# Simultaneous Brillouin and piezoelectric coupling to a high-frequency bulk
acoustic resonator: Supplemental document

Taekwan Yoon  Departments of Applied Physics and Physics, Yale University, New
Haven, CT 06520, USA  Yale Quantum Institute, New Haven, CT 06520, USA  These
two authors contributed equally  David Mason  Departments of Applied Physics
and Physics, Yale University, New Haven, CT 06520, USA  Yale Quantum
Institute, New Haven, CT 06520, USA  These two authors contributed equally
Vijay Jain  Departments of Applied Physics and Physics, Yale University, New
Haven, CT 06520, USA  Yale Quantum Institute, New Haven, CT 06520, USA  Yiwen
Chu  Department of Physics, ETH Zurich, 8093 Zurich, Switzerland  Prashanta
Kharel  Departments of Applied Physics and Physics, Yale University, New
Haven, CT 06520, USA  Yale Quantum Institute, New Haven, CT 06520, USA
William H. Renninger  Institute of Optics, University of Rochester, Rochester,
NY 14627, USA  Liam Collins  Center for Nanophase Materials Sciences, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA  Luigi Frunzio
Departments of Applied Physics and Physics, Yale University, New Haven, CT
06520, USA  Yale Quantum Institute, New Haven, CT 06520, USA  Robert J.
Schoelkopf  Departments of Applied Physics and Physics, Yale University, New
Haven, CT 06520, USA  Yale Quantum Institute, New Haven, CT 06520, USA
robert.schoelkopf.yale.edu  Peter T. Rakich  Departments of Applied Physics
and Physics, Yale University, New Haven, CT 06520, USA  Yale Quantum
Institute, New Haven, CT 06520, USA<EMAIL_ADDRESS>
###### Contents

  1. S1 Full apparatus setup
  2. S2 Individual cavity characterization
  3. S3 Coupling rate characterizations
    1. S3.1 Electromechanical coupling rate
    2. S3.2 Optomechanical coupling rate
    3. S3.3 Optomechanical coupling rate and cooperativity in single-pass configuration
  4. S4 Optical cavity enhancement of signal
    1. S4.1 Signal without optical cavity enhancement
    2. S4.2 Signal with optical cavity enhancement
    3. S4.3 Resonant enhancement of signal
  5. S5 Transduction Spectrum
    1. S5.1 X-cut quartz: transduction
    2. S5.2 State space model
    3. S5.3 Presence of electro-optic (EO) effect in transduction spectrum
  6. S6 Potential improvements in transduction
  7. S7 Understanding CaF2 data
  8. S8 Alternative piezoelectricity measurement: Piezoresponse Force Microscopy (PFM)

##  S1 Full apparatus setup

Figure S1:  Experimental apparatus PM: phase modulator, IM: intensity
modulator, AOM: acousto-optic modulator, PDH: Pound-Drever-Hall, SIG: signal
generator. For OMIT measurement, switch combination (1-a) and (2-b) (3-a) are
used (switch 3 is irrelevant). For microwave to optical conversion, switch
combination (1-b), (2-b) and (3-a) are used. For optical to microwave
conversion, switch combination (1-a), (2-a) and (3-b) are used. Part of the
apparatus shaded in green is not used for X-cut quartz measurement, but
instead only used for CaF2 measurement to achieve better signal-to-noise ratio
for highly sensitive piezoelectricity detection.

The experimental setup is illustrated in Figure S1. The laser is locked to the
optical cavity via the Pound-Drever-Hall (PDH) locking technique. The optical
sidebands required for the lock are generated by the signal generator (SIG2)
and the phase modulator (PM). For optomechanically-induced transparency (OMIT)
measurements, a sideband is generated by an intensity modulator (IM), which is
driven by a microwave signal generator (SIG1) synchronized to the readout
frequency of a spectrum analyzer. Here, the main optical tone and the sideband
act as the pump and probe, respectively. The beat note between the transmitted
pump and probe is measured via a high speed photodetector and spectrum
analyzer. The readout frequency of the spectrum analyzer can be synchronized
to the frequency of the signal generator, effectively functioning as a scalar
network analyzer. In the microwave-to-optical transduction configuration, the
drive tone from the signal generator directly drives the microwave cavity,
piezoelectrically exciting phonons from which the pump light scatters to
achieve electro-optical conversion. For optical-to-microwave transduction, the
optical probe is generated from the pump using the intensity modulator.
However, instead of measuring the optical response of the probe (as in OMIT),
the spectrum analyzer directly reads the signal leaving the microwave cavity.

For the OMIT/transduction measurements performed on quartz, the transmitted
pump serves as a local oscillator (LO) for high-frequency heterodyne detection
of the signal sideband. In this configuration, it is not easy to attenuate the
LO power (e.g. to avoid detector saturation) without also attenuating the
signal. Therefore, for enhanced sensitivity in the CaF2 measurements, we
modify the circuit (green shaded box of Figure S1), filtering out the
transmitted pump and using a separate, controllable LO derived from the
original laser. We also shift the frequency of this new LO using an Acousto-
Optic Modulator (AOM) to avoid microwave crosstalk in the detection channel.
The spectrum analyzer frequency can be offset accordingly to accommodate this
shift between drive frequency and heterodyne frequency, resulting in a unique
signal frequency to avoid any potential microwave crosstalk. Using this
separate-path LO can introduce low-frequency phase noise ($<$1kHz) in the
heterodyne beat note, which does not impact our measurements.

##  S2 Individual cavity characterization

Figure S2:  Cavity loss and tunability characterization (a) Microwave, (b)
acoustic, and (c) optical cavity spectrum with corresponding Q-factor. (d)
Illustration of optical cavity mode-spacing tunability. Red datapoints show
varying FSR at different optical wavelength whereas red arrow shows the tuning
range we could achieve with a piezo actuator controlling the optical cavity
length. Black arrow indicates one of the crossing points between optical
cavity mode-spacing and Brillouin frequency where we chose to operate at.

In section 3 of the manuscript, we list the loss rates and the Q factors
(finesse for optical cavity) of optical, acoustic, and microwave cavities
($\kappa_{\mathrm{opt}}/2\pi=2.2\mathrm{MHz},\mathcal{F}=5170,\Gamma/2\pi\approx
500\mathrm{kHz},Q_{\mathrm{m}}=22800,\kappa_{\mu}/2\pi=17.1\mathrm{MHz},Q_{\mu}=664$)
used for x-cut quartz demonstration of piezo-Brillouin spectroscopy. Figure
S2(a-c) are the cavity spectroscopy data with fits for x-cut quartz.

Figure S2(a) shows a normalized reflection measurement of the microwave
resonator, from which we find $\kappa_{\mu\mathrm{,i}}/2\pi=9.79$ MHz
$(Q_{\mu\mathrm{,i}}=1160)$, $\kappa_{\mu\mathrm{,c}}/2\pi=7.33$ MHz
$(Q_{\mu\mathrm{,c}}=1550)$, giving a total of $\kappa_{\mu}/2\pi=17.1$ MHz
$(Q_{\mu}=662)$. We use a coaxial stub cavity as the microwave cavity. The
body of cavity is made out of OFHC copper, while a brass screw with a center
through-hole is used as the center pin in order to achieve high tunability in
resonance frequency. The seam loss between the copper body and the brass
center pin results in a rather modest microwave Q factor in the current
design. Figure S2(c) shows (normalized) transmission measurements for the two
optical cavity modes used as pump and signal mode. We use optical sidebands to
calibrate a linear frequency sweep over the modes, from which we can extract
their linewidths. A broader, calibrated wavelength scan is used to identify
the mode spacing. Due to the dielectric interface within the cavity, it is
possible to have asymmetric cavity loss rates through the two mirrors, even
though they have nominally identical reflectivities. Measurements of the
resonant transmission and reflection are used to extract the relative loss
rates through each mirror, as well as the internal loss. For both pump/probe
modes of the optical cavity, we observe coupling rates of
$\kappa_{\mathrm{opt,c1}}/2\pi\approx 0.7$ MHz and
$\kappa_{\mathrm{opt,c2}}/2\pi\approx 1.2$ MHz, giving an internal loss rate
of $\kappa_{\mathrm{opt,i}}/2\pi\approx 0.3$ MHz and combined loss rate of
$\kappa_{\mathrm{opt}}/2\pi\approx 2.2$ MHz. Figure S2(b) presents
characterization data for an acoustic mode in x-cut quartz. Specifically, we
present single-pass piezo-Brillouin spectroscopy data (for which the
optomechanical cooperativity is small, ensuring that we measure the bare
mechanical linewidth in the absence of any optomechanical backaction).

### Optical cavity FSR tunability

The Fabry-Perot optical cavity used in our platform consists of two dielectric
mirrors with 99.9% reflectivity and a dielectric substrate (quartz or CaF2) in
the middle. The non-zero reflection of optical mode at the boundary of the
dielectric media produces an intrinsic modulation of optical mode spacing. For
the cavity geometry used here (11 mm vacuum, 0.5 mm quartz), we find
modulation by $\pm$ 0.5 GHz around a mean spacing of 10.9 GHz. This variation
allows us to find a mode spacing that approximately matches the Brillouin
frequency. In the experiments here, we pump the lower-frequency mode, using
the upper-frequency mode to resonantly match the anti-Stokes sideband. Since
the mode spacing variation is larger than the optical cavity linewidth, the
other (Stokes) process is strongly suppressed.

_In situ_ fine-tuning of optical cavity mode spacing at cryogenic temperatures
is achieved through the use of a piezo-actuator that allows translation of one
mirror position. With this, we can match the optical mode spacing to the
acoustic mode within the both optical and acoustic linewidths
($\Delta_{\mathrm{opt}}\approx\Omega_{\mathrm{m}}$).

The frequency tuning range is shown in Figure S2(d), where we clearly see
greater than than 10% ($>$ 1 GHz) intrinsic variation, plus fine-tuning via
the piezo-actuator voltage (0V $\sim$ 150V) to match the optical mode spacing
(red arrow) with the acoustic mode (green dashed line). In our experiment, we
operate at optical wavelength of $\sim$ 1546.4 nm, indicated in back arrow in
the figure.

##  S3 Coupling rate characterizations

###  S3.1 Electromechanical coupling rate

The electromechanical interaction Hamiltonian can be expressed as a product
between piezo-induced stress field
($\sigma(\bar{r})=\overline{\overline{c}}:\overline{\overline{d}}\cdot
E(\bar{r})$) and strain field in the substrate ($s(\bar{r})$), given as
$H_{\mathrm{int}}=\int{\sigma(\bar{r})s(\bar{r})dV}$, where
$\overline{\overline{c}}$ is the stiffness tensor, $\overline{\overline{d}}$
is the piezoelectric tensor, and $:$ is tensor product. In our system, the
coupling occurs predominantly between the z-directional electric and strain
field due to the microwave cavity design and material properties. Thus we
consider longitudinal strain fields (${s}(\bar{r})={S}_{z}(\bar{r})$)
interacting with an electric field along the z-axis
(${\sigma}(\bar{r})=c_{33}d_{33}{E}_{z}(\bar{r})$), resulting in
$H_{\mathrm{int}}=c_{33}d_{33}\int{{E}_{z}(\bar{r}){S}_{z}(\bar{r})dV}$.
Quantizing the electric
(${E}_{z}(\bar{r})=\bar{E}_{0}(\bar{r})({a}+{a}^{\dagger})$) and strain fields
(${S}_{z}(\bar{r})=\bar{S}_{0}(\bar{r})({b}+{b}^{\dagger})$), the interaction
Hamiltonian becomes

$\hbar
g_{\mathrm{em}}=c_{33}d_{33}\int_{\mathrm{pz}}{\bar{E}_{0}(\bar{r})\bar{S}_{0}(\bar{r})dV},$
(S1)

where $\int_{\mathrm{pz}}dV$ is a volume integral across the piezoelectric
substrate.

Acoustic strain field is assumed to be a standing wave in longitudinal
direction and a Gaussian shape in transverse direction with mode waist
proportional to the probing optical mode waist[1]
($r_{\mathrm{m}}=r_{\mathrm{opt}}/\sqrt{2}$, where $r_{\mathrm{m}}$ is the
acoustic mode waist and $r_{\mathrm{opt}}$ is the optical mode waist),
resulting in an expression

$\bar{S}_{0}(\bar{r})=s_{0}e^{-\frac{r^{2}}{r_{\mathrm{m}}^{2}}}\sin(\frac{2\pi
z}{\lambda_{\mathrm{m}}})\hat{z}.$ (S2)

$\lambda_{\mathrm{m}}$ is the acoustic wavelength,
$s_{0}=\sqrt{\frac{2\hbar\Omega}{c_{33}L_{\mathrm{m}}A_{\mathrm{m}}}}$ is the
zero point strain, $L_{\mathrm{m}}$ is the crystal thickness, and
$A_{\mathrm{m}}$ is the acoustic mode area.

The microwave mode is concentrated around the central pin of a coaxial
microwave cavity, resulting in a spherical mode shape with a radius of $\sim$3
mm. Meanwhile, mode waist (radius) of the cylindrical acoustic mode is $\sim$
50 um, which is substantially smaller than that of the microwave mode.
Moreover, the thickness of the crystalline substrate varies between 0.5 mm and
1 mm, which is also smaller compared to the microwave mode waist. As a result,
the electric field parallel to the phonon propagation ($\hat{z}$-direction)
within the microwave/acoustic mode overlap volume can be assumed to be
uniform,

$\bar{E}_{0}(\bar{r})=E_{0}\hat{z},$ (S3)

where $E_{0}=\sqrt{\frac{\hbar\Omega}{2\epsilon_{0}\epsilon_{r}V_{\mu}}}$ is
the uniform zero-point electric field strength within the overlapping mode
volume. We can obtain Electric field of the system through HFSS (3D high
frequency simulation software). It is convenient to output the electric field
values of the microwave mode when the E-field is at its maximum and H-field is
is zero for a stored energy of $\hbar\Omega$. E-field value from the
simulation ($\bar{E}_{\mathrm{sim}}(\bar{r})$) follows,
$\hbar\Omega=\frac{\epsilon_{0}\epsilon_{r}}{2}\int
dV|\bar{E}_{\mathrm{sim}}(\bar{r})|^{2}$. Within the microwave/acoustic mode
overlap volume, we see uniform simulated electric field
($\bar{E}_{\mathrm{sim}}(\bar{r})=E_{\mathrm{sim}}\hat{z}$). In terms of zero-
point field, simulated field amplitude can be expressed as,
$E_{\mathrm{sim}}=2E_{0}$.

Combining equations (S1), (S3) and (S2), we obtain the electromechanical
coupling rate,

$\displaystyle g_{\mathrm{em}}$
$\displaystyle=\frac{1}{\hbar}c_{33}d_{33}s_{0}E_{0}\int_{\mathrm{pz}}{e^{-\frac{r^{2}}{r_{\mathrm{m}}^{2}}}\sin(\frac{2\pi
z}{\lambda_{\mathrm{m}}})dV}$ (S4)
$\displaystyle=d_{33}E_{\mathrm{sim}}\frac{\lambda_{\mathrm{m}}}{\pi}\sqrt{\frac{\Omega
c_{33}A_{\mathrm{m}}}{2\hbar L_{\mathrm{m}}}}\sin^{2}\left(\frac{\pi
t_{\mathrm{pz}}}{\lambda_{\mathrm{m}}}\right),$

where $t_{\mathrm{pz}}$ is the piezoelectric substrate thickness.

There are three distinct cases of piezoelectric coupling that one can
consider. Firstly, piezoelectricity can be distributed evenly throughout the
entire bulk of the acoustic substrate. In this case, acoustic modes with an
even longitudinal index ($t_{\mathrm{pz}}=n\frac{\lambda_{\mathrm{m}}}{2}$
with even $n$) will have zero coupling rate ($\sin^{2}\left(\frac{\pi
t_{\mathrm{pz}}}{\lambda_{\mathrm{m}}}\right)=0$), while the odd index modes
($t_{\mathrm{pz}}=n\frac{\lambda_{\mathrm{m}}}{2}$ with odd $n$) will have
maximal coupling ($\sin^{2}\left(\frac{\pi
t_{\mathrm{pz}}}{\lambda_{\mathrm{m}}}\right)=1$). Secondly, substrates may
have surface piezoelectricity on one side of the substrate. In this case,
piezoelectricity is concentrated within a thin surface layer on one surface
with thickness $t_{\mathrm{pz}}$ ($t_{\mathrm{pz}}\ll\lambda_{\mathrm{m}}$),
while the bulk remains non-piezoelectric, resulting in a simplification of the
sin term in the coupling rate expression as $\sin^{2}\left(\frac{\pi
t_{\mathrm{pz}}}{\lambda_{\mathrm{m}}}\right)=(\frac{\pi
t_{\mathrm{pz}}}{\lambda_{\mathrm{m}}})^{2}$. Lastly, there is the case for
surface piezoelectricity on both sides of the substrate. Here, the modulation
of zero and non-zero coupling rate for even and odd indexed acoustic modes
appears as in the bulk piezoelectricity case, while the non-zero coupling rate
is proportional to $t_{\mathrm{pz}}^{2}$ as in the single-sided surface
piezoelectricity case. With these observations in mind, we examine our
experimental results in Section 5 (on X-cut quartz) and Section 7 (on CaF2) of
the supplement.

Another aspect of $g_{\mathrm{em}}$ is that it has a factor of
$1/\sqrt{L_{\mathrm{m}}}$. Although we are integrating along the full length
of the substrate, the sin term results in values equivalent to integrating
along an effective piezoelectric thickness of
$t_{\mathrm{pz}}<\lambda_{\mathrm{m}}/2$. Hence the factor,
$1/\sqrt{L_{\mathrm{m}}}$, from strain normalization term remains throughout
the expression of $g_{\mathrm{em}}$. Such dependency of $g_{\mathrm{em}}$ in
substrate thickness suggests that using thinner substrate is advantageous in
achieving larger $g_{\mathrm{em}}$, which we will later discuss the
supplement.

###  S3.2 Optomechanical coupling rate

Here, we derive the Brillouin optomechanical coupling rate, closely following
prior studies[2, 1]. Specifically, we consider phase-matched photoelastic
coupling between optical Fabry Perot modes and longitudinal bulk acoustic
resonances. The electric field profile of the $j$-th optical cavity mode is
given by

${E}_{j}(r,z,t)=E_{j,0}e^{-r^{2}/r_{\mathrm{opt}}^{2}}\mathrm{sin}(k_{j}z)({a}_{j}(t)+{a}^{\dagger}_{j}(t))$
(S5)

where
$E_{j,0}=\sqrt{\frac{2\hbar\omega_{j}}{\epsilon_{0}\epsilon_{r}A_{opt}L_{opt}}}$
is the zero-point field, $\omega_{j}$ is the $j$-th optical cavity mode
frequency, $A_{\mathrm{opt}}$ ($L_{\mathrm{opt}}$) is optical mode area
(optical cavity length), $\epsilon_{0}$ ($\epsilon_{r}$) is vacuum (relative)
permittivity, $k_{j}$ is $j$-th optical wavevector, and ${a}_{j}$ is the
anhillation operator for $j$-th optical mode. Note that for simplicity, this
model neglects modifications of the optical field due to the combined
vacuum/dielectric composition of the cavity. Full modelling of the optical
mode profile, in the presence of this dielectric interface, is possible
through use of a transfer matrix model [1]. For the acoustic modes, we write
the displacement of the $i$-th longitudinal mode as

${U}_{z}(r,z,t)=U_{i,0}e^{-r^{2}/r_{m}^{2}}\mathrm{cos}(q_{i}z)({b}_{i}(t)+{b}^{\dagger}_{i}(t)),$
(S6)

where
$U_{i,0}=\sqrt{\frac{2\hbar}{\rho\Omega_{\mathrm{m}}A_{\mathrm{m}}L_{\mathrm{m}}}}$
is the zero-point displacement, $\rho$ is crystal density, $q_{i}$ is $i$-th
acoustic wavevector, and ${b}_{i}$ is the anhillation operator for $i$-th
acoustic mode. Since we are using a plano-plano (as opposed to plano-convex)
crystal for the acoustic resonator, our acoustic mode waist is not
independently determined by the crystal geometry, but rather defined by the
optical mode waist. Specifically, $r_{\mathrm{m}}=r_{\mathrm{opt}}/\sqrt{2}$.

The photoelastic interaction Hamiltonian is given by

$H_{\mathrm{int}}=\frac{1}{2}\epsilon_{0}\epsilon_{r}^{2}p_{13}\int\left(\frac{\partial{U}}{\partial
z}\right){E}^{2}_{\mathrm{opt}}dV,$ (S7)

where $p_{13}$ is the relevant photoelastic constant. In anticipation of the
phase-matching requirement, we will consider intermodal coupling involving the
$j$ and $j+1$ optical modes with wavevectors/frequencies which satisfy
$k_{j+1}+k_{j}=q_{i}$ and $\Omega_{m}=\omega_{j+1}-\omega_{j}$. Since
$\omega_{j}\approx\omega_{j+1}$, this yields the Brillouin phase-matching
condition from the main text ($q_{i}=2n\omega_{j}/c$). Plugging in the
appropriate mode definitions and making the rotating wave approximation, the
interaction Hamltonian is

$\displaystyle H_{\mathrm{int}}$
$\displaystyle=\frac{1}{2}\epsilon_{0}\epsilon_{r}^{2}p_{13}\int
dVq_{i}U_{i,0}E_{j+1,0}E_{j,0}e^{-\frac{r^{2}}{r_{m}^{2}}}e^{-\frac{r^{2}}{r_{\mathrm{opt}}^{2}}}e^{-\frac{r^{2}}{r_{\mathrm{opt}}^{2}}}\mathrm{sin}(q_{i}z)\mathrm{sin}(k_{j+1}z)\mathrm{sin}(k_{j}z)({a}^{\dagger}_{j+1}{a}_{j}{b}_{i}+H.C.),$
(S8)

Note that here we have assumed the crystal begins precisely at $z=0$ (i.e.
coincident with the mirror). In reality, the longitudinal component of this
overlap integral will depend sensitively on the position of the acoustic
standing wave within the optical standing wave. As shown in [1], the approach
we take here offers a good estimate of the _maximum_ possible coupling rate.
Experimental values may differ by up to a factor of 2, based on this exact
longitudinal overlap. This approximation, as well as the choice to neglect
optical field redistribution due to the dielectric interface, constitute the
main sources of uncertainty in predicting the cavity optomechanical coupling
rate.

The interaction Hamiltonian can be be alternatively written in terms of the
single-photon coupling rate, $g_{\mathrm{om,0}}$, as following,

$H_{\mathrm{int}}=\hbar
g_{\mathrm{om,0}}({a}^{\dagger}_{j+1}{a}_{j}{b}_{i}+H.C.).$ (S9)

Combining Equation S8 and S9, we obtain the single-photon coupling rate in the
presence of an optical cavity,

$g_{\mathrm{om,0}}=\frac{\omega_{j}^{2}n^{3}p_{13}}{8c}\sqrt{\frac{2\hbar}{\Omega_{\mathrm{m}}\rho
A_{\mathrm{m}}L_{\mathrm{m}}}}\frac{L_{\mathrm{m}}}{L_{\mathrm{opt}}}.$ (S10)

The x-cut quartz OMIT measurements from Figure 2 of the main text offer a
direct measurement of the cavity-enhanced coupling rate. Using cavity
parameters to calculate the intracavity power, we can infer an experimental
value of $g_{\mathrm{om,0}}=5.28$ Hz which closely agrees with the analytical
value of $g_{\mathrm{om,0}}=5.89$ Hz from Equation S10. For all data
presented, experimental value is used unless specified.

Optical cavity-enhanced $g_{\mathrm{om}}$ is obtained by linearizing around a
strong pump ($H_{\mathrm{int}}=\hbar
g_{\mathrm{om,0}}{a}^{\dagger}_{j+1}{a}_{j}{b}_{i}\approx\hbar\sqrt{N_{p}}g_{\mathrm{om,0}}{a}^{\dagger}_{j+1}{b}_{i}=\hbar
g_{\mathrm{om}}{a}^{\dagger}_{j+1}{b}_{i}$), giving the expression,
$g_{\mathrm{om}}=\sqrt{N_{\mathrm{p}}}g_{\mathrm{om,0}}$, where
$N_{\mathrm{p}}$ is the inter-cavity pump photon number. In our case, the
maximum cavity-enhanced $g_{\mathrm{om}}$ achievable in our system is
$g_{\mathrm{om}}=$ 1.4 MHz from OMIT ($N_{\mathrm{p}}\sim 7\times 10^{10}$ for
112 mW of pump power)

###  S3.3 Optomechanical coupling rate and cooperativity in single-pass
configuration

In single-pass configuration without an optical cavity, we can consider an
optical cavity with $L_{\mathrm{opt}}=L_{\mathrm{m}}$ where the boundaries are
defined by the acoustic crystal surfaces. This removes the filling-factor term
($\frac{L_{\mathrm{m}}}{L_{\mathrm{opt}}}=1$) from Equation S10 in describing
the single-pass single-photon optomechanical coupling rate, resulting in an
expression,
$g_{\mathrm{om,0}}^{\prime}=\frac{\omega_{j}^{2}n^{3}p_{13}}{8c}\sqrt{\frac{2\hbar}{\Omega_{\mathrm{m}}\rho
A_{\mathrm{m}}L_{\mathrm{m}}}}$.

Free-space cooperativity of an optomechanical system with an acoustic cavity
but without an optical cavity is defined in reference [2] as

$C_{\mathrm{om,0}}^{\mathrm{sp}}=\frac{P_{\mathrm{p}}}{\hbar\omega_{\mathrm{p}}}\frac{L_{\mathrm{m}}}{v_{\mathrm{g,p}}}\frac{L_{\mathrm{m}}}{v_{\mathrm{g,s}}}\frac{g_{\mathrm{om,0}}^{\prime
2}}{\Gamma},$ (S11)

where $P_{\mathrm{p}}$ is the pump power, $\omega_{\mathrm{p}}$ is the optical
pump wavelength, $v_{\mathrm{g,p}}$ ($v_{\mathrm{g,s}}$) is the group velocity
of the pump (signal) light and $\Gamma$ is the acoustic loss rate. Introducing
group delay ($\tau=L_{\mathrm{m}}n/c$), and inter cavity photon number
($N_{\mathrm{p}}=P_{\mathrm{p}}L_{\mathrm{m}}/(\hbar\omega_{\mathrm{p}}v_{\mathrm{g,p}})$),
the single-pass cooperativity can be re-written in a simpler form,
$C_{\mathrm{om}}^{\mathrm{sp}}=\frac{g_{\mathrm{om}}^{2}}{\Gamma\tau^{-1}}$.

##  S4 Optical cavity enhancement of signal

Here, we consider the specific configuration in which one seeks to measure an
optical sideband scattered from a (piezoelectrically-driven) phonon
population. In particular, we analyze the optical power in the scattered
sideband on both single-pass and optical cavity configuration, to illustrate
the benefit of the optical cavity.

###  S4.1 Signal without optical cavity enhancement

In the absence of an optical cavity, a strong pump can still scatter off
phonons to generate a Brillouin sideband. Following reference [2], the optical
power in this sideband can be written as

$P_{\mathrm{sig}}^{\prime}=\left(\frac{g_{\mathrm{om,0}}^{\prime}L_{\mathrm{m}}}{v_{\mathrm{o}}}\right)^{2}P_{\mathrm{p}}n_{\mathrm{m}}.$
(S12)

###  S4.2 Signal with optical cavity enhancement

In the case where we do consider an optical cavity, it becomes natural to
describe the optomechanical interaction through coupled equations of motion
for the pump and signal fields. In the rotating frame, this can be written as,

$\displaystyle\dot{{a}}_{\mathrm{s}}$
$\displaystyle=i\sqrt{N_{\mathrm{p}}}g_{\mathrm{om,0}}{b}-\frac{\kappa_{\mathrm{opt}}}{2}{a}_{\mathrm{s}}$
(S13) $\displaystyle\dot{{a}}_{\mathrm{p}}$
$\displaystyle=-\frac{\kappa_{\mathrm{opt}}}{2}{a}_{\mathrm{p}}+\sqrt{\kappa_{\mathrm{opt},c}}a_{\mathrm{in}},$

where $a_{\mathrm{in}}$ is the external pump drive field
($P_{\mathrm{p}}/\hbar\omega_{\mathrm{p}}=\langle
a_{\mathrm{in}}^{\dagger}a_{\mathrm{in}}\rangle$), ${a}_{\mathrm{s}}$
(${a}_{\mathrm{p}}$) is annihilation operator for the signal (pump) mode,
${b}$ is annihilation operator for phonons, $\kappa_{\mathrm{opt}}$
($\kappa_{\mathrm{opt,c}}$) is optical cavity loss (coupling) rate. A more
extensive Hamiltonian approach will be taken in the later state space model
section of the supplement. Certain simplifications are made in this section
($\Delta_{o}=0,g_{\mathrm{eo}}=0$).

In steady state, we obtain

$\displaystyle
i\sqrt{N_{\mathrm{p}}}g_{\mathrm{om,0}}{b}=\frac{\kappa_{\mathrm{opt}}}{2}{a}_{\mathrm{s}}$
(S14)
$\displaystyle\frac{\kappa_{\mathrm{opt}}}{2}{a}_{\mathrm{p}}=\sqrt{\kappa_{\mathrm{opt},c}}a_{\mathrm{in}}.$

Inter-cavity pump photon number can then be written as

$\left(\frac{\kappa_{\mathrm{opt}}}{2}\right)^{2}N_{\mathrm{p}}=\kappa_{\mathrm{opt},c}\langle
a_{\mathrm{in}}^{\dagger}a_{\mathrm{in}}\rangle.$ (S15)

From input-output formalism, Equation S14, and Equation S15, outgoing
scattered signal field ($a_{\mathrm{out}}$) is

$\displaystyle a_{\mathrm{out}}$
$\displaystyle=\sqrt{\kappa_{\mathrm{opt},c}}{a}_{\mathrm{s}}$ (S16)
$\displaystyle=\left(\frac{\sqrt{\kappa_{\mathrm{opt},c}}}{\kappa_{\mathrm{opt}}}\right)^{2}(4ig_{\mathrm{om,0}})\sqrt{\langle
a_{\mathrm{in}}^{\dagger}a_{\mathrm{in}}\rangle}{b},$

and the corresponding scattered signal power is

$\displaystyle
P_{\mathrm{sig}}=16\left(\frac{\kappa_{\mathrm{opt,c}}}{\kappa_{\mathrm{opt}}^{2}}\right)^{2}g_{\mathrm{om,0}}^{2}P_{\mathrm{p}}n_{\mathrm{m}}.$
(S17)

###  S4.3 Resonant enhancement of signal

Comparing Equation S12 and Equation S17, we get a ratio of

$\displaystyle\frac{P_{\mathrm{sig}}}{P_{\mathrm{sig}}^{\prime}}$
$\displaystyle=\frac{16\left(\frac{\kappa_{\mathrm{opt},c}}{\kappa_{\mathrm{opt}}^{2}}\right)^{2}g_{\mathrm{om,0}}^{2}P_{\mathrm{p}}n_{\mathrm{m}}}{\left(\frac{g_{\mathrm{om,0}}^{\prime}L_{m}}{v_{\mathrm{o}}}\right)^{2}P_{\mathrm{p}}n_{\mathrm{m}}}$
(S18)
$\displaystyle=\frac{16}{\pi^{2}}\eta_{\mathrm{opt}}^{2}\mathcal{F}^{2}.$

Assuming everything was kept constant, we expect 16$\mathcal{F}^{2}$
improvement in signal power by introducing an optical cavity with with a
finesse, $\mathcal{F}$.

##  S5 Transduction Spectrum

###  S5.1 X-cut quartz: transduction

Figure S3:  Full OMIT and transduction spectrum for x-cut quartz (a) colorplot
of OMIT spectrum on x-cut quartz with varying optical pump power. Clear
splitting of the optomechanical response indicates strong coupling. (b) Linear
relation between cooperativity fitted from data in (a) and optical pump power.
(c) Measured microwave-to-optical conversion at varying pump power. Shifting
of transduction peak is clearly visible, indicating we are reaching
$C_{\mathrm{om}}>1$. EO effect induced dip on the right side of the
transduction peak is also visible. Dashed line in purple is the cross section
along which Figure 3 in the main text is plotted. (d) Expected transduction
spectrum closely following the measurement. Color scale is set to be identical
to (c). Dashed line in black is the cross section along which Figure 3 in the
main text is plotted.

Beyond what is presented in Figure 2 and Figure 3 in the main text, additional
characterization can be made for x-cut quartz, as shown in Figure S3. We
performed a series of OMIT measurement at different optical pump powers (see
Figure S3a). Here we observe normal mode splitting, which indicates
optomechanical strong coupling
($2g_{\mathrm{om}}>\kappa_{\mathrm{opt}},\Gamma$), in addition to a large
visibility in OMIT dip, indicating high optomechanical cooperativity
($C_{\mathrm{om}}>1$). We can directly extract optomechanical cooperativity
and confirm its linearity against optical pump power by fitting the OMIT data
($C_{\mathrm{om}}\propto N_{\mathrm{p}}\propto P_{\mathrm{p}}$, see Figure
S3b). Microwave-to-optical transduction spectroscopy is measured across a
range of optical pump power in Figure S3(c). The optical mode splitting from
S3(a) is also visible in the transduction spectra in S3(c), particularly as
the emergence of a left branch at high powers. The right branch is obscured
due to interference with the electro-optic effect. To accurately match this
model to our data, note that we have to use $g_{\mathrm{em}}/2\pi=347$ Hz (see
Figure S3d). This is close to the simulated coupling rate,
$g_{\mathrm{em}}/2\pi=298$ Hz, where the discrepancy can be attributed to the
difference between simulated and machined microwave cavity and the shift in
material properties in cryogenic conditions. The theoretical model is based on
a comprehensive state space model that describes piezoelectric EM, Brillouin
OM, and Pockel EO interactions, laid out in the following section.

###  S5.2 State space model

(This subsection closely follows the works of reference [3, 4].) The
Hamiltonian describing the system is

$H/\hbar=\omega_{\mathrm{p}}{a}_{\mathrm{p}}^{\dagger}{a}_{\mathrm{p}}+\omega_{\mathrm{s}}{a}_{\mathrm{s}}^{\dagger}{a}_{\mathrm{s}}+\Omega_{\mathrm{m}}{b}^{\dagger}{b}+\Omega_{\mu}{c}^{\dagger}{c}+(g_{\mathrm{om,0}}{a}_{\mathrm{p}}{a}_{s}^{\dagger}{b}+g_{\mathrm{em}}{b}^{\dagger}{c}+g_{\mathrm{eo}}{a}^{\dagger}{c}+\mathrm{H.c.}),$
(S19)

where $\omega_{\mathrm{p}}$ ($\omega_{\mathrm{s}}$) is optical wavelength of
pump (signal) mode, $\Omega_{\mu}$ is microwave cavity resonance frequency,
and ${c}$ is annihilation operator for microwave cavity mode. In the rotating
frame with
$H_{0}/\hbar=\omega_{\mathrm{p}}{a}_{\mathrm{p}}^{\dagger}{a}_{\mathrm{p}}+\omega_{\mathrm{p}}{a}_{\mathrm{s}}^{\dagger}{a}_{\mathrm{s}}$,
we obtain an effective Hamiltonian

$H_{\mathrm{eff}}/\hbar=\Delta_{\mathrm{o}}{a}_{\mathrm{s}}^{\dagger}{a}_{\mathrm{s}}+\Omega_{\mathrm{m}}{b}^{\dagger}{b}+\Omega_{\mu}{c}^{\dagger}{c}+(g_{\mathrm{om,0}}{a}_{\mathrm{p}}{a}_{\mathrm{s}}^{\dagger}{b}+g_{\mathrm{em}}{b}^{\dagger}{c}+g_{\mathrm{eo}}{a}^{\dagger}{c}+\mathrm{H.c.}),$
(S20)

where $\Delta_{\mathrm{o}}=\omega_{\mathrm{s}}-\omega_{\mathrm{p}}$ is the
detuning of optical signal frequency from optical pump. In the undepleted pump
regime, we can linearize the Hamiltonian about a strong, coherent pump, giving
the expression

$H_{\mathrm{eff}}/\hbar=\Delta_{\mathrm{o}}{a}_{\mathrm{s}}^{\dagger}{a}_{\mathrm{s}}+\Omega_{\mathrm{m}}{b}^{\dagger}{b}+\Omega_{\mu}{c}^{\dagger}{c}+(g_{\mathrm{om}}{a}_{\mathrm{s}}^{\dagger}{b}+g_{\mathrm{em}}{b}^{\dagger}{c}+g_{\mathrm{eo}}{a}^{\dagger}{c}+\mathrm{H.c.}),$
(S21)

where $g_{\mathrm{om}}=\sqrt{N_{\mathrm{p}}}g_{\mathrm{om,0}}$, and
$N_{\mathrm{p}}$ is the intracavity pump photon number. The Heisenberg
equations of motion for this Hamiltonian and the input-output relation between
the fields are given by the following equations:

$\dot{\textbf{a}}(t)=A\textbf{a}(t)+B\textbf{a}_{\mathrm{in}}(t)$ (S22)
$\textbf{a}_{\mathrm{out}}(t)=B^{T}\textbf{a}(t)-\textbf{a}_{\mathrm{in}}(t)$
(S23) $\textbf{a}=({a},{c},{b})^{T}$ (S24)
$\textbf{a}_{\mathrm{in}}=(a_{\mathrm{in}},c_{\mathrm{in}})^{T}$ (S25)
$\textbf{a}_{\mathrm{out}}=(a_{\mathrm{out}},c_{\mathrm{out}})^{T}$ (S26)
$A=\begin{pmatrix}-i\Delta_{\mathrm{o}}-\frac{\kappa_{\mathrm{o}}}{2}&ig_{\mathrm{eo}}&ig_{\mathrm{om}}\\\
ig_{\mathrm{eo}}&-i\Omega_{\mu}-\frac{\kappa_{\mu}}{2}&ig_{\mathrm{em}}\\\
ig_{\mathrm{om}}&ig_{\mathrm{em}}&-i\Omega_{\mathrm{m}}-\frac{\kappa_{\mathrm{m}}}{2}\end{pmatrix}$
(S27) $B=\begin{pmatrix}\sqrt{\kappa_{\mathrm{o,c}}}&0\\\
0&\sqrt{\kappa_{\mu\mathrm{,c}}}\\\ 0&0\end{pmatrix},$ (S28)

where a is a vector for resonator mode operators and ain(out) is a vector of
input (output) fields. This is a simplified state space model in which we only
include ports that we care about – in/output coupling ports. There are
additional internal decay channels in this process that are not included in
our model as we are assuming zero noise input.

In the frequency domain, we can reduce the expressions into a scattering
matrix,

$\textbf{a}_{\mathrm{out}}(\omega)=S(\omega)\textbf{a}_{\mathrm{in}}(\omega)$
(S29) $S(\omega)=B^{T}(-i\omega I-A)^{-1}B-I.$ (S30)

Using this scattering matrix, we can obtain various features that we can
expect from our system, including OMIT, microwave-to-optical transduction, and
optical-to-microwave transduction. Here we mainly focus on the transduction
spectrum
($\eta(\omega)=|S_{\mathrm{oe}}(\omega)|^{2}=|S_{\mathrm{eo}}(\omega)|^{2}$).

$\eta(\omega)=\frac{\kappa_{\mathrm{o,c}}}{\kappa_{\mathrm{o}}}\frac{\kappa_{\mu\mathrm{,c}}}{\kappa_{\mu}}\left|\frac{\alpha(\omega)}{\beta(\omega)}\right|^{2}$
(S31)
$\alpha(\omega)=2(-\sqrt{C_{\mathrm{em}}C_{\mathrm{om}}}+iC_{\mathrm{eo}}(1-\frac{i(\omega-\Omega_{\mathrm{m}})}{\Gamma/2}))$
(S32) $\displaystyle\beta(\omega)$
$\displaystyle=2i\sqrt{C_{\mathrm{em}}C_{\mathrm{om}}C_{\mathrm{eo}}}+C_{\mathrm{em}}(1-\frac{i(\omega-\Delta_{\mathrm{o}})}{\kappa_{\mathrm{o}}/2})+C_{\mathrm{om}}(1-\frac{i(\omega-\Omega_{\mu})}{\kappa_{\mu}/2})+C_{\mathrm{eo}}(1-\frac{i(\omega-\Omega_{\mathrm{m}})}{\Gamma/2})$
(S33)
$\displaystyle+(1-\frac{i(\omega-\Delta_{\mathrm{o}})}{\kappa_{\mathrm{o}}/2})(1-\frac{i(\omega-\Omega_{\mu})}{\kappa_{\mu}/2})(1-\frac{i(\omega-\Omega_{\mathrm{m}})}{\Gamma/2}).$

We observe peak conversion when the optical detuning, microwave mode, and the
acoustic mode are in resonance. Measuring transduction at this resonant
frequency ($\omega=\Delta_{\mathrm{o}}=\Omega_{\mu}=\Omega_{\mathrm{m}}$), we
get

$\eta=\frac{\kappa_{\mathrm{o,c}}}{\kappa_{\mathrm{o}}}\frac{\kappa_{\mu\mathrm{,c}}}{\kappa_{\mu}}\frac{4(C_{\mathrm{em}}C_{\mathrm{om}}+C_{\mathrm{eo}}^{2})}{4C_{\mathrm{em}}C_{\mathrm{om}}C_{\mathrm{eo}}+(C_{\mathrm{em}}+C_{\mathrm{om}}+C_{\mathrm{eo}}+1)^{2}}.$
(S34)

In the case of no electro-optic coupling, Equation S34 can be reduced into the
expression for a typical electro-optomechanical transduction.

$\eta=\frac{\kappa_{\mathrm{o,c}}}{\kappa_{\mathrm{o}}}\frac{\kappa_{\mu\mathrm{,c}}}{\kappa_{\mu}}\frac{4C_{\mathrm{em}}C_{\mathrm{om}}}{(C_{\mathrm{em}}+C_{\mathrm{om}}+1)^{2}}$
(S35)

###  S5.3 Presence of electro-optic (EO) effect in transduction spectrum

The expression of electro-optic (EO) coupling (Pockels effect) in materials is
studied in various sources [5, 6, 7, 8]. The single-photon electro-optic
coupling rate between optical pump, optical signal, and microwave can be
written as:

$\displaystyle\hbar g_{\mathrm{eo,0}}$
$\displaystyle=\epsilon_{0}\int_{\mathrm{eo}}{dV\epsilon_{1}E_{\mathrm{p}}\epsilon_{1}E_{\mathrm{s}}r_{13}E_{\mu}}$
(S36)
$\displaystyle=\epsilon_{0}\epsilon_{1}^{2}r_{13}E_{\mathrm{p,0}}E_{\mathrm{s,0}}E_{\mu\mathrm{,0}}\int_{\mathrm{eo}}{dVe^{-\frac{r^{2}}{r_{\mathrm{opt}}^{2}}}e^{-\frac{r^{2}}{r_{\mathrm{opt}}^{2}}}\mathrm{sin}(k_{\mathrm{p}}z)\mathrm{sin}(k_{\mathrm{s}}z)},$

where $r_{13}$ is the relevant linear electro-optic coefficient component.
Electric fields of optical pump ($E_{\mathrm{p}}$), optical signal
($E_{\mathrm{s}}$) and microwave ($E_{\mu}$) is defined as in the previous
sections, with corresponding normalization factors of
$E_{\mathrm{p,0}}=\sqrt{2\hbar\omega_{\mathrm{p}}/\epsilon_{0}\epsilon_{1}A_{\mathrm{opt}}L_{\mathrm{opt}}}$
and
$E_{\mathrm{s,0}}=\sqrt{2\hbar\omega_{\mathrm{s}}/\epsilon_{0}\epsilon_{1}A_{\mathrm{opt}}L_{\mathrm{opt}}}$.
$E_{\mu\mathrm{,0}}$ is the zero-point electric field obtained through HFSS
($E_{\mu\mathrm{,0}}=E_{\mathrm{sim}}/2$), uniform along the interaction
region. Assuming identical optical pump and signal mode profiles in both
transverse and longitudinal direction, the above expression simplifies into,

$\displaystyle g_{\mathrm{eo,0}}$
$\displaystyle=\frac{1}{4}\epsilon_{1}r_{13}E_{\mathrm{sim}}\omega_{\mathrm{p}}\frac{L_{\mathrm{m}}}{L_{\mathrm{opt}}}.$
(S37)

EO coefficient in quartz is $r_{13}$ = 0.45 pm/V from crystal symmetry [9].
Resulting single photon EO coupling rate in quartz is $g_{\mathrm{eo,0}}/2\pi$
= 1.05 mHz. Inside an optical cavity, this coupling rate is parametrically
enhanced by the intercavity photon number
($g_{\mathrm{eo}}=\sqrt{N_{\mathrm{p}}}g_{\mathrm{eo,0}}$), reaching
$g_{\mathrm{eo}}/2\pi$ = 137Hz.

Figure S4:  Microwave to optical conversion measured (purple) is fitted using
state space model. The data is taken at $P_{\mathrm{p}}=$ 23.8 mW. Separate
OMIT measurement indicates $g_{\mathrm{om}}/2\pi$ = 643 kHz and
$C_{\mathrm{om}}=1.48$. (a) Piezo-optomechanical (pOM) component (dashed
green) and EO component (dashed red) of the fitted model is separately plotted
against the conversion data. Fitted values of the coupling rates are
$g_{\mathrm{em}}/2\pi$ = 347 Hz (theory value 298 Hz) and
$g_{\mathrm{eo}}/2\pi$ = 162 Hz (theory value 137 Hz). (b) EO and pOM combined
model (dashed black) fitted to data.

Incorporating a single-photon EO coupling rate into the state space model, we
can compare our theoretical model to the transduction spectrum. In fact, we
can even observe individual contributions from piezoelectric coupling and EO
coupling to the net transduction. In Figure S4, transduction from
piezoelectricity is in dashed green and that from EO is in dashed red. The
combined net transduction is in dashed black can be fitted to the experimental
data in purple, showing good agreement. Optomechanical coupling rate used in
the fit is obtained through OMIT measurement ($g_{\mathrm{om}}/2\pi$ = 643 kHz
and $C_{\mathrm{om}}=1.48$ at $P_{\mathrm{p}}=23.8$ mW). The fitted parameters
are $g_{\mathrm{em}}/2\pi$ = 347 Hz (theory value is 298 Hz) and
$g_{\mathrm{eo}}/2\pi$ = 162 Hz (theory value is 137 Hz). Such difference can
be attributed to a discrepancy between the simulated microwave cavity and the
assembled device, and to the variation in material parameters at cryogenic
temperatures.

The apparent interference between piezoelectric and EO effect near the center
peak is because of similar EM and EO cooperativities. In the relatively lossy
acoustic cavity regime that we are operating, we can still reach
$C_{\mathrm{om}}>1$. However, we have relatively low
$C_{\mathrm{em}}=5.6\times 10^{-8}$, which is greater than
$C_{\mathrm{eo}}=2.8\times 10^{-9}$ only by an order of magnitude. This is
also observed in Figure S4, where the peak of theoretical EO transduction and
that of theoretical pOM transduction are separated by 10 dB. As a result, we
observe a fano interference in the transduction spectrum where the pOM
response crosses through EO response. This destructive interference is visible
only on one side due to the phase relation between the pOM and EO interactions
- constructive interference occurs on the other side.

One of the future paths to improve transduction, suggested in the manuscript,
is using an acoustic cavity with better Q factor. Integration of a high-Q HBAR
will bring about a significant increase in $C_{\mathrm{em}}$, while keeping
$C_{\mathrm{eo}}$ constant, thus suppressing the formation of such fano-like
feature in future implementations of this design.

##  S6 Potential improvements in transduction

Suggested improvements | $g_{\mathrm{em}}/2\pi$ | $C_{\mathrm{em}}$ |  $\eta_{\mathrm{opt}}$($\eta_{\mu}$) | $\eta$
---|---|---|---|---
Current experiment | 347 Hz | $5.6\times 10^{-8}$ | 0.53 (0.43) | $1.2\times 10^{-8}$
Optimized coupling efficiencies | - | - |  $\sim$1 ($\sim$1) | $\uparrow 4\times$
X-cut quartz $\rightarrow$ Z-cut LiNbO3 | $\uparrow 7.3\times$ | $\uparrow 50\times$ | - | $\uparrow 50\times$
Optimized acoustic mode geometry | $\uparrow 9\times$ | $\uparrow 80\times$ | - | $\uparrow 80\times$
Re-entrant cavity | $\uparrow 3\times$ | $\uparrow 10\times$ | - | $\uparrow 10\times$
Superconducting cavity | - | $\uparrow 10^{2}\times$ | - | $\uparrow 10^{2}\times$
Plano-convex hBAR | - | $\uparrow 10^{3}\times$ | - | $\uparrow 10^{3}\times$
Combined improvement factors | $\uparrow\sim 200\times$ | $\uparrow\sim 4\cdot 10^{9}\times$ |  $\uparrow 2\times$ ($\uparrow 2\times$) | $\uparrow\sim 1.6\cdot 10^{10}\times$
Combined improvement values |  $\sim$68 kHz |  $\sim$224 |  $\sim$1 ($\sim$1) |  $\sim$0.9 ($C_{\mathrm{om}}=C_{\mathrm{em}}=10$)
Table S1: Electromechanical cooperativity improvements

Recall the transduction efficiency expression

$\eta=\eta_{\mathrm{opt}}\eta_{\mu}\frac{4C_{\mathrm{om}}C_{\mathrm{em}}}{(1+C_{\mathrm{om}}+C_{\mathrm{em}})^{2}}.$
(S38)

In our system, we can readily reach $C_{\mathrm{om}}$ of 1 and
$\eta_{\mathrm{opt}}$ ($\eta_{\mu}$) of 0.53 (0.43). Unity coupling
efficiencies can be reached with ease by adjusting the microwave coupling pin
to be over-coupled to the microwave cavity, and by having asymmetric optical
cavity mirrors (i.e. creating a single-sided cavity). Also note that the
Gaussian optical cavity mode is well-suited for achieving high fiber-coupling
efficiency, a key challenge for low loss integration of a transducer. The
expression therefore simplifies into

$\displaystyle\eta$ $\displaystyle\approx C_{\mathrm{em}}$ (S39)
$\displaystyle=\frac{4g_{\mathrm{em}}^{2}}{\kappa_{\mu}\Gamma},$

in the limit of small $C_{\mathrm{em}}$ ($C_{\mathrm{em}}\ll C_{\mathrm{om}}$)
and assuming unity coupling efficiencies
($\eta_{\mathrm{opt}}\approx\eta_{\mu}\approx 1$). Thus, improving the
transduction efficiency boils down to improving $C_{\mathrm{em}}$ up to and
above unity.

Among the approaches that we can take to enhance $C_{\mathrm{em}}$, one
obvious choice is to achieve stronger $g_{\mathrm{em}}$ as it is quadratically
related to $C_{\mathrm{em}}$. Rearranging Equation S4 in the bulk
piezoelectric limit,

$g_{\mathrm{em}}=E_{0}d_{33}c_{33}\sqrt{\frac{2A_{\mathrm{m}}}{\hbar\Omega_{\mathrm{m}}\rho
L_{\mathrm{m}}}}.$ (S40)

As laid out in Table S1, one way we can improve $g_{\mathrm{em}}$ is through
choosing a stronger piezoelectric material, such as LiNbO3 or BaTiO3, instead
of quartz. In the case of Z-cut LiNbO3, piezoelectric tensor component
increases from $d_{33}=2.3$ pm/V to $d_{33}=16.2$ pm/V, and stiffness tensor
component from $c_{33}=86.6$ GPa to $c_{33}=244$ GPa. Even taking into account
the impact from the change in density ($\rho$ = 2650 kg/m3 to $\rho$ = 4630
kg/m3), Brillouin frequency ($\Omega_{\mathrm{m}}$ = 11.4 GHz to
$\Omega_{\mathrm{m}}$ = 21.0 GHz), and $E_{0}$ ($\frac{0.80}{\sqrt{2}}\times
10^{-3}$ V/m to $\frac{0.53}{\sqrt{2}}\times 10^{-3}$ V/m) due to higher
relative permittivity, we can expect a higher $g_{\mathrm{em}}$
($C_{\mathrm{em}}$) by 7.3$\times$ (50$\times$).

Another factor that influences $g_{\mathrm{em}}$ is the acoustic mode
geometry. Increasing the acoustic mode waist allows us to achiever better mode
matching between acoustic and microwave modes, while thinner substrate
thickness allows for lower mass acoustic mode (with larger zero point motion)
while maintaining same acoustic-microwave mode overlap. Reasonable
modifications can be made to the acoustic waist from 50 um to 200 um and to
the substrate thickness ($L_{\mathrm{m}}$) from 500 um to 100 um. This result
in a higher $g_{\mathrm{em}}$ ($C_{\mathrm{em}}$) by 9$\times$ (80$\times$).
Note that reducing the substrate thickness will decrease the
$g_{\mathrm{om}}$, yet it will not deter our ability to exceed unity
optomechanical cooperativity ($C_{\mathrm{om}}>1$) as the current setup
reaches $C_{\mathrm{om}}\sim 10$ with relative ease.

Similarly, we can also improve the microwave and acoustic mode-matching by
making modifications to the microwave cavity. In doing so, we will be
optimizing $E_{0}$, the remaining term in $g_{\mathrm{em}}$. In order to
reduce the microwave mode volume and concentrate the electric field, we can
explore using a smaller coaxial microwave cavity or different cavity
geometries such as a re-entrant cavity and other 2-D designs. Simulating a re-
entrant cavity, we obtain approximately $3\times$ stronger electric field
compared to the coaxial design ($\frac{0.53}{\sqrt{2}}\times 10^{-3}$ V/m
$\rightarrow\frac{1.8}{\sqrt{2}}\times 10^{-3}$ V/m), resulting in a higher
$g_{\mathrm{em}}$ ($C_{\mathrm{em}}$) by 3$\times$ (10$\times$).

Besides optimizing $g_{\mathrm{em}}$, we can also improve on other factors in
$C_{\mathrm{em}}$; namely, $\kappa_{\mu}$ and $\Gamma$. This demonstration
uses a non-superconducting cooper microwave cavity, with a modest Q $<10^{3}$.
With superconducting materials, both post and re-entrant cavities can readily
reach Q $>10^{8}$. Hence, in an ideal circumstance, $\kappa_{\mu}$ can be
boosted by $>10^{5}\times$. However, we have to consider practical limitations
such as the combination of high laser powers with superconducting resonators
and the restriction in the bandwidth of the device that high Q microwave
cavity may impose. In reality, we will consider a conservative enhancement in
$\kappa_{\mu}$ by $>10^{2}\times$ as shown in Table S1. Diffractive loss in
the acoustic cavity can be mitigated via shaping a concave surface on one side
of the cavity through reactive-ion-etching. Doing so will allow the acoustic
cavity to form stable resonances with Q up to $\sim 2.8\times 10^{7}$[10] from
the current $2.3\times 10^{4}$, allowing us a $\times 10^{3}$ improvement in
$\Gamma$ at a cost in transduction bandwidth.

##  S7 Understanding CaF2 data

Experimental parameters for CaF2 are as following: detector resolution
bandwidth 1 kHz, $\Omega_{\mathrm{m}}=$ 13.354 GHz, $\Gamma=$ 535 kHz,
$L_{\mathrm{m}}=$ 0.5mm, $A_{\mathrm{m}}=2\pi\times(64\mathrm{um})^{2}$,
$\kappa_{\mathrm{o}}=$ 2.1 MHz, $\kappa_{\mathrm{o,c}}=$ 0.598 MHz,
$E_{0}=5.1\times 10^{-4}$ V/m, $\kappa_{\mu}=$ 22.4 MHz,
$\kappa_{\mu\mathrm{,c}}=$ 10.9 MHz. For microwave driven motion measurement:
$P_{\mathrm{p}}=$ 23 mW, $g_{\mathrm{om}}=$ 385 kHz, $C_{\mathrm{om}}=$ 0.705.
For thermal phonon measurement: $P_{\mathrm{p}}=$ 30.1 mW, $g_{\mathrm{om}}=$
427 kHz, $C_{\mathrm{om}}=$ 0.914.

Conversion from detected signal power to phonon number is done following the
expression,
$P_{\mathrm{sig}}=(\hbar\omega_{\mathrm{p}})(\frac{4g_{\mathrm{om}}^{2}}{\kappa_{\mathrm{o}}\Gamma})\Gamma
n_{\mathrm{m}}$, where $n_{\mathrm{m}}$ is the number of driven phonons. In
order to obtain piezoelectric coupling rate from the number of
piezoelectrically driven phonons, we revisit the state space model for the
equation of motion regarding electromechanical interaction with
simplifications ($g_{\mathrm{eo}}=g_{\mathrm{om}}=0$ and
$\Delta_{\mu}=\Delta_{\mathrm{m}}=0$ where $\Delta_{\mu}$ and
$\Delta_{\mathrm{m}}$ are detunings from microwave and mechanical resonance
frequencies.),

$\displaystyle\dot{{c}}$
$\displaystyle=ig_{\mathrm{em}}{b}-\frac{\kappa_{\mu}}{2}{c}-\sqrt{\kappa_{\mu,\mathrm{,c}}}c_{\mathrm{in}}$
(S41) $\displaystyle\dot{{b}}$
$\displaystyle=ig_{\mathrm{em}}{c}-\frac{\Gamma}{2}{b}.$

Assuming steady state, number of piezoelectrically generated phonons can be
expressed as

$\displaystyle n_{\mathrm{m}}$
$\displaystyle=\frac{4\kappa_{\mu\mathrm{,c}}}{\kappa_{\mu}^{2}}\left(\frac{g_{\mathrm{em}}}{\frac{2g_{\mathrm{em}}^{2}}{\kappa_{\mu}}+\frac{\Gamma}{2}}\right)^{2}\langle
c_{\mathrm{in}}^{\dagger}c_{\mathrm{in}}\rangle,$ (S42)

where $\langle c_{\mathrm{in}}^{\dagger}c_{\mathrm{in}}\rangle$ denotes the
input microwave drive flux ($\langle
c_{\mathrm{in}}^{\dagger}c_{\mathrm{in}}\rangle=P_{\mu}/\hbar\Omega_{\mu}$)
and $P_{\mu}$ is the input microwave drive power. Considering the regime we
are in ($\frac{2g_{\mathrm{em}}^{2}}{\kappa_{\mu}}\ll\frac{\Gamma}{2}$), above
expression simplifies to

$n_{\mathrm{m}}=\frac{4\kappa_{\mu\mathrm{,c}}}{\kappa_{\mu}^{2}}\left(\frac{2g_{\mathrm{em}}}{\Gamma}\right)^{2}\langle
c_{\mathrm{in}}^{\dagger}c_{\mathrm{in}}\rangle.$ (S43)

In the experiment, $P_{\mu}$, measured with a spectrum analyzer, is swept from
4 dBm to 20dBm. From Equation S43, we obtain electromechanical coupling rate,
which turns out to be $g_{\mathrm{em}}/2\pi=0.03$ Hz. Combining this with
Equation S4, we can correlate a bulk piezoelectric constant of 0.083 fm/V or a
surface piezoelectric constant of 2.44 pm/V across a 1 nm thick piezoelectric
layer.

##  S8 Alternative piezoelectricity measurement: Piezoresponse Force
Microscopy (PFM)

Figure S5: PFM scan across a 50 um$\times$50 um surface of CaF2. CaF2 used
here is the same substrate used in section 5 of the main text.(a) Example
measurement at $V_{\mathrm{AC}}$ = 20 V. Histogram is shown in inset. (b) The
average of displacement data is plotted with its standard deviation at various
$V_{\mathrm{AC}}$. Displacement noise floor of $\sim$2.2 pm is indicated in
dashed grey. Minimal variability in detected displacement indicates
piezoelectricity smaller than the device’s detection sensitivity.

Piezoresponse Force Microscopy (PFM) is a method of detecting piezoelectricity
in $\sim$ 100 nm depth[11, 12, 13]. The substrate is put under an AC electric
field (usually in the range of $\sim$100 kHz), which leads to a piezoelectric
deformation due to the converse piezoelectric effect. This deformation is
mapped with a probe cantilever across the substrate. Recently an
interferometric displacement sensing approach to PFM (IDS-PFM) has been
developed which allows quantitative determination of out of plane
electromechanical responses[13, 12]. This method removes many of the artifacts
that plagues traditional PFM, and is made inherently quantitative via the
interferometric read out.

We put the same CaF2 that we study in section 5 of the main text under a IDS-
PFM as a way to alternatively detect piezoelectricity (see Figure S5). The
frequency of the AC field is $\sim$100 kHz. We measure a white noise floor of
$\sim$70 fm/$\sqrt{\mathrm{Hz}}$, which is equivalent to a displacement noise
floor of $\sim$2.2 pm in an imaging bandwidth of 1kHz. For a 1 V drive
amplitude, this yields a minimum detectable piezo-sensitivity of
$d_{\mathrm{eff}}$ = 2.2 pm/V. Minimum detectable piezo-sensitivity can be
lowered by simply applying a larger drive voltage which was allowable in the
case of CaF2, unlike thin films or materials having low dielectric breakdown
potentials. For maximum applied voltage of 40 V, this implies a minimal
detectable $d_{\mathrm{eff}}$ of 55 fm/V. From data, we barely see any
deviation from the $\sim$2.2 pm background. Even though the data may appear to
have a non-zero slope ($14\pm 40$ fm/V), it is inconclusive that we detect any
piezoelectricity since all data are within the standard deviation/error bar.
We attribute the smaller but finite slope of 14 fm/V to influence of
electrostatic forces. At high voltage especially, we can expect non-neglighble
electrostatic forces and force gradients to act between the AFM tip and
sample, which may lead to small but finite displacement of the tip position,
even for the case of IDS-PFM (which is largely insensitive to cantilever-
sample electrostatic effects[13, 12]). We further note that using softer (3
nN) cantilevers than the one used in Figure S5 ($\sim$42 nN) leads to a clear
enlargement of the measured displacement (slope of $290\pm 80$ fm/V) as would
be expected for an electrostatic effect.

Therefore, we observe no detectable piezoelectricity though IDS-PFM, meaning
that parasitic piezoelectricity in CaF2 is smaller than the detection limit of
55 fm/V. These results indicate the unlikely-hood of the parasitic
piezoelectricity coming from an intrinsic surface layer, which requires
piezoelectricity to be $\gtrsim$ 1pm/V. Instead, it is more likely that the
paracitic piezoelectricity detected in CaF2 is originated from the bulk of the
crystal.

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