# Tunable Magnon-Photon Coupling by Magnon Band Gap in a Layered Hybrid
Perovskite Antiferromagnet

Yi Li<EMAIL_ADDRESS>Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA  Timothy Draher  Materials Science Division, Argonne
National Laboratory, Argonne, IL 60439, USA  Northern Illinois University,
Department of Physics, Dekalb Illinois, 60115, USA  Andrew H. Comstock
Department of Physics and Organic and Carbon Electronics Laboratory (ORACEL),
North Carolina State University, Raleigh, NC 27695 USA  Yuzan Xiong
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27599, USA  Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA  Md Azimul Haque  Chemistry and Nanoscience Center,
National Renewable Energy Laboratory, Golden, CO 80401, USA  Elham Easy
Department of Mechanical Engineering, Stevens Institute of Technology,
Hoboken, NJ 07030, USA  Jiang-Chao Qian  Department of Materials Science and
Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61820, USA
Tomas Polakovic  Physics Division, Argonne National Laboratory, Lemont, IL
60439, USA  John E. Pearson  Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA  Ralu Divan  Center for Nanoscale
Materials, Argonne National Laboratory, Argonne, IL 60439, USA  Jian-Min Zuo
Department of Materials Science and Engineering, University of Illinois
Urbana-Champaign, Urbana, IL 61820, USA  Xian Zhang  Department of Mechanical
Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA  Ulrich
Welp  Materials Science Division, Argonne National Laboratory, Argonne, IL
60439, USA  Wai-Kwong Kwok  Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA  Axel Hoffmann  Department of Materials
Science and Engineering, University of Illinois Urbana-Champaign, Urbana, IL
61820, USA  Joseph M. Luther  Chemistry and Nanoscience Center, National
Renewable Energy Laboratory, Golden, CO 80401, USA  Matthew C. Beard
Chemistry and Nanoscience Center, National Renewable Energy Laboratory,
Golden, CO 80401, USA  Dali Sun<EMAIL_ADDRESS>Department of Physics and
Organic and Carbon Electronics Laboratory (ORACEL), North Carolina State
University, Raleigh, NC 27695 USA  Wei Zhang<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599,
USA  Valentine Novosad<EMAIL_ADDRESS>Materials Science Division, Argonne
National Laboratory, Argonne, IL 60439, USA

###### Abstract

Tunability of coherent coupling between fundamental excitations is an
important prerequisite for expanding their functionality in hybrid quantum
systems. In hybrid magnonics, the dipolar interaction between magnon and
photon usually persists and cannot be switched off. Here, we demonstrate this
capability by coupling a superconducting resonator to a layered hybrid
perovskite antiferromagnet, which exhibits a magnon band gap due to its
intrinsic Dzyaloshinskii-Moriya interaction. The pronounced temperature
sensitivity of the magnon band gap location allows us to set the photon mode
within the gap and to disable magnon-photon hybridization. When the resonator
mode falls into the magnon band gap, the resonator damping rate increases due
to the nonzero coupling to the detuned magnon mode. This phenomena can be used
to quantify the magnon band gap using an analytical model. Our work brings new
opportunities in controlling coherent information processing with quantum
properties in complex magnetic materials.

Hybrid quantum systems Kurizki  _et al._ (2015); Clerk  _et al._ (2020);
Lachance-Quirion  _et al._ (2019) offer an important pathway for harnessing
different natural advantages of complementary quantum systems, leveraging the
distinct properties of their constituent excitations. The fundamental
excitations of magnetically ordered materials, i.e. magnons, provide efficient
coupling with other excitations Li  _et al._ (2020a, 2021a), such as microwave
photons Huebl  _et al._ (2013); Goryachev  _et al._ (2014); Tabuchi  _et al._
(2014); Zhang  _et al._ (2014); Bai  _et al._ (2015); Li  _et al._ (2019); Hou
and Liu (2019), acoustic phonons Kikkawa  _et al._ (2016); Berk  _et al._
(2019); An  _et al._ (2020); Xu  _et al._ (2021a), and magnons themselves
Klingler  _et al._ (2018); Chen  _et al._ (2018); MacNeill  _et al._ (2019);
Li  _et al._ (2020b); Shiota  _et al._ (2020); Sud  _et al._ (2020); Sklenar
and Zhang (2021); Li  _et al._ (2021b), therefore holding promise for future
integration with diverse quantum modules Harder and Hu (2018); Awschalom  _et
al._ (2021); Fukami  _et al._ (2021); Yuan  _et al._ (2022). In addition,
coherent magnon interactions exhibit great controllability in different
aspects, such as polarization Li  _et al._ (2019), mode profile Shiota  _et
al._ (2020), phase Harder  _et al._ (2018); Bhoi  _et al._ (2019); Wang  _et
al._ (2019); Boventer  _et al._ (2020) and layer structure Li  _et al._
(2020b), allowing for implementation of coherent magnon operations Xu  _et
al._ (2021b). Recent demonstrations of coherent magnon-magnon coupling with
controllable coupling strength by frequency detuning Lambert  _et al._ (2016);
Li  _et al._ (2022) have further expanded the capability of distributed hybrid
magnonic networks Li  _et al._ (2020a).

Figure 1: (a) Lattice structure of layered perovskite antiferromagnet Cu-EA,
with Cu filling the octahedral sites of Cl and the antiferromagnetic layers
separated by the CH3CH2NH3 molecules. (b) Optical microscope image of a small
Cu-EA flake mounted onto a CPW superconducting resonator. (c) Raman
spectroscopy of Cu-EA showing the high-frequency octahedral modes and the low-
frequency organic structure modes. (d) Broad-band ferromagnetic resonance
spectra of a large Cu-EA crystal measured at 1.6 K, which is used to extract
the magnon band gap $2\delta$, the interlayer exchange field $2H_{E}$, and
magnon damping rate $\kappa_{m}$. The colorbar shows the signals $\Delta
S_{21}$ after background subtraction. (e) Extracted $2H_{E}$ magentaand
$2\delta$ as a function of $T$. (f) Mode anticrossing between magnons and
photons at 5.5 K, with $H_{B}\perp h_{\text{rf}}^{y}$. The colorbar shows the
signals $S_{21}$ in absolute values. The red curves are the fits with
$g/2\pi=45$ MHz. The black dashed lines denote the magnon and photon modes
without interaction. Figure 2: (a-b) Illustration of two different in-plane
field alignments and their selective mode excitations. In (a),
$\mu_{0}H_{B}\perp h_{\textrm{rf}}^{y}$ and only the acoustic mode is excited.
In (b), $\mu_{0}H_{B}\parallel h_{\textrm{rf}}^{y}$ and both the acoustic and
optical modes are excited. (c-d) Temperature dependence of the magnon-photon
coupling evolutions from 1.5 to 6 K with two different magnetic field
alignments. All the dispersion centered at the resonator photon mode
($\omega_{p}/2\pi\approx 3.5$ GHz). Dashed curves are guide to eye for the
acoustic mode crossing the resonator mode at $\mu_{0}H_{a}=95$ mT and the
optical mode crossing the resonator mode at different fields. (e-g).
Illustration of the three regimes where the resonator mode is (e) above, (f)
within, and (g) below the magnon band gap.

Despite its controllability, which is largely based on extrinsic control of
magnetic systems, the intrinsic magnetic properties are rarely explored for
manipulating coherent magnon interactions. To date, landmark demonstrations of
hybrid magnonics have been centered on the ferrimagnetic insulator yttrium
iron garnet (YIG) Huebl  _et al._ (2013); Tabuchi  _et al._ (2014); Zhang  _et
al._ (2014); Bai  _et al._ (2015); Tabuchi  _et al._ (2015) or metallic
magnets such as NiFe Li  _et al._ (2019); Hou and Liu (2019). Their relatively
simple and rigid chemical and magnetic structures limit the potential for
developing highly tunable hybrid systems. On the other hand, the recent two-
dimensional (2D) organic layered magnets Wang and et al. (2022) offer distinct
advantages in their structure-enabled topological chirality and symmetry
breaking Tang  _et al._ . One nice class of materials is 2D magnetic hybrid
organic-inorganic perovskites (HOIPs) possessing both superior structural
versatility and long-range magnetic order Saparov and Mitzi (2016); Nugroho
_et al._ (2016); Kim  _et al._ (2020); Willett  _et al._ (2012). They usually
exhibit an interlayer antiferromagnetic (AFM) coupling De Jongh  _et al._
(1972), inducing the acoustic and optical magnon modes Keffer and Kittel
(1952); MacNeill  _et al._ (2019) in the gigahertz (GHz) frequency range. In
addition, the structural symmetry breaking leads to Dzyaloshinskii–Moriya
interaction (DMI) Moriya (1960), causing a finite spin canting Bloembergen
_et al._ (1973); Bogdanov  _et al._ (2007) and creating an intrinsic magnon
band gap where the acoustic and optical modes intersect Comstock  _et al._
(2023). This is fundamentally different from the magnon band gap induced by an
external field Diederich  _et al._ (2023)MacNeill  _et al._ (2019); Shiota
_et al._ (2020); Sud  _et al._ (2020); Sklenar and Zhang (2021); Li  _et al._
(2021b); Makihara  _et al._ (2021) in that the DMI has provided an intrinsic
effective field for magnon-magnon coupling without the need of external field.
Furthermore, the large sensitivity of the magnon band gap to small temperature
change can lead to new opportunities of modulating coherent magnonic coupling.

Zollitsch  _et al._ (2022) In this Letter, we report a hybrid magnonic system
consisting of a 2D HOIPs, (CH3CH2NH3)2CuCl4 (Cu-EA) Comstock  _et al._ (2023),
coupled to a superconducting resonator. The high sensitivity of the
superconducting resonator enables coherent magnon-photon coupling and avoided
crossing with a small Cu-EA flake. By changing the temperature of the sample,
the location of the DMI-induced magnon band gap can be adjusted so that the
resonator photon mode completely falls into the gap and cancels the mode
hybridization. At the non-hybridized state, the magnetic interaction with the
resonator causes the resonator linewidth to broaden. Using our developed
analytical model, the narrow-band linewidth broadening measurements can be
used to extract the magnon band gap, which quantitatively agrees with the
broad-band FMR measurements. Our results highlight the opportunity of
manipulating coherent mode hybridization with new quantum materials and
probing their complex magnonic dispersion with narrow-band microwave
characterizations.

The chemical structure of Cu-EA features corner-sharing halogen (Cl) octahedra
with the Cu atom situated at the center, as shown in Fig. 1(a). The canted
inorganic CuCl${}_{4}^{2-}$ octahedral structures allow for intralayer long-
range magnetic order with superexchange Cu-Cl-Cu interactions, while the
interlayer organic cations modulate the interlayer antiferromagnetic (AFM)
coupling De Jongh  _et al._ (1972). Raman spectroscopy of the Cu-EA Easy  _et
al._ (2021) confirms the vibration modes of the octahedral structure (at 175,
250, and 280 cm-1) and the organic cation (at 100 cm-1) Caretta  _et al._
(2014), as shown in Fig. 1(c). We have also conducted Inductively Coupled
Plasma (IPC) spectroscopy on the sample, showing accurate stochiometry of the
elemental weight as compared with the chemical structure sm .

Fig. 1(d) shows the broad-band ferromagnetic resonance of a large Cu-EA
crystal at 1.6 K at parallel pumping condition, i.e. $\mu_{0}H_{B}\parallel
h_{\textrm{rf}}^{y}$ as illustrated in Fig. 2(b). Both the acoustic and
optical modes are measured, which can be formulated as MacNeill  _et al._
(2019):

$\omega_{a}=\mu_{0}\gamma\sqrt{2H_{E}(2H_{E}+M_{eff})}{H\over 2H_{E}}$ (1)
$\omega_{o}=\mu_{0}\gamma\sqrt{2H_{E}M_{eff}\left(1-{H^{2}\over
4H_{E}^{2}}\right)}$ (2)

where $H_{E}$ is the interlayer exchange coupling field, $M_{eff}$ is the
effective magnetization which contributes to the perpendicular demagnetization
field, and $\gamma/2\pi=(g_{e}/2)\times 28$ GHz/T is the gyromagnetic ratio,
with $g_{e}$ as the $g$-factor of the magnetization. Clear avoided crossing
gaps between the two modes show the existence of magnon band gap around 4 GHz.
The coupled magnon spectra can be fitted to the hybrid mode expression
$\omega^{mm}_{\pm}=(\omega_{a}+\omega_{o})/2\pm\sqrt{(\omega_{a}-\omega_{o})^{2}/4+\delta^{2}}$,
where $\delta$ is the magnon-magnon coupling strength. The fitting curves are
plotted in Fig. 1(d). The extracted parameters are $\mu_{0}H_{E}=0.16$ T,
$\mu_{0}M_{eff}=80$ mT, $g_{e}=2.3$, and $\delta/2\pi=150$ MHz. Note that the
actual saturation magnetization of Cu-EA can be larger than $M_{eff}$ because
the shape of the sample crystal is not a perfect two-dimension system and the
perpendicular demagnetization factor can be smaller than one. The strong
magnon-magnon coupling observed in Cu-EA at parallel pumping condition, which
is absent in other layered MacNeill  _et al._ (2019) or synthetic Shiota  _et
al._ (2020); Sud  _et al._ (2020) antiferromagnets at the same pumping
condition, is caused by the spontaneous canting of the octahedral CuCl42- spin
sites from their chiral DMI and the resultant overlap between the acoustic and
optical modes Comstock  _et al._ (2023). Fig. 1(e) shows the temperature
dependence of extracted $2H_{E}$ and the magnon band gap $2\delta$ from Fig.
1(d). With the same $y$-axis proportion ratio in Fig. 1(e), a good overlap of
$H_{E}$ and $\delta$ shows that they are proportional to each other at
different temperatures. This suggests that the DMI-induced spin canting shares
a similar mechanism with the interlayer exchange coupling in Cu-EA. The magnon
damping rate, $\kappa_{m}$, of the acoustic and optical modes are also
extracted and are found to be weakly frequency and temperature dependent. In
the range of 2-5 GHz, $\kappa_{m}/2\pi\sim 50$ MHz for the acoustic mode and
$\sim 80$ MHz for the optical mode; see the Supplemental Materials for details
sm .

To feature the sensitivity of the superconducting resonator to small magnetic
crystals, we precisely transfer a thin Cu-EA flake with lateral dimensions of
500 $\mu$m $\times$ 200 $\mu$m and a thickness of 40 $\mu$m onto the center of
a half-wavelength NbN coplanar waveguide (CPW) superconducting resonator with
a signal line width of 20 $\mu$m, as shown in Fig. 1(b). The dimension
matching between the signal line and the flake thickness allows for optimal
coupling of the magnon excitations to the resonator. The loaded
superconducting resonator exhibits a sharp peak at $\omega_{p}/2\pi=3.5$ GHz
and a zero-field half-width half-maximum linewidth of $\kappa_{p}/2\pi=0.4$
MHz at 1.6 K, which corresponds to a quality factor of
$\omega_{p}/2\kappa_{p}=4400$.

The maximum mode splitting happens at 5.5 K between the acoustic magnon mode
of Cu-EA and the resonator photon mode, shown in Fig. 1(f). The peak positions
of the avoided crossing can be fitted to the hybrid modes Huebl  _et al._
(2013):

$\omega^{mp}_{\pm}=(\omega_{m}+\omega_{p})/2\pm\sqrt{(\omega_{m}-\omega_{p})^{2}/4+g^{2}}$
(3)

where $\omega_{m}$ is the magnon frequency, $\omega_{p}$ is the photon
frequency, and $g$ is the magnon-photon coupling strength due to dipolar
interaction. The field dependence of $\omega_{p}$ can be extracted from the
linear extrapolation of the background, and the field dependence of
$\omega_{m}$ can be obtained from the broad-band FMR spectrum. Fits to Eq. (3)
yield $g/2\pi=45$ MHz. Using the damping rates of $\kappa_{p}/2\pi=2.7$ MHz
for the superconducting resonator at 5.5 K and $\kappa_{m}/2\pi=50$ MHz for
the Cu-EA acoustic mode, we obtain a cooperativity of
$C=g^{2}/\kappa_{p}\kappa_{m}=15$. We note that even though the cooperativity
becomes higher at lower temperature, e.g. 1.6 K, because of the much lower
$\kappa_{p}$, the real bottleneck of strong magnon-photon coupling is the
ratio $g/\kappa_{m}$, which is maximized as 0.95 at 5.5 K. The strong coupling
regime requires both $g/\kappa_{m}$ and $g/\kappa_{p}$ to be greater than one
Zhang  _et al._ (2014). See the Supplemental Materials for the temperature
dependence of $\kappa_{p}$ and $\kappa_{m}$ sm .

Figure 3: Extracted effective magnon-photon coupling $g$ as a function of $T$.
The resonator mode is within the magnon band gap between 3 and 3.5 K, yielding
$g=0$.

Next, we investigate the temperature dependence of the magnon-photon
interactions. Shown in Figs. 2(a) and (b), the signal line of the resonator
generates both the in-plane and perpendicular Oersted fields,
$h_{\textrm{rf}}^{y}$ and $h_{\textrm{rf}}^{z}$, respectively. In the
orthogonal pumping condition ($\mu_{0}H_{B}\perp h_{\textrm{rf}}^{y}$), the
Oersted field components $h_{\textrm{rf}}^{y}$ and $h_{\textrm{rf}}^{z}$ only
couple to the acoustic mode. In the parallel pumping condition
($\mu_{0}H_{B}\parallel h_{\textrm{rf}}^{y}$), $h_{\textrm{rf}}^{y}$ couples
to the acoustic mode and $h_{\textrm{rf}}^{z}$ couples to the optical mode.
Thus, the field alignment allows for selective excitation of the acoustic mode
in Fig. 2(c), or the mutual excitation of both modes in Fig. 2(d). The
interaction with the acoustic mode is manifested by an avoided crossing at a
constant field of $\mu_{0}H_{\text{a}}=95$ mT for both pumping geometries. The
optical mode found in Fig. 2(d) shows a large temperature-dependent drift of
its location, as marked by the dashed curves. The reversed anticrossing
compared with the acoustic mode shows that the magnon frequency decreases as
the field rises, agreeing with the feature of the optical mode as shown in
Fig. 1(d).

Due to the tunability of $H_{E}$, the center frequency of the magnon band gap
changes rapidly with temperature in the range from 1.5 to 6.0 K, therefore,
allowing the resonator mode (much less sensitive to temperature) to intercept
with the magnon band gap while maintaining a nearly constant quality factor.
Three regimes of the magnon-photon coupling between the Cu-EA flake and the
superconducting resonator are observed, with the relation between the magnon
band gap and the resonator mode shown in Figs. 2(e-g). In regime (i) ($T>3.5$
K), the magnon band gap is below the superconducting resonator frequency [Fig.
2(e)]. The resonator photon mode coherently interacts with the acoustic mode
in Fig. 2(c), and both the acoustic and optical modes in Fig. 2(d). In regime
(ii) ($3.5\geq T\geq 3$ K), the acoustic and optical magnon modes cross each
other and form the magnon band gap at the superconducting resonator mode
frequency. This causes the resonator mode to fall inside the magnon band gap,
leading to a moderate change of the peak amplitude and linewidth without peak
frequency shift around 95 mT. In regime (iii) ($T<3$ K) where the magnon band
gap is above the resonator mode, the acoustic mode resumes its anticrossing-
like interaction with the resonator mode. In addition, for the
$\mu_{0}H_{B}\parallel h_{\textrm{rf}}^{y}$ geometry where the optical mode
also interacts with the resonator mode [Fig. 2(d)], the regime between the
optical and acoustic modes are blurred, as shown in regimes (i) and (iii).
This indicates that one of the two acoustic-optical hybrid magnonic modes is
still near the resonator mode and maintains the magnon-photon interaction.

Figure 4: (a) Superconducting resonator linewidth $\kappa_{c}$ as a function
of $H_{B}$ at $T=3.3$ K, where the resonator mode is inside the magnon band
gap. (b) SC resonator linewidth change $\Delta\kappa_{c}$ as a function of
$g_{\text{eff}}^{2}$ for the CPW and LER resonator designs. Error bars denote
the uncertainty of base resonator linewidth drift under external magnetic
fields. The red line is a fit to Eq. 4, with the slope quantifying the magnon
band gap $2\delta$.

Figure 3 summarizes extracted magnon-photon coupling strength, $g$, as a
function of $T$. To verify the phenomena, we have also coupled another Cu-EA
crystal to a lumped-element resonator (LER) McKenzie-Sell  _et al._ (2019).
This allows for a larger magnon-photon coupling strength while maintaining the
same magnon band gap. For both the CPW resonator and LER, $g$ quickly
decreases in regime (ii) due to mode degeneracy breaking between the magnon
mode and the resonator photon mode. The zero coupling strength for the CPW
resonator is manifested by the continuous evolution of resonator peak without
mode anticrossing, as shown in Fig. 2(c-d). For the LER, a finite $g$ can
still be extracted in regime (ii) which is due to non-perfect centering of the
resonator mode in the magnon band gap when then magnon-photon coupling is
large. The maximal acoustic mode coupling strengths for the CPW resonator are
$g_{\text{CPW}}^{\perp}/2\pi=45$ MHz for $\mu_{0}H_{B}\perp
h_{\textrm{rf}}^{y}$ and $g_{\text{CPW}}^{\parallel}/2\pi=28$ MHz for
$\mu_{0}H_{B}\parallel h_{\textrm{rf}}^{y}$ at 5.5 K. Their difference
quantifies the coupling ratio of the acoustic magnon mode between the in-plane
($h_{rf}^{y}$) and perpendicular($h_{rf}^{z}$) Oersted fields from the CPW: at
$\mu_{0}H_{B}\perp h_{\textrm{rf}}$, both $h_{rf}^{y}$ and $h_{rf}^{z}$ couple
to the acoustic mode, while at $\mu_{0}H_{B}\parallel h_{\textrm{rf}}$, only
$h_{rf}^{z}$ couples to the acoustic mode. The ratio can be calculated as
$h_{rf}^{y}/h_{rf}^{z}=\sqrt{(g_{\text{CPW}}^{\perp})^{2}-(g_{\text{CPW}}^{\parallel})^{2}}/g_{\text{CPW}}^{\perp}=1.25$.
For the LER, the obtained ratio is 1.23. This suggests that $h_{rf}^{z}$ plays
an important role in magnon-photon coupling. When the magnon band gap is far
from the resonator mode (e.g. 1.5 K and 6 K), a reduction of $g$ from 6 K to
1.5 K reflects the change of coupling efficiency between the Oersted field and
the canted magnetization at different biasing field directions. We plot the
calculated prediction of the effective magnon-photon coupling,
$g_{\text{eff}}$, for the acoustic magnon mode without considering the magnon
band gap sm , and the trend nicely captures the experiment at low and high
temperatures.

We show that the magnon band gap of Cu-EA can be quantitatively extracted from
the modulated magnon-photon interaction. When the resonator mode is inside the
magnon band gap in regime (ii), the interaction between the magnon and photon
modes leads to a linewidth broadening of the resonator photon mode. Such an
effect has been previously observed in magnon-magnon coupled bilayers in the
Purcell regime Xiong  _et al._ (2020, 2022); Inman  _et al._ (2022); Khan  _et
al._ (2021). We develop an analytical model for quantifying the change of
photon linewidth by considering two detuned magnon mode coupled to the photon
mode. The photon damping rate $\kappa_{c}$ can be expressed as:

$\kappa_{c}=\kappa_{c0}+(g_{\text{eff}})^{2}{\kappa_{m}\over\kappa_{m}^{2}+\delta^{2}},$
(4)

where $\kappa_{c0}$ is the intrinsic photon damping rate, $g_{\text{eff}}$ is
the effective magnon-photon coupling strength as plotted in Fig. 3,
$\kappa_{m}$ is the magnon damping rate, and $2\delta$ is the magnon-magnon
band gap. The detailed derivation of the model is included in the Supplemental
Materials sm . Note that the information of $g_{\text{eff}}$ needs to be
obtained from regime (iii) where mode anticrossing between the magnon and
photon modes are resumed. Eq. (4) shows that the change of linewidth
$\Delta\kappa_{c}=\kappa_{c}-\kappa_{c0}$ is proportional to
$(g_{\text{eff}})^{2}$, with the slope determined by two intrinsic magnon
characteristics of the Cu-EA: $\kappa_{m}$ and $2\delta$. With two completely
different superconducting resonator designs, i.e., the CPW resonator and LER,
we find that the extracted $\Delta\kappa_{c}$ nicely follows the linear
dependence of $(g_{\text{eff}})^{2}$, with a slope of (210 MHz)-1. For
$\kappa$, we take the average of the acoustic and optical modes, as
$\kappa_{m}/2\pi=65$ MHz. The magnon band gap is calculated to be
$\delta/2\pi=152$ MHz, which is close to the value in Fig. 1(e) as 140 MHz
around 3.5 K. Thus, we confirm the validity of this new technique for
quantifying the magnon band gap $\delta$ of a small magnetic flake with a
highly sensitive superconducting microwave resonator, where the linewidth
change of the resonator mode acts as a probe to interact with the acoustic-
optical hybrid magnon modes.

In summary, we demonstrate tunable magnon-photon coupling by adjusting the
intrinsic magnon band gap in a layered perovskite antiferromagnet in coupling
with a superconducting resonator. The use of high-quality-factor
superconducting resonator allows for coherent interaction with the magnon
excitations and the study of unique magnon band gap in a magnetic material
with narrow-band microwave measurements. The magnon-photon coupling strength
can be tuned from a few tens of megahertz to zero by modifying the magnon band
gap location with temperature. At the zero coupling strength state where the
resonator mode falls into the magnon band gap, probing the change of photon
mode linewidth also allows one to extract the value of magnon band gap using
an analytical model. Our results provide a new idea to modify magnon-photon
interaction as well as a new approach to study the quantum properties of novel
layered magnetic materials from cavity magnonics. To improve the slow
temperature tunability of magnon band gap, we anticipate other approaches such
as strain or electric field Wang  _et al._ (2018); Yan  _et al._ (2020); Liu
_et al._ (2021) for controlling the magnetic properties with high speed and
extending the application in coherent information processing.

Acknowledgement. D.S. and M.B. acknowledge the primary financial support
through the Center for Hybrid Organic Inorganic Semiconductors for Energy
(CHOISE), an Energy Frontier Research Center funded by the Office of Basic
Energy Sciences, Office of Science within the U.S. Department of Energy
(Hybrid perovskite synthesis, crystal preparation, structural
characterization, and motivation of this work). This work was authored in part
by the National Renewable Energy Laboratory (NREL), operated by Alliance for
Sustainable Energy LLC, for the U.S. Department of Energy (DOE) under contract
no. DE-AC36-08GO28308. The views expressed in this article do not necessarily
represent the views of the DOE or the U.S. Government. Works at Argonne
National Laboratory and University of Illinois and Urbana-Champaign, including
the superconducting resonator fabrication, design, ICP chemical analysis, and
hybrid magnonics characterization, were supported by the U.S. DOE, Office of
Science, Basic Energy Sciences, Materials Sciences and Engineering Division
under contract No. DE-SC0022060. Work at UNC-CH were supported by NSF-ECCS
2246254 for the experimental design, data analysis, theoretical analysis, and
manuscript preparation. D.S. acknowledges the partial financial support from
the Department of Energy grant DE-SC0020992 and the National Science
Foundation grant DMR-2143642 for magnetic properties characterization. X.Z.
acknowledges support by the National Science Foundation CAREER Award (Grant
CBET-2145417) and LEAPS Award (Grant DMR-2137883) for the Raman
characterization. Use of the Center for Nanoscale Materials (CNM), an Office
of Science user facility, was supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, under Contract No. DE-
AC02-06CH11357.

## References

  * Kurizki  _et al._ (2015) Gershon Kurizki, Patrice Bertet, Yuimaru Kubo, Klaus Mølmer, David Petrosyan, Peter Rabl, and Jörg Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. 112, 3866–3873 (2015).
  * Clerk  _et al._ (2020) A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, “Hybrid quantum systems with circuit quantum electrodynamics,” Nature Phys. 16, 257 (2020).
  * Lachance-Quirion  _et al._ (2019) Dany Lachance-Quirion, Yutaka Tabuchi, Arnaud Gloppe, Koji Usami, and Yasunobu Nakamura, “Hybrid quantum systems based on magnonics,” Appl. Phys. Express 12, 070101 (2019).
  * Li  _et al._ (2020a) Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Hoffmann, and V. Novosad, “Hybrid magnonics: physics, circuits and applications for coherent information processing,” J. Appl. Phys. 128, 130902 (2020a).
  * Li  _et al._ (2021a) Yi Li, Chenbo Zhao, Wei Zhang, Axel Hoffmann, and Valentyn Novosad, “Advances in coherent coupling between magnons and acoustic phonons,” APL Mater. 9, 060902 (2021a).
  * Huebl  _et al._ (2013) Hans Huebl, Christoph W. Zollitsch, Johannes Lotze, Fredrik Hocke, Moritz Greifenstein, Achim Marx, Rudolf Gross, and Sebastian T. B. Goennenwein, “High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids,” Phys. Rev. Lett. 111, 127003 (2013).
  * Goryachev  _et al._ (2014) Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon, Yaohui Fan, Mikhail Kostylev, and Michael E. Tobar, “High-cooperativity cavity qed with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
  * Tabuchi  _et al._ (2014) Yutaka Tabuchi, Seiichiro Ishino, Toyofumi Ishikawa, Rekishu Yamazaki, Koji Usami, and Yasunobu Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
  * Zhang  _et al._ (2014) Xufeng Zhang, Chang-Ling Zou, Liang Jiang, and Hong X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
  * Bai  _et al._ (2015) Lihui Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, “Spin pumping in electrodynamically coupled magnon-photon systems,” Phys. Rev. Lett. 114, 227201 (2015).
  * Li  _et al._ (2019) Yi Li, Tomas Polakovic, Yong-Lei Wang, Jing Xu, Sergi Lendinez, Zhizhi Zhang, Junjia Ding, Trupti Khaire, Hilal Saglam, Ralu Divan, John Pearson, Wai-Kwong Kwok, Zhili Xiao, Valentine Novosad, Axel Hoffmann, and Wei Zhang, “Strong coupling between magnons and microwave photons in on-chip ferromagnet-superconductor thin-film devices,” Phys. Rev. Lett. 123, 107701 (2019).
  * Hou and Liu (2019) Justin T. Hou and Luqiao Liu, “Strong coupling between microwave photons and nanomagnet magnons,” Phys. Rev. Lett. 123, 107702 (2019).
  * Kikkawa  _et al._ (2016) Takashi Kikkawa, Ka Shen, Benedetta Flebus, Rembert A. Duine, Ken-ichi Uchida, Zhiyong Qiu, Gerrit E. W. Bauer, and Eiji Saitoh, “Magnon polarons in the spin seebeck effect,” Phys. Rev. Lett. 117, 207203 (2016).
  * Berk  _et al._ (2019) Cassidy Berk, Mike Jaris, Weigang Yang, Scott Dhuey, Stefano Cabrini, and Holger Schmidt, “Strongly coupled magnon–phonon dynamics in a single nanomagnet,” Nature Commun. 10, 2652 (2019).
  * An  _et al._ (2020) K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurdequint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, “Coherent long-range transfer of angular momentum between magnon kittel modes by phonons,” Phys. Rev. B 101, 060407 (2020).
  * Xu  _et al._ (2021a) Jing Xu, Changchun Zhong, Xianjing Zhou, Xu Han, Dafei Jin, Stephen K. Gray, Liang Jiang, and Xufeng Zhang, “Coherent pulse echo in hybrid magnonics with multimode phonons,” Phys. Rev. Appl. 16, 024009 (2021a).
  * Klingler  _et al._ (2018) Stefan Klingler, Vivek Amin, Stephan Geprägs, Kathrin Ganzhorn, Hannes Maier-Flaig, Matthias Althammer, Hans Huebl, Rudolf Gross, Robert D. McMichael, Mark D. Stiles, Sebastian T. B. Goennenwein, and Mathias Weiler, “Spin-torque excitation of perpendicular standing spin waves in coupled $\mathrm{YIG}/\mathrm{Co}$ heterostructures,” Phys. Rev. Lett. 120, 127201 (2018).
  * Chen  _et al._ (2018) Jilei Chen, Chuanpu Liu, Tao Liu, Yang Xiao, Ke Xia, Gerrit E. W. Bauer, Mingzhong Wu, and Haiming Yu, “Strong interlayer magnon-magnon coupling in magnetic metal-insulator hybrid nanostructures,” Phys. Rev. Lett. 120, 217202 (2018).
  * MacNeill  _et al._ (2019) David MacNeill, Justin T. Hou, Dahlia R. Klein, Pengxiang Zhang, Pablo Jarillo-Herrero, and Luqiao Liu, “Gigahertz frequency antiferromagnetic resonance and strong magnon-magnon coupling in the layered crystal ${\mathrm{crcl}}_{3}$,” Phys. Rev. Lett. 123, 047204 (2019).
  * Li  _et al._ (2020b) Yi Li, Wei Cao, Vivek P. Amin, Zhizhi Zhang, Jonathan Gibbons, Joseph Sklenar, John Pearson, Paul M. Haney, Mark D. Stiles, William E. Bailey, Valentine Novosad, Axel Hoffmann, and Wei Zhang, “Coherent spin pumping in a strongly coupled magnon-magnon hybrid system,” Phys. Rev. Lett. 124, 117202 (2020b).
  * Shiota  _et al._ (2020) Yoichi Shiota, Tomohiro Taniguchi, Mio Ishibashi, Takahiro Moriyama, and Teruo Ono, “Tunable magnon-magnon coupling mediated by dynamic dipolar interaction in synthetic antiferromagnets,” Phys. Rev. Lett. 125, 017203 (2020).
  * Sud  _et al._ (2020) A. Sud, C. W. Zollitsch, A. Kamimaki, T. Dion, S. Khan, S. Iihama, S. Mizukami, and H. Kurebayashi, “Tunable magnon-magnon coupling in synthetic antiferromagnets,” Phys. Rev. B 102, 100403 (2020).
  * Sklenar and Zhang (2021) Joseph Sklenar and Wei Zhang, “Self-hybridization and tunable magnon-magnon coupling in van der waals synthetic magnets,” Phys. Rev. Appl. 15, 044008 (2021).
  * Li  _et al._ (2021b) Mei Li, Jie Lu, and Wei He, “Symmetry breaking induced magnon-magnon coupling in synthetic antiferromagnets,” Phys. Rev. B 103, 064429 (2021b).
  * Harder and Hu (2018) M. Harder and C.-M. Hu, “Cavity spintronics: An early review of recent progress in the study of magnon-photon level repulsion,” Solid State Phys. 70, 47 (2018).
  * Awschalom  _et al._ (2021) D. D. Awschalom, C.H.R. Du, R. He, J. Heremans, A. Hoffmann, J. Hou, H. Kurebayashi, Y. Li, L. Liu, V. Novosad, J. Sklenar, S. Sullivan, D. Sun, H. Tang, V. Tyberkevych, C. Trevillian, A. W. Tsen, L. Weiss, W. Zhang, X. Zhang, L. Zhao, and Ch. W. Zollitsch, “Quantum engineering with hybrid magnonics systems and materials,” IEEE Trans. Quantum Eng. 2, 5500836 (2021).
  * Fukami  _et al._ (2021) Masaya Fukami, Denis R. Candido, David D. Awschalom, and Michael E. Flatté, “Opportunities for long-range magnon-mediated entanglement of spin qubits via on- and off-resonant coupling,” PRX Quantum 2, 040314 (2021).
  * Yuan  _et al._ (2022) H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, “Quantum magnonics: When magnon spintronics meets quantum information science,” Phys. Rep. 965, 1–74 (2022).
  * Harder  _et al._ (2018) M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, “Level attraction due to dissipative magnon-photon coupling,” Phys. Rev. Lett. 121, 137203 (2018).
  * Bhoi  _et al._ (2019) Biswanath Bhoi, Bosung Kim, Seung-Hun Jang, Junhoe Kim, Jaehak Yang, Young-Jun Cho, and Sang-Koog Kim, “Abnormal anticrossing effect in photon-magnon coupling,” Phys. Rev. B 99, 134426 (2019).
  * Wang  _et al._ (2019) Yi-Pu Wang, J. W. Rao, Y. Yang, Peng-Chao Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C.-M. Hu, “Nonreciprocity and unidirectional invisibility in cavity magnonics,” Phys. Rev. Lett. 123, 127202 (2019).
  * Boventer  _et al._ (2020) Isabella Boventer, Christine Dörflinger, Tim Wolz, Rair Macêdo, Romain Lebrun, Mathias Kläui, and Martin Weides, “Control of the coupling strength and linewidth of a cavity magnon-polariton,” Phys. Rev. Research 2, 013154 (2020).
  * Xu  _et al._ (2021b) Jing Xu, Changchun Zhong, Xu Han, Dafei Jin, Liang Jiang, and Xufeng Zhang, “Coherent gate operations in hybrid magnonics,” Phys. Rev. Lett. 126, 207202 (2021b).
  * Lambert  _et al._ (2016) N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty, and A. J. Ferguson, “Cavity-mediated coherent coupling of magnetic moments,” Phys. Rev. A 93, 021803 (2016).
  * Li  _et al._ (2022) Yi Li, Volodymyr G. Yefremenko, Marharyta Lisovenko, Cody Trevillian, Tomas Polakovic, Thomas W. Cecil, Peter S. Barry, John Pearson, Ralu Divan, Vasyl Tyberkevych, Clarence L. Chang, Ulrich Welp, Wai-Kwong Kwok, and Valentine Novosad, “Coherent coupling of two remote magnonic resonators mediated by superconducting circuits,” Phys. Rev. Lett. 128, 047701 (2022).
  * Tabuchi  _et al._ (2015) Yutaka Tabuchi, Seiichiro Ishino, Atsushi Noguchi, Toyofumi Ishikawa, Rekishu Yamazaki, Koji Usami, and Yasunobu Nakamura, “Coherent coupling between a ferromagnetic magnon and a superconducting qubit,” Science 349, 405–408 (2015).
  * Wang and et al. (2022) Q. H. Wang and et al., “The magnetic genome of two-dimensional van der waals materials,” ACS Nano 16, 6960 (2022).
  * (38) C. Tang, L. Alahmed, M. Mahdi, M. Xiong, J. Inman, N. J. McLaughlin, Ch. Zollitsch, T. H. Kim, C. R. Du, H. Kurebayashi, E. J. G. Santos, W. Zhang, P. Li, and W. Jin, “Ferromagnetic resonance in two-dimensional van der waals magnets: A probe for spin dynamics,” arXiv 2301.09822.
  * Saparov and Mitzi (2016) B. Saparov and D. B. Mitzi, “Organic–inorganic perovskites: Structural versatility for functional materials design,” Chem. Rev. 116, 4558 (2016).
  * Nugroho  _et al._ (2016) A. A. Nugroho, Z. Hu, C. Y. Kuo, M. W. Haverkort, T. W. Pi, D. Onggo, M. Valldor, and L. H. Tjeng, “Cross-type orbital ordering in the layered hybrid organic-inorganic compound $({\mathrm{c}}_{6}{\mathrm{h}}_{5}{\mathrm{ch}}_{2}{\mathrm{ch}}_{2}{\mathrm{nh}}_{3}){}_{2}{\mathrm{cucl}}_{4}$,” Phys. Rev. B 94, 184404 (2016).
  * Kim  _et al._ (2020) Ki-Yeon Kim, Garam Park, Jaehun Cho, Joonwoo Kim, June-Seo Kim, Jinyong Jung, Kwonjin Park, Chun-Yeol You, and In-Hwan Oh, “Intrinsic magnetic order of chemically exfoliated 2d ruddlesden–popper organic–inorganic halide perovskite ultrathin films,” Small 16, 2005445 (2020).
  * Willett  _et al._ (2012) R. D. Willett, C. J. Gómez-García, and B. Twamley, “Long-range order in layered perovskite salts – structure and magnetic properties of [(ch3)2chch2nh3]2cux4 (x = cl, br),” Eur. J. Inorg. Chem. 2012, 3342 (2012).
  * De Jongh  _et al._ (1972) L. J. De Jongh, W. D. Van Amstel, and A. R. Miedema, “Magnetic measurements on (c2h5nh3)2 cucl4 - ferromagnetic layers coupled by a very weak antiferromagnetic interaction,” Physica 58, 277 (1972).
  * Keffer and Kittel (1952) F. Keffer and C. Kittel, “Theory of antiferromagnetic resonance,” Phys. Rev. 85, 329–337 (1952).
  * Moriya (1960) Tôru Moriya, “Anisotropic superexchange interaction and weak ferromagnetism,” Phys. Rev. 120, 91–98 (1960).
  * Bloembergen  _et al._ (1973) P. Bloembergen, P. J. Berkhout, and J. J. M. Franse, “Static magnetic torque measurements on a system of ferromagnetic layers, coupled by feeble antiferromagnetic interactions; weak ferromagnetic behaviour,” AIP Conf. Proc. 10, 1598 (1973).
  * Bogdanov  _et al._ (2007) A. N. Bogdanov, A. V. Zhuravlev, and U. K. Rößler, “Spin-flop transition in uniaxial antiferromagnets: Magnetic phases, reorientation effects, and multidomain states,” Phys. Rev. B 75, 094425 (2007).
  * Comstock  _et al._ (2023) Andrew H. Comstock, Chung-Tao Chou, Zhiyu Wang, Tonghui Wang, Ruyi Song, Joseph Sklenar, Aram Amassian, Wei Zhang, Haipeng Lu, Luqiao Liu, Matthew C. Beard, and Dali Sun, “Hybrid magnonics in hybrid perovskite antiferromagnets,” Nature Commun. 14, 1834 (2023).
  * Diederich  _et al._ (2023) Geoffrey M. Diederich, John Cenker, Yafei Ren, Jordan Fonseca, Daniel G. Chica, Youn Jue Bae, Xiaoyang Zhu, Xavier Roy, Ting Cao, Di Xiao, and Xiaodong Xu, “Tunable interaction between excitons and hybridized magnons in a layered semiconductor,” Nature Nano. 18, 23 (2023).
  * Makihara  _et al._ (2021) Takuma Makihara, Kenji Hayashida, G. Timothy Noe II, Xinwei Li, Nicolas Marquez Peraca, Xiaoxuan Ma, Zuanming Jin, Wei Ren, Guohong Ma, Ikufumi Katayama, Jun Takeda, Hiroyuki Nojiri, Dmitry Turchinovich, Shixun Cao, Motoaki Bamba, and Junichiro Kono, “Ultrastrong magnon–magnon coupling dominated by antiresonant interactions,” Nature Commun. 12, 3115 (2021).
  * Zollitsch  _et al._ (2022) C. W. Zollitsch, S. Khan, V. T. T. Nam, I. A. Verzhbitskiy, D. Sagkovits, J. O’Sullivan, O. W. Kennedy, M. Strungaru, E. J. G. Santos, J. J. L. Morton, G. Eda, and H. Kurebayashi, “Probing spin dynamics of ultra-thin van der waals magnets via photon-magnon coupling,” arXiv (2022), 2206.02460.
  * Easy  _et al._ (2021) Elham Easy, Yuan Gao, Yingtao Wang, Dingkai Yan, Seyed M. Goushehgir, Eui-Hyeok Yang, Baoxing Xu, and Xian Zhang, “Experimental and computational investigation of layer-dependent thermal conductivities and interfacial thermal conductance of one- to three-layer wse2,” ACS Appl. Mater. Interfaces 13, 13063 (2021).
  * Caretta  _et al._ (2014) Antonio Caretta, Rany Miranti, Remco W. A. Havenith, Elia Rampi, Michiel C. Donker, Graeme R. Blake, Matteo Montagnese, Alexey O. Polyakov, Ria Broer, Thomas T. M. Palstra, and Paul H. M. van Loosdrecht, “Low-frequency raman study of the ferroelectric phase transition in a layered ${\mathbf{cucl}}_{\mathbf{4}}$-based organic-inorganic hybrid,” Phys. Rev. B 89, 024301 (2014).
  * (54) See the Supplemental Information for details.
  * McKenzie-Sell  _et al._ (2019) L. McKenzie-Sell, J. Xie, C.-M. Lee, J. W. A. Robinson, C. Ciccarelli, and J. A. Haigh, “Low-impedance superconducting microwave resonators for strong coupling to small magnetic mode volumes,” Phys. Rev. B 99, 140414 (2019).
  * Xiong  _et al._ (2020) Y. Xiong, Y. Li, M. Hammami, R. Bidthanapally, J. Sklenar, X. Zhang, H. Qu, G. Srinivasan, J. Pearson, A. Hoffmann, V. Novosad, and W. Zhang, “Probing magnon–magnon coupling in exchange coupled y3fe5o12/permalloy bilayers with magneto-optical effects,” Sci. Rep. 10, 12548 (2020).
  * Xiong  _et al._ (2022) Yuzan Xiong, Jerad Inman, Zhengyi Li, Kaile Xie, Rao Bidthanapally, Joseph Sklenar, Peng Li, Steven Louis, Vasyl Tyberkevych, Hongwei Qu, Zhili Xiao, Wai K. Kwok, Valentine Novosad, Yi Li, Fusheng Ma, and Wei Zhang, “Tunable magnetically induced transparency spectra in magnon-magnon coupled ${\mathrm{y}}_{3}{\mathrm{fe}}_{5}{\mathrm{o}}_{12}/$permalloy bilayers,” Phys. Rev. Appl. 17, 044010 (2022).
  * Inman  _et al._ (2022) Jerad Inman, Yuzan Xiong, Rao Bidthanapally, Steven Louis, Vasyl Tyberkevych, Hongwei Qu, Joseph Sklenar, Valentine Novosad, Yi Li, Xufeng Zhang, and Wei Zhang, “Hybrid magnonics for short-wavelength spin waves facilitated by a magnetic heterostructure,” Phys. Rev. Appl. 17, 044034 (2022).
  * Khan  _et al._ (2021) S. Khan, O. Lee, T. Dion, C. W. Zollitsch, S. Seki, Y. Tokura, J. D. Breeze, and H. Kurebayashi, “Coupling microwave photons to topological spin textures in ${\mathrm{cu}}_{2}{\mathrm{oseo}}_{3}$,” Phys. Rev. B 104, L100402 (2021).
  * Wang  _et al._ (2018) X. Wang, Q. Yang, L. Wang, Z. Zhou, T. Min, M. Liu, and N. X. Sun, “E‐field control of the rkky interaction in fecob/ru/fecob/pmn‐pt (011) multiferroic heterostructures,” Adv. Mater. 30, 1803612 (2018).
  * Yan  _et al._ (2020) H. Yan, Z. Feng, P. Qin, X. Zhou, H. Guo, X. Wang, H. Chen, X. Zhang, H. Wu, C. Jiang, and Z. Liu, “Electric-field-controlled antiferromagnetic spintronic devices,” Adv. Mater. 32, 1905603 (2020).
  * Liu  _et al._ (2021) C. Liu, Y. Luo, D. Hong, S. S.-L. Zhang, H. Saglam, Lin Y. Li, Y. and, B. Fisher, J. E. Pearson, J. S. Jiang, H. Zhou, J. Wen, A. Hoffmann, and A. Bhattacharya, “Electric field control of magnon spin currents in an antiferromagnetic insulator,” Sci. Adv. 7, eabg1669 (2021).