 We measure the signal photon statistics by using a Hanbury-Brown-Twiss (HBT) interferometer conditioned on the detection of the idler photon.  Figure~\ref{fig:homodyne_setup.eps} (a) illustrates the experimental arrangement for the HBT. 

The signal and idler modes are coupled into a single mode fibers.
The HBT is obtained mating  a 50/50 fiber beam splitter to the signal fiber.
The two outputs of the fiber beam splitter are sent to detectors D2 and D3.
All detectors clicks are registered on a timestamp card with a resolution of 0.125\,ps. 
The idler fiber is connected to detector D1, whose clicks provide the heralding trigger for the recording of coincidences in the HBT.
From previous measurement of  the coincidence statistics between the signal and idler arms (\citeref{bharath paper}) we know that 
a time window of 30\,ns is sufficient to consider almost all the correlated events in those two arms. We thus consider coincidence events in the two arms of the HBT only if at least one of the two detection events is detected within 30\,ns after the trigger detection. 

The three detectors used are avalanche photo detectors (APD's) with quantum efficiencies of $\approx 40\%$, dark count rates 40 to 150\,s$^{-1}$).
The combined timing uncertainty of the detectors and time-stamping is $\approx $0.6\,ns.
 We use pump powers of 3.8\,mW for 776\,nm and  400\,$\mu$W for 780\,nm, detuning
${\Delta}\approx40$\,MHz from the intermediate level and MOT of optical density  $\approx32$.

\begin{figure}
  \begin{center}
 \includegraphics[width=\columnwidth]{figures/conditional_g2_762_low.eps}
   \caption{Conditional $\mathrm{g}^{(2)}(\tau)$ measurement on heralded signal photon}
\includegraphics[width=\columnwidth]{figures/conditional_g2_795_low.eps}
   \caption{Conditional $\mathrm{g}^{(2)}(\tau)$ measurement on heralded idler photon}
   \end{center}
\end{figure}


In figure~\ref{fig:conditional_g2_762.eps} we report the normalized statistics of the heralded coincidences between the two arms of the HBT, $\mathrm{g}^{(2)}$, as function of the delay between the arrival of the photon on D2 and D3 $\Delta t_{2,3}=t_3-t_2$ sampled into time bins of width  2\,ns.
The statistics is normalised on the expected coincidence count rates for two sources of photons correlated with the trigger but independent of each other.
To obtain these rates we consider the time correlation between the idler and each of the HBT arms in the 30\,ns time window.

As expected, the value of $\mathrm{g}^{(2)}(0)$ of 0.018$\pm$0.007 is much smaller than one, indicating a strongly non classical state. 
To our knowledge this is the lowest value reported for a heralded single photon sources based on atomic ensembles \cite{Eisaman:2011}.
Assuming a lorentzian temporal shape, we can estimate the bandwidth of the photons in the signal arm from the the measured correlation function.
Fitting the experimental data we obtain a bandwidth of $\Delta \nu = 1/(2\pi\tau_0)=22.72\pm0.48$\,MHz (FWHM).

We repeated the same measurement switching the roles of the signal and idler arms. The resulting normalized correlation is presented in fig.  We obtain a minimum for the correlation of $\mathrm{g}^{(2)}(0)$ of 0.032$\pm$0.004 and an estimated bandwidth of $17.33\pm0.12$\,MHz (FWHM).






