Quantum force sensing by digital twinning of atomic Bose-Einstein condensates

High-sensitivity detection is crucial for scientific advancements and technological applications. Quantum sensing leverages quantum resources to achieve high-precision detection of physical quantities like force and electromagnetic fields at the atomic scale. While various quantum sensing techniques have been developed using platforms such as cold atoms, superconducting circuits, and solid-state spin systems, practical implementations often face challenges due to hardware and software limitations. Hardware upgrades can improve performance, but software approaches like advanced data analysis offer an alternative path to enhance sensitivity. This paper focuses on improving sensitivity using machine learning, specifically through the development of a digital twin model for an atomic Bose-Einstein condensate (BEC) based force sensor. Previous computational approaches using statistical learning or machine learning for anomaly detection often require substantial data or prior knowledge of signal and noise properties, limiting their applicability. Digital twinning, creating a virtual replica of a physical system, offers a promising approach to overcome these limitations by providing a comprehensive understanding and facilitating real-time analysis and prediction of system behavior. This research explores digital twinning for quantum force sensing in BECs by implementing a generative machine learning model to create a digital twin that accurately represents the BEC system, incorporating quantum, thermal, and technical noise channels simultaneously.

Existing quantum sensing techniques leverage many-body correlations and quantum entanglement to enhance sensitivity, but these methods often face practical challenges due to technological requirements. Previous computational approaches utilized statistical learning of signal acquisition, considering inherent noise and variations. More recently, machine learning has been applied for signal selection and anomaly detection; however, these methods usually necessitate substantial data or prior knowledge of signal and noise characteristics. This study differentiates itself by employing digital twinning, a powerful technique that enables the creation of a virtual replica mirroring the behavior and properties of a physical system, providing a novel approach for enhancing the sensitivity of quantum force sensing. The method presented here combines the power of a generative adversarial network (GAN) for digital twinning with anomaly detection, enabling the full utilization of high-dimensional experimental data to significantly enhance sensitivity beyond existing techniques.

The experimental setup utilizes a Bose-Einstein condensate (BEC) of approximately 2 × 10⁷ Rb atoms trapped in a triangular optical lattice to suppress unwanted dynamics. After preparing the BEC, the optical lattice and trapping potential are shut off, allowing the atoms to expand ballistically. Time-of-flight (TOF) imaging is used to measure the momentum distribution n(k). The experimental cycle takes approximately 38 seconds. Conventional force sensing in this setup involves analyzing the averaged center-of-mass (COM) momentum, which utilizes only the zeroth and first-order moments of the two-dimensional data, discarding higher-order correlations. The proposed method employs digital twinning using a generative adversarial network (GAN) to create a digital replica of the experimental system, capturing intricate correlations and non-linear dynamics present in the high-dimensional TOF data. This GAN consists of a generator that produces synthetic TOF images, and a discriminator that attempts to differentiate between real and synthetic images. The generator is trained to produce realistic images that can fool the discriminator. An encoder is added to the GAN architecture to create a bijective mapping, enhancing the stability and sensitivity of the anomaly score. The anomaly score A(n(k)) is calculated using a combination of residual loss (RL), the Euclidean distance between the real and generated images, and discrimination loss (DL), the difference in features extracted from the discriminator. The anomaly score serves as a highly nonlinear function of the momentum distribution, effectively incorporating higher-order correlations and improving the signal-to-noise ratio. The optimal weighting coefficient λ balancing RL and DL was determined through an optimization process. The sensitivity of the method is quantitatively assessed by comparing the sensitivity (S) obtained from the anomaly score with the sensitivity obtained from the conventional COM momentum approach, using Equation 4 from the paper, which relates sensitivity to the standard deviation of the signal and the measurement time. Gaussian processing was also applied to the COM momentum data to evaluate the extent of improvement achievable through noise reduction techniques.

The study demonstrates a significant enhancement in force sensing sensitivity using the proposed digital twinning and anomaly detection approach. The sensitivity achieved using the anomaly score (S_AS = 1.7(4) × 10⁻²⁵ N/Hz) is approximately 40 times greater than that obtained using the conventional COM momentum method (S_COM = 6.8(9) × 10⁻²⁴ N/Hz). Even after applying Gaussian processing for noise reduction to the COM momentum data (S_COM = 1.6(4) × 10⁻²⁴ N/Hz), the anomaly detection method remains an order of magnitude more sensitive. Analysis of the anomaly localization reveals that the signals contributing to the force sensing primarily originate from the large momentum region in the second Brillouin zone. This is consistent with the understanding that high-momentum components in BECs are less affected by noise sources like atomic scattering and thermal activation. The long-term stability of the anomaly detection method is comparable to the conventional method, showing a 1/√τ decay in the Allan deviation, indicating no significant long-term drifts introduced by the nonlinear data processing. This indicates that the proposed method maintains high sensitivity without sacrificing long-term stability. The achieved sensitivity surpasses other state-of-the-art experimental techniques. The authors suggest the existence of an optimal sensitivity (S_opt) for a given sensor configuration, emphasizing the importance of data processing strategies in achieving optimal performance.

The results demonstrate the significant potential of combining digital twinning and anomaly detection for enhancing the sensitivity of quantum force sensing. The order-of-magnitude improvement in sensitivity achieved in this study is noteworthy, surpassing previous techniques. The data-driven nature of the method makes it highly robust and widely applicable to other sensing applications without the need for detailed system-specific knowledge. The findings emphasize the importance of data processing strategies in maximizing the performance of physical sensors, suggesting an upper bound on achievable sensitivity. The linear relationship between the anomaly score and the applied force demonstrates the feasibility of the approach for precise force measurement. The observed anomaly localization, with contributions mainly from the high-momentum region, highlights the effectiveness of the method in distinguishing subtle signal changes from noise, leading to improved sensitivity.

This study presents a novel method for quantum force sensing that utilizes digital twinning of atomic BECs and anomaly detection. The method achieves a significant improvement in sensitivity by leveraging the information contained in high-dimensional data through a nonlinear processing approach. The data-driven nature and high sensitivity, combined with comparable long-term stability to conventional methods, make this a promising technique for various applications. Future research directions include investigating the fundamental quantum limits of the anomaly detection approach and determining the optimal sensitivity achievable for a given sensor configuration.

While the study demonstrates significant improvements in sensitivity and stability, further research is necessary to fully explore the limitations of the method. The dependence of the optimal sensitivity on the chosen hyperparameters and the robustness of the method to different noise characteristics require further investigation. The current implementation relies on a specific type of BEC and experimental setup. The generalizability of the method across different systems and noise environments needs to be explored.