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\begin{document}
% \title{Direct test of Siegert relation on bunched light}
\title{Direct test to identify a pseudothermal light source}
\author{Xi~Jie~Yeo}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Ming~Ze~Qing}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Justin~Yu~Xiang~Peh}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Darren~Ming~Zhi~Koh}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

% \author{Chang~Hoong~Chow}
% \affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

% \author{Jaesuk~Hwang}
% \affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Peng~Kian~Tan}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Christian~Kurtsiefer}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}
\email[]{christian.kurtsiefer@gmail.com}


\date{\today}
\begin{abstract}
The first-order field correlation $g^{(1)}(\tau)$ and
second-order photon correlations $g^{(2)}(\tau)$ of thermal light
are related via the equation
$g^{(2)}(\tau) = 1 + \lvert g^{(1)}(\tau) \rvert ^{2}$,
commonly referred to as the Siegert relation.
However, the Siegert relation may not hold
for a pseudothermal light source.
% A thermal light source exhibiting 
We present a technique to identify a pseudothermal light source,
by measuring timing
correlations between photoevents 
detected at the output ports of an asymmetric Mach-Zehnder interferometer.
From these correlations,
we directly extract the difference between 
the second-order intensity correlation %$g^{(2)}(\tau)$
and interferometric visibility-square
of the light source. % $g^{(2)}(\tau) - \lvert g^{(1)}(\tau) \rvert ^{2}$.% $\lvert g^{(1 of a light source.
For thermal light,
this difference is equal to one,
according to the Siegert relation.
In contrast, 
we identify a pseudothermal light source when this difference is not equal to one,
although the light source exhibits photon bunching.
% This difference is extracted from % using interferometric photon correlations;
% $g^{(2)}(\tau) - \lvert g^{(1)}(\tau) \rvert ^{2} \neq 1$.
We perform this difference measurment on two light sources exhibiting photon bunching:
laser light scattering off a rotating ground glass, and light from a mercury vapor lamp.
Our measurements
show that laser light scattering off a rotating ground glass emits pseudothermal light %violates the Siegert relation, and hence cannot be classified as thermal light.
and suggest that the mercury vapor lamp emits thermal light.
\end{abstract}
% \textbf{Pseudothermal light
% exhibits photon bunching,
% like thermal light,
% but it
% does not originate from 
% randomly phased emission.
% % Pseudothermal light
% % commonly refers to a light source that
% As photon bunching 
% is common in both pseudothermal and thermal light,
% the analysis of photon bunching
% is insufficient
% to differentiate pseudothermal light and thermal light.
% In this work,
% we present a technique to identify a pseudothermal light source
% by a single-shot measurement that 
% subtracts
% interferometric visibility-square
% from photon correlations. %$g^{(2)}(\tau)$
% % This technique was originally used to differentiate 
% % incoherent emission
% % and coherent emission with amplitude fluctuations 
% % from a lasing medium. 
% % A difference not equal to one
% % indicates a pseudothermal light source.
% % versus those that do not.
% % The Siegert relation 
% % 
% % with their interferometric visibility and second-order degree of coherence
% % simply related via the Siegert relation,
% % We extract this difference in a single measurement, 
% % by 
% % using the light source under test.
% We use this technique 
% to successfully distinguish a commonly used 
% pseudothermal light source,
% laser light scattering off a rotating ground glass, 
% from a thermal light source,
% a mercury vapor lamp.
% Our results show that 
% these two sources have different modulation mechanisms
% in their electric fields despite 
% both exhibiting photon bunching. }
% Our measurements
% suggest that the mercury vapor lamp emits thermal light
% and
% show that laser light scattering off a rotating ground glass emits pseudothermal light. %violates the Siegert relation, and hence cannot be classified as thermal light.
% % and 
% insert suggested PACS numbers in braces on next line
% \pacs{}
% \end{abstract}

\maketitle

\section{Second-order photon correlations: an inconclusive test for thermal light} 
Second-order photon correlations $g^{(2)}(\tau)$,
a modern approach to intensity interferometry by Hanbury-Brown and Twiss~\cite{HBT1954,HBT1956},
is a common technique to distinguish between light sources
of different photon statistics~\cite{Glauber1963,MandelWolf1995,Fox2006}.
A light source
% The different photon statistics include
% sub-Poissonian, Poissonian and super-Poissonian photon statistics,
with super-Poissonian photon statistics 
exhibits photon bunching,
i.e. $g^{(2)}(0)>1$~\cite{Glauber1963}.
Examples of these light sources include
blackbody radiation~\cite{PK2016,Deng2019}
discharge lamps~\cite{HBT1958,Morgan1966,Scarl1968}
lasers undergoing amplitude modulations~\cite{lebreton2013exp},%superbunching?
and light scattered off a collection of scatterers
such as rotating ground glass diffusers~\cite{MS1964,Arecchi1965,Estes1971},
or particles undergoing Brownian motion suspended in a medium~\cite{Arecchi1967,Pecora1985,Dravins2015,PK2017}.

% \textbf{(XJ: In the previous sentence, listing Poissonian 4 times seems a bit weird,
% and doesn't seem to add much value to the discussion.
% Shall we remove this sentence?)}
% A light source that exhibits photon bunching is commonly referred to as bunched light.

% so why do i want to know if the light is thermal or not
\textbf{
% Although bunched light can be generated through different mechanisms,
Amongst these light sources,
a subset classified as thermal light 
is of fundamental interest.
Thermal light originates from 
spontaneous emission by an ensemble of light emitters in thermal equilibrium. %undergoing 
These emitters would
radiate stationary light at different frequencies with no fixed phase relationship~\cite{Davis1996}.
}

Apart from exhibiting photon bunching, 
thermal light also satisfies 
the Siegert relation $g^{(2)}(\tau) = 1 + \lvert g^{(1)}(\tau) \rvert ^{2}$,
where $\lvert g^{(1)} \rvert$ is the interferometric visibility~\cite{Siegert1943,Loudon2000}.
In contrast,
there exists light sources that exhibits photon bunching,
yet violates the Siegert relation are also referred to as pseudothermal light
such as lasers undergoing amplitude modulations~\cite{lebreton2013exp,Lebreton2013Theory}.
The test for whether Siegert relation is obeyed 
is therefore a more stringent criterion for qualifying a light source as emitting thermal light,
as compared to only depending on the fact that the light sources exhibit photon bunching.

The test for Siegert relation typically requires two separate measurements to obtain $g^{(2)}(\tau)$ and $\lvert g^{(1)} \rvert$.
To obtain $\lvert g^{(1)} \rvert$, 
a scanning Michelson or Mach-Zehnder type interferometer may be used~\cite{Fox2006}.
The interferometer scans through a path difference on the order
of the coherence length of the light,
which is the coherence time multiplied by the speed of light in the interferometer medium.
However, the construction of a scanning interferometer may be tedious when the coherence length of the light source  
exceeds the size of laboratory. 
In the context of testing for Siegert relation , 
this motivates for methods that allow testing for Siegert relation  that eliminates the need for scanning interferometer.

We present a method to identify a pseudothermal light source
via testing whether the Siegert relation holds 
% without a need for a scanning interferometer,
using interferometric photon correlations,
% A violation of Siegert relation for ``bunched'' light 
a correlation of photoevents detected at the output ports 
of an asymmetric Mach-Zehnder interferometer.
Furthermore, this method directly tests the Siegert relation in a single
measurement, 
rather than obtaining $g^{(2)}(\tau)$ and $\lvert g^{(1)} \rvert$ separately.  
This method 
was originally used to differentiate chaotic light 
and a laser undergoing amplitude fluctuations~\cite{lebreton2013exp,Lebreton2013Theory}.
Here, we use it to test for a violation of Siegert relation on two commonly used light sources exhibiting photon bunching: %~\cite{HBT1958,Arecchi1965}: 
a mercury vapor lamp filtered with a 546\,nm optical bandpass,
and scattered light from a rotating ground glass illuminated by a 780\,nm laser light focused on the ground glass.
We observe a violation of Siegert relation
for light scattered off the rotating ground glass.
Conversely, light emission from the mercury vapor lamp obeys Siegert relation,
suggesting thermal light. 
This method is also an improvement over a previous technique that tests the Siegert relation directly~\cite{Ferreira2020}.
as it removes the need to interfere the light source with an external local oscillator,
but instead interferes the light source with a delayed copy of itself.
% \section{Thermal vs. pseudothermal light} 
% \textbf{A thermal light source
% can be modelled as
% an ensemble of particles emitting randomly phased light~\cite{Planck1900,Einstein1905,Glauber1963_PhotonCorrelations,Davis1996,Loudon2000}. 
% Using this model,
% thermal light is predicted to 
% exhibit photon bunching~\cite{Glauber1963_PhotonCorrelations,Glauber1963_QuantumTheory,MandelWolf1995,Fox2006};
% a more frequent than random excess of photon coincidences.
% Photon bunching has been observed in thermal light sources,
% such as 
% blackbody radiation~\cite{PK2016,Deng2019}
% and
% discharge lamps~\cite{Rebka1957,HBT1958,Morgan1966,Scarl1968}.
% Photon bunching has also been observed in light sources
% termed pseudothermal or quasithermal,
% with examples 
% such as light scattered off rotating ground glass diffusers~\cite{MS1964,Arecchi1965,Estes1971,Gonsiorowski1983,Scarcelli2004,Valencia2005,Zhou2017,Kuusela2017},
% particles undergoing Brownian motion suspended in a medium~\cite{Arecchi1967,Pecora1985,Dravins2015,PK2017},
% or a random phase screen~\cite{Ratcliffe1956},
% or modulated by noise~\cite{Kuusela2018}.}
%
% \textbf{
% To our knowledge,
% the distinction between thermal and pseudothermal light
% has never been made clear.
% However,
% the description of pseudothermal light in existing literature~\cite{MS1964,Arecchi1965,Estes1971,Gonsiorowski1983,Scarcelli2004,Valencia2005,Zhou2017,Kuusela2017,Arecchi1967,Pecora1985,Dravins2015,PK2017,Ratcliffe1956,Kuusela2018}
% suggests that the origin 
% of photon bunching
% is not from inherently random phased emission,
% but rather by external modulation of a light source.
% Since photon bunching occurs in both thermal and pseudothermal light, 
% it cannot be used to differentiate these two light sources.
% In order to differentiate the two,
% methods are needed to detect the external modulation of the light source
% or determine if the light is randomly phased.}
%
% \textbf{
% Apart from exhibiting photon bunching, 
% the randomly phased emission of thermal light
% also results in a simple relationship~\cite{Siegert1943,Glauber1963_PhotonCorrelations,Loudon2000}
% \begin{equation}
%   g^{(2)}(\tau) = 1 + \lvert g^{(1)}(\tau) \rvert ^{2},
%   \label{eqn:Siegert}
% \end{equation}
% where $g^{(2)}(\tau)$ is its second-order photon correlation 
% and $\lvert g^{(1)}(\tau) \rvert$ is its interferometric visibility. 
% % to have a simple relation
% This relationship is 
% commonly known as the Siegert relation~\cite{Pike1974,Jakeman1980,Lemieux1999,Ferreira2020,Las2022}.}
%
% \textbf{
% To experimentally test if Siegert relation holds,
% measurements to obtain $g^{(2)}(\tau)$ and $\lvert g^{(1)}(\tau) \rvert$ are needed.
% This usually requires two separate measurements:
% a Hanbury-Brown and Twiss type interferometer to obtain $g^{(2)}(\tau)$
% and
% a scanning Michelson or Mach-Zehnder type interferometer
% to obtain $\lvert g^{(1)}(\tau) \rvert$~\cite{Loudon2000,Fox2006}.
% Another method to test the Siegert relation 
% is from the intensity correlations between the detectors 
% in a heterodyne measurement,
% when the source exhibiting photon bucnhing is 
% heterodyned with a local oscillator~\cite{Ferreira2020}.}
%
% \textbf{
% We present another technique to identify 
% a pseudothermal light source 
% in a single measurement,
% by testing the Siegert relation 
% without the need of a local oscillator.
% To do this,
% we directly extract 
% the difference $g^{(2)}(\tau) - \lvert g^{(1)}(\tau) \rvert^{2}$,
% from
% the correlation of photoevents detected at the output ports 
% of an asymmetric Mach-Zehnder interferometer,
% also known as interferometric photon correlations. 
% A light source that shows a
% non-unity difference
% $g^{(2)}(\tau) - \lvert g^{(1)}(\tau) \rvert^{2} \neq1$,
% indicates a deviation from the Siegert relation,
% which means that the it is pseudothermal.
% This method 
% was used previously to 
% measure the spectral diffusion of molecules,
% differentiate thermal light 
% from a laser undergoing amplitude fluctuations~\cite{lebreton2013exp,Lebreton2013Theory},
% as well as to estimate the fraction of coherent light emitted from a laser diode~\cite{XJ2023}.
% Here, we use it to distinguish
% a commonly used thermal light source,
% a mercury vapor lamp,
% from a commonly used pseudothermal light source,
%   % filtered with a 546\,nm optical bandpass,
% laser light scattered from a rotating ground glass.}
%  % illuminated by a 780\,nm laser light focused on the ground glass.
\section{Photon bunching in thermal and pseudothermal light}

\begin{figure}
  \begin{center}
  %  \includegraphics[width=0.8\columnwidth]{./figures/g2x_setup/g2x_setup_vs2.eps}
    \includegraphics[width=\columnwidth]{g2_setup.eps}
  \end{center}
 \caption{
  Experimental set up for
  a) the thermal light source with mercury vapor lamp,
  b) the pseudothermal light source with laser light scattered off the rotating ground glass,
  c) the Hanbury-Brown Twiss type interferometer to observe photon bunching in a second-order photon correlations $g^{(2)}(\tau)$.
    (BPF: bandpass filter, LP: linear polariser, RGG: rotating ground glass BS: fibre-based beamsplitter, APD: Single photon avalanche photodetectors)
    } \label{fig:g2_setup}
\end{figure}


% In our experiment, we prepare two light sources that exhibit photon bunching:
% light from a mercury vapor lamp, 
% and light scattered off a rotating ground glass diffuser.
% % For a Doppler broadened light source,

 %, shown in Fig.\,\ref{fig:g2X_setup}, top.
\textbf{In our experiment, a mercury vapor lamp is used as a thermal light source, prepared as shown in Fig.\,\ref{fig:g2_setup}a.}
Light from the lamp is filtered with a $546\pm3$\,nm optical bandpass and a linear polariser.
The filtered light is collected into a multimode fibre,
before projecting into a single spatial mode with a single mode fibre.
\textbf{The emission profile from the filtered mercury vapor lamp is expected to follow 
a Lorentzian lineshape~\cite{Griem1974}.
The corresponding second-order correlation $g^{(2)}(\tau)$ of a Lorentzian lineshape light source
is a double exponential decay function due to the Wiener-Khintchine theorem~\cite{Wiener1930,Khintchine1934},
\begin{equation} 
    % \begin{aligned}
        g^{(2)}(\tau) = 1 + \beta_{\text{Hg}} \cdot \exp\left[-\left\lvert\frac{2\tau}{\tau_{\text{Hg}}}\right\rvert\right],
    % \end{aligned}
  \label{eqn:g2_eqn_exp}
\end{equation}
% Both light sources exhibit super-Poissonian photon statistics $g^{(2)}(0)>1$. 
where $\beta_{\text{Hg}}$ is the amplitude of the bunching peak,
and $\tau_{\text{Hg}}$ is the characteristic timescale of this bunching feature.}

\textbf{For the pseudothermal light source,}
we prepare laser light scattered off the rotating ground glass diffuser as shown in Fig.\,\ref{fig:g2_setup}b.
Light from a 780\,nm distributed feedback laser is focused on a reflective ground glass difffuser of grit 1500.
We estimate the diameter of the beam on the ground glass $W$ to be about 4\,$\mu$m,
and at a radial distance $R$ of about 10\,mm from the rotation axis of the motor.
The motor rotates the ground glass with a period $T_{0}$ of about 4\,ms.
A single mode fibre for 780\,nm was placed 19\,cm away
from the illuminated spot on the ground glass,
to sample the light scattered off the rotating ground glass.

\textbf{
Theoretical models of
laser light scattered from a rotating ground glass
predict a $g^{(2)}(\tau)$ with a Gaussian profile at a point of detection~\cite{Estes1971,Jakeman1975,Pusey1976,Kuusela2017}
}
% Both light sources exhibit super-Poissonian photon statistics $g^{(2)}(0)>1$. 
\begin{equation} 
    % \begin{aligned}
        g^{(2)}(\tau) = 1 + \beta_{\text{RGG}} \cdot \exp\left[-\left(\frac{\tau}{\tau_{\text{RGG}}}\right)^{2}\right],
    % \end{aligned}
  \label{eqn:g2_eqn_gaussian}
\end{equation}
where $\beta_{\text{RGG}}$ is the amplitude of the bunching peak,
and $\tau_{\text{RGG}}$ is the characteristic timescale of this bunching feature.
In our experiment,
as the ground glass is placed at the focus of the lens,
and the scattered light is collected at a distance significantly larger than the spot size,
the value of $\tau_{\text{RGG}}$ can be approximated using~\cite{Estes1971,Jakeman1975,Pusey1976,Kuusela2017}
\begin{equation} 
    % \begin{aligned}
       \tau_{\text{RGG}} \approx \frac{W T_{0}}{2 \pi R}.
    % \end{aligned}
  \label{eqn:coherence_time}
\end{equation}
From a substitution of our experimental settings to Eq.~\ref{eqn:coherence_time},
we predict $\tau_{\text{RGG}}\approx 200$\,ns.

\textbf{To observe photon bunching,
we measure the second-order photon correlation $g^{(2)}(\tau)$,
using a Hanbury-Brown Twiss type interferometer, shown Fig.\,\ref{fig:g2_setup}c.}
Light from the source under test is sent to a beamsplitter.
Photoevents at each output port of the beamsplitter were detected using actively quenched
silicon single photon avalanche diodes (APD).
The photoevents were timestamped over an integration time $T$.
The time differences $\tau$ between timestamped photoevents across the two APDs
were measured, and the number of coincidences $N$ for each $\tau$ is sorted into a histogram.
As the counting of $N$ returns a mean value from a Poisson distribution,
error bars of magnitude $\sqrt{N}$ were assigned for each bin of the histogram.
\textbf{The histogram is then normalised by the product of single events rates at each detector,
to obtain $g^{(2)}(\tau)$, 
shown in Fig.\,\ref{fig:g2}. 
The $g^{(2)}(\tau)$ histograms for each light source
were also fitted to their respective theoretical models in Eq.~\ref{eqn:g2_eqn_exp}-\ref{eqn:g2_eqn_gaussian}. 
 }

\begin{figure}
  \begin{center}
  %  \includegraphics[width=0.8\columnwidth]{./figures/g2x_setup/g2x_setup_vs2.eps}
    \includegraphics[width=\columnwidth]{g2_compare_truncated.eps}
  \end{center}
  \caption{Second-order photon correlations  $g^{(2)}(\tau)$ for (a) mercury vapor lamp, (b) laser light scattered off a rotating ground glass. 
    The solid lines are best-fit curves to their respective $g^{(2)}(\tau)$ models, from  Eqn.\,\ref{eqn:g2_eqn_gaussian} and \,\ref{eqn:g2_eqn_exp}. 
    For the mercury vapor lamp,  
    $\beta_{\text{Hg}} = 0.17 \pm 0.01$, 
    $\tau_{\text{Hg}}=0.41 \pm 0.03$\,ns, 
    and a reduced $\chi^{2}=1.98$.
    For laser light scattered off a rotating ground glass, 
    $\beta_{\text{RGG}}=0.869\pm0.009$,
    $\tau_{\text{RGG}}=164 \pm 1$\,ns, 
    and a reduced $\chi^{2}=2.00$.
    } \label{fig:g2}
\end{figure}

\section{Interferometric photon correlations}\label{sec:2}

\begin{figure}
  \begin{center}
  %  \includegraphics[width=0.8\columnwidth]{./figures/g2x_setup/g2x_setup_vs2.eps}
    \includegraphics[width=\columnwidth]{g2X_setup.eps}
  \end{center}
  \caption{Experimental setup for measuring interferometric photon correlations $g^{(2X)}$. 
  The propagation delay in the interferometer introduced about 2.22\,$\mu$s , 
  for testing laser light scattered off the rotating ground glass, 
  and about 10\,ns propagation delay 
  for testing light from the mecury vapor lamp.
    (BS: Beamsplitter, APD: Single photon avalanche photodetectors)
    } \label{fig:g2X_setup}
\end{figure}


The setup to measure interferometric photon correlations $g^{(2X)}(\tau)$ is shown in Fig.\,\ref{fig:g2X_setup}.
Light from the source under test is sent through an asymmetric Mach-Zehnder interferometer.
The propagation delay $\Delta$ between the two arms of the interferometer was introduced using single mode optical fibres.
In order to clearly resolve features in $g^{(2X)}(\tau)$, 
the length of optical fibres used introduced
a propagation delay at least 4 times longer than $\tau_{\text{c}}$ for the respective light sources.
Photoevents were detected at the output ports of the interferometer,
also using actively quenched silicon single photon avalanche photodiodes.
To extract $g^{(2X)}(\tau)$,
the detected photoevents were timestamped and processed similar to
how second-order photon correlation were extracted in Sec.\,\ref{sec:2}.

The respective light fields 
at the output ports of interferometer $A,B$ are 
\begin{equation}
    E_{A,B}(t) = \frac{E(t) \pm E(t+\Delta)}{\sqrt{2}}\,,
    \label{eqn:fields}
\end{equation}
where $E(t)$ is the input light field into the interferometer,
and the sign difference is a result of a relative $\pi$ phase 
acquired by one of the fields at the beamsplitter~\cite{Ou1989}.

Photoevents are detected at the output ports of the interferometer and timestamped.
The interferometric photon correlation $g^{(2X)}(\tau)$
is computed from the timestamped events
\begin{equation} 
    \begin{aligned}
        g^{(2X)}(\tau) = &\frac{\langle E^{*}_{A}(t+\tau)E^{*}_{B}(t)E_{B}(t)E_{A}(t+\tau)\rangle}{ \langle E^{*}_{A}(t)E_{A}(t)\rangle \langle E^{*}_{B}(t)E_{B}(t)\rangle}\\
        = &\frac{\langle \hat{n}_{A}(t+\tau) \hat{n}_{B}(t) \rangle}{\langle \hat{n}_{A}(t)\rangle\langle \hat{n}_{B}(t) \rangle},
    \end{aligned}
  \label{eqn:g2X}
\end{equation}
where $\langle\dots\rangle$ takes the ensemble average over the variable $t$,
\textbf{ $\hat{n}_{A,B}(t)$ is the number of photons detected by the respective photodetectors at time $t$, }
and $\tau$ is the detection time difference between timestamped events.

Upon expansion of Eqn.\,\ref{eqn:g2X} using Eqn.\,\ref{eqn:fields},
it can be shown that
the non-zero terms of $g^{(2X)}(\tau)$ can be written as  
\begin{equation} 
    \begin{aligned}
        &g^{(2X)}(\tau) = \\
        =&\frac{1}{4}\,g^{(2)}(\tau+\Delta)+ \frac{1}{4}\,g^{(2)}(\tau-\Delta) \\
        &+\frac{1}{2}[g^{(2)}(\tau) -  |g^{(1)}(\tau)|^{2}],
    \end{aligned}
  \label{eqn:zerotime}
\end{equation}
where $|g^{(1)}|$ is the interferometeric visibility,
and $g^{(2)}$ is the standard second-order photon correlation~\cite{lebreton2013exp,XJ2023}.

For a light source exhiting photon ``bunching'' $g^{(2)}(0)>1$,
a ``bunching'' feature at a same timescale $\tau_\text{c}$
but at $1/4$ amplitude
would appear at $g^{(2X)}(\tau=\pm\Delta)$.
In the interval of $\tau \in [-\tau_{c},\tau_{c}]$,
a light source obeying the Siegert relation 
would result in the $[g^{(2)}(\tau) -  |g^{(1)}(\tau)|^{2}]=1$,
resulting in $g^{(2X)}(\tau) = 1$.
In contrast, pseudothermal light sources that deviates from the Siegert relation,
would result in $g^{(2X)}(\tau) \neq 1$,
for $\tau \in [-\tau_{c},\tau_{c}]$.

\section{Identifying pseudothermal light}
Using the setup shown in Fig.\,\ref{fig:g2X_setup},
we extract the interferometric photon correlations $g^{(2X)}(\tau)$ of light from the mercury vapor lamp,
and laser light scattered off a rotating ground glass, shown in Fig.\,\ref{fig:g2X}
Asssuming that Siegert relation holds,
we predict $g^{(2X)}(\tau)$ of the two light sources under test,
using the fitted parameters of $\beta$ and $\tau_{\text{c}}$ extracted earlier.
The predicted $g^{(2X)}(\tau)$ are compared 
with $g^{(2X)}(\tau)$ extracted from measurements.

\begin{figure}[h]
  \begin{center}
  %  \includegraphics[width=0.8\columnwidth]{./figures/g2x_setup/g2x_setup_vs2.eps}
    \includegraphics[width=\columnwidth]{g2X_compare_truncated.eps}
  \end{center}
  \caption{Interferometric photon correlations for (a) mercury vapor lamp, (b) laser light scattered off a rotating ground glass. 
    The solid lines show the predicted $g^{(2X)}(\tau)$ extrapolated from the measured $g^{(2)}(\tau)$, assuming that the light source emits thermal light.
    } \label{fig:g2X}
\end{figure}


For light emitted by the mercury vapor lamp, 
$g^{(2X)}(\tau\approx 0)=1$ shows that it obeys the Siegert relation,
as expected of thermal light.
The asymmetry between the height of the two peaks at $g^{(2X)}(\tau=\pm\Delta)$
is attributed to an unequal splitting ratio of the beamsplitter recombining the two paths in the Mach-Zehnder interferometer.

For laser light scattered off the rotating ground glass,
$g^{(2X)}(\tau\approx 0)\neq1$ indicates a deviation from the Siegert relation and
therefore a pseudothermal light source.
Instead, $g^{(2X)}(\tau\approx 0)$ consists of a dip and 
peak each with a distinct characteristic timescale.
The charactersistic timescale of the dip 
is associated with the coherence time of the laser emitting coherent light,
while the peak feature suggests amplitude modulation of the laser~\cite{Lebreton2013Theory,lebreton2013exp}.


\section{Conclusion}
In summary, we performed experiments to
extract interferometric photon correlations $g^{(2X)}(\tau)$ of 
a commonly used thermal light source,
a mercury vapor lamp, 
and a pseudothermal light source,
laser light scattered off a rotating ground glass. 
Using this technique,
we found from $g^{(2X)}(\tau)$ 
that laser light scattered off a rotating ground glass 
deviates from Siegert relation despite exhibiting photon bunching
and hence a pseudothermal light source.
On the contrary,
light from the mercury vapor lamp obeys Siegert relation
suggesting thermal light.

\textbf{
Apart from positively identifying pseudothermal light,
a null identification of pseudothermal light using the technique presented here
increases the confidence that a source emits thermal light
than the observation of photon bunching alone. 
The is crucial in applications with
results relying on the assumption of a thermal light source.
For example,
the phase randmoness in thermal light is used in range sensing~\cite{Zhu2012,PK2023}
and optical coherence tomography~\cite{Fercher2000},
or inferring spectral lineshape~\cite{PK2017} or spectral broadening mechanisms~\cite{Pecora1985} from the photon correlations of thermal light.}

\textbf{
This technique may also be used to
identify laser signatures falsely identified as a thermal light source exhibiting photon bunching.
Applicable scenarios would be in identifying
laser signatures from extraterrestial signals~\cite{Howard2007}, 
technosignatures~\cite{Acharyya2023} 
and astrophysical lasers~\cite{Letokhov2002},
with its amplitude modulated by scattering media,
such as the atmosphere and interstellar dust,
along its line of sight.
}

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