Abstract Single‐beam atomic magnetometers herald a new era of high‐precision magnetic field sensing, with applications spanning fundamental physics to biomagnetism. Nevertheless, their utility is often curtailed by quantum sensor noise, encompassing both technical and quantum‐mechanical noise. This research delves into the potential of the Kalman filter as a tool to subdue quantum sensor noise, thereby augmenting the sensitivity of single‐beam atomic magnetometers. Quantum‐mechanical noise is integrated into the system model as the process noise and measurement noise, and a discrete Kalman filter equipped with a time delay variable is employed. The findings reveal that the time delay variable significantly influences temporal signal tracing, while the discrete Kalman filter enhances sensitivity performance in frequency domain analysis, bypassing the typical sensitivity and time resolution trade‐off encountered in coherent sensing strategies. Partial‐knowledge signal scenarios are also taken into account, wherein a polynomial model is proposed to expansively render the discrete Kalman filter more relevant and adaptable to real‐world situations. Collectively, through experimentation involving sine‐like, non‐Gaussian, and medical magnetocardiography (MCG) signals, our results underscore the promising potential of the Kalman filter in enhancing the sensitivity of atomic magnetometers for practical sensing applications.
Abstract Single‐beam atomic magnetometers herald a new era of high‐precision magnetic field sensing, with applications spanning fundamental physics to biomagnetism. Nevertheless, their utility is often curtailed by quantum sensor noise, encompassing both technical and quantum‐mechanical noise. This research delves into the potential of the Kalman filter as a tool to subdue quantum sensor noise, thereby augmenting the sensitivity of single‐beam atomic magnetometers. Quantum‐mechanical noise is integrated into the system model as the process noise and measurement noise, and a discrete Kalman filter equipped with a time delay variable is employed. The findings reveal that the time delay variable significantly influences temporal signal tracing, while the discrete Kalman filter enhances sensitivity performance in frequency domain analysis, bypassing the typical sensitivity and time resolution trade‐off encountered in coherent sensing strategies. Partial‐knowledge signal scenarios are also taken into account, wherein a polynomial model is proposed to expansively render the discrete Kalman filter more relevant and adaptable to real‐world situations. Collectively, through experimentation involving sine‐like, non‐Gaussian, and medical magnetocardiography (MCG) signals, our results underscore the promising potential of the Kalman filter in enhancing the sensitivity of atomic magnetometers for practical sensing applications.
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 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.
Array-form ultrahigh sensitivity single-beam atomic magnetometers are ideal for high-resolution biomagnetic imaging applications, such as magnetocardiography and magnetoence-phalography. In the low-frequency range of interest for biomagnetism, an essential reason for the deterioration of atomic magnetometer low-frequency sensitivity is the light shift noise caused by temperature fluctuations of the pump laser. Accordingly, this article proposes a new frequency response model of the atomic magnetometer sensitivity, treating light shift as a time-dependent variable. The model reveals the frequency relationship between the deterioration of atomic magnetometer sensitivity and the low-frequency light shift. A new method for suppressing the low-frequency light shift noise based on precise laser temperature control is deduced from the frequency response model, significantly improving the average sensitivity of atomic magnetometers by over 30%. This study presents a theoretical foundation for investigating the low-frequency light shift in atomic magnetometers from a novel perspective and contributes to the optimization of biomagnetic imaging systems composed of single-beam atomic magnetometers.
The extraction of weak signals plays a crucial role in quantum precision measurement, where the estimation results are often limited by low signal-to-noise ratios. Here, we demonstrate a parameter-estimation framework based on the adaptive extended Kalman filter for dynamic magnetic-field estimation in quantum systems using spin-noise measurements -- a challenging regime characterized by weak signals. By modeling the magnetic field as an unknown parameter, the proposed approach alleviates model dependence in state estimation. Furthermore, by introducing an adaptive algorithm with real-time noise estimation, our method overcomes the measurement noise intensity constraints of conventional extended Kalman filtering and enhances its practical applicability. Numerical simulations covering three representative magnetic-field dynamics validate the capability of the proposed framework, while experimental results demonstrate successful tracking of a seismo-magnetic-like signal beyond the intrinsic sensitivity of conventional spin-noise spectroscopy.
Spin estimation plays a crucial role in quantum metrology, particularly in systems like atomic magnetometers where atomic spin ensembles transduce magnetic-field information into measurable optical signals. However, precise spin estimation remains challenging, especially in weak polarization schemes where signals are often obscured by both classical and quantum noise. Conventional fitting methods for spin signal and variance estimation are computationally intensive, lack real-time capabilities, and exhibit sensitivity to stochastic fluctuations and data-acquisition parameters. To address these challenges, we propose and demonstrate the implementation of a Kalman filter (KF) combined with a Rauch-Tung-Striebel (RTS) smoother for enhanced spin estimation across various polarization conditions. Our comprehensive analysis of spin conditional variance estimation reveals that the KF method achieves a remarkable 4.92-dB noise suppression compared to conventional fitting methods, while the RTS smoother provides further enhancement with 6.94-dB suppression that enables quantum noise squeezing observation. These findings offer new tools for quantum noise analysis and signal recovery, offering significant potential for advancing the performance of quantum-enhanced sensors and other precision measurement technologies.
Quantum sensors represent a revolutionary technology with the potential to surpass the limits of classical sensors across a wide range of applications, including precision measurement, navigation, and medical diagnostics. However, the realization of this potential is significantly hindered by the pervasive presence of noise. This work addresses the critical need for accurate noise modeling and effective suppression techniques within quantum sensors. We introduce a novel noise model that incorporates both thermal and coherent noise sources, explicitly accounting for the quantum mechanical nature of these sensors. This model is then utilized to design and analyze various noise suppression strategies, including feedback control algorithms and advanced filtering methods. The theoretical framework presented here provides a foundation for improved sensor design and operation, ultimately driving advancements in quantum sensing technology. The core claim of this work is that precise noise modeling and effective suppression techniques are paramount for enhancing the precision of quantum sensors. This is achieved through the establishment of a new noise model and the subsequent design of suppression techniques such as feedback control and filtering. The novelty lies in the detailed and accurate representation of noise sources within the quantum sensor system.