Limits of absolute vector magnetometry with NV centers in diamond

TL;DR

This work tackles absolute vector magnetometry with NV centers by deriving exact, analytic formulas that map between the magnetic-field vector $\mathbf{B}$ and spin-resonance frequencies across the four NV axes, enabling rapid field reconstruction with minimal computational cost. It presents a depressed-cubic framework for the NV Hamiltonian, yielding Viète-based solutions for resonance frequencies from $\mathbf{B}$ and closed-form expressions to infer $\mathcal{B}$ and $\theta$ from measured resonances, while also reconstructing the full field using a BLUE estimator across four NV axes and exploiting $O_h$ symmetry. The study highlights the dominant role of the NV gyromagnetic ratio uncertainty $\gamma_{\text{NV}}$ (and associated $g_{\text{NV}}$) in absolute-field accuracy—often at the $\mu$T level at practical fields—plus systematic errors from slight misalignment, and shows that exact formulas substantially reduce computational time compared to optimization-based approaches. It also demonstrates that using a Voigt profile to fit ODMR spectra yields more accurate linewidths and higher-fidelity sensitivities than Gaussian or Lorentzian fits, enabling instantaneous diagnostics of power broadening from a single spectrum. Together, these results provide a practical, fast, and accurate framework for absolute NV-based vector magnetometry with openly available data and software.

Abstract

The nitrogen-vacancy (NV) center in diamond has become a widely used platform for quantum sensing. The four NV axes in mono-crystalline diamond specifically allow for vector magnetometry, with magnetic-field sensitivities reaching down to $\mathrm{fT}/ \sqrt{\mathrm{Hz}}$. The current literature primarily focuses on improving the precision of NV-based magnetometers. Here, we study the experimental accuracy of determining the magnetic field from measured spin-resonance frequencies via solving the NV Hamiltonian. We derive exact, analytical, and fast-to-compute formulas for calculating resonance frequencies from a known magnetic-field vector, and vice versa, formulas for calculating the magnetic-field vector from measured resonance frequencies. Additionally, the accuracy of often-used approximations is assessed. Finally, we promote using the Voigt profile as a fit model to determine the linewidth of measured resonances accurately. An open-source Python package accompanies our analysis.

Figures (8)

• Figure 1: Zeeman splitting of the triplet ground states of four NV axes in a mono-crystalline diamond measured from ODMR spectra (color bar) and calculated from the analytical solution of Equation (\ref{['equ:viete']}) (white dashed lines). The splitting is a function of (A) the total magnetic-field value and (B) the rotation of the magnetic-field vector. The relations between the spin-resonance frequencies of different NV axes allow for the determination of the vector components of the magnetic field. Slight deviations between the theory prediction and the measured resonances in (B) are due to alignment errors of the coil system used to generate the magnetic fields in our experiment setup. Appendix \ref{['sec:field_calcs']} contains a detailed discussion about calculating the theory curves.
• Figure 2: Visualization of the angles between magnetic-field vector and NV axes. (A) For a given magnetic-field vector (red line), the spin-resonance frequencies of the NV axis (black arrow) allow for the calculation of the angle between the two. This is akin to two cones (blue surfaces) oriented along the NV axis, with the magnetic-field vector along one side of the two cones. (B) When the resonance frequencies of four NV axes are known precisely, the cones (colored surfaces) will intersect along the magnetic-field vector (red line). However, a real-world measurement will inevitably lead to discrepancies, and the cones will not overlap perfectly. We solve this problem by computing the best linear unbiased estimator instead of the intersection in Equation (\ref{['equ:bfield']}). The inversion symmetry of the NV axes leads to the additional problem that in the calculation, one or more cones might be oriented in the opposite direction so that no intersection exists. This problem is solved by considering the sum of squared residuals in Equation (\ref{['equ:ssr']}).
• Figure 3: Relative systematic error of the approximation that the magnetic field is aligned with the NV axis in Equation (\ref{['equ:b_approx']}) when the NV axis is, in reality, slightly misaligned by an angle $\theta$. Even for small magnetic fields and small angles $\theta$, the relative error of the assumption is far more significant than the typical sensitivities of NV-based sensors. For example, when measuring an absolute magnetic field of $10mT$, at an angle of $1°$ to the NV axis, the relative error is $\approx 1.5µT$ and therefore larger than typical magnetic-field sensitivities.
• Figure 4: Benchmarks for computation time of (A) analytical approach presented here and (B) a typical numerical approach. DC readout bandwidths of NV magnetometers are in the range of $\approx 20kHz$, requiring computation times of the magnetic-field vector in the order of $50µs$. Our analytical approach outperforms a typical numerical approach and can determine the magnetic-field vector faster than the resonances are usually measured.
• Figure 5: An example of an ODMR resonance fitted with a Gaussian, a Lorentzian, and a Voigt profile. The Voigt profile achieves the highest $R^2$ value. Calculating ODMR sensitivities from these fit curves leads to significant differences, i.e., $\eta_\text{Voigt} = 2.65µT\per\sqrt{Hz}$, $\eta_\text{Lorentz} = 2.45\pm0.05µT\per\sqrt{Hz}$ and $\eta_\text{Gauss} = 3.05\pm0.08µT\per\sqrt{Hz}$. The Lorentz fit in this example overestimates the actual sensitivity by more than $8%$.