Tracking time-varying signals with quantum-enhanced atomic magnetometers

Abstract

Quantum entanglement, in the form of spin squeezing, is known to improve the sensitivity of atomic instruments to static or slowly-varying quantities. Sensing transient events presents a distinct challenge, requires different analysis methods, and has not been shown to benefit from entanglement in practically-important scenarios such as spin-precession magnetometry (SPM). Here we adapt estimation control techniques introduced in [PRX Quantum 6, 030331 (2025)] to the experimental setting of SPM and analogous techniques. We demonstrate that real-time tracking of fluctuating fields benefits from measurement-induced spin squeezing and that quantum limits dictated by decoherence are within reach of today's experiments. We illustrate this quantum advantage by single-shot tracking, within the coherence time of a spin-precession magnetometer, of a magnetocardiography signal overlain with broadband noise.

Figures (6)

• Figure 1: Real-time atomic magnetometry. The magnetometer consists of $N$ spin-1/2 atoms initially pumped (blue) into a coherent spin state along the $x$-axis. It is used to sense time-varying magnetic fields $B(t)$ along $z$, e.g., stochastic (OUP) or a cardiac-like (MCG) signals. This is possible by continuously probing (red) the $y$-component of the ensemble spin over time. In particular, an Extended Kalman Filter (EKF) is used to estimate in real time the field and dominant moments of the atomic spin from the detected photocurrent $y(t)$. The EKF's estimates $\tilde{x}(t)$ are then processed by the Linear Quadratic Regulator (LQR), which drives a feedback magnetic coil to produce an auxiliary field $u(t)$ that cancels any Larmor precession of the atomic spin. As a result, the sensitivity is quantum-enhanced in real time -- the ensemble is driven into a spin-squeezed state pointing along $x$, whose variance is reduced in the $y$-direction by the measurement Amoros-Binefa2024.
• Figure 2: Quantum-enhanced tracking of a fluctuating magnetic field.Top: Stochastic fluctuations of the field (blue) being efficiently tracked in real time by the EKF estimate (red) after gathering only $\approx\!\!0.01ms$ data of the photocurrent. The shaded area represents the confidence band limited by $\pm 2 \sqrt{\mathbb{E}\!\left[{\Delta^2 \tilde{\omega}(t)}\right]}$, i.e. the error obtained upon averaging. Middle: Evolution of the (average) spin-squeezing parameter (blue), compared with its (average) real-time prediction by the EKF (red). Bottom: Evolution of the average error (green) in estimating the fluctuating field, $\sqrt{\mathbb{E}\!\left[{\Delta^2 \tilde{\omega}(t)}\right]}$, which stabilises at the value $\approx\!\! 1rad\,s^{-1}$, as correctly predicted by the EKF covariance (dashed, yellow). The quantum limit imposed by local dephasing (black) is not attained due to insufficient measurement strength (here $M=e-8Hz$Kong2020, but see also Appendix Fig. \ref{['fig:largerM']} for larger $M$) Amoros-Binefa2024. In all plots, averaging was performed over 1000 field+atom stochastic trajectories.
• Figure 3: Tracking field fluctuations at the quantum limit (\ref{['eq:CSlimit_main']}). As in the bottom plot of Fig. \ref{['fig:oup_sq_estimation']}, the (true) average error, $\sqrt{\mathbb{E}\!\left[{\Delta^2 \tilde{\omega}(t)}\right]}$, is depicted against the error predicted by the EKF (dashed, yellow) and the limit (\ref{['eq:CSlimit_main']}) imposed by the decoherence (black), but this time including a tiny contribution of collective dephasing $\kappa_\text{coll} = 10µHz$Note3. The measurement strength achieved in Kong2020 is now sufficient for the magnetometer to operate at the quantum limit (\ref{['eq:CSlimit_main']}), no matter whether the EKF is provided with the exact OUP dynamics (\ref{['eq:oup']}) of the field (top) or its mismatched version (bottom). Although in the latter case the EKF expects fluctuations of twice smaller strength ($q_K = q_\omega/2$) and much faster decay ($\chi_K = 10\chi$), the average error ($\sqrt{\mathrm{aMSE}}$, green) still attains the quantum limit (black), while the EKF covariance (dashed, yellow) underestimates the error. Both plots were obtained by averaging over 1000 field+atom stochastic trajectories. In each case, the inset presents an exemplary field trajectory (blue) together with its EKF real-time estimate (red), which is well within the confidence interval $\pm 2\sqrt{\mathbb{E}\!\left[{\Delta^2 \tilde{\omega}(t)}\right]}$ (shaded green).
• Figure 4: Tracking a MCG-like signal. The magnetometer, under the same conditions as in Fig. \ref{['fig:oup_sq_estimation']}, driven now by the field (blue) representing a noisy magnetocardiogram (MCG) in pT-range Bison2009Kim2019, whose clean waveform (black) is to be tracked. The EKF within our estimation and control scheme is set to expect the signal as a solution of a VdP equation Kaplan2008Das2013. The EKF estimate (red) follows the waveform very well once the magnetometer stabilises over one MCG-cycle ($\approx\!20$ms), with the highest average error ($\pm 3\sqrt{\mathbb{E}\!\left[{\Delta^2\tilde{\omega}(t)}\right]}$ averaged over 1000 trajectories, green shading) observed at the R-wave already at $2.6rad\,s^{-1}$ within the third cycle.
• Figure 5: Quantum-enhanced tracking of a fluctuating magnetic field with a higher measurement strength. In the top plot, the OUP field (blue) is accurately tracked by its EKF estimate (red), remaining within within the error bounds of $\pm 2 \sqrt{\mathbb{E}\!\left[{\Delta^2 \tilde{\omega}(t)}\right]}$, which are so small compared to the fluctuating field as to be nearly imperceptible. The middle plot shows the conditional spin squeezing (blue) generated by higher measurement strength of $M = 1mHz \gg \kappa_\text{coll} = 1nHz$, and its real-time estimation by the EKF (dashed red). In the bottom plot, the error in estimating $\omega(t)$ (green) achieves a sensitivity of $\sim 0.066 \textrm{Hz}$ that matches the square-root of EKF covariance (dashed, yellow). The stronger measurement significantly reduces the error; however, the quantum limit set by the dephasing noise (black) at around $0.056 \textrm{Hz}$ is not perfectly reached. An further increase in the measurement strength $M$ would yield an error closer to the optimal limit Amoros-Binefa2024. The bottom two plots have been obtained by averaging over 1000 field+atom trajectories.