# Bidirectional microwave-optical transduction based on integration of high-
overtone bulk acoustic resonators and photonic circuits

Terence Blésin  Institute of Physics, Swiss Federal Institute of Technology
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Center of Quantum Science and
Engineering (EPFL), CH-1015 Lausanne, Switzerland  Wil Kao  Institute of
Physics, Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Center of Quantum Science and Engineering (EPFL),
CH-1015 Lausanne, Switzerland  Anat Siddharth  Institute of Physics, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center of Quantum Science and Engineering (EPFL), CH-1015 Lausanne,
Switzerland  Rui N. Wang  Institute of Physics, Swiss Federal Institute of
Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Center of Quantum
Science and Engineering (EPFL), CH-1015 Lausanne, Switzerland  Alaina
Attanasio  OxideMEMS lab, Purdue University, West Lafayette, IN, USA  Hao Tian
OxideMEMS lab, Purdue University, West Lafayette, IN, USA  Sunil A. Bhave
<EMAIL_ADDRESS>OxideMEMS lab, Purdue University, West Lafayette, IN, USA
Tobias J. Kippenberg<EMAIL_ADDRESS>Institute of Physics, Swiss
Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center of Quantum Science and Engineering (EPFL), CH-1015 Lausanne,
Switzerland

###### Abstract

Coherent interconversion between microwave and optical frequencies can serve
as both classical and quantum interfaces for computing, communication, and
sensing. Here, we present a compact microwave-optical transducer based on
monolithic integration of piezoelectric actuators atop silicon nitride
photonic circuits. Such an actuator directly couples microwave signals to a
high-overtone bulk acoustic resonator defined by the suspended silica cladding
of the optical waveguide core, which leads to enhanced electromechanical and
optomechanical couplings. At room temperature, this triply resonant piezo-
optomechanical transducer achieves an off-chip photon number conversion
efficiency of $1.6\times 10^{-5}$ over a bandwidth of
$25\text{\,}\mathrm{MHz}$ at an input pump power of $21\text{\,}\mathrm{dBm}$.
The approach is scalable in manufacturing and, unlike existing electro-optic
transducers, does not rely on superconducting resonators. As the transduction
process is bidirectional, we further demonstrate synthesis of microwave pulses
from a purely optical input. Combined with the capability of leveraging
multiple acoustic modes for transduction, the present platform offers
prospects for building frequency-multiplexed qubit interconnects and for
microwave photonics at large.

##  I Introduction

Modern data centers have seen rapidly increasing traffic that motivates an
overhaul of existing network infrastructures. As optical fibers support nearly
lossless transport and high bandwidths, twisted-pair copper deployment has
shrunk in favor of optical interconnects. Energy-efficient optical
transceivers [leeEnergyEfficiencyOptical2012] and optical network
architectures [chengRecentAdvancesOptical2018] are being explored in parallel
to accommodate emerging data- and resource-intensive applications. In an
analogous fashion, processing quantum information in superconducting circuits
and networking via photonic interconnects has been envisioned as an effective
strategy to address the scalability challenges in advancing quantum
technologies [ciracQuantumStateTransfer1997,
duanLongdistanceQuantumCommunication2001,
awschalomDevelopmentQuantumInterconnects2021]. The scheme, featuring the full
universal set of microwave quantum gates [chowUniversalQuantumGate2012] with
vanishing thermal occupancy and loss of the optical channels, calls for the
development of microwave-optical transducers to bridge the two energy scales
that differ by more than four orders of magnitude. Aside from facilitating the
networking of remote quantum processors, these transducers may enable fully
optical control and readout of microwave qubits
[lecocqControlReadoutSuperconducting2021,
youssefiCryogenicElectroopticInterconnect2021]. The subsequent replacement of
coaxial lines bridging room temperature and cryogenic environments by optical
fibers is expected to significantly ease the space and heat load constraints
in dilution refrigerators, opening up a path toward upscaling processor units
housed in a single fridge.

Efficient frequency conversion requires a nonlinear interaction stronger than
the coupling to loss channels. The highest conversion efficiency to date has
been achieved using an electro-optomechanical approach
[andrewsBidirectionalEfficientConversion2014]. There, a silicon nitride
membrane interacts with an optical mode of a free-space Fabry-Pérot cavity via
radiation pressure and simultaneously serves as the top plate of a capacitor,
parametrically coupling the mechanics to the microwave resonator. By virtue of
the high resonator quality factors and pump power handling capability, $47$%
of input photons can be interconverted between optical and microwave domains
[higginbothamHarnessingElectroopticCorrelations2018], approaching the $50$%
efficiency required for attaining finite quantum capacity
[wolfQuantumCapacitiesBosonic2007]. This highly efficient transducer has
enabled optical dispersive qubit readout with negligible excess backaction
[delaneySuperconductingqubitReadoutLowbackaction2022]. The low-frequency
($\text{\,}\mathrm{MHz}$) mechanical intermediary nevertheless limits the
transduction bandwidth and leads to appreciable added noise even at dilution
refrigerator temperature. To address these drawbacks, piezo-optomechanical
transducers based on optomechanical crystals (OMC) have been developed
[bochmannNanomechanicalCouplingMicrowave2013,
vainsencherBidirectionalConversionMicrowave2016,
jiangLithiumNiobatePiezooptomechanical2019,
forschMicrowavetoopticsConversionUsing2020,
jiangEfficientBidirectionalPiezooptomechanical2020,
mirhosseiniSuperconductingQubitOptical2020,
honlMicrowavetoopticalConversionGallium2022,
jiangOpticallyHeraldedMicrowave2023,
weaverIntegratedMicrowavetoopticsInterface2022,
meesalaNonclassicalMicrowaveopticalPhoton2023], where a tightly confined high-
frequency ($\text{\,}\mathrm{GHz}$) mechanical mode and a co-localized optical
mode can interact at a vacuum optomechanical coupling rate $g_{0}\sim
2\pi\times 500$ kHz. The trade-off lies instead in the sophistication required
for microwave-phonon wave matching that significantly increases design
complexity, as well as thermo-optic instability that constrains the intra-
cavity photon number. An on-chip efficiency of $5$% has recently been reported
on such a platform [jiangOpticallyHeraldedMicrowave2023]. However, low thermal
conductance of these suspended quasi-one-dimensional structures hinders
correlated microwave-optical photon pair generation at a practical rate.

Cavity electro-optic modulators constitute a conceptually simpler approach
where Pockels effect directly mediates microwave-optical interaction
[tsangCavityQuantumElectrooptics2010, tsangCavityQuantumElectrooptics2011,
javerzac-galyOnchipMicrowavetoopticalQuantum2016]. On-chip realizations
employing planar superconducting microwave resonators, benefited from the deep
sub-wavelength mode volume of the vacuum electric field, to reach single-
photon electro-optic coupling rates $g_{0}\sim 2\pi\times 1$ kHz
[fanSuperconductingCavityElectrooptics2018,
holzgrafeCavityElectroopticsThinfilm2020,
mckennaCryogenicMicrowavetoopticalConversion2020,
xuBidirectionalInterconversionMicrowave2021]. However, the material science of
$\chi^{(2)}$ crystals poses additional challenges. Photorefractive
effects—observed, for instance, in LiNbO3 thin films
[liPhotonlevelTuningPhotonic2019]—hamper optical power handling, while
piezoelectric loss and scattered optical photons degrade the quality factor of
superconducting resonators [dinizIntrinsicPhotonLoss2020,
xuLightInducedDynamicFrequency2022]. The difficulty of producing smooth
sidewall surfaces in the workhorse Pockels material, LiNbO3, through dry
etching results in propagation losses an order of magnitude above the
absorption limit [zhuIntegratedPhotonicsThinfilm2021]. As a result, the
maximum on-chip efficiency achieved with integrated electro-optic transducers
exceeds just $2$% [fanSuperconductingCavityElectrooptics2018]. In the spirit
of Ref. [andrewsBidirectionalEfficientConversion2014], a bulk transducer
comprising a mm-size mechanically polished LiNbO3 whispering gallery mode
(WGM) resonator coupled to a 3D superconducting microwave cavity has proved
competitive, trading $g_{0}$ for improved power handling and quality factors
[ruedaEfficientMicrowaveOptical2016,
heaseBidirectionalElectroOpticWavelength2020]. Taking one step further with
pulsed optical pumping, the system has demonstrated not only a hallmark
$14.4$% total efficiency but also electro-optic dynamical backaction
[sahuQuantumenabledOperationMicrowaveoptical2022,
qiuCoherentOpticalControl2023] and microwave-optical quadrature entanglement
[sahuEntanglingMicrowavesLight2023].

With the aforementioned design trade-offs in mind, we present a new piezo-
optomechanical transducer based solely on wafer-scale, CMOS-compatible
fabrication processes. We utilize a low-loss silicon nitride (Si3N4) photonic
molecule, as well as multiple $\text{\,}\mathrm{GHz}$ high-overtone bulk
acoustic resonances (HBAR) parametrically coupled to the optical modes.
Endowed with power handling capabilities superior to existing integrated
transducers, the device fits compactly within a
$100\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-by-$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ footprint, in contrast to
state-of-the-art bulk designs
[higginbothamHarnessingElectroopticCorrelations2018,
sahuQuantumenabledOperationMicrowaveoptical2022]. In addition, the
transduction HBAR modes are more readily coupled to the microwave signal,
unlike OMC transducers. We demonstrate bidirectional microwave-optical
transduction with a bandwidth of $25\text{\,}\mathrm{MHz}$ and total
efficiency up to $1.6\times 10^{-5}$ by pumping with $21\text{\,}\mathrm{dBm}$
of off-chip optical power in continuous-wave (CW) operation. The device is
also characterized with a pulsed optical pump, which constitutes the first
step toward optical control and readout of qubits, as well as heralded
microwave-optical photon pair generation. The simple design, ease of
fabrication, robust operation, and compact form factor anticipate wide
applicability in quantum technologies and microwave photonics at large.

##  II Results

###  II.1 Physics and design

Figure 1 delineates the principle of operation of the present transducer,
while the theoretical formalism is detailed in Appendix A. The requisite
nonlinear interaction for microwave-optical transduction is a parametric
three-wave mixing process described by standard optomechanical Hamiltonian

$\hat{\mathcal{H}}_{\mathrm{int}}=-\hbar
g_{0}\hat{a}^{\dagger}\hat{a}\left(\hat{b}+\hat{b}^{\dagger}\right).$

Through the bilinear piezoelectric interaction, the acoustic resonance
($\hat{b}$) is coupled to the itinerant transmission line mode
($\hat{c}_{\mathrm{in}}$) of the same frequency
$\omega_{\mathrm{in}}=\omega_{m}\approx$3.5\text{\,}\mathrm{GHz}$$. This
frequency matches the detuning of the optical pump from the optical resonance
($\hat{a}$). Specifically, an optical pump addressing the red (blue) side of
the resonance induces an effective beam-splitter (two-mode-squeezing)
interaction between $\hat{a}$ and $\hat{b}$, as illustrated by the signal flow
graph in Fig. 1a (1b). The primary figure of merit for a transducer is its
efficiency, defined as the number of output photon number for each input
photon. An efficient device necessitates strong nonlinear interaction between
the modes of interest, as well as ease of coupling to these modes internal to
the device. Our design addresses these two aspects. First, the internal
efficiency,

$\eta^{\mathrm{int}}_{\pm}=\frac{4C}{\left(1\pm C\right)^{2}},$ (1)

depends solely on the three-wave mixing cooperativity
$C=4g_{0}^{2}\bar{n}/\left(\kappa_{o}\kappa_{m}\right)$. Here, $\bar{n}$,
$\kappa_{o}$, and $\kappa_{m}$ denote the photon number in the optical cavity,
and the total optical and acoustic linewidths of the transduction modes
respectively. The single photon optomechanical coupling rate
$g_{0}=-\left(\partial\omega_{a}/\partial x\right)x_{\mathrm{ZPF}}$, a product
of the optical cavity frequency-pull parameter and zero-point displacement of
the acoustic wave. In Eq. 1, the plus and minus signs in the denominator
correspond to the scenario where the pump is red- and blue-detuned (“anti-
Stokes” and “Stokes” processes), respectively. Using a triply resonant
configuration as in Refs. [fanSuperconductingCavityElectrooptics2018,
holzgrafeCavityElectroopticsThinfilm2020,
mckennaCryogenicMicrowavetoopticalConversion2020,
xuBidirectionalInterconversionMicrowave2021] enhances the intracavity photon
number $\bar{n}$ for a given off-chip input pump power $P_{\mathrm{in}}$, or
corresponding photon flux
$\dot{n}_{\mathrm{in}}=P_{\mathrm{in}}/\left(\hbar\omega_{L}\right)$. The
improved photon pumping efficiency can be explicitly seen by considering a
“hot” cavity with coupling rate $\kappa_{\mathrm{ex},o}$ driven at a detuning
$\Delta=\pm\omega_{m}$, yielding

$\frac{\bar{n}}{\dot{n}_{\mathrm{in}}}=\frac{\kappa_{\mathrm{ex},a}}{\kappa_{o}^{2}/4+\Delta^{2}}.$

Introducing an additional cavity mode centered at the pump frequency
$\omega_{L}$ makes $\Delta=0$, leading to an $\bar{n}$ enhanced by
$\mathcal{O}\left(\omega_{m}^{2}/\kappa_{o}^{2}\right)=\mathcal{O}(10^{3})$
and a significantly reduced laser power overhead required for a given
$\eta^{\mathrm{int}}$.

The on-chip conversion efficiency $\eta^{\mathrm{oc}}$ is related to
$\eta^{\mathrm{int}}$ by a proportionality constant

$\eta^{\mathrm{ext}}=\frac{\kappa_{\mathrm{ex},o}}{\kappa_{o}}\frac{\kappa_{\mathrm{ex},m}}{\kappa_{m}}$
(2)

such that $\eta^{\mathrm{oc}}=\eta^{\mathrm{ext}}\eta^{\mathrm{int}}$. This
extraction efficiency denotes the fraction by which optical and microwave
photons are emitted out of the resonator modes with rates
$\kappa_{\mathrm{ex},o}$ and $\kappa_{\mathrm{ex},m}$. Our strategy of
optimizing $\eta^{\mathrm{ext}}$ is most directly seen in the device
implementation, illustrated in Fig. 1d and proposed in Ref.
[blesinQuantumCoherentMicrowaveoptical2021]. The vertical stack comprises a
piezoelectric actuator, a Si3N4 photonic waveguide layer, and a silica (SiO2)
cladding that also serves as the acoustic cavity
[tianHybridIntegratedPhotonics2020, tianMagneticfreeSiliconNitride2021]. The
actuator, deposited on a standard silicon wafer after fabrication of the
optical circuitry using the photonic Damascene process, is composed of a
$c$-cut aluminum nitride (AlN) thin film sandwiched between top and bottom
molybdenum (Mo) electrodes on which the microwave signal is applied. The
alternating electric field launches longitudinal bulk acoustic waves (BAWs)
into the cladding layer, which is suspended by etching away the silicon
substrate to confine HBARs. Since the AlN is only located above the suspended
cladding, the HBAR modes do not extend into the Si substrate (Appendices B and
D). These cavity-enhanced BAWs modify the waveguide refractive index through
photoelastic and moving boundary effects, realizing the three-wave mixing
interaction with rate $g_{0}$ [blesinQuantumCoherentMicrowaveoptical2021]. The
stack composition is illustrated in Fig. 1d. Resonances that simultaneously
satisfy the interfacial acoustic boundary conditions defined by the actuator
and the whole stack naturally feature sizable $\kappa_{\mathrm{ex},m}$. The
simplicity of our design is in stark contrast to piezo-optomechanical
transducers based on OMCs. To facilitate efficient microwave extraction,
design challenges need to be overcome to achieve strong hybridization between
the piezo mode supported by the phonon waveguide and the transduction mode of
the OMC cavity.

Central to both $\eta^{\mathrm{int}}$ and $\eta^{\mathrm{ext}}$ are the
resonator intrinsic quality factors, as seen in Eqs. 1 and 2. Unlike
superconducting microwave resonators, HBARs based on amorphous SiO2 have been
shown to exhibit sufficiently low (acoustic) loss at room temperature,
relaxing the requirement for cryogenic operation
[tianHybridIntegratedPhotonics2020]. A photonic molecule comprising a pair of
evanescently coupled micro-rings can be hybridized into a symmetric mode
($\hat{a}_{-}$) and an antisymmetric mode ($\hat{a}_{+}$), which in turn make
up the optical portion of the triple resonance system. The waveguides are
fabricated using the photonic Damascene process with wafer-scale yield
[pfeifferUltrasmoothSiliconNitride2018,
liuHighyieldWaferscaleFabrication2021]. The thick Si3N4 core reduces bending
loss, allowing us to employ micro-rings with a radius (free spectral range) of
$22\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($1\text{\,}\mathrm{THz}$). As
illustrated in Figs. 1c and 2a, the micro-ring defines the dimensions of the
piezoelectric actuator. Reducing the micro-ring size therefore results in a
reduced acoustic mode volume, which serves to increase
$x_{\mathrm{ZPF}}\propto 1/\sqrt{V_{m}}$ and hence $g_{0}$.

###  II.2 Device characterization

We perform metrological characterization to affirm the integrity of the stack.
Shown in Fig. 2a, the region of suspended cladding is first identified through
optical micrography; the released substrate below leads to a contrast in the
image. To further access the layer structure, we use focused ion beam milling
to create an opening on the device surface at the suspension site. This
opening provides sufficient clearance to directly image the cladding acoustic
resonator, pictured in Fig. 2b, confirming the removal of silicon.

We then characterize the transducer as an optoelectronic network with one
microwave port, two optical ports, and one auxiliary electrical DC port. Each
optical port comprises an edge-coupled lensed fiber and on-chip bus waveguide
terminated by 1D nanotapers designed for TE-polarized light, yielding a fiber-
to-fiber coupling efficiency of $-8\text{\,}\mathrm{dB}$. For adjusting the
frequency splitting between optical supermodes, a bias voltage is applied to
the DC port to drive either the piezoelectric actuator
[liuMonolithicPiezoelectricControl2020] or an integrated thermo-optic heater
placed in the vicinity of the micro-rings (Appendix C.2). The former affords
no additional heat load and is hence cryogenic-compatible, whereas the latter
provides an easy alternative for fast room-temperature characterization. The
avoided mode crossing characteristic of a photonic molecule is exemplified in
Fig. 2c. We access the HBARs through the microwave port. Calibrated high-
frequency probes are used to contact the top and bottom electrode pads of the
piezoelectric actuator, effectively linking the acoustic resonator with a
transmission line. The reflection spectrum (Fig. 2d) reveals predominantly one
single series of HBARs with a free spectral range (FSR) of
$320\text{\,}\mathrm{MHz}$, which corresponds to the acoustic length of the
cladding. The transduction acoustic mode at
$3.48\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ exhibits a typical total
linewidth of $13\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ with a microwave
extraction efficiency of $\kappa_{\mathrm{ex},m}/\kappa_{m}\approx 11\%$. The
microwave response is distinct from that of an identical stack composition
with unreleased substrate. There, a periodic envelope corresponding to the
cladding modes that are more strongly coupled to the microwave is superimposed
over the full-stack HBAR response (Appendix D and Ref.
[tianHybridIntegratedPhotonics2020]). The absence of these full-stack modes
provide another piece of evidence of successful cladding suspension. Finally,
we study the device acousto-optic response in the triply resonant
configuration, where the transducer effectively operates as a resonant single-
sideband modulator. The optical output containing both the pump at the
symmetric-mode frequency and generated sideband at the antisymmetric-mode
frequency are mixed by a photodetector. Such a beat-note spectrum shown in
Fig. 2e displays response peaks aligned with the HBAR frequencies in Fig. 2d,
demonstrating three-wave mixing.

###  II.3 Bidirectional microwave-optical transduction

We measure the off-chip efficiency $\eta^{\mathrm{tot}}$ of the microwave-to-
optical (up-conversion) and optical-to-microwave (down-conversion) processes
as a function of off-chip CW input optical pump power $P_{\mathrm{in}}$.
Compared to the on-chip efficiency $\eta^{\mathrm{oc}}$, this efficiency also
accounts for the loss channels of the microwave and optical ports. First, we
study the transducer in the triply resonant configuration. For up-conversion,
the optical sideband generated from the microwave input, detuned by
$\omega_{\mathrm{in}}=\omega_{m}$ from the pump frequency, is measured via a
self-calibrated heterodyne detection method. The off-chip optical output
containing both the pump ($\omega_{L}$) and the sideband ($\omega_{s}$) is
combined with a local oscillator (LO; $\omega_{\mathrm{LO}}$) and detected. By
placing the LO at a frequency
$\omega_{\mathrm{LO}}=\left(\omega_{L}+\omega_{s}\right)/2+\delta$ such that
the photodetector response is constant over a frequency span of $\delta$, we
determine the sideband power relative to the pump. Picking off a fraction of
the optical output then enables the determination of its absolute power and,
by extension, that of the sideband with a power meter. The measured optical
sideband power is compared to that of the microwave input to yield
$\eta^{\mathrm{tot}}$. In the case of down-conversion, the optical input is
generated by modulating the phase of the pump with an electro-optic modulator
(EOM) driven by a microwave source of frequency
$\omega_{\mathrm{EOM}}=\omega_{m}$. Only one of the resulting sidebands is
admitted into the photonic molecule and transduced, as the other is far off-
resonance. The converted microwave power is directly probed with an electrical
spectrum analyzer (ESA). Summarized in Figs. 3a and c, bidirectional
transduction processes corresponding to both the effective beam-splitter
($\omega_{L}=\omega_{-}$; anti-Stokes) and two-mode-squeezing
($\omega_{L}=\omega_{+}$; Stokes) interactions are investigated. We reach a
maximal $\eta^{\mathrm{tot}}$ of $-48\text{\,}\mathrm{dB}$ at
$21\text{\,}\mathrm{dBm}$ pump for each configuration. Knowing the port
losses, we estimate an on-chip efficiency
$\eta^{\mathrm{oc}}=$-41\text{\,}\mathrm{dB}$$ (Appendix E). Accounting for
the extraction efficiency $\eta^{\mathrm{ext}}$, we further obtain an internal
conversion efficiency $\eta^{\mathrm{int}}=$2\text{\times}{10}^{-3}\text{\,}$$
from Eq. 2.

Furthermore, we deviate from triple resonance to map out the transduction
bandwidth. First, for a pump still resonant with one of the optical
supermodes, the input microwave frequency $\omega_{\mathrm{in}}$ and input
optical detuning (controlled by EOM drive frequency $\omega_{\mathrm{EOM}}$)
are varied for up- and down-conversion, respectively. Shown in Fig. 3b and d,
transduction leveraging the main transduction mode exhibits a full width at
half maximum (FWHM) of $25\text{\,}\mathrm{MHz}$. The multimode nature of the
transducer is further manifest in Fig. 3e, where the pump is in the beam-
splitter configuration but slightly detuned from $\hat{a}_{-}$. An additional
transduction peak in $\eta^{\mathrm{tot}}$ with a FWHM of
$10\text{\,}\mathrm{MHz}$ is observed around
$\omega_{\mathrm{in}}=2\pi\times$3.165\text{\,}\mathrm{GHz}$$, which
corresponds to another HBAR one FSR away from the main transduction mode
$\omega_{m}=2\pi\times$3.480\text{\,}\mathrm{GHz}$$. The engineering degrees
of freedom such as cladding and actuator thickness (HBAR FSR), and optical
dispersion (supermode splitting as a function of optical wavelength) in the
present system offer possibilities for frequency-multiplexed transduction.

###  II.4 Pulsed transduction

Quantum-enabled operation, where the added noise referred to the transducer
input is less than one quanta, would require concurrently attaining high
conversion efficiency and low output noise. As the three-wave mixing
interaction is parametric in nature, the associated cooperativity and hence
efficiency can in principle be enhanced by boosting the optical pump power. A
potential trade-off is nevertheless the subsequently increased thermal noise.
While the high resonance frequency of the transduction HBAR already helps
suppressing this noise to some extent, additionally employing a pulsed pump
presents a number of utilities. Reducing the integrated optical power serves
as an effective measure to mitigate thermal load onto the cryostat as well as
pump-induced noise while maintaining high peak power. Quantum-enabled
operation has in fact been achieved with this strategy by a bulk electro-optic
transducer employing Watt-scale optical pumping
[sahuQuantumenabledOperationMicrowaveoptical2022]. Gate-based superconducting
quantum computers too function inherently in the pulsed regime. As such, we
characterize our transducer in both frequency and time domains with a pulsed
optical pump to evaluate its compatibility with these cryogenic microwave
circuits as well as its potential for quantum-enabled operation.

To study pulsed bidirectional transduction, we choose a pulse-on time
$\tau_{\mathrm{on}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and a
repetition rate $f_{\mathrm{rep}}=$100\text{\,}\mathrm{kHz}$$. We program the
pulse sequence such that the increase in temperature, and hence acoustic mode
thermal occupancy due to optical heating, is expected to be inconsequential
(Appendix C.3). First, we measure the up-conversion efficiency, summarized in
Fig. 4a, using the same heterodyne method as in the CW case (Fig. 3). The
pulsed optical pump mediates transduction of a CW microwave input into a
pulsed optical output. This optical pulse comprising both the up-converted
output and pump is then down-mixed by the LO, resulting in a microwave pulse
whose frequency content can be probed via an ESA. The measured efficiency
exhibits reasonable agreement with CW data. Additionally, the down-converted
microwave pulse envelope in the time domain provides an independent measure of
the transducer bandwidth—a key metric for qubit-photonic interconnects
[lecocqControlReadoutSuperconducting2021,
youssefiCryogenicElectroopticInterconnect2021]. With the pulsed pump set in
the two-mode squeezing configuration, an optical input pulse is converted on-
chip into a microwave pulse with a carrier frequency
$\omega_{\mathrm{EOM}}=\omega_{m}$ through difference frequency generation.
The time-domain dynamics of the pulse envelope is captured through phase-
sensitive demodulation using a digital lock-in amplifier. As shown in Fig. 4b,
we observe that the pulse envelope is consistent with the step response of the
lock-in integrator, which has a bandwidth of $5\text{\,}\mathrm{MHz}$, or an
RC time constant $\tau_{\mathrm{RC}}=$30\text{\,}\mathrm{ns}$$. The lock-in
bandwidth therefore sets a lower bound for the transducer bandwidth.

##  III Discussion

Non-classically correlated microwave-optical photon pairs can be generated
through spontaneous parametric down-conversion in our transducer. These
correlated photon pairs constitute a key ingredient toward entangling distant
quantum processors via the Duan-Lukin-Cirac-Zoller protocol
[duanLongdistanceQuantumCommunication2001,
krastanovOpticallyHeraldedEntanglement2021]. With the current device
parameters, the pair generation rate on chip amounts to
$190\text{\,}\mathrm{Hz}$, which is well above the thermal decoherence rate
for $3.5\text{\,}\mathrm{GHz}$ at $10\text{\,}\mathrm{mK}$ as explained in
Appendix A.4. However, losses in the measurement setup specific to such an
experiment, as well as the gating of the optical pump required to alleviate
the effective heat load, are expected to limit further the final heralding
rate. Quasi-free-standing structures, such as 1D OMCs, do not readily
thermalize and are thus more susceptible to heating effects. Even though our
transducer also utilizes a suspended acoustic resonator, it may be feasible to
operate at a higher duty cycle than what is presented in Fig. 4. It has been
shown that a buffer gas environment can facilitate thermalization of a 1D OMC,
at the cost of increased damping of the mechanical breathing mode
[shomroniOpticalBackactionevadingMeasurement2019]. On the contrary, the HBAR
mode of interest here should be relatively insensitive to such viscous
damping, as the acoustic wave propagates predominantly along the longitudinal
direction inside the cladding. Therefore, a sample cell affixed to the mixing
chamber flange and filled with buffer superfluid helium—an inviscid fluid that
is both an excellent thermal conductor and electrical insulator—could
potentially improve the heralding rate without hampering the conversion
efficiency [kashkanovaOptomechanicsSuperfluidHelium2017].

The present design is also suited for tasks beyond quantum state transfer. The
transducer can be a key component of photonic interconnects for
superconducting qubits [lecocqControlReadoutSuperconducting2021,
youssefiCryogenicElectroopticInterconnect2021], where laser lights routed
through optical fibers are used to encode microwave signals directly inside
the dilution refrigerator during qubit readout. It may be challenging for
traveling-wave EOMs to attain the requisite half-wave voltage to be
competitive in noise performance against conventional all-electrical readout
scheme utilizing high-electron-mobility transistor amplifiers
[youssefiCryogenicElectroopticInterconnect2021]. Our triply resonant approach
affords a path towards a scalable optically controlled cryogenic waveform
generator for both qubit readout and control. The multimode nature of the
transducer is particularly suited for frequency-multiplexed dispersive
readout. Transduction leveraging multiple HBAR modes is already demonstrated
in Fig. 3e. One can additionally utilize line-type instead of point-type dimer
coupling for the photonic molecule, which gives rise to a dispersion in the
supermode splitting [kimDispersionEngineeringFrequency2017]. In combination
with the multitude of acoustic overtones, a single transducer can therefore
support several spectrally distinguishable triply resonant systems that serve
as qubit multiplex channels. Finally, since the transducer requires no
superconducting element to function, it may be pertinent for applications in
classical microwave photonics.

Optimization of design and fabrication should engender further improvement on
device performance. The conversion efficiency, proportional to microwave
extraction efficiency (Eq. 2), can be improved by replacing AlN with scandium-
doped AlN [akiyamaEnhancementPiezoelectricResponse2009], which features a
piezoelectric coefficient $d_{33}$ about five times larger than AlN
[akiyamaInfluenceGrowthTemperature2009,
yanagitaniElectromechanicalCouplingGigahertz2014] while preserving CMOS
compatibility. There is likewise additional upside on the photonics (Appendix
C.1). Bending radiation loss can be significantly reduced by increasing the
micro-ring radius $r$. In particular, intrinsic quality factor exceeding
$10^{7}$ has been demonstrated with an $r$ approximately ten times larger
using the same fabrication process [liuUltralowpowerChipbasedSoliton2018]. It
may be anticipated that the efficiency gain here may be chiefly compensated by
the reduction in optomechanical coupling rate $g_{0}\sim 1/\sqrt{r}$.
Nevertheless, by thinning the waveguide to $200\text{\,}\mathrm{nm}$ that
approaches half of the acoustic wavelength, $g_{0}$ is expected to remain
unchanged. As in the thick-core waveguide fabricated using the photonic
Damascene process, such a thin-core waveguide on a subtractive manufacturing
platform is expected to exhibit negligible bending loss
[siddharthHertzlinewidthFrequencyagilePhotonic2023]. Assuming critical
coupling conditions (Appendix F), we therefore expect another 1.5-time gain in
$\eta^{\mathrm{tot}}$. Finally, optical insertion loss can be nearly
eliminated by employing a spot-size converter to facilitate mode-matching with
lensed fiber [zhuUltrabroadbandHighCoupling2016,
yaoBroadbandHighefficiencyTripletip2020, bhandariCompactBroadbandEdge2020].
Combining these improvements, our transducer could achieve a total efficiency
$\eta^{\mathrm{tot}}=$6\text{\times}{10}^{-3}\text{\,}$$.

In conclusion, we have designed, fabricated, and characterized a compact
piezo-optomechanical microwave-optical transducer that integrates wafer-scale,
CMOS-compatible HBAR and Si3N4 photonic technologies. Free of any
superconducting components, this triply resonant transducer attains a
bidirectional off-chip photon number conversion efficiency of $1.6\times
10^{-5}$ and a bandwidth of $25\text{\,}\mathrm{MHz}$, with an input pump
power of $21\text{\,}\mathrm{dBm}$ at room temperature. Multimode transduction
leveraging distinct HBAR modes have been demonstrated, suggesting prospects
for designing frequency-multiplexed photonic interconnects for superconducting
qubits. Realistic design and fabrication improvements may even allow access to
experiments in the quantum regime.

## Acknowledgments

The authors would like to thank Amirali Arabmoheghi, Nils Johan Engelsen and
Junyin Zhang for experimental assistance and fruitful discussions. This work
was supported by the Air Force Office of Scientific Research under award no.
FA8655-20-1-7009, Swiss National Science Foundation under grant agreement no.
204927, and European Research Council (ERC) under the EU H2020 research and
innovation programme, grant agreement no. 835329 (ExCOM-cCEO). This work was
further supported by United States Air Force Research Laboratory Award no.
FA8750-21-2-0500, as well as United States National Science Foundation’s
International Collaboration Supplements in Quantum Information Science and
Engineering Research for RAISE-TAQS Award no. 18-39164. A.S. acknowledges
support from the European Space Technology Centre with ESA Contract No.
4000135357/21/NL/GLC/my. Samples were fabricated in the Center of
MicroNanoTechnology (CMi) at EPFL and Birck Nanotechnology Center at Purdue
University.

## Author contributions

T.B. designed the device. R.N.W, A.A. and H.T. fabricated the device. T.B.,
A.S. and W.K. conducted the experiments. T.B. and W.K. wrote the paper with
input from all authors. T.J.K. and S.A.B. supervised the project.

## References

Figure 1:  Triply resonant piezo-optomechanical system. a,b Signal flow graphs
illustrating the linearized equations of motion of the system, where the
optical pump at frequency $\omega_{L}$ is set to the anti-Stokes and Stokes
configurations, respectively. The optical and acoustic modes with
susceptibilities $\chi$ are coupled with (multi-photon) rates
$g_{\pm}=g_{0,\pm}\alpha$, where $\alpha$ is the pump optical field amplitude.
Photons are emitted into (out of) the itinerant modes of input optical fiber
$\hat{a}_{\mathrm{out}}$ ($\hat{a}_{\mathrm{in}}$) and microwave transmission
line $\hat{c}_{\mathrm{out}}$ ($\hat{c}_{\mathrm{in}}$) at rates
$\sqrt{\kappa_{\mathrm{ex}}}$. The acoustic and optical modes employed for
transduction based on cavity-enhanced three-wave mixing are shown in frequency
domain on the left. c Experimental realization of the transducer (not to
scale), with only one micro-ring of the photonic molecule in view. A
piezoelectric actuator (electrodes in orange and yellow; piezoelectric thin
film in green) is integrated atop the photonic circuits (purple), allowing the
acoustic modes supported by the suspended cladding (light grey) to be coupled
to the microwave transmission line. The optical and microwave ports are
labeled by wavelet cartoons. d Stack composition and finite-element simulation
of the transduction process. Both the mechanical stress pattern of the
acoustic mode and the optical TE mode of the micro-ring are shown in the cross
section.  Figure 2:  Device metrological and network characterization. a
Optical micrograph of the transducer. The shaded region originates from the
released silicon substrate. b False-color cross section of the device imaged
by focused ion beam scanning electron microscopy showing the top and bottom
electrodes (Mo; yellow), piezoelectric layer (AlN; green), suspended cladding
(SiO2; gray), and optical waveguides (Si3N4; purple). c Optical transmission
spectra near $1550\text{\,}\mathrm{nm}$ as a function of applied DC
piezoelectric control voltage and detuning from the antisymmetric supermode.
The transmission color map follows the representative spectrum in the bottom
panel. The dashed line denotes the best-fit transmission to coupled mode
theory (Appendix A.1). d Microwave reflection spectrum indicating locations of
the HBARs. e Acousto-optic response spectrum with an off-chip optical pump
power $P_{\mathrm{in}}=$16\text{\,}\mathrm{dBm}$$ centered at the symmetric
supermode.  Figure 3:  Bidirectional, multimode transduction with a
continuous-wave pump. a, b Anti-Stokes transduction. c, d Stokes transduction.
Off-chip efficiencies as a function of off-chip input pump power in the triply
resonant configuration are shown in the left panels. Off-chip efficiencies as
a function of input microwave frequency and input optical detuning are plotted
in the right panels, which depict the transduction bandwidth. e Bidirectional
transduction with a detuned pump leveraging multiple HBAR modes, as a function
of input microwave frequency and input optical detuning. Up- and down-
conversion efficiencies are denoted by triangles and inverted triangles,
respectively, with the corresponding acoustic-optic response spectrum (as in
Fig. 2e) superimposed. The markers are empty for Stokes processes and filled
for anti-Stokes processes. The dotted lines represent the best-fit efficiency
in the low-cooperativity regime where $\eta^{\mathrm{tot}}\propto C\propto
P_{\mathrm{in}}$. The insets illustrate the respective configurations for both
the pump and input signal.  Figure 4:  Pulsed transduction. a Pulsed up-
conversion as a function of off-chip input pump power for a pulse-on time of
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and repetition rate of
$100\text{\,}\mathrm{kHz}$. The pump is in the Stokes (empty triangles) and
anti-Stokes (filled triangles) configurations. The best-fit efficiency (dotted
line) for CW operation from Fig. 3c is shown for comparison. b Down-converted
microwave pulse in the time domain with a $15\text{\text{-}}\mathrm{dBm}$ pump
set to the Stokes configuration, detected as demodulated amplitude and phase
averaged over 6000 shots. The dashed and dash-dotted lines are fits to the RC
step response function that physically corresponds to the lock-in integrator,
which has a bandwidth (RC time constant $\tau_{\mathrm{RC}}$) of
$5\text{\,}\mathrm{MHz}$ ($30\text{\,}\mathrm{ns}$). The amplitude and phase
response yields a best-fit $\tau_{\mathrm{RC}}=$35(4)\text{\,}\mathrm{ns}$$
and $27(3)\text{\,}\mathrm{ns}$, respectively. These values are consistent
with the lock-in bandwidth. The insets illustrate the respective
configurations for the optical pump (thick black), input signal (black) and
measured output (colored). The pulse sequence is shown in time domain.

##  Appendix A Theory

###  A.1 Coupled mode theory for optical modes hybridization

Using a pair of optical resonances increases the intracavity photon number for
a given input optical power, resulting in an enhanced optomechanical
interaction. Such an optical doublet is implemented here by evanescently
coupling two micro-rings, forming a photonic molecule. Consider two bare
optical modes with resonance frequencies $\omega_{l}$ and $\omega_{r}$ that
correspond to the left and right micro-rings, respectively. The two modes
couple with a rate $J$, which yields the Hamiltonian
[hausCoupledmodeTheory1991]

$\hat{\mathcal{H}}=\hbar\omega_{l}\hat{a}_{l}^{\dagger}\hat{a}_{l}+\hbar\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}-\hbar
J\left(\hat{a}_{l}\hat{a}_{r}^{\dagger}+\hat{a}_{l}^{\dagger}\hat{a}_{r}\right)$
(3)

and the equations of motion

$\displaystyle\frac{d}{dt}\hat{a}_{l}(t)$
$\displaystyle=\left(-i\omega_{l}-\frac{\kappa_{l}}{2}\right)\hat{a}_{l}(t)+iJ\hat{a}_{r}(t),$
(4) $\displaystyle\frac{d}{dt}\hat{a}_{r}(t)$
$\displaystyle=\left(-i\omega_{r}-\frac{\kappa_{r}}{2}\right)\hat{a}_{r}(t)+iJ\hat{a}_{l}(t).$

With Laplace transform, they can be rewritten in frequency domain as

$\begin{pmatrix}\chi_{l}^{-1}[s]&iJ\\\ iJ&\chi_{r}^{-1}[s]\\\
\end{pmatrix}\begin{pmatrix}\hat{a}_{l}[s]\\\
\hat{a}_{r}[s]\end{pmatrix}=\begin{pmatrix}\hat{a}_{l}(0)\\\
\hat{a}_{r}(0)\end{pmatrix},$ (5)

where $\chi_{o}[s]=\left(s+i\omega_{o}+\kappa_{o}/2\right)^{-1}$, $o=l,r$
denote the susceptibilities. Diagonalizing the matrix yields the eigenvalues

$\displaystyle\lambda_{\pm}$
$\displaystyle=\frac{\chi_{l}^{-1}+\chi_{r}^{-1}}{2}\pm\frac{1}{2}\sqrt{\left(\chi_{l}^{-1}-\chi_{r}^{-1}\right)^{2}-4J^{2}}$
$\displaystyle=\left(s+i\bar{\omega}+\frac{\bar{\kappa}}{2}\right)\pm\frac{1}{2}\sqrt{\frac{\mu^{2}}{4}-\delta^{2}+i\mu\delta-4J^{2}}.$
(6)

Here we define the average optical frequency
$\bar{\omega}=\left(\omega_{l}+\omega_{r}\right)/2$, average linewidth
$\bar{\kappa}=\left(\kappa_{l}+\kappa_{r}\right)/2$, relative detuning of the
micro-rings $\delta=\omega_{l}-\omega_{r}$, and linewidth difference
$\mu=\kappa_{l}-\kappa_{r}$. The difference in resonance frequency and
linewidth of the two supermodes can be computed as

$\displaystyle\Delta\omega$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{-\frac{\mu^{2}}{4}+\delta^{2}+4J^{2}+\sqrt{\left[\left(2J-\frac{\mu}{2}\right)^{2}+\delta^{2}\right]\left[\left(2J+\frac{\mu}{2}\right)^{2}+\delta^{2}\right]}},$
(7) $\displaystyle\Delta\kappa$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{+\frac{\mu^{2}}{4}-\delta^{2}-4J^{2}+\sqrt{\left[\left(2J-\frac{\mu}{2}\right)^{2}+\delta^{2}\right]\left[\left(2J+\frac{\mu}{2}\right)^{2}+\delta^{2}\right]}},$
(8)

where $\lambda_{+}-\lambda_{-}=\Delta\kappa+i\Delta\omega$. In the strong
coupling regime where $2J\gg\mu/2$, we simply have
$\Delta\omega\approx\sqrt{\delta^{2}+4J^{2}}$ and
$\Delta\kappa\approx\mu/\sqrt{8}$. The two supermodes are thus centered at
$\bar{\omega}$ with a splitting of $\Delta\omega$, resulting in
susceptibilites

$\chi_{\pm}[s]=\left[s+i\left(\bar{\omega}\pm\frac{\Delta\omega}{2}\right)+\frac{\bar{\kappa}\pm\Delta\kappa}{2}\right]^{-1}.$
(9)

Solving for the eigenvectors gives the participation ratios of the bare cavity
modes in the supermodes

$\displaystyle\alpha_{\pm}$
$\displaystyle=\frac{2J}{\sqrt{4J^{2}+|\mu\mp\Delta\kappa+i\left(\delta\mp\Delta\omega\right)|^{2}}}\exp\left(i\arg{\alpha_{\pm}}\right),$
(10) $\displaystyle\beta_{\pm}$
$\displaystyle=\frac{|\mu\mp\Delta\kappa+i\left(\delta\mp\Delta\omega\right)|}{\sqrt{4J^{2}+|\mu\mp\Delta\kappa+i\left(\delta\mp\Delta\omega\right)|^{2}}}\exp\left[i\arg{\alpha_{\pm}}+i\arg{\frac{\mu\mp\Delta\kappa+i\left(\delta\mp\Delta\omega\right)}{2iJ}}\right],$

where $|\alpha_{\pm}|^{2}+|\beta_{\pm}|^{2}=1$. In particular, the supermode
$\hat{a}_{-}=\alpha_{-}\hat{a}_{l}+\beta_{-}\hat{a}_{r}$ has a resonance
frequency $\omega_{-}=\bar{\omega}-\Delta\omega/2$ and linewidth
$\kappa_{-}=\bar{\kappa}-\Delta\kappa/2$. It is referred to as the symmetric
supermode. The rationale behind the nomenclature can be seen by considering
the limit of zero detuning $\delta=0$, where we simply have
$\hat{a}_{-}=\left(\hat{a}_{l}+\hat{a}_{r}\right)/\sqrt{2}$ with
$\omega_{-}=\bar{\omega}-J$ and $\kappa_{-}=\bar{\kappa}$. On the other hand,
the antisymmetric supermode
$\hat{a}_{+}=\alpha_{+}\hat{a}_{l}+\beta_{+}\hat{a}_{r}$ has a resonance
frequency $\omega_{+}=\bar{\omega}+\Delta\omega/2$ and linewidth
$\kappa_{+}=\bar{\kappa}+\Delta\kappa/2$. When $\delta=0$, this reduces to
$\hat{a}_{+}=\left(\hat{a}_{l}-\hat{a}_{r}\right)/\sqrt{2}$ with
$\omega_{+}=\bar{\omega}+J$ and $\kappa_{+}=\bar{\kappa}$.

###  A.2 Transducer Hamiltonian and input-output relations

In terms of bare micro-ring modes, the transducer Hamiltonian is given by

$\hat{\mathcal{H}}=\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{l}\hat{a}_{l}^{\dagger}\hat{a}_{l}+\hbar\omega_{r}\hat{a}_{r}^{\dagger}\hat{a}_{r}-\hbar
J\left(\hat{a}_{l}^{\dagger}\hat{a}_{r}+\hat{a}_{r}^{\dagger}\hat{a}_{l}\right)-\hbar
g_{0}\hat{a}_{l}^{\dagger}\hat{a}_{l}\left(\hat{b}+\hat{b}^{\dagger}\right)+\hat{\mathcal{H}}_{\mathrm{drive}}.$
(11)

Note that the piezo-coupled acoustic mode interacts with only one of the
micro-rings. To compute quantities directly related to the experiments, we
rewrite the Hamiltonian in terms of the supermodes (Appendix A.1) as

$\hat{\mathcal{H}}=\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{-}\hat{a}_{-}^{\dagger}\hat{a}_{-}+\hbar\omega_{+}\hat{a}_{+}^{\dagger}\hat{a}_{+}-\hbar
g_{0}\left(|x|^{2}\hat{a}_{-}^{\dagger}\hat{a}_{-}+|y|^{2}\hat{a}_{+}^{\dagger}\hat{a}_{+}+xy^{*}\hat{a}_{+}^{\dagger}\hat{a}_{-}+x^{*}y\hat{a}_{-}^{\dagger}\hat{a}_{+}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)+\hat{\mathcal{H}}_{\mathrm{drive}}$
(12)

such that $\hat{a}_{l}=x\hat{a}_{-}+y\hat{a}_{+}$. The driving term is given
by

$\displaystyle\hat{\mathcal{H}}_{\mathrm{drive}}=$ $\displaystyle
i\hbar\sqrt{\kappa_{\mathrm{ex,-}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{-}^{\dagger}e^{-i\omega_{L}t}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{-}e^{i\omega_{L}t}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,+}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{+}^{\dagger}e^{-i\omega_{L}t}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{+}e^{i\omega_{L}t}\right)$
(13) $\displaystyle+\
i\hbar\sqrt{\kappa_{\mathrm{ex,m}}}\left(\hat{c}_{\mathrm{in}}\hat{b}^{\dagger}e^{i\omega_{m}t}-\hat{c}_{\mathrm{in}}^{\dagger}\hat{b}e^{-i\omega_{m}t}\right)$

for input optical and microwave fields $\hat{a}_{\mathrm{in}}$ and
$\hat{c}_{\mathrm{in}}$. By going into the frames rotating at the laser drive
frequency $\omega_{L}$, we obtain

$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}-\hbar\Delta_{-}\hat{a}_{-}^{\dagger}\hat{a}_{-}-\hbar\Delta_{+}\hat{a}_{+}^{\dagger}\hat{a}_{+}$
(14) $\displaystyle-\hbar
g_{0}\left(|x|^{2}\hat{a}_{-}^{\dagger}\hat{a}_{-}+|y|^{2}\hat{a}_{+}^{\dagger}\hat{a}_{+}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)-\hbar
g_{0}\left(xy^{*}\hat{a}_{+}^{\dagger}\hat{a}_{-}+x^{*}y\hat{a}_{-}^{\dagger}\hat{a}_{+}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,-}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{-}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{-}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,+}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{+}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{+}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,m}}}\left(\hat{c}_{\mathrm{in}}\hat{b}^{\dagger}e^{i\omega_{m}t}-\hat{c}_{\mathrm{in}}^{\dagger}\hat{b}e^{-i\omega_{m}t}\right),$

where $\Delta_{\pm}=\omega_{L}-\omega_{\pm}$. Considering small deviations of
the supermodes around their steady-state amplitudes $\alpha_{\pm}$ and letting
the input field operators denote just the Langevin force, the Hamiltonian

$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}-\hbar\Delta_{-}\hat{a}_{-}^{\dagger}\hat{a}_{-}-\hbar\Delta_{+}\hat{a}_{+}^{\dagger}\hat{a}_{+}$
(15) $\displaystyle-\hbar
g_{0}|x|^{2}\left(|\alpha_{-}|^{2}+\alpha_{-}^{*}\hat{a}_{-}+\hat{a}_{-}^{\dagger}\alpha_{-}+\hat{a}_{-}^{\dagger}\hat{a}_{-}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle-\hbar
g_{0}|y|^{2}\left(|\alpha_{+}|^{2}+\alpha_{+}^{*}\hat{a}_{+}+\hat{a}_{+}^{\dagger}\alpha_{+}+\hat{a}_{+}^{\dagger}\hat{a}_{+}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle-\hbar
g_{0}xy^{*}\left(\alpha_{+}^{*}\alpha_{-}+\alpha_{+}^{*}\hat{a}_{-}+\hat{a}_{+}^{\dagger}\alpha_{-}+\hat{a}_{+}^{\dagger}\hat{a}_{-}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle-\hbar
g_{0}x^{*}y\left(\alpha_{-}^{*}\alpha_{+}+\alpha_{-}^{*}\hat{a}_{+}+\hat{a}_{-}^{\dagger}\alpha_{+}+\hat{a}_{-}^{\dagger}\hat{a}_{+}\right)\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,-}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{-}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{-}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,+}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{+}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{+}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,m}}}\left(\hat{c}_{\mathrm{in}}\hat{b}^{\dagger}e^{i\omega_{m}t}-\hat{c}_{\mathrm{in}}^{\dagger}\hat{b}e^{-i\omega_{m}t}\right)$

can be further simplified. In the large intracavity photon number limit, the
$\mathcal{O}\left(\hat{a}^{2}\right)$ terms are ommited as their contribution
to the system dynamics is hidden by the $\mathcal{O}\left(\hat{a}\right)$
terms. The $\mathcal{O}\left(\alpha^{2}\right)$ constant terms are removed by
shifting the position origin by the steady-state displacement

$x_{ss}=\frac{2g_{0}}{\omega_{m}}\left(|x|^{2}|\alpha_{-}|^{2}+|y|^{2}|\alpha_{+}|^{2}+xy^{*}\alpha_{+}^{*}\alpha_{-}+x^{*}y\alpha_{-}^{*}\alpha_{+}\right)x_{\mathrm{ZPF}}$
(16)

using the translation operator
$\hat{T}_{\hat{x}}(x_{ss})=\exp\left(x_{ss}\hat{p}/(i\hbar)\right)$, $\hat{p}$
being the momentum operator associated with
$\hat{x}=x_{\mathrm{ZPF}}\left(\hat{b}+\hat{b}^{\dagger}\right)$. The
resulting effective Hamiltonian is then given by

$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}-\hbar\Delta_{-}\hat{a}_{-}^{\dagger}\hat{a}_{-}-\hbar\Delta_{+}\hat{a}_{+}^{\dagger}\hat{a}_{+}$
(17) $\displaystyle-\hbar
g_{0}\left(|x|^{2}\alpha_{-}^{*}+xy^{*}\alpha_{+}^{*}\right)\hat{a}_{-}\left(\hat{b}+\hat{b}^{\dagger}\right)-\hbar
g_{0}\left(|x|^{2}\alpha_{-}+x^{*}y\alpha_{+}\right)\hat{a}_{-}^{\dagger}\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle-\hbar
g_{0}\left(|y|^{2}\alpha_{+}^{*}+x^{*}y\alpha_{-}^{*}\right)\hat{a}_{+}\left(\hat{b}+\hat{b}^{\dagger}\right)-\hbar
g_{0}\left(|y|^{2}\alpha_{+}+xy^{*}\alpha_{-}\right)\hat{a}_{+}^{\dagger}\left(\hat{b}+\hat{b}^{\dagger}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,-}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{-}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{-}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,+}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{+}^{\dagger}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{+}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,m}}}\left(\hat{c}_{\mathrm{in}}\hat{b}^{\dagger}e^{i\omega_{m}t}-\hat{c}_{\mathrm{in}}^{\dagger}\hat{b}e^{-i\omega_{m}t}\right).$

Finally, by defining effective optomechanical coupling rates

$\displaystyle g_{-}$
$\displaystyle=g_{0}\left(|x|^{2}\alpha_{-}+x^{*}y\alpha_{+}\right),$ (18)
$\displaystyle g_{+}$
$\displaystyle=g_{0}\left(|y|^{2}\alpha_{+}+xy^{*}\alpha_{-}\right),$

and going into a rotating frame defined by the operator

$\hat{U}=\exp\left(\frac{1}{i\hbar}\hbar\Delta_{-}\hat{a}_{-}^{\dagger}\hat{a}_{-}t\right)\exp\left(\frac{1}{i\hbar}\hbar\Delta_{+}\hat{a}_{+}^{\dagger}\hat{a}_{+}t\right)\exp\left(\frac{-1}{i\hbar}\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}t\right),$
(19)

we have

$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle-\hbar\left(g_{-}^{*}\hat{a}_{-}e^{i\Delta_{-}t}+g_{-}\hat{a}_{-}^{\dagger}e^{-i\Delta_{-}t}\right)\left(\hat{b}e^{-i\omega_{m}t}+\hat{b}^{\dagger}e^{i\omega_{m}t}\right)$
(20)
$\displaystyle-\hbar\left(g_{+}^{*}\hat{a}_{+}e^{i\Delta_{+}t}+g_{+}\hat{a}_{+}^{\dagger}e^{-i\Delta_{+}t}\right)\left(\hat{b}e^{-i\omega_{m}t}+\hat{b}^{\dagger}e^{i\omega_{m}t}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,-}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{-}^{\dagger}e^{-i\Delta_{-}t}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{-}e^{i\Delta_{-}t}\right)+i\hbar\sqrt{\kappa_{\mathrm{ex,+}}}\left(\hat{a}_{\mathrm{in}}\hat{a}_{+}^{\dagger}e^{-i\Delta_{+}t}-\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{+}e^{i\Delta_{+}t}\right)$
$\displaystyle+i\hbar\sqrt{\kappa_{\mathrm{ex,m}}}\left(\hat{c}_{\mathrm{in}}\hat{b}^{\dagger}-\hat{c}_{\mathrm{in}}^{\dagger}\hat{b}\right)$

in the interaction picture. The time evolution of the bosonic modes can be
computed using the Heisenberg-Langevin equations given by

$\displaystyle\frac{d}{dt}\hat{a}_{-}(t)=$
$\displaystyle\frac{-\kappa_{-}}{2}\hat{a}_{-}(t)+ig_{-}\left[\hat{b}(t)e^{-i(\omega_{m}+\Delta_{-})t}+\hat{b}^{\dagger}(t)e^{i(\omega_{m}-\Delta_{-})t}\right]+\sqrt{\kappa_{\mathrm{ex,-}}}\hat{a}_{\mathrm{in}}(t)e^{-i\Delta_{-}t},$
(21) $\displaystyle\frac{d}{dt}\hat{a}_{+}(t)=$
$\displaystyle\frac{-\kappa_{+}}{2}\hat{a}_{+}(t)+ig_{+}\left[\hat{b}(t)e^{-i(\omega_{m}+\Delta_{+})t}+\hat{b}^{\dagger}(t)e^{i(\omega_{m}-\Delta_{+})t}\right]+\sqrt{\kappa_{\mathrm{ex,+}}}\hat{a}_{\mathrm{in}}(t)e^{-i\Delta_{+}t},$
$\displaystyle\frac{d}{dt}\hat{b}(t)=$
$\displaystyle\frac{-\kappa_{m}}{2}\hat{b}(t)+ig_{-}^{*}\hat{a}_{-}(t)e^{i(\omega_{m}+\Delta_{-})t}+ig_{-}\hat{a}_{-}^{\dagger}(t)e^{i(\omega_{m}-\Delta_{-})t}$
$\displaystyle+ig_{+}^{*}\hat{a}_{+}(t)e^{i(\omega_{m}+\Delta_{+})t}+ig_{+}\hat{a}_{+}^{\dagger}(t)e^{i(\omega_{m}-\Delta_{+})t}+\sqrt{\kappa_{\mathrm{ex,m}}}\hat{c}_{\mathrm{in}}(t).$

We can move to the frequency domain by using the Fourier transform

$\hat{a}[\omega]=\mathcal{F}\left(\hat{a}(t)\right)[\omega]=\int_{-\infty}^{\infty}\hat{a}(t)e^{i\omega
t}dt$ (22)

along with the properties

$\displaystyle\mathcal{F}\left(\frac{d}{dt}\hat{a}(t)\right)[\omega]$
$\displaystyle=-i\omega\hat{a}[\omega],$ (23)
$\displaystyle\mathcal{F}\left(\hat{a}(t)e^{-i\Delta t}\right)[\omega]$
$\displaystyle=\hat{a}[\omega-\Delta],$
$\displaystyle\hat{a}^{\dagger}[\omega]\equiv\mathcal{F}\left(\hat{a}^{\dagger}(t)\right)[\omega]$
$\displaystyle=\left(\hat{a}[-\omega]\right)^{\dagger}.$

We then have

$\displaystyle\hat{a}_{-}[\omega]=$ $\displaystyle
ig_{-}\chi_{-}[\omega]\left(\hat{b}[\omega-(\omega_{m}+\Delta_{-})]+\hat{b}^{\dagger}[\omega+(\omega_{m}-\Delta_{-})]\right)+\sqrt{\kappa_{\mathrm{ex,-}}}\chi_{-}[\omega]\hat{a}_{\mathrm{in}}[\omega-\Delta_{-}],$
(24) $\displaystyle\hat{a}_{+}[\omega]=$ $\displaystyle
ig_{+}\chi_{+}[\omega]\left(\hat{b}[\omega-(\omega_{m}+\Delta_{+})]+\hat{b}^{\dagger}[\omega+(\omega_{m}-\Delta_{+})]\right)+\sqrt{\kappa_{\mathrm{ex,+}}}\chi_{+}[\omega]\hat{a}_{\mathrm{in}}[\omega-\Delta_{+}],$
$\displaystyle\hat{b}[\omega]=$ $\displaystyle
ig_{-}^{*}\chi_{m}[\omega]\hat{a}_{-}[\omega+(\omega_{m}+\Delta_{-})]+ig_{-}\chi_{m}[\omega]\hat{a}_{-}^{\dagger}[\omega+(\omega_{m}-\Delta_{-})]$
$\displaystyle+ig_{+}^{*}\chi_{m}[\omega]\hat{a}_{+}[\omega+(\omega_{m}+\Delta_{+})]+\
ig_{+}\chi_{m}[\omega]\hat{a}_{+}^{\dagger}[\omega+(\omega_{m}-\Delta_{+})]+\sqrt{\kappa_{\mathrm{ex,m}}}\chi_{m}[\omega]\hat{c}_{\mathrm{in}}[\omega],$

with the susceptibilities
$\chi_{-}[\omega]=\left(-i\omega+\kappa_{-}/2\right)^{-1}$,
$\chi_{+}[\omega]=\left(-i\omega+\kappa_{+}/2\right)^{-1}$ and
$\chi_{m}[\omega]=\left(-i\omega+\kappa_{m}/2\right)^{-1}$. The bath operators
are linked through the input-output relations

$\displaystyle\hat{a}_{\mathrm{out}}(t)e^{-i\omega_{L}t}$
$\displaystyle=\hat{a}_{\mathrm{in}}(t)e^{-i\omega_{L}t}-\sqrt{\kappa_{\mathrm{ex},-}}\hat{a}_{-}(t)e^{-i\omega_{-}t}-\sqrt{\kappa_{\mathrm{ex},+}}\hat{a}_{+}(t)e^{-i\omega_{+}t},$
(25) $\displaystyle\hat{c}_{\mathrm{out}}(t)e^{-i\omega_{m}t}$
$\displaystyle=-\hat{c}_{\mathrm{in}}(t)e^{-i\omega_{m}t}+\sqrt{\kappa_{\mathrm{ex},m}}\hat{b}(t)e^{-i\omega_{m}t}$

or

$\displaystyle\hat{a}_{\mathrm{out}}[\omega]$
$\displaystyle=\hat{a}_{\mathrm{in}}[\omega]-\sqrt{\kappa_{\mathrm{ex},-}}\hat{a}_{-}[\omega+\Delta_{-}]-\sqrt{\kappa_{\mathrm{ex},+}}\hat{a}_{+}[\omega+\Delta_{+}],$
(26) $\displaystyle\hat{c}_{\mathrm{out}}[\omega]$
$\displaystyle=-\hat{c}_{\mathrm{in}}[\omega]+\sqrt{\kappa_{\mathrm{ex},m}}\hat{b}[\omega]$

in frequency domain. They satisfy the commutation relations
$\left[\hat{a}_{\mathrm{in}}[\omega],\hat{a}_{\mathrm{in}}^{\dagger}[\omega^{\prime}]\right]=\delta[\omega-\omega^{\prime}]$
and
$\left[\hat{c}_{\mathrm{in}}[\omega],\hat{c}_{\mathrm{in}}^{\dagger}[\omega^{\prime}]\right]=\delta[\omega-\omega^{\prime}]$.

###  A.3 Conversion efficiency and added noise

For the following analysis, we consider the triply resonant configuration
where $\omega_{+}-\omega_{-}=\omega_{m}$ and the optical pump is on resonance
with one of the optical supermodes. First, we consider the anti-Stokes
configuration where $\Delta_{-}=0$ and $\Delta_{+}=-\omega_{m}$. From Eq. 26,
we have the input-output relation

$\hat{a}_{\mathrm{out}}[\omega]=\hat{a}_{\mathrm{in}}[\omega]-\sqrt{\kappa_{\mathrm{ex},-}}\hat{a}_{-}[\omega]-\sqrt{\kappa_{\mathrm{ex},+}}\hat{a}_{+}[\omega-\omega_{m}].$
(27)

The equations of motion is given by

$\displaystyle\hat{a}_{-}[\omega]$ $\displaystyle=\
ig_{-}\chi_{-}[\omega]\left(\hat{b}[\omega-\omega_{m}]+\hat{b}^{\dagger}[\omega+\omega_{m}]\right)+\sqrt{\kappa_{\mathrm{ex,-}}}\chi_{-}[\omega]\hat{a}_{\mathrm{in}}[\omega]$
(28)
$\displaystyle\approx\sqrt{\kappa_{\mathrm{ex,-}}}\chi_{-}[\omega]\hat{a}_{\mathrm{in}}[\omega],$
$\displaystyle\hat{a}_{+}[\omega]$ $\displaystyle=\
ig_{+}\chi_{+}[\omega]\left(\hat{b}[\omega]+\hat{b}^{\dagger}[\omega+2\omega_{m}]\right)+\sqrt{\kappa_{\mathrm{ex,+}}}\chi_{+}[\omega]\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]$
$\displaystyle\approx\
ig_{+}\chi_{+}[\omega]\hat{b}[\omega]+\sqrt{\kappa_{\mathrm{ex,+}}}\chi_{+}[\omega]\hat{a}_{\mathrm{in}}[\omega+\omega_{m}],$
$\displaystyle\hat{b}[\omega]$ $\displaystyle=\
ig_{-}^{*}\chi_{m}[\omega]\hat{a}_{-}[\omega+\omega_{m}]+ig_{-}\chi_{m}[\omega]\hat{a}_{-}^{\dagger}[\omega+\omega_{m}]+ig_{+}^{*}\chi_{m}[\omega]\hat{a}_{+}[\omega]+ig_{+}\chi_{m}[\omega]\hat{a}_{+}^{\dagger}[\omega+2\omega_{m}]+\sqrt{\kappa_{\mathrm{ex,m}}}\chi_{m}[\omega]\hat{c}_{\mathrm{in}}[\omega]$
$\displaystyle\approx\
ig_{+}^{*}\chi_{m}[\omega]\hat{a}_{+}[\omega]+\sqrt{\kappa_{\mathrm{ex,m}}}\chi_{m}[\omega]\hat{c}_{\mathrm{in}}[\omega].$

Here we drop the terms that contain susceptibilities and ladder operators with
an offsetted frequency dependence (with the exception of the bath operators),
since they necessarily imply an attenuation of at least
$\kappa_{+}/\left(2\omega_{m}\right)$ compared to the dominant terms. This
approximation is valid in the resolved-sideband regime
$\kappa_{-},\kappa_{+}\ll\omega_{m}$. Supplementary Figure 1a shows the signal
flow graph representing the equations of motion, from which we deduced the
transfer functions between the microwave field at $\omega_{m}$ and the optical
field at $\omega_{L}+\omega_{m}$ using Mason’s rule
[blesinQuantumCoherentMicrowaveoptical2021]

$\displaystyle
S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}=$
$\displaystyle
S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}=S_{\hat{a}_{\mathrm{out}}^{\dagger}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}^{*}=S_{\hat{c}_{\mathrm{out}}^{\dagger}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega+\omega_{m}]}^{*}$
(29) $\displaystyle=$
$\displaystyle\frac{-i\sqrt{\kappa_{\mathrm{ex,+}}}\sqrt{\kappa_{\mathrm{ex,m}}}g_{+}\chi_{+}[\omega]\chi_{m}[\omega]}{1+|g_{+}|^{2}\chi_{+}[\omega]\chi_{m}[\omega]}.$

The optical-optical and microwave-microwave transfer functions can also be
derived as

$\displaystyle
S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}$
$\displaystyle=1-\frac{\kappa_{\mathrm{ex,+}}\chi_{+}[\omega]}{1+|g_{+}|^{2}\chi_{+}[\omega]\chi_{m}[\omega]}-\kappa_{\mathrm{ex,-}}\chi_{-}[\omega+\omega_{m}],$
(30) $\displaystyle
S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}$
$\displaystyle=-1+\frac{\kappa_{\mathrm{ex,m}}\chi_{m}[\omega]}{1+|g_{+}|^{2}\chi_{+}[\omega]\chi_{m}[\omega]}.$
(31)

The other transfer functions (e.g.,
$S_{\hat{a}_{\mathrm{out}}^{\dagger}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}$)
are expected to be small given the minuscule weight of the corresponding edges
in the signal flow graph. The optical photon number at the output of the
transducer at $\omega_{L}+\omega_{m}$ is given by

$\displaystyle\hat{a}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\approx$
$\displaystyle\left(S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}^{*}\hat{a}_{\mathrm{in}}^{\dagger}[\omega+\omega_{m}]+S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}^{*}\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\right)$
(32)
$\displaystyle\left(S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]+S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}\hat{c}_{\mathrm{in}}[\omega]\right)$
$\displaystyle=$ $\displaystyle\
\eta_{+}[\omega]\left(\hat{c}_{\mathrm{in}}^{\dagger}\hat{c}_{\mathrm{in}}[\omega]+\hat{n}_{\mathrm{added,+}}^{\uparrow}[\omega]\right),$

which can be used to compute the on-chip photon number conversion efficiency

$\displaystyle\eta_{+}[\omega]$
$\displaystyle=|S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}|^{2}=|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}|^{2}$
(33)
$\displaystyle=\frac{\kappa_{\mathrm{ex,+}}\kappa_{\mathrm{ex,m}}|g_{+}|^{2}|\chi_{+}[\omega]|^{2}|\chi_{m}[\omega]|^{2}}{|1+|g_{+}|^{2}\chi_{+}[\omega]\chi_{m}[\omega]|^{2}}\approx\frac{\kappa_{\mathrm{ex,+}}}{\kappa_{+}}\frac{\kappa_{\mathrm{ex,m}}}{\kappa_{m}}\frac{4C_{+}}{(1+C_{+})^{2}}$

and the added noise for up-conversion

$\displaystyle\hat{n}_{\mathrm{added,+}}^{\uparrow}[\omega]=$ $\displaystyle\
\frac{S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}^{*}}{S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}^{*}}\hat{a}_{\mathrm{in}}^{\dagger}[\omega+\omega_{m}]\hat{c}_{\mathrm{in}}[\omega]+\frac{S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}}{S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}}\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]$
(34)
$\displaystyle+\frac{|S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}|^{2}}{|S_{\hat{a}_{\mathrm{out}}[\omega+\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}|^{2}}\
\hat{a}_{\mathrm{in}}^{\dagger}[\omega+\omega_{m}]\hat{a}_{\mathrm{in}}[\omega+\omega_{m}].$

The last approximation of Eq. 33 gives the steady-state efficiency of the
transducer ($\omega\rightarrow 0$), which depends on anti-Stokes cooperativity
$C_{+}=4g_{+}^{2}/\left(\kappa_{+}\kappa_{m}\right)$ as well as extraction
efficiencies of the antisymmetric supermode
$\kappa_{\mathrm{ex,+}}/\kappa_{+}$ and the acoustic mode
$\kappa_{\mathrm{ex,m}}/\kappa_{m}$. Additionally, the output photon number at
the microwave port is

$\displaystyle\hat{c}_{\mathrm{out}}^{\dagger}\hat{c}_{\mathrm{out}}[\omega]\approx$
$\displaystyle\
\eta_{+}[\omega]\left(\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]+\hat{n}_{\mathrm{added,+}}^{\downarrow}[\omega]\right).$
(35)

While the expressions for the conversion efficiency are identical for both up-
and down-conversion, the added noise for down-conversion is instead

$\displaystyle\hat{n}_{\mathrm{added,+}}^{\downarrow}[\omega]=$
$\displaystyle\
\frac{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}}{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}}\hat{a}_{\mathrm{in}}^{\dagger}[\omega+\omega_{m}]\hat{c}_{\mathrm{in}}[\omega]+\frac{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}^{*}}{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}^{*}}\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]$
(36)
$\displaystyle+\frac{|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}|^{2}}{|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega+\omega_{m}]}|^{2}}\
\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{c}_{\mathrm{in}}[\omega].$

We follow the same recipe to derive the conversion efficiency and added noise
for the Stokes configuration where $\Delta_{-}=\omega_{m}$ and $\Delta_{+}=0$.
We have the optical input-output relation

$\hat{a}_{\mathrm{out}}[\omega]=\hat{a}_{\mathrm{in}}[\omega]-\sqrt{\kappa_{\mathrm{ex},-}}\hat{a}_{-}[\omega+\omega_{m}]-\sqrt{\kappa_{\mathrm{ex},+}}\hat{a}_{+}[\omega]$
(37)

and the equations of motion

$\displaystyle\hat{a}_{-}[\omega]$ $\displaystyle=\
ig_{-}\chi_{-}[\omega]\left(\hat{b}[\omega-2\omega_{m}]+\hat{b}^{\dagger}[\omega]\right)+\sqrt{\kappa_{\mathrm{ex,-}}}\chi_{-}[\omega]\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]$
(38) $\displaystyle\approx\
ig_{-}\chi_{-}[\omega]\hat{b}^{\dagger}[\omega]+\sqrt{\kappa_{\mathrm{ex,-}}}\chi_{-}[\omega]\hat{a}_{\mathrm{in}}[\omega-\omega_{m}],$
$\displaystyle\hat{a}_{+}[\omega]$ $\displaystyle=\
ig_{+}\chi_{+}[\omega]\left(\hat{b}[\omega-\omega_{m}]+\hat{b}^{\dagger}[\omega+\omega_{m}]\right)+\sqrt{\kappa_{\mathrm{ex,+}}}\chi_{+}[\omega]\hat{a}_{\mathrm{in}}[\omega]$
$\displaystyle\approx\
\sqrt{\kappa_{\mathrm{ex,+}}}\chi_{+}[\omega]\hat{a}_{\mathrm{in}}[\omega],$
$\displaystyle\hat{b}[\omega]$ $\displaystyle=\
ig_{-}^{*}\chi_{m}[\omega]\hat{a}_{-}[\omega+2\omega_{m}]+ig_{-}\chi_{m}[\omega]\hat{a}_{-}^{\dagger}[\omega]+ig_{+}^{*}\chi_{m}[\omega]\hat{a}_{+}[\omega+\omega_{m}]+ig_{+}\chi_{m}[\omega]\hat{a}_{+}^{\dagger}[\omega+\omega_{m}]+\sqrt{\kappa_{\mathrm{ex,m}}}\chi_{m}[\omega]\hat{c}_{\mathrm{in}}[\omega]$
$\displaystyle\approx\
ig_{-}\chi_{m}[\omega]\hat{a}_{-}^{\dagger}[\omega]+\sqrt{\kappa_{\mathrm{ex,m}}}\chi_{m}[\omega]\hat{c}_{\mathrm{in}}[\omega].$

Using the signal flow graph shown in Supplementary Fig. 1b, the transfer
functions between the microwave field at $\omega_{m}$ and the optical field at
$\omega_{L}-\omega_{m}$ can once again be derived as

$\displaystyle
S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}$
$\displaystyle=S_{\hat{a}_{\mathrm{out}}^{\dagger}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}^{*}=\frac{-i\sqrt{\kappa_{\mathrm{ex,-}}}\sqrt{\kappa_{\mathrm{ex,m}}}g_{-}\chi_{-}[\omega]\chi_{m}^{*}[\omega]}{1-|g_{-}|^{2}\chi_{-}^{*}[\omega]\chi_{m}[\omega]},$
(39) $\displaystyle
S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]}$
$\displaystyle=S_{\hat{c}_{\mathrm{out}}^{\dagger}[\omega]\leftarrow\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]}^{*}=\frac{i\sqrt{\kappa_{\mathrm{ex,-}}}\sqrt{\kappa_{\mathrm{ex,m}}}g_{-}\chi_{-}^{*}[\omega]\chi_{m}[\omega]}{1-|g_{-}|^{2}\chi_{-}[\omega]\chi_{m}^{*}[\omega]}.$
(40)

While the microwave-microwave transfer function is identical to Eq. 31, the
optical-optical transfer function is

$S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]}=1-\frac{\kappa_{\mathrm{ex,-}}\chi_{-}[\omega]}{1-|g_{-}|^{2}\chi_{-}^{*}[\omega]\chi_{m}[\omega]}-\kappa_{\mathrm{ex,+}}\chi_{+}[\omega-\omega_{m}].$
(41)

The optical photon number at the output of the transducer at
$\omega_{L}-\omega_{m}$ can be computed as

$\displaystyle\hat{a}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\approx$
$\displaystyle\
\eta_{-}[\omega]\left(\hat{c}_{\mathrm{in}}^{\dagger}\hat{c}_{\mathrm{in}}[\omega]+\hat{n}_{\mathrm{added,-}}^{\uparrow}[\omega]\right),$
(42)

where we define the on-chip photon number conversion efficiency

$\displaystyle\eta_{-}[\omega]$
$\displaystyle=|S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}|^{2}=|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]}|^{2}$
(43)
$\displaystyle=\frac{\kappa_{\mathrm{ex,-}}\kappa_{\mathrm{ex,m}}|g_{-}|^{2}|\chi_{-}[\omega]|^{2}|\chi_{m}[\omega]|^{2}}{|1+|g_{-}|^{2}\chi_{-}[\omega]\chi_{m}[\omega]|^{2}}\approx\frac{\kappa_{\mathrm{ex,-}}}{\kappa_{-}}\frac{\kappa_{\mathrm{ex,m}}}{\kappa_{m}}\frac{4C_{-}}{(1-C_{-})^{2}}$

and the added noise for up-conversion

$\displaystyle\hat{n}_{\mathrm{added,-}}^{\uparrow}[\omega]=$ $\displaystyle\
\frac{S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]}^{*}}{S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}^{*}}\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]\hat{c}_{\mathrm{in}}^{\dagger}[\omega]+\frac{S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]}}{S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}}\hat{c}_{\mathrm{in}}[\omega]\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]$
(44)
$\displaystyle+\frac{|S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]}|^{2}}{|S_{\hat{a}_{\mathrm{out}}[\omega-\omega_{m}]\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}[\omega]}|^{2}}\
\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]+1+n_{\mathrm{m,th}}(\omega).$

In a similar fashion, we define the Stokes cooperativity
$C_{-}=4g_{-}^{2}/\left(\kappa_{-}\kappa_{m}\right)$. Importantly, for a
temperature $T$, Boltzmann’s constant $k_{B}$ and number of thermal
excitations in the microwave bath
$n_{\mathrm{m,th}}(\omega)=\left[\exp\left(\hbar\omega/(k_{B}T)\right)-1\right]^{-1}$,
the commutation relation
$\left[\hat{c}_{\mathrm{in}}[\omega],\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\right]=1+n_{\mathrm{m,th}}(\omega)$
gives a fundamental limit on the added noise during up-conversion
$\hat{n}_{\mathrm{added,-}}^{\uparrow}[\omega]\geq 1$. The output photon
number at the microwave port for the Stokes configuration is

$\displaystyle\hat{c}_{\mathrm{out}}^{\dagger}\hat{c}_{\mathrm{out}}[\omega]\approx$
$\displaystyle\
\eta_{-}[\omega]\left(\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]+\hat{n}_{\mathrm{added,-}}^{\downarrow}[\omega]\right).$
(45)

The photon number on-chip conversion efficiency is again the same for both up-
and down-conversion, but the added noise for the down-conversion process is
instead

$\displaystyle\hat{n}_{\mathrm{added,-}}^{\downarrow}[\omega]=$
$\displaystyle\
\frac{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}}{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]}}\hat{a}_{\mathrm{in}}[\omega-\omega_{m}]\hat{c}_{\mathrm{in}}[\omega]+\frac{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}^{*}}{S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]}^{*}}\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]$
(46)
$\displaystyle+\frac{|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{c}_{\mathrm{in}}[\omega]}|^{2}}{|S_{\hat{c}_{\mathrm{out}}[\omega]\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}[\omega-\omega_{m}]}|^{2}}\
\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{c}_{\mathrm{in}}[\omega]+1.$

As in the up-conversion case, the commutation relation
$\left[\hat{a}_{\mathrm{in}}[\omega],\hat{a}_{\mathrm{in}}^{\dagger}[\omega]\right]=1$
gives a fundamental limit on the added noise during down-conversion
$\hat{n}_{\mathrm{added,-}}^{\downarrow}[\omega]\geq 1$.

###  A.4 Estimation of correlated microwave-optical pair generation rates

Equations 42 and 45 suggest that entangled pairs of microwave and optical
photons can be generated in the Stokes configuration since it gives rise to an
effective two-mode squeezing Hamiltonian
[ruedaElectroopticEntanglementSource2019, zhongProposalHeraldedGeneration2020,
krastanovOpticallyHeraldedEntanglement2021]. In order to verify that the HBAR
transducer can generate entangled pairs of microwave and optical photons, we
compute the on-chip pair generation rate, expected to be equal to the output
photon flux from the lower frequency optical mode as in standard spontaneous
parametric down-conversion (SPDC) scheme. Integrating over the transduction
bandwidth, we have

$R=\int_{-\infty}^{\infty}\expectationvalue{\hat{a}_{\mathrm{out}}^{\dagger}[\omega]\hat{a}_{\mathrm{out}}[\omega]}d\omega\approx\int_{-\infty}^{\infty}\eta_{-}[\omega]d\omega\approx\frac{\pi}{2}\left(\kappa_{-}^{-1}+\kappa_{m}^{-1}\right)^{-1}\eta_{-}[0].$
(47)

In practice, the pump laser should be filtered for the optical output to be
dominated by heralding photons, and other losses from the measurement setup
can further reduce the effective rate. For
$\eta^{\mathrm{oc}}=$7.9\text{\times}{10}^{-5}\text{\,}$$,
$\kappa_{-}=2\pi\times$300\text{\,}\mathrm{MHz}$$ and
$\kappa_{m}=2\pi\times$10\text{\,}\mathrm{MHz}$$, this would corresponds to an
on-chip pair generation rate of $190\text{\,}\mathrm{Hz}$. Losses in the
measurement chain will have to be accounted for to estimate the count rate at
the detector. In order to measure entanglement between the microwave and
optical fields, the on-chip rate must be higher than the thermal decoherence
rate of the acoustic mode $\kappa_{m}n_{\mathrm{th,m}}(\omega_{m})$, where the
thermal occupancy $n_{\mathrm{th,m}}(\omega_{m})$ is given by the Bose-
Einstein distribution. For $\omega_{m}=2\pi\times$3.5\text{\,}\mathrm{GHz}$$
and $\kappa_{m}=2\pi\times$10\text{\,}\mathrm{MHz}$$, the decoherence rate at
$800\text{\,}\mathrm{mK}$ is $43\text{\,}\mathrm{MHz}$ and
$0.5\text{\,}\mathrm{Hz}$ at $10\text{\,}\mathrm{mK}$. Therefore, the
transducer needs to be operated in the mixing chamber of the dilution
refrigerator, with a pulsed pump to reduce heat load. The total rate of
optical heralding is hence gated by the pump duty cycle.

To evaluate the non-classical correlation required for the DLCZ protocol
[duanLongdistanceQuantumCommunication2001,
krastanovOpticallyHeraldedEntanglement2021], one can compute the second-order
cross-correlation function. Since no input signal needs to be sent to the
transducer for SPDC,
$\expectationvalue{\hat{a}_{\mathrm{in}}}=\expectationvalue{\hat{a}_{\mathrm{in}}^{\dagger}}=\expectationvalue{\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{\mathrm{in}}}=\expectationvalue{\hat{a}_{\mathrm{in}}\hat{a}_{\mathrm{in}}}=\expectationvalue{\hat{a}_{\mathrm{in}}^{\dagger}\hat{a}_{\mathrm{in}}^{\dagger}}=0$
and
$\expectationvalue{\hat{c}_{\mathrm{in}}^{\dagger}[\omega]\hat{c}_{\mathrm{in}}[\omega]}=n_{\mathrm{th,m}}(\omega)$,
leading to

$\displaystyle g_{ac}^{(2)}=$
$\displaystyle\frac{\expectationvalue{\hat{c}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}\hat{c}_{\mathrm{out}}}}{\expectationvalue{\hat{c}_{\mathrm{out}}^{\dagger}\hat{c}_{\mathrm{out}}}\expectationvalue{\hat{a}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}}}$
(48) $\displaystyle\approx$
$\displaystyle\frac{\left(|S_{\hat{a}_{\mathrm{out}}\leftarrow\hat{a}_{\mathrm{in}}}|^{2}+\eta_{-}\right)}{\left(\eta_{-}+|S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}}|^{2}n_{\mathrm{th,m}}\right)\left(1+n_{\mathrm{th,m}}\right)}+\frac{n_{\mathrm{th,m}}}{1+n_{\mathrm{th,m}}}$
$\displaystyle+\frac{\left(S_{\hat{a}_{\mathrm{out}}\leftarrow\hat{a}_{\mathrm{in}}}^{*}S_{\hat{a}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}}S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}}^{*}S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}}+S_{\hat{a}_{\mathrm{out}}\leftarrow\hat{a}_{\mathrm{in}}}S_{\hat{a}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}^{\dagger}}^{*}S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{a}_{\mathrm{in}}^{\dagger}}S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}}^{*}\right)n_{\mathrm{th,m}}}{\eta_{-}\left(\eta_{-}+|S_{\hat{c}_{\mathrm{out}}\leftarrow\hat{c}_{\mathrm{in}}}|^{2}n_{\mathrm{th,m}}\right)\left(1+n_{\mathrm{th,m}}\right)}$

where the frequency dependences are omitted for readability. Non-classicality
is expected to lead to violation of the Cauchy-Schwarz inequality, given by
$g_{ac}^{(2)}\leq\sqrt{g_{aa}^{(2)}g_{cc}^{(2)}}$. Therefore, it is desirable
to minimize both microwave thermal occupancy (cooling) and optical
transmission (attaining critical coupling).

##  Appendix B Fabrication process flow

The piezoelectric actuators are monolithically integrated on Si3N4 waveguides,
fabricated using the photonic Damascene process
[pfeifferUltrasmoothSiliconNitride2018,
liuHighyieldWaferscaleFabrication2021]. The
$850\text{\text{-}}\mathrm{nm}$-thick Si3N4 film is deposited using low-
pressure chemical vapor deposition into the Damascene preform, with
$2.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thermal oxide below the
waveguides. After annealing to drive hydrogen impurities out of the Si3N4
layer, $2.1\text{\text{-}}\mathrm{\SIUnitSymbolMicro m}$-thick TEOS oxide and
$1.0\text{\text{-}}\mathrm{\SIUnitSymbolMicro m}$-thick low-temperature oxide
top claddings are deposited and subsequently annealed. The metallic and
piezoelectric films—$95\text{\,}\mathrm{nm}$ of Mo,
$1.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of AlN and another
$95\text{\,}\mathrm{nm}$ of Mo—are sputtered through foundry services provided
by Plasma-Therm [tianHybridIntegratedPhotonics2020]. They are then patterned
using deep reactive ion etching (DRIE) to form the actuators, ground planes
and integrated heaters. We employ the same Band-Aid process as in Ref.
[siddharthHertzlinewidthFrequencyagilePhotonic2023] to localize the electrodes
atop the suspended cladding. In particular, when connecting the Al feedline to
the top electrode by a lift-off process, the bottom electrode is etched back
using XeF2 to avoid short-circuiting the electrodes. The process to suspend
the cladding in the vicinity of the actuator is combined with the chip
singulation steps of the Damascene process. First, a hole is opened in the
middle of the donut-shaped actuator. The chip facets are simultaneously
defined where trenches in the SiO2 are etched by C4F8 DRIE between neighboring
chips. A second photolithography is performed to protect the facets while
leaving the etched holes exposed for further processing, resulting in narrower
trenches between chips. The Si substrate is then isotropically etched with SF6
until the SiO2 cladding below the actuator is suspended
[tianMagneticfreeSiliconNitride2021]. Following cladding suspension, the Si
substrate is etched anisotropically until the desired chip thickness of
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is reached. A final isotropic Si
etch removes the parts of the substrate protruding from the SiO2 facet,
providing proper access to the bus waveguide nanotapers with lensed fibers, or
even using the butt-coupling scheme. To facilitate this last step, we employ
ion-beam etching at a $20^{\circ}$ tilt angle to remove the passivation layer
formed on the hole sidewalls during the Si DRIE. The chips are ultimately
released by grinding the back side of the wafer. This convoluted process
ensures that all the actuators on the wafer are fully suspended, while the
chip facets do not become exceedingly fragile from the undercut.

##  Appendix C Optical characterization

###  C.1 Resonance linewidths

Fitting the hybridized transmission spectrum in Fig. 2c to the coupled mode
theory presented in Appendix A.1 and in Ref.
[blesinQuantumCoherentMicrowaveoptical2021] yields
$\kappa_{r}/(2\pi)=$154\text{\,}\mathrm{MHz}$$ and
$\kappa_{l}/(2\pi)=$190\text{\,}\mathrm{MHz}$$, with
$\kappa_{\mathrm{ex},l}/(2\pi)=\kappa_{\mathrm{ex},r}/(2\pi)=$60\text{\,}\mathrm{MHz}$$.

###  C.2 Hybridization of optical resonances via integrated thermo-optic
heaters

For the purpose of rapid characterization of the photonic molecules at room
temperature, the bottom electrode of the piezoelectric actuator also serves as
an integrated heater to control the relative detuning between the micro-rings
by thermo-optic effect. As seen in Fig. 2a, we pattern Mo to make three
electrical connections to the bottom electrode. The top and bottom connections
are connected to ground, whereas the central connection is biased to a
constant voltage. The finite resistance of Mo at room temperature leads to
Joule heating as current flows through the bottom electrode, modifying the
refractive index of the Si3N4 waveguide buried directly underneath.

###  C.3 Thermal response characterization through cavity-enhanced
photothermal spectroscopy

For the purpose of programming the pulse sequence to alleviate thermal
effects, we study the thermal response of the present transducer due to
optical pump heating. Specifically, thermally induced refractive index change
causes a shift in resonance frequency of the Si3N4 micro-ring, which is
detected using cavity-enhanced pump-probe spectroscopy
[gaoProbingMaterialAbsorption2022]. The intensity-modulated pump addresses a
TE resonance, while the probe measures the resulting side-of-fringe modulation
of another TE mode. To ascertain the physical mechanism of each cross phase
modulation (XPM) process, we sweep the pump modulation frequency to leverage
the separation of time scales. Seen in Supplementary Fig. 2, the response
exhibits three plateaus. We associate the process at modulation frequencies
$\gtrsim 1$ MHz with Kerr-induced XPM. The two slower processes at $\sim 100$
kHz and $\sim 1$ kHz are characteristic of photothermal XPM in quasi-free-
standing microresonators such as spheres
[braginskyOpticalWhisperinggalleryMicroresonators1994] and toroids
[schliesserRadiationPressureCooling2006]. Here, the former “local” time scale
can be attributed to thermalization of the mode volume with the acoustic
resonator, while the thermalization of the suspended structure with the rest
of the chip constitutes the latter “global” time scale. Therefore, it is
expected that for future cryogenic operations, employing a pulse-on time
($\tau_{\mathrm{on}}$) shorter than the local time scale to gate the optical
pump will significantly reduce thermal occupancy in the cladding acoustic
mode. The pulse repetition rate ($f_{\mathrm{rep}}$) and hence pulse-off time
can then be chosen based on the available cooling power.

##  Appendix D Characterization of acoustic resonances

The microwave reflection measurements (also known as $S_{11}$) shown here were
measured at room temperature using custom probes (GGB industries nickel-alloy
40A-GS-135-PC-N Picoprobe) and a vector network analyzer (Rhode&Schwarz
ZNB-20). The spectrum shown in Fig. 2d is corrected using a calibration
substrate (GGB industries CS-8) to remove the phase delay and attenuation
introduced by coaxial cables and probes. However, this calibration procedure
was only used to get the S-parameter, and was not applied during the total
conversion efficiency measurements shown in Fig. 3 and Fig. 4. In a previous
version of the transducers and in Ref. [tianMagneticfreeSiliconNitride2021]
where the Band-Aid process (Appendix B) is not employed, the piezoelectric
actuator extends beyond the suspended cladding and covers the electrical
feedline, causing two detrimental effects for transduction. First, the
resulting stray capacitance leads to excess microwave insertion loss. Given
the small size of the suspended area
($\sim$1000\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$) compared to that of
the feedline ($\gtrsim$10\,000\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$),
this effect is readily seen in Supplementary Fig. 3, where the spectrum
without the Band-Aid process exhibits reduced background reflection. Second,
the part of the actuator on top of the unreleased cladding leads to resonant
excitation of HBAR modes extending over the entire Si substrate. These modes
have a smaller free spectral range (FSR) of around $19\text{\,}\mathrm{MHz}$
and a lower microwave extraction efficiency. The theory presented in Appendix
A does not correspond to the case of such a multimode system as it assumes
being in the sideband-resolved regime, implying the high density of substrate
modes is inappropriate for low-noise frequency conversion. The feedline Band-
Aid, discussed in Appendix B and shown in Supplementary Fig. 3, enables the
removal of the piezoelectric layer above the substrate, leaving only HBAR
modes well confined in the oxide cladding.

We fit the measured complex reflection to obtain the mechanical quality factor
$Q_{m}=284$ and microwave extraction efficiency
$\kappa_{\mathrm{ex,m}}/\kappa_{m}=0.11$ for the transduction mode
[leongPreciseMeasurementsFactor2002]. In addition, we simulate microwave
reflection of the designed stack geometry using finite-element method (FEM).
Fitting the simulated spectrum leads to $Q_{m}=205$ and
$\kappa_{\mathrm{ex,m}}/\kappa_{m}=0.61$. The discrepancy is attributed to
difference in clamping losses and piezoelectric coefficients between the
fabricated and simulated devices. We extract the effective mass of the
transduction HBAR mode from FEM simulations as

$m_{\mathrm{eff}}=\frac{1}{\max(|\mathbf{x}(\mathbf{r})|^{2})}\iiint\rho(\mathbf{r})\mathbf{x}^{*}(\mathbf{r})\mathbf{x}(\mathbf{r})d\mathcal{V}\approx$6\text{\,}\mathrm{ng}$,$
(49)

where $\rho(\mathbf{r})$ is the local mass density and
$\mathbf{x}(\mathbf{r})$ is the mechanical displacement, corresponding to
$x_{\mathrm{ZPF}}=\sqrt{\hbar/(2m_{\mathrm{eff}}\omega_{m})}\approx$2\text{\times}{10}^{-8}\text{\,}\mathrm{nm}$$.

##  Appendix E Characterization of bidirectional transduction

###  E.1 Estimation of cooperavity and single-photon coupling rate

Given the measured off-chip efficiency, the single-photon coupling rate can be
estimated. The off-chip efficiency is given by

$\eta^{\mathrm{tot}}=\eta^{\mathrm{probes}}\eta^{\text{fiber-
chip}}\eta^{\mathrm{oc}},$ (50)

where $\eta_{\mathrm{probes}}$ and $\eta_{\text{fiber-chip}}$ denote the
microwave and optical insertion losses, respectively. The on-chip efficiency
can be written as

$\eta^{\mathrm{oc}}=\eta^{\mathrm{ext}}\eta^{\mathrm{int}}=\eta_{o}\eta_{m}\eta^{\mathrm{int}}.$
(51)

The internal efficiency

$\eta^{\mathrm{int}}=\frac{4C}{(1\pm C)^{2}}\approx 4C$ (52)

in the low-cooperativity regime, where the plus and minus signs correspond to
the anti-Stokes and Stokes configurations, respectively. The optical and
microwave extraction efficiencies

$\displaystyle\eta_{o}$
$\displaystyle=\frac{\kappa_{\mathrm{ex},o}}{\kappa_{o}},$ (53)
$\displaystyle\eta_{m}$
$\displaystyle=\frac{\kappa_{\mathrm{ex},m}}{\kappa_{m}}.$ (54)

The single-photon cooperativity $C_{0}$ is enhanced by the intracavity photons

$\displaystyle C$ $\displaystyle=C_{0}\bar{n},$ (55) $\displaystyle n_{c}$
$\displaystyle=\kappa_{\mathrm{ex}}|\chi_{o}|^{2}\frac{P_{\mathrm{wg}}}{\hbar\omega_{L}}$
$\displaystyle\approx\frac{4\kappa_{\mathrm{ex}}}{\kappa_{o}^{2}}\frac{P_{\mathrm{wg}}}{\hbar\omega_{L}}=\eta_{o}\frac{4}{\kappa_{o}}\frac{P_{\mathrm{wg}}}{\hbar\omega_{L}},$
(56)

where the pump detuning is assumed to be small. The input power in the bus
waveguide $P_{\mathrm{wg}}$ is related to the power in the input fiber
$P_{\mathrm{in}}$ by the fiber-chip coupling efficiency

$\eta^{\text{fiber-chip}}=\frac{P_{\mathrm{in}}}{P_{\mathrm{wg}}}.$ (57)

The off-chip efficiency with respect to the microwave probe and optical fiber
is thus

$\displaystyle\eta^{\mathrm{tot}}$
$\displaystyle\approx\eta^{\mathrm{probes}}\eta^{\text{fiber-
chip}}\times\eta_{m}\eta_{o}\times
4C_{0}\times\eta_{o}\frac{4}{\kappa_{o}}\frac{1}{\hbar\omega_{L}}\eta^{\text{fiber-
chip}}P_{\mathrm{in}}$
$\displaystyle=16\eta^{\mathrm{probes}}\eta^{\text{fiber-
fiber}}\times\eta_{m}\eta_{o}^{2}\times\frac{C_{0}P_{\mathrm{in}}}{\hbar\omega_{L}\kappa_{o}},$
(58)

where the fiber-fiber coupling efficiency $\eta^{\text{fiber-
fiber}}=\eta^{\text{fiber-chip}}\times\eta^{\text{fiber-chip}}$. Calibration
of the RF probes informed about the probes collections efficiency
$\eta_{\mathrm{probes}}\approx$-3\text{\,}\mathrm{dB}$$. A fraction of light
is tapped just before and after the coupling lensed fibers for power
monitoring. We obtain a typical $\eta^{\text{fiber-
fiber}}=$-8\text{\,}\mathrm{dB}$$ and hence $\eta^{\text{fiber-
chip}}=$-4\text{\,}\mathrm{dB}$$. Fitting the microwave (optical) reflection
spectrum yields $\eta_{m}=11\%$ ($\eta_{o}=35\%$). Given that the measured
efficiency $\eta^{\mathrm{tot}}=$-60\text{\,}\mathrm{dB}$$ at
$P_{\mathrm{in}}=$10\text{\,}\mathrm{dBm}$$, it follows that the single-photon
cooperativity is

$C_{0}=\frac{\eta^{\mathrm{tot}}\times\hbar\omega_{L}\kappa_{o}}{16\eta^{\mathrm{probes}}\eta^{\text{fiber-
fiber}}\times\eta_{m}\eta_{o}^{2}\times
P_{\mathrm{in}}}\approx$8\text{\times}{10}^{-13}\text{\,}$.$ (59)

The single-photon coupling rate between the acoustic and bare optical modes is
then

$g_{0}=2\times\sqrt{\frac{\kappa_{o}\kappa_{m}C_{0}}{4}}\approx
2\pi\times$42\text{\,}\mathrm{Hz}$.$ (60)

Here $\kappa_{o}=2\pi\times$170\text{\,}\mathrm{MHz}$$,
$\kappa_{m}=2\pi\times$13\text{\,}\mathrm{MHz}$$ and the additional factor of
two comes from the hybridization of the optical modes (Eq. 18).

###  E.2 Estimation of on-chip and internal transduction efficiencies

We have achieved $\eta^{\mathrm{tot}}=$-48\text{\,}\mathrm{dB}$$ at an input
pump power of $21\text{\,}\mathrm{dBm}$. With the losses quoted in Appendix
E.1, the on-chip and internal efficiencies can be estimated. For conversion in
either directions, the signal goes through the respective input and output
ports exactly once, acquiring an attenuation of
$\eta^{\mathrm{probes}}\eta^{\mathrm{fiber-chip}}=$-7\text{\,}\mathrm{dB}$$.
We therefore have $\eta^{\mathrm{oc}}=$7.9\text{\times}{10}^{-5}\text{\,}$$.
Knowing the extraction efficiencies $\eta_{o}=35\%$ and $\eta_{m}=11\%$, we
further obtain $\eta^{\mathrm{int}}=$2\text{\times}{10}^{-3}\text{\,}$$.

##  Appendix F Optimization of optical extraction efficiency

An optimal optical external coupling can be chosen to maximize the total
conversion efficiency. Equation 58 can be rewritten in terms of quality
factors by using the relation $\kappa=\omega/Q$, which yields

$\displaystyle\eta^{\mathrm{tot}}$ $\displaystyle\approx
16\eta^{\mathrm{probes}}\eta^{\text{fiber-
fiber}}\times\eta_{m}\eta_{o}^{2}\times
g_{0}^{2}\frac{Q_{m}}{\omega_{m}}\frac{Q_{o}}{\omega_{o}}\times\frac{Q_{o}}{\omega_{o}}\frac{P_{\mathrm{in}}}{\hbar\omega_{L}}.$
(61)

For the optics, we have $Q_{o}=Q_{\mathrm{int}}+Q_{\mathrm{ext}}$, where we
separate the intrinsic quality factor $Q_{\mathrm{int}}$, oftentimes
fabrication-limited and hence not easily adjustable, from the external
coupling quality factor $Q_{\mathrm{ex}}$ that can be readily engineered
through coupler design. The optical extraction efficiency can thus be written
in the form

$\eta_{o}=\frac{\kappa_{\mathrm{ex}}}{\kappa_{\mathrm{ex}}+\kappa_{0}}=\frac{Q_{\mathrm{int}}}{Q_{\mathrm{ex}}}\left(1+\frac{Q_{\mathrm{int}}}{Q_{\mathrm{ex}}}\right)^{-1}=\frac{R}{1+R},$
(62)

where $R=Q_{\mathrm{int}}/Q_{\mathrm{ex}}$ and the total optical quality
factor

$Q_{o}=\frac{1}{Q_{\mathrm{int}}^{-1}+Q_{\mathrm{ex}}^{-1}}=Q_{\mathrm{int}}\frac{1}{1+R}.$
(63)

The total efficiency in the low-cooperativity regime is then given by

$\eta^{\mathrm{tot}}\approx F\frac{R^{2}}{(1+R)^{4}},$ (64)

where

$F=16\eta^{\mathrm{probes}}\eta^{\text{fiber-
fiber}}\times\eta_{m}g_{0}^{2}\frac{Q_{m}Q_{\mathrm{int}}^{2}}{\omega_{m}\omega_{o}^{2}}\times\frac{P_{\mathrm{in}}}{\hbar\omega_{L}}.$
(65)

For a given $F$, the optimal efficiency is achieved at the critical coupling
condition $Q_{\mathrm{ex}}=Q_{\mathrm{int}}$, or $R=1$.

## References

Supplementary Fig. 1:  Signal flow graphs of the frequency conversion
processes. a Effective beam-splitter process. b Effective two-mode-squeezing
process. The bath operators are indicated by rectangles, the internal modes of
the transducers by circles and the coupling rates by arrows.  Supplementary
Fig. 2:  Thermal response measurement through pump-probe spectroscopy. a
Schematics of the setup used for optical pump-probe spectroscopy. A pump
laser, addressing a micro-ring resonance and modulated by an intensity
modulator (IM), induces photothermal and Kerr cross-phase modulations. As a
result, the frequency of a nearby resonance is also modulated, which is
detected by a probe laser. The power of the probe laser is adjusted by a
variable attenuator (ATT). The pump and probe polarizations are optimized with
their respective fiber polarization controllers (FPC). The white boxes
represent fiber beam splitters. b Measured cross-phase modulation response of
a micro-ring as detected at the “Probe” photodetector with the pump filtered.
The dotted and dashed lines indicate the $3$-dB response bandwidths of the
“global” and “local” photothermal processes, respectively.  Supplementary Fig.
3:  Elimination of substrate HBAR modes. Microwave reflection spectra obtained
with and without applying the Band-Aid process. The respective stack
compositions are indicated as insets.