Phase retrieval algorithms applied to the reconstruction of photonic quantum states

Abstract

Reconstructing a wavefront is a problem that appears in many areas of physics and engineering, and leads to many important applications. X-ray crystallography, electron microscopy, femtosecond laser temporal characterization, blind deconvolution of degraded images, and tomographic imaging are examples of areas that, if not directly involving the reconstruction of a wavefront, have beneted from the techniques developed to solve this problem. In general, the measurement devices are able to record the intensities of the wavefront, but not its phases; the numerical tools used to recover them thus became known as phase retrieval algorithms. In this work we propose the use of these algorithms to reconstruct pure quantum states encoded into transverse spatial modes of single photons { the so-called spatial qudits. We made significant adaptations on the algorithms found in the literature in orderto fit experimental features of this kind of encoding. The most striking of these was to magnify Fourier-plane amplitudes in order to compensate for the small range of sampled frequencies, leading to great improvements in the quality of the results. To demonstrate this technique, we performed a proof-of-principle experiment with an optical beam mimicking spatial qudits states of dimensions D = 2; 3; 4; 7 and 9. Aftercompensating for some experimental deviations, the recovered states presented fidelities frequently above 99% with respect to the target state, showing that the phase retrieval algorithms may be a useful tool for quantum state characterization.