Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion

In this work we propose and analyze a scheme where the full spatiotemporal correlation of twin photons or beams generated by parametric down-conversion is detected by using its inverse process,i.e.,sum frequency generation. Our main result is that,by imposing independently a temporal delay t and a transverse spatial shift x between two twin components of PDC light,the up-converted light intensity provides information on the correlation of the PDC light in the full spatiotemporal domain and should enable the reconstruction of the peculiar X-shaped structure of the correlation predicted previously [A. Gatti et al. Phys. Rev. Lett. 102,223601 (2009); L. Caspani et al. Phys. Rev. A 81,033808 (2010); E. Brambilla et al. Phys. Rev. A 82,013835 (2010)]. Through both a semianalytical and a numerical modeling of the proposed optical system,we analyze the feasibility of the experiment and identify the best conditions to implement it. In particular,the tolerance of the phase-sensitive measurement against the presence of dispersive elements,imperfect imaging conditions,and possible misalignments of the two crystals is evaluated.