Biphoton State Reconstruction via Phase Retrieval Methods

The complete measurement of the quantum state of two correlated photons requires reconstructing the amplitude and phase of the biphoton wavefunction. We show how, by means of spatially resolved single photon detection, one can infer the spatial structure of bi-photons generated by spontaneous parametric down conversion. In particular, a spatially resolved analysis of the second-order correlations allows us to isolate the moduli of the pump and phasematching contributions to the two-photon states. When carrying this analysis on different propagation planes, the free space propagation of pump and phasematching is observed. This result allows, in principle, to gain enough information to reconstruct also the phase of pump and phasematching, and thus the full biphoton wavefunction. We show this in different examples where the pump is shaped as a superposition of orbital angular momentum modes or as a smooth amplitude with a phase structure with no singularities. The corresponding phase structure is retrieved employing maximum likelihood or genetic algorithms. These findings have potential applications in fast, efficient quantum state characterisation that does not require any control over the source.