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\begin{document}


%\preprint{}

\title{Direct generation of an exponentially rising single photon field from
  parametric conversion in atoms}

\author{Gurpreet Kaur Gulati}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore,
  117543}
\author{Bharath Srivathsan}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore,
  117543}
\author{Brenda Chng}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore,
  117543}
\author{Alessandro~Cer\`e}
\altaffiliation{Center for Quantum Technologies 3 Science Drive 2, Singapore,
  117543}
\author{Dzmitry Matsukevich}
\altaffiliation{Department of Physics, National University of Singapore, 2
  Science Drive 3,  Singapore, 117542}
\altaffiliation{Center for Quantum Technologies, National University of
  Singapore, 3 Science Drive 2, Singapore, 117543}
\author{Christian Kurtsiefer}
\altaffiliation{Department of Physics, National University of Singapore, 2
  Science Drive 3,  Singapore, 117542}
\altaffiliation{Center for Quantum Technologies, National University of
  Singapore, 3 Science Drive 2, Singapore, 117543}
%\homepage[]{Your web page}
%\thanks{}
%\email[]{christian.kurtsiefer@gmail.com}
\date{\today}

\begin{abstract}
We prepare heralded single photons from a photon pair source based on
non-degenerate four-wave mixing in a cold atomic ensemble via a cascade decay
scheme. Their statistics shows strong anti-bunching with $g^{(2)}(0)<0.03$,
indicating a near single photon character. In an optical homodyne experiment,
we directly measure the temporal envelope of these photons and find,
depending on the heralding scheme, an exponentially decaying or rising
profile.
\textcolor{red}{Such a rising envelope is required for the efficient interaction between
a
single 
% photons
photon and
 % any 
 a two level 
 % systems
 system}.
At the
same time, their observation  illustrates the breakdown of a realistic
interpretation of the heralding process in terms of defining an initial
condition of a physical system.

\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

\input{heralded_intro}
\begin{figure}[]
  \begin{center}
    \includegraphics[width=\columnwidth]{figures/level_scheme_HBT3.eps}
   \caption{\label{fig:level_scheme_HBT}(a) Cascade level scheme for four
     wave mixing in$^{87}${Rb}. (b) Setup of the heralded single photon
     source. Polarizers (P1,2, Pi, Ps) and interference filters (Fi, Fs)
     separate the signal and idler photons from residual pump light. 
     An etalon (E) in the idler arm removes uncorrelated photons from a decay to
     the $5\text{S}_ {1/2},\,F=1$ level. Signal photons detected by an
     avalanche photodiode (APD) trigger a Hanbury-Brown--Twiss measurement
     between idler photons distributed to two detectors, $D_{\text{i1}}$ and
     $D_{\text{i2}}$,  via a 50:50 fiber beam splitter (BS).}
 \end{center}
\end{figure}

%In the first experiment (see Fig.~\ref{fig:level_scheme_HBT}) we show that the
%heralded photons have single-photon character.
%We show in the first experiment (see Fig.~\ref{fig:level_scheme_HBT}) that the heralded photons are indeed single photons with near single photon character.
We initially demonstrate (see Fig.~\ref{fig:level_scheme_HBT}) the single photon character of our heralded photons. 
An ensemble of $^{87}\text{Rb}$ atoms is
first cooled and confined in a magneto-optical trap (MOT) reaching an
optical density of $\approx$ 32 on the $5\text{S}_{1/2},\,F=2\rightarrow
5\text{P}_{3/2},\,F=3$ transition after about 12\,ms.
In the following 1\,ms the MOT beams are switched off, and the atoms are excited to the $5\text{D}_{3/2},F=3$ level by two orthogonally linearly
polarized pump beams (780\,nm, 0.1\,mW and 776\,nm, 5\,mW, beam waists 0.45\,mm) intersecting
at an angle of 0.5$^\circ$ in the cold atomic cloud. 
The 780\,nm pump beam is red detuned by $\Delta_1=40$\,MHz from
the intermediate level $5\text{P}_{3/2},\,F=3$ to reduce
incoherent scattering. The combined detuning of both
pumps from the two-photon transition ranges from $\Delta_2=0-6$\,MHz to the blue.
Photon pairs of wavelength 762\,nm (signal) and 795\,nm (idler) are generated by a cascade decay from the $5\text{D}_{3/2},F=3$ level via $5\text{P}_{1/2},F=2$ to $5\text{S}_{1/2},F=2$. 
We use interference filters and an etalon to filter any background light generated by other processes in the ensemble. 
%We filter uncorrelated photons from other decays with interference filters and an etalon.
Energy conservation and phase matching between pump and collection modes allow efficient coupling of photon pairs with a strong temporal correlation into
single mode fibers. 

Unlike single quantum emitters
\cite{Kurtsiefer_2000,Michler_2000,Kuhn_2002}, the probability of generating
more than one photon per heralding event in a parametric process does not
vanish due to the thermal nature of the emission process from the atomic
ensemble \cite{mandel_wolf}. We consider the second order
correlation function $g^{(2)}(\Delta t_{12})$ for the probability of
observing two photons in a given mode with a time difference $\Delta t_{12}$.
Any classical light field exhibits $g^{(2)}(0)\ge1$, while
$g^{(2)}(\Delta t_{12})<1$ is referred to as photon antibunching, with an ideal
single photon source reaching $g^{(2)}(0)=0$ \cite{Glauber_1963}. 

We determine this correlation function experimentally in a
Hanbury-Brown--Twiss (HBT) geometry, where the idler light is distributed
with a 50:50 fiber beam splitter onto two single photon counting silicon
avalanche detectors (APD) $D_{\text{i1}}, D_{\text{i2}}$ (quantum efficiency
$\approx\,\,$40\%, dark count rates 40 to 150\,s$^{-1}$),  while signal photons are
detected by $D_{\text{s}}$ as heralds. The detection events are
time-stamped with 125\,ps resolution. The combined timing uncertainty
of the photodetection process is $\approx$ 600\,ps. 

\input{g2_normalization}

\begin{figure}
  \begin{center}
    \includegraphics[width=\columnwidth]{figures/conditional_g2_combined.eps}
    \caption{\label{fig:conditional_g2_762} (a) The correlation function
      $g_{i1i2|s}^{(2)} $ of idler photons separated by a time difference
      $\Delta t_{12}$, conditioned on detection of a heralding event in the
      signal mode, shows strong photon antibunching  over a time scale of
      $\pm20$\,ns, 
      indicating the single photon character of the heralded photons.
	The error bars indicate the propagated poissonian counting uncertainty from $G^{(2)}_{i1i2|s}$ and $N_{i1i2|s}$.    
	 (b) Same measurement, but with the signal and idler modes swapped.}
  \end{center}
\end{figure}

The resulting $g^{(2)}_{\text{i1i2$|$s}}(\Delta t_{12})$ is shown in
Figure~\ref{fig:conditional_g2_762}(a) as function of the
delay $\Delta t_{12}$, sampled into 2\,ns wide time bins. 
With a signal photon detection rate of 50000\,s\,$^{-1}$ (at $\Delta_2=0$), we
observe $g^{(2)}_{\text{i1i2$|$s}}(0)=0.032\pm0.004$. 
%indicating a near single photon character.
%a near single photon character of our source.

When switching the roles of the signal and idler arms, the resulting
normalized correlation function shown in
Figure~\ref{fig:conditional_g2_762}(b) has a minimum $g^{(2)}_{\text{s1s2$|$i}}$ of
0.018$\pm$0.007 with an idler photon detection rate of
13000\,s$^{-1}$.

We now proceed to determine the temporal envelope of the heralded single
photon fields by measuring the field quadrature in the time domain~\cite{Yuen:83,lvovsky:2001} via a balanced homodyne detection.
The experimental scheme is shown in Figure~\ref{fig:homodyne_setup}: the
idler mode is mixed with a local oscillator (LO)
which is frequency-stabilized to the idler transition
$5\text{S}_{1/2},\,F=2$~$\rightarrow$~$5\text{P}_{1/2},\,F=2$.
The balanced mixing is done with two polarizing beam splitters (PBS1,2)
and a half-wave plate
% with 
resulting in 
an interference visibility of~$\approx$~95\%.
%\textbf{and  with an interference visibility of $\approx$95\%. } 
The difference of photocurrents from pin silicon photodiodes (D+, D-;
quantum efficiency $\approx$~87\%) is proportional to the optical field
quadrature in the idler mode.
 With a LO power of 4.5\,mW, the electronic noise is about 6-20~dB below the
 shot noise limit over a band of 10\,kHz--210\,MHz.
We record the homodyne signal with a digital oscilloscope (analog
bandwidth 1\,GHz), with the click detection of the signal photon on the APD
triggering the acquisition.
 \begin{figure}[]
  \begin{center}
    \includegraphics[width=0.8\columnwidth]{figures/homodynesetup2.eps}
   \caption{\label{fig:homodyne_setup}
     Field measurement setup. One of the photons (idler in this figure) is
     combined with a coherent laser field as a local oscillator on a polarizing beam splitter (PBS1), and sent to a balanced pair of pin photodiodes D+, D- for a homodyne measurement. The
     photocurrent difference as a measure of the optical field strength is
     amplified and recorded with an oscilloscope on the arrival of a heralding
     event from the avalanche photodetector in the signal arm.}
 \end{center}
\end{figure}
We then calculate the variance of the optical field from $2.7\times10^5$ traces,
normalized to the shot noise, as a measure of the
temporal envelope of the photon. We also switched the
roles of the signal and idler modes for triggering and homodyne detection, this time using a local oscillator resonant with the  transition $5\text{P}_{1/2},\,F=2$$\rightarrow$
$5\text{D}_{3/2},\,F=3$ near 762\,nm.  The variance for this measurement is calculated from $5\times10^5$ traces.
Both results are shown in Figure~\ref{fig:combined_field}. In both configurations, we set
$\Delta_2\approx6$~MHz to maximise the heralding efficiency.




%In the first measurement,
In Figure~\ref{fig:combined_field}(a) the normalized field variance of the idler photon suddenly rises about 4\% above the shot noise level at the detection time of the trigger photon, and
exponentially decays back to the shot noise level, with a time constant of
$\tau_i=7.2\pm0.2$\,ns obtained from a fit. This can be easily understood by
the timing sequence of a cascade decay, where the signal photon heralds the
population of the intermediate level, which subsequently decays exponentially,
leading to the characteristic decaying envelope of the idler photon according
to the Weisskopf-Wigner solution~\cite{weisskopf:30,Franson_1989}.

\begin{figure}
  \begin{center}
	\includegraphics[width=\columnwidth]{figures/combined_field.eps}
   \caption{\label{fig:combined_field}Optical homodyne results. (a)
     Exponential decay of the field variance of heralded idler photons. (b)
     Exponential rise of the field variance of heralded signal photons}.
  \end{center}
\end{figure}

%The time constant can be increased %to match the free space single atom spontaneous decay time 
%by reducing the optical density of the atomic cloud at the expenses of a lower rate of pairs generation.}
% The rise/fall times can also be varied in the experiment by varying the optical density of the cloud as previously shown in~\cite{Srivathsan:2013}.


The probability of detecting an idler photon given the detection of a signal
photon (heralding efficiency) was independently determined with two APDs
to be $\eta_{i}\approx13\% $ (uncorrected for APD efficiency and optical
losses).

In Fig.~\ref{fig:combined_field}(b) the normalized field variance of the
signal photon exponentially increases a few 10\,ns before the trigger event to
a value of 1.06, and then quickly returns to the shot noise level, with a rise
time constant $\tau_s=7.4\pm0.2$\,ns obtained from a fit. Here, the
suppression of uncorrelated trigger (idler) photons by the etalon E
(figure~\ref{fig:level_scheme_HBT}(b)) results in a higher heralding
efficiency $\eta_{s}\approx19\%$, and therefore a higher signal to shot noise
level.
\textcolor{red}{
  % The time constants for the exponential rise or fall times  are much shorter
  % than the spontaneous decay time of 27\,ns from $5\text{P}_{1/2},\,F=2$ level.
  % The difference is
  % attributed due to  collective decay effects observed in the dense atomic ensemble~\cite{Dicke:53}.
  % The rise/fall times can also be varied in the experiment by varying the optical density of the cloud as previously shown in~\cite{Srivathsan:2013}.}
  % We have previously demonstrated to tune the 
  The time constants for the exponential rise and decay profiles 
  are shorter than the lifetime of the intermediate state.
  This is due to collective decay effects observed in dense atomic ensembles~\cite{Dicke:53}.
  The measured value is compatible with our previous measurements of
  the 
  distribution of detection time differences for this optical
  density~\cite{Srivathsan:2013}.
  }

From the results of the HBT experiment, we know that the idler detection
witnessed a single photon to a very good approximation. We therefore have to
conclude that the heralded signal field is a single photon state with an
exponentially rising temporal envelope as required for optimal absorption by a
two-level system \cite{Sondermann:07,Yimin:2011}.

In this case, however, the simple causal interpretation of the physical
process in the Weisskopf-Wigner picture does not work: the trigger
time is fixed by the herald that leaves the atoms in the ground state, but the
signal field  starts to rise to a maximum before that. So the
heralding process does not set an initial condition of a physical system that
then evolves forward in time, but marks the end of a (signal) field evolution
that is compatible with the exponential rise that started
{\em before} the heralding event. Formally this is not a problem, because the
heralding event just sets a different boundary condition.

This experiment again highlights a problem with the
definition of  ``real'' physical quantities (in the spirit of an EPR definition~\cite{einstein:35}). The physical quantity here is the electrical
field in the signal mode at any point in time. Nothing seems to set the initial
condition leading to such an increase, with a dynamics governed by
some laws of physics. Yet, when an idler event is registered in a
photodetector, the recorded field is perfectly compatible with a single photon
with an exponentially rising envelope. 
In this example, an interpretation
that is more symmetric between preparation and detection
procedures~\cite{Watanabe:55}, like the two-state vector
formalism~\cite{Reznik:95}, may be adequate.


\input{heralded_conclusions}

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

