Correlated quantum Airy photons: An analytical approach

We describe the generation of correlated photon pairs by means of spontaneous parametric down-conversion of an optical pump in the form of a finite-energy Airy beam. The optical system function, which contributes to the propagation of the down-converted beam before being registered by the detectors, is computed. The spectral function is utilized to calculate the biphoton amplitude for finding the coincidence count of the inbound Airy photons in both far-field and near-field configurations. We report the reconstruction of the finite-energy Airy beam in the spatial correlation of the down-converted beams in a near-field scenario. In the far field, the coincidence counts resemble the probability density of the biphoton in momentum space, revealing a direct mapping of the anticorrelation of the biphoton momentum. By examining the spatial Schmidt modes, we also demonstrate that longer crystals have tighter real-space correlations, but higher-dimensional angular correlations, whereas shorter crystals have fewer modes in momentum space and broader multimode correlations in position space.