Quantitative phase gradient microscopy with spatially entangled photons

We present an entanglement-based quantitative phase gradient microscopy technique that employs principles from quantum ghost imaging and ghost diffraction. In this method, a transparent sample is illuminated by both photons of an entangled pair - one detected in the near-field (position) and the other in the far-field (momentum). Due to the strong correlations offered by position-momentum entanglement, both conjugate observables can be inferred nonlocally, effectively enabling simultaneous access to the sample's transmission and phase gradient information. This dual-domain measurement allows for the quantitative recovery of the full amplitude and phase profile of the sample. Unlike conventional classical and quantum phase imaging methods, our approach requires no interferometry, spatial scanning, microlens arrays, or iterative phase-retrieval algorithms, thereby circumventing many of their associated limitations. Furthermore, intrinsic temporal correlations between entangled photons provide robustness against dynamic and structured background light. We demonstrate quantitative phase and amplitude imaging with a spatial resolution of 2.76 $μ$m and a phase sensitivity of $λ/100$ using femtowatts of illuminating power, representing the highest performance reported to date in quantum phase imaging. This technique opens new possibilities for non-invasive imaging of photosensitive samples, wavefront sensing in adaptive optics, and imaging under complex lighting environments.