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Extended Kalman Smoothing of Free Spin Precession Signals for Accurate Magnetic Field Determination

Jasper Riebesehl, Lutz Mertenskötter, Wiebke Pohlandt, Wilhelm Stannat, Wolfgang Kilian

Abstract

We present a novel application of the Extended Kalman Smoother (EKS) for highly accurate frequency estimation from free spin precession signals of polarized $^3$He. Traditional approaches often rely on nonlinear least-squares fitting, which can suffer from limited robustness to signal decay and time-dependent frequency shifts. By contrast, our EKS-based method captures both amplitude and frequency variations with minimal tuning, adapting automatically to fluctuations via an expectation-maximization algorithm. We benchmark the technique in extensive simulations that emulate realistic spin precession signals with exponentially decaying amplitudes and noisy frequency drifts. Compared to least-squares fits with fixed block lengths, EKS systematically reduces estimation errors, particularly when frequencies evolve or signal-to-noise ratios are moderate to high. We further validate these findings with experimental data from a free-precession decay $^3$He magnetometer. Our results indicate that EKS-based analysis can substantially improve precision in nuclear magnetic resonance-based magnetometry, where accurate frequency estimation underpins absolute field determinations. This versatile approach promises to enhance the stability and accuracy of future highly accurate measurements.

Extended Kalman Smoothing of Free Spin Precession Signals for Accurate Magnetic Field Determination

Abstract

We present a novel application of the Extended Kalman Smoother (EKS) for highly accurate frequency estimation from free spin precession signals of polarized He. Traditional approaches often rely on nonlinear least-squares fitting, which can suffer from limited robustness to signal decay and time-dependent frequency shifts. By contrast, our EKS-based method captures both amplitude and frequency variations with minimal tuning, adapting automatically to fluctuations via an expectation-maximization algorithm. We benchmark the technique in extensive simulations that emulate realistic spin precession signals with exponentially decaying amplitudes and noisy frequency drifts. Compared to least-squares fits with fixed block lengths, EKS systematically reduces estimation errors, particularly when frequencies evolve or signal-to-noise ratios are moderate to high. We further validate these findings with experimental data from a free-precession decay He magnetometer. Our results indicate that EKS-based analysis can substantially improve precision in nuclear magnetic resonance-based magnetometry, where accurate frequency estimation underpins absolute field determinations. This versatile approach promises to enhance the stability and accuracy of future highly accurate measurements.

Paper Structure

This paper contains 10 sections, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spin precession magnetometry
  3. Frequency estimation methods
  4. The Kalman smoother
  5. The model
  6. Least Square Analysis as Reference Method
  7. Comparing the methods on simulated data
  8. Results
  9. Application to experimental data
  10. Conclusion

Figures (4)

  • Figure 1: EKS tracking of toy example with time-dependent frequency variation chosen to illustrate the variance/stiffness trade-off. The true frequency (black dashed line) is followed by a model with artificially large $Q = 200\, Q_{opt}$ (blue) even when the frequency changes rapidly in time. Meanwhile, a model with artificially small $Q= \frac{1}{200}\, Q_{opt}$ (red) exhibits a maximum slew rate that is at some point exceeded by the rate of change of the frequency. Conversely, where the frequency is constant, the model with small $Q$ has much smaller variance and outperforms the model with large $Q$. Here $Q_{opt}$ is the optimal covariance as determined by the algorithm detailed in the supplementary material.
  • Figure 2: Quantitative comparison of the SCF against the EKS in a simulation study. a)$\log_2$ of the mean ratio of the frequency estimate errors. Blue indicates a better performance of the EKS over the SCF, red the opposite. The star marker indicates a rough estimate of our experimental conditions. b) Horizontal slice through a). The RMSEs of individual simulation runs are displayed as markers, the solid lines indicate the mean. c) displays a vertical slice.
  • Figure 3: a) Schematic of the setup as used for $^3$He spin precession measurements. For MEOP, light from a laser (CYFL) via some optics (PBS and a $\lambda$/4) is shone on the metastable $^3$He generated by a radio-frequency (RF) discharge with electrodes around the spherical gas cell. A current source (CS) attached to a coil inside the four-layer shield generates the $B_0$ field. The dual-cell OPM gradiometer sensor is used for reading out the $^3$He spin precession signal. b)/c) Experimental gradiometer signal from the dual-cell OPM sensor. b) Time domain of the signal. The inset shows the fast oscillations caused by the $^3$He spin precession. c) Power spectral density of the signal showing the dominant $\approx\,$85 Hz $^3$He signal.
  • Figure 4: Frequency tracking on experimental data, comparison of the methods. Uncertainty intervals are two standard deviations. b) and c) are zoomed insets of a).