Optimal and efficient inference tools for field tracking with precessing spins

TL;DR

This work develops a Bayesian framework for real-time field tracking with spin-precession magnetometers (SPMs) operating in free-induction decay, deriving the Bayesian Cramér-Rao bound (BCRB) for the Larmor frequency and showing that offline PEM attains this bound while online nonlinear Gaussian filters (EKF, CKF) deliver near-optimal tracking with far lower computational cost. By modeling the SPM as a stochastic spin system with measurement noise and decoherence, the authors implement continuous-discrete EKF and CKF (and a slower, offline PEM) to estimate the time-varying frequency $\omega(t)$ from discretized Faraday-rotation measurements. The results demonstrate that PEM saturates the BCRB for constant or slowly varying fields, while EKF/CKF achieve sub-$\text{Hz}$ precision and can track Ornstein-Uhlenbeck and deterministic waveform changes in real time, depending on atom number $N$, coherence time $T_2$, and sampling interval $\Delta$. These methods enable efficient, real-time sensing and control in spin-based quantum sensors, with potential generalization to other nonlinear dissipative systems and FPGA/embedded implementations for field applications.

Abstract

Precise, real-time monitoring of magnetic field evolution is important in applications including magnetic navigation and searches for physics beyond the standard model. One main field-monitoring technique, the spin-precession magnetometer (SPM), observes electron, nucleus, color center, or muon spins as they precess in response to their local magnetic field. Here, we study Bayesian signal-recovery methods for SPMs in the free-induction decay (FID) mode. In particular, we study tracking of field changes well within the coherence time of the spin system, and thus well beyond the response bandwidth, as in [Phys. Rev. Lett. 120, 040503 (2018)]. We derive the Bayesian Cramér-Rao bound that dictates the ultimate precision in estimating the Larmor frequency, which we show to be attained by the computationally-expensive prediction error method (PEM). Relative to this benchmark, we show that the extended Kalman filter (EKF) and cubature Kalman filter (CKF) offer near-optimal tracking that is also computationally efficient, with the use of the latter giving better results only for large spin number. Focusing thus on the EKF, we show that it is sufficient to accurately track fluctuating and unknown transient signals. Our methods can be easily adapted to other types of sensors undergoing non-linear dissipative dynamics and experiencing intrinsic Gaussian-like stochastic noises.

Figures (10)

• Figure 1: Setup of the simulated spin-precession magnetometer (SPM). The setup consist on a 3 cm length glass cell with an isotopically enriched $^{87}$Rb with 100 Torr of N$_2$ buffer gas at 100$\degree$C. Through the cell pass a linearly polarized probe light and a circularly polarized pump light. The rotation in the polarization angle in the probe light after the atoms is given by $\varphi\propto\bm{J}_t\cdot \bm{e}_z$. $I(t)\propto\varphi$ denotes the photocurrent measured at time $t$, which then evolves according to Eq. (\ref{['eq:discrete_meas']}). $\textrm{B}_\textrm{du}$ -- Beam dumper, $\frac{\lambda}{2}$ -- half-wave plate, WP -- Wallaston Prism, BPD -- Balanced Photo Detector.
• Figure 2: Estimation error $\Delta \tilde{\omega}$ as a function of the probing time $t$ for estimators built using PEM (orange), CKF (green), and EKF (blue) methods. PEM is optimal, as it follows the BCRB (light blue) at all timescales---in particular, in the steady-state regime applicable beyond the coherence time ($T_2\approx1ms$) of the magnetometer when the minimal error is reached, whose value is constrained by the universal lower bound (\ref{['eq:noiseless_bcrb']}) (dotted grey). Both filters (EKF and CKF), however, also attain sub-$0.01Hz$ precision, reaching spectacular relative error of $\Delta \tilde{\omega}/\bar{\omega}\lesssim e-4%$ that is only half as precise as the BCRB. The noiseless BCRB (red) that admits an analytic form (see App. \ref{['app:BCRB']}) is also shown, and is only mildly optimistic relative to the true BCRB despite assuming no atomic noise ($Q=0$). The simulation was performed with experimental parameters of Ref. Jimenez2018 stated in Table \ref{['tab:exp_pars']}, averaging over $10^4$ runs. For $t\gtrsim1ms$ we omit markers, so that PEM/CKF/EKF/BCRB curves can be easily distinguished.
• Figure 3: Estimation error $\Delta \tilde{\omega}$ as a function of the atom number $N$ for estimators based on PEM (orange), CKF (green), and EKF (blue), with probing time fixed to $t=5ms\approx 5\,T_2$. As PEM attains again the BCRB (light blue), it forms an optimal estimator across all $N$ considered. Nonetheless, the EKF and CKF are sufficient for $N\lesssim10^{11}$ and $N\lesssim10^{12}$, respectively, at which they reach their best precision. For smaller $N$, all methods follow the noiseless BCRB (\ref{['eq:noiseless_bcrb']}) (red) and, in particular, the $1/N$-scaling that it predicts. The simulation was performed for experimental parameters of Ref. Jimenez2018 stated in Table \ref{['tab:exp_pars']}, averaging over $10^4$ runs.
• Figure 4: Estimation error $\Delta \tilde{\omega}$ as a function of the sampling period $\Delta$, achieved by the inference methods after collecting measurement data over the probing time $t=1ms\approx T_2$. No significant improvement for any of the methods is observed by decreasing $\Delta$ below $5µs$, i.e., the value used in Ref. Jimenez2018. Nonetheless, the CKF is much more robust than the EKF to increasing the sampling period until becoming unstable at the Nyquist sampling period $\approx50µs$ (and half its value), dictated by the Larmor frequency $\bar{\omega}= 2\pi \times 10kHz$. The simulation was performed for experimental parameters of Ref. Jimenez2018 stated in Table \ref{['tab:exp_pars']}, averaging over $10^3$ runs.
• Figure 5: Tracking strong magnetic-field fluctuations using the EKF for measurement sampling period $\Delta=1µs$ and other parameters of the SPM set as in Table \ref{['tab:exp_pars']}. The Larmor frequency follows an OU process (\ref{['eq:nlf_ou']}) characterised by the mean reversion time $\tau=1s$ and the noise strength $d_c=e9rad\squared\per s\cubed$ (top plot). For the particular experimental shot presented, the true estimation error is also shown, together with the error predicted by the EKF (bottom plot).