Imaging walk-off-driven distortions in EPR photon-pair correlations

Spontaneous parametric down-conversion is the primary source of position-correlated and momentum-anti-correlated photon pairs that form the canonical Einstein-Podolsky-Rosen (EPR) state. Their transverse spatial correlations are usually analyzed within the thin-crystal approximation, where the two-photon wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. In practice, however, birefringence-induced transverse walk-off breaks this factorization and couples these degrees of freedom. Here, we show that this coupling persists even for nominally thin crystals once the free-space propagation of the joint spatial intensity is taken into account. This sum-difference coordinate coupling leads to a distinctive tapering of the transverse correlations near the crystal image plane - an effect that standard factorized models cannot capture. Numerical simulations and experimental data clearly confirm this novel behavior. Our findings provide a more complete description of photon-pair generation in birefringent nonlinear media and clarify fundamental limits on spatially resolved quantum imaging and spatial-mode quantum information processing with EPR states.