The spatial resolution of conventional far-field optical microscopes is fundamentally limited by the diffraction limit, approximately half the wavelength of the incident light (Rayleigh criterion or Abbe limit). This limitation significantly restricts the ability to visualize fine details in various fields, including biology, physics, chemistry, device engineering, and the semiconductor industry. Overcoming this diffraction limit has been a major goal, leading to the development of super-resolution microscopy (SRM) techniques. These techniques generally work by violating one or more assumptions underlying the diffraction limit, such as homogeneous illumination, linear optical response from a stationary object, and classical optical fields. Established classical SRM techniques include stimulated emission depletion (STED), structured illumination microscopy (SIM), photoactivated localization microscopy (PALM), and stochastic optical reconstruction microscopy (STORM). These methods achieve super-resolution by manipulating the illumination, exploiting non-linearity in the optical response, or by utilizing the stochastic nature of fluorophores. Another avenue to achieve super-resolution lies in exploiting the quantum nature of light. Several quantum schemes using multimode squeezed light and generalized quantum states have been proposed, though these often require complex quantum light sources. A more practical approach focuses on the quantum nature of the emitted light itself. Certain quantum light sources exhibit sub-Poissonian photon statistics, allowing super-resolution by analyzing the autocorrelation function of the emitted light. Measuring the *n*-th order autocorrelation function at zero time delay (g⁽ⁿ⁾(r=0)) from a point source can theoretically reduce the effective point spread function size by a factor of √n. Antibunching-based SRM, which utilizes this principle, can be combined with classical approaches for further resolution enhancement; for example, combining it with image scanning microscopy has yielded a four-fold improvement beyond the diffraction limit. The primary bottleneck in antibunching-based SRM is the time required to acquire the time-resolved photon statistics needed for accurate autocorrelation function determination at zero delay. The accuracy depends on the number of correlated photon detection events, and the time requirement scales exponentially with the order of the autocorrelation function. This makes antibunching-based SRM impractical for in situ or live tissue imaging, where the available photons are limited. To address this, fast and precise methods for determining g⁽²⁾(r=0) are crucial for realizing scalable and practical antibunching-based SRM.

Existing super-resolution techniques, such as STED, SIM, PALM, and STORM, rely on classical approaches to surpass the diffraction limit. These methods have limitations, such as the need for specialized equipment or complex image processing. Quantum approaches offer a different perspective. Previous research has explored the use of complex quantum states of light, such as squeezed light, as illumination sources for enhancing resolution. However, these methods often require highly efficient and deterministic sources of entangled photon pairs, which can be challenging to implement. Alternatively, antibunching-based super-resolution microscopy leverages the quantum nature of the emitted light itself, particularly the sub-Poissonian statistics exhibited by some quantum emitters. This approach is attractive because it relies on the properties of the quantum emitters, rather than demanding sophisticated light sources. However, the existing methods suffer from long acquisition times, limiting their applicability to fast dynamic processes or in vivo studies. The current paper builds on previous work on machine learning for rapid classification of quantum emitters by proposing a regression model that speeds up the determination of the autocorrelation function, a key factor limiting the application of antibunching-based super-resolution microscopy.

The researchers developed a machine learning (ML)-assisted approach to accelerate antibunching-based super-resolution microscopy. The core of their method is a convolutional neural network (CNN) regression model designed to rapidly estimate the second-order autocorrelation function at zero time delay, g⁽²⁾(0), from sparse data. This avoids the time-consuming Levenberg-Marquardt (L-M) fitting traditionally used. The experimental setup utilized single nitrogen-vacancy (NV) centers in nanodiamonds as single-photon emitters. A Hanbury-Brown-Twiss (HBT) interferometer measured the photon correlations. The CNN was trained on a dataset of sparse autocorrelation histograms (short acquisition times, e.g., 1 second) obtained from multiple NV centers. Data augmentation was employed to expand the training dataset by creating synthetic histograms with longer acquisition times (up to 10 seconds) by combining shorter ones, assuming the emission process is memoryless over times exceeding 1 second. The ground truth g⁽²⁾(0) values were determined using L-M fitting of longer acquisition datasets. The CNN architecture consisted of an input layer (215 nodes representing the time bins of the histogram), three convolutional layers, a max-pooling layer with dropout, three fully connected layers, and an output node predicting g⁽²⁾(0). The total number of detected events (N_events) was used as an additional input to regularize the model. The network was trained using the Adamax optimizer and mean absolute percentage error (MAPE) loss function for 100 epochs. Performance was evaluated using MAPE, the coefficient of determination (R²), and root mean square error (RMSE). In the experimental demonstration, the researchers scanned an area of interest containing single or multiple NV centers, acquiring sparse HBT data at each pixel. The CNN then rapidly estimated the g⁽²⁾(0) map, which was used to reconstruct the super-resolved image. The results were compared to those obtained using traditional L-M fitting of complete autocorrelation histograms. The spatial resolution was assessed by measuring the full width at half maximum (FWHM) of the intensity distributions.

The trained CNN regression model demonstrated significantly improved performance compared to traditional L-M fitting on sparse datasets. Specifically, for 5-second acquisition time histograms, the CNN achieved a MAPE of 5%, R² of 93%, and RMSE of 0.0018, while L-M fitting resulted in a MAPE of 32%, R² of 70%, and RMSE of 0.215. The CNN's performance remained robust even when reducing the acquisition time to 6 or 7 seconds, consistently outperforming L-M fitting. The experimental demonstration showed that the ML-assisted antibunching SRM achieved a √2 improvement in spatial resolution compared to the diffraction-limited image. For a single NV center, the FWHM of the diffraction-limited image was 310 nm, while the ML-assisted approach yielded a FWHM of 219 nm using 7-second acquisitions. The method demonstrated robustness against reducing the acquisition time to 5 and 6 seconds, maintaining a similar resolution gain. Furthermore, the ML-assisted approach successfully resolved two closely spaced NV centers (separated by ≈600 nm), demonstrating a √2 resolution improvement (FWHM of ≈465 nm for the original image, and 330 nm for the super-resolved image). Monte Carlo simulations confirmed these enhanced resolution results for both two and three closely spaced emitters. Critically, the ML-assisted approach provided a 12-times speedup compared to the conventional L-M fitting method, which required at least 1 minute per pixel.

The results demonstrate the effectiveness of the machine learning-assisted approach for significantly accelerating antibunching-based super-resolution microscopy. The CNN model successfully predicts g⁽²⁾(0) values accurately from sparse data, circumventing the limitations of conventional L-M fitting that requires extensive data acquisition. The √2 improvement in resolution, confirmed experimentally and through simulations, validates the core principle of antibunching-based super-resolution microscopy. The substantial speedup (12 times) is a crucial advance, opening the door for real-time or in vivo applications that were previously infeasible. The robustness of the method against reducing the acquisition time indicates its potential for broader applicability and adaptability to various experimental conditions and quantum light sources. This work contributes to the practical realization of scalable quantum super-resolution imaging devices, pushing the boundaries of optical microscopy and its capabilities for resolving nanoscale structures.

This research presents a significant advancement in quantum super-resolution microscopy by integrating machine learning to dramatically reduce acquisition time. The successful implementation of a CNN regression model for rapid g⁽²⁾(0) estimation leads to a 12-fold speed increase compared to conventional methods, while maintaining a √2 resolution enhancement. This breakthrough enables real-time imaging of dynamic processes and in vivo applications. Future work could explore the extension of this approach to higher-order autocorrelation functions and the application to diverse quantum emitters and biological systems. Investigating strategies to address the reduced signal-to-noise ratio associated with higher-order correlations would further expand the capabilities of this powerful technique.

The current study focuses on single and dual NV center systems. The generalizability of this machine learning model to other types of quantum emitters or imaging modalities requires further investigation. While data augmentation using memoryless assumption helped in increasing the dataset size, it is crucial to assess whether the assumption holds true across a wider range of emitters and conditions. The performance of the CNN is dependent on the quality of the training dataset; careful optimization and validation of the dataset are crucial to ensure robustness. The study assumed memoryless emission processes for data augmentation; this assumption needs further validation for different emission characteristics.