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\begin{document}

%\preprint{APS/123-QED}

\title{Efficient conversion of coherent laser light to photon bunching}

\author{Xi Jie Yeo$^{1}$}
\author{Darren Ming Zhi Koh$^{1}$}
\author{Justin Yu Xiang Peh$^{1}$}
\author{Christian Kurtsiefer$^{1,2}$}
\author{Peng Kian Tan$^{1}$}
\email{cqttpk@nus.edu.sg}
\affiliation{$^{1}$Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\affiliation{$^{2}$Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore, 117551}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
Light sources demonstrating photon bunching find practical applications in range finding, imaging, and clock synchronization.
However, existing schemes to generate detectable photon bunching subject the source light to high losses, resulting in dim photon bunching light. We present a low-loss scheme to generate ultrabright photon bunching,  by transmitting laser light with a finite coherence time through a multi-path interferometer. Using this scheme, we demonstrated a less lossy photon bunching source about 14 orders of magnitude brighter in spectral density than many photon bunching sources from down-conversion in non-linear media. 
\end{abstract}

% \keywords{Photon Bunching, Quantum Sensing, Stationary Light, Temporal Correlations, Thermal Source}%Use showkeys class option if keyword display desired

\maketitle
\section{Introduction}
There are a few ways to generate photon bunching with sufficiently long coherence timescales to be detected and timing resolved by single photon avalanche detectors.
Popular techniques transmit coherent laser light through random phase dispersion media such as a rotating ground glass, or a liquid suspension of microspheres. However, the random scattering process introduces spatial incoherence that reduces the coupled single-mode bunched light by around 10 orders of magnitude. Photon bunching can also be generated by laser light through spontaneous parametric conversion (SPDC) processes which are similarly inefficient.

\begin{figure}
\centering\includegraphics[width=\columnwidth]{comparison.eps}
\caption{\label{fig:comparison} Spectral densities of stationary light sources with thermal and SPDC correlations: STL \cite{pk:23} -- subthreshold laser diode, MS \cite{dravins:15} -- suspension of microspheres, Sunlight \cite{pk:16} -- filtered Sunlight, Hg \cite{pk:14} -- Mercury discharge lamp, RGG \cite{zhu:12} -- rotating ground glass, EDFA \cite{janassek:18} -- Erbium-doped fiber amplifier, SLD \cite{rahman:20} -- superluminescent diode, MRR \cite{steiner:21} -- cavity enhanced via microring resonator, FWM \cite{england:19} -- four-wave mixing, PPLN \cite{zhang:15} -- periodically poled Lithium Niobate, BBO \cite{lohrmann:18} -- Beta-Barium Borate, PPKTP \cite{jeong:16} -- periodically poled Potassium Titanyl Phosphate.}
\end{figure}

We present an efficient photon bunching conversion technique based on sending continuous-wave laser through a multi-path interferometer with path differences significantly longer than the coherence time of the laser light.
A photon bunching source at 1\,mW output power is demonstrated, with a characteristic photon bunching timescale of about 135\,ns which is easily resolvable by readily available photodetectors.


\section{Theory}
We present here an analytical description of a scheme 
that changes the coherent light, from a laser,
to light exhibiting photon bunching.
In this scheme,
coherent light is sent through a multi-path interferometer,
formed by a cascade of $n$ number of Mach-Zehnder interferometers.
By choosing appropriate propagation delay in each of the Mach-Zehnder interferometer,
a resulting light field appears to be a superposition of $2^n$ coherent light fields
which demonstrate some extent of phase-independence between each other.  
This superposed light field approaches the model for thermal or chaotic light.

The coherent light from the laser can be modelled 
as an electric field
\begin{equation}
  E(t)  = E_{0}e^{i\left[2\pi f t +\phi(t)\right]},
\label{eqn:E-field-laser}
\end{equation}
where $E_{0}$ is the amplitude of the electric field that is assumed to be constant,
$f$ is the frequency of the laser,
and $\phi(t)$ is a random phase fluctuation 
related to the coherence time $\tau_{c}$ of the laser light source via
\begin{equation}
\langle e^{-i\phi(t)}\,e^{i\phi(t+\Delta)} \rangle = e^{-\tau/\tau_{c}},
\end{equation} 
where $\langle\dots\rangle$ is denotes the expectation value of its arguments 
and we have assumed a Lorentzian spectral lineshape for the laser light.

The laser light is coupled into a fibre beam splitter which the light into two paths.
The light in one of the paths travels through a long optical fibre
which introduces a propagation delay $\Delta_{1}$ 
with respect to the light in the second path. 
The light from these two paths each enter separate input ports of a second beam splitter
which interferes the two light fields, forming an asymmetric Mach-Zehnder interferometer.
The resulting light emerging from the output port $A_1,B_1$ of the interferometer 
has an electric field 
\begin{equation}
  E_{A_1,B_1}(t)  = \frac{1}{\sqrt{2}}\left[E(t)\pm E(t-\Delta_{1})\right],
\label{eqn:E-1}
\end{equation}
where the sign difference is the result of a relative $\pi$ phase-shift
added to one of the fields by the beam splitter, 
here assumed to be on $E(t+\Delta_{1})$.

The light from one of the output port,
here assumed to be port $B_1$,
is sent through another long optical fibre.
This introduces another propagation delay $\Delta_{2}$ 
to the light from at output $B_1$,
with respect to the light from output $A_1$.
The light from output port $A_1$ 
and the light from output port $B_1$ delayed by $\Delta_{2}$ 
each enters separate input ports of a third beamsplitter,
interfering the two light fields.
This results in the light emerging from the output ports $A_2,B_2$
of the third beamsplitter to be
\begin{equation}
  E_{A_2,B_2}(t)  = \frac{1}{\sqrt{2}}\left[E_{A_1}(t)\pm E_{B_1}(t-\Delta_{2})\right],
\label{eqn:E-2}
\end{equation}
where the sign difference is the result of a relative $\pi$ phase-shift
added to one of the fields by the beam splitter, 
here assumed to be on $E_{B_1}(t+\Delta_{1})$.

The act of sending light from one of the beamsplitter output ports  
through a long optical fibre, and recombining at another beams splitter
may be repeated for a total of $n$ iterations.
The resultant electric field emerging from the final beamsplitter in the chain is
\begin{equation}
  E_{A_n,B_n}(t)  = \frac{1}{\sqrt{2}}\left[E_{A_{n-1}}(t) \pm E_{B_{n-1}}(t-\Delta_{n})\right],
  \label{eqn:E-n}
\end{equation}
where we have assumed the light field  
$E_{B_{n-1}}$ has been sent through the $n$-th long optical fibre, 
introducing a propagation delay $\Delta_n$,
and have picked up a $\pi$-phase at the last beamsplitter.

From Eqn. \ref{eqn:E-1} to \ref{eqn:E-n},
it can be shown that the light field $E_{A_n,B_n}(t)$ emerging from the the last beamsplitter
generally consists of $2^{n}$ electric field terms, 
which each term being the coherent light field $E(t)$
with a propagation delay formed by different combinations of $\Delta_1$ to $\Delta_n$.
Furthermore,
we impose the following condition between the propagation delays $\Delta_{i}$
and the coherence time $\tau_c$ of the coherent light
\begin{equation}
\Delta_i - \sum_{\alpha=1}^{\alpha=i-1} > \Delta_{\text{min}} \gg \tau_{c},
\end{equation}
where $\Delta_{\text{min}}$ is the smallest difference in propagation delays pairwise between the $2^{n}$
electric fields. 
This results in the difference in propagation delays between any of the $2^{n}$ fields
to be alwats significantly greater than $\tau_c$.
Effectively,
any electric field pairs $E_{\alpha,\beta}$ of the $2^{n}$ electric fields
would appear phase-independent between zero-time difference up to the vicinity of $\Delta_{\text{min}}$ i.e.
\begin{equation}
\langle E^{*}_\alpha(t)E_\beta(t+\tau) \rangle \approx 0 \text{ for } \|\tau\|\lessapprox \Delta_{\text{min}}.
\end{equation}

This approximate phase-independence condition results in the output light field to appear like a superposition of $2^{n}$ independent coherent emitters, modelled similarly for thermal light.
Following the model of an ensemble of indendent emitters,
the second-order photon correlation would similarly predict
photon bunching for this light source.


\section{Setup}
This work shows a near-unity conversion scheme to produce ultrabright, spectrally narrowband, and spatially single-mode photon bunching from a seed laser.

\begin{figure}[h!]
\centering\includegraphics[width=\columnwidth]{figures/setup/setup_simplified.eps}
\caption{\label{fig:setup} 
Multi-path interferometer to generate bunched light. Laser light is sent to an input port of a beamsplitter (BS) and splits into two paths. Light in one of the path is delayed by a propogation delay $\Delta_1$. The delayed light and non-delayed light interferes at second beamsplitter. The act of splitting the light, delaying one of them and recombining is repeated for $n$ times.
Photon bunching would be observed from light at any of the two output ports A,B of the final beamsplitter in the chain. 
The propagation delay $\Delta$ is distinct in each Mach-Zehnder interferometer satisfying the condition Eqn. \textbf{(to insert later)}.
Inset: Details of components between pairs of beamsplitters. In-line polarisation control pedals are used to match the polarisations of light in the delayed and non-delayed paths of the respective Mach-Zehnder interferometer when they recombine at the next beamsplitter.}
\end{figure}

A distributed feedback laser is used to output 30\,mW of linearly polarised coherent laser light at 780\,nm. 
This light is then coupled into a spatial single-mode through a fiber beamsplitter which splits the light into two paths.
One of the paths is then transmitted through a delay optical fiber longer than the coherence length of the laser at around 40\,m.
This is so that the two splitted beams are phase-independent when they mode-overlap and recombine and a second fiber beamsplitter.
As such, the two beams are effectively a micro-ensemble of 2 emitters, and exhibit temporal photon bunching behaviour like a thermal light source.
A pair of half-wave plate and quarter-wave plate are placed into both beam paths prior to recombination, to match their polarisation modes.
The variable attenuator in the undelayed beam path is to compensate for attenuation losses in the delay optical fiber and rebalance both beams before recombining.

\begin{figure}
  \centering
\includegraphics[width=\columnwidth]{g2-loops.eps}
  % \vspace{\BeforeCaptionVSpace}
  \caption{
   \label{fig:g2} 
   	Tunable photon bunching amplitudes $g^{(2)}(0)$ 
    using different number n of cascaded Mach-Zehnder interferometer, 
    approaching $g^{(2)}(0)=2$ for thermal light.
   	For n $=0$ (diamonds), $g^{(2)}(\tau)=1$. 
  	For n $=1$ (triangles), $g^{(2)}(0)=1.471\pm0.003$, coherence timescale of $\tau_{c}=135.6\pm0.3$\,ns. 
  	For n $=2$ (circles), $g^{(2)}(0)=1.665\pm0.003$ and $\tau_{c}=134.8\pm0.2$\,ns.
  	For n $=3$ (squares), $g^{(2)}(0)=1.805\pm0.004$ and $\tau_{c}=135.2\pm0.2$\,ns.
    The solid lines show the fits to the second-order timing correlation function $g^{(2)}(\tau) = 1 + e^{-2|\tau|/\tau_c}$.
    } 
\end{figure}

\begin{figure}
\includegraphics[width=\columnwidth]{power_coherence_time.eps}
\caption{\label{fig:power_coherence_time}
Tunable (a) output optical power and (b) coherence time $\tau_c$ of photon bunching source 
by adjusting the laser current $I_{L}$.}
\end{figure}

\section{Conclusion}
Recently, photon bunching sources have found applications in range finding, imaging, and clock synchronization.
In these applications, the temporal correlations of these sources were used to extract timing information or increase the distinction between signal and noise.
Furthermore, in contrast to light sources with timing correlations artificially encoded from external sources, such as randomly pulsed lasers, the photon bunching light sources with timing correlations from natural causes are potentially resilient to vulnerabilities such as optical cross-talk between similar devices or third-party manipulation.
This provides motivation for bright photon bunching sources with resolvable characteristic timescales $\tau_c$.

\section{Acknowledgments}
This research is supported by the Quantum Engineering Programme through NRF2021-QEP2-03-P02, the Ministry of Education, and the National Research Foundation, Prime Minister's Office, Singapore.


\bibliography{HBT}


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