# Astrophysically sourced quantum coherent photonic signals

Arjun Berera<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Edinburgh, Edinburgh, EH9 3FD, United Kingdom  Jaime Calderón-Figueroa
<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Edinburgh, Edinburgh, EH9 3FD, United Kingdom  Departamento de Física, Escuela
Politécnica Nacional, Quito, 170143, Ecuador  Liang Chen<EMAIL_ADDRESS>School
of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for
Advanced Study, UCAS, Hangzhou 310024 ,China  University of Chinese Academy of
Sciences, 100190 Beijing, China  Thomas W. Kephart<EMAIL_ADDRESS>Department of Physics and Astronomy, Vanderbilt University, Nashville, TN
37235, USA

###### Abstract

Stimulated emission is shown to be robust in stars. Through Bose enhancement
this produces quantum states of aligned, monochromatic photons somewhat
similar to a laser. The probability of creating such states is computed. We
show that from the solar corona such quantum states would propagate outside of
the solar region and through the Solar System without decoherence. For a
$1{\rm m}^{2}$ test detector at the distance of the Earth from the Sun, we
estimate rates of such quantum states in the few per second thus potentially
detectable. The same process should lead to such quantum states also arriving
from stars at interstellar distances.

##  I Introduction

Recently it was observed by one of us [1] that photons in certain frequency
bands can traverse interstellar distances, without their initial quantum state
decohering. This fundamental point has potential far reaching consequences for
astronomical observation. That paper already noted applications both for
interstellar quantum communication, with further work along that line done in
[2], and possible detection of quantum coherent signals from astrophysical
sources. By quantum coherence of photons we mean very simply N-photons in some
specified quantum state, which in this paper in particular means they are all
identical, so with the same momemtum $\vec{k}$ and polarization $s$. This,
which we will call a N-identical photons state, is suggested as a possible
type of quantum coherent photonic signal. In this paper we identify a specific
mechanism in stars and their atmosphere that can create such states and
examine the possibility for observing these quantum coherent photon signals at
distances far away from emission.

The elementary process creating a N-identical photons state is stimulated
emission. When a photon in state $\ket{1_{\vec{k},s}}$ impinges upon an atom
that is in an excited state of the same energy as itself, an emission from the
atom is possible that then leads to two photons in the same state,
$\ket{2_{\vec{k},s}}$. This two–photon quantum state will be identified as a
two–identical photons state. The process can repeat by impinging these photons
on another atom to create a three-identical photons state with a Bose
enhancing factor and so forth. We observe that in the stellar environment
there should be a ubiquitous production of such N-identical photons states,
especially the simplest two photon process.

Deep enough below the surface of the star, these states will quickly decohere
through interactions. However if such a stimulated emission event occurred
above the surface in the stellar atmosphere, then it is possible these photons
could escape with their quantum state intact.

In this paper we will examine the details of such stimulated emission events
inside the stellar environment. We will then examine the mean free path (MFP)
of photons both in the stellar environment as well as in the interplanetary
region of the Solar System and assess the flux of such photonic quantum states
that could be measured. We will then discuss ways in which such states could
be detected both directly and indirectly.

Here we identify all potential sources of decoherence and show that these
effects are small. We will take a first principles approach and look directly
at the interactions of the photons with all particles in the medium and show
the mean free paths from such interactions are much longer than the distances
the photons are required to traverse, often orders of magnitudes longer. Of
course there may be some residual decoherence and before attempting
experiments, this would need to be understood. Standard methods such as
density matrix, master equation etc., could be employed to quantify these
higher order effects. The purpose of this first paper is to highlight the
dominant physical effects and establish that to zeroth order decohering
effects are negligible.

##  II Mechanism of stimulated emission

Let us start with a brief review of the mechanism of stimulated emission. A
more detailed discussion of it can be found in textbooks such as [3]. To
calculate the transition probability of stimulated photon emission, using
Fermi’s Golden Rule gives the matrix element
$\bra{(N+1)_{\vec{k},s}}\otimes\bra{\Psi_{f}}H_{I}\ket{\Psi_{i}}\otimes\ket{N_{\vec{k},s}}$,
where $\ket{\Psi_{i}}$ and $\ket{\Psi_{f}}$ are initial and final states of
the atom, respectively, $\ket{N_{\vec{k},s}}$ is the $N$-photon state in Fock
space with momentum $\vec{k}$ and polarization $s$, and $H_{I}$ is the
interaction Hamiltonian between the electromagnetic field and atom. Such a
matrix element will lead to a quantum mechanical Bose enhancement factor such
as
$\bra{(N+1)_{\vec{k},s}}\hat{a}^{\dagger}_{\vec{k},s}\ket{N_{\vec{k},s}}=\sqrt{N+1}~{},$
which means that for a state with a high number of photons, there is an
enhanced chance of stimulated emission of one more photon in the same state.

A semiclassical approach such as the one in [4] leads to an expression for the
cross section of the stimulated emission off a single atom of the form

$\sigma=\frac{2\pi^{2}}{3n\epsilon_{0}ch}|\mu|^{2}\nu g(\nu-\nu_{0})~{}.$ (1)

Here, $n$ is the refractive index of the medium, $\mu$ is the magnitude of the
time-independent electric dipole moment between the initial and final wave
functions of the atom, $\nu$ is the frequency of the photon, and
$g(\nu-\nu_{0})$ is the line shape function with width $\Delta\nu_{0}$. In our
discussion, we will choose the Lorentzian line shape function, though note
that the results we find are comparable to those using other line shape
functions such as Gaussian.

The stimulated emission process needs the electric dipole moment, which can be
extracted from the measured spontaneous emission rate. A textbook expression
for the spontaneous emission rate reads [4]

$A=\frac{16\pi^{3}\nu_{0}^{3}n|\mu|^{2}}{3\epsilon_{0}c^{3}h}~{},$ (2)

with experimental values readily available on the NIST Atomic Spectra Database
[5]. The dipole moment $\mu$ appears both in the cross section (1) of
stimulated emission and in the rate of spontaneous emission. Thus, we can
express it in terms of the measured rates $A$ without an explicit calculation.
This allows us to write Eq. (1) now as $\sigma=\frac{\lambda^{2}}{8\pi
n^{2}}g(\nu-\nu_{0})A.$ Substituting the peak value at $\nu_{0}$ of the
Lorentzian line shape function, $g(\nu_{0}-\nu_{0})=2/(\pi\Delta\nu)\,,$ into
the expression for the cross section leads to the stimulated emission cross
section at wavelength $\lambda_{0}$ of $\sigma_{0}=(\frac{\lambda_{0}}{2\pi
n})^{2}{A\over\Delta\nu}.$ As can be seen, the stimulated emission cross
section depends on four quantities: the wavelength $\lambda_{0}$, refraction
index $n$ of the material, frequency width of the emission $\Delta\nu$, and
rate of spontaneous emission $A$. Using the NIST data for the numerical values
of $A$ allows us to calculate cross sections of any stimulated emission line
from any ions. Reexpressing the frequency width $\Delta\nu$ to wavelength
width $\Delta\lambda$, we have

$\sigma_{0}=({\lambda_{0}\over 2\pi n})^{2}({A\lambda_{0}\over
c})({\lambda_{0}\over\Delta\lambda})~{}.$ (3)

Here $A\lambda_{0}/c$ gives the ratio between the wavelength of the photon and
the length the photon can travel during the time scale of spontaneous
emission. Moreover $\lambda_{0}/\Delta\lambda$ gives the ratio between the
wavelength of the photon and the line width.

##  III N-identical photons state probability from stimulated emission

Let’s assume that we have an initial one-photon state $\ket{1}$ at one side of
a layer of medium of thickness $L$, filled with atoms of type $a$, with number
density $n_{a}$. This initial single photon state $\ket{1}$ travels through
the layer from $x=0$ to $x=L$. By using coordinate label $x$, we are not
trying to localize the photon but only want to find the probability of a
$\ket{2}$ state, a $\ket{3}$ state or in general, an $\ket{N}$ state created
through stimulated emission, if any measurement is performed at $x$.

Denote by $P(x,N)$ the probability of finding an $\ket{N}$ state at distance
$x$, then the probability of finding an $\ket{N+1}$ state at distance $x+dx$
becomes $P(x+dx,N+1)$. The relation between $P(x+dx,N+1)$ and $P(x,N)$ is

$\displaystyle P(x+dx,N+1)$ (4) $\displaystyle=$ $\displaystyle
P(x,N+1)\left[1-(N+1)n_{a}\sigma_{0}~{}dx\right]+P(x,N)(Nn_{a}\sigma_{0}~{}dx)~{}.$

On infinitesimal scale $dx$, finding $\ket{N+1}$ at $x+dx$ includes two
mutually exclusive events. The first event is finding $\ket{N+1}$ at $x$ which
carries probability $P(x,N+1)$ and then the $\ket{N+1}$ state does not induce
stimulated emission from $x$ to $x+dx$, which carries probability
$1-n_{a}\sigma_{0}(N+1)dx$. As we discussed above, the probability of a single
stimulated emission event happening after a single photon traveling the
distance $dx$ is $n_{a}\sigma_{0}dx$. Both quantum and semi-classical theory
indicate that the probability of $\ket{N}$ induces a stimulated emission that
is enhanced by a factor of $N$ compared to that of single photon state
$\ket{1}$. Thus, the probability of an $\ket{N+1}$ state inducing a stimulated
emission event from $x$ to $x+dx$ becomes $(N+1)n_{a}\sigma_{0}dx$, which
needs to be subtracted to give the probability of not inducing stimulated
emission. The second event is finding $\ket{N}$ at $x$ which carries
probability $P(x,N)$ and then the $\ket{N}$ state does induce stimulated
emission from $x$ to $x+dx$, which carries probability $Nn_{a}\sigma_{0}dx$.

After solving equation (4), the probability of finding an $\ket{N}$ state at
the other side of the layer is given by

$P(L,N)=e^{-n_{a}\sigma_{0}L}\left(1-e^{-n_{a}\sigma_{0}L}\right)^{N-1}\;.$
(5)

As a sanity check, one can sum $P(L,N)$ over all $N$ from $1$ to $\infty$ to
show that this expression is normalized to $1$, independent of $L$.

##  IV Rate estimation

Next, we estimate the rate of N-identical photons states being created at the
solar corona, to then determine the number of such states that could be
detectable, ideally in space, on a $1\ {\rm m}^{2}$ area near the Earth. We
shall consider photons in the optical, UV and X-ray regions.

For the optical, we obtain the rate of $\ket{1}$ states from experimental data
from [6]. Then, we calculate the MFP of stimulated emission, to finally
compute $P(L,2)$ using the above formula. For example, Ref. [6] tracks the
history of the intensity of the 530.3 nm line, which goes through yearly
cycles. However, going to the lower end of the reported intensities, we can
still have around $2.4\times 10^{12}$ photons arriving on an area of 1 m2 per
solid angle per second. (We conservatively picked a solar cycle of low corona
index number plotted in [6] and converted the light intensity to photon number
rate.) Hence, $N_{1}^{530.3}\sim 2.4\times 10^{12}$ would conservatively count
the rate of the $\ket{1}$ state.

Let us now determine the number of $\ket{2}$ states created from stimulated
emission. The particle density of the corona is at its lowest at the corona
holes, therefore we conservatively take the particle number density $10^{9}$
m-3 [7] of the corona holes as the density of the entire corona. Schmelz [8]
et al. gives an abundance of iron in the corona of $7.08\times 10^{-5}$ with
respect to hydrogen. This yields an iron number density of $7.08\times 10^{4}$
m-3. The ionization fraction of Fe XIV-530.3 nm depends on temperature and
peaks at around 0.2 for a temperature $T=2\times 10^{6}$ K based on [9].
Conservatively, we choose the density of Fe XIV at the excited level of 530.3
nm to be $1.0\times 10^{4}$ m-3.

Given the cross section of stimulated emission $7.585\times 10^{-25}\text{
m}^{2}~{},$ an upper limit of the MFP of stimulated emission of 530.3 nm
photons in the corona is $\ell_{mfp}^{530.3}=1.32\times 10^{20}\text{ m}~{}.$
Then, if, based on above, a number $N_{1}^{530.3}=2.4\times 10^{12}$ of 530.3
nm $\ket{1}$ states are emitted on the 1 m2 area per solid angle every second,
the number of $\ket{2}$ states emitted on the same area per solid angle every
second can be computed through (5) to be $N_{2}^{530.3}\sim~{}145.6~{}.$

Similar calculations show that states equal or higher than $\ket{3}$ are
negligibly produced. Although we have conservatively estimated the following
quantities including the number of $\ket{1}$ states of $530.3$ nm, the overall
corona density, the relative abundance of iron in the corona, and the fraction
of Fe XIV ions of the specific energy level, there are over 100 $\ket{2}$
states per second per solid angle emitted onto the 1 m2 area near the Earth.

Turning now to the UV and X-ray emission, we will first estimate the rate of
$\ket{1}$ states. Given the available experimental data, we shall use a
different approach to compute the densities of the excited ions that will lead
to the photons at those energies. For this, the following formula for the
intensity of a line will come in handy,

$I_{\rm
line}=\int_{0}^{\infty}I(\lambda)d\lambda=\frac{1}{4\pi}\frac{hc}{\lambda_{0}}AN_{k}L\;,$
(6)

where $A$ is the atomic transition probability we encountered previously, and
$N_{k}$ is the number per unit volume (number density) of excited atoms in the
upper (initial) level $k$. Based on the measurements in [10], the thickness of
the corona is about $8$ million kilometers, which we will take as the value of
$L$. Ref. [11] gives photon intensities of several lines from Fe XII and Fe
XIII in the UV range obtained via satellite instruments. With the formula
above, we seek to estimate the rate of $\ket{1}$ states from measurements made
in [11] and [12], for UV and X-ray respectively.

The intensity data of 19.664 nm line of Fe XII in [11] is over 800 erg$/$(cm2
s Å) at the peak. We pick a value of 600, (since there is a 19.654 nm peak of
600 and we are unsure of any potential overlap between the two) multiply it
with a wavelength width, convert the unit to eV$/$(m2 s), and then divide by
the energy of 19.664 nm photon in eV. This indicates that $N_{1}^{19.664}\sim
5.9\times 10^{14}$ of $\ket{1}$ states arrive on the area per second. The flux
data in [12] shows that the rate can be $N_{1}^{1.5}\sim 3.7\times 10^{10}$
for $\ket{1}$ states of the 1.5 nm line of Fe XVII on the same area.

By reversing (6) and using these intensity and flux data aforementioned [11,
12], we find the density of Fe XII and Fe XVII that are excited to the 19.664
nm and 1.5 nm level, are around $1.9\times 10^{-5}$ m-3 and $2.6\times
10^{-12}$ m-3, respectively. Following the same approach by which we estimated
the rate of $\ket{2}$ for 530.3 nm, using $P(L,2)$, we find the rate of
$\ket{2}$ for 19.664 nm and 1.5 nm to be around $N_{2}^{19.664}\sim 2.83$ and
$N_{2}^{1.5}\sim 0$ on the same area. As the frequency of the photon
increases, less $\ket{2}$ states can be created from this stimulated emission
process, which is not surprising since the cross section, Eq (3), goes as
$\nu_{0}^{-4}$.

There are many uncertainties in the understanding of the solar corona, which
have allowed us to only make rough estimates of the rates of quantum states
from this stimulated emission process. Nevertheless, we think that the key
point is we are finding rates at a few per second in our test detector, rather
than on much longer time scales, thus in the realm of being measurable.

##  V Decoherence

Having discussed a mechanism to produce this kind of N-identical photons
state, one has to wonder how likely it is for it to get to our detector intact
without decoherence. As it turns out, the possibility of decohering
interactions, in particular for the extreme ultraviolet to the X-ray range, is
basically null. In spite of the peculiarities of the corona, current
observations do allow us to put strong bounds on MFPs and interaction rates of
the potentially dangerous decohering interactions. Spectroscopic studies have
shown that there are three distinct “regions” in the corona, dubbed as K, F
and E. The K-corona shows no absorption, so there is a continuum of white
light that undergoes Thomson scattering, which could affect our quantum
coherent photonic state. The optical depth for that interaction is $\tau=\int
dr\sigma_{\rm Th}n_{e}$, where $\sigma_{\rm Th}=6.65\times 10^{-29}\ {\rm
m}^{2}$ is the Thomson cross section and $n_{e}$ is the electron density in
the corona, for which there are several models. Those by Allen and Baumbach
[13], and Edenhofer, et al. [14] respectively state

$\displaystyle n_{e}(r)$ $\displaystyle=$
$\displaystyle\left[\frac{2.99}{r^{16}}+\frac{1.55}{r^{6}}\right]10^{14}\ {\rm
m}^{-3}\;,\quad 1.2\lesssim r<3\;,$ (7) $\displaystyle n_{e}(r)$
$\displaystyle=$
$\displaystyle\left[\frac{30}{r^{6}}+\frac{1}{r^{2.2}}\right]10^{12}\ {\rm
m}^{-3}\;,\quad 3<r<65\;,$ (8)

where $r$ is in units of $R_{\odot}$. With that information, we arrive to
$\tau\simeq 5\times 10^{-7}$, which corresponds to an interaction probability
of $6\times 10^{-5}\%$, and a MFP of $10^{17}$ m, almost 6 orders of magnitude
longer than the distance between the Earth and the Sun. Notice that other
models will lead to the same order-of-magnitude densities, so our conclusions
would be the same [13].

Next, we shall consider potential interactions with the elements in the
corona. Naturally, there is less information about the distribution of the
atoms throughout the corona, but we can still (over)estimate the probability
of each interaction using available data. As previously mentioned, we know the
abundances with respect to hydrogen. Unlike before, let us now fix the
particle density of the corona to be about $10^{15}\ {\rm m}^{-3}$, which we
assign to hydrogen. In doing so, the density of each species is overinflated
as well as the interaction probabilities. These overestimates are done in
order to give us the shortest estimates for the MFPs from decohering
processes. Using the abundances reported in [8] and total attenuation cross
sections (coherent and incoherent scattering + photoelectric absorption + …)
from the XCOM software from [5], we obtained a lower value estimation of the
MFP due to photon-hydrogen interactions of $\ell=(\sigma n)^{-1}\simeq 8\times
10^{11}$ m, which is 5 times larger than the distance between the Sun and
Earth. Notice that we chose photon energies $\sim 1$ keV, where the cross
section always peaks. MFPs of the same order of magnitude are obtained for
interactions in the corona with He ($4\times 10^{11}$ m), Mg ($6\times
10^{11}$ m), Si ($9\times 10^{11}$ m), C ($10^{12}$ m), N ($2\times 10^{12}$
m), Ni ($3\times 10^{12}$ m), or S ($5\times 10^{12}$ m). The smallest MFP is
obtained for O ($2\times 10^{11}$ m), whereas for other elements they are
larger than those shown above, and among those the strongest interactions are
with Al, Cr and Mn, which in turn render $\ell\simeq 10^{13}$ m. All these
MFPs must be combined, with an effective or total MFP given by

$\frac{1}{\ell_{\rm
eff}}=\sum_{i}\sigma_{i}n_{i}=\sum_{i}\frac{1}{\ell_{i}}\,,$ (9)

where the sum is over all the atoms under consideration. Upon performing this
summation, we find:

$\displaystyle\ell_{\rm eff}=4.6\times 10^{10}\ {\rm m}\,.$ (10)

This value lies within an ${\cal O}(1)$ factor of the distance between the Sun
and the Earth, and, most importantly, it exceeds the width of the corona by at
least 2 orders of magnitude. This reaffirms the validity of our previous
conclusions that decohering effects are negligible.

Another decoherence factor could be the dust near the corona, which plays a
crucial role in the F region. The process of interest is the scattering of
photons off dust particles, which largely depends on their size. Fragments
smaller than $1\ \mu$m are not expected near the corona since they are pushed
away by radiation pressure and electromagnetic forces [15]. For this reason,
the brightness of the F-corona, mostly of a thermal nature, is dominated by
the near–IR range of the electromagnetic spectrum [16]. For radiation of
shorter wavelengths, one can compute $\ell=(\sigma_{\rm geom}n)^{-1}$, where
$\sigma_{\rm geom}$ is the geometric cross section of the dust particles. The
average flux detected by Helios [17] in a range of heliocentric distance from
$0.3$ to $1.0$ AU was of $(2.6\pm 0.3)\times 10^{-6}\ {\rm m}^{-2}\ {\rm
s}^{-1}$, with dust particles of size $0.37\ \mu$m. Taking the worst case
scenario, the flux at the photosphere is

$\Phi=2.6\times 10^{-6}\left(\frac{1\ {\rm AU}}{R_{\odot}}\right)^{2}{\rm
m}^{-2}\ {\rm s}^{-1}\approx 0.12\ {\rm m}^{-2}\ {\rm s}^{-1}\;,$ (11)

corresponding to an interaction rate $\Gamma=\Phi\sigma_{\rm geom}\simeq
5.16\times 10^{-14}\ {\rm s}^{-1}$ and a MFP of $6\times 10^{21}$ m.
Similarly, the Solar Orbiter estimated a flux of $8\times 10^{-5}\ {\rm
m}^{-2}\ {\rm s}^{-1}$ at 1 AU for particles of size of order $0.1\ \mu$m
[18]. A similar calculation leads to an interaction rate of $10^{-13}\ {\rm
s}^{-1}$, and a MFP of order $10^{21}$ m, the same as our previous estimate.

As large as these MFPs are, they are relatively short in comparison to those
associated to interactions with the blackbody radiation from the photosphere,
where $\ell\sim 10^{43}\ {\rm m}$. This is due to the weakness of the
interaction, with the cross section dependence as $\alpha^{4}\sim 137^{-4}$.

One potential decohering effect that can not be assessed through direct
particle interactions is Faraday rotation. To evaluate the impact of the
Faraday effect, we can estimate the rotation measure angle $\beta$, which
quantifies the effect for a given wavelength $\lambda$. It is given by the
expression

$\beta=\frac{e^{3}}{8\pi^{2}\epsilon_{0}m^{2}c^{3}}\lambda^{2}\int_{R_{\odot}}^{d}n_{e}(s)B_{\parallel}(s)\mathrm{d}s\,,$

where $n_{e}(s)$ represents the electron density in the medium and
$B_{\parallel}(s)$ denotes the component of the magnetic field along the
trajectory [19]. For an upper bound approximation, we can utilize equations
(7) and (8), together with the assumption of $B_{\parallel}\sim 100\ \mu$T,
which is an exaggerated approximation for the entire spatial range under
consideration [20]. This simplified calculation yields

$\beta(\lambda=100\ {\rm nm})\simeq 10^{-18}\,.$

Hence, the overall effect of Faraday rotation on the state is exceedingly
small and would impact every photon uniformly. This is due to the expectation
that the photons are part of the same wave train, with a microscopic
separation, whereas the coherence lengths of the magnetic fields involved are
macroscopic. Consequently, this effect leads to an overall rotation without
any decoherence effects.

Another effect, birefringence, though typically addressed in terms of the wave
properties of photons, ultimately still arises from the collective
interactions of the photons with the particles in the medium. The particle
analysis we conducted already accounts for these interactions, rendering long
MFPs. While our analysis employed the Thomson scattering formula instead of
the more intricate Klein-Nishina expression, our conclusions are highly
robust, because the Thomson cross section is always larger than the Klein-
Nishina one. As a result, we can confidently extrapolate our findings to
higher dispersion energies. This assertion applies equally to the Faraday
effect, as far as decoherence goes. Thus, the lack of interactions from a
particle perspective results in an effective transparency from a wave point of
view. Finally, it is important to note that the occurrence of the Faraday
effect does not necessarily imply decoherence of the state. Similar to our
study on effects from gravity [2], it suggests a change in fidelity, which
should be distinguished from the loss of quantum coherence, a crucial
differentiation for our specific objectives.

Other potential decoherence factors have been discussed elsewhere [1, 21, 2].
There, it was found that photons at the energy ranges discussed here would not
interact with the cosmological medium, or the cosmic background of photons in
the microwave, IR, and X-ray bands, with other interactions depending on the
particular local environments the photon travels through. For example, X-ray
photons could travel through the interstellar medium without interacting with
other particles for distances up to 1 Mpc depending on the region, although
this can be reduced to 100 pc for dense HII regions. Factors like the matter
environment of the Solar System have also been considered, yielding almost
null probabilities of interactions.

Finally, it is worth noting that the quantum nature of a photon state is also
unaffected by gravity (see [2] and references therein). The effects of gravity
on quantum states have been extensively explored in the literature, especially
in the context of quantum communications and the potential applications
stemming from them. Some examples include the effects of gravity on quantum
information protocols [22], frequency spectrum deformation [23], gravitational
distortion of quantum communication [24, 25], geometric phase acquired as a
wave-packet travels over a null geodesic [26, 27], and gravitational redshift
induced transformations on the photon state [28, 29]. In the context of this
work, it is crucial to recognize that while the photon state may experience
changes affecting features like the fidelity in quantum communications, its
fundamental quantum nature remains unaltered. As a result, in the case of the
N-identical photons states discussed in this paper, any phase effects induced
by gravity will be uniform among all photons within that state, ensuring the
preservation of their quantum integrity up to an overall phase. Consequently,
even though our analysis primarily focuses on the solar corona, it is
plausible that signals originating from other stars can also reach our
detectors near Earth essentially intact, especially in the UV and X-ray range.

##  VI Measurements

In principle, measuring the quantum nature of a signal from space could be
significantly more challenging than in a controlled setup. However, one could
potentially use the same principles that are common in Earth-based
experiments. For one, direct measurements of the photons could be made through
interference experiments that induce a phase between split signals leading to
characteristic correlations depending on their nature. For example, the Hong-
Ou-Mandel (HOM) effect [30] could be used to test the indistinguishability of
the two photons. The effect occurs when two identical single photons enter a
1:1 beam splitter through different input gates, and under certain temporal
conditions, the two photons will always exit the beam splitter together
through the same terminal. This effect has been used in an astrophysical
experiment by Deng et al. [31, 32] for testing quantum interference between
single optical photons from the Sun and a quantum dot on the Earth, to provide
distinct evidence for the quantum nature of thermal light. Other astrophysical
quantum experiments in the optical region have also been proposed by Dravins
et al. [33, 34]. Our above analysis has shown to minimize environmental
decoherence, the UV and X-ray are good regions for testing our proposed
mechanism. Alternatively, indirect measurements could be made through
examining the distribution probability of thermal signals, which would defer
from that of quantum signals. For example, the states emitted by a laser
follow Poisson statistics, where the probability of measuring $N$ photons in a
given coherent state peaks significantly around the mean value. On the
contrary, for thermal sources the largest probability always corresponds to
the no-photon state, with the distribution function decreasing less rapidly
than for coherent states. Thus, provided that detection at the photon level
can be performed, current technology is well suited to discern a quantum
signal from a classical one, but extending into the UV and X-ray bands and to
our particular type of N-identical photons states still needs further
development.

##  VII Conclusion

This work has identified the emission of specific types of quantum states of
photons, N-identical photons states, originating from specific processes,
stimulated emission, in the atmosphere of stars. Focusing on the Sun, we
showed that arriving at a $1{\rm m}^{2}$ detecting region near the Earth the
rate of such N-identical photons states would be a few per second thus should
be measurable. An actual experiment could of course be done at a distance
closer to the Sun, which would increase signal flux and reduce residual
decoherence and some possible experimental errors. We also showed such states
would not undergo decoherence both leaving the solar environment and
propagating from the Sun to Earth, especially in the UV and X-ray regions.
From our previous work [1, 21, 2], we expect that such states can also arrive
without decoherence from stars at interstellar distances away.

It is remarkable how the creation of such quantum states can be specifically
identified within an astrophysical body and that such states then can traverse
astronomical distances without decohereing. In fundamental terms this work has
extended the distance of measurable quantum phenomenon, and that at the
individual particle level, to astronomical scales. In practical terms these
results provide a new probe for studying stellar atmospheres.

## Acknowledgments

AB is partially funded by STFC. JCF thanks Nicolás Vásquez and Christian
Vásconez for helpful discussions about dust in the corona. This work is
supported in part by the National Key Research and Development Program of
China under Grant No. 2020YFC2201501 and the National Natural Science
Foundation of China (NSFC) under Grant No. 12147103.

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