Real-time optimal quantum control for atomic magnetometers with decoherence

TL;DR

The work tackles real-time tracking of transient magnetic fields with optical atomic magnetometers, where decoherence and measurement back-action challenge conventional sensing advantages. It develops a scalable quantum-dynamical model based on a co-moving Gaussian approximation to the stochastic master equation and couples it with an Extended Kalman Filter and Linear-Quadratic Regulator to perform real-time estimation and control. A fundamental quantum limit on sensitivity is derived, showing that the best achievable scaling is linear in the sensing time $T$ and atom number $N$, independent of the initial state, measurement, or feedback strategy. Simulations indicate that quantum-limited tracking of constant and fluctuating fields is within reach of current devices, including heartbeat-like signals, and the protocol can also prepare entangled states in real time without storing measurement data. Altogether, the framework provides a practical path to near-quantum-limited, real-time quantum sensing with atomic magnetometers and highlights avenues for biomedical and navigation applications.

Abstract

Quantum entanglement, in the form of spin squeezing, is known to improve the sensitivity of atomic sensors to static or slowly varying fields. Sensing transient events presents a distinct challenge, requires different analysis tools, and has not been shown to benefit from entanglement in practically important scenarios such as spin-precession magnetometry. To address this, we apply concepts from continuous quantum measurements and estimation theory to optical atomic magnetometers, aiming to accurately model these devices, interpret their measurement data, control their dynamics, and achieve optimal sensitivity. Quantifying this optimal performance requires determining a fundamental quantum limit on sensitivity. We derive this limit, imposed by noise, and show that it scales at best linearly with sensing time and atom number N, ruling out any super-classical scaling. This limit is independent of the initial state, measurement, estimator, and measurement-based feedback, and depends only on the decoherence model and the strength of field fluctuations. Thus, finding an estimator that attains this bound proves the sensing strategy optimal. To approach this limit, we develop a quantum dynamical model scalable with N, based on a co-moving Gaussian approximation of the stochastic master equation, which includes measurement backaction and decoherence. This enables a real-time estimation and control architecture integrating an extended Kalman filter with a linear quadratic regulator. Simulating the magnetometer with our model and EKF+LQR strategy shows that quantum-limited tracking of constant and fluctuating fields is within reach of current atomic magnetometers. Our sensing strategy can also track biologically relevant signals, such as heartbeat-like waveforms, and drive the atomic ensemble into an entangled state, even when the measurement record is used for feedback but later discarded.

Figures (32)

• Figure 1: Discrete evaluation of the function $f(t,\mathrm{X}(t))$. Visual representation of the partitioning of the time interval $[0,T\,]$ to evaluate the function $f(t,\mathrm{X}(t))$ and define the Itô integral of $f(t,\mathrm{X}(t))$ with respect to the noise process $\mathrm{W}(t)$.
• Figure 2: A continuous signal sampled using the zero-order hold assumption. To discretize a signal using zero-order hold, the time axis is divided in increments of ${\Delta t}$ in order to evaluate the function at these steps: $k{\Delta t}$, with $k \in \mathbb{Z}$. The signal is then further assumed to maintain a constant value $f(k{\Delta t})$ from time $k {\Delta t}$ to time $(k+1){\Delta t}$.
• Figure 3: Spherical coordinates conventions. The choice of rotation angles $\beta$ and $\alpha$ (see left sketch) define the rotation operator $R(\alpha,\beta)$ and thus, the axis along which the CSS is aligned (along $\pmb{k}$). Given that the rotation operator around axis $\pmb{n}$ is applied to the ground state of the ensemble $\left|j,-j\right\rangle$, by convention centered around the south pole, the rotation angle $\alpha$ is measured off the south pole Arecchi1972Dowling1994. Thus, the standard spherical coordinates parametrizing a sphere, $(\theta,\phi)$ (see right), relate to $(\alpha,\beta)$ as $\theta = \pi - \alpha$ and $\phi = \beta$Dowling1994. When we discuss Wigner functions of CSS mapped onto the Bloch sphere, it is important to note that $(\alpha,\beta)$ simply define the direction of the CSS, whereas $(\theta,\phi)$ map the Wigner function onto the 3D sphere.
• Figure 4: Comparison of the ML, MAP and MMSE estimates. Visual representation of the maximum likelihood estimator ($\;\tilde{\!\!\theta}_{\mathrm{ML}}(y)$, blue dot), the maximum a-priori estimator ($\;\tilde{\!\!\theta}_{\mathrm{MAP}}(y)$, yellow dot) and the minimum mean squared error estimator ($\;\tilde{\!\!\theta}_{\mathrm{MMSE}}(y)$, black dot).
• Figure 5: A scheme illustrating the recursive algorithm of a Bayesian filter. In the "Predict" box, a model is used to propagate the update PDF from step $k\!\shortminus\!1$ to the prediction PDF$p(\pmb{x}_k| \pmb{y}_{0:k\shortminus1})$. Then, in the "Update" box, a new measurement $\pmb{y}_k$ is incorporated into the algorithm to update the prediction PDF into a new update PDF for time $k$: $p(\pmb{x}_k|\pmb{y}_{0:k})$. Finally, before repeating this process again, an estimate at time $k$ is computed from the just-updated posterior: $\tilde{\pmb{x}}_{k|k}$.

Theorems & Definitions (49)

• Definition 1.1: Probability mass function
• Definition 1.2: Probability density function
• Definition 1.3: Multivariate PMF
• Definition 1.4: Multivariate PDF
• Definition 1.5: Joint PMF
• Definition 1.6: Joint PDF
• Definition 1.7: Independent discrete random variables
• Definition 1.8: Independent continuous random variables
• Definition 1.9: Expected value / mean
• Definition 1.10: Variance