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

TL;DR

This paper investigates how the photon statistics of down-converted fields depend on the pump field's photon statistics, deriving general expressions and analyzing different PDC processes, revealing proportionality relations and their limitations.

Contribution

The paper derives general expressions for the zero time-delay second-order correlation function in PDC processes, highlighting differences between multi-mode and single-mode cases.

Findings

In two-mode PDC, the correlation function doubles the pump's value.

In n-mode PDC, the correlation function scales as 2^{n-1} times the pump.

In single-mode PDC, the correlation function is inversely proportional to the pump's average photon number.

Abstract

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.