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%\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}}
The ability to coherently transfer quantum information between photonic qubit to a  atomic memory qubit is a prerequisite for all scalable quantum communication protocols. 
%Quantum communication promises the secure and faithful transfer of quantum information between two distant  qubits. 
At present all quantum communications protocols are based on using entangled photon pairs as information carriers to achieve a secure quantum key distribution. (Why?) 
%But because of decoherrence and other channel lossess, the communication rate decays exponentially with distance. To overcome this difficulty and ensure secure quantum communication over long distance, concept of quantum repeaters was proposed ~\cite{DLCZ:2001,vanderwal:2003}.  %lassical, narrowband, correlated photon pair source is essential for experimental realization of %many long distance quantum communication protocols with %atom~\cite{DLCZ:2001,vanderwal:2003}. 
Where as atoms are excellent candidates to be used as quantum memory elements to store and manipulate quantum information. To achieve high fidelity of quantum state transfer, one requires a source that generates photon pairs with frequency bandwidth matching the absorption profile of the atoms.  Additionally, the source should be heralded single photon source where the detection of one trigger photon indicates the presence of its pair in other arm. A high quality heralded single photon source is defined by its narrow bandwidth, high spectral brightness and a high heralding efficiency. 

(various sources in world) For many years, spontaneous parametric downconversion in non-linear crystals has become standard method for generating correlated and entangled photon pairs. Such sources of paired photons suffer serious limitations: (a) the linewidth of the generated photons is too broad ($\sim$GHz) compared to lifetime limited atomic absorptiion lines ($\sim$MHz). This problem has been overcome to some extent 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. (b) their short coherence length makes them unsuitable for the long distance quantum communication. 
%However the generated photons in such a source have linewidth ($\sim$GHz) a few orders of magnitude larger than lifetime limited atomic absorptiion lines ($\sim$MHz). This problem has been overcome to some extent 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} but 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. The two near resonant pump fields with optical modes defined by $\vec{k_1}$ and $\vec{k_2}$ are overlapped inside the atomic ensemble. An idler photon emitted in the collection mode defined by $\vec{k_i}$, results in the emission of signal photon in the phasematched direction defined by $\vec{k_s}$ = $\vec{k_1}$+$\vec{k_2}$-$\vec{k_i}$.  Under the conditions of phase matching, collective enhancement causes emission of the the signal photon in a well defined direction, with a return of the atom into the same hyperfine state from which it originated. This  results in generated signal and idler photons, strongly corelated in polarization, energy and time.
The field opening paper by Aspect, gave the first experimental generation of polarization entangled photon pairs from the dillute beams of Calcium. Later on Kimble and Lukin pioneered the field with their experiments on pair production and studying the collective effects in optically thick atomic ensemble . These experiments utilized the lambda level structure in alkali atoms to create nearly degenerate infrared biphotons . Harris and coworkers made further improvemnets in efficient photon-pair production in a cold atomic ensemble using counter-propagating write and read fields in double lambda scheme. They also introduced electromagnetic induced transparency effect in atomic ensembles that results in further narrowing down the linewidth of the photons to 1 MHz.
Recently Willis et al ~\cite{Willis:11} and Ding et al ~\cite{Ding:12}  made use of of  hot vapors as their non-linear generation media for the emission of nonclassical photon pairs. But photon pairs generated from the hot vapors are doppler broadened due to the thermal motion of the atoms. 
%(How to start)The two near resonant pump fields with optical modes defined by $\vec{k_1}$ and $\vec{k_2}$ are overlapped 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 $\vec{k_s}$ = $\vec{k_1}$+$\vec{k_2}$-$\vec{k_i}$.  Under the conditions of phase matching, collective enhancement causes emission of the the signal photon in a well defined direction, with a return of the atom into the same hyperfine state from which it originated. This  results in generated signal and idler photons, strongly corelated in polarization, energy and time.
In this paper, we report the generation of narrow bandwidth non-degenerate photon pairs from cold atomic ensemble of $\mathrm{^{87}}$Rb 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 cascade level scheme (Fig.~\ref{fig:combined_setup_levelscheme.eps}(b)) for generation of photon pairs instead of the $\lambda$ level scheme used by many groups in the past ~\cite{Du2008}. The cascade level scheme allows the generation of photon pairs quite different in frequencies from the pump beams which avoids 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.
The output of the tapered amplifier is distributed among six beams to form  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 $\mathrm{10\,mW}$  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 by 40 MHz 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 fluorescense 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 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 second-order cross correlation function $\mathrm{G^{(2)}_{SI}}(\tau)$ between signal and idler modes shown in fig.~\ref{fig:g2}(a).
The measured $\mathrm{1/e}$ decay time for heralded signal photons is $\mathrm{8.29\, ns}$ from the fit, 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 intermedieate 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. This phasing of atoms 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 the auto correlation measurements $\mathrm{G^{(2)}_{SS}}(\tau)$ and $\mathrm{G^{(2)}_{II}}(\tau)$ obtained by performing HBT type analysis independently on signal and idler modes.
A measure of purity (signal to noise) of the photon pair source can be obtained by combining the correlation measurements with the Cauchy-Schwartz inequality ~\cite{Reid1986} $[\mathrm{g}^{(2)}_{\mathrm{SI}}(\tau)]^{2} \leq \mathrm{g}^{(2)}_{\mathrm{SS}}(0) . \mathrm{g}^{(2)}_{\mathrm{II}}(0)$. 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 for measurement time of $612\mathrm{s}$  as a function of time delay between the detection of idler ($762\, \mathrm{nm}$) and signal ($795\, \mathrm{nm}$) photons with a timing resolution of $0.5\, \mathrm{ns}$. The dashed (blue) line is a fit of the form $\mathrm{B+A \times exp(\tau/\tau0)}$ where $\mathrm{B = 0.2495 , A = 1173,\, and\, \tau0 = 8.29\,ns}$. (b) Unnormalized auto correlation measurement on idler mode with timing resolution of $0.5\, \mathrm{ns}$. The dashed (blue) line is a fit of the form $\mathrm{A\times (1+B\times exp(-|\tau|/\tau0))}$ with $\mathrm{A=182}$, $\mathrm{B=0.987}$, $\mathrm{\tau0 = 22.19\, ns}$. (c) Same as (b) for signal mode with fit parameters   $\mathrm{A=48.238}$, $\mathrm{B=1}$, $\mathrm{\tau0 = 11.18\, ns}$.
}
		\end{center}
	\end{figure}

Under the optimum experimental conditions with $5 \, \mathrm{mW}$ of $776 \, \mathrm{nm}$ and $100 \, \mu W$ of $780 \, \mathrm{nm}$ pump beams and the intermediate state detuning ${\Delta}$ (\textcolor{red}{see fig}) of ${\approx}\, 40 \, {\mathrm{MHz}}$, we have heralding (pairs to singles) efficiency of $43\% $ for the idler mode (corrected for detector quantum efficiency $\approx 50\%$ , fiber coupling efficiency of $70\%$ and filtering efficiency of $92\%$). We observe photon pair rate of $1250/\mathrm{s}$ during the $1\, \mathrm{ms}$ time per duty cycle when the pump beams are on (\textcolor{red}{see Fig}). The assumed 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 ( \textcolor{red}{see FIG}). But, the decay channel to $\mathrm{F=1}$ ground state do not share correlations with idler mode, as the phase matching condition is not satisfied. We therefore use an etalon of FWHM bandwidth $375 \mathrm{MHz}$ to filter out the uncorrelated signal photons. This gives a marginally higher hearlding efficiency of $57 \%$ 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 $24000/\mathrm{s}$. 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=\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 26. 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 $21.404 \, \pm \, 0.535 \, \mathrm{MHz}$. (b) Same as (a) but without heralding. The photon (singles) bandwidth from the fit is $15.116\, \pm \, 0.209 \, \mathrm{MHz}$
		}
	\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 plots in figure ~\ref{fig:bandwidth}. 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. 

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}). %This can also be seen from the difference in decay time of signal photons from cross correlation measurement (figure ~\ref{fig:g2}(a)) and auto correlation measurement (figure ~\ref{fig:g2}(c)). The frequency spectrum of photons can be inferred from the $\mathrm{g}^{(2)}\mathrm{(\tau)}$ by fourier transform which gives bandwidth of $\Delta \nu \, = \, 1/(2 \times \pi \times \tau0)$ for exponential decay.  Since $\tau0$ from cross correlation ($8.29 \, \mathrm{ns}$) 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 observed photon pair generation from an ensemble of cold rubidium atoms with very narrow bandwidth at non-degenerate wavelengths. The source has a high signal to noise as indicated by cauchy-schwarz factor and a relatively high efficiency for such kind of system. The bandwidth measurements indicate the compatibility of photons for heralded interaction with single atom ~\cite{Syed:2011,Tey:2008}. The photons can potentially be polarization entangled by choosing appropriate pump beam polarization. Such entanglement has been observed from similar sources by Chaneli\`ere et al. ~\cite{Chaneliere:2006}, Willis et al. ~\cite{Willis:11}



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