††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: Present address: Department of Microtechnology and
Nanoscience, Chalmers University of Technology, Sweden

# Optically heralded microwave photons

Wentao Jiang  Felix M. Mayor  Sultan Malik  Raphaël Van Laer  Timothy P.
McKenna  Rishi N. Patel  Jeremy D. Witmer  Amir H. Safavi-Naeini
<EMAIL_ADDRESS>Department of Applied Physics and Ginzton Laboratory,
Stanford University, 348 Via Pueblo Mall, Stanford, California 94305, USA

###### Abstract

A quantum network that distributes and processes entanglement would enable
powerful new computers and sensors kimble2008quantum ; altman2021quantum ;
alexeev2021quantum ; awschalom2021development . Optical photons with a
frequency of a few hundred terahertz are perhaps the only way to distribute
quantum information over long distances chen2021twin . Superconducting qubits
on the other hand, which are one of the most promising approaches for
realizing large-scale quantum machines devoret2013superconducting ;
arute2019quantum ; wu2021strong ; jurcevic2021demonstration , operate
naturally on microwave photons that have roughly $40,000$ times less energy.
To network these quantum machines across appreciable distances, we must bridge
this frequency gap and learn how to generate entanglement across widely
disparate parts of the electromagnetic spectrum. Here we implement and
demonstrate a transducer device that can generate entanglement between optical
and microwave photons, and use it to show that by detecting an optical photon
we add a single photon to the microwave field. We achieve this by using a
gigahertz nanomechanical resonance as an intermediary, and efficiently
coupling it to optical and microwave channels through strong optomechanical
and piezoelectric interactions mirhosseini2020superconducting . We show
continuous operation of the transducer with $5\%$ frequency conversion
efficiency, and pulsed microwave photon generation at a heralding rate of $15$
hertz. Optical absorption in the device generates thermal noise of less than
two microwave photons. Joint measurements on optical photons from a pair of
transducers would realize entanglement generation between distant microwave-
frequency quantum nodes krastanov2021optically . Improvements of the system
efficiency and device performance, necessary to realize a high rate of
entanglement generation in such networks are within reach.

††preprint: APS/123-QED

Manipulating and transmitting quantum states with higher fidelity and at
larger scales will enable technologies that promise breakthroughs in sensing,
communication, and computation altman2021quantum ; alexeev2021quantum ;
awschalom2021development . Over the last two decades, our ability to
manipulate the states of photons in superconducting circuits has advanced
rapidly and led to demonstrations of quantum advantage for certain
computational tasks arute2019quantum ; wu2021strong ;
jurcevic2021demonstration . Separately, the first quantum networks have been
realized based on the transmission of quantum states and distribution of
entanglement over a small number of nodes using optically coupled qubits
pompili2021realization ; daiss2021quantum . Optical interconnects between
superconducting quantum machines with advanced computational and error
correction capabilities ofek2016extending ; krinner2022realizing  would
significantly accelerate the development and deployment of quantum networks.
Moreover, microwave quantum processors would be beneficiaries of quantum
networks that support distributed quantum computing serafini2006distributed ,
sensing gottesman2012longer ; brady2022entangled , and secured communications
shor2000simple . Unfortunately, in contrast to ions, atoms, and semiconductor
defect centers, superconducting circuits lack a natural optical transition
that would generate entanglement between propagating optical photons and their
internal states. To compensate for this deficiency, a highly efficient and
low-noise quantum transducer needs to be developed.

Quantum transducers that connect microwave and optical photons have used a
variety of physical processes including optomechanical, electro-optical,
magneto-optical interactions, and atomic degrees of freedom han2021microwave .
In optomechanical transducers, the interaction between light and motion via
radiation pressure and electrostriction offers the nonlinearity necessary for
frequency conversion aspelmeyer2014cavity . The mechanical vibration is then
converted to a microwave signal with either a parametric electromechanical
coupling teufel2011circuit ; andrews2014bidirectional  or the piezoelectric
effect o2010quantum ; bochmann2013nanomechanical . Steady progress in the
field has led to constant improvement of transduction efficiencies and added
noise zeuthen2020figures ; han2020cavity ; brubaker2022optomechanical ,
culminating in remarkable recent demonstrations of superconducting qubit state
readout mirhosseini2020superconducting ; delaney2022superconducting .

Here we split optical photons into correlated pairs of optical and microwave
photons in a quantum transducer and herald microwave photons by detecting
optical photons. Our demonstration is an important step towards realizing
heralded entanglement generation between distant superconducting quantum
machines that may form nodes in a quantum network kimble2008quantum . We send
a laser pulse with the sum frequency of the optical and microwave frequencies
to the transducer. Through spontaneous down-conversion, the input optical
photon is converted to an entangled pair consisting of an optical photon and a
microwave phonon. The phonon half of this pair efficiently radiates from the
device into an output line as a microwave-frequency electromagnetic signal,
generated by a strong engineered piezoelectric coupling. The light is sent
onto a single photon detector and upon detection, we herald the microwave
photon. Finally, we characterize the state of the microwave field using linear
detection tomography and verify the presence of the additional photon.

Figure 1:  Transducer schematics and implementation. a, Definition of modes in
the transducer and the coupling rates between the modes and channels. b,
Physical implementation of the coupling mechanism and the possible conversion
processes. The silicon optical mode is represented by the blue Fabry-Pérot
cavity, and the optomechanical coupling offers two types of three-wave mixing
processes between photons in the cavity and phonons in the vibrating mirror. A
blue (red) pump laser-cavity detuning allows entangled pair generation
(quantum frequency conversion). The hybrid silicon-lithium niobate (Si-LN)
mechanical mode utilizes the piezoelectric effect to convert the mechanical
vibration to a microwave field. A niobium-titanium-nitride (NbTiN) microwave
resonance formed by the standing wave in a high-impedance waveguide enhances
the external coupling rate of the mechanical mode to the microwave coaxial
cable. c, Simulated optical mode and mechanical mode of the transducer. d,e,
Optical and scanning electron microscope (SEM) images of the fabricated
transducer device and microwave resonator. SEM image of the transducer is
taken before etching the silica substrate and false-colored to highlight the
silicon (cyan), LN (purple), and aluminum electrodes (yellow). Scale bars:
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (white),
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (black). f, Top: photo of the
transducer assembly wirebonded to a printed circuit board. The transducer
(microwave) chip is highlighted with dashed blue (green) lines. The lensed
fiber is also shown as the light blue region (not to scale) Bottom: side view
of the transducer assembly as seen by the lensed fiber, showing the two chips
and the wirebonds connecting them.

The two physical processes in the transducer are the nonlinear optomechanical
interaction between the optical mode and the mechanical mode, and the linear
piezoelectric interaction between the mechanical mode and the microwave mode
(Fig. 1a). The nonlinearity that facilitates the frequency conversion process
is governed by the optomechanical interaction Hamiltonian
$\mbox{$\hat{H}_{\text{om}}$}=\hbar
g_{\text{o}}\mbox{$\hat{a}^{\dagger}$}\mbox{$\hat{a}$}(\mbox{$\hat{b}$}+\mbox{$\hat{b}^{\dagger}$})$,
where $\hat{a}$ ($\hat{b}$) is the annihilation operator for the optical
(mechanical) resonance. The optomechanical coupling rate $g_{\text{o}}$
represents the cavity frequency uncertainty from the zero-point motion in the
mechanical mode aspelmeyer2014cavity . As shown in Fig. 1b, when the system is
pumped with a red-detuned laser, the interaction can be described by the beam-
splitter Hamiltonian $\mbox{$\hat{H}_{\text{bs}}$}=\hbar
G_{\text{o}}(\mbox{$\hat{a}$}\mbox{$\hat{b}^{\dagger}$}+\mbox{$\hat{a}^{\dagger}$}\mbox{$\hat{b}$})$,
while a blue-detuned laser implements a two-mode squeezing Hamiltonian
$\mbox{$\hat{H}_{\text{tms}}$}=\hbar
G_{\text{o}}(\mbox{$\hat{a}^{\dagger}$}\mbox{$\hat{b}^{\dagger}$}+\mbox{$\hat{a}$}\mbox{$\hat{b}$})$.
We define the linearized coupling rate
$G_{\text{o}}=\sqrt{n_{\text{a}}}g_{\text{o}}$ where $n_{\text{a}}$ is the
intracavity pump photon number. The operator $\hat{a}$ now represents the
sideband component of the optical mode that is resonant with the cavity. The
piezoelectric interaction between the mechanical and microwave resonance is
described by $\mbox{$\hat{H}_{\text{pe}}$}=\hbar
g_{\upmu}(\mbox{$\hat{b}$}\mbox{$\hat{c}^{\dagger}$}+\mbox{$\hat{b}^{\dagger}$}\mbox{$\hat{c}$})$,
where $g_{\upmu}$ is the coupling rate and $\hat{c}$ is the lowering operator
of the microwave mode.

We operate our transducer in a fast-cavity limit, where the optical
(microwave) linewidth $\kappa_{\text{o}}$ ($\kappa_{\upmu}$) is much larger
than the corresponding coupling rate $\kappa_{\text{o}}\gg G_{\text{o}}$
($\kappa_{\upmu}\gg g_{\upmu}$). In this limit, both the optical and microwave
subsystems act approximately as a broad Markovian bath for the mechanical
mode. This means that the resonant phonons decay at a rate
$\gamma_{\upmu}=4g_{\upmu}^{2}/\kappa_{\upmu}$ into the microwave transmission
line. Similarly, for a red-side pump, the phonons decay directly into the
photon loss channels at a rate of
$\gamma_{\text{om}}=4G_{\text{o}}^{2}/\kappa_{\text{o}}$. The peak conversion
efficiency is given by the ratio between external coupling rates and the total
loss rates,

$\eta=\eta_{\text{o}}\eta_{\upmu}\frac{4\gamma_{\text{om}}\gamma_{\upmu}}{(\gamma_{\text{i}}+\gamma_{\text{om}}+\gamma_{\upmu})^{2}},$
(1)

where $\gamma_{\text{i}}$ is the intrinsic loss rate of the mechanical mode,
and $\eta_{\text{o}}\equiv\kappa_{\text{o,e}}/\kappa_{\text{o}}$
($\eta_{\upmu}\equiv\kappa_{\upmu,\text{e}}/\kappa_{\upmu}$) is the external
coupling efficiency of the optical (microwave) mode (Fig. 1a). For a blue-side
pump, the interaction induces a non-degenerate parametric amplification rate
of $\gamma_{\text{om}}$ for the mechanical resonator, with correlated photons
emitted at optical frequency.

To achieve higher conversion efficiency and bandwidth, larger
$\gamma_{\text{om}}$ and $\gamma_{\upmu}$ are required. This translates to
higher optomechanical and piezoelectric coupling coefficients. We found that
no single material system is optimal with respect to all the needs of the
converter. As such, we pursued a heterogeneous integration approach that
combines materials with good optomechanical and piezoelectric properties. In
addition to the materials integration challenge, this raises a challenge in
design, as we must co-design the optomechanical and piezoelectric constituents
of the transducer to maximize the modal overlaps while maintaining small mode
volumes for high interaction rates. We implement the transducer by combining
highly piezoelectric thin-film lithium niobate (LN) pop2017laterally , with
thin-film silicon (Si), which has been shown to have strong optomechanical
coupling and low loss safavi2019controlling . We use a silicon optomechanical
crystal (OMC) to co-localize the optical and mechanical resonances in a
wavelength-scale volume chan2012optimized , and engineer the mechanical mode
to be partially extended and strongly hybridized with a Si-LN hybrid
piezoelectric mode mirhosseini2020superconducting . The orientation between
the piezoelectric resonator and the silicon OMC is chosen to maximize the
mechanical hybridization. The full transducer structure is released and
supported by one-dimensional silicon phononic shields to minimize unwanted
mechanical loss (Fig. 1c,d).

To couple the phonons to microwaves, we pattern aluminum electrodes on the LN
and run these over the phononic shields that suspend the transducer. Because
of the vastly different dimensions of the piezo-optomechanical element
($\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) and the microwave circuit
($\sim$10\text{\,}\mathrm{mm}$$), we fabricate them on separate chips that we
combine by wirebonding (Fig. 1f). The microwave resonance is formed by a
standing wave in a high-impedance (high-Z) microwave coplanar waveguide (Fig.
1b,e) arising due to impedance mismatch with the output line. A niobium-
titanium-nitride (NbTiN) thin film on high resistivity silicon substrate is
patterned into nanowires to support traveling waves with characteristic
impedance $Z=$1000\text{\,}\mathrm{\SIUnitSymbolOhm}$$. The large kinetic
inductance from the nanowires enables high impedance and magnetic frequency
tunability xu2019frequency . Finally, by using NbTiN, with its short quasi-
particle lifetime, for the microwave resonator, and fabricating the microwave
subsystem on a separate chip, we mitigate some of the effects associated with
absorption of stray optical radiation mirhosseini2020superconducting .

Figure 2:  Transducer characterization. a, Electromagnetically induced
transparency (EIT). Top: schematic of the measurement allowing us to extract
the optical reflection $S_{\text{oo}}$ with a red detuned pump. Bottom:
measurement of the phase of the reflected optical sideband (blue) with the EIT
visible at the mechanical frequency
$f_{\text{m}}=$3.596\text{\,}\mathrm{GHz}$$. A close-up of the transparency
region is shown in the inset. The red line is a fit assuming the model shown
in the schematic. b, Microwave reflection. Top: schematic of the measurement
of $S_{\upmu\upmu}$. Bottom: Magnitude and phase of the measured
$S_{\upmu\upmu}$. The fit (red) is using coupled mode theory. c, Frequency
conversion. Schematic and measurement of the microwave-to-optical
($S_{\text{o}\upmu}$) and optical-to-microwave ($S_{\upmu\text{o}}$)
conversion efficiencies using a red-detuned pump. The theoretical calculation
is shown as yellow dashed line.

We characterize the transducer at the mixing chamber plate in a dilution
refrigerator where the temperature $T\lesssim$10\text{\,}\mathrm{mK}$$ – the
same environment in which superconducting circuits and qubits are operated. We
use a lensed fiber to focus light into an on-chip photonic waveguide, to which
the optical cavity is evanescently coupled. We first measure the linear
scattering parameters of the transducer with a red-detuned continuous-wave
pump. We use $4.4\text{\,}\mathrm{\SIUnitSymbolMicro W}$ on-chip pump power,
corresponding to an intracavity photon number $n_{\text{a}}=230$. We sweep a
weak probe tone across the optical resonance, generated by electro-optic
modulation of the pump light by the microwave signal from a vector network
analyzer (VNA). The reflected light is amplified and detected by a high speed
photodetector and subsequently the VNA. We obtain the scattering parameter
$S_{\text{oo}}$ and extract the phase response (Fig. 2a). We observe
electromagnetically induced transparency (EIT) aspelmeyer2014cavity  and fit
the response using input-output theory which gives a single-photon
optomechanical coupling rate of
$g_{\text{o}}/2\pi=$410\text{\,}\mathrm{kHz}$$. The mechanical mode has a
frequency $\omega_{\text{m}}/2\pi=$3.596\text{\,}\mathrm{GHz}$$. We find that
the optical cavity is nearly critically coupled with a total linewidth
$\kappa_{\text{o}}/2\pi=$1.12\text{\,}\mathrm{GHz}$$. Subsequently, we
characterize the piezoelectric interaction in the transducer by a microwave
reflection measurement. We obtain the scattering parameter $S_{\upmu\upmu}$
and plot its magnitude and phase (Fig. 2b). We find a piezoelectric coupling
rate $g_{\upmu}/2\pi=$420\text{\,}\mathrm{kHz}$$ and extract an intrinsic
mechanical linewidth of $\gamma_{\text{i}}/2\pi=$1.1\text{\,}\mathrm{MHz}$$.
During this measurement, the pump is sent to the device with
$10.1\text{\,}\mathrm{\SIUnitSymbolMicro W}$ on-chip power
($\gamma_{\text{om}}/2\pi=$324\text{\,}\mathrm{kHz}$)$. The device parameters
are summarized in Extended Data Table 1. Next, we characterize the microwave-
to-optical ($S_{\text{o}\upmu}$) and optical-to-microwave
($S_{\upmu\text{o}}$) frequency conversion using the VNA. The setup is the
same as in the reflection measurements with an on-chip pump power of
$10.2\text{\,}\mathrm{\SIUnitSymbolMicro W}$, corresponding to
$n_{\text{a}}=540$ and $\gamma_{\text{om}}/2\pi=$327\text{\,}\mathrm{kHz}$$.
The measured scattering parameters are shown in Fig. 2c. We determine the peak
conversion efficiency to be $\eta=(4.9\pm 0.5)\%$ (see Methods) with a
$3\text{\,}\mathrm{dB}$ bandwidth of $1.5\text{\,}\mathrm{MHz}$. The added
noise referred to the input for optical-to-microwave conversion is measured to
be $n_{\text{added},\upmu\text{o}}=99\pm 10$. Like state-of-the-art
demonstrations, our achieved efficiency and added noise are currently
insufficient for direct conversion of quantum states. However, the ability to
post-select quantum optical states by single photon detection obviates the
need for high efficiency by leveraging the nonlinearity afforded by strong
measurement. A more important goal for our device is then to achieve a high
rate of entanglement generation so that heralded protocols become practicable.

Figure 3:  Optically heralded microwave photon. a, Schematics of the heralding
experiment. A blue detuned pump pulse incident on the transducer is split into
a sideband photon on-resonance with the optical cavity, and a phonon resonant
with the mechanical mode. The reflected light contains the pump residual and
the sideband photon, which is filtered and detected with a single photon
detector (SPD). The phonon is converted to a propagating microwave photon, and
is amplified, downconverted by mixing with a local oscillator (LO),
demodulated, and accumulated into a phase space probabilistic distribution
$\mathcal{D}(\alpha)$. When a click is registered on the SPD, the demodulation
result is collected into a post-selected dataset
$\mathcal{D}_{\text{s}}(\alpha)$. b, Probability distribution difference
between the post-selected dataset and the full dataset
$\mathcal{D}_{\text{s}}(\alpha)-\mathcal{D}(\alpha)$. c, Probability
distribution difference of the radially binned distribution. Blue points
represent the post-selected samples, and black points are calculated from
randomly-selected samples instead of post-selected, with the same number of
samples. Errorbars represent one standard deviation according to number of
samples in each bin. Blue and grey regions are theoretical calculations with
the same uncertainty.

For heralded microwave photon generation, a blue-detuned pump laser pulse with
duration $\tau\approx$20\text{\,}\mathrm{ns}$$ is sent to the transducer. The
pulse duration is chosen so that $\tau^{-1}$ is much smaller than the optical
decay rate and laser detuning, and so the intracavity photon number follows
the pulse amplitude closely. The pulse is however much faster than the
response time of the mechanical and microwave system. As shown in Fig. 3a,
upon heralding, a phonon is effectively added to the mechanical resonator,
which then leaks out as microwave radiation at a rate on the order of
$\gamma_{\upmu}$. We integrate the optomechanical scattering rate over the
optical pulse duration to find the entangled photon-phonon-pair generation
probability $p\approx\gamma_{\text{om}}\tau$ krastanov2021optically . The
sideband photon leaks out of the optical cavity together with reflected pump
photons, most of which are filtered out by two Fabry-Pérot cavities, enabling
single photon detection (SPD) of the sideband photon. The generated microwave
photon is amplified by a near quantum-limited traveling wave parametric
amplifier (TWPA) macklin2015near  followed by a cryogenic low-noise amplifier.
At the end of a room temperature amplification chain, the microwave signal is
demodulated to obtain a quadrature sample. The samples are labeled according
to whether the microwave detection was accompanied by a photon click.

We use an on-chip peak pump power of
$\sim$5\text{\,}\mathrm{\SIUnitSymbolMicro W}$$ and a pulse duration of
$\tau\approx$20\text{\,}\mathrm{ns}$$, giving a scattering probability of
$p\approx 3.6\%$. We choose a repetition rate of $170\text{\,}\mathrm{kHz}$,
which leads to a thermal occupation of the mechanical mode prior to the
arrival of the pump pulse of $n_{\text{th}}=0.68\pm 0.08$ (see Methods). A
lower repetition rate reduces this source of noise, but makes the experiment
slower. We use a pair of filter cavities to realize
$>$90\text{\,}\mathrm{dB}$$ suppression of the pump photons with respect to
the optomechanically generated sideband photons. The probability of detecting
a photon that starts inside the optical mode is $\eta_{\text{sys}}=1\%$. This
low efficiency is caused by the optical mode external coupling efficiency
$\eta_{\text{o}}=50\%$, an insertion loss of $25.4\%$ from the on-chip
photonic waveguide to the fridge optical port, transmission through the filter
cavities ($15\%$), and quantum efficiency of the SPD ($65\%$).

Finally, we use linear detectors to characterize the microwave field emitted
from the device. Linear phase-insensitive amplification of microwave fields
effectively measures both quadratures, necessarily adding half a quantum of
noise. The resulting amplifier output corresponds to the Husimi $Q$ function
$Q(\alpha)$ of the microwave state up to a scaling eichler2011experimental ;
patel2021room , where $\alpha$ is the quadrature-phase amplitude of the
microwave mode. In practice, noise in excess of the quantum limit is added due
to the device inefficiency, amplifier noise, and demodulation inefficiency. We
denote the measured probability distribution by $\mathcal{D}(\alpha)$ for the
thermal state without the single-photon detection event, and
$\mathcal{D}_{\text{s}}(\alpha)$ for the photon added state after post-
selection with the SPD event. As a result of the excess noise, the measured
$\mathcal{D}(\alpha)$ and $\mathcal{D}_{\text{s}}(\alpha)$ look virtually
identical to the eyes but their numerical difference is clearly resolvable
eichler2011experimental . We accumulated $\sim 1.4\times 10^{6}$ post-selected
samples over the course of $\sim$35\text{\,}\mathrm{h}$$, and binned them into
a two-dimensional histogram with $41\times 41$ entries to obtain the
probabilistic distribution $\mathcal{D}_{\text{s}}(\alpha)$. During the same
experiment, $4.3\times 10^{7}$ samples of the thermal state are collected and
binned in the same way for $\mathcal{D}(\alpha)$. The difference in the
probability distributions $\mathcal{D}_{\text{s}}(\alpha)-\mathcal{D}(\alpha)$
is shown in Fig. 3b. Since the states have no well-defined phase, we bin the
samples radially and plot the results as blue points in Fig. 3c. The shaded
blue region is the resulting theoretical distribution of phonon added state
with excess noise. As a control, we randomly sampled a subset from the thermal
dataset, and obtained the probability difference shown in black in Fig. 3c. In
both the two-dimensional histogram and the radially binned data, the axes are
calibrated assuming an excess noise $n_{\text{ex}}=39\pm 6$, obtained from the
change in variance between thermal and post-selected samples (see Methods). An
independent quantum calibration of the gain and excess noise in the detection
chain uses sideband asymmetry to determine the phonon temperature, and leads
to values of gain and excess noise that are within roughly a factor of two.
However, due to the sensitive dependence of system parameters such as the
output efficiency on optical pump power, we found that other calibration
approaches, specially when they require changing optical power, are unreliable
(see Methods). The excess noise is dominated by heating after the optical
pulse, an added noise of $n_{\text{m}}=2.4\pm 0.4$ from the TWPA, and the
total measurement efficiency of $\sim 8\%$. Heating after the optical pulse
adds extra thermal noise in the output microwave photon from the transducer.
We estimate an added thermal noise $n_{\text{n}}=1.6\pm 0.5$ in the
propagating microwave photon with two different methods. First, we use the
excess noise from the heralding experiment and microwave readout efficiencies
to infer the added thermal noise. Alternatively, we calculate the overlap
between the independently measured time-domain heating after the optical pulse
and the temporal mode of the microwave field to obtain the added noise (see
Methods).

We have demonstrated the direct measurement of optically heralded microwave
added photon states. The heralding rate in our experiment is
$\sim$15\text{\,}\mathrm{Hz}$$, limited by the total optical readout system
efficiency of $\eta_{\text{sys}}=1\%$. While the heralding rate is comparable
to other quantum systems for entanglement generation hucul2015modular ;
humphreys2018deterministic ; levine2018high ; riedinger2018remote ;
van2022entangling , we can drastically improve it by increasing the fiber-to-
chip coupling efficiency groblacher2013highly  and reducing optical filter
insertion loss. These significantly improved heralding rates
($\sim$\text{\,}\mathrm{kHz}$$) would be comparable to state-of-the-art
coherence time of microwave reagor2016quantum ; romanenko2020three  and
acoustic maccabe2020nano  resonators. Another challenge is reducing the
effects of induced thermal noise. Recent designs have emerged that demonstrate
much better thermalization ren2020two  allowing lower initial thermal
occupation and added noise $n_{\text{th}},n_{\text{n}}\lesssim 0.1$. Adopting
these two-dimensional structures would also increase the achievable rates. The
piezoelectric coupling rate of our transducer is also limited by the multi-
mode nature of our microwave resonator. Moving to a single-mode microwave
system mirhosseini2020superconducting  would increase our phonon-to-microwave
photon output efficiency from $\eta_{\upmu\text{m}}\sim 35\%$ to close to
$100\%$. To entangle two distant microwave systems, we will need to implement
two copies of our transducer to produce indistinguishable optical photons
inside two separate fridges. Overcoming frequency variations in different
devices by frequency shifting in the optical domain riedinger2018remote ;
levonian2022optical  or entanglement swapping with entangled optical photons
at different frequencies from optical spontaneous parametric down conversion
krastanov2021optically  would relax the device frequency matching
requirements. Our results show that with these improvements, piezo-
optomechanical quantum frequency transducers that entangle distant quantum
microwave systems are within reach.

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## Methods

### Device parameters

Extended Data Table 1 shows the device parameters for the optical, mechanical,
microwave modes and the coupling rates. The measurement method used to obtain
each parameter is also listed. We observe that the mechanical frequency and
intrinsic loss rate are different under different optical powers. When the
optical pump is turned off, $\gamma_{\text{i}}$ is reduced by a factor of
$\sim 3$ and the mechanical frequency red-shifts by
$\sim$700\text{\,}\mathrm{kHz}$$. Variation of the mechanical frequency and
intrinsic loss rate are likely due to a combination of optically induced
heating and saturation of two-level systems. Reducing the frequency
fluctuation and improving the intrinsic mechanical quality factor will be the
subject of future effort.

Extended Data Table 1: Device parameters. Parameter | Value | method
---|---|---
$\omega_{\text{o}}/2\pi$ | $193.53\text{\,}\mathrm{THz}$ | Laser wavelength sweep
$\kappa_{\text{o}}/2\pi$ | $1.122\text{\,}\mathrm{GHz}$ | EIT ($S_{\text{oo}}$)
$\kappa_{\text{o,e}}/2\pi$ | $0.561\text{\,}\mathrm{GHz}$ | EIT
$g_{\text{o}}/2\pi$ | $413\text{\,}\mathrm{kHz}$ | EIT
$\omega_{\text{m}}/2\pi$ |  $\sim$ $3.596\text{\,}\mathrm{GHz}$ | Microwave $S_{\upmu\upmu}$
$\gamma_{\text{i}}/2\pi$ (pump off) | $0.36\text{\,}\mathrm{MHz}$ | Microwave $S_{\upmu\upmu}$
$\gamma_{\text{i}}/2\pi$ (pump on) | $1.07\text{\,}\mathrm{MHz}$ | Microwave $S_{\upmu\upmu}$
$\omega_{\upmu}/2\pi$ | $3.5958\text{\,}\mathrm{GHz}$ | Microwave $S_{\upmu\upmu}$
$\kappa_{\upmu}/2\pi$ | $3.06\text{\,}\mathrm{MHz}$ | Microwave $S_{\upmu\upmu}$
$\kappa_{\upmu,\text{e}}/2\pi$ | $3.04\text{\,}\mathrm{MHz}$ | Microwave $S_{\upmu\upmu}$
$g_{\upmu}/2\pi$ | $424\text{\,}\mathrm{kHz}$ | Microwave $S_{\upmu\upmu}$

### Tunable high-impedance waveguide

Extended Data Fig. 1:  Microwave high-impedance waveguide characterization. a,
Schematics of the high impedance (high-Z) waveguide. The mechanical external
coupling is approximately proportional to the impedance of the environment.
Thus the effective impedance $Z_{\text{eff}}$ looking from the mechanics side
into the high-Z waveguide gives us intuition on the enhanced external
coupling. b, Calculated $Z_{\text{eff}}$ and the group delay with a free
spectral range of $110\text{\,}\mathrm{MHz}$ and a waveguide characteristic
impedance $Z=$1000\text{\,}\mathrm{\SIUnitSymbolOhm}$$. c, Measured group
delay of the high-Z waveguide. The coupled mechanical modes appear as sharp
peaks in group delay. Extended Data Fig. 2:  Microwave-to-optical conversion
with different microwave resonator frequencies. Plot of normalized microwave-
to-optical conversion S parameter as a function of frequency and coil current.

The high-impedance (high-Z) waveguide is realized with a hybrid aluminum-NbTiN
coplanar waveguide (CPW). The ground plane of the CPW is made of e-beam
evaporated aluminum. The center conductor of the CPW is made of thin and
narrow NbTiN nanowires to achieve high kinetic inductance. The thickness of
the NbTiN layer is $10\text{\,}\mathrm{nm}$, deposited by StarCryo on high-
resistivity silicon
($\rho>$10\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{cm}}^{-1}$$)
substrate from WaferPro. The width of the nanowire is
$500\text{\,}\mathrm{nm}$. Square-shaped loops with
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width formed by the nanowires along
the center conductor of the CPW allow wireless tuning of the kinetic
inductance through an external magnetic field xu2019frequency . The distance
between the edges of the ground plane is chosen to be
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to reduce the capacitance. A home-
made tuning coil with $50\text{\,}\mathrm{mm}$ diameter and $\sim 2000$ turns
of NbTi wire (Supercon, Inc., SC-T48B-M-0.10mm) generates
$\sim$0.01\text{\,}\mathrm{mT}\text{\,}{\mathrm{mA}}^{-1}$$ at the chip. We
observe negligible heating from the coil with up to $200\text{\,}\mathrm{mA}$
continuous current thanks to zero heat dissipation in the superconducting NbTi
wire. The total length of the high-Z CPW is $\sim$65\text{\,}\mathrm{mm}$$,
giving rise to standing wave resonances with a free spectral range (FSR) of
$\sim$110\text{\,}\mathrm{MHz}$$ (Extended Data Fig. 1). The waveguide
resonances can be frequency-tuned by more than $150\text{\,}\mathrm{MHz}$,
larger than the FSR of the resonances, allowing us to match a waveguide
resonance to a mechanical mode over a broad frequency range.

We show the measured microwave-to-optical conversion S parameter in Extended
Data Fig. 2 as we vary the coil current. Mechanical resonances of the
transducer are not affected by the coil current, while the conversion is
enhanced when the microwave modes are resonant with the mechanical modes. The
microwave modes reach their maximal frequencies at non-zero current due to
non-zero trapped external flux in the tuning loops from a non-zero background
magnetic field during the cooldown.

### Thermal occupation measurement and time-domain heating from the optical
pulse

Extended Data Fig. 3:  Thermal occupation and heating. a, Thermal phonon
occupation of the mechanical mode versus repetition time of the optical pump
pulse. As explained in Methods, the thermal occupation is calculated from
sideband asymmetry between the blue and red detuned pump pulse. b, Time-domain
heating of the optical pulse at $6\text{\,}\mathrm{\SIUnitSymbolMicro s}$
repetition, measured with microwave readout of the mechanical mode. Variance
of the microwave noise is measured and calibrated to thermal phonon number
with optical sideband asymmetry. Dashed red line is the exponential fit of the
decaying tail of the thermal phonon occupation.

Thermal occupation of the mechanical mode $n_{\text{th}}$ is measured with
sideband asymmetry by comparing the single-photon detection rates of the
optomechanically scattered optical sidebands. Two tunable external cavity
diode lasers (Pure Photonics PPCL300) are locked to detunings
$\Delta_{\pm}\equiv\omega_{\pm}-\omega_{\text{o}}=\pm\omega_{\text{m}}$ with
respect to the optical mode $\omega_{\text{o}}$ as blue and red detuned pumps.
Due to imperfect laser detunings and the narrow linewidths of the optical
filter cavities, the count rates are recorded as the filter cavities are tuned
by $\sim$20\text{\,}\mathrm{MHz}$$ between the blue-detuned and red-detuned
sideband count rate measurement to optimize the sideband count rate. The
insertion loss between counts at the two frequencies could be different. To
calibrate the varying insertion loss, we drive a coherent phonon occupation
$n_{\text{coh}}$ with a microwave pulse and measure its sideband count rate in
addition to sideband count rate from the thermal phonons $n_{\text{th}}$ in an
interleaved fashion. Ratio of the sideband count rates between $n_{\text{th}}$
and $n_{\text{coh}}$ is independent of the filter insertion loss,

$R_{\text{r}}=\frac{n_{\text{coh}}}{n_{\text{th}}},~{}R_{\text{b}}=\frac{n_{\text{coh}}+1}{n_{\text{th}}+1},$
(2)

where the subscript denotes the laser detuning. By further comparing these
ratios between blue and red detuned pump, we find

$\frac{R_{\text{b}}-1}{R_{\text{r}}-1}=\frac{n_{\text{th}}}{n_{\text{th}}+1}\equiv
A,$ (3)

giving us the sideband asymmetry ratio $A$ and thermal phonon number
$n_{\text{th}}=1/(1/A-1)$. Extended Data Fig. 3a shows the measured thermal
occupation versus different repetition rates of the optical pulse. We find
that our repetition rate and thermal occupation are comparable to similar
optomechanical systems fiaschi2021optomechanical .

The microwave port of the transducer allow us to monitor the microwave noise
from the transducer, which is dominated by converted thermal mechanical noise
when the transducer is under pulsed optical pump. As shown in Extended Data
Fig. 3b, we measured the temporal heating of the mechanical mode by using a
series of consecutive demodulation windows, each of duration
$48\text{\,}\mathrm{ns}$. Changes in the variance of the demodulated data
correspond to varying thermal noise in the microwave signal, which can be
calibrated to a varying thermal phonon occupation by the initial thermal
occupation measured with optical sideband asymmetry.

### Conversion efficiency and added noise measurement

The conversion efficiency of the transducer is defined as

$\eta_{\upmu\text{o}}\equiv\frac{\dot{N}_{\text{out},\upmu}}{\dot{N}_{\text{in},\text{o}}},~{}\eta_{\text{o}\upmu}\equiv\frac{\dot{N}_{\text{out},\text{o}}}{\dot{N}_{\text{in},\upmu}},$
(4)

where $\dot{N}_{\text{in(out)},\upmu(\text{o})}$ is the input (output)
microwave (optical) photon flux.

The input and output optical photon flux are measured with the sideband filter
and the single photon detector (SPD). For the output photon flux from
microwave-to-optical conversion, it can be directly measured with the optical
setup shown in Extended Data Fig. 5. The system detection efficiency of the
optical detection setup, including the insertion loss of the optical switches,
the isolator, the two sideband filters, and the SPD efficiency, is measured
independently to be $\eta_{\text{SPD}}=9.9\%$. The optical insertion loss
within the dilution refrigerator, and the one-way coupling efficiency between
the lensed fiber and the on-chip waveguide, is measured to be $\eta_{\text{in-
fridge}}=25.4\%$ in total. We measure the insertion loss of the output
circulator to be $\eta_{\text{circ}}=77\%$. Together we calculate an overall
optical setup efficiency from on-chip photonic waveguide to the SPD to be
$\eta_{\text{setup}}=2\%$. Note that the optical mode external coupling
efficiency $\eta_{\text{o}}=50\%$ is not included in the setup efficiency,
while it is included in the conversion efficiency $\eta$ and system efficiency
$\eta_{\text{sys}}$. The measured count rate at the SPD and all the output
insertion losses are used to calculate the output photon flux at the device.

For the input photon flux during the optical-to-microwave conversion, the
dilution refrigerator is bypassed in the optical circuit, i.e., the output of
the circulator that is originally connected to the fridge input port is
connected to the optical switch after the circulator. As a result, the same
output optical circuit is used to measure the sideband photon flux at the
fridge input port. The in-fridge insertion loss $\eta_{\text{in-fridge}}$ is
then used to calculate the input photon flux at the device.

The microwave photon flux at the device is not directly measurable, there is
also no independent way of measuring the microwave input attenuation and the
output amplification $G_{\text{m}}$ separately in our experiment. However,
$G_{\text{m}}$ can be calculated using the measured optical photon flux and
external microwave power assuming the conversion efficiencies are equal
between the two directions. More specifically,

$\dot{N}_{\text{out},\upmu}=\frac{P_{\upmu,\upmu\text{o}}}{\hbar\omega_{\upmu}}\frac{1}{G_{\text{m}}},~{}\dot{N}_{\text{in},\upmu}=\frac{P_{\upmu,\text{o}\upmu}}{\hbar\omega_{\upmu}}\frac{1}{G_{\text{m}}|S_{\upmu\upmu}|^{2}},$
(5)

where $P_{\upmu,\upmu\text{o}(\text{o}\upmu)}$ denotes the output microwave
power measured at the real-time spectrum analyzer (RSA) for the optical-to-
microwave (microwave-to-optical) conversion. $G_{\text{m}}$ converts between
the microwave photon flux at the RSA and the flux leaving the transducer
$\dot{N}_{\text{out},\upmu}$. For the microwave input flux, the output flux is
converted to input using microwave reflection $S_{\upmu\upmu}$ independently
measured with the VNA. Substituting Eq. 5 in Eq. 4, we find that
$\eta_{\upmu\text{o}}=\eta_{\text{o}\upmu}$ leads to

$\displaystyle G_{\text{m}}$ $\displaystyle=$
$\displaystyle\left(\frac{P_{\upmu,\upmu\text{o}}P_{\upmu,\text{o}\upmu}/(\hbar\omega_{\upmu})^{2}}{\dot{N}_{\text{in},\text{o}}\dot{N}_{\text{out},\text{o}}|S_{\upmu\upmu}|^{2}}\right)^{1/2},$
(6) $\displaystyle\eta_{\upmu\text{o}}$ $\displaystyle=$
$\displaystyle\eta_{\text{o}\upmu}=\left(\frac{\dot{N}_{\text{out},\text{o}}}{\dot{N}_{\text{in},\text{o}}}\frac{P_{\upmu,\upmu\text{o}}|S_{\upmu\upmu}|^{2}}{P_{\upmu,\text{o}\upmu}}\right)^{1/2}.$
(7)

$G_{\text{m}}$ can be used to calculate the added noise in the microwave
output during the optical-to-microwave conversion, which is directly measured
by the RSA. We find it to be $4.9$ at the microwave output of the device, and
the corresponding added noise referred to the input is
$n_{\text{added},\upmu\text{o}}=99\pm 10$. Based on the relationships between
theoretical added noises and the thermal occupation of the mechanical mode
han2021microwave , we further estimate the thermal occupation of the
mechanical mode to be $n_{\text{th}}\sim 22$, and the added noise for
microwave-to-optical conversion to be $n_{\text{added},\text{o}\upmu}\sim
181$.

### Microwave readout and added noise

Extended Data Fig. 4:  Demodulation efficiency and timing. a, Calculated
intracavity phonon and output photon flux versus time of the transducer with
one initial phonon using coupled mode theory. b, Calculated mechanics-to-
microwave efficiency using different demodulation waveforms. Matched waveform
gives the highest possible efficiency and is limited by device internal loss.
c, Added noise in the microwave photon. The temporal heating is a theory fit
to the time-domain heating measurement, assuming a bath with exponentially
decaying thermal noise excited by the laser pulse. The temporal mode of the
single photon is shown in comparison. The measured temporal heating is shown
as grey dots. d, Measurement of the best demodulation timing for different
demodulation waveforms. Efficiency of the demodulation is measured by the
relative variance of post-selected state IQ data from single photon detection
events versus the thermal state IQ, explained in detail in Methods. Black
curves are control calculations using randomly selected IQ data instead of
post-selected. e, Microwave $S_{11}$ with no optical pump. $50$ measurements,
taken in quick succession of each other are plotted (blue) where one of them
is highlighted for clarity (blue dots). The red curve shows the fit result
from coupled-mode theory. For some traces the mechanical mode is undercoupled
while for others it is overcoupled due to fluctuations of the intrinsic
mechanical loss rate. The frequency is also stochastically jumping around.
These effects are possibly due to two-level systems (TLS).

The microwave readout uses a phase-insensitive amplifier with $G_{\text{m}}\gg
1$, necessarily adding noise which we represent with $\hat{h}$,

$\mbox{$\hat{S}$}=\sqrt{G_{\text{m}}}\mbox{$\hat{c}_{\text{out}}$}+\sqrt{G_{\text{m}}-1}\mbox{$\hat{h}^{\dagger}$}\approx\sqrt{G_{\text{m}}}(\mbox{$\hat{c}_{\text{out}}$}+\mbox{$\hat{h}^{\dagger}$}),$
(8)

where $\hat{c}_{\text{out}}$ is the microwave output operator. The output
operator is related to the mechanical mode operator by the external coupling
efficiency $\eta_{\upmu\text{m}}$. Note that $\eta_{\upmu\text{m}}$ is
different from the external coupling efficiency of the microwave resonator
$\eta_{\upmu}$. Assuming perfect demodulation, the temporal microwave mode
detected from the device has a ladder operator given by

$\mbox{$\hat{c}_{\text{out}}$}=\sqrt{\eta_{\upmu\text{m}}}\mbox{$\hat{b}$}+\sqrt{1-\eta_{\upmu\text{m}}}\mbox{$\hat{b}_{\text{n}}$},$
(9)

where $\hat{b}_{\text{n}}$ describes noise added from other degrees of freedom
because of the device inefficiency, and follows the relations
$\braket{\mbox{$\hat{b}_{\text{n}}$}\mbox{$\hat{b}_{\text{n}}^{\dagger}$}}=n_{\text{n}}+1$
and
$\braket{\mbox{$\hat{b}_{\text{n}}^{\dagger}$}\mbox{$\hat{b}_{\text{n}}$}}=n_{\text{n}}$.

Noise is further added by the microwave readout chain. More specifically, we
characterize the microwave readout by gain $G_{\text{m}}$, measurement noise
$n_{\text{m}}$ referred to the TWPA input, and the demodulation efficiency
$\eta_{\text{d}}$,

$\hat{I}$ $\displaystyle=$
$\displaystyle\mbox{$\hat{S}$}+\mbox{$\hat{S}^{\dagger}$}$ (10)
$\displaystyle=$
$\displaystyle\sqrt{\eta_{\text{d}}\eta_{\upmu\text{m}}G_{\text{m}}}\mbox{$\hat{X}$}$
$\displaystyle+\sqrt{\eta_{\text{d}}(1-\eta_{\upmu\text{m}})G_{\text{m}}}\mbox{$\hat{X}_{\text{n}}$}+\sqrt{G_{\text{m}}}\mbox{$\hat{X}_{\text{m}}$},$

where
$\mbox{$\hat{X}_{(\text{n})}$}=\mbox{$\hat{b}_{(\text{n})}$}+\mbox{$\hat{b}_{(\text{n})}^{\dagger}$}$.
The measurement added noise is assumed to be broadband and is independent of
the demodulation. $X_{\text{n(m)}}$ follows the noise statistics
$\braket{X_{\text{n(m)}}^{2}}=2n_{\text{n(m)}}+1$.

When the mechanical mode is in a thermal state with mean phonon number
$n_{\text{th}}$, the variance of the demodulated IQ data is

$\displaystyle\braket{I^{2}}$ $\displaystyle=$
$\displaystyle\eta_{\text{d}}\eta_{\upmu\text{m}}G_{\text{m}}(2n_{\text{th}}+1)+\eta_{\text{d}}(1-\eta_{\upmu\text{m}})G_{\text{m}}(2n_{\text{n}}+1)$
(11) $\displaystyle+G_{\text{m}}(2n_{\text{m}}+1)$ $\displaystyle=$
$\displaystyle\eta_{\text{d}}\eta_{\upmu\text{m}}G_{\text{m}}(2(n_{\text{th}}+n_{\text{ex}})+2),$

where $n_{\text{n}}$ is thermal noise from other degrees of freedom, including
heating from the optical pump pulse, emitted into the microwave channel, and
$n_{\text{m}}$ is excess noise from the microwave amplifiers. We have lumped
all excess noise into $n_{\text{ex}}$,

$\displaystyle n_{\text{ex}}$ $\displaystyle=$
$\displaystyle\frac{1-\eta_{\upmu\text{m}}}{\eta_{\upmu\text{m}}}n_{\text{n}}+\frac{1}{\eta_{\text{d}}\eta_{\upmu\text{m}}}n_{\text{m}}$
(12)
$\displaystyle+\frac{1}{2}\left(\frac{1-\eta_{\upmu\text{m}}}{\eta_{\upmu\text{m}}}+\frac{1}{\eta_{\text{d}}\eta_{\upmu\text{m}}}-1\right).$

For ideal quadrature-phase measurement,
$\eta_{\upmu\text{m}}=\eta_{\text{d}}=1$ and $n_{\text{m}}=0$, and a minimal
noise of $1/2$ is added, giving us the phase space distribution as the Husimi
Q representation of the state. Note that this minimal added noise is not
included in our definition of excess noise $n_{\text{ex}}$, and
$n_{\text{ex}}=0$ for ideal quadrature-phase measurement.

When a phonon is added to the state via post-selection with single photon
detection, the mean phonon number is $n_{\text{th, PS}}=2n_{\text{th}}+1$. As
a result,

$\displaystyle\braket{I^{2}}|_{\text{PS}}$ $\displaystyle=$
$\displaystyle\braket{I^{2}}+\eta_{\text{d}}\eta_{\upmu\text{m}}G_{\text{m}}(2n_{\text{th}}+2),$
(13) $\displaystyle\frac{\braket{I^{2}}|_{\text{PS}}}{\braket{I^{2}}}$
$\displaystyle=$ $\displaystyle
1+\frac{n_{\text{th}}+1}{n_{\text{th}}+n_{\text{ex}}+1}.$ (14)

In the actual experiment, there is a non-zero dark count rate of
$70\pm$10\text{\,}\mathrm{Hz}$$ on the SPD, resulting in a heralding
efficiency $\eta_{\text{herald}}\approx 85\%$, defined as the fraction of the
total counts that are actually from the sideband photons. The normalized post-
selected variance is then given by

$\frac{\braket{I^{2}}|_{\text{PS}}}{\braket{I^{2}}}=1+\eta_{\text{herald}}\frac{n_{\text{th}}+1}{n_{\text{th}}+n_{\text{ex}}+1}.$
(15)

To minimize added noise $n_{\text{n}}$ in the microwave tomography
measurement, matched filtering is desired and can be realized during the
digital demodulation eichler2011experimental ; patel2021room . The optimal
filter shape is given by the time domain waveform of the outgoing photon,
which has the same time-domain waveform as the classical solution of the
propagating microwave field $A(t)$. We solve the coupled-mode theory (CMT) to
calculate the propagating microwave field with one initial phonon in the
mechanical mode, as shown in Extended Data Fig. 4a. The intracavity photon
number of the mechanical and microwave modes are also shown for comparison.

We have attempted demodulation with different filtering including the matched
filtering with the numerical waveform $f_{\text{demod}}(t)=A(t)$ calculated
from CMT, and exponential waveforms
$f_{\text{d}}(t)=\sqrt{\kappa_{\text{d}}}\exp(-\kappa_{\text{d}}t/2)\theta(t)$
with decay rate of $\kappa_{\text{d}}/2\pi=$1\text{\,}\mathrm{MHz}$$ and
$5\text{\,}\mathrm{MHz}$. $\theta(t)$ is the Heaviside function. The matched
filter is plotted as a dashed black line to show the theoretical maximal
efficiency limited by the device parameters. Extended Data Fig. 4b shows the
theoretical measurement efficiency $\eta_{\text{m}}(t)\equiv|\int d\tau
A^{*}(\tau)f_{\text{d}}(\tau-t)|^{2}$, where $t$ is the demodulation delay.
The matched waveform gives the highest possible efficiency
$\eta_{\upmu\text{m}}$ which is the device efficiency, and is limited by
device internal loss. Extended Data Fig. 4c shows the measured relative change
of the IQ variance before and after post-selection as a figure of merit for
the state tomography measurement patel2021room . A higher initial thermal
phonon occupation both increases the sideband count rate, and makes the post-
selected state more distinguishable with the same excess noise from the
measurement, since the mean phonon number roughly doubles after the phonon
addition event patel2021room . Therefore, we apply a heating pulse to the
transducer $140\text{\,}\mathrm{ns}$ before the heralding pulse to increase
the initial thermal phonon occupation to $\sim 10$. We find that the two
exponential filters give a more visible change in variance, and thus lower
excess noise. We suspect this is due to two-level system induced fluctuation
of the mechanical frequency, as shown in Extended Data Fig. 4e with microwave
reflection measurement under no optical pump. As a result, the demodulation in
the heralding experiment is conducted with a $5\text{\,}\mathrm{MHz}$
exponential filter, and the theoretical total measurement efficiency is
$\approx 8\%$, where the demod efficiency $\eta_{\text{d}}\approx 24\%$ and
the device efficiency $\eta_{\upmu\text{m}}\approx 35\%$. This is larger than
the magnitude of the frequency fluctuations and therefore masks their effect
at the cost of lowered efficiency. Characterization and improvement of the
frequency fluctuation will be the subject of future study.

In order to calibrate the gain $G_{\text{m}}$ and excess noise $n_{\text{m}}$,
two different states of the microwave field are required. We carry out two
types of calibration. First, we use the laser heating to generate two thermal
states with different thermal occupations. For the thermal state with higher
$n_{\text{th}}$, a heating pulse is applied $140\text{\,}\mathrm{ns}$ prior to
the probe pulse, where the sideband asymmetry from the probe pulse is used to
obtain the thermal occupation of the mechanical mode
$n_{\text{th,high}}=5.9\pm 2$. For the thermal state with lower
$n_{\text{th}}$, we use a repetition of $10\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ and no heating pulse to obtain $n_{\text{th, low}}=0.56\pm 0.2$. The
demodulation is aligned to the probe pulse, and we measure the IQ variance
from these two different thermal states. From Eq. 11, we calculate
$n_{\text{ex}}=18\pm 8$. However, we find that the device parameters are
varying under different optical powers, which results in different
$\eta_{\upmu\text{m}}$. A $10\text{\,}\mathrm{\char 37\relax}$ change in
$\eta_{\upmu\text{m}}$ could lead to a factor of $2$ difference in
$n_{\text{ex}}$. Alternatively, $n_{\text{ex}}$ can be calculated using the
thermal and photon-added state as shown in Eq. 15. For the data shown in Fig.
3 of the main text, we find $n_{\text{ex}}=39\pm 6$.

To better understand the potential of the microwave field from the transducer,
it is important to estimate the added noise $n_{\text{n}}$ in it. Noise in the
microwave field is not directly measurable with imperfect detection.
Nevertheless, we carry out the estimation with two different methods. In
addition to the pre- and post-selected datasets, we further take a control
measurement every $500$ experimental runs, where we execute the same
demodulation with the same timing except that no optical pump pulse is sent to
the device. The system can be approximated to be in the initial thermal state
with $n_{\text{th}}=0.68$, independently measured by optical sideband
asymmetry, and no heating or microwave photon is generated. As a result, the
variance becomes

$\braket{I^{2}}|_{\text{th}}=\eta_{\text{d}}G_{\text{m}}(2n_{\text{th}}+1)+G_{\text{m}}(2n_{\text{m}}+1).$
(16)

Using $\eta_{\text{d}}$ and $\eta_{\upmu\text{m}}$ calculated from theory, the
measured $\braket{I^{2}}$, $\braket{I^{2}}|_{\text{PS}}$ and
$\braket{I^{2}}|_{\text{th}}$ allow us to calculate
$n_{\text{n}}=1.9~{}\pm~{}0.4$ and $n_{\text{m}}=2.4~{}\pm~{}0.4$.

Alternatively, we could estimate the added noise in the microwave field from
the overlap between the measured temporal heating and the temporal mode of the
microwave photon. We fit the measured temporal heating assuming the mechanical
mode is coupled to a thermal bath with an exponentially decaying population
excited by the pump pulse. Noise in the output microwave signal is calculated
with semi-classical Monte Carlo simulation of an ensemble of 3000 instances.
Extended Data Fig. 4c shows the temporal heating together with the single
photon temporal mode. Overlap between the heating and single photon temporal
mode gives $n_{\text{n}}=1.3\pm 0.2$. The uncertainty mostly comes from the
sideband asymmetry measurement of the initial $n_{\text{th}}$.

### Measurement setup

Extended Data Fig. 5 shows the optical setup used in this work. Two tunable
external cavity diode lasers (PurePhotonics PPCL300) are first intensity-
stabilized with electro-optic modulators, and then frequency-stabilized using
temperature-stabilized fiber Fabry-Pérot filters (F1 and F2). The filters also
suppress the laser phase noise at the converter frequencies. A fast
wavelength-scanning laser (Freedom Photonics FP4209) is used for fiber-to-chip
coupling optimization. Two acousto-optic modulators (AOMs) are simultaneously
pulsed to generate the optical pump pulse with high on-off ratio
($>$90\text{\,}\mathrm{dB}$$). The duration of the pulse is limited by the
rise-fall time of the $200\text{\,}\mathrm{MHz}$ AOM to be
$\gtrsim$20\text{\,}\mathrm{ns}$$. Multiple MEMS optical switches are
implemented to route the input light to either the transducer or the filter
cavities FA and FB for coarse tuning. The filter cavities have
$15\text{\,}\mathrm{MHz}$ bandwidth and $14\text{\,}\mathrm{GHz}$ free
spectral range, and provide $90\text{\,}\mathrm{dB}$ pump suppression, while
dispersion at the filter resonance delays the sideband photons by an extra
$\sim$40\text{\,}\mathrm{ns}$$ compared to the feed-through transmitted pump
photons. This allows us to further separate the pump and sideband photon
counts in time domain. The reflected light from the device can be routed to
one of the photodetectors for the EIT measurement or microwave-to-optical
conversion measurement, or to the filter cavities for single photon detection
on the sideband photons.

We show the microwave setup in Extended Data Fig. 6. An intermediate frequency
(IF) of $125\text{\,}\mathrm{MHz}$ is used from the Quantum Machine (QM OPX01)
and upconverted for the microwave input. Proper attenuations on the microwave
input line guarantee the input microwave thermal noise to be less than $0.01$.
The microwave signal from the transducer is first amplified by a traveling-
wave parametric amplifier (TWPA) with the dispersive feature around
$6.42\text{\,}\mathrm{GHz}$ and pumped at $5.027\text{\,}\mathrm{GHz}$ (not
shown) to maximize the gain at the transducer frequency. Two broadband
isolators ($3-$12\text{\,}\mathrm{GHz}$$) are installed before and after the
TWPA to minimize reflection within its gain bandwidth. The signal is then
further amplified by a high electron mobility transistor (HEMT) and two room
temperature low noise amplifiers, and measured by either the real-time
spectrum analyzer (RSA), the vector network analyzer (VNA), or downconverted
and digitized on the QM. Temporal delay in the optical setup is longer than in
the microwave setup, and the microwave signal from the transducer arrives at
the QM prior to the voltage pulse from the SPD. As a result, the demodulation
is always executed first, and then stored differently in real-time,
conditioned on the SPD event.

Extended Data Fig. 5:  Optical setup. Extended Data Fig. 6:  Microwave setup.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgment

W.J. and F.M.M. would like to thank Christopher J. Sarabalis and Haonan Xiong
for helpful discussions. A-H.S.N. acknowledges useful discussions with Oskar
Painter, Cindy Regal, Konrad Lehnert, Martin Fejer, and Simon Groeblacher. The
authors would like to thank Kevin K.S. Multani, Agnetta Y. Cleland, Oliver A.
Hitchcock, Carsten Langrock, and Matthew P. Maksymowych for fabrication
assistance, Kevin A. Villegas Rosales and Niv Drucker at Quantum Machines and
Yudan Guo for technical support, and Kyle Serniak and William D. Oliver at MIT
Lincoln Laboratory for providing the TWPA. This work was primarily supported
by the U.S. Army Research Office (ARO) Cross-Quantum Systems Science &
Technology (CQTS) program (Grant No. W911NF-18-1-0103), the National Science
Foundation CAREER award No. ECCS-1941826, the Airforce Office of Scientific
Research (AFOSR) (MURI No. FA9550-17-1-0002 led by CUNY), and the David and
Lucille Packard Fellowship. Device fabrication was performed at the Stanford
Nano Shared Facilities (SNSF) and the Stanford Nanofabrication Facility (SNF),
supported by the NSF award ECCS-2026822. A.H.S.-N. acknowledges support via a
Sloan Fellowship. The authors also wish to thank NTT Research and Amazon Web
Services Inc. for their financial support. Some of this work was funded by the
U.S. Department of Energy through Grant No. DE-AC02-76SF00515 and via the
Q-NEXT Center.

## Author contributions

W.J. designed the device with assistance from F.M.M and S.M.. W.J. and F.M.M.
fabricated the device assisted by S.M.. W.J., F.M.M. and R.V.L. developed the
fabrication process. W.J. and F.M.M measured the device with assistance from
S.M.. R.N.P., T.P.M., J.D.W. and A.H.S.-N. provided assistance with the
measurement setup. W.J., F.M.M. and A.H.S.-N. wrote the manuscript with input
from all authors. A.H.S.-N. supervised the project.

## Competing interests

A.H.S.-N. is an Amazon Scholar. The other authors declare no competing
interests.

## Supplementary information

##  Supplementary Note 1 Device design

The optomechanical design is similar to previous piezo-optomechanical
transducer works peairs2020continuous ; mirhosseini2020superconducting . We
design two different types of mirror unit cells, one optimized with both
optical and mechanical bandgaps around the frequency of the localized modes
for the optical and mechanical polarizations of interest, and one with only
the optical bandgap while still allowing the mechanical mode to propagate. We
first optimize the optomechanical crystal with a fully symmetric design using
the first type of unitcell on both ends of the OMC, and optimize the defect
cell geometries to maximize the product of the radiation-limited optical
quality factor and the optomechanical coupling rate $g_{\text{o}}$. A cubic
transition between the defect unit cell and the mirror unit cell forms the
smooth confinement of the optical and mechanical mode. Unit cells on one side
of the OMC are then replaced to be the second type, enabling the hybridization
between the mechanical mode in the OMC and the hybrid Si-LN piezoelectric
mode.

It is important to maximize the hybridization between the OMC mechanical mode
and the Si-LN piezoelectric mode, so that the device is tolerant to
fabrication variations. We note that the leaky mechanical mode from the OMC is
mostly longitudinal, while the strong piezoelectric mode in the Si-LN
structure is mostly in horizontal shear (SH) polarization. We use a
$45^{\circ}$ rotation between the OMC and the Si-LN structure to align the
polarization between the longitudinal mode and SH mode, and achieve a
hybridization with coupling strength $J/2\pi\approx$42\text{\,}\mathrm{MHz}$$.
Similar to Ref. arrangoiz2019resolving , we adopt $X$-cut thin-film LN to
utilize the strong piezoelectricity of the SH mode, where its largest
piezoelectric component
$d_{24}\approx$70\text{\,}\mathrm{pm}\text{\,}{\mathrm{V}}^{-1}$$ couples $YZ$
shear to electric field along the crystal $Y$ axis. The aluminum electrodes
are patterned to be parallel to the crystal $Z$ axis so that the electric
field between them are parallel to $Y$. To prevent the mechanical mode from
leaking into the substrate, we implement extra one dimensional phononic
shields (1DPS) with a full bandgap around the mechanical frequency. The 1DPS
is optimized separately for anchoring the Si-LN piezoelectric structure to
take into account the aluminum electrodes.

##  Supplementary Note 2 Device fabrication

Supplementary Figure 1:  Device fabrication. a, Fabrication process of the
transducer chip. b, Optical image of the transducer device after the full
fabrication process. The edge of the chip is diced to expose the on-chip
optical coupler. c, Fabrication process of the microwave chip.

Patterning of thin film LN has been done with a wide variety of underlying
substrates such as silicon, silica and sapphire. However, physical etching of
LN is not directly compatible with the LN-on-SOI platform since the etch
damages the surface of the silicon thin film and leads to significantly
reduced optical quality factors. To avoid this issue, we pattern a thin film
of LN on a separate LN-on-insulator sample and transfer print
meitl2006transfer  it onto a pristine SOI chip. The fabrication process is
summarized in Supplementary Figure 1. We start with 400 nm thick MgO-doped
X-cut LN on a $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick thermal silica
buffer layer sitting on a $400\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick
silicon handle (commercially available from NanoLN). The LN film is first
thinned to $290\text{\,}\mathrm{nm}$ with argon ion milling. The LN transducer
and alignment marks are defined with electron beam lithography (EBL) using
hydrogen silsesquioxane (HSQ) as the resist. The pattern is transferred to LN
with a $265\text{\,}\mathrm{nm}$ deep etch using argon ion milling. The HSQ is
stripped in buffered oxide etchant (BOE, 6:1) and the LN is then cleaned in
dilute hydrofluoric acid (HF, $5\text{\,}\mathrm{\char 37\relax}$). To etch
the remaining LN slab and expose the silica, we carry out another
$40\text{\,}\mathrm{nm}$ deep LN etch with blanket argon ion milling, followed
by a clean in a piranha solution (3:1 $98\%$ sulfuric acid and $40\%$ hydrogen
peroxide). The LN is released by etching the silica in 6:1 BOE for $22$
minutes which leads to a $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ undercut of
the thermal oxide. A piece of polydimethylsiloxane (PDMS) attached to a glass-
slide which is mounted on a set of translational stages is lowered until it
fully covers the LN sample. Due to its viscoelasticity meitl2006transfer ,
when rapidly lifting up the PDMS away from the sample, released LN patterns
break off from their anchors and stick to the PDMS. We then make the PDMS with
LN touch down onto a SOI piece ($220\text{\,}\mathrm{nm}$ thick silicon device
layer, $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick buried oxide,
$725\text{\,}\mathrm{\SIUnitSymbolMicro m}$ silicon handle) that has been
cleaned in a piranha solution. By slowly peeling off the PDMS from the SOI
substrate, the LN patterns are released from the PDMS and adhere to the
silicon. We could simultaneously transfer print more than 100 devices, and
typically achieve a $>$95\text{\,}\mathrm{\char 37\relax}$$ transfer print
yield. Subsequently, the chip is annealed for $8$ hours at
$500\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to improve the LN-Si adhesion.
This is followed by a clean in piranha solution. The transfer print causes a
$\sim 0.1\%$ relative displacement between separated patterns. We attach
alignment marks within $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of every
device to enable sub-$30\text{\,}\mathrm{nm}$ alignment error on EBL masks
after the transfer print.

We pattern the silicon with aligned EBL and do an inductively coupled plasma
(ICP) - reactive ion etch (RIE) using a $\text{Cl}_{2}$ and HBr gas chemistry.
The device is still attached to the transfer printed alignment mark which
would cause mechanical loss. We therefore do another aligned EBL to pattern a
small window where we argon ion mill the LN followed by ICP-RIE to etch
through the silicon device layer. Subsequently, we expose the oxide in the
areas of the chip that do not have any devices by etching the silicon device
layer with ICP-RIE after having defined etch windows with photolithography.
The sample is cleaned a final time in a piranha solution before metalization.
The transducer electrodes are patterned with aligned EBL. We evaporate
$55\text{\,}\mathrm{nm}$ aluminum at three different angles
($-66^{\circ},0^{\circ},66^{\circ}$) to help the electrodes climb the LN from
the silicon, and lift-off in N-Methyl-2-pyrrolidone (NMP) at
$80\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The wirebond pads are defined with
photolithography, and we evaporate $250\text{\,}\mathrm{nm}$ aluminum followed
by lift-off in NMP at $80\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Finally, the
chip is diced close to the edge coupler and the devices are released in HF
vapor. To accommodate fabrication uncertainties, the optomechanical crystals
and the separation of the piezoelectric resonator electrodes in different
transducer devices are scaled by $\pm 2\%$ and $\pm 5\%$ respectively.

The microwave resonator is fabricated on a separate die from the transducer to
simplify fabrication and potentially reduce optically induced quasiparticle
generation mirhosseini2020superconducting  by vertically offsetting the
resonator from the optical fiber. We start with a $10\text{\,}\mathrm{nm}$
thin layer of niobium titanium nitride (NbTiN) on high-resistivity silicon. We
use EBL to define the meander in HSQ resist and etch the NbTiN using RIE with
a $\text{CF}_{4}$ and $\text{O}_{2}$ gas chemistry. The HSQ is stripped in 6:1
BOE. Finally, a $250\text{\,}\mathrm{nm}$ thick aluminum ground plane is
defined with photolithography followed by evaporation and liftoff.

##  Supplementary Note 3 Room temperature mechanics-to-microwave external
coupling measurement

Supplementary Figure 2:  Room-temperature conversion measurement and
calibration. a, Normalized microwave-to-optical conversion S parameter.
Multiple mechanical modes are visible due to the high signal-to-noise ratio of
the measurement. Two mechanical modes are strongly hybridized with the OMC
mechanical mode and show much higher conversion efficiency than other modes.
b, Calibration of the mechanics-to-microwave external coupling. The thermal-
mechanical noise power (shaded area) is used to calibrate the optomechanical
readout of the intracavity phonon number.

The fabricated transducers can be measured at room temperature before
integrating with the microwave chip, especially for characterizing the
mechanics-to-microwave external coupling and optical quality factor. This
allows us to select the best performing transducer for packaging and cooldown.
Similar to the method described in Ref. jiang2019lithium , a microwave probe
(GGB Industries, Picoprobe model 40A-GSG) is brought in contact with the on-
chip electrodes of the piezo-optomechanical transducer. A red-detuned laser is
used to enable the microwave-to-optical conversion and readout the converted
sideband photon (Supplementary Figure 2). The coherent microwave input signal
is then fixed at the peak conversion frequency, on resonance with one of the
mechanical modes, and gives rise to a coherent intracavity phonon population

$n_{\text{coh}}=\frac{4\gamma_{\upmu}\dot{N}}{\gamma^{2}},$ (17)

where $\gamma_{\upmu}$ is the external coupling rate between the mechanical
mode and the $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ microwave line, $\gamma$
is the total mechanical linewidth, and $\dot{N}$ is the input microwave photon
flux. As shown in Supplementary Figure 2, both the coherent and the thermal
phonons are converted to sideband photons, and measured on the microwave power
spectrum from a fast photodetector output. To calculate the thermal phonon
number and calibrate the optical readout, the device temperature is assumed to
be $295\text{\,}\mathrm{K}$ . The coherent phonon number $n_{\text{coh}}$ and
the external coupling rate $\gamma_{\upmu}$ can then be calculated.

##  Supplementary Note 4 Piezoelectric coupling via cross-chip wirebond

Supplementary Figure 3:  Cross-chip wire bond model. a, Lumped element circuit
model of the cross-chip wirebonds between the Si-LN mechanical mode of the
transducer chip and the high-impedance waveguide mode of the microwave chip.
The Si-LN mechanical mode is modeled as a $RLC$ circuit (with parameters
$R_{\text{m}}$, $L_{\text{m}}$, $C_{\text{m}}$, and $C_{o}$) and the microwave
waveguide is modeled as a transmission line (with a characteristic impedance
of $Z=$1000\text{\,}\mathrm{\SIUnitSymbolOhm}$$). The signal and ground cross-
chip wirebonds are modeled as series inductance $L_{\text{wb}}$ with parasitic
capacitance to the ground plane $C_{\text{p}}$ in parallel, a contact
capacitance $C_{\text{wb}}$ and a contact resistance $R_{\text{wb}}$ in
series. There are $N_{\text{wb}}$ number of wirebonds with mutual capacitance
$C_{\text{p,wb}}$ between them. b, Coupling rate between the Si-LN mechanical
mode and the microwave waveguide as a function of wirebond length for various
values of $N_{\text{wb}}$. A shaded grey region shows the estimated coupling
rate and the estimated wirebond length in this experiment.

The transducer and the microwave resonator are fabricated on separate chips
and integrated together via cross-chip wirebonding. To understand the impact
of wirebonds on the coupling rate between the two, we consider a lumped
element circuit model of the wirebonds between the Si-LN mechanical mode of
the transducer chip and the high-impedance waveguide mode of the microwave
chip, as shown in Supplementary Figure 3. The Si-LN mechanical mode is modeled
as a $RLC$ circuit with parameters $L_{\text{m}}=$195.6\text{\,}\mathrm{nH}$$,
$C_{\text{m}}=$10.01\text{\,}\mathrm{fF}$$,
$R_{\text{m}}=$45.4\text{\,}\mathrm{M\SIUnitSymbolOhm}$$, and
$C_{o}=$0.11\text{\,}\mathrm{fF}$$, estimated from a combination of finite-
element simulation and room temperature piezo-optomechanical measurements. The
microwave waveguide is modeled as a transmission line (FSR =
$110\text{\,}\mathrm{MHz}$, $Z=$1000\text{\,}\mathrm{\SIUnitSymbolOhm}$$). We
model the signal and ground cross-chip wirebonds as series inductance
$L_{\text{wb}}$ with parasitic capacitance to the ground plane $C_{\text{p}}$
in parallel, and a contact capacitance $C_{\text{wb}}$ and a contact
resistance $R_{\text{wb}}$ in series. The wirebonds have mutual capacitance of
$C_{\text{p,wb}}$ between them. For a $25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ diameter wire, the inductance roughly scales as $L_{\text{wb}}$
$\sim$1\text{\,}\mathrm{nH}$/$\text{\,}\mathrm{mm}$$ Wenner2011 ;
huang2021microwave . The remaining wirebond parameters are estimated using
finite-element electrostatic analysis with nominal values of $C_{\text{p}}$
$\sim$20\text{\,}\mathrm{fF}$$, $C_{\text{wb}}$
$\sim$20\text{\,}\mathrm{pF}$$, $R_{\text{wb}}$
$\sim$0.4\text{\,}\mathrm{\SIUnitSymbolOhm}$$ zhong2021deterministic , and
$C_{\text{p,wb}}$ $\sim$12\text{\,}\mathrm{fF}$/$\text{\,}\mathrm{mm}$$. The
model exhibits strong reduction in the coupling rate with increasing wirebond
length and increasing number of wirebonds as seen in the plot in Supplementary
Figure 3. This is due to the increase in parasitic capacitance of the wires to
the ground plane and increase in mutual capacitance between the signal and
ground wirebonds. Hence, to maximize the coupling rate, a single signal
wirebond with length as short as possible is desirable.

We first considered placing the two chips next to each other with the optical
edge coupler side of the transducer chip facing away from the microwave chip.
However, the shortest possible wirebond in such a case is limited by the net
distance between the terminals of the devices on the two chips which turns out
to be significant. In this experiment, we utilize a different approach in
which the two chips are wirebonded together with the optical edge coupler side
of the transducer chip facing the microwave chip. To allow fiber access for
optical coupling, the transducer is then flipped over and stacked vertically,
resulting in a shorter wirebond with length on the order of
$0.75\text{\,}\mathrm{mm}$. Two small spacer chips are glued near the edge of
the microwave chip with GE varnish to support the transducer chip. Not only
does this approach give us a shorter wirebond, it also results in the
microwave chip being in a different plane than the optical fiber which likely
helps minimize optically induced quasiparticle generation.