5cm(14.5cm,-1.5cm) MIT-CTP-5111

# Color Erasure Detectors Enable Chromatic Interferometry

Luo-Yuan Qu  Shanghai Branch, National Laboratory for Physical Sciences at
Microscale and Department of Modern Physics University of Science and
Technology of China, Shanghai, 201315, P. R. China  CAS Center for Excellence
and Synergetic Innovation Center in Quantum Information and Quantum Physics,
Shanghai Branch, University of Science and Technology of China, Shanghai,
201315, P. R. China  Jinan Institute of Quantum Technology, Jinan, 250101, P.
R. China  Jordan Cotler  Stanford Institute for Theoretical Physics, Stanford
University, Stanford, CA 94305 USA  Fei Ma  Shanghai Branch, National
Laboratory for Physical Sciences at Microscale and Department of Modern
Physics University of Science and Technology of China, Shanghai, 201315, P. R.
China  CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, Shanghai Branch, University of Science and
Technology of China, Shanghai, 201315, P. R. China  Jinan Institute of Quantum
Technology, Jinan, 250101, P. R. China  Jian-Yu Guan  Shanghai Branch,
National Laboratory for Physical Sciences at Microscale and Department of
Modern Physics University of Science and Technology of China, Shanghai,
201315, P. R. China  CAS Center for Excellence and Synergetic Innovation
Center in Quantum Information and Quantum Physics, Shanghai Branch, University
of Science and Technology of China, Shanghai, 201315, P. R. China  Ming-Yang
Zheng  Xiu-Ping Xie  Jinan Institute of Quantum Technology, Jinan, 250101, P.
R. China  Yu-Ao Chen  Shanghai Branch, National Laboratory for Physical
Sciences at Microscale and Department of Modern Physics University of Science
and Technology of China, Shanghai, 201315, P. R. China  CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, Shanghai Branch, University of Science and Technology of China,
Shanghai, 201315, P. R. China  Qiang Zhang  Shanghai Branch, National
Laboratory for Physical Sciences at Microscale and Department of Modern
Physics University of Science and Technology of China, Shanghai, 201315, P. R.
China  CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, Shanghai Branch, University of Science and
Technology of China, Shanghai, 201315, P. R. China  Jinan Institute of Quantum
Technology, Jinan, 250101, P. R. China  Frank Wilczek  Center for Theoretical
Physics, MIT, Cambridge, MA 02139 USA  T. D. Lee Institute, Shanghai Jiao Tong
University, Shanghai, 200240, P. R. China  Wilczek Quantum Center, School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, P. R.
China  Department of Physics, Stockholm University, Stockholm SE-106 91 Sweden
Department of Physics and Origins Project, Arizona State University, Tempe, AZ
25287 USA  Jian-Wei Pan  Shanghai Branch, National Laboratory for Physical
Sciences at Microscale and Department of Modern Physics University of Science
and Technology of China, Shanghai, 201315, P. R. China  CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, Shanghai Branch, University of Science and Technology of China,
Shanghai, 201315, P. R. China

###### Abstract

By engineering and manipulating quantum entanglement between incoming photons
and experimental apparatus, we construct single-photon detectors which cannot
distinguish between photons of very different wavelengths. These color erasure
detectors enable a new kind of intensity interferometry, with potential
applications in microscopy and astronomy. We demonstrate chromatic
interferometry experimentally, observing robust interference using both
coherent and incoherent photon sources.

Quantum interference Hariharan (2010), which lies at the heart of quantum
theory, requires complete indistinguishability between two particles. This is
to say, as long as one can distinguish two particles even in principle,
quantum interference will not happen. Meanwhile, quantum mechanics tells us if
we can erase the two particles’ past Scully and Drühl (1982), interference
will be restored. It has been shown that path or polarization information can
be easily erased, while the frequency difference is generally hard to
eliminate. For photons, conventional optical detectors are fundamentally
photon counters, whose operation depends upon processes which are sensitive to
the photons’ energy. Thus, conventional detectors distinguish between
different wavelengths, and therefore optical interference normally involves
quasimonochromatic light Mandel and Wolf (1995). Yet relative phases between
photons of different wavelengths potentially provide a rich source of
information. It is quite astonishing that, theoretically, the frequency
information can be erased Cotler et al. (2016). Here, we leveraged frequency-
space entanglement to develop color erasure detectors and achieve intensity
interferometry Brown and Twiss (1956); Baym (1998) between light of very
different wavelengths experimentally, thus revealing new features of optical
radiation fields. This new type of interferometer might find immediate
applications in astronomy, microscopy, and metrology Monnier (2003); Shtengel
et al. (2009); Giovannetti et al. (2011).

Since the final stage of optical detection generally involves quantized
processes, i.e. absorption or inelastic scattering, it is appropriate to use
the language of photons. Consider two sources $S_{1}$, $S_{2}$ which emit
photons of different colors $\gamma_{1}$, $\gamma_{2}$ which are received at
detectors $A$, $B$. Simultaneous firing of $A$, $B$ can be achieved in two
ways: $\gamma_{1}$ excites $A$ and $\gamma_{2}$ excites $B$, or $\gamma_{2}$
excites $A$ and $\gamma_{1}$ excites $B$. If those two possibilities can be
distinguished, then there is no interference between them. But if the
detectors erase the color information, then interference will occur. Let us
emphasize that according to the principles of quantum theory, interference
only occurs if the two final states are strictly indistinguishable. Such
strict color erasureness cannot be achieved simply by ignoring color
information. Rather, one must erase it. To do that we entangle the photons to
the detectors using nonlinear processes Cotler et al. (2016).

In particular, we generate entanglement between an incoming $\gamma_{1}$ or
$\gamma_{2}$ photon and a color erasure detector. If the difference in energy
between $\gamma_{1}$ and $\gamma_{2}$ is $\Delta E$, then a color erasure
detector implements an entangling unitary of the form

$\displaystyle|\gamma_{1}\rangle|\text{detector}\rangle\longrightarrow$
$\displaystyle\frac{1}{\sqrt{2}}\,|\gamma_{1}\rangle|\text{detector, measured
}\gamma_{1}\rangle+\frac{1}{\sqrt{2}}\,|\gamma_{2}\rangle|\text{detector}-\Delta
E\,,\text{ measured }\gamma_{2}\rangle$ (1)
$\displaystyle|\gamma_{2}\rangle|\text{detector}\rangle\longrightarrow$
$\displaystyle-\frac{1}{\sqrt{2}}\,|\gamma_{1}\rangle|\text{detector}+\Delta
E,\text{ measured
}\gamma_{1}\rangle+\frac{1}{\sqrt{2}}\,|\gamma_{2}\rangle|\text{detector,
measured }\gamma_{2}\rangle$

where $|\text{detector}\rangle$ is the initial state of the detector,
$|\text{detector, measured }\gamma_{1}\rangle$ is the state of the detector
having measured a $\gamma_{1}$ photon, $|\text{detector}+\Delta E,\text{
measured }\gamma_{1}\rangle$ is the state of the detector having gained an
energy $\Delta E$ and also having measured a $\gamma_{1}$ photon, and the
other states are defined similarly. If we only consider occurrences where
$\gamma_{1}$ is measured (i.e., project onto final states with a
$|\gamma_{1}\rangle$), then we are left with either

$\displaystyle|\gamma_{1}\rangle|\text{detector, measured
}\gamma_{1}\rangle\quad\text{ or
}\quad|\gamma_{1}\rangle|\text{detector}+\Delta E,\text{ measured
}\gamma_{1}\rangle$ (2)

the first state having come from an initial $\gamma_{1}$ photon and the second
state having come from an initial $\gamma_{2}$ photon. The key point is that
the overlap of the final detector states is approximately $1$, namely

$\langle\text{detector, measured }\gamma_{1}|\text{detector}+\Delta E,\text{
measured }\gamma_{1}\rangle\approx 1\,,$ (3)

and so our two final states in Eqn. (2) are essentially indistinguishable,
regardless of whether the initial incoming photon was $\gamma_{1}$ or
$\gamma_{2}$. In other words, by generating a specific kind of entangled state
between the incoming photon and the detector, we can cause decoherence (via
our projective measurement) to quantum mechanically erase the color
information of the initial photon. More details about these entangled states
can be found in the Supplementary Materials.

Our color erasure detectors are technically and conceptually distinct from
previous experiments in frequency-space interferometry. Conventional
interferometry experiments, such as Mach-Zehnder and Hong-Ou-Mandel
interferometry, are performed with standard beamsplitters, but can equally
well be performed with light beams of distinct polarization and polarizing
beamsplitters. In this spirit, recently more sophisticated experiments
Kobayashi et al. (2016, 2017) performed Mach-Zehnder and Hong-Ou-Mandel
interferometry with light beams of distinct frequency and frequency-space
beamsplitters. By contrast, color erasure detectors retroactively recover
interference from conventional interferometry experiments performed with
standard beamsplitters but distinct frequencies of light. This is akin to
quantum eraser experiments Scully et al. (1991); Kwiat et al. (1992), but now
involving erasure of color information. An important advantage of our approach
is that only detection apparatus requires augmentation. This is convenient in
general, and essential for imaging tasks involving self-luminous sources.

Figure 1: Diagram of the chromatic intensity interferometer. VOA: variable
optical attenuator, PPLN: periodically-poled lithium niobate, BPF: an 863 nm
band pass filter, PC: polarization controller, BS: beamsplitters, WDMs: two
wavelength division multiplexers, UCSPDs: upconversion single photon
detectors.

We realize chromatic intensity interferometry with our color erasure
detectors. As shown in Fig. 1, we first choose an attenuated $1550$ nm laser
as the source of $\gamma_{1}$. With the help of an $1950$ nm pump laser, we
up-convert another independent $1550$ nm laser light into $863$ nm light via
sum-frequency generation (SFG) in a home-made straight periodically-poled
lithium niobate (PPLN) waveguide Zheng et al. (2016) (PPLN WG1). An $863$ nm
band pass filter is exploited to block the 1950 nm pump and the $863$ nm light
is taken as the source of $\gamma_{2}$. We then use beamsplitters and
wavelength division multiplexing (WDM) to divide and couple photons from both
sources to the color erasure detectors, which are composed of two integrated
PPLN waveguides (PPLN WG A,B) Ma et al. (2018), a $1950$ nm pump source, band
pass filters, and two telecom band single photon detectors Zheng et al.
(2016).

In order to observe color erasure interference, we need to change the relative
phase between the $\gamma_{1}$ and $\gamma_{2}$ photons in one arm of the
detector Cotler et al. (2016). Since the phase of a $\gamma_{2}$ photon
changes faster than that of a $\gamma_{1}$ photon with the same delay time, we
can control the relative phase by adjusting the optical fiber delay (MDL-002)
before detector $B$. We can choose the final output of the color erasure
detectors to be either $\gamma_{1}$ or $\gamma_{2}$, contingent on our choice
of band pass filters. We record the arrival time of each photon by a time-
digital converter (TDC) and a computer.

Generally, intensity interferometry is observed in terms of $g^{(2)}(\tau)$,
the second-order quantum mechanical correlation function. As we can see in the
red curve in Fig. 2(a), the correlation $g^{(2)}(\tau=3\text{ ns})$ oscillates
as we change the optical delay and detect $\gamma_{1}$ photons by filtering
out the $\gamma_{2}$ photons. Photons from lasers obey Poissonian number
statistics so that the $\tau$-average of $g^{(2)}(\tau)$ is $1$.

Figure 2: Chromatic intensity interferometry of lasers. (a)
$g^{(2)}(\tau=3\text{ ns})$ as a function of the optical delay time, where the
color erasure detectors each output 1550 nm light. The red rounded markers
display interference of different wavelengths of light due to the color
erasure detectors, whereas the blue triangle markers do not display
interference since standard detectors are used. The same color scheme is used
in (b) and (c). (b) The Fourier transform of $g^{(2)}(\tau=3\text{ ns})$ as a
function of the optical delay time. (c) $g^{(2)}(\tau=3\text{ ns})$ as a
function of the optical delay time, where the color erasure detectors each
output 863 nm light. (d) $g^{(2)}(\tau)$ as a function of the delay $\tau$
between the two detectors.

The visibility of the interference is around $0.4$, slightly less than the
theoretically expected visibility $0.5$ mainly due to the up-conversion single
photon detector’s dark counts and baseline error from imperfect devices. For
comparison, we also measure $g^{(2)}(\tau)$ without the pump light which
enables the detectors to distinguish between the incoming wavelengths, so they
are no longer color erasure. As expected, the interference pattern disappears,
as shown by the blue curve of Fig. 2(a).

Fig. 2(b) shows the Fourier transforms of the two curves in Fig. 2(a). The
location of the peak of the red curve represents the frequency of the
interference pattern, i.e. the rate of phase change as we scan the optical
delay. In our case, the rate of phase change is theoretically the frequency of
pump. The measured peak position is around $144$ THz, which well-coincides
with $1950$ nm. The blue curve in Fig. 2(b) is just noise and so has no large
peaks, demonstrating that interference does not occur in the absence of color
erasure detectors.

Instead of having each color erasure detector output 1550 nm light, we can
instead arrange that the detectors each output 863 nm light. Data for this
alternative arrangement are shown in Fig. 2(c). In the figure, we filter in
only $\gamma_{2}$ photons at the output of the waveguides, and collect
coincidence counts with and without the pumps enabling color erasure
detection. Relative to filtering in $\gamma_{1}$ photons, the visibility of
interference when filtering in $\gamma_{2}$ photons is degraded since the
photons tend to be multi-mode when propagating through the PPLN waveguides
comprising our color erasure detectors. Only photons in the lowest transverse
mode participate in interference. The photons in other modes induce noise and
thus reduce the visibility.

We also perform Hong-Ou-Mandel interference Hong et al. (1987) utilizing
standard beamsplitters and two different wavelengths of light. The
interference can only be recovered with color erasure detectors. Instead of
changing the relative time delay of the light beams, we instead observe
coincidence counts between different time slots in the TDC. In Fig. 2(d), we
observe an oscillation of $g^{(2)}(\tau)$ as a function of $\tau$, which
decays as the delay between two detectors surpasses the coherence time of the
light sources. We can produce bunching or antibunching depending on the setup
of the interferometer, and the settings of the color erasure detectors.

In a tabletop demonstration experiment, it is convenient to use lasers as
light sources. Considering future applications, we would like to observe
chromatic interferometry for incoherent or semi-incoherent sources such as
thermal light from a star or photon emission from fluorescent proteins.
Therefore, it is important to demonstrate that our chromatic intensity
interferometer can function with thermal light. Accordingly, we experimentally
performed chromatic intensity interferometry with thermal light sources. To
construct a thermal source, we prepare a C band amplified spontaneous emission
(ASE) light source with 30 nm spectral bandwidth. We first filter the ASE
light with a 100 GHz bandwidth dense wavelength division multiplexer (DWDM)
and then amplify it with an Erbium doped fiber amplifier (EDFA). The emission
of EDFA is further filtered by a 50 MHz bandwidth etalon to select out a
thermally populated mode which is then divided into two beams. One is used for
$\gamma_{1}$ and the other one is converted to $863$ nm in a PPLN waveguide to
become $\gamma_{2}$, similar to the coherent laser setting from before. In
this thermal source setup, the $\gamma_{1}$ and $\gamma_{2}$ photons are
generated from the same source and thus their phases are correlated. To
destroy these correlations, the $\gamma_{1}$ beam is sent through a 20 km
spool of fiber, and fluctuations of the fiber ruin the phase coherence between
$\gamma_{1}$ and $\gamma_{2}$. Then we send both beams to the color erasure
detectors and observe interference.

Figure 3: Chromatic intensity interferometry of thermal sources. (a)
$g^{(2)}(\tau)$ for $\tau\approx 0$ as a function of the optical delay time,
where the color erasure detectors each output 1550 nm light. The red rounded
markers show interference due to the aid of the color erasure detectors,
whereas the blue triangle markers show the null outcome in the absence of
color erasure detectors. This color scheme is also used in (b) and (c). (b)
The Fourier transform of $g^{(2)}(\tau)$ for $\tau\approx 0$ as a function of
the optical delay time. (c) $g^{(2)}(\tau)$ for $\tau\approx 0$ as a function
of the optical delay time, where the color erasure detectors each output 863
nm light. (d) $g^{(2)}(\tau)$ as a function of the delay $\tau$ between the
two detectors.

As shown in Fig. 3(a) and Fig. 3(c), we observed interference of the thermal
light in when the color erasure detectors output only
$|\gamma_{1}\gamma_{1}\rangle$ or only $|\gamma_{2}\gamma_{2}\rangle$,
respectively. We also compute the Fourier transform of the interference
pattern for the $|\gamma_{1}\gamma_{1}\rangle$ case. In the absence of color
erasure detectors (i.e., by not pumping the waveguides), we check that
interference does not occur. We have also performed chromatic Hong-Ou-Mandel
interferometry with these thermal sources, and $g^{(2)}(\tau)$ is shown in
Fig. 3(d).

One apparent difference between our experimental data for thermal sources
versus coherent lasers is the mean value of the interference patterns. In Fig.
3(a) and Fig. 3(c), the mean value is larger than $1$, which coincidences with
the super-Poissonian number statistics of thermal light. The visibility for
the thermal sources is worse than for the coherent lasers since the coherence
time of the thermal sources is much shorter. Thus every mismatch in the
optical path will lead to the loss of coherence and visibility.

Since we expect color erasure detectors to have applications in free space
imaging, we also performed chromatic interferometry in free space. As shown in
Fig. 4(a), we detect the photons from two disk-like sources emitting different
wavelengths of light.

Figure 4: Chromatic intensity interferometry in free space. (a) A diagram of
the experimental setup for the intensity interferometer in free space. Lasers
from a fiber array are utilized as sources, and the collimators are utilized
to couple light from free space into color erasure detectors. One of the
collimators is mounted on a linear translation stage to control the distance
$x$ between the two collimators. (b) The measured interference pattern when
both sources emit light of the same wavelength, reproducing standard Hanbury
Brown and Twiss interference. (c) The measured interference pattern when the
two sources emit 1550 nm and 863 nm light, respectively. The red rounded
markers show interference due to the aid of the color erasure detectors,
whereas the blue triangle markers show the absence of interference without the
color erasure detectors.

The disk-like sources are situated $125$ $\mu$m apart in a fiber array, and
color erasure detectors are placed $40$ cm away. When we move the position of
one of the detectors using a linear translation stage, we observe an
interference pattern, as shown in the red curve in Fig. 4(c). The blue curve
in Fig. 4(c) illustrates that interference is not observed in the absence of
color erasure detection. We also show in Fig. 4(b) the standard Hanbury Brown
and Twiss interference pattern when the two sources emit at the same
wavelength, utilizing standard detection apparatus. Our free space results for
chromatic interferometry demonstrate the potential application of color
erasure detection in imaging.

In conclusion, we have used our color erasure detectors to perform intensity
interferometry between photons of very different wavelengths, and to recover
their relative phase information, which is inaccessible to conventional
detectors. Since our technique does not require lenses, it could be used with
very large apertures, and in regions of the spectrum where lenses are not
readily available. This might inspire new opportunities for imaging and thus
calls for further theoretical and experimental research. As an example, color
erasure detectors can enhance the ability of fluorescent microscopes Grußmayer
and Herten (2014); Schwartz et al. (2013); Leung and Chou (2011) to resolve
nearby proteins which emit at distinct frequencies. We can also leverage a
generalization of the van Cittert-Zernike formula for sources of different
wavelengths measured with color erasure detectors Cotler et al. (2016).

If instead we had a nearly perfect single photon detector, which has no noise,
no jitter, no dead time and is very fast, we can effectively erase the
frequency of incoming photons and use it in the multi-color HBT
interferometer. However, there does not exist a photon detector or traditional
photodiode faster than 144 THz, as would be required in our experiment. What’s
more, a fast detector acts like a very narrow timing filter, which filters the
two input light pulses into a very narrow time window. This would filter out
most of the photons in the pulses. In our experiment, the linewidth for the
input laser is around 3 MHz and the detector bandwidth is around 144 THz. Only
around 0.002% (3 MHz/144THz) of the light will be detected. In this sense, it
is indeed inefficient. Meanwhile, our system can convert photons with an
efficiency of around 50% which is orders of magnitude higher than a fast
detection method. This is actually not due to a technological advance but a
difference in concept. Instead of filtering light, we coherently convert
different wavelengths of light to become indistinguishable.

Our work exploits and emphasizes the realization that detectors are themselves
quantum mechanical objects, which “measure” other systems by becoming
entangled with them Zurek (2003, 2009); Cotler et al. (2016). Indeed, the core
mechanism enabling multi-wavelength intensity interferometry is a trade-off
between coherence of multi-photon phase information and coherence of color
information, implemented by crafting and manipulating the entanglement between
source photons and the detection apparatus. (For mathematical details, see the
Supplementary Materials.) We anticipate that further analysis of the quantum
mechanics of detectors will reveal other trade-off opportunities.

Acknowledgements

We would like to thank Andreas Kaldun and Philip Bucksbaum for valuable
conversations and thank Lian-Tuan Xiao and Jian-Yong Hu for lending
equipments. This work was supported by the National Key R&D Program of China
(2018YFB0504300), the National Natural Science Foundation of China, and the
Chinese Academy of Science. JC is supported by the Fannie and John Hertz
Foundation and the Stanford Graduate Fellowship program. FW’s work is
supported by the the Swedish Research Council under Contract No. 335-
2014-7424, the U.S. Department of Energy under grant Contract No. DE-
SC0012567, and by the European Research Council under grant 742104.

L.-Y. Q., J. C., and M. F. contributed equally to this work.

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## Supplemental Materials

## Theoretical Overview

First, we review the mathematics behind color erasure detectors Cotler et al.
(2016). Suppose we have two types of photons $\gamma_{1}$ and $\gamma_{2}$,
where the wavelength of $\gamma_{2}$ is shorter than that of $\gamma_{1}$. We
also consider a third wavelength $\gamma_{3}$ whose energy is the difference
in energies between $\gamma_{2}$ and $\gamma_{1}$. We prepare a coherent state
of $\gamma_{3}$ photons, denoted by

$|\alpha,\text{coh.}\rangle_{\gamma_{3}}:=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\,|n\rangle_{\gamma_{3}}\,,$
(4)

where $\alpha$ is a complex number, and $|n\rangle_{\gamma_{3}}$ is a number
state with $n$ $\gamma_{3}$ photons in an incoming mode. The average number of
photons in the state $|\alpha,\text{coh.}\rangle_{\gamma_{3}}$ is
$\\!\,{}_{\gamma_{3}}\langle\alpha,\text{coh.}|\,\widehat{n}\,|\alpha,\text{coh.}\rangle_{\gamma_{3}}=|\alpha|^{2}$.
In our setting, the average number of photons $|\alpha|^{2}$ is large, which
holds when the pump is a strong laser.

Our PPLN will have an input state either of the form

$|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}\otimes|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
(5)

which has a single $\gamma_{1}$ photon and a coherent state of $\gamma_{3}$
photons, or

$|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}\otimes|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
(6)

which has a single $\gamma_{2}$ photon and a coherent state of $\gamma_{3}$
photons. The input state evolves with the Hamiltonian Kumar (1990)

$H=i\chi\left(a_{\gamma_{1}}\otimes a_{\gamma_{2}}^{\dagger}\otimes
a_{\gamma_{3}}-a_{\gamma_{1}}^{\dagger}\otimes a_{\gamma_{2}}\otimes
a_{\gamma_{3}}^{\dagger}\right)$ (7)

where $a^{\dagger},a$ are creation and annihilation operators. Evolving for a
time $T$, we have:

$\displaystyle
e^{-iHT}|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}\otimes|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
$\displaystyle\quad=|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}\otimes\cos\big{(}\chi\,T\sqrt{\widehat{n}_{\gamma_{3}}}\big{)}|\alpha,\text{coh.}\rangle_{\gamma_{3}}+|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}\otimes
a_{\gamma_{3}}\,\frac{\sin\big{(}\chi\,T\,\sqrt{\widehat{n}_{\gamma_{3}}}\big{)}}{\sqrt{\widehat{n}_{\gamma_{3}}}}|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
(8) $\displaystyle
e^{-iHT}|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}\otimes|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
$\displaystyle\quad=-|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}\otimes
a_{\gamma_{3}}^{\dagger}\,\frac{\sin\big{(}\chi\,T\,\sqrt{\widehat{n}_{\gamma_{3}}+1}\big{)}}{\sqrt{\widehat{n}_{\gamma_{3}}+1}}|\alpha,\text{coh.}\rangle_{\gamma_{3}}+|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}\otimes\cos\big{(}\chi\,T\sqrt{\widehat{n}_{\gamma_{3}}+1}\big{)}|\alpha,\text{coh.}\rangle_{\gamma_{3}}$
(9)

Let us call the first state
$|\Psi_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$ and the second state
$|\Psi_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$. Notice that both
states are entangled between the $\gamma_{1}$, $\gamma_{2}$ modes, and the
$\gamma_{3}$ mode. The first state is a superposition of two possibilities:
(i) the $\gamma_{1}$ remains, and (ii) the $\gamma_{1}$ absorbs a $\gamma_{3}$
and is upconverted to a $\gamma_{2}$. The second state is similarly a
superposition of the two possibilities: (i) the $\gamma_{2}$ remains, and (ii)
the $\gamma_{2}$ emits a $\gamma_{3}$ photon and is downconverted to a
$\gamma_{1}$.

Next, we put the state through a filter which only lets through $\gamma_{2}$
photons, and proceed if a $\gamma_{2}$ photon is outputted. This is equivalent
to post-selecting by projecting onto
$\textbf{1}\otimes|1\rangle_{\gamma_{2}}\,{}_{\gamma_{2}}\langle
1|\otimes\textbf{1}$ and renormalizing the residual state, as

$\displaystyle|\widetilde{\Psi}_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$
$\displaystyle=\frac{\big{(}\textbf{1}\otimes|1\rangle_{\gamma_{2}}\,{}_{\gamma_{2}}\langle
1|\otimes\textbf{1}\big{)}|\Psi_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}}{\sqrt{\,{}_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\langle\Psi_{1}|\big{(}\textbf{1}\otimes|1\rangle_{\gamma_{2}}\\!\,{}_{\gamma_{2}}\langle
1|\otimes\textbf{1}\big{)}|\Psi_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}}}$
(10)
$\displaystyle|\widetilde{\Psi}_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$
$\displaystyle=\frac{\big{(}\textbf{1}\otimes|1\rangle_{\gamma_{2}}\,{}_{\gamma_{2}}\langle
1|\otimes\textbf{1}\big{)}|\Psi_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}}{\sqrt{\,{}_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\langle\Psi_{2}|\big{(}\textbf{1}\otimes|1\rangle_{\gamma_{2}}\\!\,{}_{\gamma_{2}}\langle
1|\otimes\textbf{1}\big{)}|\Psi_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}}}\,.$
(11)

Letting $\alpha=e^{i\phi}\,\sqrt{N}$ for $N$ large, we find that

$\left|\\!\,{}_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\langle\widetilde{\Psi}_{1}|\widetilde{\Psi}_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\right|=1\,\,+\mathcal{O}(1/\sqrt{N})$
(12)

and so $|\widetilde{\Psi}_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$ and
$|\widetilde{\Psi}_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$ cannot be
distinguished. In plainer terms, this means if we evolve either $\gamma_{1}$
or $\gamma_{2}$ in the PPLN with the $\gamma_{3}$ coherent state and then
measure a $\gamma_{2}$ as the output, then the apparatus fundamentally cannot
tell us whether $\gamma_{2}$ was originally a $\gamma_{1}$ or a $\gamma_{2}$.
Hence, the detection apparatus can erase the color information.

There is another, more illuminating way of reprocessing the above analysis.
Consider again the evolved states
$|\Psi_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$ and
$|\Psi_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}$. We can write down the
corresponding density matrices

$\displaystyle\rho_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}=\big{(}|\Psi_{1}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\big{)}\big{(}\,_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\langle\Psi_{1}|\big{)}$
(13)
$\displaystyle\sigma_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}=\big{(}|\Psi_{2}\rangle_{\gamma_{1}\,\gamma_{2}\,\gamma_{2}}\big{)}\big{(}\,_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\langle\Psi_{2}|\big{)}$
(14)

and trace out the $\gamma_{3}$ photons to obtain

$\displaystyle\rho_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=\text{tr}_{\gamma_{3}}\left(\rho_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\right)$
(15) $\displaystyle\sigma_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=\text{tr}_{\gamma_{3}}\left(\sigma_{\gamma_{1}\,\gamma_{2}\,\gamma_{3}}\right)\,.$
(16)

Again letting $\alpha=e^{i\phi}\sqrt{N}$ and taking $N$ large, we can use the
Euler-Maclaurin formula and a saddle point approximation to compute the
explicit expressions of $\rho_{\gamma_{1}\,\gamma_{2}}$ and
$\sigma_{\gamma_{1}\,\gamma_{2}}$. We find that

$\displaystyle\rho_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=\big{(}|\Phi_{1}\rangle_{\gamma_{1}\,\gamma_{2}}\big{)}\big{(}\,_{\gamma_{1}\,\gamma_{2}}\langle\Phi_{1}|\big{)}$
(17) $\displaystyle\sigma_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=\big{(}|\Phi_{2}\rangle_{\gamma_{1}\,\gamma_{2}}\big{)}\big{(}\,_{\gamma_{1}\,\gamma_{2}}\langle\Phi_{2}|\big{)}$
(18)

where

$\displaystyle|\Phi_{1}\rangle_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=\cos(\chi
T\sqrt{N})\,|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}+e^{i\phi}\,\sin(\chi
T\sqrt{N})\,|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}$ (19)
$\displaystyle|\Phi_{2}\rangle_{\gamma_{1}\,\gamma_{2}}$
$\displaystyle=-e^{-i\phi}\sin(\chi
T\sqrt{N})\,|1\rangle_{\gamma_{1}}\otimes|0\rangle_{\gamma_{2}}+\cos(\chi
T\sqrt{N})\,|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}$ (20)

up to $\mathcal{O}(1/\sqrt{N})$ corrections. Notice that
$|\Phi_{1}\rangle_{\gamma_{1}\,\gamma_{2}}$ and
$|\Phi_{2}\rangle_{\gamma_{1}\,\gamma_{2}}$ are mutually orthogonal. From the
above equations, we see that the PPLN performs a rotation in color space. It
is clear that projecting either $|\Phi_{1}\rangle_{\gamma_{1}\,\gamma_{2}}$ or
$|\Phi_{2}\rangle_{\gamma_{1}\,\gamma_{2}}$ onto the
$|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}$ state and renormalizing
will yield $|0\rangle_{\gamma_{1}}\otimes|1\rangle_{\gamma_{2}}$, and so we
see more simply that the detector is blind to the initial color of the photon.

The key observation is that in the large photon limit of the coherent state,
we can truly treat $\gamma_{3}$ as a classical light field which is incapable
of recording information about individual photons. In fact, we see from Eqn.’s
(19) and (20) that the effective Hamiltonian which evolves the $\gamma_{1}$,
$\gamma_{2}$ modes is

$H_{\text{eff}}=i\chi\sqrt{N}\left(e^{i\phi}\,a_{\gamma_{1}}\otimes
a_{\gamma_{2}}^{\dagger}-e^{-i\phi}\,a_{\gamma_{1}}^{\dagger}\otimes
a_{\gamma_{2}}\right)$ (21)

which clearly performs a rotation in color space.

To perform chromatic intensity interferometry, we consider two sources $1$ and
$2$, emitting $\gamma_{1}$ and $\gamma_{2}$ photons, respectively. We also
have two color erasure detectors $A$ and $B$. Let $D_{1A}$ be the probability
amplitude that a single photon emitted from $1$ is received by $A$. The
probability amplitudes $D_{2A}$, $D_{1B}$, $D_{2B}$ are defined similarly. If
a single photon is received by each detector (this is a form of post-
selection), then we have the state

$D_{1A}D_{2B}|1\rangle_{\gamma_{1},A}\otimes|0\rangle_{\gamma_{2},A}\otimes|0\rangle_{\gamma_{1},B}\otimes|1\rangle_{\gamma_{2},B}+D_{1B}D_{2A}|0\rangle_{\gamma_{1},A}\otimes|1\rangle_{\gamma_{2},A}\otimes|1\rangle_{\gamma_{1},B}\otimes|0\rangle_{\gamma_{2},B}$
(22)

where the first term corresponds to having a $\gamma_{1}$ photon at $A$ and a
$\gamma_{2}$ photon at $B$, and the second term corresponds to having a
$\gamma_{2}$ photon at $B$ and a $\gamma_{1}$ photon at $A$. After the
detectors $A$ and $B$ process their photons and we post-select on $A$
outputting $\gamma_{2}$ and $B$ outputting $\gamma_{2}$, we are left with the
state

$e^{i\phi}\cos(\chi T\sqrt{N})\sin(\chi
T\sqrt{N})(D_{1A}D_{2B}+D_{1B}D_{2A})|0\rangle_{\gamma_{1},A}\otimes|1\rangle_{\gamma_{2},A}\otimes|0\rangle_{\gamma_{1},B}\otimes|1\rangle_{\gamma_{2},B}$
(23)

with probability

$\displaystyle\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$ $\displaystyle=\,$
$\displaystyle\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,\bigg{(}|D_{1A}D_{2B}|^{2}+|D_{1B}D_{2A}|^{2}+2\,\text{Re}\left(D_{1A}D_{2B}D_{1B}^{*}D_{2A}^{*}\right)\bigg{)}$
(24)

which contains the Hanbury-Brown Twiss interference term Mandel and Wolf
(1995); Baym (1998); Cotler et al. (2016)
$\text{Re}\left(D_{1A}D_{2B}D_{1B}^{*}D_{2A}^{*}\right)$.

As a concrete example, suppose that $\chi T\sqrt{N}=\pi/4$, and that the
length from $1$ to $A$ is $L_{1A}$. The lengths $L_{2A}$, $L_{1B}$, $L_{2B}$
are defined similarly. Denoting the wavelengths of $\gamma_{1}$ and
$\gamma_{2}$ by $\lambda_{1}$ and $\lambda_{2}$ respectively, and assuming the
sources $1$ and $2$ emit photons with equal probability, we have

$\displaystyle D_{1A}$
$\displaystyle=\frac{1}{\sqrt{2}}\,e^{i\,2\pi\,L_{1A}/\lambda_{1}+i\,\theta_{1}}\,,\quad
D_{1B}=\frac{1}{\sqrt{2}}\,e^{i\,2\pi\,L_{1B}/\lambda_{1}+i\,\theta_{1}}\,,$
$\displaystyle D_{2A}$
$\displaystyle=\frac{1}{\sqrt{2}}\,e^{i\,2\pi\,L_{2A}/\lambda_{2}+i\,\theta_{2}}\,,\quad
D_{2B}=\frac{1}{\sqrt{2}}\,e^{i\,2\pi\,L_{2B}/\lambda_{2}+i\,\theta_{2}}\,,$
(25)

where $\theta_{1},\theta_{2}$ are phases associated with the emission of
photons from sources $1$ and $2$, respectively. In this case, Eqn.
(Theoretical Overview) becomes

$\frac{1}{8}\left[1+\cos\left(2\pi\left(\frac{L_{1A}}{\lambda_{1}}+\frac{L_{2B}}{\lambda_{2}}-\frac{L_{1B}}{\lambda_{1}}-\frac{L_{2A}}{\lambda_{2}}\right)\right)\right]$
(26)

where the interference term
$\text{Re}\left(D_{1A}D_{2B}D_{1B}^{*}D_{2A}^{*}\right)$ is
$\frac{1}{4}\,\cos\left(2\pi\left(\frac{L_{1A}}{\lambda_{1}}+\frac{L_{2B}}{\lambda_{2}}-\frac{L_{1B}}{\lambda_{1}}-\frac{L_{2A}}{\lambda_{2}}\right)\right)$.
Note that the interference term is independent of $\theta_{1}$ and
$\theta_{2}$. Accordingly, we can achieve interference between two mutually
incoherent sources Baym (1998); Cotler et al. (2016). In this case,
$\theta_{1}$ and $\theta_{2}$ may be strongly time-dependent, but nonetheless
cancel out in the interference term.

In the analysis above, we have assumed that our photon sources each emit
exactly one photon within some time window. We can relax this assumption in
various ways. First, suppose that the first and second sources emit coherent
superpositions of photon number states, namely

$\displaystyle
c_{0}\,|0\rangle_{\gamma_{1}}+c_{1}\,|1\rangle_{\gamma_{1}}+c_{2}\,|2\rangle_{\gamma_{1}}+\cdots$
(27) $\displaystyle
d_{0}\,|0\rangle_{\gamma_{2}}+d_{1}\,|1\rangle_{\gamma_{2}}+d_{2}\,|2\rangle_{\gamma_{2}}+\cdots\,,$
(28)

respectively, where the $c_{i}$’s and $d_{i}$’s are complex numbers satisfying
$\sum_{i}|c_{i}|^{2}=\sum_{i}|d_{i}|^{2}=1$. Then detectors $A$ and $B$
receive the state

$\displaystyle
c_{1}\,d_{1}\,\bigg{(}D_{1A}D_{2B}|1\rangle_{\gamma_{1},A}\otimes|0\rangle_{\gamma_{2},A}\otimes|0\rangle_{\gamma_{1},B}\otimes|1\rangle_{\gamma_{2},B}+D_{1B}D_{2A}|0\rangle_{\gamma_{1},A}\otimes|1\rangle_{\gamma_{2},A}\otimes|1\rangle_{\gamma_{1},B}\otimes|0\rangle_{\gamma_{2},B}\bigg{)}$
$\displaystyle+\,$ $\displaystyle
c_{2}\,d_{0}\,D_{1A}D_{1B}|1\rangle_{\gamma_{1},A}\otimes|0\rangle_{\gamma_{2},A}\otimes|1\rangle_{\gamma_{1},B}\otimes|0\rangle_{\gamma_{2},B}+c_{2}\,d_{0}\,D_{2A}D_{2B}|0\rangle_{\gamma_{1},A}\otimes|1\rangle_{\gamma_{2},A}\otimes|0\rangle_{\gamma_{1},B}\otimes|1\rangle_{\gamma_{2},B}$
$\displaystyle+\,$ $\displaystyle\cdots$ (29)

The first line of the above equation corresponds to each source emitting a
single photon and each detector receiving a single photon, and so has the same
form as Eqn. (22). The second line corresponds to (i) the first source
emitting two photons and the second source emitting no photons, and each
detector receiving a single photon, and (ii) the first source emitting no
photons and the second source emitting two photons, and each detector
receiving a single photon. The final line with the ellipses accounts for the
remaining terms.

As before, after the detectors $A$ and $B$ process their photons and we post-
select on $A$ outputting a single $\gamma_{2}$ and $B$ outputting a single
$\gamma_{2}$, we obtain the state

$\displaystyle\bigg{(}c_{1}\,d_{1}\,e^{i\phi}\cos(\chi T\sqrt{N})\,\sin(\chi
T\sqrt{N})\,(D_{1A}D_{2B}+D_{1B}D_{2A})+c_{2}\,d_{0}\,e^{2i\phi}\sin^{2}(\chi
T\sqrt{N})\,D_{1A}D_{1B}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+c_{0}\,d_{2}\,\cos^{2}(\chi
T\sqrt{N})D_{2A}D_{2B}\bigg{)}|0\rangle_{\gamma_{1},A}\otimes|1\rangle_{\gamma_{2},A}\otimes|0\rangle_{\gamma_{1},B}\otimes|1\rangle_{\gamma_{2},B}$
(30)

with probability

$\displaystyle|c_{1}|^{2}|d_{1}|^{2}\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$ $\displaystyle+\,\,$
$\displaystyle|c_{2}|^{2}|d_{0}|^{2}\sin^{4}(\chi
T\sqrt{N})\,|D_{1A}D_{1B}|^{2}+|c_{0}|^{2}|d_{2}|^{2}\cos^{4}(\chi
T\sqrt{N})\,|D_{2A}D_{2B}|^{2}$ $\displaystyle+\,\,$ $\displaystyle 2\cos(\chi
T\sqrt{N})\sin^{3}(\chi
T\sqrt{N})\,\text{Re}\left(c_{1}\,d_{1}\,c_{2}^{*}\,d_{0}^{*}\,e^{-i\phi}(D_{1A}D_{2B}+D_{1B}D_{2A})\,D_{1A}^{*}D_{1B}^{*}\right)$
$\displaystyle+\,\,$ $\displaystyle 2\cos^{3}(\chi T\sqrt{N})\sin(\chi
T\sqrt{N})\,\text{Re}\left(c_{1}\,d_{1}\,c_{0}^{*}\,d_{2}^{*}\,e^{i\phi}(D_{1A}D_{2B}+D_{1B}D_{2A})\,D_{2A}^{*}D_{2B}^{*}\right)$
$\displaystyle+\,\,$ $\displaystyle 2\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,\text{Re}\left(c_{2}\,d_{0}\,c_{0}^{*}\,d_{2}^{*}\,e^{2i\phi}\,D_{1A}D_{1B}D_{2A}^{*}D_{2B}^{*}\right)\,.$
(31)

Several remarks are in order. First, notice that if
$|c_{0}||d_{2}|\ll|c_{1}||d_{1}|$ and $|c_{2}||d_{0}|\ll|c_{1}||d_{1}|$, then
the first term in the above equation dominates, which recovers the same
interference as in Eqn. (Theoretical Overview).

Now suppose $D_{1A},D_{2A},D_{1B},D_{2B}$ are the same as in Eqn. (Theoretical
Overview), but with $\theta_{1}=\theta_{1}(t)$ and $\theta_{2}=\theta_{2}(t)$
time-dependent and rapidly changing faster than the timescale of photon
emission from the sources. It is natural to assume that $\theta_{1}(t)$ and
$\theta_{2}(t)$ are each ergodic on $[0,2\pi]$. In this case, it is easy to
check that the time average of Eqn. (Theoretical Overview) is simply the first
two lines of the equation, i.e. the last three lines vanish under time
averaging. This is because only the terms in the first two lines are
independent of $\theta_{1}(t)$ and $\theta_{2}(t)$, whereas the remaining
terms do depend on $\theta_{1}(t)$ and $\theta_{2}(t)$ and so average to zero.
Accordingly, we can still recover the desired Hanbury-Brown Twiss interference
term contained in $|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$ since this is the only
remaining term sensitive to relative phases between the photon probability
amplitudes after time-averaging. Note that the experimental collection of data
automatically incorporates time-averaging, since one averages results over
many trials.

Instead of requiring the sources to emit coherent superpositions of photon
number states as in Eqn. (27) and (28), we can also accommodate for
arbitrarily incoherent density matrices of photon states. For instance,
suppose that the first and second sources emit completely incoherent sums of
photon number states described by the density matrices

$\displaystyle p_{0}\,|0\rangle\langle 0|_{\gamma_{1}}+p_{1}\,|1\rangle\langle
1|_{\gamma_{1}}+p_{2}\,|2\rangle\langle 2|_{\gamma_{1}}+\cdots$ (32)
$\displaystyle q_{0}\,|0\rangle\langle 0|_{\gamma_{2}}+q_{1}\,|1\rangle\langle
1|_{\gamma_{2}}+q_{2}\,|2\rangle\langle 2|_{\gamma_{2}}+\cdots$ (33)

This occurs, for instance, if the sources are thermally populating the photon
modes, and $\\{p_{i}\\},\\{q_{i}\\}$ are classical Gibbs distributions.
Running through the same analysis as above, after the detectors process their
photons and we post-select on $A$ outputting a single $\gamma_{2}$ and $B$
outputting a single $\gamma_{2}$, we are left with the density matrix

$\displaystyle\bigg{(}p_{1}\,q_{1}\,\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}+p_{2}\,q_{0}\sin^{4}(\chi
T\sqrt{N})\,|D_{1A}D_{1B}|^{2}$
$\displaystyle\qquad\qquad+p_{0}\,q_{2}\,\cos^{4}(\chi
T\sqrt{N})\,|D_{2A}D_{2B}|^{2}\bigg{)}\,|0\rangle\langle
0|_{\gamma_{1},A}\otimes|1\rangle\langle
1|_{\gamma_{2},A}\otimes|0\rangle\langle
0|_{\gamma_{1},B}\otimes|1\rangle\langle 1|_{\gamma_{2},B}$ (34)

with probability

$\displaystyle p_{1}\,q_{1}\,\cos^{2}(\chi T\sqrt{N})\sin^{2}(\chi
T\sqrt{N})\,|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$ $\displaystyle+\,\,$
$\displaystyle p_{2}\,q_{0}\sin^{4}(\chi
T\sqrt{N})\,|D_{1A}D_{1B}|^{2}+p_{0}\,q_{2}\,\cos^{4}(\chi
T\sqrt{N})\,|D_{2A}D_{2B}|^{2}\,.$ (35)

Notice that this has a similar form as Eqn. (Theoretical Overview), but
without the unwanted interference terms in the last three lines of Eqn.
(Theoretical Overview). As before, we recover the desired Hanbury-Brown Twiss
interference term contained in $|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$, which is in
fact the only interference term in Eqn. (Theoretical Overview).

## Experimental Methods

Experiment details for PPLN waveguide A key device within each color erasure
detector is an integrated PPLN waveguide. We fabricated reverse-proton-
exchange (RPE) PPLN waveguides Parameswaran et al. (2002) with a total length
of $52$ mm for both difference-frequency generation (DFG) Ikuta et al. (2011);
De Greve et al. (2012); Chou et al. (1998) between $863$ nm light and the
$1950$ nm pump, and sum-frequency generation (SFG) Vandevender and Kwiat
(2004); Ma et al. (2009); Kumar (1990) between $1550$ nm light and the $1950$
nm pump. We use an integrated waveguide structure consisting of a bent
waveguide and a straight waveguide with an entrance center-to-center
separation of $127$ $\mu$m, as shown in Fig.5. The main features of the
integrated structure are two $5.5$ $\mu$m wide mode filters, a directional
coupler used as a wavelength combiner, and a $8$ $\mu$m wide uniform straight
waveguide with $44$ mm long quasi-phase-matching (QPM) gratings for optical
frequency nonlinear mixing. $1550$ nm photons and $863$ nm photons are
combined by a $1550$ nm/$863$ nm wavelength-division multiplexer (WDM) before
they enter the straight waveguide together. $1950$ nm photons enter the bent
waveguide and pass through a $3.5$ mm long S-band before entering the
directional coupler. With a waveguide width of $5.5$ $\mu$m, an edge-to-edge
spacing of $5.5$ $\mu$m, and a length of $2.5$ mm, the directional coupler
combines the $1950$ nm pump, the $863$ nm photons and the $1550$ nm photons
into the same straight waveguide with negligible losses for both signals. The
combined photons then enter the QPM mixing region which is poled with a period
of $20$ $\mu$m. The input and output of the waveguides are fiber-pigtailed by
two polarization maintaining (PM) taper-fibers and a PM $1550$ nm fiber,
respectively. The total waveguide throughputs are $-3.5$ dB and $-4$ dB for
$1550$ nm and $863$ nm, respectively.

Figure 5: Setup for chromatic intensity interferometer for thermal light
sources.

Experiment details for coherent sources. We use a 1550 nm laser as the
$\gamma_{1}$ source, and a 1950 nm laser as the $\gamma_{3}$ source. In our
proof-of principle experiment, the $\gamma_{2}$ source is generated by up-
conversion of a separate $\gamma_{1}$ source by the $\gamma_{3}$ source.

Table 1: Key parameters for laser case Parameter | Value
---|---
Detection efficiency of UCSPD | $19.5\%$
Wavelength of $\gamma_{1}$ source | 1549.800 nm
Wavelength of $\gamma_{2}$ source | 863.344 nm
Wavelength of $\gamma_{3}$ source | 1949.157 nm
Temperature of PPLN waveguide A | $36.4\,^{\circ}C$
Temperature of PPLN waveguide B | $52.9\,^{\circ}C$
Power of pump |  $152.6$ mw

Our color erasure detector requires that
$\nu_{\gamma_{1}}+\nu_{\gamma_{3}}=\nu_{\gamma_{2}}$, and that the photons
involved be quasi-phase-matched in the PPLN waveguide. We can adjust the
temperature of the PPLN waveguide to change the refractive index so that
quasi-phase-matching can be achieved, which can be diagnosed from the count
rate of the detector. Our parameters, such as the photon wavelengths,
temperature of PPLN waveguide, and pump power, are shown in Table I. Also note
that the 1950 nm pumps for the two PPLN waveguides should be phase locked, to
enable intensity interferometry. In our experiment, the two 1950 nm pumps are
siphoned from a single source and the phase noise is weaken by insulating
fiber optical cables with cotton.

For chromatic interferometry, photons which do not participate in (partial)
frequency conversion should be taken as noise or dark counts. Only photons
with polarization parallel to the optical axis can be up-converted or down-
converted in the PPLN waveguide. Fortunately, the PPLN waveguide does not let
through photons with polarization vertical to the optical axis and thus
imperfect polarization will not contribute to the dark counts. We leverage
this feature to control the number of photons received by each detector via
polarization controllers after each light source. Similarly, photons in
higher-order spatial modes of the waveguide do not take part in the frequency-
conversion process. Therefore, the coupled fiber should be matched to the
lowest (radially symmetric) transverse mode at the input and output of PPLN
waveguide.

Experiment details for thermal sources. Our setup for chromatic interferometry
of thermal light sources is shown in Fig.6. To generate a thermal light source
in the tabletop experiment, we use an ASE source with 30 nm spectral width and
implement a 50 MHz bandwidth filter to select out one thermally populated
mode. To test the photon number distribution of our thermal source, we pass
the photons through a 50-50 beamsplitter, and record the arrival times at two
detectors placed after each out port of the beamsplitter in order to calculate
$g^{(2)}(\tau)$. As shown in Fig.7, the $g^{(2)}(\tau)$ of the thermal source
is approximately 2 within a coherence time. In this setup, the $\gamma_{1}$
and $\gamma_{2}$

Figure 6: Setup for chromatic intensity interferometer for thermal light
sources.

Figure 7: Second order coherence $g^{(2)}(\tau)$ of the thermal light source.
Table 2: Key parameters for thermal light source case
Parameter | Value
---|---
Detection efficiency of Si APD | $55\%$
Detection efficiency of UCSPD | $19.5\%$
Wavelength of $\gamma_{1}$ source | 1549.968 nm
Wavelength of $\gamma_{2}$ source | 863.396 nm
Wavelength of $\gamma_{3}$ source | 1949.157 nm
Temperature of PPLN waveguide A | $37.4\,^{\circ}C$
Temperature of PPLN waveguide B | $34.9\,^{\circ}C$
Power of pump |  $192.3$ mw
Bandwidth of optical filter |  $50$ MHz

photons are generated from the same source. In order to destroy residual phase
correlations between our $\gamma_{1}$ and $\gamma_{2}$ sources, we run the
$\gamma_{1}$ photons through a 20 km spool of fiber.

We test chromatic interferometry in four scenarios: (i) both color erasure
detectors output $\gamma_{1}$, (ii) detector $A$ outputs $\gamma_{1}$ and
detector $B$ outputs $\gamma_{2}$, (iii) detector $A$ outputs $\gamma_{2}$ and
detector $B$ outputs $\gamma_{1}$, and (iv) both detectors output
$\gamma_{2}$. The key parameters are shown in Table 2.

Calculating the second order coherence function. Let $I_{A}(t_{1})$ be the
intensity measured at detector $A$ at time $t_{1}$, and similarly let
$I_{B}(t_{2})$ be the intensity measured at detector $B$ at time $t_{2}$. The
definition of the second order coherence function $g^{(2)}(\tau)$ is:

$g^{(2)}(\tau)=\frac{\int dt\,I_{A}(t)I_{B}(t+\tau)}{\left(\int
dt\,I_{A}(t)\right)\left(\int dt\,I_{B}(t)\right)}$ (36)

where the suppressed limits of the integrals are limited by the length of our
trials. Note that in the limit of long integration time (i.e., the integrals
$\int dt$ above are essentially $\int_{-\infty}^{\infty}dt$ ), we have

$g^{(2)}(\tau=0)=|D_{1A}D_{2B}+D_{1B}D_{2A}|^{2}$ (37)

in our notation from earlier.

In our experiment, we record the arrival time of each photon detected by a
UCSPD. To analysis the data, we set a 1 ns gate time and judge the coincidence
of each count. When two counts from separate detectors fall into the same time
bin, we register a coincidence. Let $n_{\text{coincidence}}$ be the total
number of coincidence counts over the course of a run, let $n_{\text{bin}}$ be
the total number of time bins, and let $n_{A}$ and $n_{B}$ be the total counts
of detector $A$ and detector $B$, respectively. Then our calculation of
$g^{(2)}(\tau)$ amounts to

$g^{(2)}(\tau)=\frac{n_{\text{coincidence}}\cdot n_{\text{bin}}}{n_{A}\cdot
n_{B}}$ (38)

Considerations for visibility of interference. In the theoretical overview, we
wanted to choose $\theta=\chi T\sqrt{N}=\pi/4$ such that
$\cos^{2}(\theta)=\sin^{2}(\theta)=1/2$ to achieve maximum visibility of the
interference. In that analysis, we assumed that the same number of incoming
$\gamma_{1}$ and $\gamma_{2}$ photons couple to the color erasure detectors.
However, in practice, the number of $\gamma_{1}$ and $\gamma_{2}$ photons
which couple to a PPLN waveguide can be different. Suppose we are post-
selecting on $\gamma_{1}$. Then letting $N_{\gamma_{1}}$ and $N_{\gamma_{2}}$
be the number of photons which couple to a PPLN waveguide over the course of
an experimental run, to achieve maximal visibility we need to ensure that
$N_{\gamma_{1}}\cos^{2}(\theta)=N_{\gamma_{2}}\sin^{2}(\theta)$ for both color
erasure detectors $A$ and $B$.

Figure 8: Relationship between visibility and gate time. The visibilty will
not significantly increase when the gate time is less than 1 ns. Interference
will begin to vanish when the gate time is more than 200 ns.

Another parameter that influences the visibility is the gate time of the
coincidence counter, which corresponds to the time resolution of the detector.
As shown in Fig.8, the visibility decreases as the gate time increases. For
single photon detection, given the total count of each detector, increasing
the gate time increases the probability that two coincident photons come from
one light source, which will decrease the visibility of interference. On the
other hand, if the gate time is too small, then there will be fewer
coincidence counts and hence a decreased signal-to-noise ratio.


∎

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
11institutetext: Department of Mathematics,Government General Degree College,
Singur, Hooghly 712 409, West Bengal, India 22institutetext: Department of
Mathematics, Faculty of Applied Sciences, Durban University of Technology,
Durban, 4000, South Africa 33institutetext: Department of Physics, P. D.
Women’s College, Jalpaiguri 735101, India.

# Modeling Anisotropic Stars Obeying Chaplygin Equation of State

Piyali Bhare1,addr1  Megan Govendere2,addr2  Ranjan Sharmae3,addr3

(Received: date / Accepted: date)

###### Abstract

In this work we provide a framework for modeling compact stars in which the
interior matter distribution obeys a generalised Chaplygin equation of state.
The interior geometry of the stellar object is described by a spherically
symmetric line element which is simultaneously comoving and isotropic with the
exterior spacetime being vacuum. We are able to integrate the Einstein field
equations and present closed form solutions which adequately describe compact
strange star candidates like Her X-1, RX J 1856-37, PSRJ 1614-2230 and SAX
J1808.4-3658.

###### Keywords:

General Relativity, Anisotropy, Compact star, Chaplygin equation of state

††journal: Eur. Phys. J. C

##  1 Introduction

The search for exact solutions of the Einstein field equations has generated a
rich field of models describing relativistic compact objects. Since the
pioneering work of Schwarzschild who obtained the first interior solution
describing a uniform density sphere, the modeling of relativistic stars has
moved from the regime of toy models to sophisticated, realistic stellar
structures. With the discovery of pulsars, neutron stars and strange stars
there was a need to obtain relativistic analogues of Newtonian stars,
particularly when the densities of the stellar material was of the order of
$10^{14}$gm cm-3. The simplistic model of a static uniform density star has
been generalised to include the effects of pressure anisotropy, electric
charge, scalar field, dark energy and the cosmological constant, on the gross
physical properties of compact objectsrayso ; ma1 ; ma2 . Models of
relativistic fluid spheres have also been obtained within the framework of
higher order theories of gravity including the Randall-Sundrum brane scenario,
Einstein-Gauss-Bonnet gravity and Lovelock formalismssud1 ; sud2 ; abbas ;
nare . In order to close the system of equations governing the gravitational
and thermodynamical behaviour of bounded objects, various techniques were
employed by researchers working in this field of study: (i) imposition of
symmetry, (ii) adhoc assumptions of the gravitational potentials, (iii)
specific choices of the fall-off behaviour of the pressure, density or the
anisotropy, to name a fewdev1 ; dev2 ; iv1 . To construct a stellar model, a
physically motivated route, in general, is to impose an equation of state
which relates the pressure as a thermodynamical function of the density, ie.,
$p=p(\rho)$. Most of the earlier works were centered on imposing a linear
equation of state of the form $p=\alpha\rho$ where $\alpha$ is a constant.
This was later generalised to $p=\alpha\rho-\beta$, where
$\beta=\alpha\rho_{s}$ and $\rho_{s}$ is the surface densitysm1 . The
conditions were relaxed by allowing for anisotropic pressure. Note that works
in fundamental particle physics led to the MIT-bag model which hinged on an
equation of state of the form $p=\alpha\rho-4B$ where $B$ is the Bag constant.
The linear equation of state was further generalised to the quadratic equation
of state of the form $p=\alpha\rho-\beta+\sigma\rho^{2}$sifiso . One of the
first successful attempts to obtain a generalisation of the Newtonian
polytrope was achieved by Buchdahlbuch  in which he obtained a pseudo-
relativistic version of the Lane-Emden polytrope of index $5$. Herrera and
Barreto presented a general formalism to generate relativistic polytropes with
anisotropic pressure in Schwarzschild coordinatespoly . Their findings also
prompted further investigations into the origins of anisotropy, cracking in
relativistic stellar models and stabilityazam . In order to fine-tune these
models with observations, some researchers employed a mixed polytrope equation
of state in which two or more species of particles made up the stellar fluid.
The inclusion of charge within the stellar core led to a plethora of static
stellar models in which the role of the electromagnetic field on the
stability, mass-radius ratio and redshift was demonstratedr1 ; ra1 ; ra2 .

Issues surrounding the black hole horizon paradox necessitated the search for
alternative models of black holes free of horizons. The gravastar model was
first proposed by Mazur and Mottola mazur  which sought to address many of the
problems encountered during the final stages of gravitational collapse.
Dynamically the model hinged on the phenomenon that during the latter epoch of
gravitational collapse spacetime itself would undergo phase transitions which
would halt collapse. The emerging picture of a gravastar was that of a layered
composite: a de Sitter interior filled with constant positive (dark) energy
density $\rho$ featuring an isotropic negative pressure $\rho=-p>0$. This
layer is then connected via three intermediate layers to an exterior vacuum
Schwarzschild solution. The intermediate relatively thin shell is composed of
stiff matter ($p=\rho$). Stability of the composite profile is achieved by
utilising two infinitesimally-thin shells endowed with surface densities
$\sigma_{\pm}$ and surface tensions $\vartheta_{\pm}$. An interesting model
was proposed by Usmani et alra2  in which they generalised the Mazur-Mottola
gravastar picture to include charge. In addition, the interior of the
gravastar admitted conformal motion. They were able to show that charged
interior de Sitter void must generate the gravitational mass. This mass is
accountable for the attractive force that counter-balances the electromagnetic
repulsion due to the presence of charge during the collapse processrahaman1 .
In a more recent model, Banerjee et albrig  presented a Braneworld
generalisation of a gravastar admitting conformal motion. Motivated by the
existence of dark energy, Lobo and coworkerslobo1  have proposed stellar
models, the so-called ‘dark stars’ in which the equation of state of is of the
form $p=\alpha\rho$ in which $-1<\alpha<-1/3$. It has been proposed that in
the phantom regime ($\alpha=-1$), the extremely high pressures may invoke a
topological change rendering the dark energy star to a wormhole. An
interesting proposal regarding dark energy and dark matter is treating them as
different manifestations of a single entity. This proposal leads to the
Chaplygin gas model in which the equation of state derives from string theory.
Various applications of the Chaplygin gas model have been pursued in order to
account for cosmological observations such as acceleration of the cosmic fluid
and structure formation. The Chaplygin equation of state has been subsequently
modified to a more generalized Chaplygin gas equation of state. The
generalised Chaplygin equation of state has been employed to model dark stars
which are remnants of continued gravitational collapse. The idea here is that
the dark energy provides sufficient repulsion to halt collapse leading to
stable bounded configurations free of horizons and singularitiesp1 ; p2 .

This paper is structured as follows: In section $2$ we introduce the field
equations necessary for the modeling of a spherically symmetric star within
the framework of general relativity. In section $3$. we present a particular
solution describing the interior of the star in which the matter content obeys
a generalised Chaplygin equation of state. The junction conditions required
for the smooth matching of the interior spacetime to the exterior
Schwarzschild solution are worked out in section $4$. A detailed physical
analysis of the geometrical and thermodynamical behaviour of our model is
presented in section $5$. We discuss the stability, energy conditions and
mass-radius relation in sections $6$, $7$ and $8$, respectively. We conclude
with a discussion of our results in section $9$.

##  2 Spherically symmetric spacetime

We consider a model which represents a static spherically symmetric
anisotropic fluid configuration obeying a generalised Chaplygin equation of
state. The interior spacetime is described by a spherically symmetric line
element which is simultaneously comoving and isotropic

$ds^{2}=-A^{2}(r)dt^{2}+B^{2}(r)\left[dr^{2}+r^{2}d\Omega^{2}\right],$ (1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}{\theta}d\phi^{2}$ and the metric
functions, $A(r)$ and $B(r)$ are yet to be determined. For our model the
energy-momentum tensor for the stellar fluid is

$T_{ab}={\mbox{d}iag}\left(-\rho,p_{r},p_{t},p_{t}\right),$ (2)

where $\rho$, $p_{r}$ and $p_{t}$ are the proper energy density, radial
pressure and tangential pressure, respectively. The fluid four–velocity ${\bf
u}$ is comoving and is given by

$u^{a}=\displaystyle\frac{1}{A}\delta^{a}_{0}\,.$ (3)

The Einstein field equations for the line element (1) are

$8\pi\rho=-\frac{1}{B^{2}}\left(2\frac{B^{\prime\prime}}{B}-\frac{{B^{\prime}}^{2}}{B^{2}}+\frac{4}{r}\frac{B^{\prime}}{B}\right),$
(4) $8\pi
p_{r}=\frac{1}{B^{2}}\left[\frac{{B^{\prime}}^{2}}{B^{2}}+2\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}+\frac{2}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)\right],$
(5) $8\pi
p_{t}=\frac{1}{B^{2}}\left[\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}-\frac{{B^{\prime}}^{2}}{B^{2}}+\frac{1}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)\right],$
(6)

where primes denote differentiation with respect to the radial coordinate $r$.
We have utilized geometrized units in deriving the above system of equations
in which the coupling constant and the speed of light are taken to be unity.
The mass of the spherical object is given by

$m(r)=4\pi\int_{0}^{r}\omega^{2}\rho(\omega)d\omega,$ (7)

where $\omega$ is an integration variable. In order to close the system of
equations, we assume that the interior matter distribution obeys a generalised
Chaplygin equation of state of the form

$p_{r}=H\rho-\frac{K}{\rho}$ (8)

where $H$ and $K$ are positive constants. Substituting (4) and (5) in (8) we
obtain

$\frac{A^{\prime}}{A}=\frac{1}{2}\left(\frac{B^{\prime}}{B}+\frac{1}{r}\right)^{-1}[G(r)-F(r)],$
(9)

where

$G(r)=\frac{(8\pi)^{2}KB^{4}}{2\frac{B^{\prime\prime}}{B}-\frac{{B^{\prime}}^{2}}{B^{2}}+\frac{4}{r}\frac{B^{\prime}}{B}},$
(10)
$F(r)=2H\frac{B^{\prime\prime}}{B}+(1-H)\left(\frac{B^{\prime}}{B}\right)^{2}+(1+2H)\frac{2}{r}\frac{B^{\prime}}{B}.$
(11)

On integrating (9) we obtain

$A=d\exp\left[\frac{1}{2}\int H(r)dr\right],$ (12)

where

$H(r)=\left(\frac{B^{\prime}}{B}+\frac{1}{r}\right)^{-1}[G(r)-F(r)],$ (13)

and $d$ is a constant of integration. Therefore, the line element (1) can now
be written as

$ds^{2}=-d^{2}\exp\left[\int
H(r)dr\right]dt^{2}+B^{2}\left[dr^{2}+r^{2}d\Omega^{2}\right],$ (14)

where $H(r)$ is given in (13). Hence, any solution describing a static
spherically symmetric anisotropic matter distribution obeying a generalised
Chaplygin equation of state in isotropic coordinates can be easily determined
by a single generating function $B(r)$.

##  3 Generating solutions

In order to close the system of equations several choices for $B(r)$ can be
made. It is interesting to note that the choice of the metric potential $B(r)$
determines the gravitational and thermodynamical behaviour of the model. Hence
the choice of $B(r)$ must satisfy all the requirements for a realistic stellar
model. Recent work by Naidu and Govenderng2016  have shown that the end-state
of gravitational collapse resulting from a dynamically unstable static core is
‘sensitive’ to the choice of the initial metric functions. They show that for
the same $B(r)$ but with two distinct initially static cores; (i) vanishing
radial pressure within the static configuration and (ii) uniform density
interior, the final outcome of dissipative collapse leads to very different
temperature profiles. Following Govender and Thirukkaneshgov1  we utilise the
physically motivated choice for $B(r)$ as

$B(r)=\frac{a}{\sqrt{1+br^{2}}}$ (15)

where $a$ and $b$ are constants. One can easily verify that the gravitational
potential $B$ in (15) satisfies the regularity conditions, $B(0)$ = constant
and $B^{\prime}(r)=1$ at the origin. The same expression of $B(r)$ was
previously utilized to model compact objects in curvature coordinates by
Schwarzschild sch , Einstein ein  and de Sitter des  and more recently in
comoving coordinates by Govender and Thirukkanesh gov1  and Thirukkanesh et
al. thir .
With this choice of $B(r)$, we obtain from (12)

$A(r)=d\exp\left[\left\\{\frac{1+H}{4}-\frac{16a^{4}K\pi^{2}}{b^{2}}\right\\}(1+br^{2})\right](1+br^{2})^{\frac{1+5H}{4}}(6+br^{2})^{\frac{80a^{4}K\pi^{2}}{b^{2}}},$

where $d$ is a constant of integration.
Subsequently, the field equations yield

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{b(6+br^{2})}{8\pi
a^{2}(1+br^{2})},$ (16) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{bH}{8\pi
a^{2}}\left(\frac{6+br^{2}}{1+br^{2}}\right)-\frac{8\pi
a^{2}K}{b}\left(\frac{1+br^{2}}{6+br^{2}}\right),$ (17) $\displaystyle p_{t}$
$\displaystyle=$
$\displaystyle\frac{C_{1}+C_{2}r^{2}+C_{3}r^{4}+C_{4}r^{6}+C_{5}r^{8}C_{6}r^{10}}{32\pi
a^{2}b^{2}(6+br^{2})^{2}(1+br^{2})},$ (18)

where, $C_{i}^{\prime}s$ (i=1,2,…6) are given by

$C_{1}=96b(9b^{2}H-16a^{4}K\pi^{2})$
$C_{2}=16[9b^{4}\\{3+H(10+9H)\\}-96a^{4}b^{2}(5+3H)K\pi^{2}+256a^{8}K^{2}\pi^{4}]$
$C_{3}=8b[3b^{4}\\{18+H(47+36H)\\}-32a^{4}b^{2}(47+42H)K\pi^{2}+2048a^{8}K^{2}\pi^{4}]$
$C_{4}=8b^{2}[b^{4}\\{18+H(43+27H)\\}-16a^{4}b^{2}(57+61H)K\pi^{2}+3072a^{8}K^{2}\pi^{4}]$
$C_{5}=4b^{3}\\{b^{2}(1+H)-64a^{4}K\pi^{2}\\}\\{b^{2}(5+6H)-64a^{4}K\pi^{2}\\}$
$C_{6}=b^{4}\\{b^{2}(1+H)-64a^{4}K\pi^{2}\\}^{2}.$

We define the anisotropic factor as

$\Delta=p_{t}-p_{r}$ (19)

which is repulsive in nature if $\Delta>0$ and attractive if $\Delta<0$.

##  4 Matching Conditions

In this section we match the interior spacetime $({\cal M}_{-})$ to the
exterior spacetime $({\cal M}_{+})$ described by the exterior Schwarzschild
solution in comoving isotropic coordinatesbonnor1

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\left(1-\frac{M}{2r}\right)^{2}}{\left(1+\frac{M}{2r}\right)^{2}}dt^{2}+$
(20)
$\displaystyle\left(1+\frac{M}{2r}\right)^{4}[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})],$

where $M$ is the mass within a sphere of radius $R$. Matching of interior
metric (1) and exterior metric (20) at the boundary $r=R$ leads to the
constraints

$\displaystyle A(R)$ $\displaystyle=$
$\displaystyle\frac{\left(1-\frac{M}{2R}\right)}{\left(1+\frac{M}{2R}\right)},$
(21) $\displaystyle B(R)$ $\displaystyle=$
$\displaystyle\left(1+\frac{M}{2R}\right)^{2},$ (22)

where

$M=m(R)=\frac{bR^{3}+15\left(R-\frac{\arctan[\sqrt{b}R]}{\sqrt{b}}\right)}{6a^{2}}.$
(23)

The condition (21) imposes the following restriction on the constant of
integration

$\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{12a^{2}R-\left[bR^{3}+15\left(R-\frac{\arctan[\sqrt{b}R]}{\sqrt{b}}\right)\right]}{12a^{2}R+\left[bR^{3}+15\left(R-\frac{\arctan[\sqrt{b}R]}{\sqrt{b}}\right)\right]}$
$\displaystyle\times\exp\left[\left\\{\frac{16a^{4}K\pi^{2}}{b^{2}}-\frac{1+H}{4}\right\\}(1+bR^{2})\right](1+bR^{2})^{-\frac{1+5H}{4}}(6+bR^{2})^{-\frac{80a^{4}K\pi^{2}}{b^{2}}}$

The condition (22) implies

$\frac{a}{\sqrt{1+bR^{2}}}=\left[1+\frac{bR^{3}+15\left(R-\frac{\arctan[\sqrt{b}R]}{\sqrt{b}}\right)}{12a^{2}R}\right]^{2},$
(25)

which imposes a restriction on the parameters $a$ and $b$ which can be
determined if we specify the radius of the sphere.

To examine the behabiour of the model parameters like matter density, radial
and transverse pressure etc. we assume $a=1.3997$, $b=0.009$, and $H=0.294$.
By using the matching conditions together with $p_{r}(r=R)=0$, we obtain the
constant $K=2.04558\times 10^{-7}$ for a star of radius $6.7~{}$km. The mass
of the stellar configuration turns out to be $0.789~{}M_{\odot}$ which is very
close to the observed mass of the strange star candidate Her X-1rawls .

##  5 Physical Analysis

We are now in a position to discuss the the physical features of the model
generated in the preceding section. In order to describe a realistic stellar
structure our model must satisfy the following physical requirements :

  1. 1.

Regularity of the gravitational potentials at the origin:

In our model,
$A^{2}(0)=d^{2}6^{\frac{160a^{4}K\pi^{2}}{b^{2}}}e^{[b^{2}(1+{H})-64a^{4}K\pi^{2}]/2b^{2}},~{}B^{2}(0)=a^{2}$
which are constants and $(A^{2}(r))^{\prime}=(B^{2}(r))^{\prime}=0$ at the
origin $r=0$, which indicates that the gravitational potentials are regular at
the origin.

  2. 2.

Positive definiteness of the energy density and pressure at the centre:

Since $\rho(0)=\frac{3b}{4\pi a^{2}}$, the energy density is positive and
regular at the origin. We also have
$p_{r}(0)=\frac{9b^{2}H-16\pi^{2}a^{4}K}{12\pi a^{2}b}$. To ensure that the
radial pressure is positive at the center we must have
$\frac{K}{H}<\left(\frac{3b}{4\pi a^{2}}\right)^{2}$.

Moreover, $\displaystyle\frac{d\rho}{dr}=-\frac{5b^{2}r}{4\pi
a^{2}(1+br^{2})^{2}}<0$ i.e., the energy density is a decreasing function of
$r$.

We also note that $\displaystyle\frac{dp_{r}}{dr}=-\frac{80a^{2}K\pi
r}{(6+br^{2})^{2}}-\frac{5b^{2}Hr}{4\pi a^{2}(1+br^{2})^{2}}$, which implies
that $p_{r}$ is a decreasing function of $r$.

  3. 3.

Continuity of the extrinsic curvature across the matching hyper-surface,
$K^{-}_{ij}=K^{+}_{ij}$:

Continuity of the extrinsic curvature across the matching hyper-surface, $r=R$
yields

$(p_{r})_{(r=R)}=0;$ (26)

which gives

$R=\sqrt{\frac{2(32a^{4}K\pi^{2}+20a^{2}b\pi\sqrt{HK}-3b^{2}H)}{b(b^{2}H-64a^{4}K\pi^{2})}},$

which is finite for appropriate choice of parameters $a,b,H$ and $K$.

  4. 4.

Ratio of trace of stress tensor to energy density $(p_{r}+2p_{t})/\rho$:

Fulfillment of the requirement that the ratio of trace of stress tensor to
energy density $(p_{r}+2p_{t})/\rho$ should decrease radially outward is shown
graphically in Fig. (4).

  5. 5.

Velocity of sound:

For causality to be obeyed the radial and transverse velocities of sound
should be in between [$0,~{}1$]. The radial velocity $(v_{sr}^{2})$ and
transverse velocity $(v_{st}^{2})$ of sound can be obtained as

$v_{sr}^{2}=\frac{dp_{r}}{d\rho},$ (27) $v_{st}^{2}=\frac{dp_{t}}{d\rho}.$
(28)

Due to the complexity, we illustrate the causality conditions with the help of
graphical representations. Fig. (6) and Fig. (7)clearly show that
$0<v_{sr}^{2}\leq 1$ and $0<v_{st}^{2}\leq 1$ everywhere within the stellar
configuration.

  6. 6.

Stability:

Following Heintzmann and Hillebrandthein , a model of anisotropic compact star
will be stable if $\Gamma>\frac{4}{3}$ everywhere within the stellar interior
where the adiabatic index $\Gamma$ is defined as

$\Gamma=\frac{\rho+p_{r}}{p_{r}}\frac{dp_{r}}{d\rho}.$ (29)

Fig. (8) shows that values of the adiabatic index which clearly indicates that
the particular configuration developed in this paper is stable.

  7. 7.

Energy conditions:

A realistic star should satisfy the energy conditions namely, the Weak Energy
Condition (WEC), Null Energy Condition (NEC) and Strong Energy Condition (SEC)
as given below:

$(i)\,\,\mbox{Null energy condition (NEC)}:\rho\geq 0,$ (30)
$(ii)\,\,\mbox{Weak energy condition (WEC)}:\rho-p_{r}\geq 0,\rho-p_{t}\geq
0,$ (31) $(iii)\,\,\mbox{Strong energy condition (SEC)}:\rho-p_{r}-2p_{t}\geq
0.$ (32)

For the specific stellar configuration developed here, validity of the
inequalities (30)-(32) have been shown with the help of graphical
representations in Fig. (10).

Figure 1: Matter density $\rho$ is plotted against the radial distance $r$
inside the fluid for a particular configuration with $a=1.3997$, $b=0.009$,
$H=0.294$ and $K=2.04558\times 10^{-7}$

.

Figure 2: Radial pressure $p_{r}$ is plotted against the radial distance $r$
inside the stellar interior by employing the same values of the constants as
mentioned in Fig. (1). Figure 3: Transverse pressure $p_{t}$ is plotted
against the radial distance $r$ inside the stellar interior by employing the
same values of the constants as mentioned in Fig. (1). Figure 4:
$\frac{p_{r}+2p_{t}}{rho}$ is plotted against the radial distance $r$ inside
the stellar interior by employing the same values of the constants as
mentioned in Fig. (1). Figure 5: Anisotropic factor is plotted against the
radial distance $r$ inside the stellar interior by employing the same values
of the constants as mentioned in Fig. (1). Figure 6: Square of radial velocity
of sound is plotted against the radial distance $r$ inside the stellar
interior by employing the same values of the constants as mentioned in Fig.
(1). Figure 7: Square of transverse velocity of sound is plotted against the
radial distance $r$ inside the stellar interior by employing the same values
of the constants as mentioned in Fig. (1). Figure 8: The adiabatic index
$\Gamma$ is plotted against the radial distance $r$ inside the stellar
interior by employing the same values of the constants as mentioned in Fig.
(1). Figure 9: Mass function is is plotted against the radial distance $r$
inside the stellar interior by employing the same values of the constants as
mentioned in Fig. (1). Figure 10: Weak, Null and Strong energy conditions are
plotted against the radial distance $r$ inside the stellar interior by
employing the same values of the constants as mentioned in Fig. (1).

##  6 Mass-radius relation

The mass function in our model is obtained from (9) as

$m(r)=\frac{br^{3}+15\left(r-\frac{\arctan[\sqrt{b}r]}{\sqrt{b}}\right)}{6a^{2}}$
(33)

The profile of the mass function is shown in Fig. (9). Since at $r\rightarrow
0$ we have $m(r)\rightarrow 0$, it implies that the mass function is free from
any central singularity.

Buchdahlbuchdahl59  obtained an upper bound on the mass to radius ratio i.e.,
compactness $u$ of a relativistic star of a compact star such that $M/r<4/9$.
In our model, we have

$u(r)=\frac{br^{3}+15\left(r-\frac{\arctan[\sqrt{b}r]}{\sqrt{b}}\right)}{6a^{2}r}.$
(34)

The compactness $u$ for different compact star models are given in Table 2.
The table shows that compactness of configurations are wihin the Buchdahl
limitbuchdahl59 .

We have also determined the surface redshift using the formula

$1+z_{s}=\left(1-\frac{2m}{R}\right)^{-\frac{1}{2}},$

which for our model turns out to be

$z_{s}=\left(1-\frac{bR^{3}+15\left(r-\frac{\arctan[\sqrt{b}R]}{\sqrt{b}}\right)}{3a^{2}R}\right)^{-\frac{1}{2}}-1.$
(35)

The values of the surface redshift parameter for different stellar
configurations are given in Table 1. For an isotropic star, in the absence of
a cosmological constant, Buchdahlbuchdahl59  and Straumann stra  have shown
that $z_{s}\leq 2$. Böhmer and Harkobh  showed that for an anisotropic star,
in the presence of a cosmological constant, the surface redshift can take a
much higher value $z_{s}\leq 5$. The restriction was subsequently modified by
Ivanov ivanov  who showed that the maximum permissible value could be as high
as $z_{s}=5.211$. In our case, we have $z_{s}\leq 1$ for different compact
star models developed in this paper.

Table 1: Values of the constants $a$, $b$, $H$ and $K$ for different compact star models. Compact Star |  $R~{}$(km) | a | b | H | K |  $M~{}$($M_{\odot}$)
---|---|---|---|---|---|---
Her X - 1 | 6.7 | 1.3997 | 0.009 | 0.294 | 0.00012910 | 0.789
RX J 1856-37 | 6.006 | 1.6859 | 0.024 | 0.21 | 0.00020280 | 0.904
PSRJ 1614-2230 | 11.3664 | 2.0952 | 0.0138 | 0.29 | 0.000022416 | 1.86
SAX J1808.4-3658 | 7.07 | 2.1015 | 0.036 | 0.29 | 0.000149572 | 1.157
Table 2: Values of physical parameters for different compact star models.  Compact Star | central density | surface density | central pressure | $u$ | $z_{s}$
---|---|---|---|---|---
| gm cm-3 | gm cm-3 | dyne cm-2 |  |
Her X - 1 | $1.4805\times 10^{15}$ | $1.1255\times 10^{15}$ | $1.6535\times 10^{35}$ | 0.1737 | 0.2379
RX J 1856-37 | $2.7215\times 10^{15}$ | $1.6691\times 10^{15}$ | $3.2088\times 10^{35}$ | 0.2219 | 0.3409
PSRJ 1614-2230 | $1.0131\times 10^{15}$ | $0.4722\times 10^{15}$ | $2.0698\times 10^{35}$ | 0.2413 | 0.3905
SAX J1808.4-3658 | $2.6271\times 10^{15}$ | $1.2199\times 10^{15}$ | $5.3784\times 10^{35}$ | 0.2414 | 0.3906

##  7 Discussions

In this paper we have presented a new model of a compact star in isotropic
coordinates which is free from central singularity with the exterior being the
vacuum Schwarzschild spacetime. To solve the Einstein field equations we have
employed the modified Chaplygin equation of state which is inspired by the
current observation of the expanding universe and its connection to the
existence of dark energy. By considering the observed radius of the compact
star Her X-1 as an input parameter (we have assumed $R=6.7~{}$km), we have
analyzed physical viability of our model. We note that the metric coefficients
are free from any singularity. The variations of $\rho$,
$\frac{p_{r}+2p_{t}}{\rho}$ ,$p_{r},~{}p_{t}$ are plotted in Fig. (1), (4),
(2) and (3), respectively which clearly show that matter density, radial and
transverse pressure are positive inside the stellar interior and they are
monotonically decreasing functions of radial coordinate. At the boundary of
the star the matter density and transverse pressure are non-negative and the
radial pressure vanishes as expected. From the plot of
$\frac{p_{r}+2p_{t}}{\rho}$ (Fig. (4)) we note that it is non-zero and
monotonically decreasing from the center towards the stellar surface. The
anisotropic factor $\Delta=p_{t}-p_{r}$ is plotted against $r$ in Fig. (5).
Since $\Delta>0$, the anisotropic factor is repulsive in nature in our model
which is a desirable feature of a compact stargm . Moreover, at the center of
the star $\Delta$ vanishes which is also an essential feature of a realistic
star. In order to investigate the relevance of our model in the study of
compact stars, we have considered a number of compact stars, namely Her X-1,
RX J 1856-37, PSRJ 1614-2230 and SAX J1808.4-3658 and showed that for the
estimated radii the star’s masses determined from our model are very close to
the observed masses (see Ref. maurya ; tj ). This leads naturally to the
proposition that the solution obtained in this paper can be used as a viable
model for describing ultra-compact stars.

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