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

Using the zero time-delay second-order correlation function for studying the photon statistics, we investigate how the photon statistics of the field-modes generated by parametric down-conversion (PDC) process depends on the photon statistics of the pump field-mode. We derive general expressions for the zero time-delay second-order correlation function of the down-converted field-modes for both multi-mode and single-mode PDC processes. We further study these expressions in the weak down-conversion limit. We show that for a two-photon two-mode PDC process, in which a pump photon splits into two photons into two separate field-modes, the zero time-delay second-order correlation function of the individual down-converted field-modes is equal to twice that of the pump field-mode. Furthermore, for an $n$-photon $n$-mode down-conversion process, in which a pump photon splits into $n$ photons into $n$ separate field-modes, the zero time-delay second-order correlation function of the individual down-converted field-modes is equal to $2^{(n-1)}$ times that of the pump field-mode. However, in contrast to the multi-mode PDC processes, for a single-mode PDC process, in which a pump photons splits into two or more photons into a single mode, the zero time-delay second-order correlation function of the down-converted field-mode is not proportional to that of the pump in the weak down-conversion limit. Nevertheless, we find it to be inversely proportional to the average number of photons in the pump field-mode.