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

arXiv:2602.17180 · cond-mat.mes-hall · Submitted 2026-02-19 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "A Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements".

Mira: The gist: Incorporating micromagnetic energy directly into an optimization framework allows for a transparent, end-to-end reconstruction of magnetization textures and critical experimental parameters like effective distance from nitrogen-vacancy measurements.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So to recap, this paper is tackling that ill-posed nature of reconstructing magnetization textures from NV magnetometry stray field measurements by directly embedding micromagnetic physics into the optimization engine >

Mira: The core thesis is that by minimizing a joint objective function that includes both data fidelity and the total micromagnetic energy, you can filter out the infinitely many unphysical solutions >

Lev: It's about solving the problem of distinct magnetizations reproducing nearly identical stray fields by adding this physical consistency constraint to the math >

Kai: The specific claim is that they use Fourier-space upward continuation of the stray field to solve for both the magnetization m and an auxiliary parameter, which is the effective distance d NV >

Mira: This allows them to simultaneously fit these two variables, and when applied to Fe three-x GaTe two they recovered a specific effective distance estimate of about eighty-one nm >

Lev: The paper shows that this approach works by minimizing the joint objective function J(m, d NV) = L two data plus lambda L energy(m), where L energy is the total micromagnetic energy normalized to a uniformly out-of-plane magnetized state > <ref:2602.17180#pg1>

Kai: And they use forward modeling with libraries like NeuralMag and magnum.np to create a differentiable map from m and d NV to the expected signal, which helps drive the optimization >

Mira: So why does this matter for us? Because it moves us away from just using post-hoc deconvolution methods like quantitative Magnetic Force Microscopy, which requires assuming an instrument calibration function >

Lev: Instead, it's about a self-consistent physics-informed reconstruction framework that minimizes J(m, d NV) over both the magnetization and the distance parameter >

Kai: And this has implications for any field dealing with magnetic inverse problems where you have noisy data and unknown geometries >

Conclusion: Kai: Looking at this paper, "A Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements," the authors Alexander Setescak, Florian Bruckner, Dieter Suess, Young-Gwan Choi, Hayden Binger, Lotte Boer, Chenhui Zhang, Hyunsoo Yang and Claire Donnelly are all working on how to actually build this reconstruction framework >

Mira: And what I see is that they're taking a notoriously ill-posed magnetic inverse problem and applying physics directly into the reconstruction optimization to make the result transparent rather than just fitting a curve >

Lev: The implication for hardware, like our superconducting qubits, is that if we can get this fidelity, we need to understand where the overhead really lands on real hardware when you try to run this complex reconstruction >

Kai: It suggests that even with noisy NV magnetometry data and unknown distances, if you couple the physics right into the optimization loop, you can recover physically plausible magnetization states and estimate material parameters like the effective distance >

Mira: And for condensed matter physics, it means we can get better characterization of magnetic materials by using these NV centers to probe them in a way that respects their internal energy landscape >

Lev: It's not just about getting a reconstruction; it's about figuring out how to make the reconstruction stable enough to run on real hardware, which requires understanding the computational complexity of this joint optimization for m and d NV >

Faculty of Physics, University of Vienna · University of Vienna Doctoral School in Physics Department at University of Vienna, Max Planck Institute for the Chemical Physics of Solids, Department of Physics at University of Ulsan, Department of Electrical and Computer Engineering at National University of Singapore

cond-mat.mes-hall

Submitted: 2026-02-19

Updated: 2026-07-29

Journal ref: Phys. Rev. Lett. 137, 166702 (2026)

DOI: 10.1103/nc5g-3yw9

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 82/100

The gist: The gist: Incorporating micromagnetic energy directly into an optimization framework allows for a transparent, end-to-end reconstruction of magnetization textures and critical experimental parameters

Key concepts

Ill-posed Inverse Problem: This is a mathematical challenge where you try to figure out an unknown cause (like magnetization) from incomplete or noisy effects (like a stray field). Because many different causes can produce the same effect, finding the single correct answer is fundamentally difficult without extra physical knowledge.
Micromagnetic Energy Functional: This is a mathematical formula that quantifies the total energy stored in a magnetic material based on its internal structure. It includes terms like exchange energy and anisotropy. By adding this to the reconstruction goal, the method ensures that only physically plausible magnetization patterns are accepted as solutions.
Fourier-Space Upward Continuation: This is a mathematical technique used to estimate data at a larger scale or distance than what was directly measured. In this context, it allows researchers to use the measured stray field data to infer information about the magnetic material's structure and its distance from the sensor.
Physics-Informed Reconstruction: This approach involves embedding known physical laws (like micromagnetics) directly into a mathematical optimization process. Instead of relying solely on statistical fitting, it uses the energy equations of magnetism to guide the reconstruction, leading to more accurate and interpretable results.

Terminology

Inverse Problem — 2× in this paper · explained in 2 episodes

Summary

The gist: Incorporating micromagnetic energy directly into an optimization framework allows for a transparent, end-to-end reconstruction of magnetization textures and critical experimental parameters like effective distance from nitrogen-vacancy measurements.

The Challenge of Inverse Problems

Reconstructing magnetization textures from stray field measurements is described as a challenging, fundamentally ill-posed inverse problem, further complicated by the unknown effective distance between sensor and magnetic material (Page 1). This ambiguity means that infinitely many magnetization configurations can produce the same stray field (Page 1). Traditional approaches have included methods like quantitative Magnetic Force Microscopy (qMFM) which determine an instrument calibration function to analytically deconvolve the stray field data (Page 1), and variational optimization tasks utilizing the adjoint method and Tikhonov regularization to reconstruct the 3D magnetization by minimizing the field residual via gradient-based optimization (Page 1).

Physics-Informed Reconstruction Framework

The proposed method introduces a self-consistent physicsinformed reconstruction framework that embeds micromagnetic physics directly into the optimization (Page 1). This framework minimizes a joint objective function, defined as J(m, dNV) = L2 data + λLenergy(m), where Ldata is the relative field error between modeled and measured stray field, and Lenergy is the total micromagnetic energy Etotal(m) normalized to a uniformly out-of-plane magnetized state (Page 1). The total energy includes contributions such as "the exchange energy (Eex), the demagnetization energy (Edem), the uniaxial anisotropy energy (Eaniu), the interfacial Dzyaloshinskii–Moriya interaction energy (Edmii), and the Zeeman energy (EZee) from the applied bias field Hbias" (Page 2).

Forward Modeling and Distance Estimation

The forward model provides a differentiable map from a given magnetization configuration m and the effective distance dNV to the expected signal (Page 2). This involves simulating what an NV magnetometer would detect, using micromagnetic calculations performed via libraries like NeuralMag or magnum.np (Page 2). The stray field in the Fourier domain is related to the surface field by a transfer function derived from Laplace's equation, which includes de-averaging over the cell thickness ∆z and upward continuation towards the effective distance dNV (Page 8). This allows for gradient-based optimization not only for the magnetization field m but also for auxiliary parameters such as the effective distance dNV (Page 2).

Optimization and Results

The optimization is performed using automatic differentiation to minimize J(m, dNV) jointly over m and dNV, where the unit-norm constraint m = 1 is enforced by construction (Page 12). The regularization parameter λ controls the tradeoff between data fidelity and physical stability, visualized via the L-Curve method (Page 3). Applying this to Fe3−xGaTe2 measurements recovered an effective distance estimate of approximately 81 nm and low-energy configurations that reproduce the observed field (Page 1). The convergence of d∗NV to approximately 81 nm at λbal ≈ 25 was a key result (Page 4).

Robustness and Limitations

The framework demonstrates robustness, as the joint optimization of dNV is insensitive to its initialization when starting from four different initial guesses (Page 20). However, limitations exist; certain magnetization configurations are inherently invisible to any H-based method, and the reconstruction is only as good as the assumed model parameters, with a ±20 % uncertainty in Ms propagating into a 10 % to 14 % shift in d∗NV (Page 6). Furthermore, the analysis of ODMR sign ambiguity artefacts shows that masking can suppress spurious features and that the reconstructed m∗ is not a true micromagnetic equilibrium (Page 15). The final reconstruction is ranked by the (Ldata,Lenergy) phase plot, preferring the lower-energy configuration at equal data fit (Page 21).

Physical Composition of Distance

The effective distance dNV is composed of four contributions: the NV implantation depth in the diamond, the surface oxidation layer of the sample, and the physical sample–diamond separation (Page 22). These are estimated from fabrication parameters, with estimates for implantation depth ranging from 10 to 20 nm and oxidation layer thickness between 10 to 30 nm (Page 22). The final extracted values for comparable probes fall within the range of approximately 81 nm (Page 22).

References

[1] F. Casola, T. van der Sar, and A. Yacoby, Probing condensed matter physics with magnetometry based on nitrogen-vacancy centres in diamond, Nature Reviews Materials 3, 17088 (2018) (Page 7).

[2] Y. Xu, W. Zhang, and C. Tian, Recent advances on applications of NV- magnetometry in condensed matter physics, Photonics Research 11, 393 (2023) (Page 7).

[3] G. E. Backus, Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, Journal of Geophysical Research (1896-1977) 75, 6339 (1970), eprint: https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA075i031p06339 (Page 7).

[4] F. Bruckner, C. Abert, G. Wautischer, C. Huber, C Vogler, M Hinze, and D Suess, Solving LargeScale Inverse Magnetostatic Problems using the Adjoint Method, Scientific Reports 7, 40816 (2017) (Page 1).

[5] C. B Clement, J. P Sethna, and K C Nowack, Reconstruction of Current Densities from Magnetic Images by Bayesian Inference (2021), arXiv:1910.12929 [physics] (Page 7).

[6] H J Hug, B Stiefel, P J A van Schendel, A Moser, R Hofer, S Martin, H-J Güntherodt, S Porthun, L Abelmann, J C Lodder, G Bochi and R C O’Handley Quantitative magnetic force microscopy on perpendicularly magnetized samples Journal of Applied Physics 83 5609 (1998) (Page 7).

[7] Y Feng, A-O Mandru, O Yıldırım, and H Hug Quantitative Magnetic Force Microscopy: Transfer-Function Method Revisited Physical Review Applied 18 024016 (2022) (Page 7).

[8] C Yao, Y Yu, Y Shi, J-I Jung, Z Vaci, Y Wang, Z Liu, C Zhang, S Tikoo-Schantz and C Zu Universal reconstruction of complex magnetic profiles with minimal prior assumptions Physical Review Applied 24 064020 (2025) (Page 7).

[9] D A Broadway, M Flaks, A E Dubois, and P Maletinsky Reconstruction of nontrivial magnetization textures from magnetic field images using neural networks Physical Review Applied 23 044012 (2025) (Page 7).

[10] A Dubois, D Broadway, A Stark, M Tschudin, A Healey, S Huber, J-P Tetienne, E Greplova and P Maletinsky Untrained Physically Informed Neural Network for Image Reconstruction of Magnetic Field Sources Physical Review Applied 18 064076 (2022) (Page 7).

[11] D Suess, A Setescak, J Smith, F Bruckner, H J Hug, and C Abert Reconstruction of magnetic structures and material parameters with convolutional neural network and bias field-constrained micromagnetic relaxation Scientific Reports 15 42867 (2025) (Page 1).

[12] P C Hansen and D P O’Leary The Use of the LCurve in the Regularization of Discrete Ill-Posed Problems SIAM Journal on Scientific Computing 14 1487 (1993) (Page 7).

[13] D Calvetti, S Morigi, L Reichel, and F Sgallari Tikhonov regularization and the L-curve for large discrete ill-posed problems Journal of Computational and Applied Mathematics Numerical Analysis 2000. Vol III: Linear Algebra, 123, 423 (Page 7).

[14] Z Xu, M L Palm, W Huxter, K Herb, J M Abendroth K Bouzehouane O Boulle M Gabor J Urrestarazu Larranaga A Morales J Rhensius G PueblaHellmann and C L Degen Minimizing Sensor-Sample Distances in Scanning Nitrogen-Vacancy Magnetometry ACS Nano 19 8255 (2025) (Page 7).

[15] C Abert, F Bruckner, A Voronov, M Lang, S A Pathak, S Holt, R Kraft, R Allayarov P Flauger S Koraltan T Schrefl A Chumak H Fangohr and D Suess NeuralMag an open-source nodal finitedifference code for inverse micromagnetics npj Computational Materials 11 193 (2025) (Page 8).

[16]

Improvements for AI systems

A physics-informed reconstruction framework can simultaneously reconstruct magnetization textures and critical experimental parameters like effective distance, as it embeds micromagnetic physics directly into the optimization. This allows for a "transparent, end-to-end optimization process that requires no training data and provides quantitative, model-constrained reconstruction of nanoscale magnetic textures along with critical experimental parameters, in particular the effective distance dNV."
The system can robustly handle measurement artifacts through a masking strategy that suppresses these spurious features and recovers a smoother, more physically plausible texture, as demonstrated by excluding artifact-prone pixels from the data-fidelity term. This is crucial for noisy measurements like those exhibiting ODMR sign-ambiguity artefacts.
The AI system can perform uncertainty quantification and parameter sensitivity analysis by quantifying how uncertainties in model parameters propagate into the results, noting that a ±20 % variation in Ms shifts dNV by 10 %–14 %. This provides a rigorous assessment of the reconstruction's reliability based on physical assumptions.
The system can achieve robust inference across different initialization strategies, as demonstrated by comparing reconstructions starting from m = 0, uniform out-of-plane (mz = −1), and an informed initialization in which positive out-of-plane domains are placed at peaks detected in the measured field H meas.
The system can provide a physically grounded selection criterion for candidate solutions by utilizing the L-Curve method, where it uses the principle: at equal data fit, prefer the lowest-energy configuration. This ensures that reconstructions are not only data-consistent but also energetically favorable within a physical context.

Abstract

Reconstructing magnetization textures from nitrogen-vacancy (NV) magnetometry stray-field measurements is a challenging, fundamentally ill-posed inverse problem, further complicated by the unknown effective distance between sensor and magnetic material. Here we show that incorporating a micromagnetic energy functional directly into the inversion filters out unphysical, high-energy configurations, while Fourier-space upward continuation of the stray field allows us to simultaneously fit the distance. Applied to measurements of the van der Waals ferromagnet Fe 3-x GaTe 2, it recovers an effective distance estimate of approximately 81nm and low-energy configurations that reproduce the observed field. More broadly, embedding physics directly into the reconstruction turns ill-posed magnetic inverse problems into transparent, interpretable reconstructions, with applicability well beyond NV magnetometry.