#  Supplemental Material
Strongly Coupled Spin Waves and Surface Acoustic Waves at Room Temperature

Yunyoung Hwang  Institute for Solid State Physics, University of Tokyo,
Kashiwa 277-8581, Japan  CEMS, RIKEN, 2-1, Hirosawa, Wako 351-0198, Japan
Jorge Puebla<EMAIL_ADDRESS>CEMS, RIKEN, 2-1, Hirosawa, Wako
351-0198, Japan  Kouta Kondou  CEMS, RIKEN, 2-1, Hirosawa, Wako 351-0198,
Japan  Carlos Gonzalez-Ballestero  Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria  Hironari Isshiki  Institute for Solid State Physics, University of
Tokyo, Kashiwa 277-8581, Japan  Carlos Sánchez Muñoz  Departamento de Física
Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),
Universidad Autónoma de Madrid, 28049 Madrid, Spain  Liyang Liao  Institute
for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan  Fa Chen
School of Integrated Circuits, Wuhan National Laboratory for Optoelectronics,
Huazhong University of Science and Technology, Wuhan 430074, People’s Republic
of China.  Wei Luo  School of Integrated Circuits, Wuhan National Laboratory
for Optoelectronics, Huazhong University of Science and Technology, Wuhan
430074, People’s Republic of China.  Sadamichi Maekawa  CEMS, RIKEN, 2-1,
Hirosawa, Wako 351-0198, Japan  Advanced Science Research Center, Japan Atomic
Energy Agency, Tokai 319-1195, Japan  Kavli Institute for Theoretical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China  Yoshichika Otani<EMAIL_ADDRESS>Institute for
Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan  CEMS,
RIKEN, 2-1, Hirosawa, Wako 351-0198, Japan

††preprint: Supplemental Material

###### Contents

  1. S1 SAW modes in our acoustic cavity devices
  2. S2 SAW strain simulations
  3. S3 Additional measurements to confirm contributions of the strain components
    1. S3.1 Different propagation direction of SAWs
    2. S3.2 Magnon–phonon coupling of devices with thin CFB
  4. S4 Magnon–phonon coupling model
  5. S5 SAW transmission model under magnon–phonon coupling
    1. S5.1 In-plane magnetic field angular dependence of SAW-driven SWR
    2. S5.2 Reproduced anticrossings by the SAW transmission model
  6. S6 Microwave ferromagnetic resonance measurements

##  S1 SAW modes in our acoustic cavity devices

A distinctive feature of an optical or acoustic cavity is the emergence of a
set of peaks separated by the same frequency spacing representing the set of
standing wave modes inside the cavity. This particular frequency spacing is
known as free spectral range (FSR) and can be estimated by $v/(2L_{c})$, where
$v$ is the wave velocity and $L_{c}$ is the cavity length. And, it has been
observed in two-port SAW resonator devices [1, 2]. For the case of the
acoustic cavity devices of the present study, if we assume the cavity length
as $L_{c}\geq L$, where $L$ is the length of propagation, we estimate a
separation by the FSR of $\leq 10$ MHz. However, in our
transmission/absorption measurements we hardly observed this set of peaks with
a separation of $\leq 10$ MHz. We understand it as consequence of the
linewidth broadening due to the mass loading of Ti/CFB/Ti layers.

Instead, in our transmission/absorption measurements we clearly observe two
modes within our cavity [see Fig. 2(a) in the main text]. We understand it as
follows: i) first, needless to say in an ideal device we should expect a
single well-defined peak representing a single mode, however, the fabrication
resolution sets a limit, in our case, given by the resolution of the electron
beam lithography system used to fabricate our acoustic wave device. This limit
of fabrication resolution is shown as the linewidth broadening of our
transmission/absorption measurements, which is approximately equivalent to
fluctuations of IDTs dimensions of $\sim$ 8 nm. ii) Second, as the nominal
distance between our IDT sets or propagation length is a multiple of
$\lambda_{r}$, the existence of additional modes requires to fulfill the phase
shift condition given by $e^{ikL}=e^{i2\pi L/\lambda_{r}}=1$, defining the
arrival frequency of an additional mode to the detecting IDT set. The
frequency intervals can be simply related to a period of $v/L$, as shown in
Fig. S1, where $v$ is the velocity of the SAW, and $L=$ 210 $\mu$m is not the
cavity length but the propagation length from the generating IDT set to the
detecting IDT set, resulting in approximately $\sim$ 19 MHz, in proximity to
the frequency separation we observe in our transmission/absorption
measurements.

Figure S1:  SAW phase shift due to the propagation. (a) Schematic illustration
of SAWs with two different wavelengths in the same propagation length. (b)
Schematic illustration of separated SAW peaks due to the phase shift in a
frequency domain.

##  S2 SAW strain simulations

We performed simulations to confirm the existence of SAW strain components
$\varepsilon_{xx}$ and $\varepsilon_{xy}$ using COMSOL Multiphysics®. For
simplicity, 4 pairs of signal and ground lines of an IDT set are simulated. We
have set two geometries: one with two sets of 20 reflectors [Fig. S2(a)]; and
the other without reflectors [Fig. S2(b)].

Figure S2:  Simulation of the SAW strain components. (a),(b) Schematic
description of the geometry of the simulation (a) with and (b) without
reflectors. (c),(d) Simulated strain components as a function of time (a) with
and (b) without reflectors with SAWs propagating along the X-axis. (e),(f)
Same as (c) and (d), but with SAWs propagating along the Y’-axis.

The widths of one stripe of IDTs and reflectors, and the gap between them are
150 nm. In accordance with the IDT design, we set the SAW wavelength
$\lambda=600$ nm. We simulate SAWs propagating on the X- and Y’-axes of the
128∘ Y-cut LiNbO3. We set the SAW velocity $v=4,000$ m/s for the SAW
propagating along the X-axis, and $v=3,665$ m/s for the SAW propagating along
the Y’-axis, which have been calculated independently by the eigenfrequency
analyses. The distance between the two IDT sets is $10\lambda$ and the
thickness of the substrate is $3\lambda$. We set a $\lambda$-thick perfectly
matching layer at the bottom of the substrate. The mesh size for the
simulation is $\sim\lambda/8$. The input frequency for the IDT is $v/\lambda$.

Figures S2(c) and S2(d) show the strain components with and without
reflectors, respectively, with the SAW propagating along the X-axis. Figures
S2(e) and S2(f) show the same simulation as Figs. S2(c) and S2(d) but with the
SAW propagating along the Y’-axis. It is important to note that, we defined
the $x$-axis as always parallel to the SAW propagation direction, thus
$\varepsilon_{xx}$ is always the longitudinal strain of SAWs in Figs.
S2(c)–S2(f). From the simulation results, one can confirm that employing an
acoustic cavity structure significantly enhances $\varepsilon_{xy}$ when the
SAW propagates parallel to the X-axis.

To better account for the strain components, we calculate the root mean square
(RMS) of each strain component as $\varepsilon_{xx}^{\mathrm{RMS}}$ and
$\varepsilon_{xy}^{\mathrm{RMS}}$ in a time range 25–30 ns and calculate the
ratio $\eta=\varepsilon_{xy}^{\mathrm{RMS}}/\varepsilon_{xx}^{\mathrm{RMS}}$.
As a result, we obtain $\eta=0.95$ for the simulation with 20 reflectors [Fig.
S2(c)] and $\eta=0.15$ for the simulation without reflectors [Fig. S2(d)]. We
subsequently employ $\eta=0.5$ as a compromise since our simulations have been
done using a simplified geometry in comparison to our actual device design.

##  S3 Additional measurements to confirm contributions of the strain
components

###  S3.1 Different propagation direction of SAWs

In the above, in Figs. S2(c)–S2(f), we confirmed that the dominant SAW strain
is the longitudinal strain $\varepsilon_{xx}$, however, $\varepsilon_{xy}$ is
significant for SAWs propagating along the crystal X-axis of a 128∘ Y-cut
LiNbO3 substrate, whereas $\varepsilon_{xy}$ is negligible compare to
$\varepsilon_{xx}$ when SAWs propagate along the Y’-axis, which is in-plane
perpendicular to the X-axis.

To confirm the magnetoelastic coupling by $\varepsilon_{xy}$ occurred in our
device, we fabricated the same-structured devices as the one used for the
results in the main text with the different propagation direction; the
Y’-axis. The SAW transmissions at the resonant frequency at various in-plane
angles to $\mathbf{k}$ of the external magnetic field are shown in Fig. S3.
Note that since the difference in the SAW velocity for different propagation
directions, the resonant frequencies are 6.58 GHz and 6.06 GHz for the devices
of which the SAW propagates along the X- and Y’-axis, respectively. On the one
hand, when SAWs propagate along the X-axis [Fig. S3(a)], as the result shown
in the main text, one can find significant SAW absorptions at $\phi_{H}\sim
0$. On the other hand, when SAWs propagate along the Y’-axis [Fig. S3(b)], SAW
absorptions are not observed at $\phi_{H}\sim 0$, which implies the lack of
$\varepsilon_{xy}$ results in the lack of the magnetoelastic coupling at
$\phi_{H}\sim 0$. Moreover, from the device of which the SAW propagates along
the Y’-axis [Fig. S3(b)], we did not observe anticrossing at any $\phi_{H}$.

Figure S3:  In-plane magnetic field angle dependence of SAW transmissions
propagating on the X- and Y’-axes of the samples with $t_{\mathrm{CFB}}=20$
nm. (a),(b) SAW transmission spectra at the frequency of 6.58 GHz as a
function of the amplitude and in-plane angle $\phi_{H}$ of the applied
magnetic field $\mu_{0}H$ when $\mathbf{k}$ propagates along (a) the X-axis
and (b) Y’-axis of the LiNbO3 substrate.

###  S3.2 Magnon–phonon coupling of devices with thin CFB

As shown in Ref. [3], the magnetoelastic coupling by $\varepsilon_{xx}$ is
substantially suppressed even though the local value of $\varepsilon_{xx}$ is
significant due to its rapid change in a thickness direction of a
ferromagnetic film. In contrast, $\varepsilon_{xy}$ does not decrease rapidly
in the thickness direction, thus the thicker ferromagnet layer allows for more
magnon–phonon interactions by $\varepsilon_{xy}$, leading to a higher
effective coupling strength. To confirm this, we fabricated the same-
structured device as the main text but with a thinner CFB;
$t_{\mathrm{CFB}}=3$ nm.

The SAW absorptions at the frequency of 6.58 GHz at various $\phi_{H}$ of a
device including reflectors and a device in the absence of reflectors are
shown in Fig. S4(a) and S4b, respectively. As shown in Sec. S2,
$\varepsilon_{xy}$ is enhanced by reflectors, thus only the device with
reflectors shows significant SAW absorptions at $\phi_{H}\sim 0$. We note that
the result in Fig. S4(b) shows a similar behavior as our previous result of
1.6-nm-thick CFB [4].

In addition, due to the lower saturation magnetization of the thin CFB than
the devices with $t_{\mathrm{CFB}}\geq 10$ nm [5], magnon–phonon coupling can
occur at $\phi_{H}=90^{\circ}$ due to the lower dipolar field [see Eqs.
(S2–S3), Sec. S5.1, and Fig. S7], thus one observes SAW absorption. Therefore,
we confirm the typical in-plane angular dependence of the magnetoelastic
coupling by $\varepsilon_{xy}$, which is proportional to $\cos{2\phi}$ (see
the next section).

Figure S4:  SAW transmission measurements of the sample with
$t_{\mathrm{CFB}}=3$ nm. (a),(b) SAW transmission spectrum at the frequency of
6.58 GHz as a function of the amplitude and in-plane angle $\phi_{H}$ of the
applied magnetic field $\mu_{0}H$ of the samples (a) with reflectors and (b)
without reflectors. (c),(d) SAW transmission signal of the sample with
reflectors under (c) various amplitudes of the magnetic field and (d) 100 mT
of the magnetic field. The SAWs in (a) and (b) propagate along the X-axis.

##  S4 Magnon–phonon coupling model

Here, we consider a CFB layer with the thickness of $t_{\mathrm{CFB}}$
deposited on top of a LiNbO3 substrate with the thickness of $L_{z}$, as shown
in Fig. S5(a). Note that, we omitted the Ti layers for simplicity.

Consider an in-plane magnetic field $H$ is applied externally with the angle
$\phi_{H}$ to the SAW wavevector $\mathbf{k}$, as shown in Fig. S5(b). The
linearized equations of motions of the magnetization in units of magnetic
field in a cylindrical coordinate are [6]

$\begin{pmatrix}-\mu_{0}H_{z}+i\alpha\omega/|\gamma|&i\omega/\gamma&\mu_{0}h_{\mathrm{eff}}^{z}\\\
-i\omega/\gamma&-\mu_{0}H_{\phi}+i\alpha\omega/|\gamma|&\mu_{0}h_{\mathrm{eff}}^{\phi}\\\
\mu_{0}\overline{h_{\mathrm{eff}}^{z}}t_{\mathrm{CFB}}/L&\mu_{0}\overline{h_{\mathrm{eff}}^{\phi}}t_{\mathrm{CFB}}/L&\frac{\rho}{M_{s}}\left(\frac{\omega^{2}}{k^{2}}-\frac{\omega_{p}^{2}}{k^{2}}+i\mu_{v}\omega\right)\end{pmatrix}\begin{pmatrix}n_{z}\\\
n_{\phi}\\\
\varepsilon_{\mathbf{k}}\end{pmatrix}=\frac{\sigma}{M_{s}}\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix},$ (S1)

where $\sigma$ is the external stress, $\mu_{v}$ is the coefficient of
viscosity, $n_{z}$ and $n_{\phi}$ are components of the normalized spin
vector, $\omega_{p}$ the phonon frequency, and $\varepsilon_{\mathbf{k}}$ is
the amplitude of SAW strain. The overline denotes complex conjugation.
Considering that the generated spin wave by the SAW has the same wavevector
$\mathbf{k}$, the static magnetic field components are given by

$H_{z}=H\cos{(\phi-\phi_{H})}+M_{s}\frac{1-e^{-kt_{\mathrm{CFB}}}}{kt_{\mathrm{CFB}}}+H_{K}\cos^{2}{(\phi-\phi_{K})}+\frac{A_{\mathrm{ex}}^{2}}{\mu_{0}M_{s}},$
(S2)
$H_{\phi}=H\cos{(\phi-\phi_{H})}+M_{s}\left(1-\frac{1-e^{-kt_{\mathrm{CFB}}}}{kt_{\mathrm{CFB}}}\right)\sin^{2}{\phi}+H_{K}\cos{2(\phi-\phi_{K})}+\frac{A_{\mathrm{ex}}^{2}}{\mu_{0}M_{s}},$
(S3)

where $\phi$ is the in-plane angle of the ground state magnetization $M$,
shown in Fig. S5(b), $M_{s}$ the saturation magnetization, $\phi_{K}$ the in-
plane angle of the easy axis, $H_{K}$ the uniaxial anisotropy field,
$A_{\mathrm{ex}}$ the exchange stiffness, and $k=|\mathbf{k}|$. The terms on
the right-hand side in Eqs. (S2) and (S3) are, in the order from the left, the
Zeeman, dipolar, anisotropy, and exchange fields. The dynamical effective
magnetic field components generated by acoustic waves in the thin film limit
read [4, 6, 7]

$\mu_{0}h^{z}_{\mathrm{eff}}=\frac{2c}{M_{s}}\nu_{R}\sqrt{kL_{z}}\xi_{S}^{2}\sqrt{1-\xi_{P}^{2}}\cos{\phi},$
(S4)
$\mu_{0}h^{\phi}_{\mathrm{eff}}=-i\frac{2}{M_{s}}\nu_{R}\sqrt{kL_{z}}\xi_{S}^{2}\left(1-\xi_{S}^{2}\right)\left(b_{1}\sin{\phi}\cos{\phi}-\eta
b_{2}\cos{2\phi}\right),$ (S5)

where $b_{1}$, $b_{2}$ are the cubic magnetoelastic coupling coefficients, $c$
is the magnetorotation coupling coefficient, normally $c=-\mu_{0}M_{s}^{2}/2$
if it arises solely from the shape anisotropy, and
$\eta=\varepsilon_{xy}/\varepsilon_{xx}$ as described in Sec. S2. We
approximated the SAW in our experiments to the Rayleigh wave, which is
commonly used for SAWs generated on a 128∘ Y-cut LiNbO3 substrate [8, 9]. The
parameters from the SAW normalization [4, 6, 7] are $\nu_{R}\sim 1.6$,
$\xi_{P}^{2}=c_{R}^{2}/c_{P}^{2}$, and $\xi_{S}^{2}=c_{R}^{2}/(2c_{S}^{2})$,
where $c_{P}$, $c_{S}$, and $c_{R}$ are the longitudinal, transverse, and
Rayleigh sound velocities, respectively. The magnitude of these fields is a
measure of the field amplitude when a strain of order unity is excited
$|\varepsilon_{\mathbf{k}}|\sim 1$. However, $|\varepsilon_{\mathbf{k}}|$ is
the average strain across the whole elastic body, which means for surface
waves the strain near the surface would be much larger, which explains the
enhancement factor $\sqrt{kL_{z}}$ [6, 7].

Figure S5:  (a),(b) Setting for the magnon–phonon coupling model of (a) the
substrate and the CFB stack and (b) the in-plane coordinates.

The resonant frequencies of the hybridized modes can be calculated by turning
off the dampings and setting the determinant of the matrix on the left-hand
side of Eq. (S1) zero:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\omega^{2}-\gamma^{2}\mu_{0}^{2}H_{z}H_{\phi}\right)\left(\omega^{2}-\omega_{p}^{2}\right)$
(S6)
$\displaystyle-\frac{\gamma\mu_{0}^{2}M_{s}k^{2}}{\rho}\frac{t_{\mathrm{CFB}}}{L}\left\\{\gamma\mu_{0}\left(H_{\phi}\left|h_{\mathrm{eff}}^{z}\right|^{2}+H_{z}\left|h_{\mathrm{eff}}^{\phi}\right|^{2}\right)+2\omega\mathrm{Im}\left[h_{\mathrm{eff}}^{z}\overline{h_{\mathrm{eff}}^{\phi}}\right]\right\\}.$

Note that the last term in the curly bracket is related to the non-
reciprocity. Since this research mainly focuses on the coupling strength at
$\phi=0$, where the non-reciprocity becomes zero [4], we omit this term. Then,
Eq. (S6) yields

$\omega^{2}=\frac{\omega_{m}^{2}+\omega_{p}^{2}}{2}\pm\frac{1}{2}\sqrt{\left(\omega_{m}^{2}-\omega_{p}^{2}\right)^{2}+\left(2\delta\omega_{\mathrm{bare}}^{2}\right)^{2}},$
(S7)

where

$\delta\omega_{\mathrm{bare}}^{2}=2\nu_{R}\xi_{S}^{2}\sqrt{\frac{\gamma^{2}\mu_{0}k^{3}t_{\mathrm{CFB}}}{\rho
M_{s}}}\left\\{\sqrt{H_{z}}(1-\xi_{S}^{2})\left(b_{1}\sin{\phi}\cos{\phi}-\eta
b_{2}\cos{2\phi}\right)+c\sqrt{H_{\phi}}\sqrt{1-\xi_{P}^{2}}\cos{\phi}\right\\}$
(S8)

and $\omega_{m}=|\gamma|\mu_{0}\sqrt{H_{z}H_{\phi}}$. To derive the value of
the coupling $g$, we compute the level repulsion gap for
$\omega_{m}=\omega_{p}$ at which

$\omega=\sqrt{\omega_{m}^{2}\pm\delta\omega_{\mathrm{bare}}^{2}}\sim\omega_{m}\pm\frac{\delta\omega_{\mathrm{bare}}^{2}}{2\omega_{m}}=\omega_{m}\pm
g.$ (S9)

Therefore,

$g=\nu_{R}\xi_{S}^{2}\sqrt{\frac{k^{3}t_{\mathrm{CFB}}}{\rho
M_{s}}}\left\\{\frac{(1-\xi_{S}^{2})}{\sqrt{\mu_{0}H_{\phi}}}\left(b_{1}\sin{\phi}\cos{\phi}-\eta
b_{2}\cos{2\phi}\right)+c\sqrt{\frac{1-\xi_{P}^{2}}{\mu_{0}H_{z}}}\cos{\phi}\right\\}.$
(S10)

Since $\varepsilon_{xx}$, coupling with $b_{1}$, decreases abruptly away from
the surface due to a change of sign of $u_{x}$ [3], we assume
$\varepsilon_{xx}$ has the effective penetration depth $t_{\mathrm{eff}}$.
Then, Eq. (S10) is modified as

$g=\nu_{R}\xi_{S}^{2}\sqrt{\frac{k^{3}}{\rho
M_{s}}}\left\\{\frac{(1-\xi_{S}^{2})}{\sqrt{\mu_{0}H_{\phi}}}\left(b_{1}\sqrt{t_{\mathrm{eff}}}\sin{\phi}\cos{\phi}-\eta
b_{2}\sqrt{t_{\mathrm{CFB}}}\cos{2\phi}\right)+c\sqrt{t_{\mathrm{CFB}}}\sqrt{\frac{1-\xi_{P}^{2}}{\mu_{0}H_{z}}}\cos{\phi}\right\\}.$
(S11)

Especially, when $\phi_{H}=\phi=0$ as Fig. 3 in the main text, it reduces to

$g=\nu_{R}\xi_{S}^{2}\sqrt{\frac{k^{3}t_{\mathrm{CFB}}}{\rho
M_{s}}}\left\\{\eta
b\frac{1-\xi_{S}^{2}}{\sqrt{\mu_{0}H_{\phi}}}+c\sqrt{\frac{1-\xi_{P}^{2}}{\mu_{0}H_{z}}}\right\\}$
(S12)

with $b=b_{1}=b_{2}$. For the anticrossing fittings and calculations in Fig. 3
in the main text and Fig. S8, we set the parameters as $\nu_{R}=1.6$,
$\xi_{S}^{2}=0.41$, $\xi_{P}^{2}=0.32$, $\eta=0.5$, and $\rho=8,000$ kg/m3.

##  S5 SAW transmission model under magnon–phonon coupling

Figure S6:  Sketch of the system for the transmission model.

We model the system as shown in Fig. S6. There are two acoustic SAW modes
between the cavity mirrors—namely, the two modes observed in the cavity
transmission in the absence of the magnet—of frequencies $\omega_{1}$ and
$\omega_{2}$ and with intensity loss rates $\gamma_{1}$ and $\gamma_{2}$. We
assume that the intrinsic phonon loss is negligible, so that the phonon loss
is due to leakage through the mirrors outside the cavity (for an extension
including intrinsic loss, see below). We also assume both mirrors have the
same transmission coefficient. In the absence of losses, each acoustic mode
generates a displacement field

$\mathbf{u}_{j}(\mathbf{r},t)=\sqrt{A/(2\rho\omega_{j})}\left[\mathbf{f}_{j}(\mathbf{r})e^{-i\omega_{j}t}a_{j}(t)+\text{c.c.}\right]$
(S13)

with $\rho$ the material density, $\mathbf{f}_{j}(\mathbf{r})$ the normalized
acoustic mode function, $a_{j}(t)$ the normalized amplitude and $A$ an
arbitrary constant. Each acoustic mode is coupled to a single magnon mode
(frequency $\omega_{mj}$, loss rate $\gamma_{mj}$) via the magnetoelastic
interaction. In the absence of losses, the magnetization of each magnon mode
can be decomposed in a similar way,

$\mathbf{m}_{j}(\mathbf{r},t)=\sqrt{A|\gamma|M_{s}}\left[\mathbf{h}_{j}(\mathbf{r})e^{-i\omega_{mj}t}s_{j}(t)+\text{c.c.}\right]$
(S14)

with $\gamma$ the gyromagnetic ratio, $M_{s}$ the saturation magnetization,
$\mathbf{h}_{j}(\mathbf{r})$ the properly normalized mode function [10] and
$s_{j}(t)$ the mode amplitude. The losses of both magnon and acoustic modes
originate from their coupling to fluctuating reservoirs. In the case of the
magnon, we consider a single continuum reservoir labelled by an index $\omega$
and with normalized amplitudes $b_{j}(\omega,t)$. In the case of the acoustic
modes, we consider two continuum baths describing the SAW continuum outside
each of the cavity mirrors ($r$ = right, $l$ = left), and with respective
amplitudes $a_{\eta}(\omega,t)$, where $\eta=r,l$. The whole coupled dynamics
obeys the following equations of motion,

$\frac{da_{j}(t)}{dt}=-i\omega_{j}a_{j}(t)-ig^{*}_{j}s_{j}(t)-i\sum_{\omega}\sum_{\eta=r,l}\sqrt{\frac{\gamma_{j}}{8\pi}}a_{\eta}(\omega,t)$
(S15)
$\frac{ds_{j}(t)}{dt}=-i\omega_{mj}s_{j}(t)-ig_{j}a_{j}(t)-i\sum_{\omega}\sqrt{\frac{\gamma_{mj}}{4\pi}}b(\omega,t)$
(S16) $\frac{d}{dt}a_{\eta}(\omega,t)=-i\omega
a_{\eta}(\omega,t)-i\sum_{j}\sqrt{\frac{\gamma_{j}}{8\pi}}a_{j}(t)$ (S17)
$\frac{d}{dt}b_{j}(\omega,t)=-i\omega
b_{j}(\omega,t)-i\sum_{j}\sqrt{\frac{\gamma_{mj}}{4\pi}}s_{j}(t)$ (S18)

with $g_{j}$ the magnetoelastic coupling rate between magnon mode $j$ and
acoustic mode $j$.

Let us now formally integrate the last two expressions in terms of a fixed
point in the distant past $t_{0}\to-\infty$,

$a_{\eta}(\omega,t)=e^{-i\omega(t-t_{0})}a_{\eta}(\omega,t_{0})-i\sum_{j}\sqrt{\frac{\gamma_{j}}{8\pi}}\int_{t_{0}}^{t}dsa_{j}(s)e^{-i\omega(t-s)}$
(S19)
$b_{j}(\omega,t)=e^{-i\omega(t-t_{0})}b_{j}(\omega,t_{0})-i\sum_{j}\sqrt{\frac{\gamma_{mj}}{4\pi}}\int_{t_{0}}^{t}dss_{j}(s)e^{-i\omega(t-s)}$
(S20)

and introduce them in Eqs. (S15)–(S16). After some algebra, we can cast the
system evolution as

$\frac{d}{dt}\left[\begin{array}[]{c}a_{1}\\\ a_{2}\\\ s_{1}\\\
s_{2}\end{array}\right]\equiv\frac{d}{dt}\mathbf{v}=M\mathbf{v}-\mathbf{f}_{\rm
in}\equiv
M\mathbf{v}-\left[\begin{array}[]{c}\sqrt{\gamma_{1}}\sum_{\eta}a_{\eta,\rm
in}(t)\\\ \sqrt{\gamma_{2}}\sum_{\eta}a_{\eta,\rm in}(t)\\\
\sqrt{\gamma_{m1}}b_{1\rm in}(t)\\\ \sqrt{\gamma_{m2}}b_{2\rm
in}(t)\end{array}\right]$ (S21)

with dynamical matrix

$M=\left[\begin{array}[]{cccc}-i\omega_{1}-\gamma_{1}/2&-\sqrt{\gamma_{1}\gamma_{2}}/2&-ig_{1}^{*}&0\\\
-\sqrt{\gamma_{1}\gamma_{2}}/2&-i\omega_{2}-\gamma_{2}/2&0&-ig_{2}^{*}\\\
-ig_{1}&0&-i\omega_{m1}-\gamma_{m1}/2&0\\\
0&-ig_{2}&0&-i\omega_{m2}-\gamma_{m2}/2\end{array}\right]$ (S22)

and input field amplitudes are defined as

$a_{\eta,\rm
in}(t)\equiv\frac{i}{\sqrt{8\pi}}\sum_{\omega}e^{-i\omega(t-t_{0})}a_{\eta}(\omega,t_{0})$
(S23) $b_{j\rm
in}(t)\equiv\frac{i}{\sqrt{4\pi}}\sum_{\omega}e^{-i\omega(t-t_{0})}b_{j}(\omega,t_{0}).$
(S24)

It is convenient to transform to the frequency domain using the convention

$f(\omega)=\frac{1}{\sqrt{2\pi}}\int dtf(t)e^{i\omega t},$ (S25)

where the system dynamics can be solved as

$\mathbf{v}(\omega)=(M+i\omega)^{-1}\mathbf{f}_{\rm in}(\omega).$ (S26)

In order to compute transmission, we first derive input-output relations for
the SAW bath field amplitudes. First, we use Eq. (S17) to solve for these
amplitudes in terms of a time $t_{1}\to\infty$ in the distant future,
obtaining

$a_{\eta}(\omega,t)=e^{-i\omega(t-t_{1})}a_{\eta}(\omega,t_{1})+i\sum_{j}\sqrt{\frac{\gamma_{j}}{8\pi}}\int_{t}^{t_{1}}dsa_{j}(s)e^{-i\omega(t-s)}.$
(S27)

We now introduce this expression into Eq. (S15) to obtain

$\frac{da_{j}(t)}{dt}=-i\omega_{j}a_{j}(t)-ig_{j}^{*}s_{j}(t)+\frac{\gamma_{j}}{2}a_{j}(t)+\frac{\sqrt{\gamma_{1}\gamma_{2}}}{2}a_{\bar{j}}(t)-\sqrt{\gamma_{j}}\sum_{\eta}a_{\eta,\rm
out}(t)$ (S28)

where $\bar{j}$ is the opposite index of $j$ and with output amplitudes
defined as

$a_{\eta,\rm
out}(t)\equiv\frac{i}{\sqrt{8\pi}}\sum_{\omega}e^{-i\omega(t-t_{1})}a_{\eta}(\omega,t_{1}).$
(S29)

Subtracting the corresponding component from Eq. (S21), namely

$\frac{da_{j}(t)}{dt}=-i\omega_{j}a_{j}(t)-ig_{j}^{*}s_{j}(t)-\frac{\gamma_{j}}{2}a_{j}(t)\\\
-\frac{\sqrt{\gamma_{1}\gamma_{2}}}{2}a_{\bar{j}}(t)-\sqrt{\gamma_{j}}\sum_{\eta}a_{\eta,\rm
in}(t)$ (S30)

we obtain the input-output relations

$a_{\eta,\rm out}(t)=a_{\eta,\rm
in}(t)+\sum_{j}\frac{\sqrt{\gamma_{j}}}{2}a_{j}(t).$ (S31)

It is easy to see that the Fourier transform of the output amplitude is
related to the transmission, as the occupation of the bath operators in the
distant future can be cast as

$\langle
a_{\eta}^{\dagger}(\omega,t_{1})a_{\eta}(\omega,t_{1})\rangle=4\left\langle\left[a_{\eta,\rm
out}(\omega)\right]^{\dagger}a_{\eta,\rm out}(\omega)\right\rangle.$ (S32)

Here, the brackets denote an ensemble average over any potential stochastic
variables in the SAW bath fields, e.g. thermal fluctuations.

Assuming we drive the cavity from the left side, we can define the
transmission as

$T(\omega)\equiv\frac{\langle
a_{r}^{\dagger}(\omega,t_{1})a_{r}(\omega,t_{1})\rangle}{\langle
a_{l}^{\dagger}(\omega,t_{0})a_{l}(\omega,t_{0})\rangle}=\frac{\left\langle\left[a_{r,\rm
out}(\omega)\right]^{\dagger}a_{r,\rm
out}(\omega)\right\rangle}{\left\langle\left[a_{l,\rm
in}(\omega)\right]^{\dagger}a_{l,\rm in}(\omega)\right\rangle}$ (S33)

where we have used the easy-to-check identity

$\left\langle\left[a_{\eta,\rm
in}(\omega)\right]^{\dagger}a_{\eta^{\prime},\rm
in}(\omega)\right\rangle=\frac{1}{4}e^{i(\omega-\omega^{\prime})t_{0}}\langle
a_{\eta}^{\dagger}(\omega,t_{0})a_{\eta^{\prime}}(\omega^{\prime},t_{0})\rangle.$
(S34)

We thus aim at computing $a_{r,\rm out}(\omega)$, given according to the
input-output relation by

$a_{r,\rm out}(\omega)=a_{r,\rm
in}(\omega)+\sum_{j}\frac{\sqrt{\gamma_{j}}}{2}a_{j}(\omega).$ (S35)

This expression can be written in a compact form as

$\displaystyle a_{r,\rm out}(\omega)=a_{r,\rm in}(\omega)$ $\displaystyle+$
$\displaystyle\mathbf{w}\cdot(M+i\omega)^{-1}\cdot\mathbf{w}\sum_{\eta}a_{\eta,\rm
in}(\omega)$ (S36) $\displaystyle+$
$\displaystyle\mathbf{w}\cdot(M+i\omega)^{-1}\cdot\left[\mathbf{e}_{3}\sqrt{\gamma_{m1}/2}b_{1\rm
in}(\omega)+\mathbf{e}_{4}\sqrt{\gamma_{m2}/2}b_{2\rm in}(\omega)\right]$

where we have defined the vector

$\mathbf{w}\equiv\left[\sqrt{\gamma_{1}/2},\sqrt{\gamma_{2}/2},0,0\right]^{t}.$
(S37)

The initial values corresponding to our transmission experiment are

$\langle
a_{\eta}^{\dagger}(\omega,t_{0})a_{\eta^{\prime}}(\omega^{\prime},t_{0})\rangle=\delta_{\eta\eta^{\prime}}\delta(\omega-\omega^{\prime})\left[\bar{n}(\omega)+\delta_{\eta
l}|\alpha|^{2}\right]$ (S38) $\langle
b_{j}^{\dagger}(\omega,t_{0})b_{j}(\omega^{\prime},t_{0})\rangle=\delta(\omega-\omega^{\prime})\bar{n}(\omega)$
(S39)

with $\bar{n}(\omega)$ the thermal distribution at the frequency $\omega$ and
temperature $T$, that is, all the baths are on a thermal equilibrium state
except for the left-propagating SAW bath, which on top of its thermal
fluctuations contains the coherent drive amplitude $\alpha$ generated by the
SAW generator. Using these values into Eq. (S36) we can compute the
transmission as

$\displaystyle T(\omega)$ $\displaystyle=$
$\displaystyle\left|\mathbf{w}\cdot(M+i\omega)^{-1}\cdot\mathbf{w}\right|^{2}$
(S40) $\displaystyle+$
$\displaystyle\frac{\bar{n}(\omega)}{\bar{n}(\omega)+|\alpha|^{2}}\bigg{[}1+\left|\mathbf{w}\cdot(M+i\omega)^{-1}\cdot\mathbf{w}\right|^{2}+\sum_{j=1,2}\gamma_{jm}\left|\mathbf{w}\cdot(M+i\omega)^{-1}\cdot\mathbf{e}_{j+2}\right|^{2}\bigg{]}$

where the first term describes the transmission of the drive and the second
term describes both the initial thermal fluctuations of the right SAW bath and
the injection of thermal fluctuations from the system and the left reservoir.
Assuming a drive amplitude much more intense than the thermal fluctuations,
$\bar{n}(\omega)\ll|\alpha|^{2}$, we can approximate the transmission by the
following analytical expression:

$T(\omega)=\frac{1}{4}\left|\frac{A}{B}\right|^{2},$ (S41)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\gamma_{1}|g_{2}|^{2}D_{3}(\omega)+\gamma_{2}|g_{1}|^{2}D_{4}(\omega)+D_{3}(\omega)D_{4}(\omega)(\gamma_{2}D_{1}(\omega)+\gamma_{1}D_{2}(\omega)+\gamma_{1}\gamma_{2}),$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
D_{1}(\omega)D_{2}(\omega)D_{3}(\omega)D_{4}(\omega)+|g_{1}|^{2}D_{2}(\omega)D_{4}(\omega)+|g_{2}|^{2}D_{1}(\omega)D_{3}(\omega)$
(S42)
$\displaystyle-(\gamma_{1}\gamma_{2}/4)D_{3}(\omega)D_{4}(\omega)+|g_{1}g_{2}|^{2},$

with

$D_{j}(\omega)=i(\omega-\omega_{1})-\gamma_{1}/2$ (S43)

and with $g_{j}$ being in general complex numbers and with $j=1,2$ labeling
the acoustic modes and $j=3,4$ labeling the magnon modes 1, 2, respectively.
The magnon frequency is
$\omega_{mj}=\gamma\mu_{0}\sqrt{H_{z}H_{\phi}}|_{k=k_{j}}$ with
$k_{j}=\omega_{j}/v$ and the magnon linewidth is computed as

$\gamma_{mj}=\frac{2\alpha\omega_{mj}}{\mu_{0}|\gamma|}\frac{\partial\omega_{mj}}{\partial
H}=\alpha\mu_{0}|\gamma|\left(H_{z}+H_{\phi}\right)\cos{(\phi-\phi_{H})}.$
(S44)

In the above expression, we have assumed that the only mechanism for phonon
loss from the cavity is the phonons leaking out from the reflectors, and as
such the transmission reaches a value of exactly 1 at resonance, i.e. the
height of both peaks is equal. This balance can be broken if we assume that
the phonons have a total linewidth
$\gamma^{\mathrm{tot}}_{j}=\gamma_{j}=\Gamma_{j}$, where $\Gamma_{j}$
describes intrinsic losses. In this case, the expression for the transmission
is identical to Eqs. (S41)–(S5) but with the substitution

$D_{1,2}(\omega)=i(\omega-\omega_{1,2})-\gamma^{\mathrm{tot}}_{1,2}/2.$ (S45)

In addition, we have to consider that our acoustic cavity has a certain
frequency range around the main resonance $\omega_{c}$. Finally, the
transmission is expressed as

$\tilde{T}(\omega)=T(\omega)\frac{1}{\sigma\sqrt{2\pi}}e^{-(\omega-\omega_{c})^{2}/(2\sigma^{2})}.$
(S46)

###  S5.1 In-plane magnetic field angular dependence of SAW-driven SWR

Figure S7:  Magnon and phonon dispersions at the in-plane magnetic field
angles $\phi_{H}=15^{\circ},30^{\circ}$, and 45∘. The orange lines show the
fixed phonon frequency $\omega_{p}=2\pi\times$ 6.58 GHz. The blue curves show
the SWR dispersion $\omega_{m}=|\gamma|\mu_{0}\sqrt{H_{z}H_{\phi}}$

To calculate the in-plane magnetic field angle dependence of our magnon–phonon
coupling, we first calculated the ground state of the magnetization. The free
energy normalized by the saturation magnetization is

$\displaystyle G=$ $\displaystyle-$
$\displaystyle\mu_{0}H\left\\{\sin{\theta}\sin{\theta_{H}}\cos{(\phi-\phi_{H})}+\cos{\theta}\cos{\theta_{H}}\right\\}$
(S47) $\displaystyle-$
$\displaystyle\mu_{0}H_{K}\sin^{2}{\theta}\sin^{2}{\phi}+\frac{M_{s}}{2}\cos^{2}{\theta},$

where $\theta$ ($\theta_{H}$) is the polar angle of the magnetization
(magnetic field). Note that the uniaxial magnetic anisotropy is aligned to the
in-plane perpendicular to $\mathbf{k}$. By taking the $\phi$ value to make it
minimum as $\partial G/\partial\phi|_{\theta=\theta_{H}=\pi/2}=0$, we get the
angle of ground state magnetization. Through substituting this $\phi$ value to
Eq. (S11) and the transmission model above, we reproduced the angular
dependence as shown in Fig. 2(d) in the main text with $\mu_{0}H_{K}=$ 0.2 mT
and $t_{\rm eff}=$ 2 nm.

Due to the magnetic anisotropy, the SWR dispersion of CFB with $k=2\pi/$(600
nm) makes an additional crossing point with the phonon dispersion at
$\phi_{H}<30^{\circ}$, as shown in Fig. S7. This is the origin of the SAW
absorption peaks close to $\mu_{0}H=0$ at $\phi_{H}<30^{\circ}$ in Figs. 2(b)
and 2(c) in the main text and Fig. S3(a).

###  S5.2 Reproduced anticrossings by the SAW transmission model

Here, we show the measured SAW transmission spectra under an external in-plane
magnetic field at $\phi_{H}=0$, the anticrossing fittings, and reproduced SAW
transmission spectra using Eq. (S46) not shown in Fig. 3 of the main text in
Fig. S8.

Figure S8:  Measured and calculated SAW transmissions when the external
magnetic field is applied in the direction of SAW propagation; $\phi_{H}=0$.
(a),(b) SAW transmission signal ($|S_{21}|^{2}$) of the sample with
$t_{\mathrm{CFB}}=10$ nm under (a) various amplitudes of the magnetic field
$\mu_{0}H$ and (b) $\mu_{0}H=$ 100 mT. The green marker in (b) represents the
local maximum used for the anticrossing fitting, shown as the green curves in
(a). (c) Calculated SAW transmission of the sample with $t_{\mathrm{CFB}}=10$
nm as a function of the frequency and $\mu_{0}H$. (d),(e) $|S_{21}|^{2}$ of
the sample with $t_{\mathrm{CFB}}=25$ nm under (d) various $\mu_{0}H$ and (e)
$\mu_{0}H=$ 100 mT. The green marker in (e) represents the local maximum used
for the anticrossing fitting, shown as the green curves in (d). (f) Calculated
SAW transmission of the sample with $t_{\mathrm{CFB}}=25$ nm as a function of
the frequency and $\mu_{0}H$. (g),(h) $|S_{21}|^{2}$ of the sample with
$t_{\mathrm{CFB}}=35$ nm under (g) various $\mu_{0}H$ and (h) $\mu_{0}H=$ 200
mT. The green marker in (h) represents the local maximum used for the
anticrossing fitting, shown as the green curves in (g). (i) Calculated SAW
transmission of the sample with $t_{\mathrm{CFB}}=35$ nm as a function of the
frequency and $\mu_{0}H$.

##  S6 Microwave ferromagnetic resonance measurements

Figure S9:  Microwave FMR measurement of the CPW device with $t_{\mathrm{CFB}}=20$ nm. (a) SAW reflection at the frequency of 6.58 GHz as a function of the external magnetic field. The markers and curve exhibit the measured data and the fitting, respectively. (b), FMR linewidth as a function of frequency. The markers and line exhibit the obtained values from the fittings and the fitting with Eq. (S48), respectively. Table S1: Gilbert damping constant of each thickness of CFB. $t_{\mathrm{CFB}}$ (nm) | $\alpha(\times 10^{-3})$
---|---
10 | 9.21 $\pm$ 1.11
20 | 8.97 $\pm$ 0.37
25 | 11.8 $\pm$ 1.42
30 | 9.44 $\pm$ 1.20
35 | 8.20 $\pm$ 2.14

We measured the Gilbert damping constant $\alpha$ by ferromagnetic resonance
(FMR) to estimate the magnon relaxation rate. A conventional coplanar
waveguide (CPW) is used for microwave excitation. The width and length of the
Ti/CFB/Ti layers are 6 $\mu$m and 200 $\mu$m, respectively. Signal lines of
the CPW are deposited on Ti/CFB/Ti layers. The width of the signal line of the
CPW is 6 $\mu$m and the distance between the signal line and ground line is 4
$\mu$m. Note that the Ti/CFB/Ti layers are deposited at the same time as the
samples used for the magnon–phonon coupling experiments shown in the main text
on LiNbO3 substrates. By applying an in-plane magnetic field perpendicular to
the radio-frequency microwave field, we observed the absorption of the
microwave transmission $|S_{11}|^{2}$ as the example for the device with
$t_{\mathrm{CFB}}=20$ shown in Fig. S9(a). Through Lorentzian fittings as
shown as the curve in Fig. S9(a), we obtain the FMR linewidth $\mu_{0}\Delta
H$ which has a well-known dependence upon the microwave frequency $f$ as

$\mu_{0}\Delta H=\mu_{0}\Delta H_{0}+\frac{4\pi\alpha}{\gamma}f,$ (S48)

where $\mu_{0}\Delta H_{0}$ is the inhomogeneous linewidth. Figure S9(b) shows
the obtained FMR linewidth as a function of the microwave frequency and the
fitting with Eq. (S48) which yields the Gilbert damping constant. The Gilbert
damping constants estimated by the FMR linewidth fittings of all devices are
shown in Table S1.

Additionally, note that the microwave absorption by FMR, as shown in Fig.
S9(a), we observe only a single absorption shape without any split features.
Therefore, we only observe anticrossing behaviors in CFB with our two-port SAW
resonator devices.

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