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%\slugcomment{The Astrophysical Journal, 680:L87-L90, 2014 April 11}

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%\shorttitle{Measuring Temporal Photon Bunching in Blackbodies}
%\shortauthors{P. K. Tan and C. Kurtsiefer et al.}


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\title{Measuring Temporal Photon Bunching in Blackbody Radiation}

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%\author{P. K. Tan\altaffilmark{1,2,*}}
%\author{G. H. Yeo\altaffilmark{2}}
%\author{H. S. Poh\altaffilmark{1}}
%\author{A. H. Chan\altaffilmark{2}}
%\author{C. Kurtsiefer\altaffilmark{1,2,+}}
%\altaffiltext{1}{Center for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
%\altaffiltext{2}{Department of Physics, National University of Singapore,
%  Singapore 117542}
%\altaffiltext{*}{email: pengkian@physics.org}
%\altaffiltext{+}{email: phyck@nus.edu.sg} 

\author{P. K. Tan\altaffilmark{1,2,*}, G. H. Yeo\altaffilmark{2}, H. S. Poh\altaffilmark{1}, A. H. Chan\altaffilmark{2}, C. Kurtsiefer\altaffilmark{1,2,+}}
\altaffiltext{1}{Center for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\altaffiltext{2}{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}
\altaffiltext{*}{email: pengkian@physics.org}
\altaffiltext{+}{email: phyck@nus.edu.sg} 

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\begin{abstract}
Light from thermal black body radiators such as stars exhibits photon bunching
behaviour at sufficiently short time-scales. However, with available detector
bandwidths, this bunching signal is difficult observe directly. We
present an experimental technique to increase the photon bunching signal in
blackbody radiation via spectral filtering of the light source. Our
measurements reveal strong temporal photon bunching from blackbody radiation,
including the Sun. This technique allows for an absolute measurement of the
photon bunching 
signature $g^{(2)}(0)$, and thereby a direct statement on the statistical
nature of a light source. Such filtering techniques may help revive the interest
in intensity interferometry as a tool in astronomy.
\end{abstract}

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% http://aas.org/authors/astronomical-subject-keywords-update-august-2013
\keywords{instrumentation: interferometers --- radiation mechanisms: thermal --- stars: fundamental parameters --- Sun: fundamental parameters --- techniques: interferometric}

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\section{Background}

Statistical intensity fluctuations from stationary thermal light sources
formed the basis of the landmark experiments by \citet{hbt:54, hbt:57}, 
where correlation measurements between a pair of
photodetection signals were used to assess both temporal
and spatial coherence properties of light fields. The explanation for these
fluctuations as photon bunching phenomenon, as compared to a Poissonian
distribution for uncorrelated detection processes, was a
major contribution for the development of the field of quantum optics \citep{glauber:63} 

\citet{hbt:56, hbt:74} developed their observation into a spatial intensity
interferometer which allowed them to infer stellar angular radii
from the spatial coherence properties of starlight over relatively large distances of 188\,m. Since a key quality of any telescope system is its angular resolution, given by its real or virtual aperture, they managed to
carry out these measurements with an unprecedented accuracy.
Yet, the comparatively simple optical intensity interferometry technique has
not found widespread adoption in the observation of celestial objects
due to a relatively small signal-to-noise ratio (SNR) in measuring the spatial
decay of the photon bunching signature \citep{davis:99, millour:08}.

Typically, the photon bunching signature is observed in a regime where the
ratio of the coherence time, $\tau_{c}$, over the photodetector resolution,
$\tau_{t}$, is significantly smaller than 1, even after an increase of the
coherence time via optical filtering.
In such a regime, the temporal coherence properties cannot be resolved
directly, and the SNR in observing photon bunching is independent of the filter
bandwidth \citep{hbt:74a, foellmi:09, rou:12, nunez:12, malvimat:13}.

However, the temporal coherence properties may contain interesting
information about the celestial light source, and help e.g. to assess the spectra of Wolf-Rayet stars \citep{castor:72, varshni:86, hucht:01} like Eta Carinae, with suspected laser emission lines estimated to be $\approx50$\,MHz or less \citep{dravins:08} which are too narrow to resolve by conventional spectroscopy. To understand the nature of the light of these very narrow spectral features, the suppression of a wide blackbody radiation background would be helpful.

Furthermore, in the regime of $\tau_{c}/\tau_{t}>1$, one is also able to
resolve the thermal character
explicitly, and to test for the thermal photon bunching signature of
$g^{(2)}(0)=2$. This permits the distinction between thermal and laser
emission mechanisms for the discrete spectral components of Wolf-Rayet stars.


\section{The problem}
Temporal photon bunching, i.e., the tendency for photons to cluster together
on a short time-scale, has been consistently observed from narrow-band
quasi-thermal light sources such as low pressure glow discharge lamps
\citep{martienssen:64, morgan:66}, and 
laser beams \citep{asakura:70} which are passed through a rotating ground
glass to randomize the optical phase in time.

Photon bunching in time for stationary fields is quantified by a second order
correlation function $g^{(2)}(\tau)$, which is the probability of observing a
second photodetection event at a time $\tau$ after a first one, normalized to the
probability of an uncorrelated detection. Optical modes in thermal states are
the textbook example where $g^{(2)}=2$ \citep{glauber:63, mandel:95,
  salehteich:07}. Blackbody radiation would 
therefore be the obvious choice for observing photon bunching, if it were
not for the short coherence time associated with its spectral properties.

In a scenario where spatial coherence effects are neglected, the second order
temporal correlation function for light fields can be written as
\begin{equation}
g^{(2)}(\tau)=1+\left|\gamma(\tau)\right|^2\,,
\end{equation}
where $\gamma(\tau)$ is the complex degree of coherence, or normalized
self-coherence of the stationary light field with $\gamma(0)=1$, which itself is proportional to
the Fourier transform of the spectral density $S(\nu)$ of the light field
\citep{mandel:95}. For well-defined spectral densities and coherence
functions, a meaningful coherence time $\tau_c$ can be defined, which is
reciprocal to a characteristic width $\Delta \nu$ of the spectral distribution
$S(\nu)$. 

For blackbody light sources in the optical regime, the coherence time
$\tau_{c}$ given by Planck's law is on the order of $10^{-14}$\,s, which is much
shorter than the best timing resolution $\tau_{t}$ of a few
$10^{-11}$\,s for existing photodetectors. 
As shown by \citet{scarl:68}, the observable excess from uncorrelated
pair detections, $g^{(2)}(\tau = 0)-1$, scales as $\tau_{c}/\tau_{t}$
and is thus only significant when the coherence time of the
source is of the order of the detection timing response.

Therefore, empirical observation of temporal photon bunching from blackbody
light sources has been extremely difficult, with only recent breakthroughs
using two-photon absorption in semiconductors \citep{boitier:09}, and in
ghost imaging research using a narrowband Faraday anomalous dispersion optical
filter \citep{karmakar:12, liu:14}.

In this paper, we present a spectral filtering technique that significantly
increases the observed temporal photon bunching signature from a blackbody light
source.

\section{Experiment}
As with previous attempts to observe photon bunching from blackbody radiation
\citep{karmakar:12, liu:14}, we employ narrow-band optical filtering of the
light source in order to increase 
the coherence time $\tau_c$, and use single photon detectors with low
timing uncertainties. The experimental setup is shown in
Fig.~\ref{fig:setup}.

\begin{figure}[h!]
\centering
\includegraphics[width=\hsize]{apj_hbt.eps}
\caption{Setup to determine the temporal correlation $g^{{(2)}}(\tau)$ for
  wide-band thermal light. The initial spectrum gets filtered to a narrow
  optical bandwidth such that the temporal decay of the second order
  correlation function can be observed with conventional single photon
  detectors in a Hanbury-Brown--Twiss experiment. We employ a combination of
  a grating and a temperature-tuned etalon as a spectral filter, and ensure
  spatial coherence in the setup by using single mode optical fibers (SMF).}
\label{fig:setup}
\end{figure}

Firstly, light from different sources under investigation is coupled into a
single mode optical fiber to enforce spatial coherence, because the
photon bunching signature $g^{(2)}(\tau=0)-1\propto1/M$, where $M$ is the
number of transverse modes \citep{glauber:63}. 

The light is then collimated with an aspheric lens and sent through the first
spectral filter, which is either an interferometric bandpass filter with a full width at half maximum (FWHM) of 3\,nm centered at 546.1\,nm to match the
$6p7s\,^3S_1 - 6p6s\,^3P^0_2$ transition in a Mercury (Hg) discharge lamp, or a grating
monochromator for broadband continuous spectrum. While the interference filter
is sufficient to select the spectral lines of the different mercury isotopes,
the grating monochromator had to be used for the blackbody sources.

\begin{figure}[h!]
\centering
\includegraphics[width=\hsize]{transmission.eps}
\caption{(Top) Transmission profile of grating monochromator. The solid line
  is a fit to a Gaussian profile to measurements for different grating angles
  (symbols), resulting in a full width at half maximum (FWHM) of
  0.122$\pm$0.002\,nm. (Bottom) Transmission of Hg light near 546\,nm through the
  temperature-tuned etalon with 2\,GHz bandwidth. The hyperfine structure for
  different isotopes is partially resolved, as identified in
  \citep{sansonetti:10}.}
\label{fig:transmission}
\end{figure}

The core of the monochromator is a blazed reflective diffraction grating
with 1200\,lines/mm mounted on a rotation stage. Light from the
optical fiber with a numerical aperture NA = 0.13 is collimated with an achromatic doublet ($f=30\,$mm),
illuminating about $10^4$ lines of the grating. The first diffraction order is
imaged with another achromat of ($f=30\,$mm) into another single mode optical
fiber. This arrangement acts as a tunable narrow-band filter in the
visible regime, allowing to adjust for peak emissions
of blackbody light sources. A test of the transmission of a narrowband laser
source ($\lambda$ = 532\,nm) is shown in Fig.~\ref{fig:transmission} (top
part). The fit to a Gaussian transmission profile leads to an optical
bandwidth of 0.122$\pm$0.002\,nm (FWHM). We typically observe a transmission
of narrowband light from single mode fiber to single mode fiber of
$\approx$15\%.

In a second step, the light passed through a solid etalon (thickness
$d$=0.5\,mm) with flat surfaces. The etalon has a free spectral range of
$FSR=c/(2nd)=$ 205\,GHz, with a refractive index $n=1.46$ for fused
silica (Suprasil311), and the speed of light $c$ in vacuum. At
$\lambda$ = 546\,nm, the free spectral range corresponds to
$\Delta\lambda=\lambda^2/c\cdot FSR\approx$ 0.2\,nm, so that only one
transmission order should fall within the monochromator transmission width.
Both surfaces of the etalon have dielectric coatings with a
reflectivity of $R\approx$ 97\% over a wavelength range from 390\,nm to
810\,nm, resulting in a transmission bandwidth (FWHM) \citep{fox:06} of
\begin{equation}
  \Delta\nu \approx FSR\cdot{1-R\over\pi\sqrt{R}}\approx2\,\mathrm{GHz}\,.
\end{equation}

The central etalon transmission frequency is tuned by temperature, with
a scaling of -4.1\,GHz/K due to thermal expansion and temperature
dependence of the refractive index. A temperature stability of $\pm$5\,mK over the
measurement duration of several hours ensures that the etalon stays on
a specific wavelength. Temperature tuning allows the etalon to be always used
under normal incidence, which avoids frequency walk-off
\citep{green:80} and spatial distortion \citep{park:05}. To demonstrate the
resolving power of this device, we tuned the peak transmission across the
various spectral lines from a Hg gas discharge lamp at 546.1\,nm, revealing a
partially resolved hyperfine structure for different isotopes
(see Fig.~\ref{fig:transmission}, bottom part).

A Glan-Taylor polarizer (PBS) following the etalon selects
a well-defined linear polarization mode, with the same
rationale of minimizing the number of modes ending up in the temporal
correlation setup. A half-wave plate ($\lambda/2$) transforms the
polarization emerging from the grating/fiber combination such that there is
maximal transmission through the PBS. 

A second Glan-Taylor polarizer, rotated by $\approx45^\circ$  with respect to
the polarization direction of the first, distributes the light onto two
silicon avalanche photodetectors (APD) that provide detection timing signals
to a correlator. The relative orientation of the two polarizers allows for the
balancing of the photodetection rates to optimize the integration time, and
also helps to suppress  the optical crosstalk 
between the two APDs due to hot carrier recombination in the devices upon
detection which would otherwise introduce false coincidence events
\citep{kurtsiefer:01a}.

The temporal correlation was carried out with a digital sampling
oscilloscope (4\,GHz analog bandwidth, 40\,Gsamples/sec, LeCroy) that recorded
the APD signals if two detection events fall within a coarse coincidence time
window of 20\,ns. The oscilloscope then evaluated the timing difference $\tau$
between the APD events with an uncertainty below 10\,ps, and generated a
histogram of all observed $\tau$. The temporal correlation function
$g^{(2)}(\tau)$ is just a normalized version of this histogram.

\begin{figure}[h!]
\centerline{\includegraphics[width=\hsize]{apd_combined.eps}}
\caption{Comparison of timing jitter between two actively quenched thin avalanche
  photodiodes (PDF), and two passively quenched deep avalanche photodiodes
  (C90302S) by photon pair detection from light generated by spontaneous
  parametric down conversion (SPDC) with a coherence time below 10\,ps.}
\label{fig:jitter}
\end{figure}

To assess the overall timing uncertainty with the detector/correlator combination,
we exposed two different avalanche photodetector sets to photon pairs
generated by spontaneous parametric down conversion (SPDC) with a coherence time below
10\,ps. The resulting temporal correlations shown in
Fig.~\ref{fig:jitter} reflect the overall timing resolution of the system.
For APDs with a deep conversion zone (C30902S, Perkin-Elmer, passive
quenching), the coincidence distribution appears roughly Gaussian, with 1.2\,ns
FWHM. Since the photon bunching signature $g^{(2)}(\tau)$ of the light field
under investigation needs to be convolved with this detector response, we
would observe a significant reduction of the photon bunching peak with
these detectors. A second set of APDs, based on thin conversion zones (type
PDF, Micro Photon Devices, active quenching), leads to a much better localized
coincidence distribution with 40\,ps FWHM, although a significant fraction of
the distribution shows a tail with timing differences up to 500\,ps.

\section{Results}

\begin{figure}[!]
\centering
\includegraphics[width=\hsize]{spectrum_hgarsun.eps}
\caption{Spectrum of a Hg glow discharge lamp, an Ar arc lamp as a simulator of a stellar light source, and the Sun.}
\label{fig:spectrum}
\end{figure}

Measurements for $g^{{(2)}}(\tau)$ using the high bandwidth (type PDF) photon
detectors were performed on three light sources: a 
low pressure Mercury (Hg) discharge lamp as a quasi-thermal light source, a
radio frequency-driven Argon (Ar) arc discharge lamp to produce a plasma with a
blackbody temperature $\approx$ 6000\,K as the thermal light source, and the
Sun as a celestial thermal light source. Their respective spectral distributions are shown in Fig.~\ref{fig:spectrum}. The optical filters were fixed to
maximize transmission for the spectral lines around 546.1\,nm in the Hg
spectrum. The resulting timing difference histograms are shown in
Fig.~\ref{fig:g2}.

\begin{figure}[h!]
\centering
\includegraphics[width=\hsize]{g2_combined.eps}
\caption{Intensity correlation function $g^{{(2)}}(\tau)$ from the three
  thermal light sources after spectral filtering. All light sources show
  a significant bunching at $\tau=0$, and a decay time that is
  well-resolved with the avalanche photodetectors. Error bars indicate
  uncertainties due to Poissonian counting statistics.}
\label{fig:g2}
\end{figure}

To ensure proper normalization of the resulting 
$g^{(2)}(\tau)$, we independently recorded pair events  with
a resolution of about 300\,ps with a time stamp unit that, unlike the
oscilloscope, does not exhibit a significant dead time varying with
acquisition mode and event rate, and found that
$g^{(2)}(|\tau|\gtrsim2\,\textrm{ns})\rightarrow1$. To extract a meaningful
set of model parameters, we assumed a Lorentzian line shape for the spectral
power distribution; this is mostly to reflect the approximate etalon
transmission function. A corresponding model function for the obtained pair  
rates in the histograms is
\begin{equation}
N(\tau)=a+b\cdot e^{-\left|2\tau/\tau_{c}\right|}\,
\end{equation}
where $a,b$ capture the absolute event number and the degree of bunching,
respectively, while $\tau_{c}$ represents a coherence time. The $g^{(2)}$ scale
on the right side of diagrams in Fig.~\ref{fig:g2} refers to the absolute
number of events after division by fit parameter $a$. 

For the Hg emission line at 546\,nm from the low pressure glow discharge lamp,
coincidence data was obtained over an
integration time of 68 hours, with single detector rates of 115000\,s$^{-1}$. 
From the numerical fit, we find $g^{{(2)}}(\tau=0)=1.795\pm0.006$
and an exponential decay with a characteristic coherence time of $\tau_{c}$
= 0.436$\pm$0.006\,ns. 
A convolution of an ideal correlation
function from a finite-bandwidth thermal light source (where
$g^{{(2)}}(\tau=0)=2$) with the detector response recorded in
figure~\ref{fig:jitter} leads to $g^{{(2)}}(\tau=0)=1.85$. Thus, the mercury
lamp seems perfectly compatible with a thermal light source.
The Doppler-broadened line width for individual transitions in
Hg vapour at temperatures around 400\,K is on the order of 235\,MHz,
but the contributions of different hyperfine transitions from different isotopes
result in a complex overall line shape of several GHz width (see
figure~\ref{fig:transmission}b). The part of the spectrum we select with the
etalon has a resultant coherence time that is slightly longer than what we
would expect from a flat spectrum filtered by the etalon
($\tau_c\approx0.34$\,ns). 

For the Ar arc discharge lamp, a wavelength of 540\,nm was
chosen to avoid contributions of a prominent peak at 546\,nm, possibly caused by mercury traces in the plasma. We find that
$g^{{(2)}}(0) =1.45\pm0.01$ and a shorter coherence time
$\tau_{c}=0.31\pm0.01$\,ns with $\chi^{2}_{r}=1.066$.
The wider spectrum of the cut-out blackbody radiation makes the decay time
shorter, so a peak reduction due to the detector response becomes more
prominent. Furthermore, residual transmission of the grating monochromator or
stray light into other transmission peaks of the etalon may contribute to
light that is incoherent with the main peak on observable time-scales. 

For an etalon of R $\approx97$\% and so an effective finesse of about
100, we expect a minimum transmission of $\approx0.1$\% over the whole grating
transmission of $\approx240$\,GHz (for 2 FWHMs). The etalon also has a
less-than-ideal peak transmission of approximately 50\% . The off-peak
transmission of the etalon effectively serves as an additional incoherent
light source that adds about 24\% more photons than a
singly-peaked transmission with exponential decay. These additional random
correlations further reduce the measured $g^{(2)}(\tau=0)$ from 1.8 due to the
detector resolution down to $\approx1.5$, which is in good agreement with the
value of 1.45 measured for the Ar arc lamp.

We finally collected sunlight into a single mode optical fiber with a
coelostat and transferred it into the lab. At the same working wavelength of
546.1\,nm, we find $g^{{(2)}}(0)=1.37\pm0.03$ and a coherence
time $\tau_{c}= 0.32\pm0.04$\,ns, with $\chi^{2}_{r}=0.985$ over an
integration time of 85 minutes, measuring the sun near the zenith at an altitude of 77$^{\circ}$ to 64$^{\circ}$. From an altitude of 32$^{\circ}$ to 21$^{\circ}$ close to the horizon, we find $g^{{(2)}}(0)=1.3\pm0.1$, $\tau_{c}=0.26\pm0.05$\,ns, with $\chi^{2}_{r}=1.024$ over 45 minutes.
This apparent dependency of the measured visibility of the photon
bunching on the altitude of the Sun is possibly due to seeing-induced atmospheric fluctuations in optical path
length differences \citep{dravins:07a}, where studies by \citet{kral:04, blazej:08, ortolani:12} suggest that atmospheric turbulence and scintillation may introduce jitter in photon timing measurements and is strongly dependent on the altitude of the light source.


\section{Summary and Outlook}

We have successfully developed and demonstrated an experimental technique to
resolve the temporal structure of photon bunching from thermal
blackbody light sources, including both discrete spectrum and blackbody
spectrum such as the Sun. We observed a photon bunching signature, i.e., the
excess of $g^{(2)}(0)$ above 1, that exceeds the recently reported records
of $g^{(2)}(0)=1.03$ \citep{karmakar:12} and $g^{(2)}(0)=1.04$ \citep{liu:14}
by about an order of magnitude. 

Compared to photon bunching observations using two-photon absorption
\citep{boitier:09}, our method would work at astronomical
intensity levels, and not be constrained to cases with the simultaneous arrivals of two photons at
the same detector.

We believe that with better timing control of the photodetectors and a better
match of the etalon with the grating monochromator as a first stage, it should
be possible to reach the theoretical limit $g^{(2)}(0)=2$
for blackbody radiators.

The ability to partially resolve temporal photon bunching
information directly may allow for additional probing into the emission
properties of light sources, and extend the traditional scope of
intensity interferometry of investigating the spatial coherence of a
source. Even there, our filtering technique may help to address detector
saturation and scintillation issues.

This filtering technique may also be
used to measure the visibility of starlight at various redshifts, providing
information on possible photon decoherence across extended distances
unaccessible in Earth-based
laboratory conditions. Such experiments may provide complementary
observational evidence that help to set constraints in theoretical models of
quantum gravity \citep{lieu:03, ng:03, ragazzoni:03, maziashvili:09}. 


\acknowledgments
We like to thank the referee for his helpful comments on the
earlier version of this paper. We acknowledge the support of this work by the
National Research Foundation and the Ministry of Education in Singapore.

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