Propagation of the space-time entanglement of twin photons and twin beams

We investigate how the space-time entanglement of photon pairs generated by multimode parametric down-conversion (PDC) propagates away from the source. We show that the entangled two-photon probability, tightly confined in space and time at the source, quickly broadens and decays out of a focal plane, behaving as a focused coherent beam of approximately twice the wavelength of the twin photons. This behavior is in strong contrast with the much slower decay rate of the classical mutual coherence of the PDC radiation, which follows the prescription of the generalized Van Cittert-Zernicke theorem. Besides their fundamental interest, our results could be relevant for the design of experiments, such as quantum-enhanced two-photon processes, that exploit the spatiotemporal entanglement of the pairs.