Universal relation between the conditional auto-correlation function and the cross-correlation function of biphotons.

We systematically studied the relation between the conditional auto-correlation function (CACF) and cross-correlation function (CCF) of biphotons or pairs of single photons. The biphotons were generated from a heated atomic vapor via the spontaneous four-wave mixing (SFWM) process. In practical usage, one single photon of a pair is utilized as the heralding photon, and another is employed as the heralded photon. Motivated by the data of CACF of the heralded photons versus CCF, we proposed a universal formula to predict the CACF. The derived formula was based on general theory and is also valid for the biphoton generation process of spontaneous parametric down-conversion (SPDC). With the formula, we utilized the experimentally determined parameters to predict CACFs, which can well agree with the measured CACFs. The proposed formula enables one to quantitatively know the CACF of heralded single photons without the measurement of Hanbury-Brown-Twiss-type three-fold coincidence count. This study provides a better understanding of biphoton generation using the SFWM or SPDC process. Our work demonstrates a valuable tool for analyzing a vital property of how the heralded photons are close to Fock-state single photons.