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arXiv:2512.05265

Real-time optimal quantum control for atomic magnetometers with decoherence

Julia Amoros-Binefa

quant-phSubmitted Dec 4, 2025• Announced Dec 8, 2025• 0 citations
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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.

Comments
253 pages, 32 figures. Final doctoral thesis version containing modifications suggested by the referees: Marco G. Genoni, Madalin Guta, and Lukasz Cywinski

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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.
Sensing a magnetic field with an atomic magnetometer operated in real time presents significant challenges, primarily due to sensor nonlinearity, the presence of noise, and the need for one-shot estimation. To address these challenges, we propose a comprehensive approach that integrates measurement, estimation, and control strategies. Specifically, this involves implementing a quantum nondemolition measurement based on continuous light probing of the atomic ensemble. The resulting photocurrent is then directed into an extended Kalman filter to produce instantaneous estimates of the system’s dynamical parameters. These estimates, in turn, are utilized by a linear quadratic regulator, whose output is applied back to the system through a feedback loop. This procedure automatically steers the atomic ensemble into a spin-squeezed state, yielding a quantum enhancement in precision. Furthermore, thanks to the feedback proposed, the atoms exhibit entanglement even when the measurement data are discarded. To prove that our approach constitutes the optimal strategy in realistic scenarios, we derive ultimate bounds on the estimation error applicable in the presence of both local and collective decoherence, and show that these are indeed attained. Additionally, we demonstrate that, for large ensembles, the extended Kalman filter not only reliably predicts its own estimation error in real time, but also accurately estimates spin squeezing at short timescales.
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.
Abstract Continuously monitored atomic spin-ensembles allow, in principle, for real-time sensing of external magnetic fields beyond classical limits. Within the linear-Gaussian regime, thanks to the phenomenon of measurement-induced spin-squeezing, they attain a quantum-enhanced scaling of sensitivity both as a function of time, t, and the number of atoms involved, N. In our work, we rigorously study how such conclusions based on Kalman filtering methods change when inevitable imperfections are taken into account: in the form of collective noise, as well as stochastic fluctuations of the field in time. We prove that even an infinitesimal amount of noise disallows the error to be arbitrarily diminished by simply increasing N, and forces it to eventually follow a classical-like behaviour in t. However, we also demonstrate that, ‘thanks’ to the presence of noise, in most regimes the model based on a homodyne-like continuous measurement actually achieves the ultimate sensitivity allowed by the decoherence, yielding then the optimal quantum-enhancement. We are able to do so by constructing a noise-induced lower bound on the error that stems from a general method of classically simulating a noisy quantum evolution, during which the stochastic parameter to be estimated—here, the magnetic field—is encoded. The method naturally extends to schemes beyond the linear-Gaussian regime, in particular, also to ones involving feedback or active control.
Integrating classical estimation and control theory with quantum systems holds potential for quantum measurement improvement. Here, we report a real-time linear quadratic regulator (LQR) optimal control for spin damping in electron-nuclear spin-coupling regime. The aim is a quicker response and enhanced noise suppression for precision quantum measurements. Using a Kalman quantum observer with electron spin data, we estimate the quantum state of nondirectly observable nuclear spin. LQR full-state feedback optimal control is applied to achieve faster spin-coupling damping. Demonstrated in an atomic spin gyroscope, our method significantly enhances dynamic response with a 4.3-fold improvement in rotational bandwidth. It also boosts long-term stability and measurement sensitivity by 62.9%, presenting a promising avenue to elevate spin-coupling ensemble performance in quantum measurements.
Summary form only given. Continuous monitoring or tracking of a quantum system is essential to high-sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors Fαat different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below δFα~ √N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1, 2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.
Measurements with quantum limited resolution have important applications in metrology and sensing, including atomic clocks, atom interferometry, and magnetometry. We perform a quantum-non-demolition measurement on a collective angular momentum of a million spin-4 Cs atoms with high optical depth in our work. Quantum mechanics limits the measurement's resolution to what is known as the standard quantum limit (SQL) when there is no entanglement between the individual atoms. We can generate entanglement between atoms to create spin squeezed states through quantum backaction, improving the measurement sensitivity by 5.5 dB relative to the SQL. We have recently rebuilt the entire experiment inside a state-of-the-art magnetic shield of mu-metal and aluminum. This upgrade has drastically mitigated classical control errors in the spin squeezing experiment. Furthermore, the ensemble's sensitivity to magnetic fields allows us to make precise measurements of the magnetic field environment. Using closed-loop feedback, we use our atomic magnetometer to measure magnetic fields 4.3 dB below the SQL. The shielded environment will allow us to optimize the magnetometer and sets the stage for future experiments to use the closed-loop control to explore more complicated quantum dynamics.