%!TEX root = hom_main.tex
\section{Theory - Time envelope matching}
	We now derive the expected visibility as function of the time constants of the exponential decay time envelopes of the interfering photons
	% where the latter equality holds 
	for $\Delta T\,=\,t_s-t_f\,=\,0$, i.e. when the heralding time and the single atom excitation time are synchronized.
	% 
	% from 
	The probability of coincidence counts for unit time
	% as a function of the detection time difference
	$G(\Delta t_{ab})$
	in the non-interfering case, i.e. photons with orthogonal  polarization,
	is given by:
	\begin{align}\label{eq:Gperp}
		G_{\perp}(\Delta t_{ab})\,=\,\frac{1}{4}\int\limits^{\infty}_{-\infty} &|\psi_{f}(t)\,\psi_{s}(t+\Delta t_{ab})|^2 \nonumber \\ 
		& + |\psi_{f}(t+\Delta t_{ab})\,\psi_{s}(t)|^2\,dt \,.
	\end{align} 
	When the two incident photons have identical polarizations instead:
	% , the degree of interference is determined by their temporal overlap with the $G$ given by
	\begin{align}\label{eq:Gpll}
		G_{||}(\Delta t_{ab})\,=\,\frac{1}{4}\int\limits^{\infty}_{-\infty} 
		&|\psi_{f}(t) \psi_{s}(t+\Delta t_{ab})\nonumber \\
		&-\,\psi_{f}(t+\Delta t_{ab}) \psi_{s}(t)|^2\,dt\,.
	\end{align}
	The total probability $P$
	is be obtained by integrating over time
	$P=\int G(\Delta t_{ab})d(\Delta t_{ab})$
	% , where $\Delta t_{ab}$ is the time difference between detection events at $D_a$  and $D_b$.
	% at the two output arms of the BS.
	% When the two incident photons have orthogonal polarizations, they do not interfere and the $G$ is given by
	% We evaluate Eq.~\eqref{eq:Gperp} and Eq.~\eqref{eq:Gpll} under this condition as
	% \begin{align}\label{eq:Gperp_0}
	% 	G_{\perp}(\Delta t_{ab})\,=\,&A\,
	% 	\left[
	% 	\left(
	% 		e^{-\frac{\Delta t_{ab}}{\tau_{f}}}+e^{-\frac{\Delta t_{ab}}{\tau_{s}}}
	% 	\right) \,\Theta(\Delta t_{ab}) \nonumber
	% 	\right.\\
	% 	&\left. +\,
	% 	\left(
	% 		e^{\frac{\Delta t_{ab}}{\tau_{f}}}+e^{\frac{\Delta t_{ab}}{\tau_{s}}}
	% 	\right)\,\Theta(-\Delta t_{ab})
	% 	\right]\,,
	% \end{align}
	% \begin{align}\label{eq:Gpll_0}
	% 	G_{||}(\Delta t_{ab})\,=\,&A\,
	% 	\left[
	% 	\left(
	% 		e^{-\frac{\Delta t_{ab}}{2\tau_{f}}}-e^{-\frac{\Delta t_{ab}}{2\tau_{s}}}
	% 	\right)^2\,\Theta(\Delta t_{ab}) \nonumber
	% 	\right.\\
	% 	&\left.
	% 	+\, 
	% 	\left(
	% 		e^{\frac{\Delta t_{ab}}{2\tau_{f}}}-e^{\frac{\Delta t_{ab}}{2\tau_{s}}}\right)^2\,\Theta(-\Delta t_{ab})
	% 		\right]\,,
	% \end{align}
	% where the constant $A\,=\,\eta_{f}\eta_{s}/(4\tau_{f}+4\tau_{s})$.
	% We define the visibility of HOM interference in terms of $G_{\perp}$ and $G_{||}$ as
	Replacing Eq.~\eqref{eq:Gperp} and \eqref{eq:Gpll} into 
	%the definition of visibility given in
	Eq.~\eqref{eq:vis} we obtain:
	\begin{equation} %\label{eq:vis}
		V\,=\,1- \frac{\int G_{||}(\Delta t_{ab}) d(\Delta t_{ab})}
		{\int G_{\perp}(\Delta t_{ab}) d(\Delta t_{ab})}
		=\,1-\frac{\left(\tau_{f}-\tau_{s}\right)^2}{\left(\tau_{f}+\tau_{s}\right)^2} \,,
	\end{equation}
	%EOF