11institutetext: CONICET-Universidad Nacional de Córdoba, Instituto de
Astronomía Teórica y Experimental (IATE), Laprida 854, X5000BGR,Córdoba,
Argentina 22institutetext: Observatorio Astronómico, Universidad Nacional de
Córdoba, Laprida 854, X5000BGR, Córdoba, Argentina 33institutetext: Instituto
de Astronomía y Física del Espacio (IAFE), Ciudad Universitaria - Pabellón 2,
Intendente Güiraldes 2160 (C1428EGA), Ciudad Autónoma de Buenos Aires -
Argentina 44institutetext: Remote Sensing Division, Naval Research Laboratory,
Code 7213, 4555 Overlook Ave SW, Washington, DC, 20375, USA 55institutetext:
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA
91106, USA

# One size fits all: Insights on Extrinsic Thermal Absorption from Similarity
of Supernova Remnant Radio Continuum Spectra

Mario G. Abadi  E-mail<EMAIL_ADDRESS>Gabriela Castelletti 33
Leonardo Supan 33 Namir E. Kassim 44 Joseph W. Lazio 55

(Received May XX, 2023; accepted Month XX, 2023)

Typically, integrated radio frequency continuum spectra of supernova remnants
(SNRs) exhibit a power-law form due to their synchrotron emission. In numerous
cases, these spectra show an exponential turnover, long assumed due to thermal
free-free absorption in the interstellar medium. We use a compilation of
Galactic radio continuum SNR spectra, with and without turnovers, to constrain
the distribution of the absorbing ionized gas. We introduce a novel
parameterization of SNR spectra in terms of a characteristic frequency
$\nu_{\ast}$, which depends both on the absorption turnover frequency and the
power-law slope. Normalizing to $\nu_{\ast}$ and to the corresponding flux
density, $S_{\ast}$, we demonstrate that the stacked spectra of our sample
reveal a similarity in behavior with low scatter (r.m.s. $\sim$15%), and a
unique exponential drop-off fully consistent with the predictions of a free-
free absorption process. Observed SNRs, whether exhibiting spectral turnovers
or not, appear to be spatially well mixed in the Galaxy without any evident
segregation between them. Moreover, their Galactic distribution does not show
a correlation with general properties such as heliocentric distance or
Galactic longitude, as might have been expected if the absorption were due to
a continuous distribution of ionized gas. However, it naturally arises if the
absorbers are discretely distributed, as suggested by early low-frequency
observations. Modeling based on H II regions tracking Galactic spiral arms
successfully reproduces the patchy absorption observed to date. While more
extensive statistical datasets should yield more precise spatial models of the
absorbing gas distribution, our present conclusion regarding its inhomogeneity
will remain robust.

###### Key Words.:

ISM: supernova remnants; ISM: H II regions, radio continuum: general; radio
continuum: ISM

##  1 Introduction

Continuum radio emission from shell-type supernova remnants (SNRs) is due to
synchrotron emission from relativistic electrons spiraling in compressed
circumstellar and interstellar magnetic fields, inferred to be a major source
of Galactic cosmic ray electrons. Their intrinsic nonthermal continuum spectra
typically reflect a power-law dependence of the flux density on the frequency,
$S\propto\nu^{\alpha}$, the spectral slope of which is shaped by particle
acceleration processes associated with their blast wave. Both, linear and
nonlinear diffusive shock acceleration models have been proposed to explain
the production of the energetic particles and the synchrotron emission (see
e.g., Blandford & Eichler 1987, Malkov & Drury 2001 for reviews).

Free-free thermal absorption from either internal or external ionized gas can
impact the integrated radio spectra of SNRs by causing turnovers at the lowest
frequencies ($\nu<$ 100 MHz). In the former case, for shell-type SNRs the
absorption is concentrated near the center of the remnant. It is caused by the
ionization of cool, unshocked ejecta lying interior to the reverse shock and
ionized by its X-ray emission. It is has been detected in the bright, young
SNRs Cas A (Kassim et al., 1995; DeLaney et al., 2014; Arias et al., 2018) and
Tycho (Arias et al., 2019). Hints of interior absorption attributed to its
thermal filaments has also been discerned in the Crab nebula (Bietenholz et
al., 1997). These special cases of interior or intrinsic thermal absorption in
young SNRs remain observationally rare and will no longer be considered in
this paper.

The much more common source of turnovers in SNR continuum spectra is exterior
or extrinsic absorption from ionized gas outside the main boundaries of the
remnant and located anywhere along the line of sight. Although it is customary
to distinguish two different cases, local and non-local, the spectra by
themselves do not give any information about the distance between the emitter
and the absorber. The local case can occur near the periphery of the SNR as
the blast wave interacts with the surrounding interstellar medium (ISM) or
from foreground H II regions within the same complex. On the other hand, the
non-local case refers to absorption by unrelated more distant ionized gas.
Evidence for spectral turnovers under 100 MHz with thermal absorption
signatures dates to the classic works of Dulk & Slee (1972, 1975) and Kassim
(1989), but were significantly limited due to poor angular resolution imposed
by the Earth’s ionosphere on instruments at the time. Kassim (1989) suggested
at least some absorption occurs non-locally in extended ($\sim$100 pc), low-
density envelopes (EHEs) associated with normal Galactic H II regions along
the line of sight. EHEs had been inferred earlier from widespread so-called
Galactic ridge recombination lines detected at meter wavelengths (325 MHz)
(Anantharamaiah, 1985).

Synchrotron emission combined with thermal absorption produce integrated radio
continuum spectra for most Galactic SNRs that can be fit by a power-law plus
an exponential turnover at low frequencies ($\nu<100$ MHz). Turnover spectra
offer crucial insights into the ionized gas properties of the absorbers in our
Galaxy. In particular, the presence and locations of SNRs with turnover
spectra, or limits implied by those without turnovers at currently accessible
frequencies, offer unique constrains on the spatial distribution of this gas,
improving our understanding of the interstellar medium. However, the use of
different reference frequencies ($\nu_{0}$) to parameterize the optical depth
($\tau_{0}$) associated with these turnovers has made it awkward to compare
spectra from different SNRs and formulate a clear picture of the underlying
absorbing ionized medium. Therefore, in order to make the comparison of
optical depths easier and ensure comparability between published values, it is
beneficial to convert all values to a common reference frequency. To address
this issue, we propose an alternative parameterization of turnover spectra
that avoids the need for multiple reference frequencies, streamlining the
analysis and facilitating the comparison of different SNR samples.

Early sub-arcminute resolution imaging below 100 MHz of SNRs W49B (Lacey et
al., 2001) and 3C 391 (Brogan et al., 2005) began resolving thermal absorption
towards SNRs. They foreshadowed emerging arc-second resolution, mJy
sensitivity capabilities to expand thermal absorption studies to a larger
population of sources for constraining the ionized gas, diagnosing direct
interactions, and constraining the relative radial superposition of emitters
and absorbers. More recently, Castelletti et al. (2021) significantly expanded
the sample, by performing a detailed analysis of 9 SNRs showing turnovers on
the basis of radio, infrared, and molecular data. Four of these have been
attributed to the discrete but non-local extrinsic scenario proposed by Kassim
(1989) (see ”Absorption from extended envelopes of normal H II regions (EHEs)”
in their Table 3). The remaining 5 are far better explained by local extrinsic
absorption by foreground ionized gas in the SNR neighbourhood (see
”Absorption: Special cases” in their Table 3). Comparing these two tables, the
main parameter distinguishing between the non-local and local extrinsic
scenarios seems to be the best-fit free-free optical depth $\tau_{0}$ (at a
reference frequency $\nu_{0}$=74 MHz) in the integrated continuum spectra: a
low one, $\tau_{0}\sim 0.1$, for the case of external H II regions versus a
high one, $\tau_{0}\sim 1$, in the case of local H II regions or interactions
at the ionised interface between SNR blast wave with their host complex. The
only exception seems to be G39.2$-$0.3 (3C 396), which has a low $\tau_{0}\sim
0.063$ but is classified as local rather than non-local absorption.
Additionally, best-fit models including measurements in the low frequency
portion of the spectrum are reported in Kovalenko et al. (1994b), and Kassim
(1989).

Here we explore whether sufficient “trees” now exist to begin visualizing the
“forest”. Is it possible to use the properties of the observed (or inferred)
turnovers, coupled with improving knowledge of SNR distances and H II region
distributions, to constrain the spatial distribution of the absorbers? We hope
posing this question here can stimulate forthcoming statistical and targeted
studies of the seemingly ubiquitous phenomena of thermal absorption towards
Galactic SNRs at low frequencies. The organization of this paper is as
follows. Section 2 describes our sample of integrated SNR radio continuum
spectra with or without turnovers at low frequencies. Section 1 presents our
analysis of all spectra in context of external thermal absorption. In Sect. 4
we present our simple Galactic distribution model for absorption of SNR
emission. The main results and future work are summarized in Sect. 5.

##  2 Data collection

Our analysis is based on 129 integrated SNR radio continuum spectra. Among
them, 57 exhibit a turnover at frequencies below 100 MHz, while the remaining
72 are well fit by simple power law spectra. The former group includes 9 SNR
spectra taken from Castelletti et al. (2021), 3 constructed for this paper, 31
from Kovalenko et al. (1994b), and 14 from Kassim (1989) (see Table 1; note
that certain sources appear in more than one reference). The group with pure
power law spectra comprise 2 sources from Castelletti et al. (2021), 15
remnants whose spectra were constructed for this work, 45 from Kovalenko et
al. (1994b), and 10 from Kassim (1989) (for reference see the labels in Figs.
4, 5, and 6).

While constructing the spectra for this study, encompassing both those with
and without turnovers, our main goal was to minimize the typical scatter at
the lower frequencies. To achieve this, we conducted new flux density
measurements using available images from two resources: the Very Large Array
Low-Frequency Sky Survey Redux (VLSSr, at 74 MHz, resolution 75′′; Lane et al.
2004) and the Galactic and Extragalactic All-sky Murchison Widefield Array
survey (GLEAM, at 88, 118, 155, and 200 MHz, resolutions from
$\sim$$6^{\prime}$ to $\sim$$2^{\prime}$ Hurley-Walker et al. 2019). These
fluxes are critical for constraining the low-frequency portion of the spectra,
which for many years have remained problematic due to the relatively poor
angular resolution and sensitivity of legacy instruments, such as Culgoora (80
MHz, Slee & Higgins 1973, 1975; Slee 1977), Clark Lake TPT (30.9 MHz, Kassim
1988), and the Pushchino telescopes (83 MHz, Kovalenko et al.
1994a).111Additional flux density measurements below 100 MHz, which were
included in our spectral analysis and obtained from the literature, are as
follows: 10 MHz Bridle & Purton (1968), 19 MHz Rishbeth (1958), 10-25 MHz
Braude et al. (1969, 1979), 22 MHz Roger et al. (1969, 1986), 26 MHz Erickson
& Cronyn (1965); Viner & Erickson (1975), 29.9 MHz Jones & Finlay (1974), 38
MHz Williams et al. (1966), and 86 MHz Mills & Slee (1957); Mills et al.
(1958, 1960).  It is worth noting that at the time of our analysis, no images
below 100 MHz were available for the SNRs in our sample from surveys conducted
with modern instruments like the LOw Frequency ARray (LOFAR) or the Long
Wavelength Array (LWA). At higher frequencies, depending on the position of
the SNR in the sky, the spectra incorporate new flux density measurements from
the Southern Galactic Plane Survey (SGPS, McClure-Griffiths et al. 2005) at
1420 MHz, the S-band Polarisation All Sky Survey (S-PASS, Carretti et al.
2019) at 2303 MHz, and the Parkes 6 cm survey at 5000 MHz (Haynes et al.,
1978). All our new flux estimates have been combined with previous
measurements from the literature. The criteria adopted for collecting the data
for the new spectra are consistent with those reported in Castelletti et al.
(2021). This includes encompassing only measurements with error estimates less
than 30%, ensuring there are no significant deviations from the best-fit
model, using interferometer data that adequately recover short spacings
information at frequencies above 1000 MHz, and single-dish data with
appropriate resolution to prevent flux density overestimates due to high
confusion levels. In total, considering the spectra from Castelletti et al.
(2021) and those constructed for this work, we have collected around 570 flux
density measurements. The improvements in the accuracy of the radio spectral
index determinations for the SNRs in our sample are about one order of
magnitude when compared to previously published estimates.

##  3 Analysis

Figure 1: Radio flux density measurements (colored filled circles) as a
function of frequency for 12 SNRs spectra with turnovers (see labels). This
sample is build up by 9 SNR spectra reported in Castelletti et al. (2021) plus
3 new spectra for the SNRs G189.1+3.0, G290.1$-$0.8, and G316.3+0.0, which
were specifically constructed for this work. The solid colored lines shows
each corresponding best fit given by Eq. 2 with parameters included in our
Table 1. The asterisk symbol indicates the individual characteristic frequency
$\nu_{\ast}$.

SNRs that exhibit thermal turnovers are typically characterized by fitting the
integrated continuum flux density $S(\nu)$ at a frequency $\nu$ using a power-
law plus exponential cutoff equation:

$S(\nu)=S_{0}\Big{(}\frac{\nu}{\nu_{0}}\Big{)}^{\alpha}\mathrm{exp}\Big{[}-\tau_{0}\Big{(}\frac{\nu}{\nu_{0}}\Big{)}^{-2.1}\Big{]},$
(1)

where $\alpha$ is the power-law spectral index, $S_{0}$ is a flux density
normalization, and $\tau_{0}$ is the free-free optical depth, both measured at
a reference frequency $\nu_{0}$. The power-law term in this equation reflects
the intrinsic synchrotron emission of the SNR, while the exponential term
accounts for absorption due to ionized gas along the line of sight. However,
because authors adopt different reference frequencies, typically related to
their specific observing frequencies, it is often necessary to apply the
scaling : $\nu^{\prime}_{0}/\nu_{0}=(\tau_{0}/\tau^{\prime}_{0})^{1/2.1}$ and
$S^{\prime}_{0}/S_{0}=(\nu^{\prime}_{0}/\nu_{0})^{\alpha}$ where primed
variables correspond to a reference frequency $\nu^{\prime}_{0}$. Some authors
(e.g., Castelletti et al. 2021) use the same reference frequency ($\nu_{0}=74$
MHz) for both physical process, i.e. in the power-law and in the exponential;
others (e.g., Kassim 1989) present their data using two different reference
frequencies, one for the emission $\nu_{0}=408$ MHz and one for the absorption
$\nu_{0}=30.9$ MHz.

Figure 2: Normalized radio flux density
$S_{n}(x)=S(\nu)/S_{\ast}=x^{\alpha}\mathrm{exp}(-x^{-2.1})$ measurements
(colored filled circles) as a function of the normalized frequency
$x={\nu}/{\nu_{\ast}}$ for 12 SNRs spectra with turnover (see labels). This
sample comprises 9 SNR spectra reported in Castelletti et al. (2021), with the
addition of 3 new spectra for the SNRs G189.1+3.0, G290.1$-$0.8, and
G316.3+0.0 included for this work. Solid lines show their corresponding fit
with parameters presented in our Table 1. This normalized form sorts SNRs
spectra in increasing order accordingly to their slope from bottom
$\alpha=-0.89$ to top $\alpha=-0.35$. The black asterisk symbol marks the
characteristic frequency $\nu_{\ast}$ and highlight how the normalization is
done.

Moreover, others (e.g., Kovalenko et al. 1994b) use a unique reference
frequency for absorption $\nu_{0}=100$ MHz but different ones for each
individual SNR emission over a broad range from $\nu_{0}=175$ MHz to
$\nu_{0}=6000$ MHz (see Table 1). Although it is straightforward to convert
values from one frequency to the other, the discordant parametrization can
confuse comparison. To avoid such complications we re-write Eq. 1 independent
of the reference frequency as:

$S(\nu)=S_{\ast}\Big{(}\frac{\nu}{\nu_{\ast}}\Big{)}^{\alpha}\mathrm{exp}\Big{[}-\Big{(}\frac{\nu}{\nu_{\ast}}\Big{)}^{-2.1}\Big{]},$
(2)

where $\nu_{\ast}$ is a characteristic frequency and $S_{\ast}$ is a
characteristic flux density. Equation 2 is identical to Eq. 1 but expressed by
removing the explicit dependency of $\nu_{0}$, $\tau_{0}$, and $S_{0}$, i.e.,
independent of the adopted reference frequency $\nu_{0}$. Notice that
$\nu_{\ast}=\tau_{0}^{1/2.1}\nu_{0}$ and
$S_{\ast}=S_{0}(\nu_{\ast}/{\nu_{0}})^{\alpha}$. Computing these
characteristic parameters using any other reference frequency gives the same
values of $\nu_{\ast}$ and $S_{\ast}$. Moreover, the physical meaning of these
two parameters is straightforward: $\nu_{\ast}$ is related to the position of
the turnover frequency $\nu_{\rm to}$, i.e., where $S(\nu)$ has its maximum,
and $S_{\ast}$ is the spectrum normalization height. The turnover frequency
can be computed by deriving the flux density and equalizing it to zero to
yield $\nu_{\rm to}/\nu_{\ast}=(-2.1/\alpha)^{1/2.1}$. Then, the
characteristic frequency $\nu_{\ast}$ is determined by both the turnover
frequency $\nu_{\rm to}$ and the power-law slope $\alpha$. Only in the
critical case where $\alpha=-2.1$ we have $\nu_{\ast}=\nu_{\rm to}$ while for
$\alpha>(<)-2.1$, we have $\nu_{\ast}<(>)\nu_{\rm to}$. For a fiducial value
of $\alpha\sim-0.5$ we obtain $\nu_{\ast}\sim\nu_{\rm to}/2$, while for
typical values of SNRs power-law slopes $-0.89\lesssim\alpha\lesssim-0.35$
(see Table 1) we have $1.5\lesssim\nu_{\rm to}/\nu_{\ast}\lesssim 2.5$. It is
worth noting that while Eq. 1 has been widely adopted by many authors, to the
best of our knowledge, the simple parametrization given by Eq. 2 has not been
proposed before.An additional advantage of this parametrization in terms of
$\nu_{\ast}$ is that low values of $\nu_{\ast}$ indicate low $\nu_{\rm to}$ or
lower frequency turnovers, while high $\nu_{\ast}$ indicate high $\nu_{\rm
to}$ with absorption commencing at higher frequencies.

To illustrate our approach to constrain the properties of the absorbing gas,
in the next two subsections we apply these ideas to a sample of 129 SNRs
spectra with and without observed turnovers. For the latter group, we utilize
upper limits for $\nu_{\ast}$ based on the lowest frequency measured.

Figure 3: Normalized radio flux density measurement as a function of frequency
for 9 SNR spectra presented in Castelletti et al. (2021) plus 3 new spectra
for the SNRs G189.1+3.0, G290.1$-$0.8, and G316.3+0.0, constructed for this
work (colored filled circles), along with data from Kovalenko et al. (1994b)
(black filled squares) and Kassim (1989) (black filled triangles). In order to
highlight the drop-off behavior, the data are normalized to the power-law
emission $E(x)=x^{\alpha}$, where $x=\nu/\nu_{\ast}$. The curved solid line is
the normalized absorption $A(x)=\mathrm{exp}(-x^{-2.1})$. The total r.m.s. of
the colored circles is less than $11\%$. At $\nu\sim 2.75\nu_{\ast}$
approximately 50% of the emission is absorbed.

###  3.1  SNRs with spectral turnovers

In Fig. 1 we display the measurements of 12 SNRs with turnovers, 9 of them
reported by Castelletti et al. (2021) plus 3 constructed for this work.
Accordingly, Fig. 1 illustrates the complex superposition of spectra
reflecting the relatively large range in spectral index
($-0.89\lesssim\alpha\lesssim-0.35$) and optical depth
($0.016\lesssim\tau_{0}\lesssim 1.21$ at $\nu_{0}=74$ MHz) listed in Table 1.

In Fig. 2 we show the first advantage of the new parametrization offered
through Eq. 2. Here, we plot the normalized flux density
$S_{n}(x)=S(\nu)/S_{\ast}$ as a function of the normalized frequency
$x=\nu/\nu_{\ast}$ for the same 12SNRs plotted in Fig. 1. The dynamic range of
$\sim 700$ in $S(\nu)$ seen in Fig. 1 is reduced by at least a factor $\sim
10$ using the normalized form also showing a significantly smaller scatter.
Moreover, the observational data at low frequencies, i.e. lower than the
characteristic frequency $\nu<\nu_{\ast}$ of the 12 SNRs when stacked
together, are well fit by a unique drop off curve
$A(x)=\mathrm{exp}(-x^{-2.1})$ that is more clearly traced than in the
individual curves. At high frequencies, i.e. higher than the characteristic
frequency $\nu>\nu_{\ast}$, the solid curves differ only due to their slopes
$\alpha$. Power-law emissions are ordered by their slope from a steep
$\alpha=-0.89$ value for G316.3+0.0( bright pink filled circles and solid
curve) to a flat $\alpha=-0.35$ for G39.2$-$0.3 (red filled circles and solid
curve).

In Fig. 3 we applied an additional normalization step by dividing the flux
density $S_{n}(x)$ by the power-law emission factor of each SNR,
$E(x)=x^{\alpha}$. Another advantage of the new normalization is that any SNR
spectrum can be fit by a unique absorption equation
$A(x)=\mathrm{exp}(-x^{-2.1})$, as depicted by the solid black curve.A reduced
r.m.s. scatter of $\sim 11\%$ is apparent for these 12 SNRs stacked together
indicating the quality of the fits. To enlarge our sample, we have also
compiled spectra with a low-frequency turnover of 14 SNRs reported in Kassim
(1989) and 31 from Kovalenko et al. (1994b). These dataset are represented by
filled small black squares and triangles, respectively. The inclusion of these
two additional samples reinforces our results showing a similar trend, albeit
with a higher r.m.s. scatter.

###  3.2 Pure power-law SNRs spectra

Pure power-law spectra can be interpreted as absorbed spectra as well, but
absorbed at such low frequencies that the turnover has not been
observationally detected yet. For example, if the turnover frequency is at
$\nu_{\ast}=30$ MHz, but there are no quality measurements at comparable
frequencies, the spectrum is usually fit by a pure power-law. Indeed, if a
turnover has not been detected, the lowest measured frequency
$\nu_{\mathrm{low}}$ of a spectrum can be considered as an upper limit for the
undetected turnover frequency. We therefore expanded our sample of turnover
spectra by assigning a turnover to the pure power law fits, allowing a maximum
of a 10% difference between the extrapolated power-law and the absorbed fit at
the lowest frequency. Using Eq. 2, we have $\nu_{\ast}=\nu_{\mathrm{low}}(\log
f)^{1/2.1}$ where $f$ is a parameter that fixes the ratio between the flux
density power-law and the turnover fit at $\nu_{\mathrm{low}}$. We have
adopted a value of $f=1.1$, which means that the difference is 10% yielding
$\nu_{\ast}\sim 0.33\nu_{\mathrm{low}}$.

Figs. 4, 5, and 6 are analogous to Figs. 1, 2, and 3, respectively, but
constructed for SNRs whose integrated radio continuum spectra are fitted by
pure power laws. Two of the plotted spectra are taken from Castelletti et al.
(2021), and the other 15 were constructed for this work as explained in Sect.
2. In Fig. 4 colored solid lines are the corresponding pure power-law best fit
to the data points of each SNR, while dotted lines are the putative turnover
spectra given by Eq. 2. The asterisk symbols indicate the maximum
characteristic frequency $\nu_{\ast}$ allowed by the lowest frequency observed
$\nu_{\mathrm{low}}$. Again, the large dynamic range in measured flux
densities of $\sim 300$ is reduced by a factor $\sim 7$ when the normalized
form is adopted (see Fig. 5). A final normalization to the individual power
law slope $\alpha$ shows good quality fits with r.m.s. $\sim 15\%$ as seen in
the colored open circles of Fig. 6. These fits are better quality than the
pure power-law samples of Kovalenko et al. (1994b) and Kassim (1989) depicted
by open small black squares and triangles, respectively.

###  3.3 Characteristic frequency $\nu_{\ast}$ distribution

In Fig. 7 we show as a filled histogram the distribution of $\nu_{\ast}$
values, computed for our three samples (Castelletti et al. (2021) plus this
work, Kovalenko et al. (1994b), and Kassim (1989) listed in Table 1) ranging
from $\sim$10 MHz to 100 MHz.

Figure 4: Radio flux density measurements (colored open circles) as a function
of frequency for 17 SNRs spectra without turnovers (see labels). This sample
includes the spectra of G4.5+6.8 and G28.6$-$0.1 SNRs published in Castelletti
et al. (2021) plus 15 new ones constructed for this work. The solid colored
lines shows each corresponding power-law best fit while the dotted ones
corresponds to Eq. 2 with a characteristic frequency $\nu_{\ast}$ (see
asterisk symbol) assigned as an upper limit according to the lowest measured
frequency $\nu_{\mathrm{low}}$.

In the upper panel, $\nu_{\ast}\sim 40$ MHz separates two populations with low
($\tau\sim 0.1$) and high ($\tau\sim 1$) optical depths reported in Table 3 of
Castelletti et al. (2021); these two subsamples were labeled as “Absorption
from extended envelopes” (non-local) and “Absorption: Special cases” (local),
respectively. The classification was made by additional information obtained
from low-frequency radio recombination lines, infrared fine-structure, and
molecular line emission to distinguish between spectral turnovers attributed
to ionized material in the ISM unrelated to the SNR (non-local) and an ionized
component in the immediate SNR’s surroundings (local). These low
($\nu_{\ast}\lesssim 40$ MHz) and high ($\nu_{\ast}\gtrsim 40$ MHz)
populations might hint at non-local vs. local absorbers in the other two
samples, Kassim (1989) and Kovalenko et al. (1994b), shown in the middle and
bottom panels of Fig. 7. However, due to the incompleteness of our samples, it
is not possible to definitively determine yet whether these populations
represent truly distinct groups or merely the low and high tails of a single
population with intrinsic dispersion. The open histograms show the
characteristic frequency $\nu_{\ast}$ of power-law spectra assigned according
to their observed lowest frequency $\nu_{\mathrm{low}}$. In the upper panel,
the observed peak at $\nu_{\ast}\sim 30$ MHz is a direct consequence of the
GLEAM survey cutoff frequency at $\nu_{\mathrm{low}}\sim 88$ MHz (see asterisk
symbols in Fig. 4). At $\nu_{\ast}\sim 10$ MHz we have measurements obtained
at lower frequencies by the Clark Lake TPT (30.9 MHz), Culgoora (80 MHz), and
Pushchino (83 MHz) telescopes (see Sect. 2).

Notice that the upper color bar indicates the color coding adopted in Fig. 8
and Fig. 9 based on $\log(\nu_{\ast}/\mathrm{MHz})$ value instead of using the
SNRs’ names, as done in Fig. 1 through Fig. 6, as well as in Fig. 10 and Fig.
11. On the other hand, we have also searched for, and did not find, any
significant correlations between the three fitted parameters $S_{\ast}$,
$\nu_{\ast}$, and $\alpha$. This result indicates that the emission and
absorption processes in our limited SNR sample are independent of each other.

Figure 5: Normalized radio flux density
$S_{n}(x)=S(\nu)/S_{\ast}=x^{\alpha}\mathrm{exp}(-x^{-2.1})$ measurements
(colored open circles) as a function of the normalized frequency
$x={\nu}/{\nu_{\ast}}$ for 17 SNRs spectra without low frequency turnover (see
labels). This sample includes the spectra of G4.5+6.8 and G28.6$-$0.1 SNRs
published in Castelletti et al. (2021) plus 15 new collected for this work.
Extended lines show their power-law fit while dotted ones correspond to Eq. 2
with characteristic frequency $\nu_{\ast}$ (asterisk symbol) assigned as an
upper limit according to the lowest measured frequency $\nu_{\mathrm{low}}$.
This normalized form sorts SNRs spectra in increasing order accordingly to
their slope from bottom $\alpha=-0.69$ to top $\alpha=-0.32$. The black
asterisk symbol marks the characteristic frequency $\nu_{\ast}$ and highlights
how the normalization is done.

A useful and direct application of the characteristic frequency $\nu_{\ast}$
parametrization is the possibility to provide insights concerning the
properties of the absorbing thermal gas. A complete mapping of the
$\nu_{\ast}$ distribution in the Galaxy should offer strong and unique
constraints on the distribution of the ionized absorbing gas. Despite the
incomplete and non-uniform nature of current surveys on SNRs, we have utilized
our compiled sample to extract fundamental and widespread characteristics from
the presently available data. Additionally, we have compared our findings with
ad-hoc constructed models in order to draw initial conclusions regarding the
absorber population. In Fig. 8 we show the projected spatial distribution onto
the Galactic plane for all the SNRs with turnovers (filled symbols) in our
three samples. We have also included SNRs from Castelletti et al. (2021) plus
this work, Kovalenko et al. (1994b), and Kassim (1989) whose spectra are well-
fitted by a pure power-law (open symbols) colored according to their upper
limit characteristic frequency assigned (see Sect. 3.2 and Fig. 4). At first
glance, it does not appear that there is a significant segregation between
SNRs with and without spectral turnovers, or an apparent gradient in
$\nu_{\ast}$ with global properties like heliocentric distance or Galactic
longitude. This qualitative result indicates that the absorbing media is
probably made of discrete ionized regions rather than a continuous gas
distribution. Likewise, the absence of spectral turnover in some distant SNRs,
coupled with significant absorption in some nearby ones, was first interpreted
by Kassim (1989) as a patchy distribution of low-frequency absorbing gas, and
not caused by a broadly distributed component of the ISM. Here, we test this
original idea by adding the spectra presented by Castelletti et al. (2021),
those constructed for this work, along with those included in Kovalenko et al.
(1994b).

To facilitate the interpretation of these observational findings, in the
following section we developed a two-dimensional toy model that simulates the
power-law emission from SNRs and takes into account the absorption caused by
intervening ionizing gas.

Figure 6: Normalized radio flux density measurements as a function of
frequency for the SNRs G4.5+6.8 and G28.6$-$0.1 presented in Castelletti et
al. (2021) plus 15 new SNRs spectra constructed for this work (colored open
circles). These datasets are complemented by flux density estimates from
Kovalenko et al. (1994b) (black open squares) and Kassim (1989) (black open
triangles). In order to highlight the drop-off behavior, the data are
normalized to the power-law emission $E(x)=x^{\alpha}$, where
$x=\nu/\nu_{\ast}$. The curved solid line is the normalized absorption
$A(x)=\mathrm{exp}(-x^{-2.1})$. The total r.m.s. of the colored open circles
is less than $15\%$.

##  4 Absorption Model

We assumed that the SNRs are distributed on the Galactic plane following an
exponential surface number density profile:
$\Sigma(R)=\Sigma_{0}\exp(-R/R_{d})$, where $\Sigma_{0}$ is the central
surface density and $R_{d}$ is the scale-length. We tied $\Sigma_{0}$ to a
mock assumption of 20,000 SNRs sufficiently large to mimic the statistical
behavior of a generic population of remnants distributed on the Galactic plane
with $R_{d}=3.5$ kpc following the Milky Way stellar scale-length. We started
by attributing the absorption to ionized gas characterized by an emission
measure (see Eq. 6 and discussion below) predicted by the continuum model
NE2001, at fixed temperature (Cordes & Lazio, 2002). We found that the NE2001
model predicts a correlation between absorption and distance, a correlation
that is not observed. We attribute the discrepancy to the fact that the NE2001
model assumes a nearly continuous distribution of ionized gas, together with
its reliance on measurements of relatively nearby pulsars (PSRs). Recognizing
that H II regions are discrete structures, we then considered a model in which
a set of discrete absorbing regions, representing H II regions or H II region
complexes, are distributed throughout the disk of the Galaxy.

We place clumps of ionized gas preferentially along the Galactic spiral arms
following the Hou & Han (2014) four-arm logarithmic model (see their Table 1).
Clumps are circular with radius $r$ drawn from a Maxwell-Boltzmann probability
density distribution:

Figure 7: Histograms showing the distribution of characteristic frequencies
$\nu_{\ast}$ for the three samples analyzed here. Each panel corresponds to a
different sample, as indicated by the labels. The filled histograms show the
distribution of $\nu_{\ast}$ for SNRs spectra with turnovers, while the open
ones depict the estimated upper limit of $\nu_{\ast}$ based on the lowest
measured frequency $\nu_{\mathrm{low}}$ for spectra without turnovers.

$p(r)=\sqrt{\frac{2}{\pi}}\frac{r^{2}\;\mathrm{exp}(-r^{2}/a^{2})}{a^{3}},$
(3)

with a parameter $a=50$ pc. Initially, we distribute homogeneously clumps
along the spiral arms; then we add a random noise of $5\%$ in their polar
radius to mimic the fact that H II regions are not perfectly distributed along
the spiral arms and also that the spiral arms themselves are not perfectly
delineated in real galaxies. We also fix the number of absorbers in a way that
the initial (i.e. before adding the random noise) mean separation between them
is 5 times their typical size. All together these conditions result in 981
simulated H II regions that trace approximately the spiral arms and that they
do not overlap each, consistent with the number and distribution of Galactic H
II regions revealed by observational surveys that typically point out hundreds
to thousands of these objects (Anderson et al. 2018; Wenger et al. 2021, and
references therein).

The predicted characteristic frequency $\nu_{\ast}$ can be estimated using its
definition, introduced in Sect. 1, $\nu_{\ast}=\tau_{0}^{1/2.1}\nu_{0}$ where
the optical depth $\tau_{0}$ can be computed from the physical properties of
the absorbing ionized gas using the following equation (Wilson et al., 2009)

$\tau_{0}=3.014\times 10^{-2}\Big{(}\frac{\rm{T_{e}}}{\rm
K}\Big{)}^{-3/2}\Big{(}\frac{\nu_{0}}{\rm GHz}\Big{)}^{-2}\,g_{\rm
ff}\,\Big{(}\frac{\rm{EM}}{\rm pc\;cm^{-6}}\Big{)},$ (4)

where $\rm{T_{e}}$ is the gas electron temperature,

$\rm g_{\rm ff}=ln\Big{[}4.955\times 10^{-2}\Big{(}\frac{\nu_{0}}{\rm
GHz}\Big{)}^{-1}\Big{]}+1.5\,ln\Big{(}\frac{T_{e}}{\rm K}\Big{)}$ (5)

is the Gaunt factor, and

${\rm{EM}}=\int_{0}^{L}n_{\rm{e}}^{2}\;dx$ (6)

Figure 8: Projected spatial distribution onto the Galactic plane of SNRs in
the three samples analyzed here with distances taken from Ranasinghe & Leahy
(2022) and colored according to ${\log}(\nu_{\ast}/\rm{MHz})$ using the color
bar code. Filled symbols show the location of SNRs with turnover spectra,
while the open symbols indicate those sources not showing turnovers. Circles
correspond to Castelletti et al. (2021) plus this work data, squares
correspond to Kovalenko et al. (1994b), and triangles to Kassim (1989). As a
guidance the solid black lines show the position of the spiral arms taken from
the Hou & Han (2014) model. The central plus symbol indicates the position of
the Galactic center, the solar symbol shows the Sun’s position, and the dotted
lines the Galactic quadrants. The colored points do not show any strong
dependence on heliocentric distance or Galactic longitude (see Fig. 10 and11).

is the emission measure depending on the electron density, $n_{\rm{e}}$,
integrated along the linear length, $L$, of the intervening medium. Assuming a
constant electron density, the emission measure is
$\mathrm{EM}=n_{\mathrm{e}}^{2}\,L$. Therefore, we have $\nu_{\ast}\sim
n_{\mathrm{e}}^{0.95}L^{0.47}/\rm{T_{e}}^{0.71}$, i.e. the characteristic
frequency depends more strongly on the electron density adopted, less on the
electron temperature, and more weakly on the path $L$. We assume constant
electron temperature $\rm{T_{e}}=5000\,\leavevmode\nobreak\ \mathrm{K}$ and
density $n_{\rm e}=2.0\leavevmode\nobreak\ \mathrm{cm}^{-3}$ (Castelletti et
al., 2021).

By adopting these values, our model is calibrated to reproduce an average
value $\nu_{\ast}\sim$30 MHz observed in the combined sample of Castelletti et
al. (2021) plus this work, Kovalenko et al. (1994b), and Kassim (1989) (see
Fig. 7). The linear length $L$ is computed as the cumulative sum of all
segments along the integral that intersect circular ionized clumps of radius
$r$ along the line of sight from the solar position. Notice that the NE2001
model cannot predict such characteristic frequencies $\nu_{\ast}\sim 30$ MHz
for SNRs that are located nearby (i.e. at heliocentric distances of $\sim 1$
kpc) due to the low electron density assumed ($n_{\mathrm{e}}\sim 0.2$ cm-3),
which is approximately one order of magnitude lower than that inferred from
the SNR turnover spectra by Castelletti et al. (2021).

The prediction of our model is shown in Fig. 9, where the spatial distribution
of the emitters (SNRs) are plotted as colored dots (using the color coding
shown in the upper bar), while the absorbers are displayed as black open
circles. To make the plot more clear the scale of each open circle has been
expanded by an arbitrary factor of 3. The observer is assumed to be at
Cartesian coordinates (0,8.5) kpc from the Galactic centre mimicking the Sun’s
position. Colored points correspond to SNRs that are absorbed by ionized gas
along the line of sights. In a model of sparse and discrete absorbers, like
this one, SNRs with and without spectral turnover are predicted to be mixed,
i.e. not spatially segregated as expected in a model with absorbers
continuously distributed.

Figure 9: Spatial distribution of an assumed mock sample of 20,000 SNRs
(points colored by ${\log}(\nu_{\ast}/\mathrm{MHz})$ using the color bar
code). Absorbers (open circles) are distributed along the Galactic spiral arms
taken from the Hou & Han (2014) model (solid black curves). Each absorber is
assumed to have a constant low density $n_{\mathrm{e}}=2.0$ cm-3, a
temperature $\rm{T_{e}}=5000$ K, and radius $r$ drawn from a Maxwell-Boltzmann
probability density distribution with a median of 50 pc. The scale of each
open circle has been expanded by an arbitrary factor 3 in order to make the
plot more clear. The central plus symbol indicates the position of the
Galactic center, the solar symbol shows the Sun’s position, and the dotted
lines the Galactic quadrants. The colored points do not show any strong
dependence on heliocentric distance or Galactic longitude (see Fig. 10 and11).

In a continuous model, SNRs with non-thermal turnovers are expected to be
located closer to the observer while those showing turnovers should exhibit
characteristic frequencies $\nu_{\ast}$ increasing with distance. This is
because the radiation passes through more gas as it travels towards the
observer, leading to increased intersection and thus a higher predicted
frequency. Notice that the asymmetric coil of spiral arms, i.e. that its
Galacto-centric distance increases with its polar angle, makes absorption
asymmetric between the 1st and 4th and between the 2nd and 3rd quadrants. The
strongest absorption is predicted in the fourth quadrant tangential direction
(see magenta dots) of the most internal spiral arm.

We conducted a series of tests on our model using different values of the
scale-length parameter, $R_{d}$, for the exponential distribution of SNRs. We
also experimented with using a power-law profile,
$\Sigma(R)=\Sigma_{0}(R/R_{d})^{\gamma}$. Despite these changes, we found that
our results remained consistent and robust.

To compare observational results against the predictions of our simple model,
in Figs. 10 and 11 we plot the characteristic frequency $\nu_{\ast}$ as a
function of the heliocentric distance and the Galactic longitude,
respectively. Observational values for spectra with turnovers, indicated by
colored filled circles (Castelletti et al. 2021 plus this work), small black
filled squares (Kovalenko et al., 1994b) and small black filled triangles
(Kassim, 1989), show neither a positive gradient with distance nor a negative
gradient with Galactic longitude, as expected in a model of continuously
distributed absorbing gas. Theoretical models featuring discrete absorbers, as
depicted in Fig. 9 reveal that the values of $\nu_{\ast}$ remain approximately
constant. This indicates that absorption is independent of heliocentric
distance or Galactic longitude, as illustrated by the black continuous solid
line. Besides, we have also included observed SNRs with no measured spectral
turnovers (i.e. sources with pure power-law continuum spectra) with
$\nu_{\ast}$ computed as the upper limit allowed by the lowest observed
frequency $\nu_{\mathrm{low}}$ (see Sec. 3.2). These remnants are located not
only quite close, $d\sim 2$ kpc, to the solar position, but also at distances
as large as $d\sim 10$ kpc. Similarly, they can be found either quite close
the Galactic center or in the anticenter direction.

Based on our assumed distribution of H II regions following the spiral arms,
we find that the absorption of our mock SNR population is consistent with the
patchy distribution of absorption indicated by observations to date. A much
larger sample of low frequency spectra are needed to verify if our model of
absorbing gas due to H II regions is the correct approach.

Figure 10: Characteristic frequency $\nu_{\ast}$ as a function of the
heliocentric distance for SNRs with low frequency turnovers (filled symbols)
and SNRs spectra without turnovers (open symbols). Circles correspond to
Castelletti et al. (2021) plus this work data, while squares to Kovalenko et
al. (1994b), and triangles to Kassim (1989). The solid line corresponds to the
median prediction of the model presented in Fig. 9 not showing a systematic
gradient with heliocentric distance.

##  5 Conclusions

We have analyzed a sample of 129 radio continuum spectra of Galactic SNRs from
this work and the literature (including Castelletti et al. 2021, Kovalenko et
al. 1994b, and Kassim 1989). 57 show low frequency turnovers characteristic of
extrinsic thermal absorption, while 72 remain straight power laws (special
cases of intrinsic absorption by unshocked ejecta within SNRs is not
considered in this paper). Since the frequencies sampled vary across the
spectra, we take care to interpret the latter as limits on absorption in
context of their lowest frequency measurement. Our main conclusions are:

  1. 1.

We introduced a new parametrization of SNR integrated radio continuum spectra
in terms of characteristic frequency $\nu_{\ast}$ and corresponding flux
density $S_{\ast}$ independent of arbitrary survey reference frequencies.
$\nu_{\ast}$ is directly related both to the power-law emission slope and the
turnover frequency including the density, temperature, and path through the
absorbing medium.

  2. 2.

Normalizing the SNR spectra to the frequency $\nu_{\ast}$ and to the flux
density $S_{\ast}$, we confirmed with our larger sample that while all SNRs
emit energy with independent synchrotron emission power-law indices, turnovers
are all well fit by thermal absorption.

  3. 3.

For SNRs not showing turnovers we derive an upper limit for $\nu_{\ast}$ by
considering the lowest sampled frequency yielding a range of
$4\leavevmode\nobreak\ \mathrm{MHz}\lesssim\nu_{\ast}\lesssim
40\leavevmode\nobreak\ \mathrm{MHz}$. This result should be tested and
improved by forthcoming surveys pushing SNR spectra to lower frequencies,
especially $<$ 50 MHz where sensitivity to absorption is markedly increased.

  4. 4.

Emission ($\alpha$) and absorption ($\nu_{\ast}$) spectral parameters do not
show any significant correlation indicating two unrelated, independent
processes. For example, absorption beyond the nominal boundaries of the
synchrotron emitting plasma seem oblivious to the details and history of the
shock acceleration processes within the remnant. For our three samples of SNR
spectra with turnovers we found the characteristic frequency ranges
$10\leavevmode\nobreak\ \mathrm{MHz}\lesssim\nu_{\ast}\lesssim
100\leavevmode\nobreak\ \mathrm{MHz}$.

Figure 11: Characteristic frequency $\nu_{\ast}$ as a function of the Galactic
longitude for SNRs with low frequency turnovers (filled symbols) and SNRs
spectra without turnovers (open symbols). Circles correspond to Castelletti et
al. (2021) plus this work data, while squares to Kovalenko et al. (1994b), and
triangles to Kassim (1989). The solid line corresponds to the median
prediction of the model presented in Fig. 9 not showing a strong dependence
with the Galactic longitude.

  5. 5.

There is no evidence for a correlation between absorption and global Galactic
geometric parameters such as heliocentric distance and Galactic longitude, as
would be expected if the distribution of the absorbing gas were continuous.
Our result is also valid for pure power-law SNR spectra. This is consistent
with Kassim (1989) who inferred that SNRs were being absorbed by an
intervening patchy distribution of H II regions or their envelopes whose
distances were unknown.

  6. 6.

Modelling based on a distribution of H II regions tracking Galactic spiral
arms produces patchy absorption towards a mock SNR population consistent with
the limited observations to date. One prediction of this scheme is that
strongly absorbed SNRs should be located preferentially towards the fourth
Galactic quadrant relative to the first.

  7. 7.

Emission measures predicted from NE2001 could not match the observations. We
attribute the discrepancy to the fact that the NE2001 model assumes a nearly
continuous distribution of ionized gas, and relies predominantly on a
relatively nearby population of PSRs. The NE2001 model predicts a correlation
between absorption and distance, particularly for greater distances, that is
not observed.

Unfortunately, the incompleteness of current SNR catalogs, and absence of low
frequency spectra even for those that are known, preclude making a much more
detailed comparison between models and observations. This motivates the
completion of deeper, global surveys at both MHz and GHz frequencies, to push
all SNR spectra to lower frequencies and to fully sample the intrinsic
properties like density, temperature, size, and distance, of intervening
Galactic thermal gas. Arc-second resolution observations below 100 MHz, as
provided by LOFAR and emerging low frequency instruments like LWA and SKA-Low,
are critical for discerning the relative radial superposition of SNRs and H II
regions along complex lines of sight. Given readily available kinematic
distances to H II regions, this can offer powerful physical constraints for
SNRs whose distances are far more problematic.

###### Acknowledgements.

This work has been partially supported by the Consejo de Investigaciones
Científicas y Técnicas de la República Argentina (CONICET), the Secretaría de
Ciencia y Técnica de la Universidad Nacional de Córdoba (SeCyT), Agencia
Nacional de Promoción Científica y Tecnológica (PICT 2017-3320, PICT
2019-1600), Argentina. MA acknowledges the hospitality of the Astronomical
Observatory of Trieste where part of the work was done in the framework of the
European Union LACEGAL program. Basic research in radio astronomy at the Naval
Research Laboratory is funded by 6.1 Base funding. Part of this research was
carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space
Administration.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Fitted parameters of SNRs spectra sample

Table 1: Each group of rows, separated by a horizontal line, corresponds to Castelletti et al. (2021) and this work data, Kovalenko et al. (1994b), and Kassim (1989) sample, respectively. Column 1 is the SNR identification name while columns 2 to 6 are published parameter for Eq. 1 taken from these references. For SNRs G189.1+3.0, G290.1$-$0.8, and G316.3+0.0, we derived the parameters through spectral fits to the datasets constructed for this study. Column 7 and 8 display our computed values for $\nu_{\ast}$ and $S_{\ast}$, respectively, using $\nu_{\ast}=\tau_{0}^{1/2.1}\nu_{0}$ and $S_{\ast}=S_{1}(\nu_{\ast}/\nu_{1})^{\alpha}$ (see Eq. 2). SNR | $\alpha$ | $\nu_{0}$ | $\tau_{0}$ | $\nu_{1}$ | $S_{1}$ | $\nu_{\ast}$ | $S_{\ast}$
---|---|---|---|---|---|---|---
|  | [MHz] |  | [MHz] | [Jy] | [MHz] | [Jy]
G18.8+0.3 (Kes 67) | $-0.373$ | 74 | 0.043 | 74 | 79.75 | 16.54 | 139.46
G21.8$-$0.6 (Kes 69) | $-0.505$ | 74 | 0.088 | 74 | 208.00 | 23.26 | 373.16
G23.3$-$0.3 (W41) | $-0.628$ | 74 | 1.214 | 74 | 347.20 | 81.16 | 327.62
G27.4+0.0 (Kes 73) | $-0.690$ | 74 | 0.878 | 74 | 34.72 | 69.55 | 36.24
G29.7$-$0.3 (Kes 75) | $-0.659$ | 74 | 0.176 | 74 | 50.87 | 32.36 | 87.74
G31.9+0.0 (3C 391) | $-0.521$ | 74 | 1.210 | 74 | 90.70 | 81.03 | 86.51
G39.2$-$0.3 (3C 396) | $-0.351$ | 74 | 0.063 | 74 | 41.65 | 19.84 | 66.11
G41.1$-$0.3 (3C 397) | $-0.356$ | 74 | 0.141 | 74 | 75.30 | 29.11 | 104.96
G43.3$-$0.2 (W49B) | $-0.461$ | 74 | 0.580 | 74 | 111.80 | 57.09 | 125.96
G189.1+3.0 (IC 443) | $-0.388$ | 74 | 0.016 | 74 | 436.90 | 10.24 | 941.26
G290.1$-$0.8 (MSH 11$-$61A) | $-0.421$ | 74 | 0.258 | 74 | 137.80 | 38.82 | 180.80
G316.3+0.0 | $-0.893$ | 74 | 0.215 | 74 | 166.20 | 35.59 | 319.52
G0.9+0.1 | $-0.64$ | 100 | 0.820 | 2000 | 6.9 | 90.98 | 49.86
G1.05$-$0.1 | $-0.66$ | 100 | 2.000 | 1700 | 13.0 | 139.10 | 67.83
G6.4$-$0.1 (W28) | $-0.42$ | 100 | 0.080 | 450 | 440.0 | 30.04 | 1371.00
G8.7$-$0.1 (W30) | $-0.48$ | 100 | 0.150 | 940 | 80.0 | 40.52 | 361.80
G9.8+0.6 | $-0.56$ | 100 | 0.670 | 740 | 4.1 | 82.64 | 13.99
G11.2$-$0.3 | $-0.49$ | 100 | 0.240 | 6000 | 9.2 | 50.68 | 95.43
G11.4$-$0.1 | $-0.46$ | 100 | 0.430 | 1000 | 6.0 | 66.91 | 20.82
G15.9+0.2 | $-0.63$ | 100 | 0.160 | 1100 | 4.2 | 41.78 | 32.97
G16.7+0.1 | $-0.53$ | 100 | 0.530 | 960 | 2.9 | 73.91 | 11.29
G18.8+0.3 (Kes 67) | $-0.42$ | 100 | 0.030 | 580 | 37.0 | 18.83 | 156.10
G20.0$-$0.2 | $-0.00$ | 100 | 0.05 | 180 | 10.0 | 24.01 | 10.00
G21.8$-$0.6 (Kes 69) | $-0.50$ | 100 | 0.050 | 600 | 77.0 | 24.01 | 384.90
G22.7$-$0.2 | $-0.40$ | 100 | 0.560 | 1850 | 59.0 | 75.87 | 211.70
G23.3$-$0.3 (W41) | $-0.48$ | 100 | 0.550 | 190 | 190.0 | 75.23 | 296.40
G24.7$-$0.6 | $-0.59$ | 100 | 0.760 | 800 | 8.0 | 87.75 | 29.47
G24.7+0.6 | $-0.14$ | 100 | 0.220 | 1600 | 18.0 | 48.63 | 29.36
G27.4+0.0 | $-0.71$ | 100 | 0.380 | 560 | 9.4 | 63.08 | 44.30
G29.7$-$0.3 (Kes 75) | $-0.59$ | 100 | 0.060 | 3100 | 4.4 | 26.19 | 73.56
G31.9+0.0 (3C 391) | $-0.51$ | 100 | 0.520 | 2200 | 15.5 | 73.24 | 87.89
G39.2$-$0.3 (3C 396) | $-0.48$ | 100 | 0.060 | 840 | 20.0 | 26.19 | 105.70
G39.7$-$2.0 (W50) | $-0.50$ | 100 | 0.055 | 760 | 90.0 | 25.13 | 495.00
G40.5$-$0.5 | $-0.51$ | 100 | 0.220 | 410 | 19.5 | 48.63 | 57.84
G41.1$-$0.3 (3C 397) | $-0.46$ | 100 | 0.070 | 10000 | 7.0 | 28.19 | 104.30
G78.2+2.1 ($\gamma$-Cygni) | $-0.73$ | 100 | 0.280 | 1480 | 40.0 | 54.54 | 2671.00
G82.2+5.3 (W63) | $-0.63$ | 100 | 0.270 | 1900 | 73.0 | 53.61 | 691.10
G89.0+4.7 (HB21) | $-0.39$ | 100 | 0.012 | 800 | 240.0 | 12.17 | 1228.00
G111.7$-$2.1 | $-0.75$ | 100 | 0.010 | 180 | 11500.0 | 10.94 | 93922.52
G120.1+1.4 | $-0.62$ | 100 | 0.010 | 520 | 86.0 | 11.16 | 930.89
G189.1.1$-$3.0 | $-0.42$ | 100 | 0.010 | 700 | 190.0 | 11.16 | 1080.67
G205.5+0.5 (Monoceros) | $-0.47$ | 100 | 0.080 | 1300 | 135.0 | 30.04 | 793.20
G206.9+2.3 (PKS 0646+06) | $-0.51$ | 100 | 0.060 | 360 | 10.0 | 26.19 | 38.06
G6.4$-$0.1 (W28) | $-0.4$ | 30.9 | 0.9 | 408 | 424.0 | 29.39 | 1214.00
G11.2$-$0.3 | $-0.5$ | 30.9 | 2.1 | 408 | 30.0 | 43.99 | 91.36
G11.4$-$0.1 | $-0.4$ | 30.9 | 3.7 | 408 | 7.1 | 57.61 | 15.53
G15.9+0.2 | $-0.7$ | 30.9 | 2.1 | 408 | 8.5 | 43.99 | 40.41
G18.8+0.3 (Kes 67) | $-0.5$ | 30.9 | 0.6 | 408 | 49.0 | 24.23 | 201.10
G21.8$-$0.6 (Kes 69) | $-0.5$ | 30.9 | 0.9 | 408 | 95.0 | 29.39 | 354.00
G22.7$-$0.2 | $-0.2$ | 30.9 | 3.5 | 408 | 65.0 | 56.11 | 96.66
G23.3$-$0.3 (W41) | $-0.4$ | 30.9 | 7.2 | 408 | 119.0 | 79.11 | 229.40
G29.7$-$0.3 (Kes 75) | $-0.6$ | 30.9 | 0.8 | 408 | 15.0 | 27.79 | 75.20
G39.2$-$0.3 (3C 396) | $-0.4$ | 30.9 | 0.5 | 408 | 25.0 | 22.21 | 80.09
G40.5$-$0.5 | $-0.5$ | 30.9 | 2.1 | 408 | 16.0 | 43.99 | 48.73
G41.1$-$0.3 (3C 397) | $-0.4$ | 30.9 | 0.9 | 408 | 39.0 | 29.39 | 111.70
G43.3$-$0.2 (W49) | $-0.4$ | 30.9 | 0.9 | 408 | 46.0 | 29.39 | 131.80
G355.9$-$2.5 | $-0.5$ | 30.9 | 1.1 | 408 | 12.0 | 32.33 | 42.63


# Microwave-optics Entanglement via Cavity Optomagnomechanics

###### Abstract

Microwave-optics entanglement is a vital component for building hybrid quantum
networks. Here, a new mechanism for preparing stationary entanglement between
microwave and optical cavity fields in a cavity optomagnomechanical system is
proposed. It consists of a magnon mode in a ferrimagnetic crystal that couples
directly to a microwave cavity mode via the magnetic dipole interaction, and
indirectly to an optical cavity through the deformation displacement of the
crystal. The mechanical displacement is induced by the magnetostrictive force
and coupled to the optical cavity via radiation pressure. Both the opto- and
magnomechanical couplings are dispersive. Magnon-phonon entanglement is
created via magnomechanical parametric down-conversion, which is further
distributed to optical and microwave photons via simultaneous optomechanical
beamsplitter interaction and electromagnonic state-swap interaction, yielding
stationary microwave-optics entanglement. The microwave-optics entanglement is
robust against thermal noise, which will find broad potential applications in
quantum networks and quantum information processing with hybrid quantum
systems.

###### keywords:

cavity magnonics, cavity magnomechanics, optomechanics, microwave-optics
entanglement, quantum network

Zhi-Yuan Fan, Liu Qiu, Simon Gröblacher, Jie Li*

Z.-Y. Fan, J. Li
Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics
Zhejiang University
Hangzhou 310027, China
Email Address<EMAIL_ADDRESS>
L. Qiu
Institute of Science and Technology Austria
am Campus 1, 3400, Klosterneuburg, Austria

S. Gröblacher
Kavli Institute of Nanoscience, Department of Quantum Nanoscience
Delft University of Technology
2628CJ Delft, The Netherlands

##  1 Introduction

Optical entanglement is a crucial resource and finds a wide range of
applications in quantum information science, such as quantum teleportation,[1,
2, 3] quantum networks,[4, 5] quantum logical operations,[6] quantum
metrology,[7] and fundamental tests of quantum mechanics.[8, 9, 10] Entangled
optical fields can be produced in various different ways, such as by
exploiting parametric down-conversion in nonlinear crystals,[11, 12] four-wave
mixing in optical fibers[13, 14] and atomic vapors,[15, 16] semiconductor
quantum dots,[17, 18] and periodically poled lithium niobate waveguide.[19] In
the microwave domain, entangled fields are normally generated using Josephson
parametric amplifiers.[20, 21, 22] They can also be created by reservoir
engineering of an atomic beam,[23] and injecting a squeezed vacuum through a
microwave beamsplitter.[24] In addition, optical (microwave) fields can be
entangled in an optomechanical system by coupling them to a common mechanical
resonator via radiation pressure.[25, 26, 27, 28, 29, 30]

Despite the above mentioned approaches, there are much fewer efficient ways to
produce microwave-optics entanglement due to large frequency mismatch. This
special type of optical entanglement finds particularly important applications
in quantum information processing with hybrid quantum systems,[31, 32, 33] and
hybrid quantum networks with distinct quantum nodes working in the microwave
and optical frequency ranges.[34, 35, 36, 37, 38] It can be generated by
exploiting optomechanical interactions, where both optical and microwave
cavity modes couple to a common mechanical oscillator (forming the
optoelectromechanical system[39, 40, 41]), and by activating the parametric
down-conversion and state-swap interactions in the electro- and optomechanical
subsystems, respectively.[25, 26, 27, 28] Alternatively, it can also be
produced by the parametric down-conversion in directly coupled electro-optic
systems.[42, 43, 44]

In this work, we present a new mechanism to prepare stationary microwave-
optics entanglement in a cavity optomagnomechanical (OMM) configuration
exploiting nonlinear magnetostrictive and radiation-pressure interactions. The
OMM system consists of a magnon mode in a ferrimagnetic crystal, e.g.,
yttrium-iron-garnet (YIG), that is dispersively coupled to a phonon mode of
the magnetostrictively induced mechanical vibration,[45, 46, 47, 48, 49] and
the mechanical motion further couples to an optical cavity via radiation
pressure.[50] The magnon mode further couples to a microwave cavity mode via
the magnetic dipole interaction,[51, 52, 53] forming the cavity-OMM system. We
show that by simultaneously driving the magnon mode (the optical cavity) to
activate the magnomechanical parametric down-conversion (optomechanical
beamsplitter) interaction, magnon-phonon entanglement is created. The
entanglement is further transferred to optical and microwave cavity photons
due to the effective optomechanical and electromagnonic state-swap
interactions, and consequently, a stationary microwave-optics entangled state
is prepared. Our system is one of the configurations of hybrid quantum systems
based on magnons[54], which find many promising applications, such as magnon
laser and optomagnonic frequency combs etc[55, 56, 57, 58].

##  2 System and Model

We consider a cavity-OMM system, as depicted in Figure 1a,b, which consists of
a microwave cavity mode, a magnon mode (e.g., the Kittel mode[59]) in a YIG
crystal, a mechanical vibration mode, and an optical cavity mode. The magnon
mode is embodied by the collective motion (spin wave) of a large number of
spins in the YIG crystal, and is activated by placing the crystal in a uniform
bias magnetic field and applying a microwave drive field with its magnetic
component perpendicular to the bias field. The magnon mode couples to a
microwave cavity field via the magnetic dipole interaction[51, 52, 53] by
putting the crystal near the maximum magnetic field of the cavity mode. The
magnetostrictive interaction of the ferrimagnet leads to a dispersive coupling
between magnons and lower-frequency vibration phonons for a relatively large
size of the crystal.[45, 46, 47, 48, 49] The magnomechanical displacement
further couples to an optical cavity via radiation pressure.[60, 61] The
magno- and optomechanical dispersive couplings play a key role in cooling the
mechanical motion and generating entanglement in the system. The YIG crystal
can be either a sphere[45, 48, 49] or a micro bridge structure.[61, 62] For
the convenience of optical cavity fabrication, we adopt the latter and the
optical cavity can be realized by attaching a small highly-reflective mirror
pad to the surface of the micro bridge.[50, 63] The attached mirror should be
fabricated sufficiently small and light, such that it will not appreciably
affect the mechanical properties of the micro bridge, e.g. the mechanical
displacement. The deformation displacement can be approximately regarded as
uniform in the direction perpendicular to the attached surface (with
negligible bending displacement), such that the YIG bridge and the mirror pad
can stick together tightly and are integrated into one body, which oscillate
approximately with the same frequency. Alternatively, one may consider using
the ‘membrane-in-the-middle’ approach[64] to achieve the optomechanical
dispersive coupling. Besides, the micro bridge can be made small to obtain a
relatively large magnomechanical bare coupling. We, however, do not use thin
YIG films, where high-frequency (GHz) phonon modes couple linearly (and
strongly) to the magnon mode[65, 66]. The linear coupling is not desired for
our purpose of producing entanglement. One can also consider using the direct
mechanical contact approach to realize the optomagnomechanical coupling.[67]
The red-detuned driven optical cavity is used to cool the lower-frequency
mechanical motion, while the activated magnomechanical Stokes scattering (by
driving the magnons with a blue-detuned microwave field) is sufficient to
create entanglement. This is a key difference from the entanglement mechanism
in ref.[46], where the mechanical motion is cooled by a red-detuned microwave
drive field.

The Hamiltonian of such a hybrid microwave cavity-OMM system reads[46, 50]

$\displaystyle\begin{split}H/\hbar&=\\!\sum_{j=a,m,c}\\!\\!\omega_{j}j^{\dagger}j+\frac{\omega_{b}}{2}\left(q^{2}+p^{2}\right)+H_{\rm
dri}/\hbar\\\
&+g_{a}\left(a^{\dagger}m+am^{\dagger}\right)+g_{m}m^{\dagger}mq-
g_{c}c^{\dagger}cq,\end{split}$ (1)

where $j=a,m,c$ ($j^{\dagger}$) are the annihilation (creation) operators of
the microwave cavity mode, the magnon mode, and the optical cavity mode,
respectively, satisfying the canonical commutation relation
$[j,j^{\dagger}]=1$. $q$ and $p$ ($[q,p]=i$) denote the dimensionless position
and momentum of the mechanical vibration mode, and $\omega_{k}$
($k\,\,{=}\,\,a,m,c,b$) are the resonant frequencies of the four modes.
$g_{a}$ is the electromagnonic coupling strength, responsible for the
microwave-magnon state swapping, and $g_{m}$ ($g_{c}$) is the vacuum
magnomechanical (optomechanical) coupling strength, which can be significantly
enhanced by driving the magnon mode (optical cavity) with a strong field. The
driving Hamiltonian $H_{\rm
dri}/\hbar=i\Omega\big{(}m^{\dagger}e^{-i\omega_{0}t}-{\rm
H.c.}\big{)}+iE\big{(}c^{\dagger}e^{-i\omega_{L}t}-{\rm H.c.}\big{)}$, where
the Rabi frequency $\Omega=\frac{\sqrt{5}}{4}\gamma\sqrt{N}H_{d}$[46]
describes the coupling strength between the magnon mode and the microwave
drive field, with the gyromagnetic ratio $\gamma$, the total spin number $N$
of the ferrimagnet, and the amplitude of the drive magnetic field $H_{d}$.
$E=\sqrt{2\kappa_{c}P_{L}/(\hbar\omega_{L})}$ denotes the coupling strength
between the optical cavity and the driving laser, where $P_{L}$ ($\omega_{L}$)
is the power (frequency) of the laser field, and $\kappa_{c}$ is the cavity
decay rate.

Including the dissipation and input noise of each mode and working in the
interaction picture with respect to
$\hbar\omega_{0}(a^{\dagger}a+m^{\dagger}m)+\hbar\omega_{L}c^{\dagger}c$, we
obtain the quantum Langevin equations (QLEs) of the system, given by

$\displaystyle\begin{split}\dot{a}=&-i\Delta_{a}a-\kappa_{a}a-ig_{a}m+\sqrt[]{2\kappa_{a}}a_{in},\\\
\dot{m}=&-i\Delta_{m}m-\kappa_{m}m-ig_{a}a-ig_{m}mq+\Omega+\sqrt{2\kappa_{m}}m_{in},\\\
\dot{q}=&\
\omega_{b}p,\,\,\dot{p}=-\omega_{b}q-\gamma_{b}p+g_{c}c^{\dagger}c-g_{m}m^{\dagger}m+\xi,\\\
\dot{c}=&-i\Delta_{c}c-\kappa_{c}c+ig_{c}cq+E+\sqrt{2\kappa_{c}}c_{in},\end{split}$
(2)

where $\Delta_{a(m)}=\omega_{a(m)}-\omega_{0}$,
$\Delta_{c}=\omega_{c}-\omega_{L}$, and $\gamma_{b}$ and $\kappa_{j}$
$(j=a,m,c)$ are the dissipation rates of the mechanical, microwave cavity,
magnon, and optical cavity modes, respectively. $j_{in}$ are the corresponding
input noise operators, which are zero-mean and obey the following correlation
functions: $\langle
j_{in}(t)j^{\dagger}_{in}(t^{\prime})\rangle=\big{[}N_{j}(\omega_{j})+1\big{]}\delta(t-t^{\prime})$,
and $\langle
j^{\dagger}_{in}(t)j_{in}(t^{\prime})\rangle=N_{j}(\omega_{j})\delta(t-t^{\prime})$.
$\xi(t)$ is a Brownian stochastic force acting on the mechanical oscillator,
which is non-Markovian by nature, but can be assumed Markovian for a large
mechanical quality factor $Q_{b}=\omega_{b}/\gamma_{b}\gg 1$.[68, 69] It then
takes a $\delta$-autocorrelation:
$\left\langle\xi(t)\xi(t^{\prime})+\xi(t^{\prime})\xi(t)\right\rangle/2\simeq\gamma_{b}\big{[}2N_{b}(\omega_{b})+1\big{]}\delta(t-t^{\prime})$.
The mean thermal excitation number of each mode
$N_{k}(\omega_{k})=\big{[}\mathrm{exp}(\hbar\omega_{k}/k_{B}T)-1\big{]}^{-1}$($k=j,b$),
with $T$ being the environmental temperature.

The creation of microwave-optics entanglement requires sufficiently strong
magno- and optomechanical coupling strengths. To this end, we apply strong
drive fields onto the magnon and optical cavity modes, respectively, which
lead to large coherent amplitudes $\left|\left\langle
m\right\rangle\right|,\left|\left\langle c\right\rangle\right|\gg 1$ at the
steady state. This enables us to linearize the system dynamics around the
large average values by neglecting small second-order fluctuation terms.[70]
We therefore obtain a set of linearized QLEs for the quantum fluctuations of
the system, which can be written in the matrix form of

$\displaystyle\dot{u}(t)=Au(t)+n(t),$ (3)

where $u(t)=\big{[}\delta X_{a}(t),\delta Y_{a}(t),\delta X_{m}(t),\delta
Y_{m}(t),\delta q(t),\delta p(t),\delta X_{c}(t)$, $\delta
Y_{c}(t)\big{]}^{T}$ is the vector of the quadrature fluctuations of the
system,
$n(t)=[\sqrt{2\kappa_{a}}X_{a}^{in}(t),\sqrt{2\kappa_{a}}Y_{a}^{in}(t),\sqrt{2\kappa_{m}}X_{m}^{in}(t),\sqrt{2\kappa_{m}}Y_{m}^{in}(t),$
$0,\xi(t),\sqrt{2\kappa_{c}}X_{c}^{in}(t),\sqrt{2\kappa_{c}}Y_{c}^{in}(t)]^{T}$
is the vector of the input noises, where the quadratures are defined as
$X_{j}(t)=\frac{1}{\sqrt{2}}(j+j^{\dagger})$ and
$Y_{j}(t)\,{=}\frac{i}{\sqrt{2}}(j^{\dagger}{-}j)$, and $\delta X_{j}(t)$ and
$\delta Y_{j}(t)$ ($X_{j}^{in}(t)$ and $Y_{j}^{in}(t)$) represent the
corresponding fluctuations (input noises), and the drift matrix $A$ is given
by

$\displaystyle A=\begin{pmatrix}-\kappa_{a}&\Delta_{a}&0&g_{a}&0&0&0&0\\\
-\Delta_{a}&-\kappa_{a}&-g_{a}&0&0&0&0&0\\\
0&g_{a}&-\kappa_{m}&\tilde{\Delta}_{m}&G_{m}&0&0&0\\\
-g_{a}&0&-\tilde{\Delta}_{m}&-\kappa_{m}&0&0&0&0\\\
0&0&0&0&0&\omega_{b}&0&0\\\ 0&0&0&-G_{m}&-\omega_{b}&-\gamma_{b}&0&-G_{c}\\\
0&0&0&0&G_{c}&0&-\kappa_{c}&\tilde{\Delta}_{c}\\\
0&0&0&0&0&0&-\tilde{\Delta}_{c}&-\kappa_{c}\end{pmatrix}.$ (4)

We define the effective magno- and optomechanical coupling strengths as
$G_{m}=-i\sqrt{2}g_{m}\left\langle m\right\rangle$ and
$G_{c}=i\sqrt{2}g_{c}\left\langle c\right\rangle$, where the steady-state
amplitudes are

$\displaystyle\left\langle
m\right\rangle=\frac{\Omega(\kappa_{a}+i\Delta_{a})}{g_{a}^{2}+(\kappa_{m}+i\tilde{\Delta}_{m})(\kappa_{a}+i\Delta_{a})},\,\,\,\,\left\langle
c\right\rangle=\frac{E}{\kappa_{c}+i\tilde{\Delta}_{c}},$ (5)

and the effective detunings $\tilde{\Delta}_{m}=\Delta_{m}+g_{m}\left\langle
q\right\rangle$ and $\tilde{\Delta}_{c}=\Delta_{c}-g_{c}\left\langle
q\right\rangle$, including the frequency shifts due to the mechanical
displacement, which is jointly induced by the magnetostriction and radiation
pressure interactions. The steady-state displacement $\left\langle
q\right\rangle=\big{(}g_{c}\left|\left\langle
c\right\rangle\right|^{2}-g_{m}\left|\left\langle
m\right\rangle\right|^{2}\big{)}/\omega_{b}$. Note that the above drift matrix
$A$ is derived under the optimal conditions for the microwave-optics
entanglement, i.e.,
$\left|\Delta_{a}\right|,\left|\tilde{\Delta}_{m}\right|,\tilde{\Delta}_{c}\simeq\omega_{b}\gg\kappa_{j}$
(see Figure 1c). This yields approximate expressions of the amplitudes,
$\left\langle m\right\rangle\simeq
i\Omega\Delta_{a}/\big{(}g_{a}^{2}-\tilde{\Delta}_{m}\Delta_{a}\big{)}$ and
$\left\langle c\right\rangle\simeq-iE/\tilde{\Delta}_{c}$, which are pure
imaginary and thus give rise to approximately real coupling strengths $G_{m}$
and $G_{c}$.

We aim to prepare stationary entanglement between microwave and optical cavity
fields. The steady state of the system is a four-mode Gaussian state, because
of the linearized dynamics and the Gaussian input noises. A Gaussian state is
fully characterized by the first and second moments of the quadrature
operators. The first moments can be adjusted by local displacement operations
in phase space and such operations do not change quantum correlations like
entanglement. The state of interest is therefore characterized by an $8\times
8$ covariance matrix (CM) $V$, of which the entries are defined as
$V_{ij}=\langle u_{i}(t)u_{j}(t^{\prime})+u_{j}(t^{\prime})u_{i}(t)\rangle/2\
$($i,j=1,2,...,8$). The CM in the steady state can be obtained by solving the
Lyapunov equation[70]

$\displaystyle AV+VA^{T}=-D,$ (6)

where
$D=\mathrm{diag}\,[\,\kappa_{a}(2N_{a}+1),\kappa_{a}(2N_{a}+1),\kappa_{m}(2N_{m}+1),$
$\kappa_{m}(2N_{m}+1),\,\,0,\,\,\gamma_{b}(2N_{b}+1),\ \kappa_{a}(2N_{c}+1),\
\kappa_{a}(2N_{c}+1)]$ is the diffusion matrix and defined by
$D_{ij}\delta(t-t^{\prime})=\langle
n_{i}(t)n_{j}(t^{\prime})+n_{j}(t^{\prime})n_{i}(t)\rangle/2$. We adopt the
logarithmic negativity[71, 72, 73] to quantify the microwave-optics
entanglement, i.e.,

$\displaystyle E_{ca}=\mathrm{max}\big{[}0,-\mathrm{ln}(2\eta^{-})\big{]},$
(7)

where $\eta^{-}\equiv
2^{-1/2}\big{[}\Sigma-\big{(}\Sigma^{2}-4\,\mathrm{det}V_{4}\big{)}^{1/2}\big{]}^{1/2}$,
and $V_{4}$ is the $4\times 4$ CM associated with the microwave and optical
modes. $V_{4}=\big{[}V_{a},V_{ac};V_{ac}^{T},V_{c}\big{]}$, with $V_{a}$,
$V_{c}$, and $V_{ac}$ being the $2\times 2$ blocks of $V_{4}$, and
$\Sigma\equiv\mathrm{det}V_{a}+\mathrm{det}V_{c}-2\mathrm{det}V_{ac}$.
Similarly, we can calculate the magnon-optics entanglement ($E_{cm}$) and the
microwave-mechanics entanglement ($E_{ab}$).

##  3 Results and Discussions

The mechanism of producing microwave-optics entanglement is as follows.
Firstly, the highly thermally populated mechanical mode should be cooled
(close) to its quantum ground state in order to generate entanglement in the
system. This can be achieved by driving the optical cavity with a red-detuned
laser field ($\tilde{\Delta}_{c}\simeq\omega_{b}$) to activate the
optomechanical anti-Stokes scattering, optimally in the resolved sideband
limit $\omega_{b}\gg\kappa_{c}$ (Figure 1c).[60] The effective optomechanical
coupling is then a beamsplitter type, which takes the same form as the
electromagnonic coupling. As is known, such beamsplitter couplings solely will
not create any entanglement, and the system requires a parametric down-
conversion process to generate entanglement. This can be realized by driving
the magnon mode with a blue-detuned microwave field
($\tilde{\Delta}_{m}\simeq-\omega_{b}$) to activate the magnomechanical Stokes
scattering, which yields magnon-phonon entanglement. Further due to the
effective beamsplitter (state-swap) interaction in the optomechanical and
electromagnonic couplings, the optical and microwave cavity photons therefore
get entangled.

The optimal detunings
$\tilde{\Delta}_{c}\simeq-\tilde{\Delta}_{m}\simeq\omega_{b}$ adopted in the
tripartite OMM system are analogous to those used to produce optical
entanglement in a tripartite optomechanical system, where two cavity fields,
either optical or microwave, couple to a common mechanical resonator.[25, 26,
27, 28] The optical entanglement is the result of the combination of
simultaneous parametric down-conversion and state-swap interactions associated
with the two cavity fields and the mechanical oscillator being an intermediary
to distribute the entanglement. This is the fundamental mechanism of producing
optical entanglement in optomechanical systems.[25, 26, 27, 28, 74, 75] Due to
their similar Hamiltonians and scattering processes, magnons and optical
photons are expected to be entangled in our OMM system. This is confirmed by
Figure 2a. Further, the microwave-magnon state-swap interaction enables the
distribution of the entanglement to the microwave photons, thus establishing
the microwave-optics entanglement, as shown in Figure 2b. The complementary
distribution of the entanglement in Figure 2a,b is a clear sign of the
entanglement transfer from the magnon-optics subsystem ($E_{cm}$) to the
microwave-optics subsystem ($E_{ca}$). Such entanglement transfer is optimally
achieved in the strong-coupling regime, where the microwave-magnon coupling
rate is larger than their dissipation rates,
$g_{a}>\kappa_{a},\kappa_{m}$,[76, 77] see Figure 2c. The microwave-optics
entanglement $E_{ca}$ achieves its maximum around $g_{a}/2\pi\simeq 4$ MHz and
starts to decline as $g_{a}$ continues to increase. This is because the
entanglement is further distributed to the microwave-mechanics subsystem
(Figure 2c, the dot-dashed line). The distribution of entanglement among
different subsystems is a typical feature of the entanglement in multipartite
systems[46] where entanglement is considered as a finite quantum resource.

Figure 2 is plotted with experimentally feasible parameters[45, 46, 47, 48,
49, 62]: $\omega_{a}/2\pi=10$ GHz, $\omega_{b}/2\pi=40$ MHz, an optical
wavelength $\lambda_{c}=1550$ nm, $\kappa_{a(m)}/2\pi=1.5$ MHz,
$\kappa_{c}/2\pi=2$ MHz, $\gamma_{b}/2\pi=10^{2}$ Hz, $g_{a}/2\pi=4$ MHz,
$G_{m}/2\pi=2$ MHz, $G_{c}/2\pi=8$ MHz, $T=10$ mK, and the magnon frequency is
tuned to be resonant with the microwave cavity
($\Delta_{a}=\tilde{\Delta}_{m}$). All the entanglements are in the steady
state confirmed by the negative eigenvalues (real parts) of the drift matrix.
Under these parameters and the optimal detunings, the effective mean phonon
number is $\bar{n}_{b}^{\rm eff}\simeq 0.17$, indicating that the mechanical
mode is significantly cooled to its ground state. The microwave-optics
entanglement is robust against thermal noises, and is present at an
environmental temperature up to $T\simeq 380$ mK (Figure 2d). Note that a
relatively large optomechanical coupling $G_{c}/2\pi=8$ MHz is used for
cooling, under which the system remains stable due to the multi dissipation
channels in our quadripartite system. This corresponds to a laser power
$P_{L}\simeq 2.56$ mW for $g_{c}/2\pi=2$ kHz. The magnomechanical coupling
$G_{m}/2\pi=2$ MHz corresponds to a microwave drive power $P_{0}\simeq 0.91$
mW (the drive magnetic field $H_{d}\simeq 8.7\times 10^{-4}$ T) for a 5
$\times$ 2 $\times$ 1 $\mu$m3 YIG cuboid with $g_{m}/2\pi=20$ Hz. Note that a
much stronger $g_{m}$ is used than $\sim$10 mHz for a large-size YIG sphere
with the diameter in the $10^{2}$ $\mu$m range,[45, 48, 49] because we adopt a
much smaller micron-sized YIG crystal, which permits a much stronger yet
dispersive magnon-phonon coupling. To determine the power, we have used the
relation of the drive magnetic field $H_{d}$ and the power $P_{0}$, i.e.,
$H_{d}=\sqrt{2\mu_{0}P_{0}/(lwc)}$, where $\mu_{0}$ is the vacuum magnetic
permeability, $c$ is the speed of the electromagnetic wave propagating in
vacuum, and $l$ and $w$ are the length and width of the YIG crystal. This is
derived as follows. The time average of energy density is
$\rho_{e}=H_{d}^{2}/(2\mu_{0})$ and thus the microwave drive power
$P_{0}=\rho_{e}Ac$, where $A=l\times w$ is the cross-sectional area of the YIG
cuboid. The drive magnetic field is applied in the direction perpendicular to
the cross-section. This then yields the relation between the drive magnetic
field $H_{d}$ and the power $P_{0}$, i.e., $H_{d}=\sqrt{2\mu_{0}P_{0}/(lwc)}$.
We remark that a stronger magnomechanical coupling strength yields a greater
microwave-optics entanglement but also a narrower parameter regime of the
system being stable. For example, under the parameters of Figure 2b and the
optimal detunings, the coupling can be as high as $G_{m}/2\pi=4.6$ MHz while
keeping the system stable, which yields $E_{ca}\simeq 0.4$.

Having determined the optimal working conditions of the OMM system for the
entanglement (i.e.,
$\tilde{\Delta}_{c}\simeq-\tilde{\Delta}_{m}\simeq\omega_{b}$, and
$\omega_{b}\gg\kappa_{m,c}$ for resolved sidebands), we optimize the
entanglement over two parameters of the microwave cavity, i.e., the cavity-
magnon detuning $\Delta_{am}\equiv\Delta_{a}-\tilde{\Delta}_{m}$ and the
cavity decay rate $\kappa_{a}$, as shown in Figure 3a. Clearly, the resonant
case $\Delta_{am}=0$ maximizes the microwave-optics entanglement, indicating
that the entanglement is transferred from the magnomechanical subsystem
(explicitly, the magnon-phonon parametric down-conversion process) via the
microwave-magnon and optomechanical beamsplitter couplings (c.f. Figure 1b).
It also shows that small cavity loss is always helpful as long as the
stability is maintained. Figure 3b studies the effect of the temperature and
mechanical damping on the optical entanglement. The entanglement is robust
against temperature: For moderate damping rates $\gamma_{b}/2\pi<10^{3}$ Hz,
the entanglement can survive up to $\sim 0.3$ K, whereas for a significantly
large damping rate $\gamma_{b}/2\pi=10^{5}$ Hz (quality factor $Q_{b}=400$),
the entanglement can still be present up to $\sim 0.1$ K.

##  4 Conclusion

We have presented a feasible and efficient approach to generate stationary
microwave-optics entanglement in a microwave cavity-OMM system. The
entanglement is achieved by exploiting dispersive magnetostriction and
radiation pressure interactions. Specifically, by realizing the effective
magnomechanical parametric down-conversion and optomechanical beamsplitter
interactions, magnon-phonon entanglement is created and the entanglement is
further distributed to optical and microwave cavity photons via the
optomechanical and electromagnonic state-swap interactions, yielding
stationary microwave-optics entanglement. Optimal working conditions are
analyzed and the entanglement is robust against temperature even for
significantly large mechanical damping. Such microwave-optics entanglement
will find many distinctive applications in quantum networks and quantum
information processing with hybrid quantum systems.

## Acknowledgements

This work is supported by National Key Research and Development Program of
China (Grant No. 2022YFA1405200) and National Natural Science Foundation of
China (Nos. 92265202).

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Figure 1. a,b) Sketch of the cavity-OMM system. A magnon mode $m$ in a YIG
crystal couples to a microwave cavity mode $a$ and to an optical cavity mode
$c$ via the mechanical vibration $b$ induced by the magnetostriction. The blue
(red) line in (b) denotes the effective parametric down-conversion
(beamsplitter) interaction. c) Mode frequencies and linewidths used in the
protocol. When the optical cavity is resonant with the (blue) anti-Stokes
sideband of the driving laser at frequency $\omega_{L}+\omega_{b}$, and the
magnon and microwave cavity modes are resonant with the (red) Stokes sideband
of the microwave drive field at frequency $\omega_{0}-\omega_{b}$, and when
the mechanical sidebands are resolved ($\omega_{b}\gg\kappa_{j}$), a
stationary microwave-optics entangled state is generated. Figure 2. a,b)
Density plot of the steady-state magnon-optics entanglement $E_{cm}$ and
microwave-optics entanglement $E_{ca}$ versus effective detunings
$\tilde{\Delta}_{m}$ and $\tilde{\Delta}_{c}$. c) $E_{cm}$ (red dashed),
$E_{ca}$ (black solid), and microwave-mechanics entanglement $E_{ab}$ (blue
dot-dashed) versus the microwave-magnon coupling rate $g_{a}$ at optimal
detunings $\tilde{\Delta}_{c}=-\tilde{\Delta}_{m}=\omega_{b}$, and
$\Delta_{a}=\tilde{\Delta}_{m}$. d) $E_{ca}$ versus temperature $T$ at the
optimal detunings. $g_{a}/2\pi=4$ MHz is used in (a), (b) and (d). See the
text for the rest of the parameters. Figure 3. Stationary microwave-optics
entanglement $E_{ca}$ versus a) the cavity-magnon detuning $\Delta_{am}$ and
the cavity decay rate $\kappa_{a}/\kappa_{m}$ (with fixed $\kappa_{m}$ and
$\tilde{\Delta}_{m}=-\tilde{\Delta}_{c}=-\omega_{b}$); b) temperature $T$ and
mechanical damping rate $\gamma_{b}$. The other parameters are the same as in
Figure 2b.