% ****** Start of file apssamp.tex ******
%
%   This file is part of the APS files in the REVTeX 4.2 distribution.
%   Version 4.2a of REVTeX, December 2014
%
%   Copyright (c) 2014 The American Physical Society.
%
%   See the REVTeX 4 README file for restrictions and more information.
%
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed
% as well as the rest of the prerequisites for REVTeX 4.2
%
% See the REVTeX 4 README file
%
% \documentclass[aps,pra,twocolumn,superscriptaddress]{revtex4-2}
% \usepackage{mathtools}
% \usepackage{hyperref}
% \usepackage{enumerate}
\documentclass[
reprint,
%superscriptaddress,
%groupedaddress,
%unsortedaddress,
%runinaddress,
%frontmatterverbose, 
%preprint,
%preprintnumbers,
%nofootinbib,
%nobibnotes,
%bibnotes,
amsmath,
amssymb,
aps,
% pra,
%prb,
%rmp,
%prstab,
%prstper,
%floatfix,
showkeys,
]{revtex4-2}

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{natbib}%\citestyle{nature}
\usepackage{upgreek}
%\usepackage{hyperref}% add hypertext capabilities
%\usepackage[mathlines]{lineno}% Enable numbering of text and display math
%\linenumbers\relax % Commence numbering lines

%\usepackage[showframe,%Uncomment any one of the following lines to test 
%%scale=0.7, marginratio={1:1, 2:3}, ignoreall,% default settings
%%text={7in,10in},centering,
%%margin=1.5in,
%%total={6.5in,8.75in}, top=1.2in, left=0.9in, includefoot,
%%height=10in,a5paper,hmargin={3cm,0.8in},
%]{geometry}
\usepackage{xcolor}

\begin{document}

%\preprint{APS/123-QED}

%\title{Efficient conversion of coherent laser light to photon bunching}
\title{Near-losslesss method for generating thermal photon-bunched light}

\author{Xi Jie Yeo$^{1,2}$}
\author{Darren Ming Zhi Koh$^{1}$}
\author{Justin Yu Xiang Peh$^{1}$}
\author{Christian Kurtsiefer$^{1,3}$}
\author{Peng Kian Tan$^{1}$}
\email{cqttpk@nus.edu.sg}
\affiliation{$^{1}$Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\affiliation{$^{2}$DSO National Laboratories, 12 Science Park Drive, Singapore 118225}
\affiliation{$^{3}$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. 
Thermal light sources exhibiting photon bunching have been suggested for sensing applications that exploit timing correlations of stationary light, including range finding, clock synchronization, and non-line-of-sight imaging.
However, these proposals have remained unrealized in practice because available sources of photon bunching either possess coherence times too short to be timing resolved by photodetectors, or produce brightness levels too low to tolerate realistic return losses.
In this work, we demonstrate a low-loss method for generating photon bunching with a conversion efficiency nearly 9 orders of magnitude higher than that achieved by many other bunching processes.
\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.
Many sensing methods with non-classical light generated  by parametric
conversion mostly rely on the tight temporal correlation of photon
pairs (citation maybe \cite{Clark:21}?). However, a significant temporal correlation between
photodetection times can also be found in thermal light (citation maybe \cite{loudon:00}?).
\begin{figure}
\centering\includegraphics[width=\columnwidth]{conversion.eps}
\caption{\label{fig:conversion}
Power relations of various methods and for generating photon-bunched light
from laser light: MS \cite{dravins:15} -- liquid suspension of microspheres,
RGG \cite{zhu:12} -- rotating ground glass, EDFA \cite{janassek:18} --
Erbium-doped fiber amplifier. Methods based on parametric conversion: MRR
\cite{steiner:21} -- cavity enhanced via microring resonator, FWM
\cite{england:19} -- parametric conversion in four-wave mixing.}
\end{figure}
Blackbody radiation provides a natural source of thermal photon bunching, with Sunlight filtered at 546\,nm exhibiting a spectral density comparable to that of a low-pressure Mercury vapor discharge lamp, on the order of $10^{7}$ photoevents per second per nanometer per mode.
Much brighter sources of photon-bunched light are based on conversion of
coherent laser light \cite{dravins:15, zhu:12, janassek:18, steiner:21,
  england:19, lohrmann:18, jeong:16} (see Fig.~\ref{fig:conversion}).
Already established methods, commonly referred to as sources of `pseudothermal
light', rely on scattering coherent laser light from random phase-dispersive media, such as a rotating ground glass, or a liquid suspension of microspheres in Brownian motion.



\begin{figure}
\centering\includegraphics[width=\columnwidth]{comparison.eps}
\caption{\label{fig:comparison} 
Spectral densities of stationary light sources exhibiting photon-bunching
correlations. Apart from the demonstrations referenced in
Fig.~\ref{fig:conversion}, other demonstrations shown are: STL \cite{pk:23} --
subthreshold laser diode, \cite{pk:16} -- filtered Sunlight, Hg \cite{pk:14}
-- Mercury discharge lamp, SLD \cite{rahman:20} -- superluminescent diode,
PPLN \cite{zhang:15} -- parametric down conversion in periodically poled Lithium Niobate.}
\end{figure}

The scattering processes in these methods introduce speckle patterns.
These scattering-induced fluctuations lead to highly inefficient coupling to
single spatial optical modes, reducing the intensity of bunched light by approximately 10 orders of magnitude relative to the input laser.
Photon bunched light can also be prepared through spontaneous parametric down-conversion (SPDC) processes which are similarly inefficient.

Erbium-doped fiber amplifiers (EDFA) operating below their lasing threshold
can reach a relatively high conversion efficiency, transforming about 25\% of
the 980\,nm pump laser into amplified spontaneous emission centered around
1550\,nm. However, the resulting output thermal light is spectrally broad,
spanning several tens of nanometers, with a correspondingly short
sub-picosecond coherence time that cannot be readily resolved by conventional
photodetectors for observing photon bunching.
Moreover, the accessible conversion wavelengths are inherently restricted by
the amplification process. A chart of the relationship between spectral
bandwidth and photon putput per optical mode for demonstrated photon-bunched
light sources is shown in Fig.~\ref{fig:comparison}.

%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.
Here, we demonstrate a wavelength-independent technique for generating
spectrally narrowband photon-bunched light from laser light.
In principle, the process can approach a near-unity conversion efficiency, as
it is free from the fundamental losses typically associated with mode mismatch
associated with scattering processes.


\section{Idea}
\begin{figure}
\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 propagation 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.
Scheme to efficiently generate thermal photon-bunched light from coherent laser light by using a cascaded series of asymmetric Mach-Zehnder fiber interferometers.
The input field is first split by a beamsplitter (BS), with one arm acquiring a propagation delay $\Delta_{1}$ before recombination at a subsequent beamsplitter with the non-delayed arm.
The interferometric loop can be repeated $n$ times, with each stage
introducing progressively longer fiber delays $\Delta_{n}$, providing two single-mode outputs ($A_n, B_n$).
%This scheme transforms input light from a laser into phase-randomized optical
%modes whose recombination now yields two single-mode outputs ($A_n, B_n$)
%exhibiting thermal photon bunching.
}
\end{figure}
%Our scheme generates photon bunching by superposing a collection of phase-independent coherent fields, similar to how thermal light is modeled \cite{loudon:00}. Our scheme works by sending coherent light through a multi-path interferometer. This can be formed by cascading $n\geq 1$ Mach-Zehnder interferometers each with a different propagation delay, $\Delta_i$, in one of the two paths. The multiple paths create the superposition while phase-independence is created by our choice of propagation delay.
Thermal light can be modeled as an ensemble of phase-independent coherent fields \cite{loudon:00}.
To generate thermal photon-bunched light from laser light, a scheme is
required to transform the laser emission into a micro-ensemble of mutually
phase-independent fields. It is starting with the random phase relation of
typical laser light between time differences outside the coherence time of the
laser.  
The conversion can be realized by cascading a series of asymmetric
Mach-Zehnder interferometers, each introducing an additional propagation delay
$\Delta_n$ in one arm (see Fig.~\ref{fig:setup}).
The accumulated delays randomize the relative phases of the component fields without introducing scattering losses.
This scheme can be implemented in single-mode optical fibers, suppressing the
loss by eliminating mode mismatch at each stage in a free space implementation.

The electric field amplitude of the input laser light can be described by
\begin{equation}            
  E(t) = E_0e^{i\left[2\pi f t +\phi(t)\right]}\,,
  \label{eqn:E-field-laser}
\end{equation}
where $E_{0}$ denotes the constant electric field amplitude, $f$ the laser
frequency, and $\phi(t)$ the time-dependent phase fluctuation capturing the
finite coherence time of the laser \cite{mandel:59}.

After the light passes through the first interferometer with a additional
delay of $\Delta_1$ in one arm, the electric field amplitudes (for matching
polarizations at the second beamsplitter) at outputs A$_1$, B$_1$ are
\begin{equation}
  E_{A_1,B_1}(t)=\frac{1}{2}[E(t) \pm E(t-\Delta_1)]\,,
  \label{eqn:first-splitter-output}  
\end{equation}
with the difference between the two outputs caused by the relative $\pi$ phase
shift imparted by any beamsplitter. 

Passing both output fields through the next interferometer with a
corresponding delay $\Delta_2$ produces the following field amplitudes (for
matching polarizations) at outputs $A_2$, $B_2$:
\begin{align}
	E_{A_2,B_2}(t) =& \frac{1}{2\sqrt{2}}[E(t) - E(t-\Delta_1) \mp E(t-\Delta_2) 		\nonumber \\
	&\mp E(t-\Delta_1-\Delta_2)]\,.
\end{align}
These light fields are a superposition of the initial laser field $E(t)$ with delayed copies of itself through a combination of delays $\Delta_1, \Delta_2$.

Repeating this procedure through $n$ cascaded interferometers leads to output fields that are superpositions of the original coherent field and all delayed copies generated by the full set of delays $\{\Delta_{1},\Delta_{2},...,\Delta_{n}\}$,
%The act of sending light from one of the beam splitter output ports through a long optical fiber, and recombining at another beams splitter may be repeated for a total of $n$ iterations. The resultant electric field emerging from the final beam splitter in the chain is
\begin{align}
  E_{A_n}(t) = \frac{1}{\sqrt{2}^{\,n+1}} & \Big[E(t) + \sum_{j=1}^{n}\alpha_{j}E(t-\Delta_j)\nonumber\\
  &+\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}\alpha_{jk}E(t-\Delta_j-\Delta_k) 
   \nonumber\\
  &+\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}\sum_{l=k+1}^{n}\alpha_{jkl}E(t-\Delta_j-\Delta_k-\Delta_l) 
	\nonumber\\
	&+\cdots 
	\nonumber\\
	&+E(t-\Delta_1-\cdots-\Delta_n)\Big]\,,
\end{align}
with all coefficients $\alpha \in \{1,-1\}$. A similar expansion holds for
$E_{B_n}$. So the resulting output fields are superpositions of the initial laser light field $E(t)$ together with the $2^n-1$ delayed copies of itself that went through different combinations of delays $\Delta_1$ to $\Delta_n$.

%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 interferometer consists of $2^{n}$ terms, each being the coherent light field $E(t)$ with a propagation delay formed by a different combinations of $\Delta_1$ to $\Delta_n$.
To ensure phase independence between the various delayed copies of the initial light field $E(t)$, the delays $\Delta_n$ in each interferometric loop $n$ are chosen to satisfy
\begin{equation}
	\Delta_{n}-\Delta_{n-1} \geq \tau_c, 
	\label{eqn:delta}
\end{equation}
such that each successive delay exceeds the laser coherence time $\tau_{c}$.
According to the Wiener-Khinchin Theorem, the autocorrelation function of a stationary signal is given by the Fourier transform of its power spectral density.
A laser with a Lorentzian spectral line shape \cite{loudon:00} therefore exhibits a first-order coherence function $g^{(1)}(\tau)$ proportional to $e^{-\tau/\tau_{c}}$, whereby $1/\tau_{c}$ is the corresponding laser linewidth.

The output is therefore an ensemble of $2^{n}$ phase-independent coherent fields.
As a result, the light exhibits thermal photon bunching characterized by the second-order coherence function:
\begin{equation}
	g^{(2)}(\tau) - 1 = (1 - 1/2^{n})e^{-2|\tau|/\tau_{c}}.
	\label{eqn:g2}
\end{equation}
The photon bunching amplitude increases with the number of interferometric
loops $n$, converging to $g^{(2)}(0)-1=1$ for an infinite number of
phase-independent light fields, matching the correlation of an ideal thermal
light source.


\section{Experimental implementation}
%This work shows a near-unity conversion scheme to produce ultrabright, spectrally narrowband, and spatially single-mode photon bunching from a seed laser.
% The setup consists of $n$ asymmetric Mach-Zehnder fiber interferometers arranged in series. 
% By introducing optical delays in each interferometer, the input coherent laser light is randomized into $2^{n}$ mutually phase-independent beams. 
% These beams are then recombined into 2 single-mode outputs that exhibit photon bunching.



%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.

To demonstrate the idea, we inject light from a distributed feedback laser
providing linearly polarized light with a center wavelength of 780\,nm into a single spatial mode and couple this to a fiber beamsplitter
separating it into two modes. The first asymmetric Mach-Zehnder
interferometer passes light through an additional optical fiber
to introduce a propagation delay $\Delta_{1}$ in one arm with an additional
optical fiber of $l_1=100$\,m length (corresponding to $\Delta_{1}\approx 495$\,ns),
exceeding the laser coherence time $\tau_{c}$, and another fiber beam splitter
closing the interferometer.

A paddle polarization a controller (not shown in Fig.~\ref{fig:setup}) ensures the simple form of adding amplitudes in
Eq.~(\ref{eqn:first-splitter-output}).
Additionally, variable attenuators are inserted in the short leg of the interferometer to compensate for losses in the delay fibers, thereby balancing light intensities just prior to mode-overlap at the beamsplitters. 
These parameters are optimized to increase the interferometric visibility; the
maximum of the photon bunching signature $g^{(2)}(\tau=0)$ is taken as
an optimization criterion for polarization and intensity balancing control.


To chain the delay combination, more asymmetric Mach-Zehnder interferometers are
added, as well as polarization and intensity balance components. The delay
fiber lengths ($l_2=186$\,m, $l_3=500$\,m) are chosen to introduce
propagation delays $\Delta_{n}$ in each loop that closely satisfy the condition Eq.~(\ref{eqn:delta}).

% In-line fiber pedal controllers are adjusted to align the polarization planes of the beam pairs recombining at each fiber beamsplitter.
% This ensures that the two beams are identical in polarization when they mode-overlap, while remaining mutually phase-independent due to the propagation delays $\Delta_{n}$. 
%Both beams are also identical in their spectral profiles as they are emitted by the same laser.

To perform the $g^{(2)}(\tau)$ timing correlation measurements, light from
either output $A_n$ or $B_n$ is  collected by a pair of actively quenched
Silicon avalanche photodetectors. The resulting photodetection events are
timestamped with a time resolution of 2\,ns. % resolution using a field programmable gate array device. 

\section{Results}
\begin{figure}
  \centering
\includegraphics[width=\columnwidth]{g2-loops.eps}
  % \vspace{\BeforeCaptionVSpace}
  \caption{\label{fig:g2}
Photon bunching signatures from a laser light (center wavelength
$\lambda=780$\,nm)  passing through a number $n$ of
asymmetric Mach-Zehnder interferometers.
A numerical fit to a double-exponential decay reaches a
peak value of the second order correlation function at time delay $\tau=0$. We
find for $n=0$ (no interferometer): $g^{(2)}(\tau)=1$, %(diamonds)
for $n=1$: $g^{(2)}(0)=1.471\pm0.003$, %(triangles)
for $n=2$: $g^{(2)}(0)=1.665\pm0.003$,  %(circles)
and for $n=3$: $g^{(2)}(0)=1.805\pm0.004$. %(squares)
The coherence times extracted for all traces is around $\tau_c\approx135$\,ns
(see text for details). } 
\end{figure}
To show the photon bunching signature, the temporal correlation functions
$g^{(2)}(\tau)$ for a different number $n$ interferometers were numerically
extracted from the stream of detector time stamps, and are shown in
Fig.~\ref{fig:g2}.

As a reference, a flat $g^{(2)}(\tau)=1$ is observed for $n=0$ (i.e., no
interferometer) which is consistent for the un-bunched Poisson statistics
expected from a laser emitting coherent light.
%The recombined light beams at outputs A and B can be considered as originating from an effective ensemble of $2^{n}$ random emitters.
The resultant photon bunching peaks for $n>0$ were fitted to two-sided
exponential decays (as expected for Lorentzian line shapes of the laser
light), using a model function $g^{(2)}(\tau) = 1 + b\cdot
e^{-2|\tau|/\tau_c}$.
For $n=1$, we obtain $g^{(2)}(0)=1.471\pm0.003$ with a coherence time of
$\tau_{c}=135.6\pm0.3$\,ns. For $n=2$, we obtain $g^{(2)}(0)=1.665\pm0.003$
and $\tau_{c}=134.8\pm0.2$\,ns, for $n=3$, $g^{(2)}(0)=1.805\pm0.004$ and
$\tau_{c}=135.2\pm0.2$\,ns, respectively.
This suggests that the correlation function converges towards
$g^{(2)}(\tau=0)=2$ with increasing number $n$ of interferometers (providing
an increasing number of phase-independent effective light sources with the
same spectrum).
This behavior closely approximates the statistics of an ideal thermal light source with photon bunching characteristics  $g^{(2)}(\tau) = 1 + e^{-2|\tau|/\tau_c}$.

For $n=1$, we observe a conversion efficiency from laser light to
photon-bunched light at both output ports together of 40\%. This loss is
mostly contributed by the losses in the delay fiber at 780\,nm wavelength,
and the corresponding attenuation to balance the intensity in both
interferometer arms. A significantly higher conversion efficiency should be
obtained when moving to a wavelength where optical fibers have a lower loss.

%A conversion efficiency of 92.5\% is achieved, yielding two outputs of 1.69\,mW and 1.67\,mW of thermal photon bunching from 3.63\,mW of laser light input at 1550\,nm.
%The generated photon bunching exhibits a long coherence timescale in the several tens of ns, which is inversely proportional to the linewidth of the input laser, and can be readily resolved by standard photodetectors. 

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

The coherence time $\tau_c$ of the laser we used was long conveniently allowed
to resolve the exponential decay of the correlation function from the thermal
bunching peak. However, by adjusting the injection current to the laser diode
we used, we could vary the coherence time. In
Fig.~\ref{fig:power_coherence_time}, we show the dependency of the laser power
and coherence time $\tau_c$.
For injection currents shortly above the lasing threshold (around 40\,mA), the
laser emits very little power, and shows a short coherence time. With
increasing injection current, the output power of the laser increases, as well
as the coherence time extracted from the decay of $g^{(2)}(\tau)$. Therefore, 
the presented technique also allows control over the temporal coherence
time, complementing the control over the photon bunching amplitude
$g^{(2)}(\tau=0)$ with the number $n$ of asymmetric Mach-Zehnder interferometers.

\section{Conclusion and Outlook}
%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 $\tau_c$.

We demonstrated a technique to generate thermal photon bunching that is about
9 orders of magnitude more efficient than many other thermal or parametric
down conversion processes, showing a photon pair timing correlation that could
have applications that rely on time correlations between two photons with an
intrinsic randomness of single-photon timings. Moreover, this method also
allows for control over the peak photon bunching signature and temporal
coherence. If optical wavelengths are used that lead to negligible losses
in the asymmetric interferometers, a unit conversion efficiency of laser light
to photon bunched light should be reachable, provided an unprecedented
brightness of photon-bunched light. 
This efficiency and tunability enable the sensing system to tolerate high return losses in realistic environments, thereby opening a pathway toward practical sensing implementations such as range finding, clock synchronization, and non-line-of-sight imaging.

\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}


\end{document}
%
% ****** End of file apssamp.tex ******
