Machine learning (ML) techniques, particularly deep learning, offer efficient data processing capabilities with applications in diverse fields, including pattern recognition and classification of quantum many-body phases. In experimental quantum physics, ML's strength lies in extracting information from data by simplifying it to essential features. Examples include classifying topological phases using limited experimental observables, investigating Fermi-Hubbard model phase diagrams using density snapshots, and optimizing experimental cooling techniques. These examples showcase ML's potential for faster and more accurate data analysis, even with noise or imperfections. The use of ML has been theoretically explored in quantum gas microscopy to reduce stringent experimental imaging requirements. This work demonstrates a deep-learning architecture for reconstructing optical lattice occupation from fluorescence images, benchmarked with experimental data. Quantum gas microscopy provides unprecedented control and observation of quantum many-body systems using neutral atoms in optical lattices. Fluorescence imaging is commonly used, where atoms pinned in deep optical lattices scatter photons recorded by a high-resolution imaging system. Site-resolved reconstruction of lattice occupation is essential for extracting observables like counting statistics and correlation functions. Reconstruction accuracy directly impacts observable accuracy, fundamentally limited by the signal-to-noise ratio (SNR) and the ratio (β) of imaging resolution to lattice spacing. Most quantum gas microscopes operate with resolutions close to or slightly worse than the lattice spacing. Previous reconstruction techniques include iterative least-squares approaches (computationally expensive and limited fidelity at low SNR), deconvolution with linear kernels (fast but limited performance due to assumptions violated at short lattice spacings), and image restoration techniques like Wiener filtering and Richardson-Lucy deconvolution (rapidly decreasing fidelity for β ≥ 2). These algorithms aim to invert the convolution of the atomic distribution with the point spread function (PSF), which is ill-conditioned with experimental noise. Neural networks offer advantages in solving such inverse problems due to their nonlinearity and low computational complexity compared to iterative algorithms. A supervised neural network approach has been proposed, but its performance is limited by the simulation accuracy required for training. This paper presents an unsupervised deep-learning algorithm for reconstructing lattice occupation from fluorescence images. The unsupervised nature allows training directly with experimental data, eliminating the need for simulated data. The algorithm's fidelity is benchmarked using data from a cesium quantum gas microscope with a resolution-to-spacing ratio of β = 2.2. The algorithm achieves reconstruction fidelities ≥96% across fillings, reconstructing large images (several thousand sites) in under a second. This enables high-fidelity reconstruction at shorter lattice spacings than previous methods. This is beneficial for advanced lattice configurations and experiments involving dipolar interactions.

Several techniques have been previously developed to reconstruct the lattice occupation from fluorescence images in quantum gas microscopy experiments. Early experiments used iterative least-squares methods, which are computationally expensive and suffer from low fidelity at low SNR. Deconvolution with a linear kernel, designed to minimize overlap with adjacent atoms, is computationally efficient but its performance is limited because it assumes a single kernel that is independent of the number of neighboring atoms. This assumption, however, is violated, especially for short lattice spacings, where density-dependent effects such as superradiance become significant. Image restoration techniques, including Wiener filtering and Richardson-Lucy deconvolution, have also been employed. These methods, however, show a marked decrease in fidelity when the ratio between imaging resolution and lattice spacing (β) exceeds 2. Adding information, such as the discrete lattice grid and a sophisticated noise model, can improve unconstrained deconvolution methods, as demonstrated in a one-dimensional system. All these conventional algorithms aim to invert the convolution of the atomic distribution with the point spread function (PSF) of the imaging system. This deconvolution is inherently ill-conditioned in the presence of noise. Recent work has shown the potential of neural networks to overcome these limitations. Deep learning methods can effectively approximate nonlinear relationships and solve these inverse problems efficiently. Previous studies have proposed supervised neural network approaches for this reconstruction task. However, the supervised nature of these methods requires training data generated from simulations. The accuracy of these simulations ultimately limits the performance of the network when reconstructing experimental data. In contrast, this research introduces an unsupervised approach that avoids this limitation.

This research employs a convolutional autoencoder, a type of artificial neural network, for unsupervised learning and dimensionality reduction. The autoencoder consists of an encoder and a decoder network, connected by a bottleneck layer. The encoder compresses the high-dimensional input image (a section of the lattice) into a lower-dimensional representation in the bottleneck layer, which represents the site occupation. The decoder then reconstructs the input image from this compressed representation. The network architecture is a regularized convolutional autoencoder specifically designed for this task. The input layer receives a 16x16 lattice section (256x256 pixels), which is processed through four convolutional layers with a stride of two, reducing the image size. A final convolutional layer with a stride of one implements the bottleneck layer, producing a 16x16 output matrix representing the binarized lattice occupancy. This concludes the encoder. The decoder then upsamples this representation using three transposed convolutional layers and a final layer to reproduce the original image size. Each convolutional layer utilizes a set of learned kernels, performing discrete convolutions and applying nonlinear activation functions (ReLU except for tanh in the bottleneck and last decoder layer). The multiple convolutional layers create a deeper network, enhancing the network's ability to learn complex features at different length scales. The stride of two in the convolutional layers ensures efficient downsampling and upsampling during encoding and decoding, while maintaining spatial information. The tanh activation function in the bottleneck layer is crucial for achieving a bimodal distribution of values representing empty and occupied lattice sites. The network is trained using a composite loss function that balances reconstruction loss (L1 norm between input and output images) and bottleneck regularization loss. The regularization term penalizes non-binary values in the bottleneck layer, encouraging the network to produce a clear binary representation of site occupation. The optimal regularization strength (λ) and kernel sizes were determined through hyperparameter tuning. The experimental data for training and benchmarking was obtained from a cesium quantum gas microscope. A single layer of ultracold cesium atoms was prepared in a 2D square optical lattice. Fluorescence imaging was performed by increasing the lattice depth and adding optical molasses. Images were taken with an sCMOS camera through a high-resolution microscope. The experimental SNR was approximately 5.2. The lattice vectors were extracted from the Fourier transform of dilute atom images, and the absolute lattice phase was determined by fitting isolated atom positions. The training dataset consisted of approximately 100,000 crops from homogeneous clouds with varying average fillings. The trained encoder is then used for reconstruction. Input images are divided into overlapping crops, each processed by the encoder to obtain deconvolved site counts. These counts are then reassembled and averaged across overlapping regions. A threshold (zero in this case, due to the symmetry of the tanh activation function) is applied to obtain the final binary site occupation. The decoder's output is analyzed by applying different occupation matrices as inputs. This analysis reveals the ability of the network to approximate the experimentally measured PSF and to capture density-dependent effects like superradiance.

The unsupervised deep learning algorithm demonstrates high-fidelity reconstruction of site-resolved lattice occupation in a challenging regime where the lattice spacing is significantly smaller than the imaging resolution (β = 2.2). The algorithm's performance was evaluated through two methods: 1. **Analysis of Deconvolved Site Count Distributions:** The distributions of deconvolved site counts from the bottleneck layer before binarization show a clear bimodal distribution across various fillings, indicating effective separation of empty and occupied sites. The overlap between the two distributions is minimal, especially at low fillings. At half-filling, the data is well-fitted by two Gaussian distributions, yielding an estimated reconstruction fidelity of ~99%. 2. **Double-Exposure Imaging:** By taking two consecutive images of the same atomic sample and comparing the reconstructed occupations, the algorithm's robustness is assessed. A formula is derived to calculate the reconstruction fidelity (F) considering the probability (β) of finding different reconstruction results between the two images and the probability (ps) of hopping and atom loss during imaging. This analysis yields a reconstruction fidelity exceeding 96% across different fillings, reaching nearly 99% at low and high fillings.

The results demonstrate the effectiveness of the proposed unsupervised deep-learning algorithm for single-site reconstruction in quantum gas microscopy, even under challenging conditions of short lattice spacing and limited resolution. The high reconstruction fidelities achieved (≥96%) across various fillings surpass the capabilities of traditional deconvolution methods. The algorithm's unsupervised nature eliminates the need for precise simulations, making it a more practical and versatile tool for experimentalists. The ability to directly train on experimental data significantly simplifies the process and reduces the reliance on accurate parameter estimations of imaging characteristics. The algorithm's speed and efficiency allow for the rapid processing of large datasets, opening new possibilities for real-time analysis and feedback control in quantum gas microscopy experiments. The network's capacity to capture density-dependent effects, such as superradiance, is a significant advancement, allowing for a more accurate representation of the underlying atomic distribution. The development of methods for estimating reconstruction fidelity directly from experimental data, without relying on simulated data, is a valuable contribution to the field. The algorithm's applicability extends beyond cesium atoms and can be readily adapted to other quantum systems.

This work introduces a highly effective unsupervised deep-learning algorithm for reconstructing site-resolved lattice occupation in quantum gas microscopy. The algorithm achieves high fidelity (≥96%) in a challenging experimental regime, outperforming conventional methods. Its unsupervised nature and efficient implementation make it a valuable tool for researchers. Future work could explore the application of this algorithm to other quantum platforms and investigate its performance with even shorter lattice spacings and different atom species.

The current study primarily focuses on a specific experimental setup with cesium atoms and a particular lattice geometry. While the algorithm's unsupervised nature enhances its adaptability, further testing with different experimental parameters (e.g., various lattice types, atom species, and imaging systems) is necessary to fully assess its generalizability. Additionally, although methods are implemented to estimate the reconstruction fidelity from experimental data, there is still a need for a more robust method for quantifying systematic errors. The effect of systematic errors on the fidelity might also need further investigation.