# Locating Red Supergiants in the Galaxy NGC 6822

Tzvetelina A. Dimitrova Earth and Space Exploration, Arizona State University,
P.O. Box 871404, Tempe, AZ 85287-1404, USA  Department of Astronomy,
University of Washington, 3910 15th Ave NE, Seattle, WA 98105, USA  Kathryn F.
Neugent Department of Astronomy, University of Washington, 3910 15th Ave NE,
Seattle, WA 98105, USA  Dunlap Institute for Astronomy & Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
Lowell Observatory, 1400 W Mars Hill Road, Flagstaff, AZ 86001, USA  Philip
Massey Lowell Observatory, 1400 W Mars Hill Road, Flagstaff, AZ 86001, USA
Department of Astronomy and Planetary Sciences, Northern Arizona University,
PO Box 6010, Flagstaff, AZ 86011-6010, USA  Emily M. Levesque Department of
Astronomy, University of Washington, 3910 15th Ave NE, Seattle, WA 98105, USA

###### Abstract

Using archival near-IR photometry, we identify 51 of the K-band brightest red
supergiants (RSGs) in NGC 6822 and compare their physical properties with
stellar evolutionary model predictions. We first use Gaia parallax and proper
motion values to filter out foreground Galactic red dwarfs before constructing
a $J-K$ vs. $K$ color-magnitude diagram to eliminate lower-mass asymptotic
giant branch star contaminants in NGC 6822. We then cross-match our results to
previously spectroscopically confirmed RSGs and other NGC 6822 content studies
and discuss our overall completeness, concluding that radial velocity alone is
an insufficient method of determining membership in NGC 6822. After
transforming the $J$ and $K$ magnitudes to effective temperatures and
luminosities, we compare these physical properties with predictions from both
the Geneva single-star and Binary Population and Spectral Synthesis (BPASS)
single and binary star evolution tracks. We find that our derived temperatures
and luminosities match the evolutionary model predictions well, however the
BPASS model that includes the effects of binary evolution provides the best
overall fit. This revealed the presence of a group of cool RSGs in NGC 6822,
suggesting a history of binary interaction. We hope this work will lead to
further comparative RSG studies in other Local Group galaxies, opportunities
for direct spectroscopic follow-up, and a better understanding of evolutionary
model predictions.

##  1 Introduction

Red supergiants (RSGs) are the coolest of the evolved massive stars, with K
and M spectral types and temperatures below $4500$K. They start their lives on
the main sequence, spending $10^{6}-10^{7}$ years as hydrogen fusing OB-type
stars with initial masses of 8-30$M_{\odot}$. Once the core hydrogen supply of
these progenitor stars is depleted, they dramatically expand in size over a
short period of time as they leave the main sequence and move across the HR-
diagram, evolving into a phase of helium core fusion. After briefly passing
through a yellow supergiant (YSG) phase, the expansion increases their radius
up to a thousand times that of the Sun, and they are now cool RSGs with
effective temperatures of $\sim$3500-4500 K. RSGs continue on to fuse heavier
elements in their cores before reaching the end of their life cycles with a
core collapse death that leaves behind either neutron stars or black holes
(see Levesque 2017 and references therein for an overview of RSG evolution).

A number of recent studies have been made of the RSG content of nearby
galaxies, thanks both to Gaia and improved near-IR (NIR) surveys. In the last
few years, both Massey et al. (2021b) and Ren et al. (2021) used this method
to identify RSGs in the high metallicity spiral galaxies of M31 and M33. The
Large and Small Magellanic Clouds (LMC, SMC) provide low-metallicity
complements in the Local Group and their RSG contents were recently determined
by Yang et al. (2021a) and Neugent et al. (2020a) for the LMC and Yang et al.
(2020) and Massey et al. (2021a) for the SMC. Additionally, Dimitrova et al.
(2020) located 138 RSGs in the starburst galaxy IC 10 following a similar
process. Here we extend these RSG content studies to the low-metallicity
environment of NGC 6822 with the goal of identifying nearby RSG populations
and testing stellar evolution models by comparing them against our
observational findings.

NGC 6822, also known as Barnard’s Galaxy, was discovered by Barnard (1884) and
is a barred irregular member of the Local Group located at a distance of
around 0.5 Mpc (van den Bergh, 2000). Searching for RSGs in NGC 6822 was
particularly appealing in part because of the recently available NIR
photometric data from the United Kingdom Infra-Red Telescope (UKIRT), and
parallax and proper motion values of its stars from the Gaia data releases
(Gaia Collaboration et al., 2018, 2021). With these datasets, we can use the
same techniques applied in the Local Group galaxies of M31, M33, LMC, SMC, and
IC 10 to identify RSGs in NGC 6822. NGC 6822 also has a low metallicity of
$0.4Z\odot$ (see further discussion in Levesque & Massey 2012) which will
allow for future studies of the RSG content over a range of different
metallicities.

NGC 6822 has been the subject of several previous RSG content studies
including Massey (1998), Levesque & Massey (2012), Patrick et al. (2015), and
Yang et al. (2021b). The largest sample was identified by Yang et al. (2021b),
which identified 234 RSGs, but the survey suffered from a discrepancy with the
photometry (see Section 3.2) that introduces problems in their selection
criteria. Here we propose a sample of the $K$-band brightest 51 RSGs in NGC
6822 (many, but not all, of these RSGs are in common with those previously
discovered) and additionally compare our observational findings to theoretical
models of single and binary star evolution.

In Section 2 we identify NGC 6822 RSGs after removing both foreground Galactic
red dwarfs and asymptotic giant branch (AGB) contaminants from our initial
sample. We additionally compare our final list of RSGs with results from
previous work and discuss our overall completeness. In Section 3 we then
determine their physical properties such as temperature and luminosity before
comparing these observed properties to evolutionary model predictions in
Section 4. Finally, in Section 5 we briefly summarize our research and discuss
next steps.

##  2 Identifying RSGs

###  2.1 NIR Photometry

To locate RSGs in NGC 6822, we relied on the archival NIR dataset taken by
Irwin (2013) using the Wide Field Camera (WFCAM) on the 3.8-m UKIRT on Mauna
Kea. These data were taken as part of a survey of the Local Group galaxies. We
began with the pipeline reduced source catalogs in $J$ and $K$, and identified
sources that were flagged as “stellar” or “marginally stellar” that appeared
in both filters, following the same procedure used in our study of IC 10
(Dimitrova et al., 2020) and M33 (Massey et al., 2021b). Comparison with 2MASS
photometry (Skrutskie et al., 2006) of the same stars led us to revise the
UKIRT photometry by +0.023 mag in $J$ and -0.016 mag in $K$, consistent with
our previous studies; thus the correction in $J-K$ colors are 0.039 mags. The
comparison also confirmed that the UKIRT photometry became unreliable brighter
than $J=K=12$. We therefore added stars from the 2MASS catalog covering the
same area. All together, our initial source list consisted of 213,697 stars,
of which 211,473 were identified from the UKIRT data, and the remainder from
2MASS.

We cross-matched this list with the Gaia DR2 (Gaia Collaboration et al.,
2018), finding 115,106 (54%) stars with complete Gaia data. We will use this
below to establish membership in NGC 6822.

###  2.2 Removing Contaminants

While the UKIRT NIR data allows us to identify red stars in the direction of
NGC 6822, foreground Galactic nearby red dwarfs (and the occasional halo red
giant) fall into the same magnitude range as distant, but much more luminous,
RSGs in NGC 6822. To remove the red foreground stars, we relied on high-
precision parallax ($\pi$) and proper motion values ($\mu_{\alpha}$,
$\mu_{\delta}$) from the Gaia Data Releases to assign membership in NGC 6822.
However, as is well known, there are spatially dependent zero-point issues
with Gaia parallaxes and proper motions values, and in order to account for
these, we used the same process described in Neugent et al. (2020a) and Massey
et al. (2021b), based upon the recommendations given in Gaia Collaboration et
al. (2018). We first identified a set of stars that were near-certain members
of NGC 6822. For this, we selected probable OB stars from the Local Group
Galaxy Survey (LGGS; Massey et al. 2007). A sample of blue stars was defined
using the $B-V$, $U-B$, and $V$ bands, with values from $B-V<0.1$, $U-B<-0.7$,
and $V=18$ to 20 based on examining spectroscopically confirmed OB stars in
NGC 6822 (Massey et al., 2007). While our NIR photometric catalog was cross-
matched against Gaia DR2, we relied on the newly released Early Gaia Data
Release 3 (EDR3; Gaia Collaboration et al. 2021) to determine the zero-points,
as this resulted in more accurately determined zero-point values. By averaging
the blue star members’ proper motion and parallax values, we determined the
following Gaia zero points for NGC 6822 members: $\pi=0.16\pm 0.07$ mas,
$\mu_{\alpha}=-0.30\pm 0.11$ mas yr-1, and $\mu_{\delta}=-0.39\pm 0.14$ mas
yr-1, where the quoted errors are the standard deviations of the mean. Note
that all three values differ by multiple sigma from zero. We then used the
Gaia DR2 proper motion and parallax values, along with their estimated
uncertainties, to identify near-certain members of NGC 6822. For this, we
assigned membership status if all three values were within three times their
uncertainties of the expected zero-points. (We were careful not to exclude
stars whose only problem was that their parallax values were too small
compared to this criterion.) In all, 20,729 (18%) of the 115,106 with Gaia
data met these criteria for membership.

Figure 1: CMD constructed from the $K$ and $J-K$ values of NGC 6822’s stellar
members with foreground stars removed using Gaia. The approximate regions of
different stellar populations are as labeled above, revealing the presence of
both YSGs on the left side of the approximate RSGs on the CMD, and AGB stars
to their right on the CMD.

After the removal of foreground stars, we next needed to remove non-RSG
contaminants such as lower mass AGBs and cooler YSGs, also residing in NGC
6822, as shown in Figure 1. To do this, a color magnitude diagram (CMD) was
constructed from the $J-K$ and $K$ values of the remaining sample. Originally,
we attempted to apply $J-K$ and $K$ cuts used previously to identify RSGs in
the LMC by Neugent et al. (2020a) since the metallicity of the LMC
($0.5Z_{\odot}$, Russell & Dopita 1990) is comparable to that of NGC 6822
($0.4Z_{\odot}$). However, even after accounting for NGC 6822’s higher
reddening compared to the LMC ($E(B-V)=0.25$ for NGC 6822 vs. 0.13 for the
LMC; Massey et al. 2007) and distance (0.5 Mpc vs. 0.05 Mpc, respectively; van
den Bergh 2000), the cuts did not adequately select the RSGs and instead
preferentially selected YSGs, suggesting our $J-K$ cutoffs were too low. Given
this method worked with great success in both M31 and M33 (Massey et al.,
2021b), as well as in the SMC (Massey et al., 2021a) and IC 10 (Dimitrova et
al., 2020), we do not have a sound explanation for why a transformation of the
equations was not successful in NGC 6822. A few possibilities include
difficulties with the higher reddening value, the slight metallicity
difference between the LMC and NGC 6822, or problems arising from the small
number of candidate RSGs in NGC 6822.

Instead of attempting to add an offset to the poorly-matched $J-K$ cuts, the
photometric selections of the most luminous RSGs within NGC 6822 were instead
made by eye. Due to our inability to use the methodology described above, our
list of RSGs is not complete down to a limiting luminosity. However, our $J-K$
and $K$ selections were made to instead select the most luminous RSGs without
running the risk of selecting any high-luminosity AGB stars or cooler YSGs.
The selected 51 of the most $K$-band brightest RSGs in NGC 6822 are shown in
Figure 2 and their coordinates, magnitudes, and colors are listed in Table 1.

Figure 2: CMD of the final 51 $K$-band brightest RSGs in NGC 6822. The RSGs
are plotted in red and the galaxy’s other stellar members are shown in black.
The limits are defined in (J-K, K) space as: (0.9, 15.2), (1.2, 14), (0.9,
11), and (1.2, 11).

###  2.3 Comparative Studies and Completeness

We have identified the 51 $K$-band brightest RSGs in NGC 6822. However, given
our difficulties photometrically differentiating the RSGs from the YSGs and
AGBs, we are not able to claim completeness down to a limiting luminosity. To
get a better sense of our completeness, we can compare our sample to previous
studies of RSGs in NGC 6822 to further test various steps of our method.

The first study of RSGs in NGC 6822 was done by Massey (1998), which probed
the evolution of massive stars as a function of metallicity in various Local
Group galaxies and investigated RSGs within a small subset of NGC 6822. This
study was conducted prior to Gaia and contained both RSG candidates and
foreground Galactic dwarfs that were then assigned membership through radial
velocity measurements and the strength of the Ca II triplet. Massey (1998)
observed 199 candidate RSGs in NGC 6822 and classified 77 of them as RSGs and
the remaining 122 stars as foreground red dwarfs. We first compared our list
of NGC 6822 members to their list of 77 RSGs and found that 50 of them are
indeed likely RSGs while the remaining 27 are foreground red dwarfs. Of the
122 foreground stars classified by Massey (1998), 16 of them are actually
members based on the Gaia proper motion and parallax data. We then compared
our list of 51 RSGs to their initial list of 199 stars and identified 19
matches. Of these, 16 of them were identified as RSGs by Massey (1998) while
the remaining 3 were identified as foreground stars. Of the remaining 34 RSGs
identified by Massey (1998) but not in our list of 51 RSGs, 32 of them were
simply too faint with $K$ magnitudes dimmer than 15, and 2 are in the region
of bright AGBs. The discrepancy in membership status between this survey and
that of Massey (1998) is due to his lack of Gaia data and reliance on radial
velocities to define membership. The radial velocity of NGC 6822 varies across
the disk of the galaxy between -40 km s-1 and -90 km s-1, which are also
reasonable radial velocities for Galactic stars. Thus, this overlap can lead
to confusion when determining membership. We plot the magnitudes of the stars
that matched between our sample and that of Massey (1998) over the CMD of our
candidates in Figure 3.

We next compared our sample to the work done by Levesque & Massey (2012) which
was also done before the release of Gaia and relied on membership
determination through a combination of $B-V$ vs. $V-R$ color-cuts and radial
velocity measurements. Their data contained a sample of 16 RSGs in NGC 6822
that were observed spectroscopically and analyzed to determine their spectral
types. Of these 16 RSGs, only 8 were identified as members of NGC 6822 after
being filtered through Gaia. Of the remaining 8, only 4 were within our list
of 51. One of the remaining stars is fainter than our sample and the other
three fall into the regime of bright AGBs. We plot the matching stars in
Figure 3.

Our candidate sample was then compared to the spectroscopic observations of 18
RSGs in NGC 6822 by Patrick et al. (2015). Of these 18 RSGs, 11 of them appear
to be members based on the Gaia data while the remaining 7 are likely
foreground red dwarfs (including two of the stars they used for abundance
analysis). Of the remaining 11, 6 stars are in common with our list of 51
RSGs. The remaining 5 stars were not included in our sample because they were
too dim in $K$. We again plot the matching stars in Figure 3.

Figure 3: Color Magnitude Diagram summarizing all of the sample matches with
the comparative data sets. The Massey (1998) comparison is shown in dark blue
stars and produced the most matches within our data set, with 16 matching
RSGs. The Patrick et al. (2015) comparison is shown in light blue triangles
and contains 6 matching RSGs. The Levesque & Massey (2012) comparison is shown
in green circles and contains 4 matching RSGs.

We additionally attempted to compare our sample of RSGs to the most recent RSG
study in NGC 6822 done by Yang et al. (2021b) which also relied on the same
archival UKIRT data. As part of their study, Yang et al. (2021b) identified
234 RSGs in NGC 6822 using both the stars’ $1.6\mu m$ “H-bump” as well as
color and magnitude cuts, and we attempted to use their $J-K$ and $K$ cuts to
select our own sample of RSGs. However, after further investigation, we found
that their $J$ magnitudes have a serious discrepancy with 2MASS. For the 42
stars in common with their RSG catalog and 2MASS, the $J$ magnitudes disagree
by $\\-0.23\pm 0.01$ mag (in the sense of Yang-2MASS). Indeed, the issue can
be seen within their own table when alternative sources for $J$-band
photometry are included. There is no similar problem for their $K$-band
photometry. This results in a large error in the $J-K$ colors, and thus their
resulting CMD and selection criteria. Given this, we do not compare our sample
to theirs.

After comparing our list of 51 RSGs to the work done by previous surveys (a
summary of which can be found in Table 1), we can draw a few conclusions both
about previous studies as well as the completeness of our own. The first is
that due to the overlap in the expected radial velocities between NGC 6822 and
Galactic halo stars, radial velocities alone are not enough to determine
membership in the galaxy. The second is that, while we have identified the
brightest RSGs in NGC 6822, there are many fainter RSGs with $K$ magnitudes
dimmer than 15th magnitude that likely still exist. However, given our
inability to separate these RSGs from the bright AGBs, we cannot use this
method to identify them. Finally, separating RSGs from bright AGBs, even with
high resolution spectra, is still an ongoing area of research. While we have
classified a few of the stars observed by Levesque & Massey (2012) and Patrick
et al. (2015) as bright AGBs rather than RSGs, their evolutionary status is
still an open question. Overall, we emphasize that this study lacks
completeness but instead presents a survey of the brightest RSGs in NGC 6822.

##  3 Determining Temperatures and Luminosities

The observed physical properties of NGC 6822’s RSGs can ultimately be compared
to the theoretical work of stellar evolutionary models by first determining
their effective temperatures and luminosities. Starting with the $K$ and $J-K$
UKIRT photometry, these colors and magnitudes were transformed to the Bessell
& Brett photometric system (Bessell et al., 1998) using the following
equations from Carpenter (2001):

$K_{B}=K+0.044$ $(J-K)_{B}=(J-K+0.11)/0.972$

Where $(J-K)_{B}$ and $K_{B}$ are the converted $K$ and $J-K$ bands in this
photometric system. Next the photometry was de-reddened using $E(B-V)=0.25$ as
was determined for NGC 6822 by Massey et al. (2007). The $K$ and $J-K$
photometry underwent de-reddening transformations using the reddening
relations derived from Schlegel et al. (1998) as given in Neugent et al.
(2020b):

$E(B-V)=0.25$ $K_{0}=K_{B}-0.367\times E(B-V)$ $(J-K)_{0}=(J-K)_{B}-0.54\times
E(B-V)$

With $K_{0}$ and $(J-K)_{0}$ being the de-reddened photometry. We then
computed the absolute $K$ magnitudes corresponding to a true distance modulus
of 23.49 based upon an adopted distance to NGC 6822 of 0.50 Mpc (van den
Bergh, 2000).

The transformed and de-reddened $J-K$ band values were then used to calculate
the effective temperatures of the stars in NGC 6822, following the equation
from Neugent et al. (2020a) with values adjusted for NGC 6822’s distance:

$T_{\rm{eff}}=5643.5-(1807.1\times(J-K)_{0})$

Next, the bolometric correction was determined from the effective temperature
values calculated above, to convert the stars’ K magnitudes to bolometric
magnitudes. This was done with an equation similar to that in Neugent et al.
(2020a).

$BC_{K}=5.567-(7.5686\times 10-4)\times T_{\rm{eff}}$

Where $BC_{K}$ is the bolometric correction. The bolometric corrections were
then applied and a bolometric magnitude was used to determine the luminosity
values of the stars in NGC 6822. As described in Neugent et al. (2020a),
uncertainties in $\log T_{\rm{eff}}$ and $\log\frac{L}{L_{\odot}}$ are 0.02
dex and 0.05 dex, respectively.

We list the derived temperatures and luminosities of our 51 RSGs in Table 1
and compare these values to the theoretical predictions of binary and single
star evolutionary models in the next section.

##  4 Comparing our Observations with Evolutionary Model Predictions

We compared our observations with theoretical predictions from two different
stellar evolutionary codes: the Geneva evolutionary models (Eggenberger et
al., 2021), which focus on single-star evolution as well as the Binary
Population and Spectral Synthesis (BPASS) models (Eldridge et al., 2017;
Stanway & Eldridge, 2018), which include both single and binary star
evolution. For both of these evolutionary codes, we used the LMC metallicity
($z=0.006$) models.

Figure 4: HR Diagram showing the Geneva evolutionary tracks compared to our
proposed RSGs in red, and the remainder of NGC 6822’s stellar population in
black. Uncertainty values for all points are shown in the upper right corner
with further discussion in the text. The tracks displayed represent a stellar
mass range of 9 to 25$M_{\odot}$ for $z=0.006$ (Eggenberger et al., 2021).
Rotating models are shown in solid lines and non-rotating models are shown in
dashed lines. The locations of the tracks generally fit the observed
distribution of our RSGs well and reveal our sample of RSGs to be within a
mass range of 9 to 20$M_{\odot}$. Note, the group of RSGs located in the
upper-right that fall outside these tracks, which is discussed in greater
detail below.

A comparison between our observations and the Geneva models for stars with
initial masses between $9-25M_{\odot}$ is shown in the HRD in Figure 4. As
expected, the majority of the identified RSGs fall along the upturn at the
cool edge of each evolutionary path, otherwise known as the Hayashi track.
These tracks additionally suggest that the brightest RSGs in NGC 6822 have
initial masses between $9-20M_{\odot}$. This lower limit of $9M_{\odot}$
suggests that we have done an adequate job selecting only the more massive
RSGs as opposed to the lower mass AGBs which would come from stars with
initial masses below $9M_{\odot}$. For the upper limit, while the Geneva
models predict that RSGs can form from stars with initial solar masses as high
as $30M_{\odot}$ at LMC-like metallicities, we do not find any of these more
massive RSGs in NGC 6822. This is likely due to the small number of RSGs
present in NGC 6822, as the lifetimes of the RSG stage will be shorter with
increasing mass. It is additionally important to note that some of our
observed RSGs extend to cooler temperatures than what the Geneva models
predict for the metallicity regime. An explanation for this small disagreement
between the observations and the models is offered below.

The locations of the BPASS evolutionary tracks were also compared to the
physical properties of our final candidates for both single and binary star
evolution, as is shown in the HRD in Figure 5. These BPASS models produce
density maps indicating predictions of effective temperature and luminosity
positions on the HRD through a stellar lifetime, with shaded regions
corresponding to theoretical population density. The single star evolution
model shows a very similar prediction to the Geneva tracks shown in Figure 4,
partially excluding the region of RSGs that skew to the right. Interestingly,
the BPASS model for a mixture of single and binary RSGs predicts a bump of
RSGs to the right, aligning with our population of cooler RSGs. This suggests
that our sample includes either current or previous binary RSGs, which would
theoretically affect the RSG distribution.

One way to determine whether the cooler stars represent current RSG binaries
or just the products of past binary interactions is to compare the HRD shown
in Figure 5 with the HRD shown in Figure 6 of Neugent et al. (2020a). Their
HRD of RSGs in the LMC is color-coded by the probability of binarity for each
star with blue dots representing binary RSGs and red dots representing single
RSGs. Note that the binary RSGs are _not_ clustered in a singular area of the
HRD and thus the presence of a binary companion does not appear to be
correlated with the temperature of the RSG. This suggests that this cooler
population of RSGs we observe in the HRD of NGC 6822 is not due to the
presence of currently binary RSGs. Instead, it is perhaps the end result
caused by previous stages of binary evolution.

An alternate explanation to this cooler population of RSGs is that higher
luminosity RSGs are known to be both more variable and also have higher mass-
loss rates than lower luminosity RSGs (Meynet et al., 2015; Humphreys et al.,
2020). It is possible that these high luminosity RSGs have recently gone
through eruptive mass-loss events similar to the recent outburst by Betelgeuse
(Levesque & Massey, 2020; Montargès et al., 2021) and thus appear cooler since
much of their light has been blocked by dust. However, as discussed briefly by
Massey et al. (2021a), when a similar group of cooler RSGs was discovered in
the SMC, spectroscopy confirmed them to be cool M-type RSGs as opposed to
highly reddened and dusty K-type stars. While it is impossible to
differentiate any of these effects from the intrinsic uncertainties in our
temperature calculations, the good agreement between our observations and the
BPASS binary evolutionary models is still striking even if alternative
explanations for this cooler population of RSGs exist.

Figure 5: HR Diagrams comparing the BPASS single (left) and binary (right)
compared to our proposed RSGs in red, and the remainder of NGC 6822’s stellar
population in black. Uncertainty values for all points are shown in the upper
right corner with further discussion in the text. In both diagrams, population
density is indicated by the shaded regions (darker colors indicate a denser
population, and lighter colors indicate a less dense population). Note the
agreement between the BPASS single and Geneva evolutionary tracks but the
improved agreement between our observations and the models once binaries are
included in the BPASS models. This BPASS model that includes binaries better
models the “bump” of cooler RSGs and follows the overall distribution pattern
more accurately. This BPASS model for a mixture of single and binary stars is
the best-fit model for our proposed RSGs, thereby suggesting the importance of
binary interactions.

##  5 Summary, Conclusions, and Next Steps

We present a list of the 51 $K$-band brightest RSGs in NGC 6822. These RSGs
were discovered after using Gaia parallax and proper motion values to
determine membership, archival NIR photometry from UKIRT to constrain stellar
characteristics, and $K$ and $J-K$ cut-offs from the constructed CMD to
distinguish RSGs from AGBs. This sample was studied comparatively alongside
several other RSG searches in NGC 6822, revealing our results to be in general
agreement with these previous studies and additionally discovering that radial
velocity is not an adequate method of determining membership in NGC 6822 due
to Galactic halo star contamination. Our RSGs were compared with the Geneva
and BPASS stellar evolutionary models for LMC-like metallicities as a means of
testing theoretical predictions to observational findings in the galaxy,
confirming the models as a good fit for the RSGs in NGC 6822 and in particular
identifying the BPASS model for a mixture of binary and single stars as the
best-fit scenario. This result underscores the importance of binary
interactions both before and during the RSG phase when studying the RSG
population of a galaxy.

This proposed RSG sample has broader applications for the overall study of RSG
populations in the Local Group, as it provides a low-metallicity complement to
nearby galaxies in which previous RSG searches have occurred. This research is
also complementary to our previous work, in which we located RSGs in the
starburst galaxy IC 10. Future work comparing our results in IC 10 and NGC
6822 with the RSG searches done by our collaborators in M31, M33, and the
Magellanic Clouds, could be beneficial to further testing the existing stellar
evolution models and theoretical predictions. Additionally, now that a new
population of RSGs in NGC 6822 has been located, future investigations into
these massive stars with direct spectroscopic follow-up, and studies into the
binary frequency of the RSGs in this low metallicity galaxy, are supported.
Furthermore, our work reveals an interesting group of cool RSGs, potentially
indicating the presence of RSGs effected by binary evolution and creating an
opportunity for future RSG binary related probes within this and other
galaxies.

We thank both Ming Yang and the anonymous referee for their helpful comments
that improved the paper. TAD would like to acknowledge financial support from
both the William Goddard Family through Lowell Observatory and the Mary Gates
Foundation through the University of Washington. This work was additionally
supported in part by the Dunlap Institute for KFN, the National Science
Foundation under grant AST-1612874 awarded to PM, and a Cottrell Scholar Award
from the Research Corporation for Scientific Advancement granted to EML.

Table 1:  $K$-Band Brightest RSGs in NGC 6822 $\alpha_{\rm 2000}$ | $\delta_{\rm 2000}$ | $K$ | $\sigma K$ | $J-K$ | $\sigma J-K$ | $\log T_{\rm{eff}}$ | $\log L/L_{\odot}$ | Other Survey**1 = Massey 1998; 2 = Levesque & Massey 2012; 3 = Patrick et al. 2015.
---|---|---|---|---|---|---|---|---
19 42 21.59 | -14 45 11.9 | 14.122 | 0.005 | 0.934 | 0.009 | 3.60 | 4.64 |
19 43 30.67 | -14 32 35.7 | 14.807 | 0.007 | 0.962 | 0.011 | 3.59 | 4.35 |
19 44 05.19 | -14 11 25.0 | 13.619 | 0.003 | 0.935 | 0.005 | 3.60 | 4.84 |
19 44 29.74 | -14 52 36.9 | 14.635 | 0.007 | 1.013 | 0.011 | 3.58 | 4.39 |
19 44 30.35 | -14 48 35.9 | 13.549 | 0.004 | 0.990 | 0.006 | 3.58 | 4.83 |
19 44 32.09 | -14 49 26.7 | 13.178 | 0.003 | 1.059 | 0.005 | 3.57 | 4.94 |
19 44 35.29 | -14 40 45.7 | 14.426 | 0.005 | 1.083 | 0.009 | 3.56 | 4.43 | 1
19 44 43.27 | -14 51 51.5 | 14.786 | 0.007 | 0.953 | 0.011 | 3.59 | 4.36 |
19 44 43.80 | -14 46 11.0 | 13.080 | 0.003 | 1.106 | 0.005 | 3.56 | 4.96 | 1,3
19 44 43.83 | -14 51 31.3 | 14.232 | 0.005 | 1.098 | 0.009 | 3.56 | 4.50 |
19 44 45.04 | -14 51 18.6 | 13.530 | 0.003 | 1.052 | 0.006 | 3.57 | 4.81 |
19 44 45.76 | -14 52 21.7 | 12.812 | 0.002 | 1.080 | 0.004 | 3.57 | 5.08 | 2
19 44 45.80 | -14 51 08.1 | 14.829 | 0.008 | 0.978 | 0.012 | 3.59 | 4.33 |
19 44 45.98 | -14 51 02.8 | 14.505 | 0.006 | 1.052 | 0.010 | 3.57 | 4.42 | 3
19 44 47.13 | -14 35 06.9 | 13.871 | 0.004 | 1.095 | 0.006 | 3.56 | 4.65 |
19 44 47.80 | -14 50 52.9 | 13.287 | 0.003 | 1.112 | 0.005 | 3.56 | 4.87 | 2,3
19 44 48.09 | -14 45 18.4 | 14.019 | 0.005 | 1.020 | 0.008 | 3.58 | 4.63 | 1
19 44 50.42 | -14 44 10.4 | 14.122 | 0.005 | 0.999 | 0.008 | 3.58 | 4.60 | 1,2
19 44 51.09 | -14 43 55.8 | 12.577 | 0.002 | 1.011 | 0.004 | 3.58 | 5.21 | 1
19 44 51.56 | -14 43 20.5 | 13.479 | 0.002 | 1.087 | 0.004 | 3.56 | 4.81 | 1
19 44 53.36 | -14 45 24.7 | 14.141 | 0.005 | 1.069 | 0.009 | 3.57 | 4.55 | 1
19 44 54.17 | -14 53 04.1 | 14.239 | 0.005 | 1.034 | 0.009 | 3.58 | 4.53 |
19 44 54.24 | -14 49 35.0 | 14.053 | 0.064 | 1.097 | 0.084 | 3.56 | 4.57 |
19 44 54.46 | -14 48 06.5 | 13.329 | 0.003 | 1.131 | 0.005 | 3.55 | 4.84 | 2,3
19 44 54.71 | -14 52 46.9 | 14.298 | 0.005 | 1.010 | 0.009 | 3.58 | 4.52 |
19 44 54.80 | -14 43 48.1 | 12.796 | 0.002 | 1.104 | 0.004 | 3.56 | 5.07 | 1
19 44 54.86 | -14 52 16.7 | 14.252 | 0.005 | 1.032 | 0.009 | 3.58 | 4.53 |
19 44 55.77 | -14 45 30.2 | 13.671 | 0.004 | 1.160 | 0.006 | 3.55 | 4.69 | 1
19 44 56.14 | -14 43 21.0 | 14.370 | 0.004 | 1.022 | 0.006 | 3.58 | 4.49 | 1
19 44 56.73 | -14 44 25.5 | 13.811 | 0.004 | 1.052 | 0.006 | 3.57 | 4.69 | 1
19 44 57.39 | -14 54 08.4 | 13.998 | 0.005 | 1.028 | 0.008 | 3.58 | 4.63 |
19 44 57.61 | -14 48 28.3 | 13.904 | 0.004 | 0.975 | 0.006 | 3.59 | 4.70 |
19 44 57.70 | -14 49 17.3 | 14.349 | 0.006 | 1.024 | 0.009 | 3.58 | 4.49 |
19 44 57.97 | -14 44 09.4 | 14.118 | 0.005 | 0.996 | 0.008 | 3.58 | 4.60 | 1
19 44 58.16 | -14 44 56.4 | 13.657 | 0.004 | 1.042 | 0.006 | 3.57 | 4.76 | 1
19 44 58.25 | -14 45 51.0 | 13.247 | 0.003 | 1.064 | 0.005 | 3.57 | 4.91 | 1
19 44 58.31 | -14 44 47.2 | 13.184 | 0.003 | 1.035 | 0.005 | 3.58 | 4.95 | 1
19 44 59.41 | -14 42 17.9 | 14.205 | 0.005 | 1.086 | 0.008 | 3.56 | 4.52 | 1
19 44 59.56 | -14 52 30.6 | 14.685 | 0.007 | 0.976 | 0.011 | 3.59 | 4.39 |
19 45 00.32 | -14 41 05.0 | 13.851 | 0.004 | 1.101 | 0.006 | 3.56 | 4.65 |
19 45 00.54 | -14 48 26.8 | 13.599 | 0.004 | 1.131 | 0.006 | 3.55 | 4.73 | 3
19 45 01.12 | -14 54 38.1 | 14.938 | 0.008 | 0.934 | 0.013 | 3.60 | 4.31 |
19 45 03.10 | -14 32 03.6 | 14.618 | 0.006 | 0.923 | 0.009 | 3.60 | 4.44 |
19 45 05.74 | -14 54 58.1 | 14.087 | 0.005 | 1.030 | 0.008 | 3.58 | 4.60 |
19 45 05.83 | -14 55 17.7 | 14.925 | 0.008 | 0.939 | 0.013 | 3.60 | 4.31 |
19 45 06.97 | -14 50 31.6 | 14.772 | 0.007 | 1.005 | 0.011 | 3.58 | 4.34 | 3
19 45 08.21 | -14 55 17.0 | 13.408 | 0.003 | 1.084 | 0.005 | 3.56 | 4.84 |
19 45 09.54 | -14 53 45.0 | 14.823 | 0.008 | 0.956 | 0.012 | 3.59 | 4.34 |
19 45 13.20 | -14 58 43.9 | 13.478 | 0.004 | 0.959 | 0.006 | 3.59 | 4.88 |
19 45 21.23 | -14 59 05.0 | 14.675 | 0.007 | 0.943 | 0.011 | 3.59 | 4.41 |
19 45 47.99 | -14 41 52.7 | 15.063 | 0.008 | 0.932 | 0.013 | 3.60 | 4.26 |

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# Quantum state transfer in atom-cavity systems with uncolored Cayley
interacting networks

R. Sufiania,b
a Department of Theoretical Physics and Astrophysics, University of Tabriz,
Tabriz 51664, Iran.
b Institute for Studies in Theoretical Physics and Mathematics, Tehran
19395-1795, Iran<EMAIL_ADDRESS>
###### Abstract

Considering the two-photon exchange interaction between $n$ coupled cavities
each of them containing a two level atom, the atomic and photonic state
transfer is investigated. In fact, $n$ atom-cavity systems are considered to
be distributed on the nodes of an uncolored Cayley network and interact with
each other via the adjacency matrix of the corresponding network. Then, by
employing the photon-excitation conservation and also the algebraic structure
of the networks, such as irreducible characters of the groups associated with
the networks, some suitable basis for the atom-cavity state space is
introduced based on the corresponding generalized Fourier transform, so that
the Hamiltonian of the whole system, is block diagonalized with two-
dimensional blocks. Then, by solving the corresponding Schrödinger equation
exactly, quantum state transfer and also entanglement generation between the
atoms or the photons are discussed. For instance, the probability amplitudes
associated with the photon transition between the cavities or excitation
transition between the atoms are obtained in terms of the irreducibles
characters of the corresponding network and the hopping parameter $\xi$
between the cavities.

Keywords: coupled cavities, two-photon exchange, hopping strength, two-level
atoms, generation of entanglement, excitation and photon transfer, uncolored
Cayley graphs, Generalized Fourier transform

PACs Index: 03.65.Ud

##  1 Introduction

In the past years, several schemes for quantum information transition and the
generation and distribution of entanglement have been designed and implemented
in a number of physical systems (see for example [1]-[10]). Atoms and ions are
particularly considered as tools for storing quantum information in their
internal states. On the other hand, photons represent the best qubit carrier
for the fast and reliable communication over long distances [11, 12].
Recently, using photons in order to achieve efficient quantum information
transmission between spatially distant atoms has investigated in several works
[13, 14, 15, 16, 17, 18]. The main idea, is to utilize two photon exchange
hopping between optical cavities containing the atoms. One of the known models
in quantum optics describing the atom-field interaction is the Jaynes-Cummings
Hamiltonian [19, 20]. In Refs. [13, 16], entanglement properties of two and
three atom-cavities coupled via two-photon exchange interaction, was studied
in detail. The motivation of interest to such systems is their ability to be
used in quantum switching, quantum communication and computation, and quantum
phase transition applications.

In the recent work [18], the approach of Alexanian, et. al has generalized to
the case of $n$ atom-cavities - instead of two or three cavities- all of them
interacting with each other by hopping strength $\xi$ (in that case the
cavities were distributed on a complete network $K_{n}$ at which all of the
vertices connected to each other). In this paper, I consider that the atom-
cavity systems are interacting with each other via the more general
connectivity networks (graphs) called uncolored Cayley networks. Due to the
fact that these graphs are constructed via finite groups, one can naturally
use their algebraic structure in order to state transfer or entanglement
generation. In fact, I consider the Cayley graphs associated with abelian
groups and employ the group theoretical properties of these networks such as
generalized Fourier transform associated with the corresponding groups (the
unitary transform which diagonalizes the regular representation of the
groups). Then, by using the photon-excitation conservation symmetry of the
Hamiltonian, some suitable basis for the atom-cavity state space is
introduced, so that the Hamiltonian of the whole system, is block diagonalized
with two dimensional blocks. Then, by solving the corresponding Schrödinger
equation exactly, state transfer between atoms or photons is discussed. For
instance, the probability amplitudes associated with the photon transition
between the cavities or excitation transition between the atoms are obtained
in terms of the irreducibles characters of the groups associated with the
corresponding networks, and the hopping parameter $\xi$ between the cavities.

The paper is organized as follows. In section 2, a preliminary is given in
order to recall the definition of uncolored Cayley networks. In section $3$,
the model describing the system of $n$ identical atom-cavities is introduced.
The block-diagonalization of the Hamiltonian of the system, solving the
corresponding Shrödinger equation for time dependent probability amplitudes of
the state of the system, and entanglement generation between atoms or photons
are discussed in this section. Section $4$ is devoted to discussions about
state transfer (excitation or photon transfer between the atoms or the
cavities). In sections $5$, two examples of Cayley networks (Cycle network
$C_{n}$ and the hypercube network $Q_{d}$) are discussed in detail, in order
to clarify the results of the paper. The paper is ended with a brief
conclusion.

##  2 Preliminary: Uncolored Cayley networks

For a given finite group $G$, we have always a set of group elements $S$ so
that all of the group elements can be generated via the possible
multiplications of the elements of $S$ and their powers. The set $S$ is called
the generating set of $G$. The Cayley graph $\Gamma(G,S)$ is a colored
directed graph constructed as follows: Each element $g$ of $G$ is assigned a
vertex: the vertex set $V(\Gamma)$ of $\Gamma$ is then identified with $G$.
Each generator $s$ of $S$ is assigned a color $c_{s}$. For any $g\in G$ and
$s\in S$, the vertices corresponding to the elements $g$ and $gs$ are joined
by a directed edge of color $c_{s}$. Thus the edge set $E(\Gamma)$ consists of
pairs of the form $(g,gs)$ with $s\in S$ providing the color.

The set $S$ is usually assumed to be finite, symmetric (i.e. $S=S^{-1}$ ) and
not containing the identity element of the group. In this case, the uncolored
Cayley graph is an ordinary graph: its edges are not oriented and it does not
contain loops. Then, the adjacency matrix $A$ corresponding to the Cayley
graph $\Gamma(G,S)$ is defined as:

$A=\sum_{s\in S}R(s),$ (2-1)

where $R(s)$ is the regular representation of the group element $s\in S$.

We will assume the Cayley graphs corresponding to the Abelian group, so that
the adjacency matrix $A$ can be diagonalized via generalized Fourier transform
$P$ defined by $P_{ij}=\frac{1}{\sqrt{n}}\chi_{i}(j)$ where, $\chi_{i}(j)$,
for $i,j=1,2,\ldots,n\;(=|G|)$, is the $i$-th irreducible character of the
$j$-th group element, and $|G|$ is the cardinality of the group. Then, the
eigenvalues of the adjacency matrices $A_{k}$ are given by

$x_{i}=\sum_{s\in S}\chi_{i}(s).$ (2-2)

It could be noticed that, from the orthogonality relation for the irreducible
characters [21],

$\frac{1}{n}\sum_{k}\chi_{i}(k)\chi^{*}_{j}(k)=\delta_{ij},$ (2-3)

one can see that $(P^{-1})_{ij}=\frac{1}{\sqrt{n}}\chi^{*}_{j}(i)$ and so we
have $P^{-1}=P^{{\dagger}}$.

##  3 The Model: $n$ coupled cavities via two-photon exchange interaction

We will consider $n$ identical cavities each containing a two-level atom,
where the cavities are coupled via two photon hopping between them. In fact,
we consider that the cavities are located at the nodes of an uncolored Cayley
network with $n$ nodes and each cavity interacts with all of the adjacent
cavities, via two-photon exchange.

Let us first introduce the Hamiltonian of one individual atom-cavity as a two-
photon JCM [22] (assume $\hbar=1$):

$H^{(i)}=\omega_{a}s^{(i)}_{z}+\omega_{c}a^{{\dagger}}_{i}a_{i}+\lambda(\sigma^{(i)}_{eg}a^{2}_{i}+\sigma^{(i)}_{ge}a^{{\dagger}2}_{i}),$
(3-4)

where, the operators $a_{i}$ and $a_{i}^{{\dagger}}$ are photonic annihilation
and creation operators of the $i$-th cavity,
$\sigma^{(i)}_{ab}=|a\rangle^{{{(i)}{(i)}}}\langle b|$, for $i=1,2,\ldots,n$
denote the atomic transition operators for the $i$-th cavity referring to
either the ground (g) or excited (e) state; $\omega_{a}$ is the energy
separation between two levels $|g\rangle$ and $|e\rangle$,
$s^{(i)}_{z}=\frac{1}{2}\sigma^{(i)}_{z}=\frac{1}{2}(|e\rangle^{{{(i)}{(i)}}}\langle
e|-|g\rangle^{{{(i)}{(i)}}}\langle g|)$, and $\omega_{c}$ is the resonance
frequency of the cavity.

In the Hamiltonian (3-4), the excitation-number operator

$\hat{N}^{(i)}=a_{i}^{{\dagger}}a_{i}+s^{(i)}_{z}$

is a constant of motion for the $i$-th atom-cavity subsystem, i.e, we have
$[H^{(i)},\hat{N}^{(i)}]=0$ for each $i=1,2,\ldots,n$. Now, the Hamiltonian
for the $n$ cavities is given by:

$H=\sum_{i=1}^{n}H^{(i)}+\xi\sum_{i,j=1;i\sim
j}^{n}(a_{i}^{2{{\dagger}}}a_{j}^{2}+a_{j}^{2{{\dagger}}}a_{i}^{2}),$ (3-5)

where, $i\sim j$ means that the cavities $i$ and $j$ are adjacent on the
network. The last term in the Hamiltonian (3-5) is the two-photon exchange
interaction between the cavities, characterized by the hopping rate $\xi$.

The operator $\hat{N}=\sum_{i=1}^{n}{\hat{N}}^{(i)}$ commutes with the
Hamiltonian (3-5) and so we can reduce the Hamiltonian to the subspace spanned
with the eigenstates of $\hat{N}$ and consider the time evolution of the
states in this subspace. For a given eigenspace of $\hat{N}$ with eigenvalue
$N$, the maximum possible number of photons in a cavity is $N$ when the
corresponding atom is in the ground state, which occurs when there are no
photons present in the other cavities and the atoms are also in the ground
state. Then, the total number of photons in the system will be $N$. The
constant number of total photons determines the subspace or the manifold in
which the states evolve in time (the initial state of the system determines
the constant number $N$). For simplicity, we will consider the manifold with
$N=\frac{2-n}{2}$. In this case, each single atom-cavity system can take one
of the three possible states $|g,0\rangle$, $|g,2\rangle$ or $|e,0\rangle$,
and so, the total possible states that the system of $n$-cavities can take,
are $3^{n}$ states. Due to the consistency of total $N=\frac{2-n}{2}$, the
only possible states which we can have, are $2n$ states instead of $3^{n}$
ones. In fact, these $2n$ states are eigenstates of $\hat{N}$ with eigenvalue
$2$, and the $3^{n}$-dimensional Hamiltonian $H$ is reduced to
$2n$-dimensional one in the bases which span the eigenspace of $\hat{N}$ with
the corresponding eigenvalue $2$. The bases states that span this subspace or
manifold, are given by:

$|c_{i}\rangle=|g,0\rangle\ldots|g,0\rangle\underbrace{|g,2\rangle}_{i-th}|g,0\rangle\ldots|g,0\rangle,$
$|a_{i}\rangle=|g,0\rangle\ldots|g,0\rangle\underbrace{|e,0\rangle}_{i-th}|g,0\rangle\ldots|g,0\rangle,$

for $i=0,1,\ldots,n-1$. Indeed, these bases span the eigenspace of $\hat{N}$
with eigenvalue $2$, i.e., we have
$\hat{N}(\alpha|c_{i}\rangle+\beta|a_{i}\rangle)=\frac{2-n}{2}(\alpha|c_{i}\rangle+\beta|a_{i}\rangle)$.
Therefore, the general time dependent state of the $n$-cavity system is given
by

$|\psi(t)\rangle=\sum_{i=0}^{n-1}(C_{i}(t)|c_{i}\rangle+A_{i}(t)|a_{i}\rangle).$
(3-6)

Then, one can easily show that

$\sum_{i=1}^{n}H^{(i)}|a_{k}\rangle=\frac{2-n}{2}\omega_{a}|a_{k}\rangle+\sqrt{2}\lambda|c_{k}\rangle,$
$\sum_{i=1}^{n}H^{(i)}|c_{k}\rangle=\frac{4\omega_{c}-n\omega_{a}}{2}|c_{k}\rangle+\sqrt{2}\lambda|a_{k}\rangle,$
$H_{{}_{int.}}|c_{k}\rangle\equiv\xi\sum_{i,j=1;i\sim
j}^{n}(a_{i}^{2{{\dagger}}}a_{j}^{2}+a_{j}^{2{{\dagger}}}a_{i}^{2})|c_{k}\rangle=2\xi\sum_{i\sim
k}|c_{i}\rangle,\;\;\;\ H_{{}_{int.}}|a_{k}\rangle=0;\;\;\;\
k=0,1,\ldots,n-1.$

Now, by considering the order of bases as
$|c_{0}\rangle,|a_{0}\rangle,\ldots,|c_{n-1}\rangle,|a_{n-1}\rangle$, and
defining $\Delta=\omega_{a}-2\omega_{c}$, the Hamiltonian $H$ takes the
following direct product form

$H=I_{n}\otimes\left(\begin{array}[]{cc}\omega+\Delta/2&\sqrt{2}\lambda\\\
\sqrt{2}\lambda&\omega-\Delta/2\\\ \end{array}\right)+2\xi
A\otimes\left(\begin{array}[]{cc}1&0\\\ 0&0\\\ \end{array}\right),$ (3-7)

where, $\omega:=\frac{1}{2}[(2-n)\omega_{a}-\Delta]$ and $A$ is the adjacency
matrix of the corresponding network containing the atom-cavities on its nodes.
We will assume that the cavities are interacting with each others via an
uncolored Cayley network as connectivity network, so that $A\equiv A_{1}$ is
given by (2-1). That is, the photons only hope between the cavities which are
adjacent in the network.

Now, due to the fact that the adjacency matrix $A$ is diagonalized by the
generalized Fourier transform
$P_{kl}:=\frac{1}{\sqrt{n}}\chi_{k}(\alpha_{l})$, one can naturally consider
the new transformed bases
$\\{|c_{j}\rangle^{\prime},|a_{j}\rangle^{\prime}\\}_{j=0}^{n-1}$ as:

$|c_{j}\rangle^{\prime}:=\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\chi_{j}(\alpha_{i})|c_{i}\rangle,$
$|a_{j}\rangle^{\prime}:=\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\chi_{j}(\alpha_{i})|a_{i}\rangle.$
(3-8)

Then, by considering the ordering
$\\{|c_{0}\rangle^{\prime},|a_{0}\rangle^{\prime};\ldots;|c_{n-1}\rangle^{\prime},|a_{n-1}\rangle^{\prime}\\}$,
the Hamiltonian (3-7) takes the following block diagonalized form:

$H=I_{n}\otimes\left(\begin{array}[]{cc}\omega+\Delta/2&\sqrt{2}\lambda\\\
\sqrt{2}\lambda&\omega-\Delta/2\\\ \end{array}\right)+2\xi\;\
diag{(x_{0},x_{1},\ldots,x_{n-1})}\otimes\left(\begin{array}[]{cc}1&0\\\
0&0\\\ \end{array}\right),$ (3-9)

where, $diag{(x_{0},x_{1},\ldots,x_{n-1})}$ is the $n\times n$ diagonal matrix
with diagonal entries as $x_{i}=\chi_{i}(\alpha_{1})$ with $i=0,1,\ldots,n-1$.
Therefore, the $i$-th block of the Hamiltonian is given by

$H^{(i)}=\left(\begin{array}[]{cc}\omega+\Delta/2+2\xi
x_{i}&\sqrt{2}\lambda\\\ \sqrt{2}\lambda&\omega-\Delta/2\\\
\end{array}\right).$ (3-10)

with eigenvalues $E_{\pm}^{(i)}=(\omega+\xi
x_{i})\pm\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}$. Now, by using
the Schrödinger equation of motion $i\hbar\frac{\partial}{\partial
t}|\psi\rangle=H|\psi\rangle$, the equations of motion are given by:

$i\dot{C^{\prime}_{i}}=[\omega+\Delta/2+2\xi
x_{i}]C^{\prime}_{i}+\sqrt{2}\lambda A^{\prime}_{i};$
$i\dot{A^{\prime}_{i}}=\sqrt{2}\lambda
C^{\prime}_{i}+(\omega-\Delta/2)A^{\prime}_{i},$ (3-11)

for $i=0,1,\ldots,n-1$. Solving the corresponding differential equations, one
can obtain

$C^{\prime}_{i}(t)=\frac{e^{-i(\omega+\xi
x_{i})t}}{\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}}}\\{[\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}\cos
t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}-$
$i(\frac{\Delta}{2}+\xi x_{i})\sin t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}}]C^{\prime}_{i}(0)-i\sqrt{2}\lambda\sin
t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}A^{\prime}_{i}(0)\\},$
$A^{\prime}_{i}(t)=\frac{e^{-i(\omega+\xi
x_{i})t}}{\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}}}\\{[\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}\cos
t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}+$
$i(\frac{\Delta}{2}+\xi x_{i})\sin t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}}]A^{\prime}_{i}(0)-i\sqrt{2}\lambda\sin
t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi x_{i})^{2}}C^{\prime}_{i}(0)\\},$
(3-12)

In the resonance case $\Delta=0$, by considering $\lambda=\frac{1}{\sqrt{2}}$,
the coefficients (3-12) read as

$C^{\prime}_{i}(t)=\frac{e^{-i(\omega+\xi x_{i})t}}{\sqrt{1+(\xi
x_{i})^{2}}}\\{[\sqrt{1+(\xi x_{i})^{2}}\cos t\sqrt{1+(\xi x_{i})^{2}}-i\xi
x_{i}\sin t\sqrt{1+(\xi x_{i})^{2}}]C^{\prime}_{i}(0)-i\sin t\sqrt{1+(\xi
x_{i})^{2}}A^{\prime}_{i}(0)\\},$ $A^{\prime}_{i}(t)=\frac{e^{-i(\omega+\xi
x_{i})t}}{\sqrt{1+(\xi x_{i})^{2}}}\\{[\sqrt{1+(\xi x_{i})^{2}}\cos
t\sqrt{1+(\xi x_{i})^{2}}+i\xi x_{i}\sin t\sqrt{1+(\xi
x_{i})^{2}}]A^{\prime}_{i}(0)-i\sin t\sqrt{1+(\xi
x_{i})^{2}}C^{\prime}_{i}(0)\\}.$ (3-13)

By using (3-8), one can obtain the time dependence of the coefficients
$C_{i}(t)$ and $A_{i}(t)$ of the state of the system in (3-6) via the inverse
$P$ transform as,

$C_{i}(t)=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}\chi^{*}_{j}(\alpha_{i})C^{\prime}_{j}(t),$
$A_{i}(t)=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}\chi^{*}_{j}(\alpha_{i})A^{\prime}_{j}(t),$
(3-14)

where, $*$ denotes the complex conjugate. It should be pointed out that, for
the probabilities associated with the state of the system as a superposition
of atomic states $|a_{i}\rangle$, and that of photonic states $|c_{i}\rangle$
(denoted by $P_{a}(t)$ and $P_{c}(t)$, respectively), the equation (3-14)
gives

$P_{a}=\sum_{i=0}^{n-1}|A_{i}(t)|^{2}=\frac{1}{n}\sum_{j,k}\underbrace{\sum_{i}\chi^{*}_{j}(\alpha_{i})\chi_{k}(\alpha_{i})}_{n\delta_{jk}}A^{\prime}_{j}(t)A^{\prime}_{k}(t)=\sum_{j=0}^{n-1}|A^{\prime}_{j}(t)|^{2}$
$P_{c}=\sum_{i=0}^{n-1}|C_{i}(t)|^{2}=\frac{1}{n}\sum_{j,k}\underbrace{\sum_{i}\chi^{*}_{j}(\alpha_{i})\chi_{k}(\alpha_{i})}_{n\delta_{jk}}C^{\prime}_{j}(t)C^{\prime}_{k}(t)=\sum_{j=0}^{n-1}|C^{\prime}_{j}(t)|^{2}$

For instance, considering the initial state

$|\psi(0)\rangle=\frac{1}{\sqrt{n}}(|g,2\rangle|g,0\rangle...|g,0\rangle+|g,0\rangle|g,2\rangle|g,0\rangle...|g,0\rangle+\ldots+|g,0\rangle...|g,0\rangle|g,2\rangle)=$
$|gg\ldots
g\rangle\otimes\frac{1}{\sqrt{n}}(|200...0\rangle+|020...0\rangle+\ldots+|0...002\rangle),$
(3-15)

in which the atoms are initially unentangled, whereas the photons are in a
multipartite entangled $W$-state. Then, the initial conditions are given by
$A_{j}(0)=0$ and $C_{j}(0)=\frac{1}{\sqrt{n}}$ for all $j=0,1,...,n-1$, and we
obtain via equation (3-8), $A^{\prime}_{i}(0)=0$, and
$C^{\prime}_{i}(0)=\frac{1}{n}\sum_{j}\chi_{i}(\alpha_{j})$ for all $i$. Then,
with the aid of Eq. (3-12), we obtain

$P_{a}(t)=\sum_{i=0}^{n-1}|A^{\prime}_{i}(t)|^{2}=\sum_{i}\sum_{j,k}\frac{2\lambda^{2}\sin^{2}t\sqrt{2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}}}{n^{2}[2\lambda^{2}+(\frac{\Delta}{2}+\xi
x_{i})^{2}]}\chi^{*}_{i}(\alpha_{j})\chi_{i}(\alpha_{k}),$
$P_{c}(t)=1-P_{a}(t).$ (3-16)

The above result indicates that, in the limit of large $\xi\rightarrow\infty$,
we have $P_{c}(t)\simeq 1$ for every time $t$, i.e., for large enough $\xi$,
all of the atoms will be at their ground state $|g\rangle$ at every time $t$.

In particular, at the resonance case $\Delta=0$, we obtain

$P_{a}(t)=\sum_{i}\sum_{j,k}\frac{2\lambda^{2}\sin^{2}t\sqrt{2\lambda^{2}+\xi^{2}x_{i}^{2}}}{n^{2}[2\lambda^{2}+\xi^{2}x_{i}^{2}]}\chi^{*}_{i}(\alpha_{j})\chi_{i}(\alpha_{k}),$

where by taking $\lambda=\frac{1}{\sqrt{2}}$, we have

$P_{a}(t)=\sum_{i}\sum_{j,k}\frac{\sin^{2}t\sqrt{1+\xi^{2}x_{i}^{2}}}{n^{2}[1+\xi^{2}x_{i}^{2}]}\chi^{*}_{i}(\alpha_{j})\chi_{i}(\alpha_{k}).$

From the above result it can be seen that for small hopping strength $\xi\ll$,
we have

$P_{a}(t)=\frac{\sin^{2}t}{n^{2}}\sum_{j,k}\underbrace{\sum_{i}\chi^{*}_{i}(\alpha_{j})\chi_{i}(\alpha_{k})}_{n\delta_{jk}}=\sin^{2}t,$

which indicates that for times $T=\frac{(2k+1)\pi}{2}$, we have $P_{a}(t)=1$,
whereas for times $T=l\pi$, we have $P_{c}(t)=1$, for any $k,l\in{Z}$. In
other words, for small hopping $\xi$, the initial state (3-15) evolves as

$|\psi(\frac{\pi}{2})\rangle=(\frac{-i}{\sqrt{n}}\sum_{i=1}^{n}|g\ldots
g\underbrace{e}_{i}g\ldots g\rangle)\otimes|00\ldots 0\rangle$

after time $T=\frac{\pi}{2}$ for which we have
$A_{i}(\frac{\pi}{2})=\frac{-i}{\sqrt{n}}$ for all $i$, and so the initially
unentangled atoms become entangled and the initial photonic $W$-state
oscillates between atomic and photonic $W$-states periodically, with time
period $\frac{\pi}{2}$. In order to evaluate the amount of quantum
entanglement between two atoms labeled by $l$ and $m$, one can calculate the
state of the corresponding two atoms as

$\rho^{lm}_{{}_{at}}(\frac{\pi}{2})=Tr_{{}_{ph;k\neq
l,m}}(|\psi(\frac{\pi}{2})\rangle\langle\psi(\frac{\pi}{2})|)=(1-\frac{2}{n})|gg\rangle\langle
gg|+\frac{1}{n}(|eg\rangle\langle eg|+$ $|ge\rangle\langle
ge|+|eg\rangle\langle ge|+|ge\rangle\langle eg|),$

where the partial trace is taken over photonic states and also over the atomic
states corresponding to the atoms other than the atoms $l$ and $m$. Now, the
Peres-Horodecki criteria [23, 24] known also as positive partial transpose
(PPT) criteria can be checked. The partial transpose of
$\rho^{lm}_{{}_{at}}(\frac{\pi}{2})$ with respect to the first subsystem is
given by

$[\rho^{lm}_{{}_{at}}(\frac{\pi}{2})]^{T_{1}}=\frac{1}{n}\left(\begin{array}[]{cccc}n-2&0&0&1\\\
0&1&0&0\\\ 0&0&1&0\\\ 1&0&0&0\\\ \end{array}\right),$

so that the corresponding eigenvalues $\lambda_{i}$ are given by $\frac{1}{n}$
(with double degeneracy) and
$\lambda_{\pm}=\frac{1}{2n}(n-2\pm\sqrt{(n-2)^{2}+4})$, and the negativity is
then
$N(\rho^{lm}_{{}_{at}}(\frac{\pi}{2}))=\sum_{i}|\lambda_{i}|-1=-2\lambda_{-}=\frac{1}{n}[\sqrt{(n-2)^{2}+4}-(n-2)]$.
Then, for the case of two cavities, $n=2$, we have
$N(\rho_{{}_{at}}(\frac{\pi}{2}))=1$ and the initially unentangled atoms,
become maximally entangled at times $T=\frac{(2k+1)\pi}{2}$, $k\in Z$.

##  4 Photon and excitation transfer between the atom-cavity systems

###  4.1 Photon transition

Hereafter, we consider the resonance case $\Delta=0$ and take
$\lambda=\frac{1}{\sqrt{2}}$. In order to investigate photon transition
between the cavities, we consider the initial state
$|\psi(0)\rangle=|g,2\rangle|g,0\rangle...|g,0\rangle$, with initial
conditions $A_{j}(0)=0$, $C_{0}(0)=1$ and $C_{l}(0)=0$ for all $j=0,1,...,n-1$
and $l=1,\ldots,n-1$. Then, one can obtain via equation (3-8),
$A^{\prime}_{j}(0)=0$, and
$C^{\prime}_{j}(0)=\frac{1}{\sqrt{n}}\chi_{j}(\alpha_{0})=\frac{d_{j}}{\sqrt{n}}=\frac{1}{\sqrt{n}}$
for every $j$. Now, with the aid of Eqs. (3-13) and (3-14), we obtain

$C_{i}(t)=\frac{1}{n}\sum_{j}\frac{e^{-i(\omega+\xi x_{j})t}}{\sqrt{[1+(\xi
x_{j})^{2}]}}[\sqrt{1+(\xi x_{j})^{2}}\cos t\sqrt{1+(\xi x_{j})^{2}}-i\xi
x_{j}\sin t\sqrt{1+(\xi x_{j})^{2}}]\chi^{*}_{j}(\alpha_{i}),$
$A_{i}(t)=\frac{-i}{n}\sum_{j}\frac{e^{-i(\omega+\xi x_{j})t}}{\sqrt{[1+(\xi
x_{j})^{2}]}}\sin t\sqrt{1+(\xi x_{j})^{2}}\chi^{*}_{j}(\alpha_{i}).$ (4-17)

A. Small hopping $\xi\ll$
The above result indicates that for small hopping $\xi\ll$, we have

$C_{i}(t)\simeq\frac{e^{-i\omega t}\cos
t}{n}\sum_{j}\chi^{*}_{j}(\alpha_{i}),$ $A_{i}(t)\simeq\frac{-ie^{-i\omega
t}\sin t}{n}\sum_{j}\chi^{*}_{j}(\alpha_{i}).$ (4-18)

It would be noticed that from the orthogonality relation
$\sum_{i}\chi^{*}_{j}(\alpha_{i})\chi_{k}(\alpha_{i})=n\delta_{j,k}$, we have
clearly

$\sum_{i}(|C_{i}(t)|^{2}+|A_{i}(t)|^{2})=\frac{1}{n^{2}}\sum_{j,k}\sum_{i}\chi^{*}_{j}(\alpha_{i})\chi_{k}(\alpha_{i})=1.$

The result (5-21) indicates that, at times $T=k\pi$, with $k\in Z$, we have
$A_{i}(T)=0$ and the probability of transfer of two photons from the first
cavity to the $i$-th cavity is given by

$P_{i}(T)=|C_{i}(T)|^{2}=\frac{1}{n^{2}}\sum_{j,k}\chi^{*}_{j}(\alpha_{i})\chi_{k}(\alpha_{i}).$

B. Large hopping $\xi\gg$
For large hopping strength $\xi\rightarrow\infty$, the result (4-17) leads to

$C_{i}(t)\simeq\frac{e^{-i\omega t}}{n}\\{\cos
t\sum_{j;x_{j}=0}\chi^{*}_{j}(\alpha_{i})+\sum_{j;x_{j}\neq 0}e^{-2i\xi
tx_{j}}\chi^{*}_{j}(\alpha_{i})\\},$ $A_{i}(t)\simeq\frac{-ie^{-i\omega
t}}{n}\\{\sin t\sum_{j;x_{j}=0}\chi^{*}_{j}(\alpha_{i})+\sum_{j;x_{j}\neq
0}\frac{e^{-i\xi tx_{j}}\sin(\xi tx_{j})}{\xi
x_{j}}\chi^{*}_{j}(\alpha_{i})\\}\simeq\frac{-ie^{-i\omega t}}{n}\sin
t\sum_{j;x_{j}=0}\chi^{*}_{j}(\alpha_{i}).$ (4-19)

Again, at times $T=k\pi$, with $k\in Z$, we have $A_{i}(T)\simeq 0$ and the
initially unentangled atoms remain unentangled.

###  4.2 Excitation transition

The excitation transfer between the atoms, can be achieved by preparing the
system initially at the state
$|\psi(0)\rangle=|e,0\rangle|g,0\rangle...|g,0\rangle$, with initial
conditions $C_{j}(0)=0$, $A_{0}(0)=1$ and $A_{l}(0)=0$ for all $j=0,1,...,n-1$
and $l=1,\ldots,n-1$. Then, one can obtain via equation (3-8),
$C^{\prime}_{j}(0)=0$, and $A^{\prime}_{j}(0)=\frac{1}{\sqrt{n}}$ for every
$j$. Again, with the aid of Eqs. (3-13) and (3-14), one can obtain

$C_{i}(t)=\frac{-i}{n}\sum_{j}\frac{e^{-i(\omega+\xi x_{j})t}}{\sqrt{[1+(\xi
x_{j})^{2}]}}\sin t\sqrt{1+(\xi x_{j})^{2}}\chi^{*}_{j}(\alpha_{i}),$
$A_{i}(t)=\frac{1}{n}\sum_{j}\frac{e^{-i(\omega+\xi x_{j})t}}{\sqrt{[1+(\xi
x_{j})^{2}]}}[\sqrt{1+(\xi x_{j})^{2}}\cos t\sqrt{1+(\xi x_{j})^{2}}+i\xi
x_{j}\sin t\sqrt{1+(\xi x_{j})^{2}}]\chi^{*}_{j}(\alpha_{i}).$ (4-20)

Similar to the photon transition (as argued in the previous subsection), one
can discuss excitation transfer for small and large hopping limits, where it
is seen that at the limiting cases $\xi\ll$ and $\xi\gg$, we have
$C_{i}(T)\simeq 0$ for times $T=l\pi$, with $l\in Z$; The probability of
transfer of atomic excitation from the first atom to the $i$-th atom, i.e.
$P_{i}(T)=|A_{i}(T)|^{2}$, can be evaluated similarly.

##  5 Examples

In this section we give two examples of uncolored Cayley networks for which
the photonic and atomic state transfer between atom-cavities is investigated
in details.
1\. The Cycle network $C_{n}$
If $G=Z_{n}$ is the finite cyclic group of order n and the set $S$ consists of
two elements, the standard generator of $G$ and its inverse, i.e., $a,a^{-1}$
with $a^{n}=1$, then the Cayley graph is the cycle $C_{n}$. In this case the
irreducible characters of the group are given by
$\chi_{k}(a^{l})=\omega^{kl}$, for $k,l=0,1,\ldots,n-1$, where
$\omega:=e^{\frac{2\pi i}{n}}$ is the $n$-th root of identity.
The adjacency matrix of the network is $A=S+S^{-1}$, where $S$ is the shift
matrix of order $n$ so that we have $S^{n}=I$. Then, the corresponding
eigenvalues of $A$ are given by
$x_{l}=\chi_{l}(a)+\chi_{l}(a^{-1})=\omega^{l}+\omega^{-l}=2\cos(\frac{2\pi
l}{n})$, $l=0,1,\ldots,n-1$. Now, one can evaluate the probability amplitudes
corresponding to the excitation or photon transition via the results (4-17).
For instance, in the case of even $n=2m$ with odd $m=2l+1$, we have
$x_{j}=\cos\frac{\pi j}{m}\neq 0$ for $j=0,1,\ldots,2m-1$, so that
$x_{j}=x_{2m-j}=-x_{m-j}$. Then, for the photon transition process, the
results (5-21) and (5-22) lead to

$C_{i}(t)\simeq\frac{e^{-i\omega t}\cos t}{2m}\sum_{j}\omega^{-ij}=e^{-i\omega
t}\cos t\;\ \delta_{i,0},$ $A_{i}(t)\simeq\frac{-ie^{-i\omega t}\sin
t}{2m}\sum_{j}\omega^{-ij}=e^{-i\omega t}\sin t\;\ \delta_{i,0}$ (5-21)

and

$C_{i}(t)\simeq\frac{e^{-i\omega t}}{2m}\sum_{j=0}^{2m-1}e^{-2i\xi
tx_{j}}\omega^{-ij}=\frac{e^{-i\omega t}}{2m}\\{e^{-4i\xi t}+(-1)^{i}e^{4i\xi
t}+2\sum_{j=1}^{m-1}e^{-2i\xi x_{j}t}\cos\frac{\pi ji}{m}\\},$
$A_{i}(t)\simeq\frac{-ie^{-i\omega t}}{2m}\sum_{j}\frac{e^{-i\xi
tx_{j}}\sin(\xi tx_{j})}{\xi x_{j}}\omega^{-ij}\simeq 0$ (5-22)

for the asymptotic limits $\xi\ll$ and $\xi\gg$, respectively. The above
relations indicate that for small hopping $\xi\rightarrow 0$, we have

$C_{0}(t)\simeq e^{-i\omega t}\cos t,\;\;\ A_{0}(t)\simeq e^{-i\omega t}\sin
t,\;\;\;\ C_{i}(t)=A_{i}(t)=0;\;\ \mbox{for}\;\ i\neq 0,$

so that after time $t$, the initial state
$|\psi(0)\rangle=|g,2\rangle|g,0\rangle\ldots|g,0\rangle$ evolves as

$|\psi(t)\rangle\simeq e^{-i\omega t}(\cos t|g,2\rangle+\sin
t|e,0\rangle)|g,0\rangle\ldots|g,0\rangle.$

Therefore, the photon transition can not be achieved and a quantum correlation
is generated between the atomic and photonic states of the first cavity. On
the other hand, for large hopping $\xi\rightarrow\infty$, one obtains
$A_{i}(t)\simeq 0$ for all times and

$C_{i=odd}(t)\simeq\frac{-ie^{-i\omega t}}{m}\\{\sin 4\xi
t+2\sum_{j=1}^{\frac{m-1}{2}}\sin(2\xi x_{j}t)\cos\frac{\pi ji}{m}\\},$
$C_{i=even}(t)\simeq\frac{e^{-i\omega t}}{m}\\{\cos 4\xi
t+2\sum_{j=1}^{\frac{m-1}{2}}\cos(2\xi x_{j}t)\cos\frac{\pi ji}{m}\\}.$

For example, for $n=6$ with $m=3$, we have $x_{0}=2$, $x_{1}=x_{5}=1$,
$x_{2}=x_{4}=-1$, $x_{3}=-2$ and the above amplitudes are given by

$C_{0}(t)=\frac{e^{-i\omega t}}{3}\\{\cos 4\xi t+2\cos 2\xi t\\},$
$C_{1}(t)=C_{5}(t)=\frac{-ie^{-i\omega t}}{3}\\{\sin 4\xi t+\sin 2\xi t\\},$
$C_{2}(t)=C_{4}(t)=\frac{-ie^{-i\omega t}}{3}\\{\cos 4\xi t-\cos 2\xi t\\},$
$C_{3}(t)=\frac{-ie^{-i\omega t}}{3}\\{\sin 4\xi t-2\sin 2\xi t\\},$

Then, after time $T=\frac{3\pi}{8\xi}$, we obtain the probabilities associated
with two photon transition from the first cavity to the other cavities as
follows

$P_{0}(T)=\frac{2}{9}\simeq 0.22,\;\;\ P_{1}(T)=P_{5}(T)=0.01,\;\;\
P_{2}(T)=P_{4}(T)=0.05,\;\;\ P_{3}(T)=\frac{5.76}{9}\simeq 0.64.$

2\. The Hypercube network $Q_{d}$
The Cayley graph of the direct product of groups $Z_{2}$, $d$ times (with the
cartesian product of generating sets as a generating set) is the cartesian
product of the corresponding Cayley graphs, which gives the hypercube graph
$Q_{d}$ (called also the Hamming network $H(d,2)$). Thus, the Cayley graph of
the abelian group $G=\underbrace{Z_{2}\times Z_{2}\times\ldots\times
Z_{2}}_{d-times}$ with the set of generators consisting of $n$ elements
$(1,0,0,\ldots,0),(0,1,0,\ldots,0),\ldots,(0,0,\ldots,0,1)$, is the hypercube
network $Q_{d}$ with $n=2^{d}$ nodes and the adjacency matrix

$A=\sum_{i=1}^{d}I\otimes\ldots\otimes
I\otimes\underbrace{\sigma_{x}}_{i-th}\otimes I\ldots\otimes I.$

The irreducible characters of the group $G$ are direct products of the
irreducible characters of the group $Z_{2}$, i.e., by denoting $k,l\in G$ in
the binary representation $k\equiv k_{d-1}\ldots k_{1}k_{0}$ and $l\equiv
l_{d-1}\ldots l_{1}l_{0}$, with $k_{i},l_{i}\in\\{0,1\\}$ (so that
$k=k_{0}+2^{1}k_{1}+\ldots+2^{d-1}k_{d-1}$), we have $\chi_{k}(l)=(-1)^{k\cdot
l}$, where $k\cdot l=\sum_{i=0}^{d-1}k_{i}l_{i}$. Also, one can easily show
that the eigenvalues of the adjacency matrix $A$ with the corresponding
degeneracy degrees are given by $x_{i}=d-2i$ and
$D(x_{i})=\frac{d!}{i!(d-i)!}$ for $i=0,1,\ldots,d$, respectively.

For instance, for $d=2$, we have $x_{0}=2$, $x_{1}=x_{2}=0$ and $x_{3}=-2$.
Then, the Eq. (4-17) leads to

$C_{i}(t)=\frac{e^{-i\omega
t}}{4\sqrt{1+4\xi^{2}}}\\{\sqrt{1+4\xi^{2}}(e^{-2i\xi t}+\chi_{3}(i)e^{2i\xi
t})\cos t\sqrt{1+4\xi^{2}}-2i\xi(e^{-2i\xi t}-\chi_{3}(i)e^{2i\xi t})\sin
t\sqrt{1+4\xi^{2}}+$ $\sqrt{1+4\xi^{2}}(\chi_{1}(i)+\chi_{2}(i))\cos t\\},$
$A_{i}(t)=\frac{-ie^{-i\omega t}}{4\sqrt{1+4\xi^{2}}}\\{(e^{-2i\xi
t}+\chi_{3}(i)e^{2i\xi t})\sin
t\sqrt{1+4\xi^{2}}+\sqrt{1+4\xi^{2}}(\chi_{1}(i)+\chi_{2}(i))\sin t\\}.$

The above results are respectively simplified for the small and large hopping
parameter $\xi$, as follows

$C_{i}(t)\simeq\frac{e^{-i\omega t}\cos t}{4}\\{e^{-2i\xi
t}+\chi_{3}(i)e^{2i\xi t}+\chi_{1}(i)+\chi_{2}(i)\\},$
$A_{i}(t)\simeq\frac{-ie^{-i\omega t}\sin t}{4}\\{e^{-2i\xi
t}+\chi_{3}(i)e^{2i\xi t}+\chi_{1}(i)+\chi_{2}(i)\\};\;\;\;\;\ \mbox{for}\;\;\
\xi\ll$

and

$C_{i}(t)\simeq\frac{e^{-i\omega t}}{4}\\{e^{-4i\xi t}+\chi_{3}(i)e^{4i\xi
t}+(\chi_{1}(i)+\chi_{2}(i))\cos t\\},$ $A_{i}(t)\simeq\frac{-ie^{-i\omega
t}}{4}(\chi_{1}(i)+\chi_{2}(i))\sin t,\;\;\;\;\;\ \mbox{for}\;\;\ \xi\gg.$

By substituting $\chi_{1}(0)=\chi_{1}(2)=1,\;\ \chi_{1}(1)=\chi_{1}(3)=-1$,
$\chi_{2}(0)=\chi_{2}(1)=1,\;\ \chi_{2}(2)=\chi_{2}(3)=-1$ and
$\chi_{3}(0)=\chi_{3}(3)=1,\;\ \chi_{3}(1)=\chi_{3}(2)=-1$ in the above
relations, the corresponding amplitudes are determined for every $i=0,1,2,3$.

##  6 Conclusion

In summery, quantum state transfer and entanglement generation between $n$
coupled atom-cavity systems with uncolored Cayley interacting networks, was
analyzed. By employing the excitation-photon conservation symmetry of the
Hamiltonian and introducing some suitable ‘generalized’ Fourier transformed
basis states for the state space of the system, the corresponding Hilbert
space was block-diagonalized with $2$ dimensional blocks. Due to this useful
reduction, the corresponding Shrödinger equation was solved exactly, and state
transfer (excitation or photon transition between the atoms or the cavities)
and entanglement generation between the atoms were discussed. The large and
small hopping limits was discussed where, it was shown that for large hopping
strength, the initially unentangled atoms remain effectively unentangled
forever whereas for small hopping, the initially unentangled atoms become
entangled and the initial photonic $W$-state oscillates between atomic and
photonic $W$-states periodically.

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