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

The study addresses improving sensitivity in quantum sensing, where precise detection of small forces, fields, and time is crucial but often limited by hardware constraints and conventional data-processing methods. While advances across cold atoms, superconducting circuits, and solid-state systems have achieved high precision, practical deployment is hindered by noise, drift, and the need for substantial prior modeling. Previous computational and machine learning efforts for sensing and anomaly detection typically require extensive labeled datasets or prior knowledge of signal/noise characteristics. The authors propose a data-driven, system-agnostic strategy leveraging digital twinning via generative models to capture the full statistical structure of high-dimensional measurements from a Bose-Einstein condensate (BEC). By constructing a digital twin trained on force-free time-of-flight (TOF) images and applying anomaly detection, they aim to exploit higher-order correlations in the data to enhance force detection beyond conventional center-of-mass (COM) analysis, without sacrificing long-term stability.

The paper situates its contribution within quantum sensing progress across platforms (cold atoms, superconducting circuits, solid-state spins) and applications (navigation, gravimetry, dark matter searches). Prior approaches have improved sensitivity via hardware (quantum correlations, entanglement, squeezing) and via software (statistical learning for signal acquisition, anomaly detection in physics contexts). However, these often rely on strong prior knowledge, large datasets, or simplified statistics (e.g., low-order moments). The concept of digital twinning has emerged as a powerful paradigm for mirroring complex physical systems for analysis and prediction. The authors leverage this concept using generative models (GANs) to replicate the distribution of BEC TOF images including multiple noise channels, enabling anomaly detection that utilizes high-dimensional, higher-order correlations. They compare their approach against conventional COM-based sensing and prior methods such as phase-coherent velocimetry with trapped ions and ultracold atoms in cavities, highlighting potential for broad applicability.

Physical system and data acquisition: A Bose-Einstein condensate of approximately 2 × 10^7 Rb atoms at around 50 nK is prepared in a triangular optical lattice to suppress unwanted real-space dynamics. Following preparation, the lattice and trapping potentials are switched off and the atoms expand ballistically. Time-of-flight (TOF) imaging along z records the momentum distribution nα(k) per experimental run (2D images). Each cycle takes T0 ≈ 38 s. Shot-to-shot fluctuations arise from quantum shot noise (superposition of momentum eigenstates), thermal noise (residual thermal atoms), and technical noise (drifts in atom number, trapping potential, lattice depth). Conventional sensing uses the averaged COM momentum, which consumes only zeroth and first moments of the high-dimensional data. Digital twin via GAN: A generative adversarial network (GAN) is trained on force-free TOF images to learn the full probability distribution, implicitly capturing higher-order correlations and the combined noise channels. Dataset: 3600 independent force-free TOF images (64×64, pixel values normalized to [−1,1]); ~90% used for training, remainder for testing. Architecture: Discriminator D(x) has six fully connected layers with Leaky ReLU, outputting a scalar probability p ∈ [0,1]. Generator G(z) has five blocks (fully connected + batch normalization + ReLU), with latent vector dimension z = 100. Training uses the standard min-max adversarial loss V(D,G) = Ex~pdata[log D(x)] + Ez~pz[log(1 − D(G(z)))]. Convergence is assessed by the generator successfully fooling the discriminator and by visual similarity to real data without mode collapse. Anomaly detection pipeline: After adversarial training on force-free data only, an encoder E(x) is introduced to invert G, producing a latent vector z for a given input image x, improving stability of the anomaly score by approximating a bijective mapping. The anomaly score A(x) combines residual loss (RL) and discrimination loss (DL): RL AR = ||x − G(E*(x))||2 and DL AD = ||h(x) − h(G*(E*(x)))||2, where h(·) is a feature layer (last fully connected layer) of the discriminator. The overall anomaly score is A(x) = AR(x) + λ AD(x), with λ a hyperparameter chosen to optimize sensitivity; λ is tuned in [−1,1] using the force-free dataset, with an optimal value identified empirically for best sensitivity. The encoder is trained by minimizing the residual loss while G and D are fixed. The anomaly score distribution for force-free data is approximately normal. Force sensing protocol: Applying a weak external force F to the BEC modifies the distribution of TOF images and thus the anomaly score A(n(k)). Compared with COM-based linear processing, A(n(k)) is a nonlinear function of the image that captures higher-order correlations, leading to improved sensitivity. The response of A is observed to be linear vs impulse I = F ΔT for a fixed optical force Fo, validated by varying force application time ΔT in accordance with the impulse theorem. Anomaly localization analysis (examining n_g(k) = [n(k) − G*(E*(n(k)))]) reveals that the most force-informative features arise in higher-momentum regions (e.g., second Brillouin zone), consistent with reduced influence of interaction and imaging artifacts at high momenta. Sensitivity and stability estimation: Sensitivity S is defined for a signal q (COM or anomaly score A) as S = sqrt(T0) × |dν_q| / σ_q, where σ_q is the standard deviation of q over independent shots and dν_q is the response slope vs force. Using identical experimental datasets, sensitivities are computed for COM momentum and anomaly score. Allan deviation analysis versus integration time τ confirms 1/√τ scaling for both methods, indicating white-noise-dominated fluctuations and no added long-term drift from the nonlinear processing.

- A purely data-driven digital twin using a GAN, trained on 3600 force-free TOF images (~40 hours of data), reproduces the distribution of BEC momentum images and enables an anomaly score that captures higher-order correlations. - The anomaly score distribution is approximately Gaussian and displays a clear shift in response to a weak applied force, while COM momentum distributions show minimal separation under the same conditions. - Linear response: The anomaly score scales linearly with impulse I = F ΔT, enabling quantitative sensitivity extraction. - Sensitivity improvement: Using the same data, anomaly detection achieves S_AS = 1.7(4) × 10^−25 N/Hz. Conventional COM analysis yields S_COM = 6.8(9) × 10^−24 N/Hz (raw). With Gaussian processing-based denoising, COM improves to S_COM = 1.6(4) × 10^−24 N/Hz. The anomaly detection is about 40× more sensitive than raw COM and roughly an order of magnitude better than denoised COM. - Weak-force detection: Demonstrated detection around Fo = 7.81 × 10^−26 N with clear anomaly-score distributional separation relative to force-free data. - Stability: Allan deviation for the anomaly score decays as 1/√τ, matching COM’s scaling and indicating white-noise-limited performance without introducing long-term drifts. The approach maintains improved long-term stability compared to prior work where Allan deviation bent upward around τ ≈ 10^6 s; here it continues to decrease up to ~4 × 10^6 s. - Interpretability: Anomaly localization identifies dominant sensing features in high-momentum peaks (e.g., second Brillouin zone), aligning with physical intuition about reduced noise impacts at higher energies.

The work demonstrates that digital twinning combined with anomaly detection can greatly enhance quantum force sensing by exploiting the full high-dimensional structure of TOF images rather than relying on low-order statistics. The GAN-based digital twin learns the underlying distribution of force-free measurements, enabling an anomaly score that functions as a data-driven, nonlinear signal. This yields linear response to force impulse and substantial sensitivity gains while preserving long-term stability. The authors highlight that sensing performance depends on the data-processing functional f_NN, suggesting an optimization landscape with a theoretical upper bound S_opt = max_f S[f_NN] for a given sensor configuration. The digital twin provides a framework to approach this bound by guiding the choice and optimization of anomaly scoring. The observed anomaly localization in high-momentum regions corroborates domain understanding and provides insight into which image features contribute most to sensitivity. The method’s generality and lack of reliance on system-specific modeling or prior knowledge suggest applicability across sensing platforms with high-dimensional observables.

The study introduces a machine-learning-assisted quantum force sensing method that constructs a digital twin of BEC TOF measurements and uses anomaly detection to harness higher-order correlations in the data. This approach achieves a sensitivity of 1.7(4) × 10^−25 N/Hz, improving over COM-based methods by up to ~40×, while maintaining 1/√τ long-term stability. The framework is model-agnostic, purely data-driven, and easily transferable to other sensing systems with high-dimensional outputs. The authors propose viewing sensitivity as a functional S[f_NN] of the data-processing pipeline, motivating the search for an optimal processing strategy and the theoretical characterization of an upper bound S_opt. Future work should determine S_opt, investigate fundamental quantum limits (e.g., shot-noise constraints, scaling with atom number and imaging resolution), and explore enhanced generative models and anomaly metrics to further improve sensitivity.

- Training data requirement: Constructing the digital twin required 3600 force-free images (~40 hours), which may be resource-intensive for some systems. - Hyperparameter/model dependence: Sensitivity depends on the anomaly score design and hyperparameter λ; identifying globally optimal configurations remains open. - Theoretical bounds: The maximal achievable sensitivity S_opt and fundamental quantum limits (e.g., shot-noise scaling) are not yet established. - Domain scope: Validation is performed on BEC TOF images; extension to other sensing modalities, while promising, remains to be experimentally demonstrated. - Non-injectivity mitigation: An encoder is added to stabilize the anomaly score due to GAN non-injectivity; residual uncertainties from model imperfections may affect sensitivity.