% !TEX root = homodyne.tex

From our previous characterization of the source~\cite{Srivathsan:2013}, 
we know that
the correlation function between the signal and idler $g^{(2)}_{si}(\Delta
t_{si})$ has the shape of a decreasing exponential, with more than 98\% of the coincidences occuring within a time window $T_c$=30\,ns. 
We record a histogram $G^{(2)}_{\text{i1i2$|$s}}(\Delta t_{12})$ of
idler detection events on $D_{\text{i1}}$ and $D_{\text{i2}}$ with
a time difference $\Delta t_{12}=t_2-t_1$ if
one of them occurs within a coincidence time window $T_c$
after the detection of a heralding event in the signal mode. 
The normalized correlation function of heralded coincidences between
the two idler modes is
\begin{equation}
g^{(2)}_{\text{i1i2$|$s}}(\Delta t_{12}) =G^{(2)}_{\text{i1i2$|$s}}(\Delta
t_{12}) / N_{i1i2|s}(\Delta t_{12}),
\end{equation}
where $N_{i1i2|s}(\Delta t_{12})$ is the estimated number of
accidental coincidences. 
Due to the strong temporal correlation between signal and idler photons, the probability of accidental coincidences is not uniform.
We thus estimate $N_{i1i2|s}(\Delta t_{12})$ for every $\Delta t_{12}$ by
integrating the time difference histograms between the signal and each arm of
the HBT, $G^{(2)}_{si1}\left(\Delta t_{si}\right)$ and
$G^{(2)}_{si2}\left(\Delta t_{si}\right)$ within $T_c$
\textcolor{red}{
	normalized to the total number of triggers $N_s$.
}
Due to the time ordering of the cascade  process, it is only meaningful to consider positive time delays after the detection of the heralding photon,
thus splitting $N_{i1i2|s}$ into two cases. For $\Delta t_{12}\geq0$, we use
\begin{equation}
  N^{(+)}_{i1i2|s}(\Delta t_{12})= \frac{1}{N_s}\int\limits_0^{T_c}
  G^{(2)}_{si1}\left(\Delta t_{si}\right)
  G^{(2)}_{si2}\left(\Delta t_{si}+\Delta t_{12}\right)\; d\Delta t_{si}
\end{equation}
while for $\Delta t_{12}<0$, we use
\begin{equation}
  N^{(-)}_{i1i2|s}(\Delta t_{12})=\frac{1}{N_s}\int\limits_0^{T_c}
  G^{(2)}_{si1}\left(\Delta t_{si}+\Delta t_{12}\right)
  G^{(2)}_{si2}\left(\Delta t_{si}\right)\; d \Delta t_{si}.
\end{equation}
