Coincidence measurement, based on measuring the second-order coherence g₂, has emerged as a powerful technique in optical imaging, enabling extraction of sample features like reflection/transmission amplitude and phase delay. However, accurate g₂ measurement requires many photons, posing challenges under low-light conditions. Two-photon imaging using coincidence measurement has found applications in distinguishing between classical and quantum light sources, identifying entangled pairs, quantum state characterization, and various imaging modalities including hologram generation and super-resolution. Metasurfaces, offering fine control over light's degrees of freedom, are valuable platforms in imaging. In quantum imaging, metasurfaces facilitate high-dimensional entangled state generation, control over two-photon interference, and quantum state tomography. Jones matrix imaging, crucial for studying light polarization, traditionally necessitates numerous measurements. Recent work simplifies this using Fourier ptychography and Fourier space sharing, while metasurface-driven methods leverage Hong-Ou-Mandel (HOM) interference to extract unknown object polarization responses. Machine learning (ML) techniques, particularly deep learning (DL), have proven beneficial in quantum optics, offering higher efficiency and accuracy than conventional methods in classifying light sources and extracting spatial modes, even under low-light conditions. A DL-assisted approach for antibunching super-resolution imaging, predicting g₂(0), demonstrates significant speed enhancements. However, limited photon collection makes accurate g₂(0) identification challenging. This research addresses these limitations by developing a DL-assisted approach for coincidence imaging under low-light conditions, enabling direct extraction of Jones matrix profiles from photon arrival data without manually derived algorithms.

The paper extensively reviews existing methods in coincidence imaging and Jones matrix characterization. It highlights the use of second-order coherence (g₂) measurements in two-photon imaging, discussing its applications in various imaging techniques such as hologram generation and super-resolution microscopy. The role of metasurfaces in manipulating light's degrees of freedom and their application in quantum imaging is detailed, including examples of image edge detection and imaging metasurface polarization responses. The limitations of traditional Jones matrix characterization techniques and recent advancements using Fourier ptychography, Fourier space sharing, and HOM interference are also discussed. Finally, the paper explores the application of machine learning in quantum optical applications, including quantum device optimization, automation of quantum experiments, and quantum state tomography, emphasizing the benefits of deep learning in improving efficiency and accuracy, particularly under low-light conditions.

The proposed deep learning methodology combines a variational autoencoder (β-VAE) with a regression network. The experimental setup involves illuminating a sample (a metasurface with reference and object regions) with a photon pair with orthogonal circular polarizations and an optical delay (τ). The transmitted photons are detected by a single-photon avalanche diode (SPAD) camera. The reference region comprises four panels with known polarization responses, while the object region has unknown Jones matrices to be extracted. The SPAD camera records binary photon arrival images, with coincidence events occurring when two pixels detect photons within the same timeframe. The second-order coherence g²(τ) is obtained by scanning the optical delay. The previous semi-analytic algorithm for Jones matrix imaging, based on fitting g²(τ), is described, highlighting its limitations, particularly under low-light conditions. The deep learning approach uses numerical simulation to generate training data. Various configurations of the three degrees of freedom (DOFs) of the Jones matrix (transmission amplitude, two angles characterizing the off-diagonal elements) are randomly generated. For each configuration, the probabilistic photon arrival events, including two-photon interference, are simulated, resulting in a correlation matrix C that records coincidence events between the object pixel and each reference pixel in each timeframe. The correlation matrix is flattened and input into the β-VAE. The β-VAE, consisting of an encoder and a decoder, aims to find a minimal representation of the input data in the form of latent variables. The loss function combines mean squared error (MSE) and Kullback-Leibler divergence, balancing reconstruction quality and disentanglement of latent variables. A regression network is subsequently trained to map the meaningful latent variables (identified by their distribution parameters) to the three DOFs of the Jones matrix. Experimental data from the SPAD camera is then used to test the trained networks and extract Jones matrix images.

The proposed deep learning approach significantly outperforms the previous semi-analytic algorithm derived from g₂(0) in both efficiency and accuracy. The β-VAE successfully identifies the number of degrees of freedom (DOFs) inherent in the data, enabling assessment of information sufficiency in the experimental procedure. In a single DOF case, the deep learning method achieves lower error and faster convergence than the fitting algorithm, extracting a clear image with only 30 timeframes (14 photons per pixel). In the three DOF case, the deep learning approach produces a clear image with approximately 400 timeframes (88 photons per pixel), demonstrating a substantial improvement over the fitting algorithm. The analysis of error components reveals that the deep learning method significantly reduces noise, especially in the brightness channel (representing the overall transmission amplitude). The deep learning approach provides an upper bound for performance achievable by analytical algorithms, guiding the design of imaging equipment and algorithms.

The deep learning approach addresses limitations of previous methods by directly extracting Jones matrix information from photon arrival data, eliminating the need for complex analytical models and reducing the number of required photons. The ability of the β-VAE to assess information sufficiency is a crucial advantage, aiding in the design of optimal experimental procedures. The superior performance compared to the semi-analytic algorithm suggests potential improvements in the analytical model itself. The method's applicability extends beyond quantum imaging, suggesting potential use in various fields such as medical imaging and ultrasound.

This paper presents a novel deep learning approach for coincidence imaging that significantly improves the efficiency and accuracy of Jones matrix extraction under low-light conditions. The use of a β-VAE enables automatic assessment of experimental information sufficiency, guiding the design of optimal imaging systems. Future research could focus on further improving the accuracy and robustness of the method by incorporating more sophisticated deep learning architectures and exploring its applicability to other imaging modalities and sample types.

The current study relies on numerical simulations for training data, which may not perfectly reflect real-world experimental conditions. The presence of dead pixels in the SPAD camera was addressed through simple averaging of neighboring pixels, which could be improved by more advanced techniques. The generalization capability of the trained model to samples with significantly different characteristics needs further investigation.