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\title{Controlling the interference of single photons emitted by independent atomic sources}

%>>>> The author is responsible for formatting the
%  author list and their institutions.  Use  \skiplinehalf
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\author{
Alessandro Cer\`{e}\supit{a},
Victor Leong\supit{a,b},
Sandoko Kosen\supit{a},
Bharath Srivathsan\supit{a},
Gurpreet Kaur Gulati\supit{a},
and Christian Kurtsiefer\supit{a,b}
\skiplinehalf
\supit{a}Centre for Quantum Technologies, National University of Singapore,\\3 Science Drive 2, Singapore
  117543\\
\supit{b}Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore  117542
}

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\authorinfo{Further author information: (Send correspondence to A.C. and C.K.)\\A.C. e-mail: cqtac@nus.edu.sg, C.K. e-mail: christian.kurtsiefer@gmail.com}


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\begin{document}
\maketitle

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\begin{abstract}
	Hong-Ou-Mandel interference between independent sources is a fundamental primitive of many quantum communication and computation protocols.
	We present a study of the Hong-Ou-Mandel interference of single photons generated via two different physical processes by two independent atomic systems:
	scattering by a single atom, and parametric generation via four-wave mixing
	in a cloud of cold atoms.
	By controlling the coherence time and central frequency of the heralded single photons generated by four-wave mixing we observe quantum beat and a varying degree of interference.
\end{abstract}

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\keywords{quantum optics, cold atoms, hong ou mandel interference, four wave mixing, single atom}

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\section{Introduction}
	\label{sec:intro}
	Quantum networks promise to provide the infrastructure for distributed quantum computing and secure quantum communication~\cite{kimble:2008}.
	Any network is based on nodes and the connections between them
	and,
	in a quantum network, both the nodes and the connections
	must be sufficiently resilient to the decohering effects of the environment.
	% must beat the decoherence of the environment
	and show quantum properties.
	% and be able to distribute the coherence.
	Many protocols propose nodes composed of single or an ensemble of neutral atoms, and single photons connecting them~\cite{DLCZ:2001}.
	Hong-Ou-Mandel~(HOM) interference~\cite{hong:87} takes place when two
	indistinguishable photons arrive simultaneously at the two inputs of a 50:50
	beam splitter, making them leave together from the same output
	port~\cite{Loudon:579300}. It provides a fundamental primitive for the
	coherent interfacing of separate quantum systems via their emitted
	photons~\cite{Briegel:1998} as an alternative to their direct
	interaction~\cite{Dyckovsky:2012ea,Meyer:2015}.  It is the basis of quantum
	teleportation~\cite{Bennett:1992tv,bouwmeester:97,DeMartini:2002vf} and
	entanglement swapping~\cite{Zukowski:1993fs,pan:98}.
	In a previous work~\cite{Leong:2015eb} we demonstrated how the heralded single photons generated by four-wave mixing in a cloud of $^{87}$Rb can interfere with photons scattered by a single $^{87}$Rb atom in a Hong-Ou-Mandel configuration.
	We now present how adjusting the coherence time and central frequency of the heralded single photon leads to changes in the observed visibility and the rise of quantum beat.
	%
	\begin{figure}
		\begin{center}
			% \begin{tabular}{c}
			\includegraphics[width=\textwidth]{figures/idea.eps}
			% \end{tabular}
		\end{center}
		\caption{\label{fig:idea}
			(a)~Schematic representation of the Hong-Ou-Mandel
		    experiment. Heralded photons from pairs generated by four-wave mixing
		    in an atomic ensemble interfere with single photons generated by a
		    single atom after heralding on a 50:50 beam splitter, and are detected by
		    avalanche photodetectors at the
		    outputs. (b)~Simplified level scheme of the FWM process. (c)~Level scheme
		    for the single atom in the dipole trap and electronic
		    transition used for exciting the single atom.}
	\end{figure}
	%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Sources of single photons}
\subsection{Photon Pairs From Four-wave Mixing In Cold $^{87}$Rb Atoms}
	Four-wave mixing (FWM) is a parametric process that, as the name implies, mixes four different wavelengths.
	In Figure~\ref{fig:idea} we find the energy level scheme that provides the necessary third-order non linearity $\chi^{(3)}$.
	\begin{figure}
	 \includegraphics[width=\columnwidth]{figures/joint_setup_v5.eps}
	  \caption{\label{fig:schematic}
	    (Top left) Four-wave mixing setup: Pump\,1 (795\,nm) and Pump\,2 (762\,nm)  are overlapped in a copropagating geometry inside the cold cloud of  $^{87}${Rb} atoms in a Magneto-Optical Trap (MOT), generating signal (776\,nm) and idler (780\,nm) photon pairs. The detection of a signal photon heralds the presence of a single photon in idler mode, and is used to trigger the excitation of the single atom.
	    (Bottom left) Single atom setup: A $^{87}${Rb} atom is trapped in
	    free space between two confocal aspheric lenses (AL; numerical aperture
	    0.55) with a far-off-resonant optical dipole trap
	    ($\lambda=980\,\text{nm}$). After an adjustable delay time $\Delta T$
	    from the trigger, an electro-optic modulator (EOM) generates an optical
	    pulse to efficiently excite the single atom. The presence of an atom in the
	    trap is periodically checked using APD $D_f$.
	    (Right) HOM interferometer: single photons from both sources interfere at a 50:50 beam splitter (BS). An acousto-optic modulator (AOM) matches the central frequencies of both photons.
	    P: polarizer, $F$: interference filters,  $\lambda$/2, $\lambda$/4: half- and quarter-wave plates, PBS: polarizing beam splitter, BS: non-polarizing beam splitter, $D_{a}$, $D_{b}$, $D_{f}$, $D_{t}$: avalanche photodetectors.}
	\end{figure}
	%
	Two pumps of wavelength 780\,nm (pump$_1$) and 776\,nm (pump$_2$) excite the atoms from
	$5\text{S}_{1/2},F=2$ to $5\text{D}_ {3/2},\,F=3$ via a two-photon transition.
	The 780\,nm pump beam is red detuned by 60\,MHz from the intermediate $5\text{P}_{3/2},\,F=3$ level
	to reduce the rate of incoherent scattering.
	From the excited $5\text{D}_{3/2},F=3$ level there are many possible decay paths.
	We select photons generated from the cascade decay to
	$5\text{S}_{1/2},F=2$ via $5\text{P}_{1/2},F=2$ using narrowband filters.
	The time correlation of the generated photons is ensured by the
	momentum conservation of the four participant modes, enforced by the choice of pumping and collection modes via single mode fibers.
	Using all four modes in a collinear geometry makes the alignment simpler and allows for an efficient coupling  of the generated photons into a single mode fiber.
	A schematic of the setup is shown in Figure~\ref{fig:schematic}.
	Using this setup we have already demonstrated
	the generation of correlated photon pairs~\cite{Srivathsan:2013},
	the temporal shape of the emitted photons~\cite{Gulati:2014} and its manipulation~\cite{Srivathsan:2014jx},
	and polarization entanglement~\cite{Gulati:2015wi}.


\subsection{Single Photons Scattered By A Single Atom In Free Space}
	The single atom (SA) source generates single photons by optically exciting the electronic transition of interest and
	collecting the consequent photon emitted by spontaneous decay~\cite{Darquie:2005fr}.
	A single atom is trapped at the focus of
	a far-off-resonant optical dipole trap (FORT)
	obtained by focusing a Gaussian beam ($\lambda=980\,\text{nm}$) to a waist of
	1\,$\mu$m using an aspheric lens (numerical aperture 0.55).
	Further details of the trapping are described in our previous works~\cite{Syed:2013, Tey:2008}.
	The trapped atom undergoes
	molasses cooling and is optically pumped to the 5\hflev{S}{1}{2}{2}, $m_F$=-2 state.
	To ensure a sufficiently long coherence time
	of the prepared state,
	we apply a bias magnetic field of 2\,gauss along the optical axis.
	After the atom is prepared in the initial state, it can be excited to
	5\hflev{P}{3}{2}{3}, $m_F$=-3 [see Fig.~\ref{fig:idea}(c)] by a short resonant optical pulse generated using a fast electro-optic modulator (EOM).
	The beams used for optical pumping and excitation are collinear with the dipole trap, and are focused onto the atom by the same aspheric lens.
	The excitation pulse duration $\tau_e=3$\,ns is
	much shorter than the excited state lifetime $\tau_{s}=26\,\text{ns}$,
	and its amplitude is set to
	maximize the excitation probability.

	The aspheric lens is also used to
	collect the spontaneously emitted single photons.
	The collection mode is separated from the excitation mode using a 99:1 beam splitter and is
	then coupled into a single mode fiber.
	The overall generation, collection and detection efficiency is
	$\approx$\,0.5\%.
	We periodically check for the presence of the atom in the FORT by monitoring fluorescence with detector $D_f$; if the atom is lost, a new atom is loaded from a MOT.

\section{HOM interference}
	%
	\begin{figure}
	 \includegraphics[width=\columnwidth]{figures/g2_hom.eps}
	  \caption{\label{fig:g2_hom}
	  Coincidence probability between $D_{a}$ and $D_{b}$.
	    The filled and open circles represent the cases where photons have
	    perpendicular (non-interfering) and
	    parallel (interfering) polarizations, respectively.
	    The data is sorted into 10\,ns wide time bins and normalized to
	    the total number of trigger events $N_t$.
	    % For an integration window of $T_c = -25\,\text{ns}\,\leq\,\Delta
	    % t_{ab}\,\leq\,25\,\text{ns}$, the interference visibility
	    % $V=62\pm4\%$.
	    The upper solid line represents $G_{\textrm{acc}}\,+\,A\cdot G_{\perp}(\Delta t_{ab})$ [see
	    Eq.~(\ref{eq:Gperp})], and the lower solid line represents
	    $G_{\textrm{acc}}\,+\,A\cdot G_{||}(\Delta t_{ab})$ [see
	    Eq.~(\ref{eq:Gpll})].
	    $G_{\textrm{acc}}$ is a constant offset,
	    while $A$ is a scaling factor.}
	\end{figure}
	%
	The timing characteristics of the two sources are determined by the generation
	processes.  The FWM process generates photon pairs with Poissonian statistics,
	and we obtain a heralded single photon by detecting one photon of the
	pair~\cite{Clauser:1974hc,Grangier:2007ge,Hong:1986zz}, while the emission of
	a single photon from the single atom is triggered by an excitation pulse.
	The detection of the heralding photons from  the FWM also serves as the
	trigger for the excitation pulse of the single atom source,
	effectively synchronizing the whole experiment.

	Both sources generate single photons with a decaying exponential temporal envelope:
	\begin{equation}\label{eq:time_envelope}
		\psi_i(t)\,=\,\sqrt{\frac{1}{\tau_i}}\,e^{-\frac{t-t_i}{2\tau_i}}\,\Theta(t-t_i) \textrm{ with } i=f,s\,,
	\end{equation}
	where $\tau_{f, s}$ are the coherence times from FWM and SA sources respectively,
	$t_s$ is the single atom excitation time after a heralding event at $t_f$,
	and $\Theta(t)$ is the Heaviside step function.
	For the SA source, the time constant is given by the natural linewidth of the transition~\cite{Syed:2013}: $\tau_{s}=26.18\pm0.11\,\text{ns}$,
	while for the FWM source it is determined by the
	optical density of the atomic ensemble~\cite{Jen:2012,Srivathsan:2013}.


	The HOM interference can be observed by comparing the probability of
	coincidence $P$ between detectors $D_a$ and $D_b$ for interfering~($P_{||}$)
	and non-interfering~($P_\perp$) photons.
	Experimentally we estimate probabilities $P$ by counting the number of coincidences.
	%
	All detection events are timestamped with a temporal resolution of
	125\,ps. We offset the detection times of all detectors to account for the
	delays introduced by the electrical and optical delay lines, and
	we only consider a detection sequence valid if either $D_a$ or $D_b$
	clicks within 85\,ns of a trigger from $D_{t}$.
	%
	In Figure~\ref{fig:g2_hom} we show the distribution of the probability of a coincidence $G$ as a function of the delay of $\Delta t_{ab}$
	for the interfering and non-interfering cases.
	We sort the time delay between detection events $\Delta t_{ab}$ into time bins of width 10\,ns
	and normalize the distribution by dividing by the total number of trigger
	events $N_t$ over the measurement time.
	Apart from a constant offset due to accidental coincidences, the two probability distributions are well described by:
	\begin{equation}\label{eq:Gperp}
	  G_{\perp}(\Delta t_{ab})\,=\,\frac{1}{4}\int\limits^{\infty}_{-\infty} |\psi_{f}(t)\,\psi_{s}(t+\Delta t_{ab})|^2 % \nonumber \\ 
	   + |\psi_{f}(t+\Delta t_{ab})\,\psi_{s}(t)|^2\,dt \,,
	\end{equation}
	% When the two incident photons have identical polarizations, their pair amplitudes interfere (with the minus sign determined by one of the reflections on the beam splitter):
	and
	\begin{equation}\label{eq:Gpll}
	  G_{||}(\Delta t_{ab})\,=\,\frac{1}{4}\int\limits^{\infty}_{-\infty}
	  |\psi_{f}(t) \psi_{s}(t+\Delta t_{ab})  %\nonumber \\
	  -\,\psi_{f}(t+\Delta t_{ab}) \psi_{s}(t)|^2\,dt\,.
	\end{equation}
%%%%%%%%%%%%%%
\section{Quantum beat of interfering single photons}
	%
	\begin{figure}
	 \includegraphics[width=\columnwidth]{figures/beat_fit.eps}
	  \caption{\label{fig:beat}
	  Quantum beat between single photons.
	  Coincidence probability between $D_{a}$ and $D_{b}$
	  The data is sorted into 2\,ns wide time bins and normalized to
	  the total number of trigger events $N_t$.
	  The finer time binning has been chosen to being able to resolve the temporal structure of the quantum beat.
	  The solid line is a fit obtained from Equation~(\ref{eq:Gbeat}).
	  }
	\end{figure}
	%
	The single atom experiences an AC Stark shift from the dipole trap and a Zeeman shift from a bias magnetic field,
	resulting in a detuning of $\delta_s$
	% =76\,\text{MHz}$
	from the natural transition frequency for the emitted photon.
	We measured $\delta_s$ by observing the extinction of light caused by the SA as a function of the probe frequency~\cite{Tey:2008}, obtaining
	$\delta_s=76\pm1$\,MHz.
	In order to obtain a high visibility in the HOM interference, as seen in Figure~\ref{fig:g2_hom}, we compensated this frequency shift in the central frequency of the heralded photon coming from FWM using
	an acousto-optic modulator (AOM).
	Without this compensation, it is possible to observe the HOM interference for single photons of different frequency generated by independent sources, an effect first reported by Legero \textit{et al.}~\cite{Legero:2004vk}.
	Adapting their derivation~\cite{Legero:2004vk} to the temporal waveform of Equation~(\ref{eq:time_envelope}) we obtain:
	\begin{align}\label{eq:Gbeat}
	  G_\textrm{beat}(\Delta t_{ab})\,&=\,
	  \frac{1}{4}\int\limits^\infty_{-\infty} \left|\psi_f(t)\,\psi_s(t+\Delta t_{ab})\right|^2
	  + \left|\psi_f(t+\Delta t_{ab})\,\psi_s(t)\right|^2\,dt\nonumber \\
	  - &
	  \frac{\cos(\delta_s \Delta t_{ab})}{2}\int\limits^\infty_{-\infty} \left|\psi_f(t)\,\psi_s(t+\Delta t_{ab}) \psi_f(t+\Delta t_{ab})\,\psi_s(t)\right|\,dt\,.
	\end{align}
	In Figure~\ref{fig:beat} we report the experimental data and a fit obtained using Equation~(\ref{eq:Gbeat}) using as free parameters a scaling factor, the accidental count rate, and the detuning $\delta_s$.
	We obtain a $\delta_s=75.7\pm0.66$\,MHz, compatible with the value measured independently.

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\section{HOM interference and photon coherence time}
	%
	\begin{figure}
	 \includegraphics[width=\columnwidth]{figures/vis_decay.eps}
	  \caption{\label{fig:vis_vs_decay}
	  HOM interference as a function of the coherence time of the FWM photon.
	  The interference is characterized by the ratio $R=P_{(1,1)}/P_{(2,0)}$.
	  The continuous line represents Equation~(\ref{eq:vis}) with $\tau_s = 26.18$\,ns.
	  The coherence time $\tau_f$ is determined by the optical density of the atomic cloud.}
	\end{figure}
	%
	We also studied how the HOM interference changes for different coherence times of the FWM photons.
	A signature of the HOM is the change of probability of two photons being detected at the same output of the beam-splitter~\cite{hong:87,Loudon:579300}.
	In order to estimate this probability, we replaced detector $D_b$ with a 50:50 fiber beam-splitter and two similar APD, $D_{b1}$ and $D_{b2}$.
	We can now compare the probability of a coincidence between $D_a$ and either $D_{b1}$ or $D_{b2}$, $P_{(1,1)}$, with the probability of coincidences between $D_{b1}$ and $D_{b2}$, $P_{(2,0)}$.
	We define the ratio $R=P_{(1,1)}/P_{(2,0)}$.
	We obtain the expected behavior of $R$ by integrating Equation~(\ref{eq:Gpll}):
	\begin{equation}\label{eq:vis}
		R(\tau_f, \tau_s)\,=\,\left(\frac{\tau_f - \tau_s}{\tau_s + \tau_f} \right)^2\,.
	\end{equation}
	Experimentally, we can choose the coherence time of the FWM photons $\tau_f$ by adjusting the optical density of the atomic cloud~\cite{Srivathsan:2013}.
	In Figure~\ref{fig:vis_vs_decay} we can observe the probability ratio $R$ as a function of $\tau_f$.
	We compare it with the expected behavior, predicted by Equation~(\ref{eq:vis}), represented by the continuous line.

\section{Conclusion}
	We have presented a study of the amplitude interference of two single photons generated by two independent atomic sources.
	We have shown how the difference of central frequency of the two photons gives rise to quantum beat, and
	how the control of the temporal shape of the photon generated by FWM can be used to modulate the visibility of the interference.



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\acknowledgments
We acknowledge the support of this work by the National Research Foundation (partly under grant No. NRF-CRP12-2013-03) \& Ministry of Education in Singapore.
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