Fast, accurate, and predictive method for atom detection in site-resolved images of microtrap arrays

We introduce a new method, rooted in estimation theory, to detect individual atoms in site-resolved images of microtrap arrays, such as optical lattices or optical tweezers arrays. Using labelled test images, we demonstrate drastic improvement of the detection accuracy compared to the popular method based on Wiener deconvolution when the inter-site distance is comparable to the radius of the point spread function. The runtime of our method scales approximately linearly with the number of sites, and remains well below 100 ms for an array of 100 x 100 sites on a desktop computer. It is therefore fully compatible with a real-time usage. Finally, we propose a rigorous definition for the signal-to-noise ratio of the problem, and show that it can be used as a predictor for the detection error rate. Our work opens the prospect for future experiments with increased array sizes, or reduced inter-site distances.

💡 Research Summary

In this work the authors present a novel, estimation‑theory‑based algorithm for detecting individual atoms in site‑resolved images of micro‑trap arrays such as optical lattices and tweezer arrays. The problem is cast as a linear measurement model y = Mx + k + n, where M encodes the point‑spread function (PSF) of each lattice site, x contains the unknown site brightnesses (zero for empty sites, a Gaussian‑distributed value for occupied sites), k is a uniform background, and n represents shot‑noise and camera read‑noise. By assuming known statistical properties of the signal (occupancy probability p, mean brightness μ, variance σ²) and the noise (Poisson shot‑noise plus Gaussian read‑noise), the authors derive the optimal linear estimator (OLE), also known as the generalized Wiener filter. The OLE matrix H_opt = (MᵀΣ_n⁻¹M + Σ_x⁻¹)⁻¹MᵀΣ_n⁻¹ minimizes the mean‑squared error between the estimated and true brightness vectors.

Two variants are introduced. The “a‑priori” OLE uses the global parameters p, μ, σ² directly, while the “a‑posteriori” OLE updates the covariance matrices using site‑specific posterior occupancy probabilities p_i obtained from the image itself. The posterior probabilities are derived from a Gaussian‑mixture fit to the brightness distribution produced by the a‑priori OLE, allowing each site to be effectively decoupled from its neighbours when its occupancy is near certain.

Parameter estimation proceeds without external calibration. The total photon count yields ⟨x⟩ = pμ, and a hyper‑parameter γ = Σ_n/Σ_x is tuned by maximizing the kurtosis of the estimated brightness distribution, which empirically coincides with the theoretical ratio Σ_n/Σ_x. After finding the optimal γ, a Gaussian‑mixture model provides estimates of p, μ and σ², which are then fed back into the a‑posteriori OLE.

Performance is benchmarked on synthetic images generated with a realistic model: a square lattice of N_s sites, inter‑site spacing a (in pixels), Gaussian PSF with half‑width at half‑maximum r_psf, background k, and read‑noise r. Two regimes are examined: (i) a ≫ r_psf, where sites are optically resolved, and (ii) a ≲ r_psf, where PSFs strongly overlap. In the challenging overlapping regime (a ≈ r_psf, signal‑to‑noise ratio ≈ 15 dB), the standard Wiener deconvolution yields a detection error rate (DER) of 1.2 % ± 0.2 %, whereas the proposed OLE achieves 0.2 % ± 0.1 %, a six‑fold improvement. In the resolved regime the OLE matches or slightly exceeds the Wiener method.

Computationally, the OLE requires solving a linear system A Δx = b with A = MᵀΣ_n⁻¹M + Σ_x⁻¹. The authors employ a conjugate‑gradient solver with an incomplete LU preconditioner, achieving near‑linear scaling with the number of sites. For a 100 × 100 array (10⁴ sites) the total runtime on a standard desktop is below 100 ms (≈ 80 ms on average), making the method suitable for real‑time feedback and adaptive experiments.

A further contribution is a rigorous definition of the signal‑to‑noise ratio (SNR) tailored to the atom‑detection problem. This SNR, derived from the statistical model of x and n, correlates strongly with the DER and can be used during experimental design to predict the required numerical aperture (NA) or minimal inter‑site spacing to achieve a target error rate. For example, the authors show how to compute the minimal NA that guarantees DER < 0.5 % for given atom brightness and background conditions.

In summary, the paper introduces an optimal linear‑filter approach that dramatically improves atom‑detection accuracy in densely packed micro‑trap arrays while maintaining sub‑100 ms processing times. By integrating Bayesian updates of site occupancies and providing a quantitative SNR‑error relationship, the method offers both practical speed and a principled framework for designing next‑generation quantum‑simulation and quantum‑computing platforms based on large‑scale atomic arrays.