van der Waals ferromagnet $Fe_{3-x}GaTe_2$, the method reconstructs the previously unknown NV-sample distance and physically plausible magnetization textures, which accurately reproduce the experimental observations.

💡 Research Summary

The paper presents a physics‑informed variational framework for reconstructing complex magnetization textures from stray‑field measurements obtained with nitrogen‑vacancy (NV) center magnetometry. Traditional inverse‑problem approaches either rely on heuristic regularization (e.g., Tikhonov) or on deep‑learning models that require training data and separate data‑driven and physics‑driven stages. In contrast, the authors embed the full micromagnetic energy—comprising exchange, demagnetization, uniaxial anisotropy, and interfacial Dzyaloshinskii‑Moriya interaction (DMI)—directly into the loss functional as a physically meaningful regularizer. The loss is defined as

 J(m, d_NV) = L_data(m, d_NV) + λ̂ E_total(m),

where L_data is the mean absolute error between the simulated stray field (generated from a candidate magnetization m at sensor‑sample distance d_NV) and the measured field, and E_total is the total micromagnetic energy. The regularization weight λ̂ is chosen via an L‑curve analysis, which balances data fidelity against physical plausibility.

The forward model is fully differentiable and implemented in PyTorch. Magnetization is discretized on a 1 nm × 1 nm × 100 nm grid using the open‑source finite‑difference library magnum.np. Stray‑field calculation employs FFT‑based convolution with the demagnetization tensor, yielding O(N log N) scaling. Because the magnetic field above the sample satisfies Laplace’s equation, the authors derive an exact 2‑D Fourier‑space transfer function that enables upward continuation from the simulation plane to any sensor height d_NV. After inverse FFT, the field is projected onto the NV axis to produce a scalar map directly comparable to experiment. This pipeline allows simultaneous gradient‑based optimization of both m and d_NV via automatic differentiation.

Optimization proceeds with a Riemannian Adam optimizer that enforces the unit‑norm constraint |m| = 1 on the magnetization manifold, avoiding the need for additional penalty terms. The magnetization is initialized to zero, ensuring that the solution is driven solely by the measured data and the physics regularizer. The algorithm iterates forward evaluation, loss computation, back‑propagation of gradients ∇_m J and ∇_d J, and manifold‑aware updates until convergence.

Validation on synthetic data uses a ground‑truth magnetization with a known sensor height of 80 nm and adds Gaussian noise at 3 % of the RMS signal. The L‑curve identifies an optimal λ_opt ≈ 2.2 × 10¹⁷ A·J⁻¹·m⁻¹. With this regularization the reconstructed magnetization matches the reference texture closely, reproducing domain‑wall chirality and Néel‑type structure, while the estimated sensor height converges to the true 80 nm. In the absence of regularization (λ = 0) the solution becomes high‑energy and physically implausible; excessive regularization pushes the estimated height upward, illustrating the importance of the L‑curve selection.

The method is then applied to experimental NV magnetometry of a 100 nm‑thick Fe₃₋ₓGaTe₂ van‑der‑Waals ferromagnet measured under a 3.2 kA m⁻¹ bias field. ODMR sign ambiguity introduces artifacts where the stray field exceeds the bias. Using the same framework, the L‑curve yields λ_opt ≈ 2.8 × 10¹⁷ A·J⁻¹·m⁻¹ and an optimized sensor height of ≈ 80 nm. Low λ values produce overly fine, noisy textures, while high λ values overly smooth the reconstruction. At λ_opt the reconstructed magnetization faithfully reproduces the observed domain patterns, DMI‑induced chirality, and matches the measured stray field with minimal residual error.

Key contributions of the work are:

1. Integration of full micromagnetic energy as a physics‑based regularizer, eliminating the need for ad‑hoc mathematical penalties.
2. Joint estimation of the sensor‑sample distance, removing a major source of experimental uncertainty.
3. A fully differentiable, FFT‑accelerated forward model that scales efficiently to large 2‑D/3‑D datasets.
4. Use of Riemannian optimization to enforce magnetization normalization without additional constraints.
5. Demonstrated high‑fidelity reconstruction on both synthetic and real data, establishing a new benchmark for quantitative NV magnetometry.

The authors argue that this approach can be generalized to other quantum‑sensor modalities, multi‑physics inverse problems, and potentially to real‑time spin‑dynamics imaging, offering a versatile, data‑efficient pathway to extract physically consistent magnetic information from nanoscale field measurements.