Wide-field quantum imaging, leveraging the nonclassical properties of entangled photons, holds promise for surpassing the limitations of classical microscopy. Techniques like ghost imaging, quantum holography, and quantum optical coherence tomography have been explored, but have been hindered by low spatial resolution, slow acquisition speeds, and poor contrast-to-noise ratios (CNRs). This limitation stems from challenges in effectively utilizing entangled photon sources and the complexities of coincidence detection with high-resolution imaging systems. While entangled biphoton sources offer nonclassical advantages, previous attempts to implement wide-field quantum imaging at the microscopic level have faced challenges with existing detectors. Single-photon avalanche diodes (SPADs) offer direct coincidence measurements but lack spatial resolution, while SPAD arrays have limited pixel counts. Electron multiplying charge-coupled devices (EMCCDs) provide high pixel counts but have low frame rates, leading to impractically long acquisition times for coincidence measurements. Previous methods for extracting coincidence counts from EMCCDs required millions of frames, resulting in acquisition times exceeding 17 hours. This significantly hampered the application of quantum imaging techniques to high-resolution microscopy. In contrast to classical wide-field imaging, quantum imaging offers benefits such as enhanced stray light resistance, increased two-photon absorption, and improved resolution due to quantum correlations. However, these advantages have not been fully realized in high-resolution wide-field microscopy due to low light intensities and associated difficulties in coincidence measurements. Quantum entanglement theoretically allows for resolution improvements beyond the classical diffraction limit, with N entangled photons potentially offering N-fold resolution enhancement (Heisenberg limit). NOON states and spontaneous parametric down-conversion (SPDC) sources have demonstrated two-fold resolution enhancements, but these methods typically require co-propagating photons through the object, potentially causing damage to delicate biological samples. Recent work suggests that the Heisenberg limit could be reached even without both photons traversing the object. Building on these prior efforts, this study presents Quantum Microscopy by Coincidence (QMC) with balanced pathlengths, addressing the limitations of existing techniques to achieve high-speed, high-resolution, and stray light resistant quantum microscopy, pushing quantum imaging to the microscopic level.

Existing wide-field quantum imaging methods, such as those using EMCCDs, face significant limitations in speed and resolution. Previous studies, while demonstrating macroscopic quantum imaging, suffered from low numerical apertures (NAs) and slow acquisition rates, rendering them unsuitable for practical microscopy. Methods utilizing polarization entanglement and quantum holography have shown promise but lacked the high-resolution capabilities necessary for microscopy. The challenge lies in balancing the need for high-resolution imaging with the demands of efficient coincidence detection using low-intensity light sources. Theoretical work has explored achieving Heisenberg-limited resolution using entangled photons, demonstrating two-fold resolution improvements with biphoton NOON states and SPDC sources. However, these methods often require both photons to interact with the object, potentially causing damage or alteration, particularly for sensitive biological samples. This research builds upon the concept of coincidence detection using an EMCCD camera but introduces critical advancements to overcome the limitations of previous work, notably in terms of imaging speed and resistance to stray light.

The researchers developed Quantum Microscopy by Coincidence (QMC) using a high-NA microscopy design and an improved algorithm for coincidence detection. The experimental setup employs a spontaneous parametric down-conversion (SPDC) source to generate entangled photon pairs. A prism splits the signal and idler photons into two arms of the microscope, ensuring balanced optical pathlengths. This symmetry is crucial for achieving Heisenberg-limited resolution. A high-NA objective lens is integrated into each arm to achieve high spatial resolution. The signal photon arm is directed towards the sample, while the idler photon arm passes to the detector. An EMCCD camera simultaneously detects both photons. A crucial aspect of the methodology is a novel covariance algorithm for efficient coincidence intensity estimation. This algorithm leverages the correlation between the signal and idler photon intensities to distinguish true coincidences from noise and stray light. The algorithm processes data from the EMCCD camera, where the left and right regions detect the signal and idler photons, respectively. The coincidence intensity (Icoin) is related to the total intensity (I) and noise intensity (Inoise). The covariance between the left (IL) and right (IR) regions is calculated, and under the assumption that noise is uncorrelated between the regions, this covariance directly estimates the coincidence intensity. This significantly improves the speed and accuracy of coincidence detection compared to previous methods that required extensive frame averaging. The spatial resolution enhancement is quantified by measuring the full width at half maximum (FWHM) of the line spread functions (LSFs) in QMC and classical images of a USAF resolution target. The theoretical framework supports the experimental findings by modeling the QMC image as a function of object transmission, illumination distribution, and point spread function (PSF) with an effective wavelength of λ/2, contrasted with classical imaging using a wavelength of λ. This theoretical analysis demonstrates that the balanced pathlength configuration in QMC leads to a two-fold resolution improvement compared to classical microscopy, achieving the Heisenberg limit. Cancer cells were imaged using both classical and QMC methods to demonstrate the effectiveness of the technique for bioimaging. Data acquisition and processing were handled by custom LabVIEW and MATLAB scripts.

The key findings demonstrate significant improvements in quantum microscopy compared to existing wide-field quantum imaging techniques. The QMC method achieved a spatial resolution of 1.4 µm, a two-fold improvement over the classical resolution of 2.9 µm, demonstrating Heisenberg-limited resolution. This improvement is attributed to the balanced optical pathlengths in the symmetric design, effectively halving the equivalent wavelength of the biphoton. The QMC method also shows substantial advantages in speed and CNR. To reach a CNR of 3, QMC required 10⁵ frames (10 ms/frame), about 40% and 20% of the frames needed by previous methods. With 2 × 10⁶ frames, QMC outperformed those methods with 1.5x and 2.6x higher CNRs. The covariance algorithm proves effective in suppressing uncorrelated noise, including stray light. While existing methods were sensitive to stray light, QMC maintained a CNR > 1 even with stray light 120 times stronger than the classical signal (10⁵ frames). With 2 × 10⁶ frames, QMC suppressed stray light up to 155 times stronger than the classical signal. The QMC images of cancer cells revealed cellular structures that were not resolvable in the classical images. This highlights the potential of QMC for non-destructive bioimaging due to its low-intensity illumination. The combination of improved speed, CNR, stray light resistance, and super-resolution positions QMC as a powerful tool for bioimaging applications.

The results demonstrate that QMC successfully achieves Heisenberg-limited resolution in microscopy, surpassing the limitations of previous wide-field quantum imaging methods. The balanced pathlength configuration is crucial for this improvement, ensuring that the entangled photon pairs maintain their positional and momentum correlations throughout the optical path. The symmetry of the setup ensures that the phases of the paired photons are combined constructively. This path symmetry is a uniquely quantum phenomenon, as classical sources cannot correlate both position and momentum due to the uncertainty principle. The covariance algorithm effectively extracts the coincidence intensity, efficiently filtering out uncorrelated noise and enhancing the CNR. The enhanced resolution, speed, and stray light resistance of QMC, validated by both theoretical modeling and experimental imaging of cancer cells, make it a promising technique for non-destructive bioimaging. While current implementation does not yet surpass state-of-the-art classical microscopy in terms of CNR due to the SPDC source's efficiency limitations, future advancements in quantum light sources could significantly enhance performance, enabling QMC to outperform classical methods.

This study successfully demonstrates quantum microscopy of cancer cells at the Heisenberg limit using a novel QMC technique with balanced pathlengths. QMC exhibits significant advantages over existing methods in terms of resolution (1.4 µm), speed (up to 5x faster), CNR (2.6x higher), and stray light resistance (10x more robust). Low-intensity illumination makes QMC suitable for non-destructive bioimaging at the cellular level. Future work should focus on improving the efficiency of the entangled photon source to further enhance the CNR and reduce acquisition time, enabling QMC to compete with and surpass the capabilities of state-of-the-art classical microscopy techniques. Exploring QMC's applicability to other types of biological samples and imaging modalities represents a promising avenue for future research.

The current implementation of QMC is limited by the low efficiency of the SPDC source. This limitation results in longer acquisition times compared to classical imaging. Improvements in quantum light sources are needed to fully realize the potential of QMC to surpass the speed and CNR of classical techniques. The study's experimental setup and sample preparation methods might also influence the obtained results, and further optimization might be needed for broader applications. The current model might need refinement to better account for the complexities of the experimental setup and the effects of scattering.