🔬 atomic physics

Optimal and efficient inference tools for field tracking with precessing spins

This paper derives the Bayesian Cramér-Rao bound for spin-precession magnetometers and demonstrates that while the prediction error method achieves optimal precision, the extended Kalman filter offers a computationally efficient, near-optimal alternative for real-time magnetic field tracking well beyond the system's response bandwidth.

Original authors: Klaudia Dilcher, Piotr Bania, Diana Mendez-Avalos, Aleksandra Sierant, Morgan W. Mitchell, Jan Kolodynski

Published 2026-08-11

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CC BY 4.0

Original authors: Klaudia Dilcher, Piotr Bania, Diana Mendez-Avalos, Aleksandra Sierant, Morgan W. Mitchell, Jan Kolodynski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a whisper in a room that is shaking, vibrating, and filled with static. This is the daily reality for scientists trying to measure magnetic fields using tiny particles called "spins." Think of these spins not as tiny magnets, but as microscopic tops spinning on a table. When you place them in a magnetic field, they don't just spin; they wobble, or "precess," like a spinning top that is about to fall over. The speed of this wobble tells you exactly how strong the magnetic field is. This is the core of a device called a spin-precession magnetometer (SPM).

These devices are incredibly sensitive, capable of detecting magnetic fields so weak they would be invisible to almost anything else. They are the heroes behind technologies that could help navigate without GPS, find hidden underground structures, or even map the electrical activity of a human brain. However, there is a catch: the "tops" are fragile. They wobble for a short time before they get tired and stop (a process called decoherence), and the act of listening to them introduces a bit of static noise. The big question for scientists has been: How do we listen to these wobbly tops in real-time, track their changing speed, and figure out the magnetic field before the tops stop spinning, all without getting lost in the noise?

This paper tackles that exact challenge. The authors, a team of physicists and engineers, set out to find the best "listening tools" (mathematical algorithms) to track these spinning tops. They compared three different methods: a super-accurate but incredibly slow method called the Prediction Error Method (PEM), and two faster, "smart guess" methods called the Extended Kalman Filter (EKF) and the Cubature Kalman Filter (CKF).

Here is what they found. First, they confirmed that the slow, super-accurate method (PEM) is indeed the "gold standard." It hits the theoretical limit of how precise you can possibly be, known as the Bayesian Cramér-Rao bound. However, it's like trying to solve a massive puzzle by looking at every single piece from the beginning every time you add a new one; it's too slow for real-time use.

The exciting news is that the faster methods are almost as good. The authors ran detailed computer simulations to test these filters under various conditions. They discovered that the Extended Kalman Filter (EKF) is a "sweet spot" solution. It is computationally efficient, meaning it can run fast enough to update the magnetic field reading in real-time, yet it remains incredibly accurate. In their simulations, the EKF could track a constant magnetic field with a precision better than 0.01 Hz (a tiny fraction of a spin per second), reaching a relative error of just 0.0001%.

The team also tested how these tools handle messy, real-world scenarios. They simulated magnetic fields that were fluctuating randomly (like a shaky hand) and fields that changed suddenly (like a step function). Even when the magnetic field was jumping around or the number of atoms in the sensor was huge (up to $10^{12}$ atoms), the EKF held its ground. The more complex CKF filter performed slightly better for very large groups of atoms, but the EKF was sufficient for most practical situations and much less demanding on computer power.

Crucially, the paper shows that you don't need to sample the data at lightning speeds to get good results. As long as the sampling period is around 5 microseconds (or faster), the EKF works beautifully. If you sample too slowly, the filter gets confused, but the authors found a clear "safe zone" for the sampling speed.

In short, this paper demonstrates that we don't need the slow, perfect method to get near-perfect results. By using the Extended Kalman Filter, we can track magnetic fields in real-time with high precision, even when the signal is noisy or changing rapidly. This suggests that future devices for medical imaging, navigation, or physics research can be built to be both incredibly sensitive and fast enough to react instantly to the world around them. The authors simulated these results extensively, showing that these methods are robust and ready to be adapted for other types of sensors that face similar noisy, spinning challenges.

Technical Summary: Optimal and Efficient Inference Tools for Field Tracking with Precessing Spins

Problem Statement
Precise, real-time monitoring of magnetic field evolution is critical for applications such as magnetic navigation in GPS-denied environments and searches for physics beyond the standard model. Spin-precession magnetometers (SPMs), utilizing electron, nuclear, or color-center spins, offer high sensitivity without cryogenic cooling. However, the stochastic nature of spin dynamics and the light-detection process necessitates recursive signal-processing tools to interpret data sequentially. While Bayesian inference provides a natural framework for estimating time-varying fields, existing methods face a trade-off: optimal estimators often require computationally prohibitive re-evaluation of the entire measurement history, whereas efficient recursive filters may lack optimality, particularly when dealing with the nonlinear dynamics inherent in spin precession (where precession speed and spin magnitude must be estimated simultaneously).

Methodology
The authors investigate Bayesian signal-recovery methods for SPMs operating in the free-induction decay (FID) mode. They employ a stochastic dynamical model of the SPM based on a Lindblad master equation, incorporating intrinsic atomic noise and measurement shot noise. The study focuses on three specific inference approaches:

Prediction Error Method (PEM): A computationally expensive, offline method that utilizes the full history of measurement data to maximize the posterior distribution. It serves as a benchmark for optimality.
Extended Kalman Filter (EKF): A recursive, nonlinear Gaussian filter that performs time-local linearization of the dynamics. It updates estimates based only on the current measurement and the most recent parameter estimates.
Cubature Kalman Filter (CKF): A more advanced nonlinear Gaussian filter that generalizes the EKF by using spherical cubature integrals (up to the third order) to match moments within the Gaussian approximation, offering higher accuracy at increased computational cost.

The authors derive the Bayesian Cramér-Rao Bound (BCRB), which sets the ultimate lower limit on the mean squared error (MSE) for estimating the Larmor frequency, given a prior distribution. They compare the performance of PEM, EKF, and CKF against this bound across varying experimental parameters: probing time ( $t$ ), number of atoms ( $N$ ), and measurement sampling period ( $\Delta$ ).

Key Contributions and Results

Derivation of the BCRB: The paper derives the BCRB for estimating the Larmor frequency in the presence of atomic and measurement noise. It demonstrates that the PEM can attain this bound, confirming its optimality, though it is practically intractable for real-time applications due to the need to recompute the estimator using the full data history for every new measurement.
Near-Optimal Efficiency of EKF: The study shows that the EKF offers near-optimal tracking performance that is computationally efficient. For typical Larmor frequencies ( $\omega \approx 2\pi \times 10$ kHz) and atomic densities ( $N \lesssim 10^{11}$ ), the EKF achieves sub-0.01 Hz precision within the coherence time of the spin system. Its relative error reaches $\approx 10^{-4}\%$ , which is approximately half the precision of the optimal PEM but achieved with significantly lower computational complexity.
Role of CKF: The CKF provides performance nearly identical to the optimal PEM for large atomic ensembles ( $N \gtrsim 10^{11}$ ), where nonlinear effects become more pronounced. However, for smaller ensembles, the EKF is sufficient, making the CKF's higher computational cost unnecessary in many practical scenarios.
Sampling Period Constraints: The authors find that the sampling period $\Delta$ must be less than approximately 10 $\mu$ s to avoid performance degradation. Specifically, the EKF requires $\Delta \lesssim 5$ $\mu$ s for optimal performance at 10 kHz, adhering roughly to the condition $30\Delta \lesssim 2\pi/\bar{\omega}$ .
Robustness to Time-Varying Fields: The EKF is demonstrated to be robust in tracking fluctuating signals (modeled as Ornstein-Uhlenbeck processes) and unknown transient signals, including sinusoidal oscillations and sudden step-function jumps. Even when the filter is designed for a Wiener process but applied to deterministic signals, it successfully tracks variations of up to 10% of the nominal frequency.

Significance and Claims
The paper claims that the proposed inference tools, particularly the EKF, provide a practical solution for real-time magnetic field estimation in spin-based quantum sensors. By balancing computational efficiency with near-optimal precision, these methods enable the tracking of fluctuating and unknown transient signals well beyond the response bandwidth of the sensor.

The authors emphasize that their approach is adaptable to other types of sensors undergoing nonlinear dissipative dynamics and experiencing intrinsic Gaussian-like stochastic noises. They suggest that these methods can facilitate real-time active feedback and control schemes in sensing scenarios involving spin-based quantum devices, potentially implemented on field-programmable gate arrays (FPGAs) or micro-controllers. The work does not propose new experimental hardware but rather provides the algorithmic framework to maximize the utility of existing SPM platforms, such as hot-vapor atomic ensembles and crystal color centers.

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