%!TEX root = hom_main.tex
\section{HOM dip}
	\begin{figure}
		\includegraphics[width=0.9\columnwidth]{figures/HOM_dip/hom_dip_150ns.eps}
		\caption{\label{fig:hom_dip}Normalized coincidence probability $P_{||}/P_{\perp} = 1-V$, corrected for accidental coincidences, showing the ``HOM dip'' with a visibility of %$92\pm3\%$ 
		$86\pm2\%$ at $\Delta T=0$.
		The solid line shows expected values obtained from the Eqs.~\ref{eq:vis} and \ref{eq:Pperp}.
		%	The dashed line at $P_n=1$ shows the expected value for non-interfering photons, while $P_n=0.5$ indicates the quantum limit.\textbf{where do we want to allude to the quantum limit of 0.5 in the text?}
		}
	\end{figure}
	We can also vary the degree of interference by changing the delay $\Delta T$ between the heralding time $t_f$ and the single atom excitation time $t_s$.
	To maintain a constant rate of
	two photon events as we vary $\Delta T$,
	we chose an integration window $T_c=150\,\text{ns}$,
	much larger than maximum value of $\Delta T$ used in the experiment.
	To better see the effect of HOM interference with a large $T_c$,
	we have subtract the rate of accidental counts $G_{\textrm{acc}}$ from the measured $G_{\perp}$ and $G_{||}$.
	Plotting the ratio $P_{||}/P_\perp$ we obtain the familiar HOM dip~\cite{hong:87},
	as shown in Fig.~\ref{fig:hom_dip}.

	From Eqs.~\ref{eq:Gperp} and~\ref{eq:Gpll} we can derive the shape of the dip:
	\begin{equation}\label{eq:dip}
		\frac{P_{||}}{P_\perp} = \left(\frac{\tau_s-\tau_f}{\tau_s+\tau_f}\right)^2
		\exp\left(\frac{\Delta T}{\tau}\right)
		\text{ with }
			\begin{cases}
   				\tau = \tau_f & \text{if } x \geq 0 \\
   				\tau = \tau_s & \text{if } x < 0.
  			\end{cases}
	\end{equation}
	The asymmetric shape of the dip is due to the difference in coherence times of the two incident single photons
	and their strongly asymmetric shape.
	Using Eq.~\ref{eq:dip} and the measured values for $\tau_f$ and $\tau_s$
	we obtain the solid line plotted in Fig.~\ref{fig:hom_dip}.
	All the measured points lay within one standard deviation of this predicted line.
	% The visibility $V$ is then calculated for every point using Eq.~\ref{eq:norm_P_discrete}.
	% Using $T_c=150\,\text{ns}$ and correcting for accidental counts, the measured results produce the familiar HOM dip~\cite{hong:87} (see Fig.~\ref{fig:hom_dip}).
	%The visibility $V$ is then calculated for every point using Eq.~\ref{eq:norm_P_discrete}.
	% Each point is obtained using Eq.~\ref{eq:norm_P_discrete} from the measured $P_{||}$ for every $\Delta T$ and $P_{\perp}$ after subtracting the background $bg$.
	%We also obtain theoretically expected values using Eqs.~\ref{eq:Gperp} , \ref{eq:Gpll} and the measured coherence times (Fig.~\ref{fig:hom_dip}, solid line).
	

%EOF