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%       submission to OSA meetings and conferences    %
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\begin{document}

\title{Spectral Compression of Narrowband Single Photons with a Near Resonant Cavity}

\author{Mathias~A.~Seidler$^1$,
  Xi~Jie~Yeo$^1$,
  Alessandro~Cer\`{e}$^1$,
  and Christian~Kurtsiefer$^{1, 2}$}
\address{1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543\\
  2. Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542}
\email{mathias.seidler@gmail.com}

%% Uncomment the following line to override copyright year from the default current year.
\copyrightyear{2019}


\begin{abstract}
We compress the spectrum of narrowband heralded single photons generated by four-wave mixing in cold $^{87}$Rb atoms
using a near-resonant cavity as dispersion medium, 
without reducing the brightness
and
almost matching the atomic linewidth.
\end{abstract}

\ocis{270.5530, 020.1670}

\section{Introduction}
Efficient atom-light interactions at the single quantum level requires matching the bandwidth of the traveling photon with the atomic absorption linewidth.
A common way to match the two bandwidths is to narrow the single photon spectrum with a filter, 
with the unavoidable consequence of reducing the brightness. 
Inspired by techniques to compress ultra-fast pulses~\cite{Lavoie2013,Karpinski2017} we developed a method to compress the bandwidth of already narrowband photons 
using an asymmetric cavity as dispersion medium 
without, in principle, reducing the brightness.


\section{Experiment}
\begin{figure}[htpb]
\centering
\includegraphics[width=0.38\textwidth]{figures/energy_levels.pdf}
\includegraphics[width=0.6\textwidth]{figures/g2s_photon_shape/photon_g2s.eps}
% \includegraphics[width=0.48\textwidth]{figures/spectra/spectra.eps}
\caption{\label{fig:g2}
(Left)
Cascade level scheme for the generation of photon pairs in $^{87}$Rb.
(Right)
Second order time correlation of the photon pairs generated by four-wave mixing before (red)
and after (orange) the dispersion cavity.
}
\end{figure}
We obtain heralded single photons at 780\,nm  
from the photon pairs 
generated by four-wave mixing based on a cascade decay in a cold cloud of $^{87}$Rb (Fig.~\ref{fig:g2}(Left)).
% the 776\,nm photon generated by the first decay to herald a 780\,nm photon.
% We measure the bandwidth of the heralded photons using a fabry-perot cavity with linewidth of 2.7\,MHz.
% we record the coincidences when the idler mode is coupled to a cavity with a linewidth 
% we use a tunable cavity of linewidth 2.7\,MHz.
Due to collective effects in the atomic cloud ~\cite{Srivathsan:2013},
the spectrum of the heralded photon,
measured with a fabry-perot cavity with linewidth of 2.7\,MHz and reported in Fig.~\ref{fig:spectrum},
has a bandwidth of $\approx 20$\,MHz, broader than the natural atomic linewidth of 6\,MHz.

% To measure the bandwidth of the heralded photons we use a tunable cavity of linewidth 2.7\,MHz.
% We measure the bandwidth of the heralded photons recording the transmission rate through a scanning symmetric cavity of linewidth ??MHz.
% The measured spectrum (Fig.~\ref{fig:spectrum}) is broader than the natural atomic linewidth of 6\,MHz due to collective effects in the atomic cloud~\cite{Srivathsan:2013}.

The spectral compression of a single photon requires 
% Information about the time localization of the photon.
a large dispersion of the spectrum and
the time modulation of the phase envelope of the wavepacket.
A near-resonant asymmetric cavity with a bandwidth of 7.3\,MHz 
introduces the large dispersion necessary for the compression,
followed by an electro-optical modulator (EOM) to reshape the phase envelope.
We use the heralding signal to synchronize the single photon wave packet with the EOM modulation.
\newpage

\begin{figure}[!htbp]
\centering
% \includegraphics[width=0.48\textwidth]{figures/g2s_photon_shape/photon_g2s.eps}
\includegraphics[width=0.6\textwidth]{figures/spectra/spectra.eps}
\caption{\label{fig:spectrum}
Spectral profile of heralded photons with (orange) and without (red) compression.
% We measure with respect to the $^{87}$Rb D2 ground state transition.
The solid line is a Lorentzian with full-width half maximum of 8\,MHz.
}
\end{figure}

\section{Conclusion}
After the compression we observe a narrower heralded photon spectrum (Fig.~\ref{fig:spectrum}).
We fit it to a Lorentzian profile and estimate a full-width half maximum of 8\,MHz almost matching 6 MHz, the natural linewidth of the D2 transition of $^{87}$Rb.


We believe that this technique is a useful resource to relax the bandwidth requirements on the narrowband single photon sources.




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