$\displaystyle=-\frac{16\Omega^{4}\left(\Gamma^{2}+6\delta^{2}\right)}{\chi}+\frac{8\Omega^{2}\left(\Gamma^{2}+6\delta^{2}\right)\left(\delta^{2}+2\Omega^{2}\right)}{\chi}\rho_{A,A}(t),$
(122) $\displaystyle\rho_{S,S}^{SS}[\rho_{A,A}(t)]$
$\displaystyle=-\frac{8\Omega^{2}\left(\delta^{2}-2\Omega^{2}\right)\left(\Gamma^{2}+2\delta^{2}+8\Omega^{2}\right)}{\chi}+\frac{4\left(\delta^{4}-4\Omega^{4}\right)\left(\Gamma^{2}+2\delta^{2}+8\Omega^{2}\right)}{\chi}\rho_{A,A}(t),$
(123) $\displaystyle\rho_{S,ee}^{SS}[\rho_{A,A}(t)]$
$\displaystyle=-\frac{16i\sqrt{2}\Gamma\Omega^{3}\left(\delta^{2}-2\Omega^{2}\right)}{\chi}+\frac{8i\sqrt{2}\Gamma\Omega\left(\delta^{4}-4\Omega^{4}\right)}{\chi}\rho_{A,A}(t),$
(124) $\displaystyle\rho_{ee,ee}^{SS}[\rho_{A,A}(t)]$
$\displaystyle=-\frac{64\Omega^{4}\left(\delta^{2}-2\Omega^{2}\right)}{\chi}+\frac{32\Omega^{2}\left(\delta^{4}-4\Omega^{4}\right)}{\chi}\rho_{A,A}(t),$
(125)

with

$\chi\equiv\Gamma^{4}\left(\delta^{2}+2\Omega^{2}\right)+\Gamma^{2}\left(6\delta^{4}-4\delta^{2}\Omega^{2}+64\Omega^{4}\right)+8\left(\delta^{6}-4\delta^{4}\Omega^{2}+4\delta^{2}\Omega^{4}+48\Omega^{6}\right).$
(126)

Figure 15:  (a) Liouvillian gap in terms of the cooperativity $C$ and qubit-
qubit detuning $\delta$. Solid black lines are exact computations, dashed
black lines are analytical prediction from the relaxation rate, Eq. (130), and
dashed white lines are analytical predictions from the approximated relaxation
rate, Eq. (131). (b) Dynamics of the antisymmetric state for two different
values of cooperativity: $C=10^{4}$ (blue) and $C=10^{6}$ (red). Solid lines
correspond to exact computations and blue dashed lines are analytical
predictions from the hierarchical adiabatic elimination considering the full
relaxation rate. Parameters: (a-b) $r_{12}=50\ \text{nm}$, $k=2\pi/780\
\text{nm}^{-1}$, $J=10.65\gamma$, $\gamma_{12}=0.967$, $\Delta=0$,
$\Omega=10^{4}\gamma$, $g=10^{-1}\kappa$; (b) blue curve $C=10^{4}$,
$\delta=10^{3}\gamma$; red $C=10^{6}$, $\delta=10^{3}\gamma$.

Now, we can substitute the pseudostationary value of $\rho_{gg,S}$ back into
Eq. (113), and obtain the effective differential equation for the
antisymmetric state

$\dot{\rho}_{A,A}(t)=\frac{8\Gamma\delta^{2}\Omega^{2}\left(\Gamma^{2}+2\delta^{2}+8\Omega^{2}\right)}{\chi}-\frac{4\Gamma\delta^{2}\left(\delta^{2}+2\Omega^{2}\right)\left(\Gamma^{2}+2\delta^{2}+8\Omega^{2}\right)}{\chi}\rho_{A,A}(t).$
(127)

Solving this equation, we obtain the analytical expression for the evolution
of $\rho_{A,A}(t)$

$\rho_{A,A}(t)=\rho_{A,A}^{SS}(1-e^{-\Gamma_{\text{eff}}t}),$ (128)

where $\rho_{A,A}^{SS}$ is the steady-state value

$\rho_{A,A}^{SS}=\frac{2\Omega^{2}}{\delta^{2}+2\Omega^{2}},$ (129)

and $\Gamma_{\text{eff}}$ stands for the effective relaxation rate,

$\Gamma_{\text{eff}}=\frac{4\Gamma\delta^{2}\left(\delta^{2}+2\Omega^{2}\right)\left(\Gamma^{2}+2\delta^{2}+8\Omega^{2}\right)}{\Gamma^{4}\left(\delta^{2}+2\Omega^{2}\right)+\Gamma^{2}\left(6\delta^{4}-4\delta^{2}\Omega^{2}+64\Omega^{4}\right)+8\left(\delta^{6}-4\delta^{4}\Omega^{2}+4\delta^{2}\Omega^{4}+48\Omega^{6}\right)},$
(130)

which serves as an analytical estimation of the Liouvillian gap. We can
simplify this expression further under the assumption $\Omega\gg\delta$, which
is a natural condition for this mechanism to take place.The relaxation rate
then reduces to

$\Gamma_{\text{eff}}=\frac{4\Gamma_{S}\delta^{2}}{\Gamma_{S}^{2}+24\Omega^{2}},$
(131)

which is the expression shown in the main text.

###  E.3 Validity of the relaxation rate $\Gamma_{\text{eff}}$

Figure 16:  Entanglement detection and control for Mechanism II: stationary
concurrence and optical properties of the emitted light by the cavity. (a),
(b), (c) Steady-state concurrence, transmission intensity and photon
correlations versus the Rabi frequency of the laser $\Omega$ and the cavity-
laser detuning $\Delta_{a}$. Top panels show a cut along $\Omega/\gamma$,
marked by the horizontal white-dashed lines in the density plots. Parameters:
$r=50$ nm, $k=2\pi/780\ \text{nm}^{-1}$, $J=10.65\gamma$,
$\gamma_{12}=0.967\gamma$, $\Delta=0$, $\delta=10^{2}J$, $R=10^{3}\gamma$,
$\kappa=10^{4}\gamma$, $g=10^{-1}\kappa$.

Finally, we analyze the validity of the analytical relaxation rates in Eq.
(130) and Eq. (131) obtained under the application of the HAE method to
estimate the Liouvillian gap.

In Fig. 15(a) we show the exact computation of the Liouvillian gap in terms of
the cooperativity and the qubit-qubit detuning $\delta$. Through the different
contour lines included, we compare the result obtained with the full model Eq.
(4) (black straight lines) with the analytical estimations of the Liouvillian
gap given by the full expression in Eq. (130) (black dashed lines), and the
simplified expression in Eq. (131) (white dashed lines). Both analytical
expression feature a good agreement with the numerical computation provided
that $\kappa\geq\Omega$, which is precisely the condition required for
Mechanism II to take place. Therefore, the deviations below this limit are
expected.

In addition, we may observe the agreement between the exact (straight lines)
and the analytical (dashed lines) computations in Fig. 15(b), where we depict
the time evolution of the antisymmetric state for two values of the
cooperativity: (i) $C=10^{4}$ (blue lines), and (ii) $C=10^{6}$ (red lines).
We note that both the analytical expressions for the density matrix element
for $\rho_{A,A}(t)$ and the relaxation rate $\Gamma_{\text{eff}}$ provide a
good approximation to exact computation. We note that
$\gamma/\lambda_{2}\approx 1$ approximately coincides with the onset of
entanglement generation.

##  APPENDIX F ADDITIONAL OBSERVATIONAL TESTS

In this appendix we show the same optical magnitudes presented in Fig. 10,
measured in the regime where Mechanism II is active. As we noted, in this
particular regime the generation of stationary entanglement is independent of
the cavity frequency for values $\Delta_{a}\ll\kappa$, resulting in optical
observables that are nearly featureless. This is evident in Fig. 16, where we
depict the stationary concurrence (a), the mean intensity (b), and the second
order correlations (c).

We observe, as $\Omega$ is increased, the concurrence increases from zero,
reaching a maximum value and eventually vanishing. Contrary to the case of
Mechanism I, these features are not correlated with any of the observables
shown here, and therefore we conclude that, by themselves, they do not serve
as any type of indicator of the stabilization of entanglement. This motivates
a proposal of an alternative approach based on the measurement of correlations
between different spectral lines, see Fig. 11.

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1 1 3<EMAIL_ADDRESS>11institutetext:  Instituto de Astronomía y Física
del Espacio, CONICET–UBA, Argentina

# The Fe i lines and the chromospheric activity

M. C. Vieytes 11

(…; …)

Las líneas de Fe neutro son las más abundantes en el espectro de estrellas
tardías. Se utilizan para obtener diferentes parámetros estelares básicos como
abundancias relativas, gravedad superficial, y temperatura efectiva. Sin
embargo, los centros de muchas de estas líneas se forman en la cromósfera
estelar, viéndose afectados por los cambios producidos por la actividad
magnética. Calculando modelos semiempíricos de la atmósfera estelar fuera del
equilibrio termodinámico local (NLTE por sus siglas en inglés), es posible
estudiar la formación de estas líneas en detalle. En este trabajo presentamos
un nuevo modelo actualizado del átomo de Fe i, que nos permite calcular 1715
líneas de Fe i entre 3000 y 7000 Å en estrellas con diferente nivel de
actividad cromosférica. Los resultados preliminares muestran la presencia de
rangos espectrales más sensibles al calentamiento cromosférico, sugiriendo que
las líneas con mayores variaciones deben ser tenidas en cuenta como fuentes de
error, e incluso deben ser excluídas, al momento de ser utilizadas para los
fines descriptos anteriormente.

###### Abstract

Neutral Fe lines are the most abundant in the spectrum of late stars. They are
used to obtain different basic stellar parameters such as relative abundances,
surface gravity, and effective temperature. However, the centers of many of
these lines are formed in the stellar chromosphere, being affected by the
changes produced by magnetic activity. Calculating non-local thermodynamic
equilibrium (NLTE) semi-empirical models of the stellar atmosphere, it is
possible to study the formation of these lines in detail. We present a new
updated model of the Fe i atom, which allows us to calculate 1715 lines of Fe
i between 3000 and 7000 Å in stars with different levels of chromospheric
activity. Preliminary results show the presence of spectral ranges more
sensitive to chromospheric heating, suggesting that the lines with greater
variations must be taken into account as a source of uncertainty, and could
even be excluded, when be used for the purposes described above.

###### keywords:

stars: solar-type — stars: chromospheres — stars: activity

##  1 Introduction

The Fe i lines in the stellar spectra of late-type stars are used to
characterize them. Basic parameters such as effective temperature,
metallicity, or surface gravity can be calculated from these lines. They are
also used in the detection of extrasolar planets by the radial velocity
method. For these reasons, an understanding of their behavior under physical
processes acting on line formation, such as the chromospheric activity, is
crucial. Any associate change due to these processes on the line profile can
produce miscalculations if it is not properly understood and described.

Previous observational studies have shown that some spectral Fe i lines
present changes in their depth and width due to chromospheric stellar activity
(Wise et al., 2018; Spina et al., 2020). In some cases, the changes in the
line profile can track the magnetic cycle (Livingston et al., 2007; Flores et
al., 2016).

Semi-empirical NLTE models of the stellar atmospheres can be used as a tool to
study the formation of line profiles in detail. But to obtain reliable
synthetic spectra, it is necessary to build atomic and molecular models of the
species that form the atmosphere, considering the different processes that can
take place between them and the radiation field.

The goal of the present work is to update the data and expand the level
structure of the Fe i atomic model. Further on, using the new atomic model, we
are able to calculate the spectra of a solar-type star at two different
atmospheric thermal structures that differ in the level of chromospheric
activity, which was estimated by the flux in the Ca K line profile. These
synthetic spectra allow us to study the influence of activity on the Fe i line
profiles.

##  2 the new Fe i atomic model

The previous Fe i atomic model is described by Fontenla et al. (2015) and
references therein. The energy level structure contained the first 119 energy
levels, and their corresponding 381 sublevels from the National Institute of
Standards and Technology (NIST) database version 2004 (Kramida et al., 2004).
The maximum energy reached using this atomic structure was 54386.189 cm-1.
This atomic model permitted the calculation, in NLTE, of 1624 Fe I lines.

We spanned and updated the atomic structure, using the latest version 2022 of
the NIST database (Kramida et al., 2022).The new Fe I model was built
considering levels and sublevels until reaching a similar energy to the
previous model, but the number of levels was increased to 141, and their
corresponding sublevels to 444. We also selected from the same version of
NIST, 2397 lines with well-determined oscillator strength, and 1715 lines in
the range between 3000 and 7000 Å. This portion of the spectra is mostly used
to characterize the stars and for exoplanet detection. Considering that the
previous atomic model included 1624 lines, we increased the calculated lines
by 47%. The distribution of these line centers in the visible spectrum for
both atomic models is displayed in the histogram of Figure 1. The new
structure of energy levels and subleves allows to take into account an
increased number of Fe i lines mainly in the range between 3000 to 4250 Å than
the older model.

The line broadening processes for the line profiles considered were the usual,
radiative, Stark, and van der Waals. The approximate values where taken from
Kurucz & Bell (1995) when the line was present, otherwise only the radiative
coefficient was considered.

Figure 1: Comparison of the distribution of the Fe I lines calculated in NLTE
with the new (green) and old (orange) atomic model between 3000 and 7000 Å.

Regarding the rest of the atomic parameters, the electron collisions were
obtained from the semi-empirical approximation of Seaton (1962) for allowed
transitions, and van Regemorter (1962) when transitions were forbidden. For
ionizing collisions, the NRL Plasma Formulary (2005) ionization rate equation,
adapted from Earth atmospheric calculations, was used. We neglected the H i
inelastic collisions because we didn’t consider metal-poor stars.

The photoionization cross-sections were taken from TOPbase3, the Opacity
Project atomic database (Cunto et al., 1993). The radiative recombination was
included as the inverse of the photoionization process.

##  3 NLTE calculations

There are several previous works that show the importance of an NLTE
calculation for the level populations of Fe i. The more comprehensive is the
one by Mashonkina et al. (2011). Holzreuter & Solanki (2013) showed that 3D
NLTE calculation could be important for some regions with strong horizontal
temperature gradients such as the boundaries of granules, but in spatial
averaged profiles calculations, as is the case for stars, the difference
between 1D and 3D NLTE is less than 1%.

In this work, the calculations were done using the Solar-Stellar Radiation
Physical Modeling (SSRPM) system (Fontenla et al., 2015). This system allows
us to self-consistently solve the equations of radiation transport and
statistical equilibrium for a plane-parallel atmosphere, assuming hydrostatic
equilibrium.

The SSRPM computes in NLTE the populations of H, H-, H2, and 52 neutral and
lowly ionized atomic species, considering partial redistribution (PRD). These
species are shown in Table 2 of Fontenla et al. (2015), although in this work
we modify the number of levels and sublevels of Fe i. Additionally, the NLTE
effectively thin approximation is used for highly ionized species. Besides,
the most abundant diatomic molecular species are calculated in LTE. A more
detailed explanation of the code can be found in Fontenla et al. (2016) and
references therein.

We selected the atmospheric model corresponding to the solar magnetic inter-
network model 1401 built by Fontenla et al. (2015) to calculate the spectra
representing a low chromospheric active G2 star. This feature is the most
abundant in the atmosphere of the quiet Sun (Fontenla et al., 2011), and for
that reason, it is a good model for a low-activity star.

To generate a more chromospheric active G2 star, we used the same technique as
in Vieytes & Mauas (2004) and Andretta & Giampapa (1995) by shifting the
thermal structure inward, i.e. toward higher mass column densities. Figure 2
shows the thermal structure of the two models calculated, and their resulting
Ca ii K profiles. In black the model 1401, and in magenta a more active model
built as it was previously described.

The Ca ii H&K doublet is widely used as a proxy of chromospheric activity due
to its response to the chromospheric heating produced by stellar activity. For
this reason, an increasing Ca ii K profile ensures an increasing chromospheric
activity. It can be seen in the figure that the more active model gives a
higher Ca ii K profile, as is expected for a more chromospherically active
star.

It is important to note that both atmospheric models have the same
photospheric thermal structure, as can be seen in Figure 2. We are interested
only in studying the effect of a changing chromosphere due to the heating
produced by the stellar activity. on the same star.

Figure 2: Left: Atmospheric models with different chromospheric level of
activity. In black the model for the Quiet Sun 1401, and in magenta a more
active model built from it. Right: Ca II K profiles calculated from each
model, plotted with the same line colour that the corresponding model.

##  4 Results

We computed the NLTE population for all the species compounding the
atmosphere, and the resulting synthetic visible spectra.

To determine the change in the Fe i lines, we calculate the relative
difference (in percentage) between both spectra. We identified the lines
cross-crossing the wavelengths in the spectra with the line centers of the
1715 lines in our database belonging to the spectral range under study.

Figure 3 shows the result of the calculations for the spectral range from 3000
to 7000 Å.

38 % of the Fe i lines calculated with these models present a change in their
line depths between 5 and 497 %.

It is possible to distinguish two main regions severally affected by the
change in the thermal structure of the model, one between 3000 and 4550 Å, and
the other between 4780 to 5500 Å approximately.

The change shown in the figure means that these lines are affected in their
line center by the level of chromospheric activity in the star. The main
reason these lines are changing is that their formation region move, totally
or partially, outward to chromospheric regions as the temperature increases.

62 % of the lines are affected by less than 5 %, showing that the peak of the
contribution function remains at approximately the same photospheric height,
where both models have the same thermal structure.

Although we were interested in studying the change in Fe i lines with
chromospheric activity, this methodology permits us to track the change in any
line present in our synthetic spectra, for all the species from which the
stellar atmosphere is made up. In a forthcoming paper, Vieytes et al. (2024)
will address this idea, increasing the number of stars with different levels
of chromospheric activity.

Figure 3: Relative difference in percentage between the Fe i line centers
calculated with the two models for the visible range.

##  5 Conclusions

Considering the importance of Fe i in the stellar spectra of solar-type stars,
we built a more comprehensive Fe i atomic model. This new model increases 47%
of the number of lines that we can calculate with a 1D full NLTE semiempirical
stellar model atmosphere. Using this new atomic model, we computed two
atmospheric models representing dG2 stars with different levels of
chromospheric activity. Calculating the relative difference in the percentage
of the two synthetic spectra, we identified 652 (38 %) Fe i lines that are
affected by chromospheric activity. These lines should be not considered in
the stellar parameter calculations and be also excluded from the stellar mask
used for exoplanet detections.

This research was supported by grants PICT 2018-2895 and PICT 2019-4342 from
the Agencia Nacional de Promoción Científica y Tecnológica (MINCyT,
Argentina).

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