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Using the zero time-delay second-order correlation function for studying thephoton statistics, we investigate how the photon statistics of the field-modesgenerated by parametric down-conversion (PDC) process depends on the photonstatistics of the pump field-mode. We derive general expressions for the zerotime-delay second-order correlation function of the down-converted field-modesfor both multi-mode and single-mode PDC processes. We further study theseexpressions in the weak down-conversion limit. We show that for a two-photontwo-mode PDC process, in which a pump photon splits into two photons into twoseparate field-modes, the zero time-delay second-order correlation function ofthe individual down-converted field-modes is equal to twice that of the pumpfield-mode. Furthermore, for an $n$-photon $n$-mode down-conversion process, inwhich a pump photon splits into $n$ photons into $n$ separate field-modes, thezero time-delay second-order correlation function of the individualdown-converted field-modes is equal to $2^{(n-1)}$ times that of the pumpfield-mode. However, in contrast to the multi-mode PDC processes, for asingle-mode PDC process, in which a pump photons splits into two or morephotons into a single mode, the zero time-delay second-order correlationfunction of the down-converted field-mode is not proportional to that of thepump in the weak down-conversion limit. Nevertheless, we find it to beinversely proportional to the average number of photons in the pump field-mode.

Dependence of the photon statistics of down-converted field-modes on the photon statistics of pump field-mode