# Quantum-enabled continuous microwave-to-optics frequency conversion

Han Zhao  Moore Laboratory of Engineering, California Institute of Technology,
Pasadena, California 91125  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125  William David
Chen  Moore Laboratory of Engineering, California Institute of Technology,
Pasadena, California 91125  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125  Abhishek
Kejriwal  Moore Laboratory of Engineering, California Institute of Technology,
Pasadena, California 91125  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125  Mohammad
Mirhosseini  Moore Laboratory of Engineering, California Institute of
Technology, Pasadena, California 91125  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, California 91125

###### Abstract

A quantum interface between microwave and optical photons is essential for
entangling remote superconducting quantum processors. To preserve fragile
quantum states, a transducer must operate efficiently while generating less
than one photon of noise referred to its input. Here, we present a platform
that meets these criteria, utilizing a combination of electrostatic and
optomechanical interactions in devices made entirely from crystalline silicon.
This platform’s small mechanical dissipation and low optical absorption enable
ground-state radiative cooling, resulting in quantum-enabled operation with a
continuous laser drive. Under the optimal settings for high efficiency (low
noise), we measure an external efficiency of $2.2\%$ ($0.47\%$) and an input-
referred added noise of $0.94$ ($0.58$) in microwave-to-optics conversion. We
quantify the transducer throughput using the efficiency-bandwidth product,
finding it exceeds previous demonstrations with similar noise performance by
approximately two orders of magnitude, thereby paving a practical path to
interconnecting remote superconducting qubits.

## Introduction

Photons, the quantum particles of electromagnetic waves, are ideal carriers of
information due to their high propagation speed and minimal interaction with
the environment. A multitude of technologies, including global internet
infrastructure, take advantage of optical links for remote communication. With
progress in our abilities to compute and sense quantum signals, a growing
range of quantum information systems also require long-distance and multi-node
connectivity [1]. Quantum communication, much like its classical counterpart,
often relies on photons [2]. Interfaces between qubits and photons are thus
actively pursued for a variety of physical platforms [3]. Notably,
superconducting qubits – electrical circuits developed for quantum computing –
require microwave-optical interfaces for room-temperature connectivity [4].

To effectively transmit quantum states, microwave-to-optical transducers have
to possess multiple characteristics. A transducer must exhibit high
efficiency, ideally nearing $100\%$, for deterministic quantum state transfer
[5]. A more relaxed set of conditions can be defined for non-deterministic
communication, which can be achieved with no fundamental threshold for
efficiency so far as the noise is sufficiently low. While the exact noise
threshold for operation in this so-called _quantum-enabled_ regime [6] is
protocol-specific, an input noise of below one photon is a widely used
condition [7, 8]. In this regime, a practical metric for the transducer
throughput is set by the product of efficiency and the state-conversion
bandwidth, which bounds the output photon flux and subsequently the remote
entanglement generation rate [9, 10].

Achieving both high throughput and low-noise transduction has been challenging
despite steady progress in the past [11, 12, 13, 14, 15, 6, 16, 17, 18, 8, 19,
20, 21, 22, 23, 24, 25, 26, 27]. Initial approaches focused on maximizing
conversion efficiency through strong direct interactions between microwaves
and light [11, 12, 13, 14, 15, 6, 16] or via intermediaries like atoms [17,
18, 8, 19], magnons [28], or mechanical oscillators [20, 21, 22, 23, 24, 25,
26, 27]. However, it was soon found that the required strong optical and
electrical pumps – which are essential to bridge the vast frequency gap
between microwaves and light – result in thermal noise. To combat this effect,
experiments have turned to pulsed operations where strong pumps are supplied
only for short bursts to minimize heating. This approach has led to successful
demonstrations in the quantum-enabled regime, including state conversion from
qubits as well as microwave-optics quantum entanglement [22, 16, 27, 25].
Despite these successful demonstrations, the prolonged idling intervals in
pulsed operation have significantly reduced the output photon flux,
undermining the feasibility of networking operations with two or more nodes.

Figure 1: Electro-optomechanical transduction. a Schematic diagram of an
electro-optomechanical transducer. b Simulated displacement profile of the
mechanical mode, showcasing its extended energy distribution. The insets show
the microwave and optical field profiles. c Optical microscope image of a
fabricated chip containing an array of transducer devices. The optical
coupling port is laterally offset by $\sim$1.2 mm to avoid stray light
illumination of the microwave resonator. d Scanning electron microscope image
of a transducer. The nanobeam is surrounded by acoustic shields with a bandgap
at the mechanical frequency. The capacitor electrodes are shown in false
color. The letters designate the connection to the rest of the system as shown
in panel (a). Optical access is provided by evanescent coupling to a nearby
optical waveguide. e A close-up view of the microwave resonator meander
section (scanning electron microscope image, left), which serves to increase
the impedance of the microwave resonator for an enhanced electromechanical
coupling rate ($g_{\text{em}}\propto\sqrt{Z}$). The square loop array section
(optical microscope image, right) enables tuning of the microwave resonance
frequency via the dependence of the kinetic inductance on an external magnetic
field. f Power spectral density (PSD) of mechanical thermal motion measured
via optomechanical scattering of a red-detuned pump. g Coherent microwave-to-
optics scattering spectrum at $V_{\text{DC}}=20$ V.

Here, we introduce an integrated electro-optomechanical platform capable of
quantum-enabled performance under continuous operation. In this platform, we
eliminate the need for electrical pumping by employing a form of
electromechanical interaction that is based on electrostatic fields [29]. This
process’s material-agnostic nature allows us to construct the mechanical
oscillator entirely from single-crystal silicon, a material with exceptionally
low acoustic loss and low optical absorption [30, 31, 32]. These inherent
properties significantly increase cooperativities and reduce heating, enabling
ground-state radiative cooling of the mechanical oscillator. Consequently, the
device achieves microwave-to-optical conversion with an input-referred added
noise well below 1 photon. By operating continuously, our experiment harvests
the entire efficiency-bandwidth product, unlike pulsed schemes where the pulse
duty cycle reduces the throughput. As a result, we achieve a range of
throughputs ($0.47-1.9\text{ kHz}$) that exceeds previous demonstrations with
similar noise performance by about two orders of magnitude. Our experiment
thus establishes a new standard for microwave-to-optical quantum frequency
conversion and offers a viable path toward the near-term realization of remote
entanglement between distant superconducting qubits.

Figure 2: Transducer characterization a Measured electromechanical coupling
rate vs DC voltage bias. The coupling rate is determined at each voltage by
fitting the external decay rate of the mechanical mode. b Measured
optomechanical coupling rate versus intracavity photon number. The coupling
rate is derived at each photon number from the difference in the mechanical
linewidth under a red-detuned and a resonant pump with the same intracavity
photon number. c Reflection spectrum of the mechanical resonance. The
electromechanical external decay rate is minimized by choosing a large
microwave-mechanics detuning of $31\text{ MHz}$ at $V_{\text{DC}}=40\text{
V}$. The laser power is turned off, and the mechanical resonator is strongly
driven with a microwave tone (see Section D.4 for more details). d Reflection
spectrum of the microwave resonator on-resonance with the mechanical mode. e
Overview diagram of the 4 port measurements for calibrating the conversion
efficiency. The microwave measurement setup loss (gain) is denoted by
$\alpha_{\text{e}}$ ($\beta_{\text{e}}$). The optical measurement loss (gain)
is given by $\alpha_{\text{o}}$ ($\beta_{\text{o}}$). f Measured microwave-to-
optical transduction efficiency versus microwave drive frequency at
$V_{\text{DC}}=50\text{ V}$ and $n_{\text{c}}=232$. The solid line depicts the
Lorentzian fit of the measured transduction spectrum. The error bars in a and
b represent the 95% confidence interval.

## Transducer design

Our electro-optomechanical transducer features a gigahertz-frequency
mechanical oscillator that is simultaneously coupled to a microwave resonator
and an optical cavity (Fig. 1a). We achieve the electromechanical coupling
using electrostatic actuation, where the mechanical vibration of a DC-voltage-
biased capacitor creates a time-dependent dipole moment oscillating at the
mechanical frequency [29]. The charged moving capacitor is connected to a
microwave resonator that is resonant with the mechanical motion to facilitate
efficient inter-conversion between the microwave photons and phonons. The
mechanical oscillator also interacts with an optical resonator as part of a
cavity optomechanical system [33], where a red-detuned optical pump converts
phonons to optical photons. The system’s response can be described via the
Hamiltonian

$\begin{split}\hat{H}/\hbar=&\omega_{\text{e}}\hat{a}_{\text{e}}^{\dagger}\hat{a}_{\text{e}}+\omega_{\text{m}}\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{m}}+\omega_{\text{o}}\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{o}}\\\
+&G_{\text{em}}(\hat{a}_{\text{e}}\hat{a}_{\text{m}}^{\dagger}+\hat{a}_{\text{e}}^{\dagger}\hat{a}_{\text{m}})+G_{\text{om}}(\hat{a}_{\text{o}}\hat{a}_{\text{m}}^{\dagger}+\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{m}}).\end{split}$
(1)

Here, $\omega_{{j}}/2\pi$ ($j=$ e,m,o) are the resonance frequencies of the
microwave, mechanical and optical resonators, respectively. The
electromechanical and optomechanical couplings
($G_{\text{em}}=g_{\text{em}}V_{\text{DC}}$,
$G_{\text{om}}=g_{\text{om}}\sqrt{n_{\text{c}}}$) are enhanced from their
‘vacuum’ values ($g_{\text{em}}$, $g_{\text{om}}$) via a biasing DC voltage
($V_{\text{DC}}$) and an intracavity photon number ($n_{\text{c}}$) created by
the laser pump.

We realize this system in an integrated device based on a suspended nanobeam
of single-crystal silicon. The nanobeam is patterned with an air-hole phononic
crystal structure containing an electromechanical (EMC) and an optomechanical
crystal (OMC) cavity. The EMC section of the nanobeam supports a ‘breathing’
mechanical mode and is situated between the electrodes of a parallel-plate
capacitor that is connected to an on-chip microwave resonator. The OMC section
of the nanobeam supports an optical mode with a strong overlap to the
mechanical mode, providing optomechanical coupling through the photoelastic
response in silicon [34]. To keep the optical components distant from the
light-sensitive superconducting circuits, we separate the EMC and OMC sections
by extending the mechanical mode across the nanobeam using a phononic
waveguide (Fig. 1b) [35]. We fabricate the transducer on a silicon-on-
insulator (SOI) substrate along with a high-impedance flux-tunable microwave
resonator made from niobium titanium nitride (NbTiN) (Fig. 1c,d,e) [36, 37].

Figure 3: Mechanical quantum ground-state cooling. a Thermometry measurements
of the mechanical occupancy (blue circles) and microwave occupancy (red
circles) at varying intracavity photon numbers. The shaded region idnicates
the range of photon numbers in which the transducer is operated. The
mechanical occupancy is measured with $\Gamma_{\text{em}}/2\pi=2.1$ kHz, which
is attained with $V_{\text{DC}}=40$ V and a deliberate detuning of the
microwave resonator from the mechanical resonance frequency by 4.4 MHz. b
Diagram of the heating and damping model for the mechanical mode. c
Electromechanical cooling as a function of DC bias voltage. The red curve is
the theoretical prediction (see Section A.4). The measurements are performed
with a continuous optical pump power corresponding to $n_{\text{c}}=2.3$.
Inset emphasizes the dominating cooling path via the electromechanical decay
channel ($\Gamma_{\text{em}}>>\Gamma_{\text{f}},\Gamma_{\text{om}}$). Error
bars in a and c represent the 95$\%$ confidence interval. In a and c, the
mechanical occupancy is measured under the red-detuned optical pump. Due to
the small $\Gamma_{\text{om}}$ compared to $\Gamma_{\text{i}}$, the
optomechanical backaction cooling is negligible.

## Transducer efficiency characterization

We characterize the transducer by mounting the device to the mixing stage of a
dilution refrigerator. We first probe the mechanical response using a pump
laser red-detuned from the optical cavity. Figure 1f shows the power spectral
density of the optomechanical scattering, from which we identify two
mechanical modes. To confirm the mechanical modes provide sufficient
electromechanical coupling for transduction, we apply a DC bias voltage to the
device and measure the coherent microwave-to-optical scattering. As shown in
Fig. 1g, we find two transduction peaks with the frequencies aligning with the
mechanical modes in the optomechanical spectrum. To achieve single-mode
operation, we tune the microwave resonator onto resonance with the mechanical
mode at $\omega_{\text{m}}/2\pi=5.074$ GHz, which has a better balance of
electromechanical and optomechanical coupling rates.

Our transducer operates in the Purcell regime where the microwave and optical
resonator linewidths are greater than their respective coupling rates to
mechanics [38]. In this regime, the electrical and optical resonators serve as
external radiation ports for the mechanical mode with the decay rates given by
the parametrically enhanced coupling rates and the resonators’ total decay
rates: $\Gamma_{\text{em}}=4G_{\text{em}}^{2}/\kappa_{\text{e}}$ and
$\Gamma_{\text{om}}=4G_{\text{om}}^{2}/\kappa_{\text{o}}$. The dependence of
the electromechanical (optomechanical) decay rate on the DC voltage (laser
power) provides a means of characterizing the interaction rate. Using this
approach (see Fig. 2a,b), we measure an electromechanical (optomechanical)
interaction rate of $g_{\text{em}}/2\pi=3.81\text{ kHz/V}$
($g_{\text{om}}/2\pi=343\text{ kHz}$), in reasonable agreement with device
modeling [35]. The extracted interaction rates and the mechanical internal
linewidth can be used to calculate the bi-directional conversion efficiency.
The transducer external efficiency, defined by the photon flux ratio between
the microwave and optical waveguides on the chip, is given by (see Section
A.3)

$\eta_{\text{ext}}=\eta_{\text{e}}\eta_{\text{o}}\frac{4\Gamma_{\text{em}}\Gamma_{\text{om}}}{(\Gamma_{\text{i}}+\Gamma_{\text{em}}+\Gamma_{\text{om}})^{2}}.$
(2)

Here, $\eta_{\text{e}}=\kappa_{\text{e,ext}}/\kappa_{\text{e}}$
($\eta_{\text{o}}=\kappa_{\text{o,ext}}/\kappa_{\text{o}}$) is the microwave
(optical) resonator external coupling efficiency, and $\Gamma_{\text{i}}$
denotes the intrinsic decay rates of the mechanical resonator. Figure 2c shows
the reflection spectrum of the mechanical resonator measured via the
electrical port with the laser drive turned off. The fitted intrinsic
linewidth, $\Gamma_{\text{i}}/2\pi=892\text{ Hz}$ ($Q_{\mathrm{m}}=5.7\times
10^{6}$), provides an estimate for the mechanical decay rate in our system,
and its exceptionally small value is a key feature of our platform. However,
this value cannot be used to directly calculate the transduction efficiency
due to the dependence of the mechanical linewidth on the laser drive in our
system [30] (see Section D.4).

As an alternative, we directly characterize the efficiency by measuring the
reflectance ($R_{\text{ee}}$, $R_{\text{oo}}$) and transmittance
($T_{\text{oe}}$, $T_{\text{eo}}$) between the transducer’s microwave and
optical ports [20]. The calculated quantity from this experiment
$\eta_{\text{ext}}=\sqrt{T_{\text{oe}}T_{\text{eo}}/(R_{\text{oo}}R_{\text{ee}})}$
provides the transducer external efficiency with no need for calibrating the
gains and losses in the measurement paths (Fig. 2e) (see Section D.5.1). Using
this method, we obtain a maximum external efficiency of
$\eta_{\text{ext,max}}=20.0\%\pm 0.9\%$ (see Fig. 2f). In our system, this
value is limited by the maximum DC voltage allowed by the measurement
apparatus ($V_{\text{DC}}=50$ V). We also calculate the efficiency by directly
measuring the mechanical linewidth with the laser power on and using Eq. 2.
The resulting value $\eta_{\text{ext,pred}}=22.0\pm 1.5\%$, agrees with the
result from four-port characterization, which verifies our model of the device
efficiency. As detailed in the next section, achieving the quantum-enabled
regime in our device requires a different set of conditions, resulting in
smaller efficiencies. Regardless, we point out that the maximum efficiency is
quite large compared to other integrated mechanics-mediated transducers [24,
26], despite our system’s lower electromechanical interaction rates, which
signifies the benefits of the low mechanical loss.

Figure 4: Quantum-enabled microwave-to-optical transduction. a Electrically
(blue points) and optically (orange points) measured input-referred added
noise of the transducer with varying optical intracavity photon number. b
Electrically measured input-referred added noise of the transducer with
varying optical intracavity photon numbers. c Corresponding transduction
efficiencies for the same photon numbers and bias voltages as in b from four-
port measurements. The shaded region represents the quantum-enabled regime at
50 V biasing DC voltage. d Comparison of the transducer performance with the
literature. Red dots are the measured performance of our transducer.
References: Meesala, 2024 [27]; Sahu, 2022 [6]; Mirhosseini, 2020 [22]; Jiang,
2023 [24]; Brubaker, 2022 [23], Weaver, 2023 [26]. Error bars in all panels
represent the 95$\%$ confidence interval. Shaded regions in b and c denote the
quantum-enabled regime, where $n_{\text{add}}<1$.

## Quantum ground-state cooling

Thermal occupation in the microwave and mechanical resonators is the primary
source of noise in an electro-optomechanical transducer [30, 31, 22, 24, 27,
6, 26]. To characterize the effective mode temperature of these components in
our device, we perform thermometry measurements via the electrical port (see
Section D.6.1). Figure 3a shows the measurement results. We observe that the
microwave thermal population remains low at $n_{\text{mw}}\lesssim 0.1$ and
approximately independent of the laser power for $n_{\text{c}}\lesssim 200$
(equivalent to $3.3\text{ }\mu\text{W}$ at the optical waveguide). We
attribute the microwave resonator’s robustness against laser drive to the fast
quasiparticle recombination rate of NbTiN [39] and the optical waveguide
design that we have used to minimize direct illumination of the superconductor
(Fig. 1c). In contrast, we observe a much larger mechanical occupation with a
pronounced dependence on laser power. This observed heating of the mechanical
mode is consistent with previous measurements in a variety of integrated
optomechanical devices [30, 31, 40, 41, 42] and is the major roadblock to
achieving low-noise transduction.

The mechanical heating can be understood via a phenomenological model (see
Fig. 3b), where a laser-power-dependent hot bath (at the occupation number
$n_{\text{p}}$) is coupled to the mechanical mode (at the rate
$\Gamma_{\text{p}}$) [30]. The heating from this hot bath competes with the
cooling rate of the mechanical oscillator by the fridge’s cold bath
($\Gamma_{\text{f}}$) and the electrical and optical output ports
($\Gamma_{\text{em}}+\Gamma_{\text{om}}$). At equilibrium, the mechanical
occupation can be found as 111In this expression, we neglect the contribution
to mechanical heating from the nonzero microwave occupation to simplify the
presentation. The full model used for all our data analysis captures this
effect and is described in supplementary section ID.

$n_{\text{m}}\approx\frac{\Gamma_{\text{p}}n_{\text{p}}}{\Gamma_{\text{i}}+\Gamma_{\text{em}}+\Gamma_{\text{om}}}.$
(3)

Here, $\Gamma_{\text{i}}=\Gamma_{\text{f}}+\Gamma_{\text{p}}$ is the intrinsic
mechanical decay rate. The dependence of mechanical occupation on the
electromechanical and optomechanical decay rates provides a means of
mitigating thermal noise via these interactions. In our system, the
electromechanical decay rate can greatly exceed all the other rates, acting as
the dominant source of cooling. Figure 3c shows the results of an experiment
where the mechanical occupation is measured for a fixed laser power
(corresponding to $n_{\text{c}}=2.3$) as we gradually increase the
electromechanical decay rate by increasing $V_{\text{DC}}$. It can be seen
that the mechanical mode, which is initially quite hot, enters the quantum
ground state ($n_{\text{m}}<1$) for sufficiently large $V_{\text{DC}}$.

We observe that the ground-state electromechanical cooling is only possible
because of the absence of any observable heating from the electrostatic field.
This observation is supported by the good agreement between our measurements
and theoretical predictions, as shown in Fig. 3c. This contrasts with
processes that utilize radio-frequency pumps, which can create significant
heating [44]. Moreover, the effectiveness of electromechanical cooling in
mitigating heating is largely due to the inherently low heating rate in our
system. This rate, determined by the product $\Gamma_{\text{p}}n_{\text{p}}$,
depends on both the optical absorption coefficient and the mechanical loss
tangent, properties that are strongly material-dependent [45, 40, 46, 47, 42].
Previous studies have identified silicon as having low optical absorption as
well as exceptionally low acoustic loss [41, 32]. The material-independent
nature of electrostatic coupling in our approach thus allows us to leverage
these advantageous properties, together with radiative cooling, to achieve
low-noise operation as we elaborate on in the next section.

## Quantum-enabled operation

Although ground-state cooling is a necessary condition for quantum-enabled
operation, it is not sufficient for it. To find the added noise for operation
in up- or down-conversion (i.e., microwave-to-optics or optics-to-microwave),
the thermal occupations from all components need to be referred to the
corresponding input port. In our system, the much larger electromechanical
decay rate $\Gamma_{\text{em}}\gg\Gamma_{\text{om}}$ creates asymmetric
implications for the added noise in these two operation modes, where we can
only reach the quantum-enabled regime in upconversion. This asymmetric
performance can be understood by the fact that electromechanical cooling
channels thermal noise primarily to the microwave port, with the optical
output port remaining largely noise-free.

We utilize two independent methods for measuring the input-referred added
noise $n_{\text{add}}$ in microwave-to-optics conversion. In the first
approach, the noise in the optical output port ($n_{\text{out}}$) is measured
directly in a heterodyne measurement setup and then referred to the input
using the external efficiency found from the four-port measurements described
above, $n_{\text{add}}={n_{\text{out}}}/{\eta_{\text{ext}}}$ (see Section
D.7). In the second approach, we electrically measure the mechanical thermal
occupancy and convert it to $n_{\text{add}}$ via the relation
$n_{\text{add}}=n_{\text{m}}{\Gamma_{\text{tot}}}/({\eta_{\text{e}}\Gamma_{\text{em}}})$
(see Section A.5). Figure 4a compares the results from these two measurement
methods. Although we find good agreement between the two methods, we use the
electrical measurements in the low-noise regime because of its much higher
signal-to-noise ratio, which results from the much larger readout rate in the
electrical port. Figure 4b plots the electrical measurement results for
various values of laser power and voltage bias. As evident, for
$V_{\text{DC}}=$ 50 V, the input noise drops below $n_{\text{add}}=1$ for an
extended range of laser powers.

Figure 4c shows the transduction efficiencies from four-port measurements for
the voltage and photon number values used in Fig. 4b. The shaded region in the
figure depicts the parameter space where the added noise remains below one for
$V_{\text{DC}}=$ 50 V. We observe that increasing $\Gamma_{\text{em}}$ at a
fixed optical pump power results in reducing the noise as well as the
efficiency, where the increasing imbalance between the electromechanical and
optomechanical decay rates causes the latter. However, the transduction
efficiency increases with increasing $\Gamma_{\text{em}}$ for any fixed value
of added noise due to increased cooperativities. Constrained to operate in the
quantum-enabled regime, our transducer achieves a maximum external efficiency
of $\eta_{\text{ext}}=2.2\%$ at $n_{\text{add}}=0.94$. At this operation
setting, the transducer bandwidth, given by the total mechanical linewidth, is
measured as $\text{B}=\Gamma_{\text{tot}}/2\pi=88.9\pm 2.1$ kHz.

## Discussion

Our demonstration of quantum-enabled performance in continuous operation
contrasts all previous demonstrations, which have reached this regime via
pulsed operation. Conceptually, continuous operation has the advantage of
utilizing the entire efficiency-bandwidth product, whereas pulsed operation is
penalized by the duty cycle. Nevertheless, this conceptual distinction does
not provide a complete picture, as different platforms provide vastly
different bandwidths. To provide a quantitative measure of our results and
provide proper context with respect to previous work, we use the transducer
throughput, defined as the product of efficiency $\eta_{\text{ext}}$,
bandwidth B, and duty cycle D, where $\text{D}=1$ for continuous operation. In
Fig. 4d, we plot this product versus the input referred added noise from our
measurements and previous experiments. In the quantum-enabled regime, our
transducer achieves $\eta_{\text{ext}}\times\text{B}\times\text{D}=1.9\text{
kHz}$ with $n_{\text{add}}=0.94$. We also observe a trade-off between noise
and throughput in our data. At our lowest noise setting, we obtain
$n_{\text{add}}=0.58$ and
$\eta_{\text{ext}}\times\text{B}\times\text{D}=470\text{ Hz}$, which is about
two orders of magnitude larger than reported values in [6]. Although lowering
the optical pump power is expected to reduce noise further, accurately
characterizing the reduced device efficiency in this regime was prohibitively
long with heterodyne measurements in our experimental setup. Further
investigation of our platform in this regime with single-photon detectors will
be the subject of future work.

## Conclusions and outlook

In summary, we demonstrate quantum-enabled microwave-to-optical frequency
conversion on an integrated silicon platform. Utilizing electromechanical
ground-state cooling, our transducer operates continuously and achieves a
throughput that surpasses previous work significantly. The enhanced photon
emission rates resulting from this improvement hold great promise for future
demonstrations of optical remote entanglement with superconducting qubits.
Moreover, the simplicity of fabricating our devices, which rely solely on
standard silicon-on-insulator substrates and metal deposition, is promising
for scalable adoption, potentially via foundry services. Importantly, our
demonstrated bandwidth adequately matches the coherence of standard planar
qubits, and continuous operation simplifies integration with a qubit source,
which can be placed on a separate chip and connected by low-loss coaxial
cables [48]. A technical consideration for these future experiments would be
isolating the qubit source from the thermal noise emanating from the
electrical port of the transducer, which can be achieved either by low-loss
bulk circulators or by engineering directional qubit-waveguide interfaces [49,
50]. Lastly, we expect our system to improve in noise performance by about an
order of magnitude using optomechanical cavities with better thermal contact
[41, 51]. With reduced noise, the demonstrated kilohertz-level throughputs
would enable quantum networking operations with superconducting qubits at
rates comparable to optical qubit platforms [52, 53].

## Acknowledgments

We acknowledge S. Meesala, O. Painter, A. Safavi-Naeini, A. Bozkurt, and M
Kalaee for helpful discussions. This work was supported by the ARO/LPS Cross
Quantum Technology Systems program (grant W911NF-18-1-0103), the US Department
of Energy Office of Science National Quantum Information Science Research
Centers (Q-NEXT, award DE-AC02-06CH11357), and National Science Foundation
(awards 2137645 and 2238058). W.D.C gratefully acknowledges support from the
National Science Foundation Graduate Research Fellowship.

##  Appendix A Theoretical model of electro-optomechanical transducers

###  A.1 Hamiltonian

Our electro-optomechanical transducer is comprised of a mechanical resonator
that is simultaneously coupled to a microwave and optical resonator. The bare
Hamiltonian is given by

$\frac{\hat{H}_{\text{o}}}{\hbar}=\omega_{\text{e}}\hat{a}_{\text{e}}^{\dagger}\hat{a}_{\text{e}}+\omega_{\text{m}}\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{m}}+\omega_{\text{o}}\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{o}},$
(4)

where $\omega_{j}$ , with $j=\text{e,m,o}$, are the microwave, mechanical, and
optical resonator frequencies, respectively, and $\hat{a}_{j}^{\dagger}$
($\hat{a}_{j}$) are the corresponding raising (lowering) operators. Note that
$\hat{a}_{\text{o}}$ represents fluctuations about the coherent optical pump
field.

To simplify the following analysis, we work with the optical mode in a
rotating frame at the pump laser frequency $\omega_{\text{L}}$. The bare
Hamiltonian becomes

$\frac{\hat{H}_{\text{o}}}{\hbar}=\omega_{\text{e}}\hat{a}_{\text{e}}^{\dagger}\hat{a}_{\text{e}}+\omega_{\text{m}}\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{m}}+\Delta_{\text{o}}\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{o}},$
(5)

where $\Delta_{\text{o}}=\omega_{\text{o}}-\omega_{\text{L}}$ is the pump
laser detuning.

The electromechanical interaction term is given by [29]

$\begin{split}\frac{\hat{H}_{\text{em}}}{\hbar}&=\frac{\partial{C_{\text{m}}}}{\partial{x}}x_{\text{zpf}}V_{\text{zpf}}V_{\text{DC}}(\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{e}}+\hat{a}_{\text{m}}\hat{a}_{\text{e}}^{\dagger})\\\
&=G_{\text{em}}(\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{e}}+\hat{a}_{\text{m}}\hat{a}_{\text{e}}^{\dagger}),\end{split}$
(6)

where $C_{\text{m}}$ is the capacitance of the moving capacitor,
$x_{\text{zpf}}$ is the zero point motion of the mechanical resonator,
$V_{\text{zpf}}$ is the zero point voltage of the microwave resonator, and
$V_{\text{DC}}$ is the DC bias applied to the capacitor.
$G_{\text{em}}=\frac{\partial{C_{\text{m}}}}{\partial{x}}x_{\text{zpf}}V_{\text{zpf}}V_{\text{DC}}$
is the electromechanical coupling rate.

The optomechanical interaction term is given by [33]

$\begin{split}\frac{\hat{H}_{\text{om}}}{\hbar}=g_{\text{o}}\sqrt{n_{\text{c}}}(\hat{a}_{\text{o}}^{\dagger}+\hat{a}_{\text{o}})(\hat{a}_{\text{m}}^{\dagger}+\hat{a}_{\text{m}})\approx
G_{\text{om}}(\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{m}}+\hat{a}_{\text{o}}\hat{a}_{\text{m}}^{\dagger}),\end{split}$
(7)

where $G_{\text{om}}=g_{\text{o}}\sqrt{n_{\text{c}}}$ is the optomechanical
coupling rate. The interaction takes on a beam splitter form since we operate
with the pump laser red-detuned from the optical cavity
($\Delta_{\text{o}}=\omega_{\text{m}}$) in the sideband resolved regime
($\omega_{\text{m}}>>\kappa_{\text{o}}$).

Therefore, the full Hamiltonian is given by

$\begin{split}\frac{\hat{H}}{\hbar}=\omega_{\text{e}}\hat{a}_{\text{e}}^{\dagger}\hat{a}_{\text{e}}+\omega_{\text{m}}\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{m}}+\Delta_{\text{o}}\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{o}}+G_{\text{em}}(\hat{a}_{\text{m}}^{\dagger}\hat{a}_{\text{e}}+\hat{a}_{\text{m}}\hat{a}_{\text{e}}^{\dagger})+G_{\text{om}}(\hat{a}_{\text{o}}^{\dagger}\hat{a}_{\text{m}}+\hat{a}_{\text{o}}\hat{a}_{\text{m}}^{\dagger}).\end{split}$
(8)

###  A.2 Heisenberg-Langevin equations of motion

By introducing noise operators into the Heisenberg equations of motion, we can
write down a system of first-order differential equations that fully describe
the response of the electro-optomechanical transducer in the presence of
thermal baths. The noise correlations are given by

$\begin{split}\langle\hat{a}_{j,\text{ext}}^{\dagger}(t)\hat{a}_{j,\text{ext}}(t^{\prime})\rangle&=\langle\hat{a}_{j,\text{ext}}(t)\hat{a}_{j,\text{ext}}^{\dagger}(t^{\prime})\rangle-\delta(t-t^{\prime})=n_{j,\text{ext}}\delta(t-t^{\prime}),\\\
\langle\hat{a}_{j,\text{int}}^{\dagger}(t)\hat{a}_{j,\text{int}}(t^{\prime})\rangle&=\langle\hat{a}_{j,\text{int}}(t)\hat{a}_{j,\text{int}}^{\dagger}(t^{\prime})\rangle-\delta(t-t^{\prime})=n_{j,\text{int}}\delta(t-t^{\prime}),\\\
\langle\hat{a}_{\text{f}}^{\dagger}(t)\hat{a}_{\text{f}}(t^{\prime})\rangle&=\langle\hat{a}_{\text{f}}(t)\hat{a}_{\text{f}}^{\dagger}(t^{\prime})\rangle-\delta(t-t^{\prime})=n_{\text{f}}\delta(t-t^{\prime}),\\\
\langle\hat{a}_{\text{p}}^{\dagger}(t)\hat{a}_{\text{p}}(t^{\prime})\rangle&=\langle\hat{a}_{\text{p}}(t)\hat{a}_{\text{p}}^{\dagger}(t^{\prime})\rangle-\delta(t-t^{\prime})=n_{\text{p}}\delta(t-t^{\prime}),\end{split}$
(9)

where $\hat{a}_{j,\text{ext}}$, with $j=$ e,o are the electrical and optical
waveguide baths with damping rate $\kappa_{j,\text{ext}}$ and thermal
occupancy $n_{j,\text{ext}}$ ; $\hat{a}_{j,\text{int}}$ are the intrinsic
resonator baths with damping rate $\kappa_{j,\text{int}}$ and thermal
occupancy $n_{j,\text{int}}$ ; $\hat{a}_{\text{f}}$ is the intrinsic fridge
mechanics bath with damping rate $\Gamma_{\text{f}}$ and thermal occupancy
$n_{\text{f}}$ ; $\hat{a}_{\text{p}}$ is a hot mechanics bath due to optical
absorption heating with damping rate $\Gamma_{\text{p}}$ and thermal occupancy
$n_{\text{p}}$.

The corresponding Heisenberg-Langevin equations of motion are given by

$\begin{split}&\dot{\hat{a}}_{\text{o}}=-(i\Delta_{\text{o}}+\kappa_{\text{o}}/2)\hat{a}_{\text{o}}-iG_{\text{om}}\hat{a}_{\text{m}}+\sqrt{\kappa_{\text{o,ext}}}\hat{a}_{\text{o,ext}}+\sqrt{\kappa_{\text{o,int}}}\hat{a}_{\text{o,int}},\\\
&\dot{\hat{a}}_{\text{e}}=-(i\omega_{\text{e}}+\kappa_{\text{e}}/2)\hat{a}_{\text{e}}-iG_{\text{em}}\hat{a}_{\text{m}}+\sqrt{\kappa_{\text{e,ext}}}\hat{a}_{\text{e,ext}}+\sqrt{\kappa_{\text{e,int}}}\hat{a}_{\text{e,int}},\\\
&\dot{\hat{a}}_{\text{m}}=-(i\omega_{\text{m}}+\Gamma_{\text{i}}/2)\hat{a}_{\text{m}}-iG_{\text{om}}\hat{a}_{\text{o}}-iG_{\text{em}}\hat{a}_{\text{e}}+\sqrt{\Gamma_{\text{f}}}\hat{a}_{\text{f}}+\sqrt{\Gamma_{\text{p}}}\hat{a}_{\text{p}},\end{split}$
(10)

where $\kappa_{j}=\kappa_{j,\text{ext}}+\kappa_{j,\text{int}}$ is the
resonator total linewidth, and
$\Gamma_{\text{i}}=\Gamma_{\text{f}}+\Gamma_{\text{p}}$ is the mechanical
intrinsic linewidth.

From input-output theory, the output microwave and optical fields are given by
$\hat{a}_{j,\text{out}}=\sqrt{\kappa_{j,\text{ext}}}\hat{a}_{j}-\hat{a}_{j,\text{ext}}$.

This system of first-order differential equations can be described using a
state-space model [20, 21]

$\begin{split}\dot{a}(t)=Aa(t)+Ba_{\text{in}}(t),\\\
a_{\text{out}}(t)=Ca(t)+Da_{\text{in}}(t),\end{split}$ (11)

where
$a_{\text{out}}=[\hat{a}_{\text{e,out}},\hat{a}_{\text{o,out}},\hat{a}_{\text{e,out}}^{\dagger},\hat{a}_{\text{o,out}}^{\dagger}]^{\intercal}$
contains the output fields,
$a_{\text{in}}=[\hat{a}_{\text{e,ext}},\hat{a}_{\text{e,int}},\hat{a}_{\text{o,ext}},\hat{a}_{\text{o,int}},\hat{a}_{\text{f}},\hat{a}_{\text{p}}\\\
,\hat{a}_{\text{e,ext}}^{\dagger},\hat{a}_{\text{e,int}}^{\dagger},\hat{a}_{\text{o,ext}}^{\dagger},\hat{a}_{\text{o,int}}^{\dagger},\hat{a}_{\text{f}}^{\dagger},\hat{a}_{\text{p}}^{\dagger}]^{\intercal}$
contains the input fields, and
$a=[\hat{a}_{\text{e}},\hat{a}_{\text{o}},\hat{a}_{\text{m}},\hat{a}_{\text{e}}^{\dagger},\hat{a}_{\text{o}}^{\dagger},\hat{a}_{\text{m}}^{\dagger}]^{\intercal}$
contains the resonator fields. The matrices are given by

$A=\begin{pmatrix}-(i\omega_{\text{e}}+\kappa_{\text{e}}/2)&0&-iG_{\text{em}}&0&0&0\\\
0&-(i\Delta_{\text{o}}+\kappa_{\text{o}}/2)&-iG_{\text{om}}&0&0&0\\\
-iG_{\text{em}}&-iG_{\text{om}}&-(i\omega_{\text{m}}+\Gamma_{\text{i}}/2)&0&0&0\\\
0&0&0&-(-i\omega_{\text{e}}+\kappa_{\text{e}}/2)&0&iG_{\text{em}}\\\
0&0&0&0&-(-i\Delta_{\text{o}}+\kappa_{\text{o}}/2)&iG_{\text{om}}\\\
0&0&0&iG_{\text{em}}&iG_{\text{om}}&-(-i\omega_{\text{m}}+\Gamma_{\text{i}}/2)\\\
\end{pmatrix},$ (12)

$B=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}\sqrt{\kappa_{\text{e,ext}}}&\sqrt{\kappa_{\text{e,int}}}&0&0&0&0&0&0&0&0&0&0\\\
0&0&\sqrt{\kappa_{\text{o,ext}}}&\sqrt{\kappa_{\text{o,int}}}&0&0&0&0&0&0&0&0\\\
0&0&0&0&\sqrt{\Gamma_{\text{f}}}&\sqrt{\Gamma_{\text{p}}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&\sqrt{\kappa_{\text{e,ext}}}&\sqrt{\kappa_{\text{e,int}}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\sqrt{\kappa_{\text{o,ext}}}&\sqrt{\kappa_{\text{o,int}}}&0&0\\\
0&0&0&0&0&0&0&0&0&0&\sqrt{\Gamma_{\text{f}}}&\sqrt{\Gamma_{\text{p}}}\\\
\end{pmatrix},$ (13)

$C=\begin{pmatrix}\sqrt{\kappa_{\text{e,ext}}}&0&0&0&0&0\\\
0&\sqrt{\kappa_{\text{o,ext}}}&0&0&0&0\\\
0&0&0&\sqrt{\kappa_{\text{e,ext}}}&0&0\\\
0&0&0&0&\sqrt{\kappa_{\text{o,ext}}}&0\\\ \end{pmatrix},$ (14)

$D=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}-1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&-1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&-1&0&0&0\\\ \end{pmatrix}.$ (15)

The transfer function of the electro-optomechanical transducer relates the
input to output fields in the Fourier domain:
$a_{\text{out}}(\omega)=\xi(\omega)a_{\text{in}}(\omega)$ where
$\xi(\omega)=C(-i\omega I-A)^{-1}B+D$ and $I$ is the identity matrix.

###  A.3 Efficiency and bandwidth

The microwave-to-optical transmission coefficient $s_{\text{oe}}(\omega)$ can
be calculated from the transfer function matrix element that maps input field
$\hat{a}_{\text{e,ext}}$ to output field $\hat{a}_{\text{o,out}}$:

$s_{\text{oe}}(\omega)=-\frac{\sqrt{\eta_{\text{e}}}\sqrt{\eta_{\text{o}}}\sqrt{\kappa_{\text{e}}}\sqrt{\kappa_{\text{o}}}G_{\text{em}}\chi_{\text{e}}(\omega)G_{\text{om}}\chi_{\text{o}}(\omega)}{G_{\text{em}}^{2}\chi_{\text{e}}(\omega)+G_{\text{om}}^{2}\chi_{\text{o}}(\omega)+\chi_{\text{m}}^{-1}(\omega)},$
(16)

where $\eta_{j}=\kappa_{j,\text{ext}}/\kappa_{j}$ for $j=\text{e,o}$, and the
susceptibilities are given by
$\chi_{\text{o}}^{-1}=i(\Delta_{\text{o}}-\omega)+\kappa_{\text{o}}/2$,
$\chi_{\text{e}}^{-1}=i(\omega_{\text{e}}-\omega)+\kappa_{\text{e}}/2$, and
$\chi_{\text{m}}^{-1}=i(\omega_{\text{m}}-\omega)+\Gamma_{\text{i}}/2$.

Since the device operates in the weak-coupling regime
($G_{\text{em}}<<\kappa_{\text{e}}$ and $G_{\text{om}}<<\kappa_{\text{o}}$),
we can express the transmission coefficient around the mechanical mode in
terms of the resonator-mediated damping rates
$\Gamma_{\text{em}}=\frac{G_{\text{em}}^{2}\kappa_{\text{e}}}{(\omega_{\text{e}}-\omega_{\text{m}})^{2}+(\kappa_{\text{e}}/2)^{2}}$
and
$\Gamma_{\text{om}}=\frac{G_{\text{om}}^{2}\kappa_{\text{o}}}{(\Delta_{\text{o}}-\omega_{\text{m}})^{2}+(\kappa_{\text{o}}/2)^{2}}$:

$s_{\text{oe}}(\omega)=-\frac{\sqrt{\eta_{\text{e}}}\sqrt{\eta_{\text{o}}}\sqrt{\Gamma_{\text{em}}}\sqrt{\Gamma_{\text{om}}}}{i(\omega_{\text{m}}-\omega)+(\Gamma_{\text{em}}+\Gamma_{\text{om}}+\Gamma_{\text{i}})/2}.$
(17)

Note that $|s_{\text{oe}}(\omega)|=|s_{\text{eo}}(\omega)|$ since the beam-
splitter interaction is symmetric with respect to microwave-to-optical and
optical-to-microwave state conversion.

Thus, the external microwave-optics conversion efficiency is given by

$\eta_{\text{ext}}(\omega)=|s_{\text{oe}}(\omega_{\text{m}})|^{2}=|s_{\text{eo}}(\omega_{\text{m}})|^{2}=\eta_{\text{e}}\eta_{\text{o}}\frac{\Gamma_{\text{em}}\Gamma_{\text{om}}}{(\omega_{\text{m}}-\omega)^{2}+\left((\Gamma_{\text{em}}+\Gamma_{\text{om}}+\Gamma_{\text{i}})/2\right)^{2}},$
(18)

and the internal transduction efficiency is
$\eta_{\text{int}}(\omega)=\eta_{\text{ext}}(\omega)/(\eta_{\text{e}}\eta_{\text{o}})$.

In the on-resonance condition
($\omega_{\text{e}}=\Delta_{\text{o}}=\omega=\omega_{\text{m}}$), the peak
external efficiency is as reported in the main text:

$\eta_{\text{ext}}=\eta_{\text{e}}\eta_{\text{o}}\frac{4\Gamma_{\text{em}}\Gamma_{\text{om}}}{(\Gamma_{\text{i}}+\Gamma_{\text{em}}+\Gamma_{\text{om}})^{2}}=\eta_{\text{e}}\eta_{\text{o}}\frac{4\Gamma_{\text{em}}\Gamma_{\text{om}}}{\Gamma_{\text{tot}}^{2}},$
(19)

where $\Gamma_{\text{em}}=4G_{\text{em}}^{2}/\kappa_{\text{e}}$,
$\Gamma_{\text{om}}=4G_{\text{om}}^{2}/\kappa_{\text{o}}$, and
$\Gamma_{\text{tot}}=\Gamma_{\text{i}}+\Gamma_{\text{em}}+\Gamma_{\text{om}}$.

From Eq. 18, it is clear that the transducer bandwidth is given by the total
mechanical linewidth:
$\text{B}=\Gamma_{\text{tot}}/2\pi=(\Gamma_{\text{em}}+\Gamma_{\text{om}}+\Gamma_{\text{i}})/2\pi$.

###  A.4 Microwave and mechanical resonator thermal occupancy

The microwave and mechanical resonator thermal occupancy can be derived from
the Heisenberg-Langevin equations of motion Eq. 10. In the Fourier domain, the
equations of motion become

$\begin{split}&\hat{a}_{\text{o}}(\omega)=(-iG_{\text{om}}\hat{a}_{\text{m}}(\omega)+\sqrt{\kappa_{\text{o,ext}}}\hat{a}_{\text{o,ext}}(\omega)+\sqrt{\kappa_{\text{o,int}}}\hat{a}_{\text{o,int}}(\omega))\chi_{\text{o}}(\omega),\\\
&\hat{a}_{\text{e}}(\omega)=(-iG_{\text{em}}\hat{a}_{\text{m}}+\sqrt{\kappa_{\text{e,ext}}}\hat{a}_{\text{e,ext}}(\omega)+\sqrt{\kappa_{\text{e,int}}}\hat{a}_{\text{e,int}}(\omega))\chi_{\text{e}}(\omega),\\\
&\hat{a}_{\text{m}}(\omega)=(-iG_{\text{om}}\hat{a}_{\text{o}}(\omega)-iG_{\text{em}}\hat{a}_{\text{e}}(\omega)+\sqrt{\Gamma_{\text{f}}}\hat{a}_{\text{f}}+\sqrt{\Gamma_{\text{p}}}\hat{a}_{\text{p}})\chi_{\text{m}}(\omega).\end{split}$
(20)

Since the mechanical resonator operates in the weak coupling regime
($G_{\text{em}}<<\kappa_{\text{e}}$), $\hat{a}_{\text{e}}(\omega)$ can be
written as

$\hat{a}_{\text{e}}(\omega)=(\sqrt{\kappa_{\text{e,ext}}}\hat{a}_{\text{e,ext}}(\omega)+\sqrt{\kappa_{\text{e,int}}}\hat{a}_{\text{e,int}}(\omega))\chi_{\text{e}}(\omega).\\\
$ (21)

For our device, the thermal baths that contribute to the resonator thermal
occupancies are $n_{\text{e,int}}$, $n_{\text{f}}$, and $n_{\text{p}}$ since
$n_{\text{e,ext}}\approx n_{\text{o,ext}}\approx n_{\text{o,int}}\approx 0$
(see comments in Section D.6.1). Therefore, the microwave resonator thermal
occupancy is given by [54]

$\begin{split}n_{\text{mw}}&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\omega\int_{-\infty}^{+\infty}d\omega^{\prime}\langle\hat{a}^{\dagger}_{\text{e}}(\omega^{\prime})\hat{a}_{\text{e}}(\omega)\rangle=\frac{\kappa_{\text{e,int}}n_{\text{e,int}}}{\kappa_{\text{e}}}.\end{split}$
(22)

Similarly, the mechanical resonator thermal occupancy is given by

$\begin{split}n_{\text{m}}&=\frac{1}{2\pi}\int_{-\infty}^{+\infty}d\omega\int_{-\infty}^{+\infty}d\omega^{\prime}\langle\hat{a}^{\dagger}_{\text{m}}(\omega^{\prime})\hat{a}_{\text{m}}(\omega)\rangle=\frac{\Gamma_{\text{em}}n_{\text{mw}}+\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}}{\Gamma_{\text{tot}}}.\end{split}$
(23)

###  A.5 Added noise

The transducer’s added noise in microwave-to-optical and optical-to-microwave
frequency conversion due to thermal baths can be calculated from the transfer
function matrix elements that map thermal bath inputs to output field
$\hat{a}_{\text{o,out}}$ and $\hat{a}_{\text{e,out}}$, respectively.

Using the transfer matrix $\xi(\omega)$, we find the output optical field
$\hat{a}_{\text{o,out}}(\omega)$ in the presence of all thermal baths to be

$\begin{split}\hat{a}_{\text{o,out}}(&\omega)=-\frac{\sqrt{\eta_{\text{e}}}\sqrt{\eta_{\text{o}}}\sqrt{\Gamma_{\text{em}}}\sqrt{\Gamma_{\text{om}}}}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\left(\hat{a}_{\text{e,ext}}(\omega)+\sqrt{\frac{\kappa_{\text{e,int}}}{\kappa_{\text{e,ext}}}}\hat{a}_{\text{e,int}}(\omega)\right)-\frac{i\sqrt{\eta_{\text{o}}}\sqrt{\Gamma_{\text{om}}}}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\left(\sqrt{\Gamma_{\text{f}}}\hat{a}_{\text{f}}(\omega)+\sqrt{\Gamma_{\text{p}}}\hat{a}_{\text{p}}(\omega)\right)\\\
&+\left(\frac{i(\omega_{\text{m}}-\omega)+(\Gamma_{\text{em}}+\Gamma_{\text{i}})/2}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\kappa_{\text{o,ext}}\chi_{\text{o}}(\omega)-1\right)\hat{a}_{\text{o,ext}}(\omega)+\frac{i(\omega_{\text{m}}-\omega)+(\Gamma_{\text{em}}+\Gamma_{\text{i}})/2}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\sqrt{\kappa_{\text{o,ext}}}\sqrt{\kappa_{\text{o,int}}}\chi_{\text{o}}(\omega)\hat{a}_{\text{o,int}}(\omega).\end{split}$
(24)

Similarly, we can find the output microwave field
$\hat{a}_{\text{e,out}}(\omega)$ to be

$\begin{split}\hat{a}_{\text{e,out}}(&\omega)=-\frac{\sqrt{\eta_{\text{e}}}\sqrt{\eta_{\text{o}}}\sqrt{\Gamma_{\text{em}}}\sqrt{\Gamma_{\text{om}}}}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\left(\hat{a}_{\text{o,ext}}(\omega)+\sqrt{\frac{\kappa_{\text{o,int}}}{\kappa_{\text{o,ext}}}}\hat{a}_{\text{o,int}}(\omega)\right)-\frac{i\sqrt{\eta_{\text{e}}}\sqrt{\Gamma_{\text{em}}}}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\left(\sqrt{\Gamma_{\text{f}}}\hat{a}_{\text{f}}(\omega)+\sqrt{\Gamma_{\text{p}}}\hat{a}_{\text{p}}(\omega)\right)\\\
&+\left(\frac{i(\omega_{\text{m}}-\omega)+(\Gamma_{\text{om}}+\Gamma_{\text{i}})/2}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\kappa_{\text{e,ext}}\chi_{\text{e}}(\omega)-1\right)\hat{a}_{\text{e,ext}}(\omega)+\frac{i(\omega_{\text{m}}-\omega)+(\Gamma_{\text{om}}+\Gamma_{\text{i}})/2}{i(\omega_{\text{m}}-\omega)+\Gamma_{\text{tot}}/2}\sqrt{\kappa_{\text{e,ext}}}\sqrt{\kappa_{\text{e,int}}}\chi_{\text{e}}(\omega)\hat{a}_{\text{e,int}}(\omega).\end{split}$
(25)

We analyze the electrical output noise in our discussion of electrical
thermometry (see Section D.6.1). For the rest of this section, we focus on the
noise at the optical output for microwave-to-optical conversion. The optical
noise spectral density $n_{\text{o,out}}(\omega)$ is related to the output
optical field by

$n_{\text{o,out}}(\omega)=\int_{-\infty}^{\infty}d\tau
e^{i\omega\tau}\langle\hat{a}_{\text{o,out}}^{\dagger}(0)\hat{a}_{\text{o,out}}(\tau)\rangle=\int_{-\infty}^{\infty}d\omega^{\prime}\langle(\hat{a}_{\text{o,out}}(\omega^{\prime}))^{\dagger}\hat{a}_{\text{o,out}}(\omega)\rangle.$
(26)

For our device, the thermal baths that contribute to the added noise are
$n_{\text{e,int}}$, $n_{\text{f}}$, and $n_{\text{p}}$ since
$n_{\text{e,ext}}\approx n_{\text{o,ext}}\approx n_{\text{o,int}}\approx 0$
(see comments in Section D.6.1). Therefore, the thermal noise at the optical
output is

$\begin{split}n_{\text{o,out}}(\omega)=\frac{\eta_{\text{e}}\eta_{\text{o}}\Gamma_{\text{em}}\Gamma_{\text{om}}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}\frac{\kappa_{\text{e,int}}}{\kappa_{\text{e,ext}}}n_{\text{e,int}}&+\frac{\eta_{\text{o}}\Gamma_{\text{om}}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}\Gamma_{\text{f}}n_{\text{f}}\\\
&+\frac{\eta_{\text{o}}\Gamma_{\text{om}}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}\Gamma_{\text{p}}n_{\text{p}}.\end{split}$
(27)

The up-conversion input-referred added noise is

$\begin{split}n_{\text{add}}=n_{\text{o,out}}(\omega)/\eta_{\text{ext}}(\omega)=\frac{\kappa_{\text{e,int}}}{\kappa_{\text{e,ext}}}n_{\text{e,int}}+\frac{\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}}{\eta_{\text{e}}\Gamma_{\text{em}}}.\end{split}$
(28)

We can express the input-referred added noise in terms of the mechanical mode
occupancy:

$\begin{split}n_{\text{add}}=\frac{\Gamma_{\text{em}}n_{\text{mw}}+\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}}{\eta_{\text{e}}\Gamma_{\text{em}}}=n_{\text{m}}\frac{\Gamma_{\text{tot}}}{\eta_{\text{e}}\Gamma_{\text{em}}}.\end{split}$
(29)

where
$n_{\text{mw}}=\frac{\kappa_{\text{e,int}}}{\kappa_{\text{e}}}n_{\text{e,int}}$
is the microwave resonator thermal occupancy (see Eq. 22), and
$n_{\text{m}}=\frac{\Gamma_{\text{em}}n_{\text{mw}}+\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}}{\Gamma_{\text{tot}}}$
is the mechanical mode thermal occupancy (see Eq. 23).

##  Appendix B Fabrication

We fabricate the transducers starting with a high resistivity ($>3\text{
k}\Omega\cdot\text{cm}$) 220-nm silicon-on-insulator substrate sputtered with
a 12-nm thin film of niobium titanium nitride (NbTiN) with sheet inductance of
$\sim 36$ pH/sq. NbTiN is used as the superconductor for microwave circuits
due to its large kinetic inductance and fast quasi-particle recombination rate
[36, 39]. First, we selectively remove the NbTiN film from the optical and
mechanical structures and define the microwave superconducting circuit by
electron-beam lithography (EBL), followed by a pseudo-Bosch dry etch with
SF6/Ar chemistry. This step is followed by a second aligned EBL and SF6/C4F8
dry etch to define the silicon nanobeams, phonon shields, release holes, and
optical waveguides. To create space for lensed-fiber end-fire coupling, we
etch a $\sim 120$ $\mu$m deep trench area at the end of the optical waveguide
using photolithography and SF6 chemical etching. The devices are finally
released with hydrofluoric (HF) acid and wire-bonded to a printed circuit
board for electrical contact.

##  Appendix C Measurement setup

Figure 5: Measurement setup. Acronyms: Isolator (Iso), beam splitter (BS),
wavelength meter ($\lambda$-meter), optical switch (OSw), fiber polarization
controller (FPC), band-pass filter (BPF), photodetector (PD), data-acquisition
module (DAQ), amplitude modulator (a-m), signal generator (SG), variable
optical attenuator (VOA), power meter (p-meter), circulator (Circ), device
under test (DUT), phase modulator ($\phi$-m), Erbium-doped fiber amplifier
(EDFA), delay line (DL), retro-reflector (RR), variable coupler (VC), high-
speed photodetector (HSPD), balanced photodetector (BPD), spectrum analyzer
(SA), switch (Sw), vector network analyzer (VNA), eccosorb filter (Ecco), low-
pass filter (LPF), bias-tee (BT), microwave switch (MSw), high-electron-
mobility transistor (HEMT), room-temperature amplifier (RT amplifier).  Figure
6: Fridge setup. A tuning coil is placed above the device chip to generate
magnetic flux that tunes the on-chip microwave resonators. The device chip
sits on a printed circuit board (PCB), with the on-chip coplanar waveguides
wire-bonded to the PCB traces. The microwave lines are connected to the PCB
via MMPX connectors.

Figure 5 illustrates the optics and microwave setup used for all measurements.
A tunable laser provides the optical pump for the transducer and is frequency-
stabilized using a wavelength meter. For the transduction measurements, the
optical pump beam is filtered by a 50-MHz tunable bandpass filter (BPF) to
suppress laser phase noise at the $\sim$5-GHz sidebands of the laser frequency
[55]. An amplitude modulator (a-m) is used to generate a probe tone at the
optical resonance frequency as needed. The optical pump power is controlled by
a variable optical attenuator (VOA) and is sent to the device under test (DUT)
at the mixing stage of the dilution refrigerator after an optical circulator
(Circ1). A lensed optical fiber mounted on a set of cryogenic x-y-z
nanopositioners is used to couple to the optical waveguides on the DUT (Fig.
6).

The optical reflection from the DUT contains the reflected pump, the
transducer’s output signal, and the added noise photons. We characterize the
optical reflection from the DUT using both homodyne and heterodyne detection
setups. In the homodyne setup, the reflected signal is amplified by an Erbium-
doped fiber amplifier (EDFA) and then detected by a high-speed photodetector
(HSPD). Before the HSPD, a tunable bandpass filter (BPF3) is used to reject
the excessive spontaneous emission from the EDFA. The beating notes of the
pump reflection and the transduction output generate photocurrents at the
mechanical frequencies. We send the photocurrents to a spectrum analyzer (SA)
and a vector network analyzer (VNA) to characterize the optomechanical
spectrum and the microwave-to-optical transduction spectrum, respectively. The
sensitivity of the homodyne detection, however, is too low for accurate
characterizations of the transducer’s efficiency and noise performance as the
optical pump power decreases.

For the measurements of the external efficiency and added noise, we switch to
the heterodyne setup where the transduction signal can be amplified by a
higher-intensity optical local oscillator (LO) independent of the optical pump
power. In this setup, the LO is split from the laser power using a 90:10 beam
splitter (BS2), with 90% of the power directed to the LO arm. The LO beats
with the optical reflection from the DUT via a variable coupler (VC), which is
subsequently converted to photocurrents by a balanced photodetector (BPD). To
generate the beating notes within the bandwidth of the BPD (400 MHz) for the
transduction signal, the LO is blue-shifted by a frequency 50 MHz lower than
the mechanical resonances. We achieve the frequency shift by a chain of phase
modulator ($\phi$-m), EDFA and another 50-MHz tunable bandpass filter (BPF2).
The optical length of LO is matched to the reflection signal from the DUT by a
round-trip delay line (DL) to minimize the decoherence of the beating notes.
We use the SA to analyze the transduction and the added noise contained in the
photocurrents from the BPD.

The microwave input signal is provided by either a VNA or a signal generator
(SG). To minimize the room-temperature microwave noise, the input signal is
attenuated by a total of $\sim$ 80 dB, with the attenuators thermalized to
different stages of the dilution refrigerator. The attenuated microwave input
passes through an Eccosorb filter (Ecco1) and a low-pass filter (LPF) to
filter out high-frequency noise. A microwave circulator (Circ3) directs the
input signal to the DUT and the reflection to the output line. The biasing DC
voltage is applied to the DUT along with the microwave input via a bias-tee
(BT). We use a microwave switch (MSw) to toggle the connections to different
devices on the chip. An additional circulator (Circ4) and Eccosorb filter
(Ecco2) are used at the output line to prevent the thermal noise of higher
temperature stages from reaching the DUT. A high-electron-mobility transistor
(HEMT) thermalized to the 4K stage, along with a low noise amplifier at room
temperature (RT Amplifier), amplifies the microwave output of the DUT. The
output signal after this amplification chain is detected by the SA or the VNA
based on the type of measurements performed.

##  Appendix D Transducer characterization

###  D.1 Device parameters

Symbol | Description | Value
---|---|---
$\omega_{\text{e}}/2\pi$ | Microwave resonator frequency | $5.0745\text{ GHz}$
$\kappa_{\text{e,ext}}/2\pi$ | Microwave resonator extrinsic linewidth |  $1.33\text{ MHz}$*
$\kappa_{\text{e,int}}/2\pi$ | Microwave resonator intrinsic linewidth | $330\text{ kHz}$
$\omega_{\text{o}}/2\pi$ | Optical resonator frequency | $192.9263\text{ THz}$
$\kappa_{\text{o,ext}}/2\pi$ | Optical resonator extrinsic linewidth | $1.35\text{ GHz}$
$\kappa_{\text{o,int}}/2\pi$ | Optical resonator intrinsic linewidth | $404\text{ MHz}$
$\omega_{\text{m}}/2\pi$ | Mechanical resonator frequency | $5.0745\text{ GHz}$
$\Gamma_{\text{i}}/2\pi$ | Mechanical mode saturated intrinsic linewidth | $892\text{ Hz}$
$g_{\text{em}}/2\pi$ | Electromechanical coupling rate | $3.81\text{ kHz/V}$
$g_{\text{om}}/2\pi$ | Optomechanical single-photon coupling rate | $343\text{ kHz}$
$\eta_{\text{f}}$ | Optical fiber-waveguide coupling efficiency | $35\%$

* See Section D.2 for comments.

Table 1: Device parameters

###  D.2 Microwave resonator linewidth

The microwave resonator’s external coupling rate, $\kappa_{\text{e,ext}}$,
exhibits a weak, nonmonotonic dependence on the applied flux, with noticeable
hysteresis. We attribute this to impedance mismatches caused by parasitic
coupling between the microwave waveguide and neighboring resonators, which
undergo phase slips at strong flux biases due to their large kinetic
inductance [56]. As a result of this behavior and the significant flux biasing
applied before taking the dataset at $V_{\text{DC}}=50\text{ V}$, the measured
value of $\kappa_{\text{e,ext}}/2\pi=1.33\text{ MHz}$ at this voltage differs
slightly from the value in all other experiments
($\kappa_{\text{e,ext}}/2\pi=1.5\text{ MHz}$).

###  D.3 Microwave resonator under optical illumination

Optical laser illumination of superconductors can break Cooper pairs via the
absorption of high-energy optical photons. The resulting non-equilibrium
quasiparticle population is believed to cause the increased microwave thermal
occupancy, degradation of the resonator Q-factor, and frequency shift observed
in previous transduction works [22, 26, 57]. These effects on the microwave
resonator are undesired since they directly lead to increased noise and lower
transduction efficiency. Our transducer’s NbTiN nanowire microwave resonator
is shown to remain robust under high intracavity optical pump photon numbers,
which we attribute to the superconductor’s fast quasiparticle recombination
rate [39] and an optical waveguide design that minimizes direct illumination
of the superconductor (see Figure 2c of main text). Figure 7 shows the
microwave thermal occupancy, frequency shift, and intrinsic Q-factor of the
microwave resonator as a function of the intracavity optical pump photon
number. For $n_{\text{c}}<200$, we measure the microwave resonator thermal
occupancy to be $n_{\text{mw}}\approx 0.1$ . The corresponding microwave
temperature ($\sim 100$ millikelvin) is reasonably close to the fridge
temperature ($\sim 30$ millikelvin), which is likely caused by reduced
infrared shielding in our measurement setup. Significant heating of the
microwave resonator, linewidth broadening, and frequency shift only occur for
$n_{\text{c}}>200$, a range that is two orders of magnitude greater than the
pump photon numbers used for operating in the quantum-enabled regime.

Figure 7: Microwave resonator under optical illumination. a) Microwave
resonator thermal occupancy vs intracavity photon number. The laser frequency
is detuned by the mechanical frequency to the red sideband of the optical
resonance for $n_{\text{c}}<400$. The laser is tuned on-resonance with the
optical resonator for $n_{\text{c}}>400$ due to limited cooling power at the
mixing plate. b) Microwave resonator frequency shift vs intracavity photon
number. c) Microwave resonator intrinsic quality factor vs intracavity photon
number. All data in (b,c) is collected with the laser on-resonance with the
optical cavity. Error bars denote 95% confidence intervals.

###  D.4 Mechanical resonator linewidth

The laser power strongly modifies the mechanical linewidth in our device (see
Fig. 8). At low optical powers (corresponding to $n_{\text{c}}\lesssim 1$),
the mechanical linewidth narrows as the optical pump power is increased. In
contrast, at higher optical power (corresponding to $n_{\text{c}}\gtrsim
5-10$), we observe a broadening of the mechanical linewidth concomitant with
increased mechanical occupation. This behavior is consistent with previous
observations in silicon phononic crystal resonators [58]. The linewidth
narrowing at low power has been attributed to the saturation of frequency
jitter caused by two-level system (TLS) defects [59, 32, 29]. The broadening
at higher laser powers is a signature of optical absorption-induced mechanical
decay, a process that has been attributed to 3-phonon mixing [58, 32].

Figure 8: Mechanical linewidth versus optical pump power. The intrinsic
linewidth, calculated by subtracting the electromechanical and optomechanical
decay rates from the measured total linewidth
($\Gamma_{\text{i}}=\Gamma_{\text{tot}}-\Gamma_{\text{om}}-\Gamma_{\text{em}}$).
The microwave drive has been set to zero in this measurement, and the
mechanical linewidth is measured from the thermal emission spectrum collected
via the electrical port. The DC bias voltage is set at $V_{\text{DC}}=30$ V.
Error bars denote 95% confidence intervals.

In the low optical power regime, where the mechanical linewidth is dominated
by interaction with the TLS bath, the mechanical linewidth can also change as
a function of the microwave drive [29]. This behavior can be utilized to
estimate the decay rate by saturating the frequency jitter via a microwave
drive tone. Figure 2c of the main text shows the results of such a
measurement, where we populate the mechanical mode with $9.3\times 10^{5}$
phonons in the absence of any optical pump. From a Lorentzian fit to the data,
we measure a linewidth of $\Gamma_{\text{i}}/2\pi=892\text{ Hz}$. This
measurement provides a conservative estimate for the decay rate
($\Gamma_{\text{f}}$) to the cold fridge bath ($n_{\text{f}})$. For
transduction efficiency and noise measurements presented in the main text, we
operate in the regime where the frequency jitter has been nearly completely
saturated with the laser power, and consequently, the device response becomes
independent of the microwave drives (see Section D.5.2 for details).

###  D.5 Measurement of transduction efficiency

####  D.5.1 Four-port measurements

To measure transduction efficiency, we employ a four-port measurement scheme
following Ref. [20], as described below. This method requires four separate
transmittance and reflectance measurements between the microwave and optical
ports (see Fig. 9a). The measured transmittance in up- and down-conversion can
be written in terms of the transducer efficiency

$T_{\text{oe}}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{P_{\text{oe}}/(\hbar\omega_{\text{o}})}{P_{\text{e,in}}/(\hbar\omega_{\text{e}})}=\alpha_{\text{e}}\eta_{\text{ext}}\beta_{\text{o}},$
(30)
$T_{\text{eo}}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{P_{\text{eo}}/(\hbar\omega_{\text{e}})}{P_{\text{o,in}}/(\hbar\omega_{\text{o}})}=\alpha_{\text{o}}\eta_{\text{ext}}\beta_{\text{e}}.$
(31)

Here, $\alpha_{\text{e}}$ ($\alpha_{\text{o}}$) is the power loss from the
microwave (optical) input port to the transducer; $\beta_{\text{e}}$
($\beta_{\text{o}}$) is the power gain from the transducer to the microwave
(optical) output port; $\eta_{\text{ext}}$ is the bidirectional conversion
efficiency.

To obtain the transduction efficiency from the transmission coefficients
$T_{\text{eo}}$ and $T_{\text{oe}}$, it is necessary to calibrate the
microwave/optical input loss and detection gain. To this end, we measure the
microwave-to-microwave (optical-to-optical) reflected power $P_{\text{ee}}$
($P_{\text{oo}}$) at a frequency point that is multiple linewidths away from
the microwave (optical) resonator. The reflectance at such a frequency point
will not be affected by the transducer response and will only be a function of
the gains and losses in the measurement setup

$R_{\text{ee}}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{P_{\text{ee}}/(\hbar\omega_{\text{e}})}{P_{\text{e,in}}/(\hbar\omega_{\text{e}})}=\alpha_{\text{e}}\beta_{\text{e}},$
(32)
$R_{\text{oo}}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{P_{\text{oo}}/(\hbar\omega_{\text{o}})}{P_{\text{o,in}}/(\hbar\omega_{\text{o}})}=\alpha_{\text{o}}\beta_{\text{o}}.$
(33)

We can combine the results of these four measurements to find the transduction
efficiency as

$\eta_{\text{ext}}=\sqrt{\frac{T_{\text{eo}}T_{\text{oe}}}{R_{\text{ee}}R_{\text{oo}}}}.$
(34)

The calculated result from this expression can, in turn, be used to find the
product of microwave loss and optical gain of the measurement setup as
($\alpha_{\text{e}}\beta_{\text{o}}=T_{\text{oe}}/\eta_{\text{ext}}$). Using
this method, we extract ($\alpha_{\text{e}}\beta_{\text{o}}$) from a set of
four-port measurements at a fixed optical power and DC bias. We subsequently
measure the transduction efficiency via Eq. 30 as we sweep the optical pump
power and bias voltage. During these experiments, we periodically repeat the
calibration of ($\alpha_{\text{e}}\beta_{\text{o}}$) to ensure we capture
slight variations in the optical and microwave gains.

In measuring microwave-to-optical transmittance ($T_{\text{oe}}$), the input
signal at the microwave port is supplied by an RF signal generator. In this
experiment, we measure the power of the optical output with a spectrum
analyzer in a heterodyne measurement setup (described in Appendix C). The
resolution bandwidth for detecting this signal (RBW = 2 kHz) was chosen to be
much smaller than the mechanical linewidth while sufficiently large to capture
the optical beat note. In measuring optical-to-microwave transmittance
($T_{\text{eo}}$), we generate the input optical signal by modulating the
intensity of the optical pump with a microwave tone created by an RF signal
generator. In this experiment, we amplify the transducer’s microwave output
with a chain of amplifiers and detect the output power with a spectrum
analyzer. Figure 9c shows examples of these measurements where the input
signal frequency is swept across the mechanical resonance.

Figure 9: Characterization of the transduction efficiency. a Schematic of the
four-port measurement. The microwave measurement setup loss (gain) are denoted
by $\alpha_{\text{e}}$ ($\beta_{\text{e}}$). Likewise, the optical measurement
loss (gain) are given by $\alpha_{\text{o}}$ ($\beta_{\text{o}}$).
$\eta_{\text{ext}}$ is the transducer bidirectional microwave-optical
transduction efficiency. $T_{\text{oe}}$ ($T_{\text{eo}}$) are the microwave-
to-optical (optical-to-microwave) transmittance. $R_{\text{ee}}$
($R_{\text{oo}}$) are the microwave-to-microwave (optical-to-optical)
reflectance. b Measured spectrum of the microwave-to-optical transduction
power ($P_{\text{oe}}$) as the microwave input frequency sweeps across the
mechanical frequency. c Measured spectrum of the optical-to-microwave
transduction power ($P_{\text{eo}}$) as the microwave drive frequency on the
optical amplitude modulator sweeps across the mechanical frequency. In b and
c, blue dots are measured data whereas red curves are the corresponding
Lorentzian fits. We use the peak powers of the fits to calculate the external
transduction efficiency in Eq. 34.

####  D.5.2 Effects of mechanical frequency jitter

As detailed in Section D.4, the mechanical resonator at low optical pump
powers experiences frequency jitter. The saturable nature of this effect as a
function of microwave drive can create a challenge for precise efficiency
characterization. To avoid this challenge, we operate within the range of
optical laser powers where this effect is nearly absent. Further, to quantify
any residual effects on our measurements, we compare the directly measured
microwave-optics transduction efficiency (which requires microwave drives) to
the theoretically calculated values using the measured mechanical linewidth in
the absence of any microwave drive.

Figure 10 shows the two data sets. The purple dots depict the direct
measurement using the four-port method detailed in Section D.5.1. In these
experiments, we have used an input microwave power of $P_{\text{e,in}}=-58$
dBm. The green dots in Fig. 10 depict the efficiency values calculated using
the expression
$\eta_{\text{ext}}=\eta_{\text{e}}\eta_{\text{o}}({4\Gamma_{\text{em}}\Gamma_{\text{om}}})/({\Gamma_{\text{tot}}^{2}}$).
For this calculation, we use the measured values of $\Gamma_{\text{em}}$,
$\Gamma_{\text{om}}$, $\eta_{\text{o}}$ and $\eta_{\text{e}}$, which are
extracted via the methods described in the main text. To find the mechanical
linewidth ($\Gamma_{\text{tot}}$), we collect the thermal emission spectrum
via the electrical port while setting the microwave input to zero. We observe
that for higher optical powers ($n_{\text{c}}\gtrsim 1$), we observe a very
good agreement between the two datasets, pointing to near-complete saturation
of the frequency jitter by the optical pump in this regime. However, we
observe a small difference ($<22\%$) between the two datasets for the lower
end of optical powers in our measurements.

Figure 10: Comparison of measured and predicted external transduction
efficiencies versus intracavity optical photon number at
$V_{\text{DC}}=50\text{ V}$. Purple data points are the transduction
efficiencies obtained via four-port measurements. Green data points are
calculated efficiencies using Eq. 19 where the total mechanical linewidth is
measured without any external microwave drive. Error bars denote 95%
confidence intervals.

###  D.6 Measurement of mechanical and microwave resonator occupancy

####  D.6.1 Electrical thermometry

We use electrical thermometry measurements to determine the mechanical and
microwave resonator thermal occupancies. In these experiments, a spectrum
analyzer is used to measure the symmetrized noise power spectral density
$S(\omega)$. It can be shown that this quantity is related to the output
microwave field $\hat{a}_{\text{e,out}}$ by [60]

$\frac{S(\omega)}{\hbar\omega}=\frac{1}{2}\int_{-\infty}^{\infty}d\tau
e^{i\omega\tau}\langle\hat{a}_{\text{e,out}}^{\dagger}(0)\hat{a}_{\text{e,out}}(\tau)+\hat{a}_{\text{e,out}}(\tau)\hat{a}_{\text{e,out}}^{\dagger}(0)\rangle=\frac{1}{2}\int_{-\infty}^{\infty}d\omega^{\prime}\langle(\hat{a}_{\text{e,out}}(\omega^{\prime}))^{\dagger}\hat{a}_{\text{e,out}}(\omega)+\hat{a}_{\text{e,out}}(\omega)(\hat{a}_{\text{e,out}}(\omega^{\prime}))^{\dagger}\rangle.$
(35)

For our devices, we take $n_{\text{o,ext}}\approx n_{\text{o,int}}\approx 0$
due to thermal bath occupancies being negligibly small at optical frequencies.
We also take $n_{\text{e,ext}}\approx 0$ since we do not observe any change in
the electrical thermal noise background due to the optical pump or microwave
drive in any of our experiments. Therefore, the relevant thermal baths for
thermometry measurements are $n_{\text{f}}$, $n_{\text{p}}$, and
$n_{\text{e,int}}$. By using $\hat{a}_{\text{e,out}}$ derived in Eq. 25, the
symmetrized noise power spectral density becomes

$\frac{S(\omega)}{\hbar\omega}=\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})+\frac{(\omega_{\text{m}}-\omega)^{2}+\left((\Gamma_{\text{om}}+\Gamma_{\text{i}})/2\right)^{2}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}|\chi_{\text{e}}(\omega)|^{2}\kappa_{\text{e,ext}}\kappa_{\text{e,int}}n_{\text{e,int}},$
(36)

where
$|\chi_{\text{e}}(\omega)|^{2}=1/\left((\omega_{\text{e}}-\omega)^{2}+(\kappa_{\text{e}}/2)^{2}\right)$
and
$\Gamma_{\text{tot}}=\Gamma_{\text{em}}+\Gamma_{\text{om}}+\Gamma_{\text{i}}$.

In an actual experiment, we obtain the power spectral density after amplifying
the microwave field, which requires correcting this expression to account for
the gain ($G$, in dB units) and noise ($n_{\text{amp}}$) of the amplification
chain [61]

$\frac{S(\omega)}{\hbar\omega}=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})+\frac{(\omega_{\text{m}}-\omega)^{2}+((\Gamma_{\text{om}}+\Gamma_{\text{i}})/2)^{2}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}|\chi_{\text{e}}(\omega)|^{2}\kappa_{\text{e,ext}}\kappa_{\text{e,int}}n_{\text{e,int}}\right).$
(37)

To measure the microwave thermal occupancy, we eliminate the interactions with
the mechanical resonator by setting the DC bias to zero, resulting in
$\Gamma_{\text{em}}=0$. Subsequently, the microwave thermal bath occupancy
$n_{\text{e,int}}$ can be obtained from the resulting power spectral density

$\frac{S(\omega)}{\hbar\omega}\Bigr{\rvert}_{\Gamma_{\text{em}}=0}=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\kappa_{\text{e,ext}}\kappa_{\text{e,int}}}{(\omega_{\text{e}}-\omega)^{2}+(\kappa_{\text{e}}/2)^{2}}n_{\text{e,int}}\right).$
(38)

Using the microwave resonator linewidth, the microwave cavity thermal
occupancy $n_{\text{mw}}$ is thus determined by Eq. 22

$n_{\text{mw}}=\frac{\kappa_{\text{e,int}}n_{\text{e,int}}}{\kappa_{e}}.$ (39)

The mechanical thermal occupancy can be measured via electromechanical decay
into the microwave waveguide set by $\Gamma_{\text{em}}$. In the vicinity of
the mechanical mode frequency
($|\omega_{\text{m}}-\omega|<<\kappa_{\text{e}}$), we can express the power
spectral density as

$\begin{split}\frac{S(\omega)}{\hbar\omega}&=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})+\frac{(\omega_{\text{m}}-\omega)^{2}+((\Gamma_{\text{om}}+\Gamma_{\text{i}})/2)^{2}}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}|\chi_{\text{e}}(\omega)|^{2}\kappa_{\text{e,ext}}\kappa_{\text{e,int}}n_{\text{e,int}}\right)\\\
&=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})+\left(1+\frac{\Gamma_{\text{em}}^{2}/4-\Gamma_{\text{em}}\Gamma_{\text{tot}}/2}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}\right)|\chi_{\text{e}}(\omega_{\text{m}})|^{2}\kappa_{\text{e,ext}}\kappa_{\text{e,int}}n_{\text{e,int}}\right).\end{split}$
(40)

For thermometry measurements where the electromechanical cooperativity is
purposely made small ($\Gamma_{\text{em}}<<\Gamma_{\text{i}}$) by using a
large detuning between the microwave and mechanical resonators
($|\omega_{\text{e}}-\omega_{\text{m}}|>>\kappa_{\text{e}}$), the intrinsic
mechanical heating rate
$\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}$ can be obtained
from the resulting power spectral density

$\frac{S(\omega)}{\hbar\omega}=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})\right).$
(41)

For thermometry measurements where the microwave and mechanical resonators are
on-resonance ($\omega_{\text{e}}=\omega_{\text{m}}$) and the electromechanical
cooperativity is no longer small, the power spectral density becomes

$\begin{split}\frac{S(\omega)}{\hbar\omega}&=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}})+\left(1+\frac{\Gamma_{\text{em}}^{2}/4-\Gamma_{\text{em}}\Gamma_{\text{tot}}/2}{(\omega_{\text{m}}-\omega)^{2}+(\Gamma_{\text{tot}}/2)^{2}}\right)*4\eta_{\text{e}}\frac{\kappa_{\text{e,int}}n_{\text{e,int}}}{\kappa_{\text{e}}}\right)\\\
&=10^{G/10}\left(n_{\text{amp}}+\frac{1}{2}+\frac{\eta_{\text{e}}\Gamma_{\text{em}}(\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}+(\Gamma_{\text{em}}-2\Gamma_{\text{tot}})n_{\text{mw}})}{(\omega_{\text{m}}-\omega)^{2}+\left(\Gamma_{\text{tot}}/2\right)^{2}}+4\eta_{\text{e}}n_{\text{mw}}\right),\end{split}$
(42)

where the effects of noise squashing due to the term containing
$n_{\text{e,int}}$ have been fully included in the analysis.

For both cases of thermometry measurements, the independently measured
$n_{\text{mw}}$ from Eq. 38 can be used to obtain the mechanical mode thermal
occupancy by Eq. 23

$n_{\text{m}}=\frac{\Gamma_{\text{em}}n_{\text{mw}}+\Gamma_{\text{f}}n_{\text{f}}+\Gamma_{\text{p}}n_{\text{p}}}{\Gamma_{\text{tot}}}.$
(43)

Using Eq. 29, we can also express the input-referred added noise in terms of
the mechanical thermal occupancy

$\begin{split}n_{\text{add}}=n_{\text{m}}\frac{\Gamma_{\text{tot}}}{\eta_{\text{e}}\Gamma_{\text{em}}}.\end{split}$
(44)

####  D.6.2 Gain calibration by temperature sweep

To perform thermometry via the electrical port, we must calibrate the net
microwave gain from the transducer device to the spectrum analyzer. This
calibration involves measuring the output noise from the amplification chain
while varying the temperature of the mixing chamber (MXC) stage in the
cryostat ($T_{\text{MXC}}$), where our device is mounted. The measured output
noise, integrated over an IF bandwidth $f_{\text{IF}}$ at the spectrum
analyzer, comprises amplified thermal noise from the MXC stage and a constant
term

$P=f_{\text{IF}}\hbar\omega
10^{G_{\text{A}}/10}\left(\frac{1}{e^{\hbar\omega/(k_{\text{B}}T_{\text{MXC}})}-1}+n_{\text{amp}}\right),$
(45)

where $k_{\text{B}}$ is the Boltzmann constant, and $G_{\text{A}}$ is the net
gain from the MXC stage to the spectrum analyzer. In this expression, the
constant term $n_{\text{amp}}$ denotes the total added noise of the amplifier
chain, which is dominated by the contribution from a high electron mobility
transistor (HEMT) amplifier located at the 4K stage of the cryostat.

We control the MXC temperature by reducing the 3He/4He mixture flow and
applying a heater with varying power at the MXC stage. This method keeps the
HEMT amplifier’s temperature nearly constant. Consequently, any observed
changes in noise with temperature are attributed to thermal noise from the
MXC, which is multiplied by the net gain $G_{\text{A}}$ before being detected.
Figure 11 shows the measured noise power spectral density versus the MXC stage
temperature, from which we extract $G_{\text{A}}=56.59\pm 0.29$ dB.

Despite the relatively small statistical uncertainty in this value, this
method of calibrating the gain suffers from systematic error due to its
inability to account for any loss at the MXC temperature that happens between
the device and the HEMT amplifier. As an alternative, we employ an optical
method for calibrating the electrical gain, which is free from this systematic
error and detailed in the following section.

Figure 11: Detected power spectral density vs MXC temperature. Error bars
denote the measured power spectral density with 95% confidence interval. Red
line is the fit from which the electrical gain $G_{\text{A}}$ is obtained.

####  D.6.3 Gain calibration by optical sideband asymmetry

In our devices, optical sideband asymmetry provides a calibration-free means
of determining mechanical thermal occupancy. The mechanical thermal occupation
found from this method can, in turn, be used as a calibrated source for the
electromechanical readout to provide the net microwave gain $G$ as detailed
below.

In the low optomechanical cooperativity limit ( i.e., so long as the back-
action does not influence $n_{\text{m}}$), the difference between the red- and
blue-sideband scattering rates corresponds to the vacuum contribution (that
is, spontaneous Stokes scattering), which can serve as a reference for finding
the mechanical thermal occupation as [62, 58]

$\frac{1}{n_{\text{m}}}=\frac{\int_{-\infty}^{+\infty}S_{\text{b}}(\omega)\,d\omega}{\int_{-\infty}^{+\infty}S_{\text{r}}(\omega)\,d\omega}-1.$
(46)

Here, $S_{\text{r}}(\omega)$ and $S_{\text{b}}(\omega)$ represent the noise
power spectral density of the mechanical mode under the red- and blue-sideband
pumps, respectively.

Figure 12 shows the optically measured power spectral density of the
mechanical motion for blue- and red-pumps for the same intracavity photon
number. Using Eq. 46, we calculate the mechanical thermal occupancy for these
measurements as $n_{\text{m}}=1.81\pm 0.35$. We subsequently measure the
mechanical thermal emission via the microwave port under the same optical pump
conditions. The microwave power detected in this measurement is then referred
to the mechanical occupation from the optical sideband asymmetry
($n_{\text{m}}=1.81\pm 0.35$) to find the microwave output line gain as
$G=55.83\pm 0.84$ dB. We have used this calibrated gain in all the
measurements reported in the main text.

Figure 12: Spectra of mechanical mode noise power density under red- and blue-
sideband pumps. In these measurements, we use $n_{\text{c}}=9.2$ and
$V_{\text{DC}}=40$ V, corresponding to an optomechanical cooperativity of
$C_{\text{om}}\approx 0.04$. The noise floor corresponds to $16.4$ noise
photons (referred to the optical waveguide at the chip) by the optical
heterodyne detection setup and is removed from the plots. Solid curves are
Lorentzian fits to the corresponding data. To eliminate any uncertainty from
wavelength-dependent optical waveguide transmission [26], we ensure the
optomechanical back-action rate remains equal for these two measurements by
monitoring the microwave-to-optical transduction efficiency and the mechanical
linewidth.

###  D.7 Optical measurements of input-referred added-noise

The input-referred added noise of microwave-to-optical transduction is defined
as the number of microwave photons at the input port of the transducer that
would result in the same optical noise measured at the transducer optical
output. A straightforward approach of characterizing the input-referred added
noise is to optically measure the transducer noise and normalize it to the
input microwave photon number. In our experiments, we use optical heterodyne
detection, as described in Appendix C, to measure the transducer’s output
optical noise. Under a red-detuned optical pump without any external microwave
input, the optical noise photon quanta at the photodetector is given by

$N_{\text{o,noise}}=10^{G_{\text{o}}/10}n_{\text{o,out}}(\omega_{\text{m}})=\frac{P_{\text{o,noise}}}{f_{\text{IF}}\hbar\omega_{\text{o}}},$
(47)

where $G_{\text{o}}=7.9\text{ dB}$ is the optical detection gain,
$n_{\text{o,out}}(\omega_{\text{m}})$ is the optical noise spectral density at
the waveguide (see Eq. 27), and $P_{\text{o,noise}}$ is the detected optical
noise power at the mechanical resonator frequency within an IF bandwidth
$f_{\text{IF}}$ = 2 kHz. Due to the low optomechanical readout rate
($C_{\text{om}}<<1$), measuring the optical noise power via continuous
heterodyne detection took $\sim 17$ hours for the lowest intracavity photon
number reported in the main text ($n_{\text{c}}=5.8$) and becomes
impractically long for lower photon numbers.

To obtain the input-referred added noise, we normalize $N_{\text{o,noise}}$ to
an equivalent microwave photon number at the transducer microwave waveguide.
For this purpose, we measure the total conversion efficiency at the mechanical
resonator frequency from the transducer microwave waveguide to the
photodetector, which can be measured as

$\eta_{\text{tot}}=10^{G_{\text{o}}/10}\eta_{\text{ext}}(\omega_{\text{m}})=\frac{P_{\text{oe}}/\left(f_{\text{IF}}\hbar\omega_{\text{o}}\right)}{P_{\text{e}}/\left(f_{\text{IF}}\hbar\omega_{\text{e}}\right)},$
(48)

where $\eta_{\text{ext}}$ is the transducer conversion efficiency between the
on-chip microwave and optical waveguides, $P_{\text{e}}$ is the input
microwave power at the transducer microwave waveguide, and $P_{\text{oe}}$ is
the detected power of the coherent transducer output optical signal. Dividing
the detected optical noise by $\eta_{\text{tot}}$, we obtain the input-
referred added noise as

$n_{\text{add}}=N_{\text{o,noise}}/\eta_{\text{tot}}=n_{\text{o,out}}(\omega_{\text{m}})/\eta_{\text{ext}}(\omega_{\text{m}}).$
(49)

##  Appendix E Performance comparison

Device | $\mathbf{n_{\text{\bf{add}}}}$ |  $\mathbf{\eta_{\text{\bf{ext}}}\times B\times D}$ (Hz)
---|---|---
This work | 0.94 | 1900
This work | 0.58 | 470
Weaver et al. Nat Nanotech (2024) | 6.2 | 5.2
Brubaker et al. PRX (2022) | 3.2 | 130
Jiang et al. Nat Phys (2023) | 1.6 | 1100
Mirhosseini et al. Nature (2020) | 0.57 | 0.075
Sahu et al. Nat Comm (2022) | 0.4 | 5.4
Meesala et al. Nat Phys (2024) | 0.14 | 3.1
Table 2: Transducer Comparison

To compare different transducers with each other (Table 2), we consider two
figures of merit: 1) input-referred added noise $n_{\text{add}}$, where
$n_{\text{add}}<1$ defines the quantum-enabled regime [7, 6, 8], and 2) the
product of external efficiency $\eta_{\text{ext}}$, bandwidth $B$, and duty
cycle $D=T_{\text{d}}\times R_{\text{p}}$, where $T_{\text{d}}$ = pulse
duration and $R_{\text{p}}$ = pulse repetition rate.

In Ref. [23], the lowest achieved added noise is $n_{\text{add}}=3.2$ with a
reported external efficiency of $\eta_{\text{ext}}=0.47$ and a bandwidth of
$B=220\text{ Hz}$. To separate out the device properties from those of the
characterization setup, we make the conservative choice of excluding the
effects of the reported coupling loss from spatial mode mismatch in this
experiment and use $\eta_{\text{ext}}=0.59$ for calculations. This transducer
operates using continuous pumps, so $D=1$. In Ref. [6], the highest efficiency
operation point in the quantum-enabled regime is reported with the added noise
of $n_{\text{add}}=0.4$ and efficiency of $\eta_{\text{ext}}=0.15$. To
separate out the device properties from those of the characterization setup,
we make the conservative choice of excluding the effects of the reported
coupling loss from spatial mode mismatch in this experiment and use
$\eta_{\text{ext}}=0.25$ for calculations. In this experiment, the transducer
bandwidth is reported as $B=18\text{ MHz}$ and the pulse duration is
$T_{\text{d}}=300$ ns with a repetition rate of $R_{\text{p}}=4$ Hz. The
reported efficiency in [26, 22, 24, 27] is defined for a temporal mode
matching the pulse shape, which equals
$\eta_{\text{ext,p}}=\eta_{\text{ext}}\times B\times T_{\text{d}}$. In Ref.
[26], the lowest added noise and the external photon conversion efficiency are
reported as $n_{\text{add}}=6.2$ and $\eta_{\text{ext,p}}=5.2\times 10^{-5}$.
The pulse repetition rate for this experiment is $R_{\text{p}}=100\text{
kHz}$. In Ref. [22], the lowest achieved added noise is $n_{\text{add}}=0.57$
with $\eta_{\text{ext,p}}=7.5\times 10^{-4}$. The pulse repetition rate is
$R_{\text{p}}=100\text{ Hz}$. In Ref. [24], the added noise in the microwave
waveguide is $n_{\text{add}}=1.6$ with $\eta_{\text{ext,p}}=6.3\times
10^{-3}$, which is calculated by the product of the optical photon-phonon
scattering probability ($0.036$), the phonon-microwave photon conversion
efficiency ($0.35$), optical external coupling efficiency ($0.5$), and
microwave external coupling efficiency ($0.993$). The pulse repetition rate is
$R_{\text{p}}=170\text{ kHz}$. In Ref. [27], the added noise in the microwave
waveguide is $n_{\text{add}}=0.14$, which is calculated as
$\eta_{\text{mw}}\times n_{\text{i}}+n_{\text{d}}$ according to the numbers
reported in its supplement section 13. The external conversion efficiency at
the low noise operation point is $\eta_{\text{ext,p}}=6.1\times 10^{-5}$,
which is calculated by the product of the optical photon-microwave photon
scattering probability ($1.8\times 10^{-4}$), optical external coupling
efficiency ($0.5$), and microwave external coupling efficiency ($0.68$). The
pulse repetition rate is reported as $R_{\text{p}}=50\text{ kHz}$.

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