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\begin{document}


%\preprint{}

\title{Narrowband source of correlated photon pairs via four-wave mixing in atomic vapor }

\author{Bharath Srivathsan}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\author{Gurpreet Kaur Gulati}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\author{Chng Mei Yuen Brenda} 
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\author{Gleb Maslennikov}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\author{Dzmitry Matsukevich}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\altaffiliation{Department of Physics, National University of Singapore, 3 Science Drive 2,  Singapore, 117543}
\author{Christian Kurtsiefer}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543}
\altaffiliation{Department of Physics, National University of Singapore, 3 Science Drive 2,  Singapore, 117543}
%\author{Bharath Srivathsan, Gurpreet Kaur Gulati, Chng Mei Yuen Brenda, Gleb Maslennikov,}
%\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore, 117543},Dzmitry Matsukevich
%\altaffiliation{Center for Quantum Technologies and Department of Physics,
%  National University of Singapore, 3 Science Drive 2,  Singapore, 117543} , Christian Kurtsiefer
%\altaffiliation{Center for Quantum Technologies and Department of Physics,
%  National University of Singapore, 3 Science Drive 2,  Singapore, 117543}}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
%\email[]{christian.kurtsiefer@gmail.com}
%\date{\today}

\begin{abstract}
%In our setup, a cold Rb^{87} atomic ensemble is pumped by two laser beams (780nm and 776nm) resonant with the $5S_{1/2}\rightarrow 5P_{3/2} \rightarrow 5D_{3/2}$ transition.
%This generates time-correlated photon pairs (776nm and 795nm) by nondegenerate four-wave mixing via the decay path $5D_{3/2}\rightarrow 5P_{1/2} \rightarrow 5S_{1/2}$.
% Coupling the photon pairs into single mode fibres and using silicon APDs, we observe $g^(2)$ of about $2000$ and pairs to singles ratio of 11.2% (2800 photon pairs per second) with an optical bandwidth < $30$ MHz. 

In this Letter we present an experiment to generate time correlated photon pairs via nondegenerate fourwave mixing in a cold $\mathrm{Rb^{87}}$ atomic ensemble.  Coupling generated photon pairs into single mode fibres and using silicon APDs, we observe a $\mathrm{g^{(2)}}(0)$ of about 4750 (?) and pairs to single ratio of 43\% (1200 photon pairs per second). By filtering out uncorrelated 795nm photons, we obtain pairs to 795nm singles ratio of 53\%. Higher pair rate of around 24000 pairs per second is obtained by further increasing 776 pump power. The optical bandwidth of heralded 795nm photons is measured by a cavity (linewidth 2.8 MHz) to be 21.4 MHz for a MOT of optical density 26.
%This Letter presents measurements of the linewidth and photon statistics of photon pairs generated via nondegenerate fourwave mixing in a cold $\mathrm{Rb^{87}}$ atomic ensemble. 
%Many quantum communication protocols require entangled states of distant qubits which can be implemented using photons. To efficiently transfer entanglement from photons to stationary qubits such as atoms, one requires entangled photons with a frequency bandwidth matching absorption profile of atoms. In our setup, a cold $\mathrm{Rb^{87}}$ atomic enemble is pumped by two laser beams (780nm and 776nm) generates time correlated photon pairs (776nm and 795nm) by nondegenrate fourwave mixing. Coupling photon pairs into single mode fibres and using silicon APDs, we observe $\mathrm{G^{(2)}_{SI}}(\tau)$ of about 4750 and pairs to single ratio of 43\% (1200 photon pairs per second). By filtering out ucorrelated 795nm photons, we obtain pairs to 795nm singles ratio of 53\%. Higher pair rate of around 24000 pairs per second is obtained by further increasing 776 pump  power. The optical bandwidth of heralded 795nm photons is measured by a cavity (linewidth 2.8 MHz) to be 21.4 MHz for a MOT of optical density 26. 

\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{37.10.Gh, % Atom traps and guides 
42.50.Ct,       % Quantum description of interaction of light and matter;
                % related experiments 
32.90.+a        % Other topics in atomic properties and interactions of atoms;
                % with photons (restricted to new topics in section 32) 
}

\maketitle

\section{Introduction\label{intro}}
A non-classical, narrowband, correlated photon pair source is essential for experimental realization of many long distance quantum communication protocols with atoms ~\cite{DLCZ:2001,vanderwal:2003}. The basic requirement of such experiments is to have photons with bandwidth compatible with atomic absorption line for efficient atom-photon coupling. Additionally, the source should be heralded single photon source where detection of one trigger photon indicates or heralds the presence of the other photon. A high quality heralded single photon source should have high spectral brightness and a high heralding efficiency. Conventional sources for generating nonclassical photon pairs are based on spontanous parametric downconversion (SPDC) process  in a non-linear crystal. The disadvantage of such process is that the generated photons have linewidth ($\sim$GHz) a few orders of magnitude larger than lifetime limited atomic absorptiion lines ($\sim$MHz). This problem can be overcome by enhancing the down converted mode in SPDC using a high-Q resonator ~\cite{Wolfgramm:08,Fortsch:12} or by spectral filtering of downconverted photons ~\cite{Mitchell:2009} at the cost of reduced generation efficiencies.

Another approach to generate narrowband correlated photon pairs  exploits third order non-linear four wave mixing (FWM) process in atomic ensembles This scheme was first experimentally implemented by Van der wal et al. ~\cite{vanderwal:2003} and Kuzmich et al. ~\cite{Kuzmich:2003}. %It occurs when the two fields  of  different frequencies  are incident on the atoms in the directions defined by $\vec{k_1}$ and $\vec{k_2}$ respectively. 
The two near resonanat pump fields with optical modes defined by $\vec{k_1}$ and $\vec{k_2}$ are mixed inside the atomic ensemble. When idler photon is emitted in the collection mode defined by $\vec{k_i}$, the signal photon is emitted in the phasematched direction defined by $\vec{k_s}$ = $\vec{k_1}$+$\vec{k_2}$-$\vec{k_i}$.  The generated signal and idler photons are also strongly corelated in time, energy and polarization. Recently Willis et al ~\cite{Willis:11} and Ding et al ~\cite{Ding:12}  made use of  hot vapor cell as their non-linear generation media for the emission of nonclassical photon pairs. The disadvantage of using hot vapors is that the photons generated are doppler broadened due to thermal motion of the atoms. The photons of around 400 MHz linewidth are generated in the experiment by Ding et al. 

In this paper, we report the generation of narrow bandwidth non-degenerate photon pairs from cold atomic ensemble of $\mathrm{Rb^{87}}$ atoms in a Magneto-optical trap (MOT). Due to negligible doppler broadening in the MOT, the bandwidth of the photons are further reduced. We work with diamond level scheme (Fig.~\ref{fig:combined_setup_levelscheme.eps}(b)) for generation of photon pairs instead of the double $\lambda$ scheme used by many groups in the past ~\cite{Du2008}. The diamond level scheme allows the generation of photon pairs quite different in frequencies from the pump beams which makes the generated photons spatially seperated. Also It becomes easier to avoid their contamination with pump beams by using narrowband interference filters. We also measure the bandwidth of the generated photons with a scanning fabry-perot cavity.  

The experimental setup is shown in figure~\ref{fig:combined_setup_levelscheme.eps}. % of optical %density of 25 with a total number of atoms $1.3\times10^8$ as measured by absorption %imaging . 
 A tapered amplifier seeded with external cavity diode laser is used to derive the MOT cooling beams. Each cooling beam has an intensity of $\mathrm{20mW/cm^2}$ and is red detuned by 24 MHz from the $\ket{5S_{1/2},F=2\rightarrow 5P_{3/2},F=3}$ transition.%cool down the atoms. The output power from tapered amplifier is  distributed %amongsix MOT beamswith$\mathrm{40mW/cm^2}$ intensity per beam. The MOT beams are %further red detuned by 24 Mhz from$\ket{5S_{1/2},F=2\rightarrow 5P_{3/2},F=3}$ transition.
A repumper laser of power $\mathrm{10\,mW}$ locked to the $\ket{5S_{1/2},F=1\rightarrow 5P_{3/2},F=2}$ transition optically pumps the atoms back into $5S_{1/2},F=2$ level. With an axial quadrupole field of 30 Gauss/cm we obtain a MOT with an optical density of 25 with a total number of atoms $1.3\times10^8$ as measured by absorption imaging.
\begin{figure}[h]
		\begin{center}
			\includegraphics[width=\columnwidth]{figures/combined_setup_levelscheme.eps}
			\caption{\label{fig:combined_setup_levelscheme.eps}(a) Schematic diagram of the experimental set up. (b) The diamond level configuration in $\mathrm{Rb^{87}}$. (c) Timing sequence for one experimental cycle.(d) Scanning fabry perot cavity used to measure the spectral profile of idler 795nm photons.}

		\end{center}
	\end{figure}
Two co-propagating linear orthogonal polarized pump beams of wavelength 780 nm %from  $5S_{1/2}\rightarrow 5P_{3/2}$
 and 776nm %from $5P_{3/2}\rightarrow 5D_{3/2}$  
are incident at a small angle of $0.5^\circ$ onto MOT with directions defined by $\vec{k_1}$ and $\vec{k_2}$ respectively. The pump beams (780nm and 776nm) have optical powers of   $100 \, \mu W$ and $5 \, \mathrm{mW}$  and ($\mathrm{1/e^2}$) beam waists of $\mathrm{0.5\, mm}$ respectively.  %The atomic population is transferred to $5D_{3/2}$ level from the ground state $5S_{1/2}$ via two photon transition.
 The 780 nm pump beam is red detuned from the intermediate state $5P_{3/2}$ to avoid populating this state. The 776 nm pump detuning is chosen to satisfy two-photon resonance with the $5S_{1/2} \rightarrow 5D_{3/2}$ transition. Decay via $5D_{3/2} \rightarrow 6P_{3/2} \rightarrow 5S_{1/2}$ manifests in the emission of blue light at 420 nm. The presence of blue ﬂuorescence is used to check the probability of transferring the atom into the $5D_{3/2}$ state. The experiment is run periodically with a MOT trapping time of 12ms and photon generation window of 1 ms. The repump laser and quadrupole field are always switched on during the whole experiment.
\\We observe the generation of correlated signal and Idler photon pairs at wavelength 762nm and 795nm respectively from MOT in the directions that satisfy the phase matching condition. A 795 nm alignment beam is used to determine the phase matched direction of 762 nm emission. The generated signal and idler photons are filtered out from the pump beams via narrowband interference ﬁlters (F1,F2) in the collection arms. A polarizing beam splitter (PBS) and polarizer (P3) is used to select the polarization of photons that satisfies phase matching condition. A solid etalon (E) of linewidth 375 MHz is used to filter out uncorrelated 795 nm photons from $5P_{1/2}F=2 \rightarrow 5S_{1/2}F=1$  decay path. The generated signal and idler photons are coupled into single mode fibers with coupling efficiency of around 75\% and are sent through fiber 50/50 beam splitter. The beam waists of the signal and idler collection modes are $\mathrm{0.4\, mm}$ and $\mathrm{0.5\, mm}$ respectively inside the MOT. %We perform a Hanbury Brown Twiss experiment in each of the generated photon arms by sending the photons through a fiber 50/50 beamsplitter. The four output ports of the beam splitters are sent to silicon avalanche photodiodes (D1,D2,D3,D4)  with suitable delays to compensate for the deadtime of timestamp unit (125ns) used for coincidence detection. The avalanche photodiodes (Perkin Elemer) have quantum efficiency of ≈ 40\% with dark count rates of 13 per 1 ms gate interval.  
%We measured cross correlation function between generated signal and idler photons as well as auto corelation function in each of the generated photon arm. 
The photons from the four output ports of the beam splitters are detected using silicon avalanche photodiodes (APDs) D1,D2,D3,D4 (Fig ~\ref{fig:combined_setup_levelscheme.eps}(a)). The signal from each APD is recorded with a timestamp unit with suitable delays to compensate for its deadtime (125ns). The avalanche photodiodes (Perkin Elemer) have quantum efficiency of ≈ 40\% with dark count rate of 0.15 counts per ms gate interval. We also measured the spectral profile of generated idler photons with a scanning fabry perot cavity of linewidth 2.8MHz and finesse 4500. The linewidth of the cavity is measured by cavity ringdown experiment ~\cite{Rempe1992}. The cavity is kept inside vacuum under $6\times10^{-6}$mbarr and temperature stabilized with feedback loop.

%begin{figure}
%  \begin{center}
 %   \includegraphics[width=\columnwidth]{figures/setup.eps}
 %  \caption{\label{fig:setup}Experimental setup.}
%\end{center}
%\end{figure}


%\section{Time correlation measurements \label{correlation}}
We measure the two photon coincidence events $\mathrm{G^{(2)}_{SI}}(\tau)$ between signal and idler modes as a function of time delay as shown in fig.~\ref{fig:g2}(a). The peak normalized cross-correlation coefficient ($\mathrm{g^{(2)}_{SI}}$) of $5650 \, \pm \, 300$ indicates a strong temporal correlation. 
The measured $\mathrm{1/e}$ decay time for heralded $795 \, \mathrm{nm}$ photons from the fit is $\mathrm{7.2 \, \pm \, 0.1 \, ns}$, which is lower than the natural lifetime of the atom of $\mathrm{27\, ns}$ in $\mathrm{5P_{1/2},F=2}$ state. This is due to superradiance effect in an optically thick atomic ensembles.
The emission of a 762 nm photon in signal mode puts the ensemble of atoms in intermediate state $5P_{1/2},F=2$. The atoms at different positions in the ensemble along the idler mode $\vec{k_I}$ have strong phase correlations due to the phase-matching condition. This results in collective intensive emission of 795 nm photons from the ensemble with decay time proportional to inverse of optical thickness.~\cite{Dicke:1954,Rehler:1971}
 
 Figs.~\ref{fig:g2}(b) and ~\ref{fig:g2}(c) shows independent auto correlation measurements on signal and idler modes respectively. $\mathrm{G^{(2)}_{SS}}(\tau)$  ($\mathrm{G^{(2)}_{II}}(\tau)$) is the number of coincidence clicks between detectors D1 and D2 (D3 and D4) at the two output ports of a 50 / 50 beam-splitter in signal (idler) mode. The normalized correlation coefficient $\mathrm{g^{(2)}_{II}}(0)$ of $1.99 \, \pm \, 0.02$ for idler and  $\mathrm{g^{(2)}_{SS}}(0)$ of $2.02 \, \pm \, 0.05$ for signal reveals the thermal nature of the generated photons.

The temporal correlations of the photon pairs can be quantified by the factor R by which Cauchy-Schwarz inequality ~\cite{Reid1986} is violated.
$$
   \mathrm{R} = \frac{[\mathrm{g}^{(2)}_{\mathrm{SI}}(\tau)]^{2}} {\mathrm{g}^{(2)}_{\mathrm{SS}}(0) . \mathrm{g}^{(2)}_{\mathrm{II}}(0)} \leq 1 
$$
We observe a strong violation by a factor of $5.5\times10^6$ which also demonstrates a highly non-classical nature of the photons. In other similar experiments, Du et al.~\cite{Du2008} have observed a violation factor of 11600 and Willis et al.~\cite{Willis:11} have observered 495 in warm atoms.

	\begin{figure}[]
		\begin{center}
			\includegraphics[width=\columnwidth]{figures/g2_combined.eps}
			\caption{\label{fig:g2}(a) Coincidence counts (left) for measurement time of $612\mathrm{s}$ as a function of time delay between the detection of idler and signal photons. The dashed (blue) line is a fit of the form $\mathrm{B+A \times exp(\tau/\tau_0)}$ where $\mathrm{B = 0.21 \, \pm \, 0.01 , \, A = 1192 \, \pm \, 7, \, and\, \tau_0 = 7.2 \, \pm \, 0.1 \,ns}$. The optical density of the MOT $\approx 32$. (Right) Normalized cross correlation coefficient. (b) Coincidence counts (left) and normalized auto correlation coefficient (right) vs time delay on idler mode. The dashed (blue) line is a fit of the form $\mathrm{A\times (1+B\times exp(-|\tau|/\tau_0))}$ with $\mathrm{A=183 \, \pm \, 1}$, $\mathrm{B=0.99 \, \pm \, 0.02}$, $\mathrm{\tau_0 = 21.7 \, \pm \, 0.9 \, ns}$. (c) Same as (b) for signal mode with fit parameters   $\mathrm{A=48 \, \pm \, 0.4}$, $\mathrm{B=1.02 \, \pm \, 0.05}$, $\mathrm{\tau0 = 11.2 \, \pm \, 0.8 \, ns}$.
}
		\end{center}
	\end{figure}

Under the optimum experimental conditions with $5 \, \mathrm{mW}$ of $776 \, \mathrm{nm}$ and $103 \, \mu W$ of $780 \, \mathrm{nm}$ pump beams and the intermediate state detuning ${\Delta}$ (Fig. ~\ref{fig:combined_setup_levelscheme.eps}(b)) of ${\approx}\, 40 \, {\mathrm{MHz}}$, we have instantaneous pair rate of $400 \, /\mathrm{s}$ and heralding (pairs to singles) efficiency of $14.9\% $ for the idler mode. The coincidence time window for the pairs is $30 \, \mathrm{ns}$. For the signal mode, transition from $5\mathrm{P}_{1/2}, \mathrm{F=2}$ state to both $\mathrm{F=2}$ and $\mathrm{F=1}$ hyperfine ground states are dipole allowed (Fig. ~\ref{fig:combined_setup_levelscheme.eps}(b)). But, the decay channel to $\mathrm{F=1}$ ground state do not share correlations with idler mode, as the energy conservation is not satisfied ($\mathrm{E_{S} \, \neq \, E_1 \, + \, E_2 \, - \, E_{I}}$). We therefore use an etalon of FWHM bandwidth $375 \, \mathrm{MHz}$ to filter out the uncorrelated signal photons. This gives a higher hearlding efficiency of $23 \%$ for the signal mode. Higher pair generation rates can be obtained by using higer powers of pump beam or by increasing the optical thickness of the cloud. By increasing the $\mathrm{776\,nm}$ pump power to $\mathrm{14\,mW}$ we observed a pair rate of $7700\,/\mathrm{s}$. All efficiencies and rates reported above are uncorrected for losses due to non-unit detector efficiency ($\approx 50 \%$), filtering efficiency ($92 \%$) and fiber coupling efficiency ($75 \%$).
%Increasing the optical density of the cloud also results in higher heralding efficiency until saturation is reached. Zhang et al. ~\cite{Zhang:11} have observed such saturation effect in double-$\Lambda$ type scheme at opical density of about 60.

%\section{Spectral properties\label{spectrum}}

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=0.95\columnwidth]{figures/bandwidth.eps}
		\caption{\label{fig:bandwidth}(a) Spectral profile of signal photons heralded by the detection of idler photons with MOT optical density of $31.9 \, \pm \, 0.5$. The error in frequency come from the uncertainty in voltage used to drive the cavity piezo. The fit function (blue dashed curve) is convolution of cavity transmission (airy function of FWHM $2.8 \, \mathrm{MHz}$) and photon spectrum (assumed to be lorentzian function). The photon FWHM bandwidth obtained from the fit is $24.7 \, \pm \, 1.4 \, \mathrm{MHz}$. (b) Same as (a) but without heralding. The photon (singles) bandwidth from the fit is  $18.3 \, \pm \, 1.3 \, \mathrm{MHz}$
		}
	\end{center}
\end{figure}
To measure the spectral properties of the signal photons, we use a scanning fabry-perot cavity.  
 The $795 \, \mathrm{nm}$ photons from the signal mode are coupled through the fundamental $TEM_{00}$ mode of the cavity to the detector. The cavity is then scanned across $5\mathrm{P}_{1/2},\, F=2$ $\rightarrow$ $5\mathrm{S}_{1/2},\, F=2$ atomic resonance to obtain the spectral profile of the photons. A $795 \, \mathrm{nm}$ diode laser locked to the above mentioned atomic resonance line is used as absolute frequency reference for the photons through the cavity. For MOT optical density of $31.9 \, \pm \, 0.5$, we measure the bandwidth of heralded signal photons (pairs) to be $24.7 \, \pm \, 1.4 \, \mathrm{MHz}$ and that of unheralded signal photons (singles) to be $18.3 \, \pm \, 1.3 \, \mathrm{MHz}$. The frequency spectrum of photons can also be inferred from the $\mathrm{g}^{(2)}\mathrm{(\tau)}$ by fourier transform which gives bandwidth of $\Delta \nu \, = \, 1/(2 \times \pi \times \tau_0)$ for exponential decay. For MOT optical density $\approx 32$, bandwidth of pairs inferred from decay time is $\Delta \nu \, = 22.1 \, \pm \, 0.3 \, \mathrm{MHz}$. 
The discrepancy between the two methods indicates that a better theoritical model is required to describe the relation between $\mathrm{g}^{(2)}$ decay time and the photon bandwidth for this system.

%We osberve that the bandwidth of the signal photons heralded by detection of idler photons (pairs) is higher than the bandwidth of signal photons without heralding (singles) (figure ~\ref{fig:bandwidth}). Since $\tau0$ from cross correlation ($7.2 \, \mathrm{ns}$) is lower compared to $\tau0$ form auto correlation ($11.18 \, \mathrm{ns}$), pairs have a higher bandwidth compared to singles.

\begin{figure}
	\begin{center}
		\includegraphics[width=\columnwidth]{figures/od_vs_bandwidth.eps}
		\caption{\label{fig:od_bandwidth} (Red, dots) The bandwidth of signal photons with herlding (pairs) at different optical densities of the MOT. The solid line shows the theoritical prediction. (Blue, square) The bandwidth of signal photons without heralding (singles) at different optical densities. }
	\end{center}
\end{figure}

We measure the bandwidth of the photons for different optical densities of the MOT (figure ~\ref{fig:od_bandwidth}). The variation of bandwidth with optical density can be modelled by the relation $\mathrm{OD} \propto (\gamma / \gamma_0) - 1$ ~\cite{Walther:2009,Jen:2012}, where $\mathrm{OD}$ is the optical density of the MOT measured on the cooling transition $5\mathrm{S}_{1/2},\, F=2$ $\rightarrow$  $5\mathrm{P}_{3/2},\, F=3$,  $\gamma$ is the broadened bandwidth of the photons due to superradiance, $\gamma_0$ is the natural linewidth of the $5\mathrm{P}_{1/2},\, F=2$ $\rightarrow$ $5\mathrm{S}_{1/2},\, F=2$ transition ($2\pi \times 5.746\, \mathrm{MHz}$) ~\cite{steck_2010}. It can be seen that the bandwidth of the photons increase with increasing optical density of the MOT.

In summary, we have an efficient spectrally bright source of correlated photon pairs with pair rate of $7700 \, \mathrm{s^{-1}}$ and bandwidth of $20 \, \mathrm{MHz}$ for pump powers of $14 \, \mathrm{mW}$ and $100 \, \mathrm{\mu W}$. The strong temporal correlations between the photons can be exploited with suitable interferometers to generate time bin entanglement. Also, the photon decay channels via different zeeman sublevels makes polarization entanglement possible, by appropriate choice of pump polarizations  ~\cite{Chaneliere:2006, Willis:11}. Such entanglements can be used to implement various quantum communication protocols. Recently, experiments on transfer of entanglement between single atom-photon to photon-photon have been reported ~\cite{Wilk2007}. The narrow bandwidth of our photon pairs can be utilized to perform the reverse operation i.e., transfer of photon-photon entanglement to atom-photon entanglement. Our photons can also be used for absorption of heralded single-photon by single atom ~\cite{Piro2011}.



\section{Acknowledgment}
We acknowledge the support of this work by the National Research Foundation \&
Ministry of Education in Singapore.
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