On the Accuracy of Measuring the Initial Size of a Biphoton

Some nontrivial aspects of the accuracy of measuring the diameter of the biphoton birth zone based on the positions of detections of its components are discussed within the framework of an idealized experimental setup. It is assumed that biphotons are born in a plate of nonlinear medium during spontaneous parametric down-conversion of a normally incident beam of initial radiation. The desired diameter determines the entanglement of the biphoton wave function and the pattern of spatial correlations of its components detected in the observation plane. The transformation of the biphoton states by the optical system (in this work, use is made of the simplest version—a thin lens), prior to photodetection, is of fundamental importance. The Fisher information for the problem of determining the size of the biphoton birth zone as a function of the position of the photodetection plane exhibits two maxima of the same magnitude, but in general, of different widths. The correct choice of one of them ensures the highest possible accuracy in measuring the size of the biphoton birth zone with the lowest sensitivity to the position of the photodetection plane. Results are presented for the range from 10–2 to 10–3 cm of the initial biphoton sizes with a realistic choice of the focal length of the lens, the transverse size of the initial radiation beam, and its wavelength. It is shown that, for measuring the initial biphoton size, it is preferable to use data carrying information about the transverse momenta of the biphoton components, rather than on their birth positions, as might be expected. Even more surprisingly, the accuracy of measuring the initial size of a biphoton is generally unrelated to the presence or absence of correlations between the positions where its components are detected.