Quantum enhanced phase retrieval

The retrieval of phases from intensity measurements is important in many fields in science, from optical microscopy to x-ray crystallography. In its most common form, phases should be retrieved from the intensity of the far-field diffraction, yet it is known that this is not always possible. For example, for one-dimensional objects, there are many ambiguous phase distributions leading to the same intensity pattern. Here, we present a theoretical and numerical study which shows that nonclassical states of light can be advantageous for phase retrieval. We generalize the wellknown iterative Gerchberg-Saxton algorithm to photon correlation measurements in the output plane rather than the standard intensity measurements. We compare simulations of phase retrieval of a one-dimensional object from its farfield diffraction using classical and quantum light. While the classical algorithm was ambiguous and often converged to incorrect solutions, quantum light produced a unique reconstruction with smaller errors and faster convergence. We attribute these improvements to a larger Hilbert space that constrains the algorithm. Nonclassical states of light, previously known to give better estimation in single-phase measurements, therefore also have an unexpected advantage in retrieving phases of objects from their far-field diffraction.