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\begin{document}

%\preprint{APS/123-QED}

\title{Practical Photonic Quantum Sensing with Thermal Light}
%\title{Optical Timing Reference with Thermal Light using Sub-Threshold Laser}

\author{Peng Kian Tan$^{1}$}
\email{cqttpk@nus.edu.sg}
\author{Xi Jie Yeo$^{1}$}
\author{Alvin Zhen Wei Leow$^{2}$}
\author{Lijiong Shen$^{1}$}
\author{Christian Kurtsiefer$^{1,2}$}
\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, 117542}

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

%\keywords{Suggested keywords}%Use showkeys class option if keyword
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\maketitle
\textbf{Quantum sensing exploits quantum phenomena
  to improve the measurements of physical parameters
  and can be implemented in photonic, atomic or solid-state systems \cite{pirandola:18}.
 Photonic quantum sensing techniques include ghost imaging and
 super-resolution imaging \cite{moreau:19}.
% other proposals explore entanglement at the protocol-level \cite{lloyd:08}.
% PK: original intent was to compare quantum radar proposals using entanglement\cite{lloyd:08}, against actual experiments using spdc but without entanglement. But maybe this draws too much attention to the specific case of ranging, which is not the main story here
 % it remains difficult to implement \cite{sihui:08, shapiro:09}.
 Many photonic quantum sensing
 schemes rely on photon pairs generated in spontaneous parametric down
 conversion (SPDC)\cite{ghosh:87} that can be entangled in several degrees of
 freedom, but most often make use of the temporal correlation between the
 photons\cite{clark:21}. Examples are range finding \cite{lopaeva:13,
   barzanjeh:15} and clock  synchronization \cite{ho:09, jianwei:19} schemes,
 where quantum light sources have an advantage of being stationary, and
 therefore carrying no obvious timing structure that may be subject to
 manipulation or eavesdropping.
 The luminosity of SPDC-based photon pair sources is typically limited to
 sub-nanowatts. 
 In this work, we consider thermal light an alternative resource of
 time-correlated photons by exploiting its photon bunching property. We
 demonstrate quantum sensing using a relatively simple thermal light source
 with an optical power of about 12.5\,$\upmu$W and a spectral linewidth around
 43\,MHz in a single  optical mode. The resulting spectral density of this
 source exceeds that of  SPDC sources by
 approximately 10 orders of magnitude. We demonstrate its usefulness in an
 optical ranging experiment over about 1.8\,km.
 }\\


\noindent
\textbf{Time-correlated photon pairs}
Photonic sensing applications often make use of modulated light sources and
seek for correlations of a returned signal with the modulation. In an attempt
of moving to low light levels, one can make use of inherent temporal
correlations found in photon pairs emerging from a spontaneous parametric down
conversion (SPDC) process in crystals with second-order non-linear optical
response, or through four-wave mixing (FWM) in optical fibers or atomic
vapours. These processes generate pairs of photons that exhibit
a strong correlation of photodetection events, manifesting in
a strongly peaked second order correlation function
$g^{(2)}(\tau)=f(\tau/\tau_c)$, which characterizes a probability to observe a
pair at a time difference $\tau$. The function $f$ is strongly peaked around
$\tau=0$ (with a spread on the order of a coherence time $\tau_c$), and is of
inherent quantum nature, and can be observed in specialized quantum light
sources. Sensing applications based on this effect are carried out by
measuring detection time differences between one photon acting as a reference,
and the other one acting as a probe.


A more natural type of light is thermal light. Thermal light, such as blackbody radiation, exhibits a
characteristic temporal photon bunching behavior \cite{glauber:63a,
  glauber:63b}, also known as the Hanbury-Brown--Twiss effect
\cite{hbt:56a}. This is also described by a peaked second-order timing correlation,
\begin{equation}
\label{eqn:g2}
	g^{(2)}(\tau) = 1 + e^{-2|\tau|/\tau_c}\,,
\end{equation} %Eqn.~\ref{eqn:g2}
where $\tau$ is again the timing separation of the two photodetection events,
and $\tau_c$ is the coherence timescale of the temporal photon bunching where
thermal photons have a tendency to propagate closer together than described by
Poissonian statistical timing distribution. Similar to light generated by
SPDC, the coherence timescale $\tau_{c}$ is inversely proportional to the spectral width $\Delta\lambda$ of the thermal light, which is given by the Fourier transform of the source power spectrum \cite{fox:06}, such that $\Delta \lambda=1/\tau_{c}$ for single-line Gaussian spectrum.

% Light from a thermal source is therefore time-correlated within the coherence
% timescale $\tau_{c}$, such that $2\geq g^{(2)}(|\tau|<\tau_{c})>1$, similarly to light emerging from SPDC. The
% photoevents are otherwise randomly distributed in time such that
% $g^{(2)}(|\tau|>>\tau_{c})=1$.

\begin{figure}
\centering\includegraphics[width=\columnwidth]{comparison.eps}
\caption{\label{fig:comparison} Spectral densities of thermal and SPDC light sources based on: Microspheres\cite{dravins:15} -- suspension of microspheres, Hg\cite{pk:16} -- Mercury discharge lamp, RGG\cite{zhu:12} -- rotating ground glass, EDFA\cite{janassek:18} -- Erbium-doped fiber amplifier, SLD\cite{rahman:20} -- superluminescent diode, PPLN\cite{zhang:15} -- periodically poled Lithium Niobate, FWM\cite{england:19} -- four-wave mixing, BBO\cite{lohrmann:18} -- Beta-Barium Borate, PPKTP\cite{jeong:16} -- periodically poled Potassium Titanyl Phosphate.}
\end{figure}

An important practical consideration for sensing applications is the
brightness of the correlated light source.
As shown in Fig.~\ref{fig:comparison}, SPDC light sources generate an output
power below a nanowatt, or in the range of 10$^{4}$ to 10$^{9}$ photoevents
per second. This
limits the practicality of SPDC light-based sensing in environments with high
attenuation or loss.
%
Another important property in a temporal correlation measurement is the
accuracy 
that can be practically used infer e.g. a time-of-flight for one of the
photons.
Timing uncertainties of semiconductor-based single-photon detectors are
somewhere below a nanosecond, but more recent nanowire-based detectors may
reach a few picoseconds. When identifying the temporal correlation
feature in thermal light, however, it is necessary that the correlation peak
is still detectable. If the coherence time of the thermal light is
significantly smaller than the detector timing uncertainty, the visibility of
the temporal correlation washes out and may make it impossible to identify it on
top of the Poissonian background. It is therefore desirable to use thermal
light sources with a spectral width below $\approx1$\,GHz.

Thermal light with such a narrow optical bandwidth has been generated in many
different ways. Early examples include single emission lines of gas discharge
lamps. Other methods involve transmitting laser light through random
dispersion media such as suspension of microspheres, or a rotating ground
glass plate. These sources, however, have either relatively low output power
due to the spatial incoherence of the randomization mechanism, or (e.g. in the
case of rotating ground glass modulators) a relatively long coherence time.


In this work, we use a laser diode operating below the lasing threshold
\cite{bernard:61, lasher:64} to generate thermal light \cite{shockley:52,
  cassidy:91}. This amplified spontaneous emission process generates
significantly higher output power in the range of 10\,$\mu$W to 100\,mW. Light
sources of a similar category include superluminescent diodes, and Erbium-doped fiber amplifiers. These examples tend to have spectral densities above milliwatts per nanometer.\\

\noindent
\textbf{Quantum sensing setup}
The stationary thermal light generated from a subthreshold laser diode is
implemented into an optical ranging setup based on time-of-flight
measurements, commonly known as light distance and ranging (lidar).

While conventional lidar introduces timing modulation \cite{beheim:86,
  royo:19} into the intensity, amplitude, or phase of the light source, to
provide timing correlations, this work exploits the temporal photon bunching of thermal light to provide the timing correlations.


The lidar configuration illustrated in Fig.~\ref{fig:setup} is based on
increasing the coherence timescale of the thermal light by narrowband spectral
filtering \cite{pk:17}, and then performing coincidence measurements with high
 timing resolution to record the temporal photon bunching signature
 $g^{(2)}(\tau)$.

\begin{figure}
\centering\includegraphics[width=\columnwidth]{setup.eps}
\caption{\label{fig:setup}The experimental setup using thermal light to perform ranging measurements. LD: laser diode, Pol: polariser, E: etalon, BP: bandpass filter, BS: beamsplitter.}
\end{figure} %Fig.~\ref{fig:setup}. 

The $g^{(2)}$ photon bunching peak is shifted by a time $\tau_0$ in its timing
position, corresponding to the optical path length difference between probe and
reference beam, thus allowing to infer the distance of a target retroreflector
from the peak position of $g^{(2)}$ and the speed of light.

The stationary light source in our demonstration is a laser diode operating at a subthreshold current of 32.9\,mA which is determined in Fig.~\ref{fig:spectra} via the output power and spectrum, and therefore exhibits the photon bunching behaviour that is characteristic of thermal light. 

The thermal light is first sent through a single-mode optical fiber to select
for the fundamental Gaussian mode to enforce spatial coherence,
then transmitted through a polarizer to obtain a well-defined linear
polarization,  and spectrally filtered by an etalon (see Methods section for
details).

An asymmetric beamsplitter directs 92\,\% of the filtered thermal light into
the probe beam and retains 4\,\% as a local reference beam sent to a first
single photon detector.
% , and the remaining 4\,\% lost.
After some spatial beam conditioning (see Methods section), the probe beam
propagats to the retroreflector target, and back to the sensing setup onto the
second single photon detector.

Both single photon detectors are actively quenched Silicon avalanche
photodiodes with a quantum efficiency about $50\,\%$ at 550\,nm and a timing
jitter around 40\,ps. The detected photoevents are timestamped either by an
oscilloscope with (sampling rate 40\,GSPS), and time differences
were histogrammed into 40\,ps wide time bins, through an FPGA-based timestamp
device into 2\,ns wide time bins.\\

\noindent
\textbf{Range sensing demonstration}
Figure~\ref{fig:precision} shows two time difference histograms together with
a fitted second-order timing correlation function g$^{(2)}(\tau)$ according to
Eqn.~\ref{eqn:g2} that allows to determine the positions $\tau_0$ of their
respective bunching peaks, and the corresponding ranging distance $d$ extracted
from the round trip time of the probe beam for a set of target placement positions(Fig.~\ref{fig:precision}, bottom trace). The resulting ranges are in good
agreement with their corresponding target placement positions, and compatible
with a constraint in the detector timing jitter (about 40\,ps FWHM).

\begin{figure}
\centering\includegraphics[width=\columnwidth]{multiplot_g2precision.eps}
\caption{\label{fig:precision} [Top]: The photon bunching $g^{(2)}(\tau)$
  measurements with the target reflector placed at 0\,mm (black) and 90\,mm
  (red). The solid line represents a fit to Eqn.~\ref{eqn:g2} resulting in
  $\tau_{c}=1.03\pm$0.03\,ns, a peak displacement of $\tau_0=0.606\pm$0.008\,ns corresponding to a ranging distance of $d$=91.0$\pm$1.2\,mm, with a
  reduced $\chi^{2}$ of 1.19. [Bottom]: The fitted ranging distances are plotted against their placement positions to test for dstance resolution.}
\end{figure} %Fig.~\ref{fig:precision}. 


To demonstrate the robustness of the ranging setup in a real-world
environment, we conducted two outdoor field measurements in Downtown
Singapore.
%, located at 1$^{\circ}$17'\,N 103$^{\circ}$52'\,E. 

%The ranging measurements took place across the water body known as Marina Bay, Singapore, with the launch optics installed on the 47th storey of the Marina Bay Sands Hotel. The ranging vector was between an elevated position towards ground level located on the other side of the bay, as indicated in by the red line.

%The humidity in the outdoor air was high as expected for the tropical climate of equatorial Singapore, and was further increased by the site proximity to the sea and bay area, as well as the light rainy weather on that day. This humid condition introduced scattering in the transmission air column, and attenuation of the probe beam.

%Top plot [blue] results are $\chi^{2}/doF=1.09$, $g^{(2)}(\tau=0)=1.572\pm0.007$, $t_{c}=31.3\pm0.5$\,ns, time of flight distance of $96529\pm2$\,cm.  

\begin{figure}
\centering\includegraphics[width=\columnwidth]{g2lidar2km.eps}
\caption{\label{fig:g2} Optical ranging measurements to the reference zero distance position with the retroreflector placed at the telescope aperture, and to the target retroreflector located at kilometric distances.}
\end{figure} %Fig.~\ref{fig:g2}. 


Time-of-flight measurements in Fig.~\ref{fig:g2} are compared between a
reference zero position (black) and outfield locations, resulting in the
ranging distances of $96529\pm2$\,cm (blue) and $185148\pm2$\,cm (red) fitted
to Eqn.~\ref{eqn:g2} with reduced $\chi^{2}$ of 1.09 and 1.05,
respectively, under the assumption of a unit refractive index of air. The
increased uncertainty compared to the short range measurements shown in
\ref{fig:precision} are due to the more coarse histogramming for this
experiment. 

For these measurements, the etalon temperature tuning and stability was improved relative to the measurements in Fig.~\ref{fig:precision}, increasing the coherence timescale $\tau_{c}=23.2\pm0.4$\,ns (red), corresponding to a spectral linewidth $\Delta \lambda=43$\,MHz. 

The temporal photon bunching peak (red) is slightly reduced to
$g^{(2)}(\tau=0)=1.591\pm0.009$ (red) due to an increase of the bin width from
40\,ps to 2\,ns for the time differences, and by noise contribution from ambient light to the probe detector.

The very high spectral density of our light source of 12.5\,$\mu$W in a
43\,MHz spectral band increases the signal-to-noise ratio \cite{hbt:74,
  foellmi:09} of a bunching peak detection significantly. Even for an
attenuation or loss of 65\,dB in the probe path, the actively quenched Silicon
avalanche photodetectors could be saturated (see Methods section for more
details).\\

\noindent
\textbf{Quantum sensing using thermal light}
This work explored the use of thermal light for applications where
measurements (like range finding) are based on detecting correlations in time
are at the core of the sensing task. Sub-threshold lasers and a
narrowband filtering technique helped to provide a powerful alternative to
light sources based on spontaneuous parametric down conversion in quantum
sensing applications that may offer superior signal-to-noise ratios, at a
much reduced system complexity.

With such light sources, we envision that a technique originally used for
estimating the size of stars half a century ago can boost a wide range of
practical quantum sensing applications that mostly rely on temporal
correlations.\\

\noindent
\textbf{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.
\clearpage

\bibliography{HBT}

\noindent
\textbf{Methods}\\

\noindent
\textbf{Subthreshold laser diode}
Laser diodes generate coherent laser light with Poissonian timing statistics
when operating above the lasing threshold current. Below threshold, the optical
spectrum and power resemble that of light emitting diodes, with a thermal
photon statistics.

For our device (nominal lasing wavelength $\lambda=518$\,nm, single-mode spatial output), the lasing threshold current was found to be around 33\,mA
(see Fig.~\ref{fig:spectra}), where the output power exhibits a sharp increase
of 3 orders of magnitude, and the spectrum narrows to a single emission line,
limited by the grating spectrometer to about 0.3\,nm. 

%It is a commercial readily-available (Thorlabs L515A1) anode grounded laser diode with a typical output power P = 10\,mW at injection current I = 100\,mA maximum. The laser diode is temperature stabilized by a thermo-electric peltier module at $20.000\pm0.005\,^{\circ}$C.

The thermal light generated at a subthreshold current of 32.9\,mA is coupled
into a single spatial mode of an optical fiber, resulting in a very bright spectral density of around $12.5\,\mu$W, corresponding to approximately $10^{13}$ photoevents per second, within a spectral width $\Delta\lambda=43$\,MHz.\\


\noindent
\textbf{Thermal light preparation}
To select a relatively narrow window of thermal light from the spectrum
emitted by the sub-threshold laser source, a combination of a band passfilter
(BP in Fig.~\ref{fig:setup}) and a etalon is used. The etalon is based  on a
Fused Silica (Suprasil311) substrate with reflective coatings of $97\,\%$ on
both sides for a corresponding reflectivity finesse of 103. The plano-parallel
substrate has a thickness of 0.5\,mm, resulting in a reciprocal free spectral
range of about 205\,GHz, to suppress the laser diode chip modes of about
50\,GHz separation.
% as given by the GaN refractive index of 2.4 and a cavity length of 1.35\,mm.

The passband of the etalon has a full width at half maximum of
2\,GHz~\cite{pk:14}. While the subthreshold spectral linewidth of 43\,MHz is
already narrower than the etalon passband, the subthreshold spectrum shows
multiple laser cavity modes, which we reduce to one with the etalon, which is
temperature-tuned with a slope of 4\,GHz/K.
% due to temperature depencency of length and refractive index.
The bandpass interference filter BP has a 2\,nm wide
passband centred at $\lambda=518$\,nm to suppress source light beyond the free spectral range of the etalon, as well as to block ambient light from illuminating the probe detector.\\


\noindent
\textbf{Spatial light conditioning}
The spectrally filtered thermal light beam passes through a 50:50 beamsplitter
and a telescope formed by a lens pair (f=50\,mm, f=300\,mm)  around a spatial filter,
and is expanded to a diameter of about 50\,mm. The probe beam returns from the
target reflector through the same telescope and beam splitter onto the probe
photodetector. The spatial filter cleans up the returning probe beam and
reduces ambient light contribution from reaching the detectors, and the use of
a second beam splitter ensures that no breakdown flash light from the target
detector can reach the reference photodetector.\\

\noindent
\textbf{Signal-to-noise ratio}
In the regime whereby the photodetectors are fast enough to resolve the temporal coherence $\tau_{c}$ of the photon bunching signal, the measurable $g^{(2)}(\tau)$ signal-to-noise ratio (SNR) can be described by:
\begin{equation}
\label{eqn:snr}
	\mathrm{SNR} = r \cdot V^{2} \sqrt{\tau_{c} \cdot \Delta T} \,,
\end{equation} %Eqn.~\ref{eqn:snr}
with the rate $r$ in photoevents per second, the interferometric visibility $V=\sqrt{g^{(2)}(0)-1}$, the coherence time $\tau_{c}$ in seconds, and the integration time $\Delta T$ in seconds.

Considering the output power of 12.5\,$\mu$W of the subthreshold laser diode at 32.9\,mA, an attenuation loss of -65\,dB will result in the photodetectors still being saturated at about $r=10^{7}$ photoevents per second. 

Using a photon bunching peak value $V^{2}=0.6$ and a coherence timescale $\tau_{c}=23\,$ns, which corresponds to the measured values in Fig.~\ref{fig:g2}, an upper-bound signal-to-noise ratio of around 30 may be achieved within just 1\,millisecond of integration time $\Delta T$.

This high tolerance to attenuation upwards of -65\,dB provides the thermal light source an advantage over SPDC light sources in practical sensing use-cases whereby significant losses can be expected.\\

\newpage
\noindent
\textbf{Extended data}
\begin{figure}[h!]
\centering\includegraphics[width=9cm]{multiplot_powerfwhm.eps}
\caption{\label{fig:spectra} The output power and spectral width $\Delta \lambda$ of a laser diode are measured against its injection current to determine the lasing threshold.}
\end{figure} %Fig.~\ref{fig:spectra}. 

\clearpage

\end{document}
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