††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: These authors contributed equally

# Supplementary Information for “Non-classical microwave-optical photon pair
generation with a chip-scale transducer”

Srujan Meesala  Kavli Nanoscience Institute and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena,
California 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA  Steven
Wood  Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of
Applied Physics, California Institute of Technology, Pasadena, California
91125, USA  Institute for Quantum Information and Matter, California Institute
of Technology, Pasadena, California 91125, USA  David Lake  Kavli Nanoscience
Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California
Institute of Technology, Pasadena, California 91125, USA  Institute for
Quantum Information and Matter, California Institute of Technology, Pasadena,
California 91125, USA  Piero Chiappina  Kavli Nanoscience Institute and Thomas
J. Watson, Sr., Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA  Institute for Quantum Information
and Matter, California Institute of Technology, Pasadena, California 91125,
USA  Changchun Zhong  Pritzker School of Molecular Engineering, The University
of Chicago, Chicago, IL 60637, USA  Andrew D. Beyer  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena,
California 91109, USA  Matthew D. Shaw  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr, Pasadena, California 91109, USA
Liang Jiang  Pritzker School of Molecular Engineering, The University of
Chicago, Chicago, IL 60637, USA  Oskar Painter<EMAIL_ADDRESS>http://copilot.caltech.edu Kavli Nanoscience Institute and Thomas J. Watson,
Sr., Laboratory of Applied Physics, California Institute of Technology,
Pasadena, California 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA  Center
for Quantum Computing, Amazon Web Services, Pasadena, California 91125, USA

##  1 Notation

Symbol |  Meaning
---|---
$\hat{a},\hat{b},\hat{c}$ |  operators for optical, acoustic, and microwave modes of the transducer

$\hat{c}_{+},\hat{c}_{-}$ |  operators for hybridized electromechanical modes

$\hat{a}_{\mathrm{in}},\hat{c}_{\mathrm{in}},(\hat{a}_{\mathrm{out}},\hat{c}_{\mathrm{out}})$ |  operators for optical and microwave input (output) modes in coupling waveguides

$\hat{A},\hat{C}$ |  operators for optical and microwave temporal modes in coupling waveguides

$\bar{C}_{mn}$ |  moments of the temporal microwave mode

$\bar{C}_{mn}|_{\mathrm{click}}$ |  moments of the temporal microwave mode conditioned on an optical click
Table S1: Summary of the various modes and related quantities defined in the
main text.

##  2 Device fabrication process

The fabrication process for the transducer chip is illustrated in Fig. S1 and
described in the caption. Masks for all steps are patterned in ZEP-520A resist
via electron-beam lithography on a Raith EBPG 5200 tool. All dry etching is
performed in Oxford Plasmalab 100 inductive coupled plasma reactive ion
etching (ICP RIE) tools. The process flow can be sub-divided into sections
used to define various portions of the transducer device. Steps (i)-(vi)
complete the definition of the AlN box essential for the piezoacoustic cavity.
The combination of dry and wet etch steps ensures that the dimensions of the
AlN box are precisely defined while the silicon device layer is undamaged on
most of the chip. This is important to achieve optical, mechanical, and
microwave modes with high quality factors. Steps (vii)-(ix) define the NbN
resonator and step (x) defines the OMC. Steps (xi)-(xiii) are use to define
aluminum electrodes on the piezo-resonator and galvanically connect them to
the NbN resonator using bandage steps. The bandaid steps involve in situ Ar
milling for two minute and six minute durations respectively to clear the
surface of NbN and Al prior to Al bandaid evaporation. To provide optical
fiber access to coupler sections at the end of the silicon photonic
waveguides, we clear a portion of the SOI substrate up to a depth of 150
$\mathrm{\mu}$m using a deep reactive ion etch at the edge of the chip.
Finally, the buried oxide (BOX) layer is etched in anhydrous vapor HF to
release the device membrane.

Figure S1: Device fabrication process. Images are not to scale. (i) Sputter
deposition of 300 nm thick c-axis AlN piezoelectric film (grown by OEM group;
stress $T=+55$ MPa; (002) XRD peak of full-width at half-maximum = 1.79°) on a
silicon on insulator substrate (Si device layer [float zone grown, 220 nm
thick, $\rho\geq 5\text{k}\Omega\text{-cm}$]; buried oxide layer [3 µm thick,
silicon dioxide]; Si handle [Czochralski grown, 750 µm thick, $\rho\geq
5\text{k}\Omega\text{-cm}$]). (ii) AlN trench etch with Ar/Cl2 chemistry to
define the perimeter of the piezo resonator via a small trench of width
$\sim$100 nm. (iii) Conformal deposition of a SiOx hard mask via plasma
enhanced chemical vapor deposition. (iv) Patterning of the SiOx mask via dry
etching with C4F8/SF6 chemistry. (v) Removal of the remaining AlN on the chip
with a H3PO4 wet etch. (vi) SiOx mask removal with 10:1 buffered oxide
etchant. (vii) Protection of the piezo-acoustic and OMC regions with a resist
mask prior to deposition of NbN. (viii) Deposition of a 10 nm thick film of
NbN via an RF sputtetring process. (ix) Dry etch of NbN with SF6/Ar chemistry
to define the high-impedance microwave resonator. (x) Dry etch of silicon with
C4F8/SF6 chemistry to define OMC, acoustic shield and optical waveguide. (xi)
Deposition of Al electrodes for the piezo-acoustic resonator using angled
electron-beam evaporation. (xii, xiii) Bandage steps with Ar milling followed
by Al evaporation. (xiv) Etching of the buried oxide (BOX) layer using
anhydrous vapor HF to release the device membrane.

##  3 Microwave circuit design

The kinetic inductance resonator used in our transducer is fabricated from an
NbN film of 10 nm thickness and 50 pH/sq sheet inductance. The meandering
ladder geometry described in the main text is comprised of 2 $\mathrm{\mu
m}\times 1~{}\mathrm{\mu m}$ rectangular loops formed from traces of width 130
nm. We use extended electrical terminals of length 200 $\mathrm{\mu m}$ and
width 1$~{}\mathrm{\mu m}$ to spatially separate the high kinetic inductance
section from the OMC. The resonator is designed to achieve a target frequency
of 5.0 GHz for the fundamental mode with a capacitance, $C_{\mathrm{res}}=7.1$
fF, which includes a small contribution of 0.27 fF from the electrodes on the
piezo-acoustic resonator. Nearly the entire inductance of the resonator mode
is due to the kinetic inductance of the superconducting film. The use of
closed superconducting loops in the resonator allows for tuning of the kinetic
inductance, $L_{k}$, via a DC supercurrent, $I$, induced by an external
magnetic field according to the relation $L_{k}\approx
L_{k,0}[1+(I/I_{*})^{2}]$ as shown in [1]. Here $L_{k,0}$ is the kinetic
inductance at zero magnetic field. $I_{*}\gg I$ is a characteristic current on
the order of the critical current of the nanowire [2]. This relation leads to
quadratic tuning of the resonator frequency in response to an external
magnetic field as observed in Fig. 2b of the main text. The resonator is
capacitively coupled to an on-chip 50 $\mathrm{\Omega}$ coplanar waveguide
(CPW) patterned in NbN to achieve an external coupling rate,
$\kappa_{e,c}/2\pi=1.3$ MHz. On the chip used for the experiments in this
work, sixteen transducer devices were laid out in groups of four with each
group addressed by a separate CPW. Adjacent resonators on the same CPW were
designed to be detuned from each other by a frequency spacing of 100 MHz, much
larger than the cross-coupling, which was estimated to be less than 1 MHz from
simulation. This frequency multiplexing approach allows us to increase the
number of transducers available for testing on the chip while ensuring minimal
microwave crosstalk.

##  4 Piezo-optomechanics design

The piezo-optomechanical transducer in our work is realized by attaching a
half-wavelength AlN piezo-acoustic cavity to a silicon OMC as shown in Fig.
S2a. Figure S2b,c shows the optical and acoustic modes used for transduction.
The energy of the acoustic mode is split between the piezo-acoustic and OMC
regions, which are designed independently to support acoustic resonances at 5
GHz with large piezo-electric and optomechanical coupling respectively.
Similar to the device demonstrated in [3], hybridization of acoustic modes in
these two portions is achieved through the connecting OMC section, whose band
structure is designed to provide a bandgap for optical photons and waveguiding
for acoustic phonons at the frequencies of interest. The primary difference
compared to the previous design is the change to an ultra-low mode volume
piezo-acoustic cavity to reduce energy participation in the piezo region,
which is lossier for acoustics when compared to silicon. This design change is
possible without sacrificing piezoelectric coupling since the impedance of the
kinetic inductance resonator is roughly an order of magnitude larger than that
of the transmon qubit used in [3].

Figure S2: Piezo-optomechanics design. a. Geometry and material composition of
the piezo-optomechanical transducer. The black scale bar corresponds to 2 µm
b,c. Top-down views detailing the displacement profile of a hybridized
acoustic mode and the transverse electric field of the fundamental optical
mode of the transducer respectively. d,e. Simulated optomechanical coupling,
$g_{\text{om}}$, and piezoelectric coupling, $g_{\text{pe}}$, of the acoustic
modes of the transducer as a function of length of the piezo-acoustic cavity.
Dashed lines in both panels are shown as a guide to follow the modes with
maximum $g_{\text{om}}$.

Figures S2d,e show the hybridization of the piezo-acoustic and OMC modes via a
sweep of the length of the piezo box. We observe strong hybridization of two
distinct branches that are piezo-like and OMC-like as highlighted by the
dashed lines with maximal hybridization occurring at a piezo length of 880 nm.
On the transducer chip used in our experiments, we fabricate devices with
piezo length swept over a range of 20 nm about this nominal value to
compensate for fabrication disorder. Clamping loss of the piezo-acoustic
resonator is minimized by using tethers with modulated width as shown in Figs.
S2a,b. The tethers are designed to support a 1.5 GHz wide acoustic bandgap
centered around the transducer acoustic modes.

The OMC is evanescently coupled to a suspended silicon waveguide, which
terminates in a coupler section at the edge of the chip. We use a millimeter
length scale waveguide to distance the NbN circuit from scattering of pump
light generated near the optical coupler section. The waveguide is designed
with alternating curved sections in order to increase robustness against
buckling from intrinsic stress in the silicon device layer. Adiabatic tethers
are used to anchor the waveguide in order to reduce scattering loss and
maintain high optical collection efficiency.

##  5 Measurement Setup

Figure S3: Experimental setup. a. Optical pulse synthesis and detection. An
external cavity diode laser is wavelength locked to a wavemeter, and a 50 MHz
tunable narrowband filter is used to remove spontaneous emission noise.
Optical pulse shaping is achieved by modulating the laser with two acousto-
optic modulators (AOMs) connected in series. The envelope for the optical
pulses is generated by an arbitrary waveform generator (AWG). A variable
optical attenuator (VOA) is used to adjust the optical power level sent to the
device. Excitation of the device is performed via an optical circulator
through which the device reflection and emission can be routed to two separate
paths to perform either continuous wave (CW) spectroscopy or pulsed single
photon counting. For CW spectroscopy, the reflected optical pump along with
the optomechanically transduced signal is amplified with an erbium doped fiber
amplifier (EDFA) and detected on a high-speed photodetector. On the single
photon counting path, pump light is filtered using two tunable Fabry–Perot
filter cavities (Stable Laser Systems) before directing the optical signal to
the single photon detector (SPD) in the dilution fridge. b. Microwave pulse
synthesis and detection. For measurements that require electrical excitation
of the transducer, pulsed input signals to the fridge are generated at the
intermediate frequency (IF) by an arbitrary waveform generator (AWG) and
subsequently upconverted to $\sim$5GHz on an IQ mixer. Likewise, for CW
microwave spectroscopy, a vector network analyzer (VNA) may be switched into
the setup. The microwave output from the fridge passes though a tunable
bandstop filter to remove the TWPA pump tone, and is down-converted, amplified
and filtered before acquisition on an analog to digital converter (ADC). For
unconditional microwave readout, the master delay generator for the experiment
generates the trigger signal for the ADC. For microwave readout conditioned on
an optical click generated from SPDC, the trigger signal for the ADC is
generated by performing a logical AND operation between optical click events
from the SPD and a trigger bit from the master delay generator spanning the
gating window for collection of the optical signal. c. Dilution refrigerator
configuration. The microwave (MW) input line contains attenuators on the 4K,
cold, and mixing chamber plates for thermalization of the coaxial cables. The
MW output line passes through an amplification chain consisting of a
Traveling-Wave Parametric Amplifier (TWPA) mounted on the mixing plate and a
High Electron Mobility Transistor (HEMT) amplifier mounted on the 4K plate.
The TWPA is driven by a strong pump tone sent along a separate line and
inserted via a directional coupler placed before the amplifier input (not
shown). The superconducting nanowire single photon detector (SNSPD) is mounted
to the still plate and the electrical output signal is amplified on the 4K
plate before being sent to room temperature electronics.

The measurement setup used in this work is detailed in Fig. S3. For SPDC
experiments, trigger signals from a master digital delay generator are used to
synchronize optical pump pulses with the timing window used for optical and
microwave readout. Optical pump pulses with $>120$ dB extinction are generated
using two analog acousto-optic modulators (G&H Photonics) and are routed via a
circulator into the ‘Optics in/out’ path towards the device in the dilution
fridge as shown in Fig.S3a. The optical emission from the device passes
through the same circulator and is directed to a pump filtering setup prior to
single photon detection (SPD) along the ‘SPD in’ path in Fig. S3a. The
filtering setup comprises two tunable Fabry–Perot filter cavities (Stable
Laser Systems) in series and provides 104 dB extinction for a pump detuning of
5 GHz along with a transmission bandwidth of 2.7 MHz. This transmission
bandwidth naturally excludes emission due to optomechanical scattering by
other transducer modes besides the two hybridized electromechanical modes of
interest, $\hat{c}_{\pm}$. During the experiment, transmission through the
filters is checked every four minutes and a lock sequence is initiated if the
transmission drops below a set threshold. At the beginning of the transmission
check, the optical path is set to bypass the fridge via a MEMS switch. An
electro-optic phase modulator ($\mathrm{\phi}$-m in Fig. S3a) is used to
generate a sideband on the pump tone at the target frequency to which the
filters are locked. In our SPDC experiments, this target frequency corresponds
to a pump detuning, $\Delta_{\mathrm{a}}=(\omega_{+}+\omega_{-})/2$. If
required, the locking algorithm adjusts the filter cavities to maximize
transmission at the target frequency by monitoring the output of each cavity
on a separate photodetector. Additionally, during long measurements, we
periodically monitor the polarization of the pump light sent to the device and
compensate for long term polarization drifts along the excitation path. Active
polarization control is preformed by using an electronic polarization
controller (Phoenix Photonics) and maximizing the optical pump power reflected
by the device.

Optical photon counting is achieved using a tungsten silicide (WSi)
superconducting nanowire single photon detector (SNSPD) mounted on the still
plate of the dilution refrigerator (BlueFors LD-250) maintained at a
temperature of 770 mK. As shown in Fig. S3b, electrical pulses from the SNSPD
generated by single optical photon detection events are amplified and split
into two separate paths to record the detection time on a time correlated
single photon counter (TCSPC) and to trigger microwave readout conditioned on
an optical click. For conditional microwave readout, we first generate a
logical bit for every optical click by performing a logic level translation of
the SNSPD electrical pulse to a TTL signal. We then perform a logical AND
operation between this ‘optical click bit’ and a trigger bit provided by the
master delay generator defining the duration of the optical gating window. On
the other hand, unconditional microwave readout is triggered directly by the
master delay generator. An RF switch (MiniCircuits ZASWA-2-50DRA+) is used to
switch between unconditional and conditional readout of the microwave output
signal.

The microwave output chain shown along the ‘MW out’ path in Fig. S3c begins
with a Josephson traveling-wave parametric amplifier (TWPA, MIT Lincoln Labs)
[4] mounted on the mixing plate as the first amplification stage. For TWPA
operation, we use a CW pump tone at a frequency of $\sim$6.07 GHz added to the
amplifier input using a 20 dB directional coupler (not shown in the figure). A
dual junction circulator with 40 dB isolation is placed between the
directional coupler and the sample to shield the transducer from back-
reflected pump. The TWPA is followed by a high mobility electron transistor
(HEMT, Low Noise Factory LNF-LNC4_8C) amplifier mounted on the 4K plate. In
the setup outside the fridge as shown in Fig. S3b, we perform additional
amplification of the signal and filtering of the TWPA pump tone using a
tunable notch filter (Micro Lambda Wireless MLBFR-0212). The microwave signal
is then down-converted to an intermediate frequency (IF) of $\sim$100 MHz
after mixing with a local oscillator on an IQ mixer (Marki IQ-4509). The I and
Q outputs of the mixer are subsequently bandpass-filtered, amplified and
recorded independently on two channels of an analog to digital converter (ADC,
Alazartech ATS 9360). These two digitized voltage signals correspond to
measurement of the real and imaginary quadratures of the output mode of the
amplification chain, $\hat{s}_{\mathrm{out}}$. Based on the calibrated gain of
our amplification chain, we measured that our heterodyne detection setup has
an added noise of $\sim$2.5 quanta referred to the output of the transducer at
a signal frequency of $\sim$5.0 GHz. This near-quantum-limited heterodyne
readout enabled by the TWPA is a key enabler for measurements of microwave-
optical correlations from our transducer on a reasonable experimental
timescale. The setup shown in Fig. S3b additionally allows for pulsed
microwave excitation of the transducer as well as spectroscopy with a vector
network analyzer (VNA).

The transducer sample is wire-bonded to a printed circuit board (PCB) with
coaxial connectors and is housed in an oxygen-free high thermal conductivity
(OFHC) copper package. The microwave resonance frequency of the NbN resonators
can be tuned by an external magnetic field generated from an Nb-Ti coil
mounted over the sample package. Individual devices on the chip are optically
addressed using a lensed fiber (OZ Optics) affixed to the top of a three-axis
piezo stepper stack (Attocube Systems) placed in line with the on-chip tapered
optical couplers. The entire assembly is enclosed in a cylindrical magnetic
shield and is mounted to the mixing plate of the dilution refrigerator cooled
to a base temperature $T_{\text{f}}\approx 20~{}\text{mK}$.

##  6 Optical heralding rate contributions

Figure S4: a. Optical photon count rate vs time as measured with blue detuned
pump pulses, red detuned pump pulses and in calibration measurements for the
dark count rate (DCR) and pump leakage through the filter setup. b.
Contributions to the optical click probability during the gating window
extracted from the data in a.

The optical heralding events in our SPDC experiments have a finite noise
contribution from pump-induced heating of the hybridized electromechanical
modes as well as technical sources in our experimental setup, namely detector
dark counts and pump laser leakage. In Fig. S4a, we plot the optical photon
count rate due to these noise sources as determined from independent
measurements. The blue trace corresponds to the total optical signal as shown
in Fig. 3a of the main text. Dashed lines bound the gating duration used to
select photon clicks that trigger conditional microwave readout. The dark
count rate (DCR) of the SNSPD in our experiment is 1.5 Hz and is limited by
black-body radiation at wavelengths outside the telecom band guided through
the optical fiber. Stray photon flux from laser leakage through the pump
filtering setup as indicated by the purple trace in Fig. S4 is determined by
detuning the filter cavity by 100 MHz from the acoustic mode used in the
transduction experiment. To determine the noise contribution from the thermal
occupation of the acoustic mode, we perform a measurement of optomechanical
sideband asymmetry [5]. We excite the transducer with the same pump pulse
sequence used in the SPDC experiment but with the pump laser tuned to the red
mechanical sideband of the optical cavity
($\Delta_{\mathrm{a}}=-(\omega_{+}+\omega_{-})/2$). The resultant photon flux
from this measurement is shown by the red trace in Fig. S4a. Integrating the
photon counts from these independent measurements over the gating window, we
obtain the contributions of each noise source to the total probability of a
heralding event, $p_{\mathrm{click}}=2.7\times 10^{-6}$ as shown in Fig. S4b.
The corresponding fractional contributions are enumerated in Tab. S2. Further,
the difference between the count rate under red and blue detuned laser
excitation allows us to infer an average thermal occupation of $0.097\pm
0.019$ for the acoustic mode over the duration of the optical gating window.

Source of a photon click | Fraction of clicks
---|---
SPDC signal | 0.727
Thermal | 0.069
DCR | 0.171
Pump leakage | 0.033
Table S2: Contributions to the optical photon count rate.

##  7 Microwave moment inversion

To measure the statistical moments of the microwave emission from the
transducer, we adopt the method originally demonstrated in Ref. [6], which has
been widely applied to perform tomography of non-classical microwave radiation
emitted by superconducting qubits [7, 8, 9]. The microwave signal emitted from
the device undergoes a series of amplification steps before a record is
captured as a digitized, complex-valued voltage signal on the heterodyne
setup. This phase-insensitive amplification chain inevitably adds noise and is
modelled as [10, 11],

$\hat{s}_{\text{out}}=\sqrt{G}\hat{c}_{\text{out}}+\sqrt{G-1}\hat{h}^{\dagger},$
(1)

where $\hat{c}_{\text{out}}$, $\hat{s}_{\text{out}}$ and $\hat{h}^{\dagger}$
are the bosonic mode operators corresponding to the transducer output, the
heterodyne setup output and added noise respectively, with dimensions of
$T^{-1/2}$. In the limit of large amplifier gain ($G\gg 1$), this relation may
be approximated as

$\hat{s}_{\text{out}}\approx\sqrt{G}\left(\hat{c}_{\text{out}}+\hat{h}^{\dagger}\right),$
(2)

Once recorded, we take the inner product of the heterodyne traces with an
emission envelope function, $f$ in software. A particular choice of the
envelope function corresponds to a measurement of the quadratures of the
temporal mode
$\hat{S}(t):=\int\hat{s}_{\text{out}}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}$.
With this choice Eq. (2) becomes,

$\hat{S}\approx\sqrt{G}\left(\hat{C}+\hat{H}^{\dagger}\right),$ (3)

where
$\hat{C}(t):=\int\hat{c}_{\text{out}}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}$
and $\hat{H}(t):=\int\hat{h}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}$. Note
that $\hat{C}$ is a temporal mode in the output waveguide, not the internal
microwave mode, $\hat{c}$ of the transducer. Thus from a sequence of $N$
heterodyne measurements, a dataset of complex voltage samples
$\\{S_{1},S_{2},...S_{N}\\}$ is generated and the statistical moments are
numerically calculated as [11, 6],

$\bar{S}_{mn}:=\langle\hat{S}^{\dagger
m}\hat{S}^{n}\rangle=\frac{1}{N}\sum_{i=1}^{N}(S_{i}^{*})^{m}(S_{i})^{n}.$ (4)

The results in the main text require us to determine the statistical moments
of $\hat{C}$, which we organize into a moments matrix with elements
$\bar{C}_{mn}=\langle\hat{C}^{\dagger m}\hat{C}^{n}\rangle$. This is
accomplished using additional reference measurements in which the transducer
is not optically pumped so that the input to the amplifier is a vacuum state.
This reference measurement directly gives the anti-normally ordered moments of
the added noise as $\bar{H}_{mn}:=\langle\hat{H}^{m}\hat{H}^{\dagger
n}\rangle=\bar{S}_{mn}/G^{\frac{n+m}{2}}$.

We experimentally verified that $\bar{H}_{mn}$ is consistent with a thermal
state for $m,n\leq 2$. Knowledge of the mean occupancy ($n_{\text{th},H}$) of
the state constitutes full knowledge of the statistics for a thermal state.
Accordingly, we extract $n_{\text{th},H}$ from the reference measurements and
calculate $\bar{H}_{mn}$ as [12],

$\bar{H}_{mn}=\begin{cases}m!(n_{\text{th},H}+1)^{m}&m=n,\\\
0&\text{otherwise}.\end{cases}$ (5)

Finally, under the assumption that there are no correlations between $\hat{C}$
and $\hat{H}$, Eq. (3) can be used to derive the following relation between
the moment matrix elements [6],

$\bar{S}_{ij}=\bar{T}_{ijmn}\bar{C}_{mn},$ (6)

where

$\bar{T}_{ijmn}=\begin{cases}G^{\frac{i+j}{2}}{i\choose m}{j\choose
n}\bar{H}_{i-m,j-n}&m\leq i,n\leq j,\\\ 0&\text{otherwise}.\end{cases}$ (7)

By inverting Eq. (6), we recover $\bar{C}_{mn}$.

##  8 Microwave emission envelope function

Figure S5: a. Simulated temporal profile of microwave emission intensity from
the transducer heralded by an optical detection event. b. Comparison of
normalized power spectral density (PSD) of the emission envelope function, $f$
(orange trace) and of the experimentally measured transducer microwave output
conditioned on an optical click (blue data points). Detuning is defined with
respect to the mean frequency of the hybridized electromechanical modes.

We define the two hybridized electro-mechanical modes in our experiment as

$\hat{c}_{\pm}=\frac{1}{\sqrt{2}}\left(\hat{b}\pm\hat{c}\right),$ (8)

with center frequencies $\omega_{\pm}=\omega_{b}\pm g_{\text{pe}}$.

The ideal two-mode squeezed state generated under constant optical pumping on
the blue optomechanical sideband ($\Delta=\omega_{a}-\omega_{p}=\omega_{c}$)
can be written in the form [13],

$\ket{\psi}=\left[1+\sqrt{p}\left(\hat{a}_{+}^{{\dagger}}\hat{c}_{+}^{{\dagger}}+\hat{a}_{-}^{{\dagger}}\hat{c}_{-}^{{\dagger}}\right)+\mathcal{O}(p)\right]\ket{\mathrm{vac}},$
(9)

up to a normalization factor. Here $p$ is the probability of an optomechanical
scattering event, $\hat{a}_{\pm}$ are the optical modes which match the
frequencies of the hybridized modes $\hat{c}_{\pm}$ as required by energy
conservation. Performing single photon detection on the optical state coupled
out of the cavity naturally allows us to discard the vacuum component of
$\ket{\psi}$. Further, we can neglect higher order terms, $\mathcal{O}(p)$ in
the expansion since $p\ll 1$ for the optical pump power level used in our
experiments. In our experiment, the SNSPD has nanosecond timing jitter, which
is much smaller than the period of the beat note between the modes
$\hat{a}_{\pm}$ given by $2\pi/2g_{\text{pe}}=625$ ns. As a result, detection
of an SNSPD click in an experimental trial erases frequency information of the
optical state. This measurement can be described as a projective measurement
onto the state,
$\left(e^{i\phi(t)/2}\hat{a}_{+}+e^{-i\phi(t)/2}\hat{a}_{-}\right)\ket{\text{vac}}$,
where the relative phase $\phi(t)=\phi_{o}+2g_{\text{pe}}t$ for an optical
click received at time $t$. Here $\phi_{o}$ is the constant relative phase
acquired between the two frequency bins along the optical path and $t$ is the
emission time of the photon pair defined relative to the beginning of the pump
pulse. An optical click at time $t$ can therefore be used to herald the
microwave state
$\ket{\psi}_{\text{click}}=\left(e^{-i\phi(t)/2}\hat{c}_{+}^{{\dagger}}+e^{i\phi(t)/2}\hat{c}_{-}^{{\dagger}}\right)\ket{\mathrm{vac}}$.

This picture becomes slightly more complicated when considering the pulsed
optical drive used in this work as well as the parameters of the device. The
large bandwidth of the pump pulse of 2.75 MHz in comparison to the mode
splitting of 1.6 MHz means that the frequency bins cannot be individually well
resolved in the optical emission. Further, given the splitting between the
frequency bins versus their individual linewidths of 1.0 MHz, our device is
not deep in the strong piezoelectric coupling regime. This is evinced by the
two peaks with finite overlap in the experimentally measured power spectrum of
the conditional microwave emission,
$\langle\hat{c}^{\dagger}_{\text{out}}(\omega)\hat{c}_{\text{out}}(\omega)\rangle|_{\text{click}}$
shown with the blue trace in Fig. S5b. However, the theoretical picture
discussed above can guide the parametrization of the envelope function used
for microwave readout. Accordingly, we define the microwave emission envelope
function as a coherent superposition of two frequency bins centered at
$\omega_{\pm}$,

$f(t)=\frac{g(t)}{\sqrt{2}}\left(e^{-i(\omega_{+}t+\phi_{o}/2)}+e^{-i(\omega_{-}t-\phi_{o}/2)}\right).$
(10)

The relative phase, $\phi_{o}$ is assumed to be fixed since the optical pulse
duration used in our experiment is short compared to the beat period between
the frequency bins. The function, $g(t)$ accounts for the finite linewidth of
the frequency bins and is obtained by numerically solving the master equation
for our system using the QuTiP software package [14]. For this calculation, we
use the Hamiltonian in Eq. (1) of the main text along with experimentally
determined coupling and decay rates for the transducer modes. The resulting MW
emission intensity is shown in Fig. S5a. $g(t)$ is parameterized as a skewed
Gaussian to capture the faster timescale of the rising edge relative to the
decay. The corresponding voltage envelope is given by the square root of a
skewed Gaussian and is written as

$g(t)=\frac{1}{\sqrt[4]{{2\pi}T_{\text{g}}^{2}}}\mathrm{exp}\left(-\frac{t^{2}}{4T_{\text{g}}^{2}}\right)\left[1+\mathrm{erf}\left(\frac{\alpha
t}{\sqrt{2}T_{\text{g}}}\right)\right]^{1/2},$ (11)

up to a normalization factor involving the gain of the microwave amplification
chain. Here the Gaussian standard deviation, $T_{\text{g}}$ = 230 ns and skew
factor, $\alpha=2.0$ are obtained from fitting to the simulated result. We
note that for a microwave photon instantaneously loaded into the resonator,
$g(t)$ is expected to follow an exponential decay. However, in our transducer,
due to the finite conversion rate of the phonon from SPDC into a microwave
photon, $g(t)$ has a finite rise time. As a result, we find that choosing the
skewed Gaussian parametrization instead of an exponential decay provides $\sim
4\%$ higher overlap between the emission envelope function and the simulated
wavepacket as quantified by their normalized inner product. The power spectrum
of the resultant envelope function, $|f(\omega)|^{2}$ is shown as the orange
trace in Fig. S5b. This is observed to be well-matched to the power spectrum
of the conditional microwave emission from the transducer obtained from
experimentally recorded heterodyne voltage traces.

##  9 Microwave gain calibration

We calibrate the gain of our microwave heterodyne detection setup by operating
the transducer as a frequency converter with the optical pump laser on the red
mechanical sideband. The calibration procedure is performed at zero magnetic
field, when the acoustic and microwave modes, $\hat{b},\hat{c}$ are detuned by
a frequency separation, $\omega_{b}-\omega_{c}=2\pi\times 12~{}\mathrm{MHz}\gg
2g_{\mathrm{pe}}$ and are weakly hybridized. An independent measurement of
optomechanical sideband asymmetry provides a meter for phonon occupation in
the acoustic mode, $\hat{b}$ via the single phonon count rate, $R_{o}$ [5]. We
then resonantly excite the acoustic mode via the microwave port and measure
the transduced optical photon count rate, $R$. Simultaneously, we record the
reflected microwave signal from the transducer, which produces an output
power, $P_{\mathrm{det}}$ at the output of the heterodyne setup. The gain, $G$
of the amplification chain can be determined using the series of equations
below.

$\displaystyle R$ $\displaystyle=$ $\displaystyle n_{b,\mathrm{sig}}R_{o},$
(12) $\displaystyle n_{b,\mathrm{sig}}\hbar\omega_{b}$ $\displaystyle=$
$\displaystyle P_{\mathrm{in}}\frac{4\kappa_{e,b}}{\kappa_{i,b}^{2}},$ (13)
$\displaystyle P_{\mathrm{det}}$ $\displaystyle=$ $\displaystyle
G\left(\frac{\kappa_{e,b}-\kappa_{i,b}}{\kappa_{e,b}+\kappa_{i,b}}\right)^{2}P_{\mathrm{in}}.$
(14)

Here, $n_{b,\text{sig}}$ is the occupation of the acoustic mode due to the
microwave drive obtained after subtracting the thermal component of the
transduced signal. $P_{\text{in}}$ is the input microwave power to the
transducer. $\kappa_{b,\text{e}}$ is the coupling rate of the acoustic mode to
the microwave waveguide due to weak hybridization with the microwave resonator
and $\kappa_{b,\text{i}}$ is the intrinsic linewidth of the acoustic mode.
Both rates are determined via a VNA measurement of the acoustic mode in the
presence of the optical pump pulses. Following this procedure, we estimate
gain, $G$ = 103 dB between the microwave output port of the transducer and the
output of the heterodyne setup. While knowledge of the gain allows us to
estimate the microwave intensity at the transducer output, we note that the
values of the normalized correlation functions measured in our experiment are
independent of the absolute accuracy of this calibration.

##  10 TWPA optimization

The operating point of the TWPA was optimized prior to data acquisition. In
order to ensure fast acquisition and unbiased statistics, our operational
criterion were (1) maximize the gain and (2) ensure linearity in the response.
In our optimization procedure, the frequency and power of the pump tone
driving the TWPA were swept, and coherent pulses with amplitude matched to the
conditional microwave signal in our experiment were sent to the transducer
input port. The result of the heterodyne measurement of the reflected signal
from the transducer was used to compute $g^{(2)}_{CC}$. To ensure linearity,
we chose a region in the TWPA pump frequency and power space where
$g^{(2)}_{CC}=1$ to within one standard deviation. The final operating point
was then chosen as the one with maximum gain from this cohort. Over the course
of the experiment, linearity of the TWPA was periodically verified by
measuring $g^{(2)}_{CC}$ for a weak coherent input. In case of a deviation of
$g^{(2)}_{CC}$ from unity by one standard deviation, the experiment was halted
and the pump optimization routine was re-run.

##  11 Transducer heating dynamics

To investigate the heating dynamics of the transducer, we performed a series
of microwave measurements under pulsed optical excitation. In Fig. S6a, we
show the time-resolved power spectrum of the unconditional microwave emission
at the transducer output port. For this measurement, we use the optical pulse
sequence ($T_{\mathrm{p}}$ = 160 ns, $50$ kHz repetition rate) used in the
SPDC experiments in the main text albeit at a higher pump power, $n_{a}=12$,
which generates higher thermal noise. We observe that majority of the heating
occurs in the emission bandwidth of the hybridized electro-mechanical modes.
The peak of this thermal emission is noticeably delayed from the optical pump
pulse. Additionally, we observe a small but non-zero noise contribution far
from the transducer resonances, which we attribute to heating of the microwave
waveguide. These resonator and waveguide contributions to the added noise are
separated by fitting the power spectrum at each delay to a double Lorentzian
function with a constant floor. The fit result is shown in Fig. S6b, and
indicates that resonator heating significantly dominates waveguide heating and
has a stronger time dependence. This points to parasitic optical absorption in
the intrinsic baths of the acoustic and microwave modes as the dominant source
of transducer heating. Additionally, we measured the scaling of the intensity
of this thermal emission with optical pump power and observed it to be sub-
linear as shown by the results plotted in Fig. S6c. Finally, from the time
dynamics of heating in Fig. S6a, we observe that while most of the thermal
emission decays on the timescale of a few microseconds, a small component
persists as steady state heating and is observed prior to the arrival of the
optical pump pulse. We attribute this to a slower component of the hot bath
generated by optical absorption and characterize it by varying the repetition
rate of the optical pulses. The result shown in Fig. S6d confirms slow decay
of this component of heating with an exponential decay time of $33\pm
6\mathrm{\mu}$s. The measurements described in this section were performed on
the same device used for the SPDC experiments in the main text, but after a
partial warm up and cooldown to base temperature. After this thermal cycle, we
observed frequency shifts of the microwave and acoustic modes by a few MHz and
an increase in the normal mode splitting. However, the thermal noise quanta
generated by the optical pulse sequence used for the experiments in the main
text remained nearly the same.

Figure S6: Transducer heating dynamics. a. Power spectral density of the
unconditional microwave signal as a function of delay, $\tau$ from the center
of the optical pump pulse. Detuning is defined with respect to the mean
frequency of the hybridized electromechanical modes. Legend indicates
microwave quanta emitted at the output port in a 100 kHz bandwidth.
Measurement is performed with optical pump pulses of peak power, $n_{a}$ = 12,
pulse duration of 160 ns, and 50 kHz repetition rate. b. Time dependence of
waveguide and resonator components of heating extracted by fitting the data in
panel a.. c. Thermal emission in the transducer microwave output as a function
of optical pump power. Pump power is plotted in terms of peak intra-cavity
occupation of the optical mode, $n_{a}$ and microwave intensity is plotted in
terms of occupation of the temporal mode, $\bar{C}_{11}$ measured at the
delay, $\tau$ at which it is maximized. All measurements were performed with
the 160 ns pulse duration and 50 kHz repetition rate used for data in the main
text. Dashed red box denotes the power used for the experiment in the main
text ($n_{a}=0.8$) while the dashed black box denotes the power used for
measurements in panels a,b. Solid line is a power law fit revealing an
exponent, 0.58 $\pm$ 0.03. d. Decay of the slow component of the thermal
emission probed by varying the repetition period, $T_{r}$ of the pump pulse
sequence. Microwave quanta in the unconditional state, $\bar{C}_{11}$ are
measured versus prior to the pump pulse at the delay $\tau_{-}$ as defined in
the main text. Dashed line is an exponential fit revealing a decay time of
$33\pm 6~{}\mathrm{\mu}$s. Dashed red box denotes the repetition rate used for
the experiment in the main text.

##  12 Simulation of the conditional microwave state

As detailed in Eq. 1 of the main text, the dynamics of the transducer are
governed by the Hamiltonian,

$\hat{H}_{abc}(t)=\hat{H}_{o}+\hat{H}_{\text{om}}(t)+\hat{H}_{\text{pe}},$
(15)

where $\hat{H}_{o}$, $\hat{H}_{\text{om}}(t)$, and $\hat{H}_{\text{pe}}$
contain the mode frequencies, optomechanical interaction, and piezoelectric
interaction terms respectively. These are defined as

$\displaystyle\hat{H}_{o}$
$\displaystyle=-\hbar\Delta_{a}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{b}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{c}\hat{c}^{\dagger}\hat{c},$
(16a) $\displaystyle\hat{H}_{\mathrm{om}}(t)$ $\displaystyle=\hbar
G_{\mathrm{om}}(t)(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b}),$ (16b)
$\displaystyle\hat{H}_{\mathrm{pe}}$ $\displaystyle=\hbar
g_{\mathrm{pe}}(\hat{b}^{\dagger}\hat{c}+\hat{b}\hat{c}^{\dagger}).$ (16c)

We have explicitly included the time dependence of the optomechanical coupling
$G_{\mathrm{om}}(t)=\sqrt{n_{a}(t)}g_{\mathrm{om}}$ due to the temporal shape
of the intra-cavity pump photon number,
$n_{a}(t)={\kappa_{\text{e},a}}/({\Delta_{a}^{2}+\kappa_{a}^{2}/4})P_{\text{in}}(t)$.
$P_{\text{in}}(t)$, the optical pump power at the device follows a Gaussian
shape with parameters described in the main text. In our experiment, we use a
blue detuned pump with $\Delta_{a}=\omega_{b}$. Further, our device is in the
sideband resolved regime with $\omega_{b}\gg\kappa_{a}$.

In addition to unitary evolution due to couplings among the internal modes of
the transducer, the system is also subject to dissipation and heating due to
coupling to the environment. This is captured by the master equation,

$\dot{\hat{\rho}}(t)=\mathcal{L}(t)\hat{\rho}(t).$ (17)

Defining the Lindblad superoperator as
$\mathcal{D}(\hat{o})\hat{\rho}=\hat{o}\hat{\rho}\hat{o}^{\dagger}-\frac{1}{2}\left\\{\hat{o}^{\dagger}\hat{o},\hat{\rho}\right\\}$
the action of $\mathcal{L}(t)$ on the density matrix is written as

$\mathcal{L}(t)\hat{\rho}(t)=-\frac{i}{\hbar}\left[\hat{H}_{abc},\hat{\rho}(t)\right]+\left(\mathcal{L}_{a}+\mathcal{L}_{b}(t)+\mathcal{L}_{c}(t)\right)\hat{\rho}(t),$
(18)

where the superoperators, $\mathcal{L}_{a},\mathcal{L}_{b},\mathcal{L}_{c}$
describe the coupling of the respective transducer modes to the environment
according to the relations below.

$\displaystyle\mathcal{L}_{a}\hat{\rho}(t)=$
$\displaystyle\,\kappa_{a}\mathcal{D}(\hat{a})\hat{\rho}(t),$ (19a)
$\displaystyle\mathcal{L}_{b}(t)\hat{\rho}(t)=$
$\displaystyle\,n_{\text{th},b}(t)\kappa_{\text{i},b}\mathcal{D}(\hat{b}^{\dagger})\hat{\rho}(t)$
$\displaystyle\,+\left[n_{\text{th,b}}(t)+1\right]\kappa_{i,b}\mathcal{D}(\hat{b})\hat{\rho}(t),$
(19b) $\displaystyle\mathcal{L}_{c}(t)\hat{\rho}(t)=$
$\displaystyle\,n_{\text{th},c}(t)\kappa_{\text{i},c}\mathcal{D}(\hat{c}^{\dagger})\hat{\rho}(t)$
$\displaystyle\,+\left[n_{\text{th},c}(t)+1\right]\kappa_{i,c}\mathcal{D}(\hat{c})\hat{\rho}(t)$
$\displaystyle\,+n_{\text{th},w}\kappa_{\text{e},c}\mathcal{D}(\hat{c}^{\dagger})\hat{\rho}(t)$
$\displaystyle\,+\left[n_{\text{th},w}+1\right]\kappa_{e,c}\mathcal{D}(\hat{c})\hat{\rho}(t).$
(19c)

Here the total dissipation rate of the optical mode,
$\kappa_{a}=\kappa_{e,a}+\kappa_{i,a}$. To capture heating of the transducer
modes due to parasitic absorption of the optical pump, we assume that the
acoustic and microwave modes are coupled to intrinsic baths with time
dependent thermal occupation, $n_{\text{th},b}(t)$ and $n_{\text{th},c}(t)$
respectively. $n_{\text{th},w}$, the thermal occupation of the microwave
waveguide is set to a constant value much smaller than $n_{\text{th},b}(t)$
and $n_{\text{th},c}(t)$ based on the measurements detailed in Section 11. We
note that while these baths may be significantly more complex and involve
several components possessing different, time-dependent coupling strengths,
our model is aimed at capturing the total influx of thermal excitations into
the transducer modes generated by the optical pulse. With this endeavor in
mind, we assume that the acoustic (microwave) mode couples to a single
intrinsic bath at a fixed dissipation rate, $\kappa_{i,b}$ ($\kappa_{i,c}$)
and ascribe the time-dependence of the heating dynamics entirely to the
thermal occupation of the bath. This phenomenological approach to heating
induced by optical absorption has previously been used to model photon
correlations in optomechanics experiments [15].

While Eq. (17) can be numerically solved in principle, this is computationally
intensive owing to the large state space involving the three modes of the
transducer. Instead, we take advantage of the fact that $\kappa_{a}\gg
G_{\text{om}}$ to adiabatically eliminate the optical mode [16]. Moving into a
frame rotating with the mechanical and microwave modes and defining the
optomechanical scattering rate,
$\Gamma_{\text{om}}(t)=4|G_{\text{om}}(t)|^{2}/\kappa_{a}$, we arrive at the
simplified master equation,

$\dot{\hat{\rho}}_{r}(t)=\mathcal{L}_{r}(t)\hat{\rho}_{r}(t),$ (20)

where $\hat{\rho}_{r}=\mathrm{Tr}_{a}\\{\hat{\rho}\\}$ denotes the reduced
density matrix spanning only the acoustic and microwave modes.
$\mathcal{L}_{r}$ is defined by its action on the reduced density matrix,

$\displaystyle\mathcal{L}_{r}(t)\hat{\rho}_{r}(t)=$
$\displaystyle\,-\frac{i}{\hbar}\left[\hat{H}_{\text{pe}},\hat{\rho_{r}}(t)\right]$
$\displaystyle\,+\left(\mathcal{L}_{\text{om}}(t)+\mathcal{L}_{b}(t)+\mathcal{L}_{c}(t)\right)\hat{\rho}_{r}(t),$
(21)

where we have introduced optomechanical scattering as a coupling of the
reduced system to an effective bath according to the relation,

$\mathcal{L}_{\text{om}}(t)\hat{\rho}_{r}(t)=\Gamma_{\text{om}}(t)\mathcal{D}(\hat{b}^{\dagger})\hat{\rho}_{r}(t).$
(22)

The Lindblad superoperator, $\mathcal{D}(\hat{b}^{\dagger})$ contains a non-
number preserving quantum jump operator which adds a phonon to the system due
to optomechanical SPDC. This process is naturally correlated with the creation
of an optical photon, which is routed with finite collection efficiency to a
single photon detector. Given that optical decay ($\kappa_{a}/2\pi=1.3$ GHz)
occurs on a much faster timescale than the dynamics of the rest of the system
($\kappa_{a}\gg\kappa_{b},g_{\text{pe}},\kappa_{c}$), optical detection is
assumed to be instantaneous. Under this approximation, in the event of a click
on the optical detector caused by a quantum jump at time $t_{\text{J}}$, the
resultant conditional state of the reduced system is,

$\hat{\rho}_{\text{J}}(t_{\text{J}})=\frac{\hat{b}^{\dagger}\hat{\rho}_{r}(t_{\text{J}})\hat{b}}{\mathrm{Tr}\left\\{\hat{b}^{\dagger}\hat{\rho}_{r}(t_{\text{J}})\hat{b}\right\\}}.$
(23)

This state then evolves according to Eq. (20). Since we operate in the weak
pump regime where the integrated jump probability over the optical pulse
duration, $\int\Gamma_{\text{om}}(t)dt\ll 1$, we neglect two-fold SPDC events.

While the treatment above describes the evolution of the internal modes of the
transducer, we experimentally measure the emission in the temporal mode,
$\hat{C}$ in the transducer microwave output port. This temporal mode is
linearly related to the internal microwave mode of the transducer as
$\hat{C}(t):=\int\hat{c}_{\text{out}}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}=\sqrt{\kappa_{e,c}}\int\hat{c}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}$.
We can thus compute the occupation of the temporal mode as

$\displaystyle\langle\hat{C}^{\dagger}(t)\hat{C}(t)\rangle$ $\displaystyle=$
$\displaystyle\kappa_{e,c}\langle\int\hat{c}^{\dagger}(t+t^{\prime\prime})f(t^{\prime\prime})dt^{\prime\prime}\int\hat{c}(t+t^{\prime})f^{*}(t^{\prime})dt^{\prime}\rangle$
$\displaystyle=$
$\displaystyle\kappa_{e,c}\int\langle\hat{c}^{\dagger}(t+t^{\prime\prime})\hat{c}(t+t^{\prime})\rangle
f(t^{\prime\prime})f^{*}(t^{\prime})dt^{\prime\prime}dt^{\prime}$
$\displaystyle=$
$\displaystyle\kappa_{e,c}\int\langle\hat{c}^{\dagger}(t^{\prime}+\tau^{\prime})\hat{c}(t^{\prime})\rangle
f(t^{\prime}+\tau^{\prime}-t)f^{*}(t^{\prime}-t)dt^{\prime}d\tau^{\prime},$

where the correlator,
$\langle\hat{c}^{\dagger}(t^{\prime}+\tau^{\prime})\hat{c}(t^{\prime})\rangle$
inside the integral on the final line can be numerically computed using the
quantum regression theorem [17]. Further, to model the conditional state due
to a quantum jump at time $t_{\text{J}}$, this correlator is computed using
$\hat{\rho}_{\text{J}}(t_{\text{J}})$ in Eq. (23) for times,
$t>t_{\mathrm{J}}$. For times $t<t_{\mathrm{J}}$, we use the value for the
unconditional state. Performing this calculation for a sequence of jump times,
$\\{t_{\mathrm{J}}\\}$ associated with detector clicks received within the
optical gating window, we obtain a sequence of conditional time traces for the
occupation of the temporal mode,
$\\{\langle\hat{C}^{\dagger}\hat{C}\rangle|_{t_{\mathrm{J}}}\\}$. The final
conditional trace, $\langle\hat{C}^{\dagger}\hat{C}\rangle|_{\mathrm{click}}$
is given by a weighted average over this sequence with weights determined by
the infinitesimal jump probability, $\delta
p_{\mathrm{J}}\propto\Gamma_{\text{om}}(t_{\mathrm{J}})dt$ in an interval $dt$
at time $t_{\text{J}}$. The effects of detector dark counts as well as the
optical pump filter in our experiment are incorporated via appropriate
modifications to these weights.

Figure S7: Simulation of transducer heating dynamics a. Time trace of the
optical pump pulse plotted in terms of the intra-cavity occupation of the
optical mode, $n_{a}$. b. Model used for time dependent thermal occupation of
the intrinsic baths of the acoustic and microwave modes,
$n_{\mathrm{th},b}(t)$, $n_{\mathrm{th},c}(t)$. c. Time dependent occupation
of the acoustic mode, $\langle\hat{b}^{\dagger}\hat{b}\rangle$ obtained from
the master equation simulation. Gray data point corresponds to experimentally
recorded thermal occupation of the acoustic mode as inferred from an
optomechanical sideband asymmetry measurement. The vertical bar on the point
represents a confidence interval spanning two standard deviations while the
horizontal bar indicates the FWHM duration of the optical pulse. c. Time
dependent occupation of the temporal mode,
$\langle\hat{C}^{\dagger}\hat{C}\rangle$ in the microwave output port of the
transducer from the master equation simulation (solid traces) and from
experimental heterodyne data (dotted traces). The condition where the acoustic
mode is far detuned (on resonance) with respect to the microwave mode is shown
with pink (red) traces. Figure S8: Simulation of the conditional microwave
state. Occupation of the temporal mode, $\hat{C}$ in the microwave output port
of the transducer from simulation (solid traces) and from experimental
heterodyne data (dotted traces). The occupation of the microwave state
conditioned on an optical click,
$\langle\hat{C}^{\dagger}\hat{C}\rangle|_{\mathrm{click}}$ is shown with
purple traces. The occupation of the unconditional microwave state,
$\langle\hat{C}^{\dagger}\hat{C}\rangle$ is shown with red traces for
comparison.

We perform a numerical simulation of our system by implementing the above
model using the QuTiP software package [14]. Solutions to the master equation
as well as the required correlators are evaluated in a $10\times 10$ Fock
space of the acoustic and microwave modes. To incorporate effects of heating
of the microwave and acoustic baths separately in our model, we measured the
unconditional microwave emission from the transducer in response to optical
pump pulses under two conditions (i) mechanics far detuned from the microwave
mode, $\omega_{c}-\omega_{b}\gg 2g_{\text{pe}}$, (ii) mechanics on resonance
with the microwave mode, $\omega_{b}=\omega_{c}$. The results of the
measurement are shown with the dotted traces in Fig. S7d. Under condition (i),
which is achieved at zero magnetic field, the microwave emission is
approximately entirely due to heating of the microwave bath alone. On the
other hand, condition (ii) corresponds to maximal electro-mechanical
hybridization and is relevant to the SPDC experiments described in the main
text. In this setting, heating of both microwave and acoustic baths is
expected to contribute to the measured microwave output signal. Using
measurements of the microwave emission under both conditions, we invert the
master equation to determine $n_{\text{th,b}}(t)$ and $n_{\text{th,c}}(t)$
approximated as a piece-wise linear function over coarse samples as shown in
Fig. S7b. The inversion is performed by choosing an ansatz for
$n_{\text{th,b}}(t)$ and $n_{\text{th,c}}(t)$, iteratively solving the master
equation and performing least mean square optimization with respect to the
experimentally measured thermal microwave emission. The resulting occupations
of the acoustic mode, $\hat{b}$ and of the unconditional temporal microwave
mode, $\hat{C}$ obtained by solving the master equation are shown in Fig. S7c
and with the solid traces in Fig. S7d respectively. While the model for the
hot baths is determined entirely using microwave measurements, we observe that
the simulated occupation of the acoustic mode is in reasonable agreement with
an experimental measurement shown by the gray data point in Fig. S7c. This
data point corresponds to the optomechanical sideband asymmetry measurement
described in Section 6, and represents the thermal occupation of the acoustic
mode averaged over the duration of the optical pump pulse. Finally, the
conditional microwave emission determined using our model shown with the solid
purple trace in Fig. S8 is in good agreement with the corresponding
experimental result shown via the dotted purple trace. The simulated
conditional and unconditional intensity traces plotted in this figure predict
a value of $g^{(2)}_{AC}(\tau_{o})=3.97$, which is in good agreement with the
experimentally observed value of 3.90.

Simulation of $g^{(2)}_{CC}|_{\text{click}}$ requires evaluation of
$\langle\hat{C}^{{\dagger}2}\hat{C}^{2}\rangle|_{\text{click}}$. Extending the
approach of Eq. (LABEL:eq:CdagC) to the second order moment is a nontrivial
computational endeavor as it requires calculation of the four-time correlator,
$\langle\hat{c}^{\dagger}(t+t^{\prime\prime\prime\prime})\hat{c}^{\dagger}(t+t^{\prime\prime\prime})\hat{c}(t+t^{\prime\prime})\hat{c}(t+t^{\prime})\rangle$.
For the purpose of this work, we pursue the more tractable task of placing
bounds on $g^{(2)}_{CC}|_{\text{click}}$. First, we consider the conditional
state of the acoustic mode immediately after the addition of a phonon as
triggered by an SPDC event. Since the extraction of this acoustic state into
the microwave output port is accompanied by loss and addition of noise, we
expect the function,
$g^{(2)}_{bb}|_{\text{click}}=\langle\hat{b}^{{\dagger}2}b^{2}\rangle|_{\text{click}}/(\langle\hat{b}^{{\dagger}}b\rangle|_{\text{click}})^{2}$
to provide a lower bound for $g^{(2)}_{CC}|_{\text{click}}$. Our model
estimates $g^{(2)}_{bb}|_{\text{click}}(0)=0.24$. Next, we consider a choice
of the emission envelope function, $f(t)=\delta(t-T_{o}-\tau_{o})$, where
$T_{o}$ is the time corresponding to the peak intensity of the pump pulse.
Physically, employing this function results in collecting the microwave
emission at all frequencies from the transducer at a fixed time. This is
strictly less optimal than the choice in our measurements, namely a function
coherently matched to the theoretically expected single photon wavepacket as
discussed in Sec. 8. In the event of this sub-optimal choice, moments of the
temporal mode, $\hat{C}$ are proportional to the corresponding moments of the
internal microwave mode, $\hat{c}$ and the function,
$g^{(2)}_{cc}|_{\text{click}}=\langle\hat{c}^{{\dagger}2}c^{2}\rangle|_{\text{click}}/(\langle\hat{c}^{{\dagger}}c\rangle|_{\text{click}})^{2}$,
serves as an upper bound for $g^{(2)}_{CC}|_{\text{click}}$. Our model
estimates $g^{(2)}_{cc}|_{\text{click}}(\tau_{o})=0.77$. In summary, the above
arguments allow us to use numerical simulations to bound
$g^{(2)}_{CC}|_{\text{click}}(\tau_{o})$ to the interval (0.24, 0.77), which
has substantial overlap with the experimental observation of
$g^{(2)}_{CC}|_{\text{click}}(\tau_{o})=0.42^{+0.27}_{-0.28}$.

##  13 Data analysis for correlation functions

For measurements of $g^{(2)}_{AC}$ and $g^{(2)}_{CC}|_{\text{click}}$
presented in figure 3 of the main text, $9.1\times 10^{4}$ heterodyne voltage
traces were acquired over a one month period. For every $\sim$16 minutes of
acquisition of conditional microwave data, we interleaved the acquisition of
voltage samples of amplifier noise and the unconditional microwave signal. The
unconditional microwave intensity signal, $\bar{C}_{11}(\tau)$ used to
evaluate the normalized cross-correlation function, $g^{(2)}_{AC}$ was
obtained from $1.4\times 10^{7}$ heterodyne voltage traces. Likewise, the
measurement of the normalized second order intensity correlation function,
$g^{(2)}_{CC}(\tau_{\mathrm{o}})$ used $3.2\times 10^{7}$ heterodyne voltage
samples.

Figure S9: a. Moments matrix for the unconditional microwave output of the
transducer, $\bar{C}_{mn}$. b. Moments matrix for the microwave output of the
transducer conditioned on receipt of an optical click,
$\bar{C}_{mn}|_{\text{click}}$. Error bars calculated via bootstrapping with
replacement are indicated by the black pins.

To minimize the effect of long term fluctuations in the added noise and gain
of the TWPA, we divide the dataset into chunks corresponding to acquisition
over one day, and invert each separately to extract the moments
$\bar{C}_{mn}^{(k)},\bar{C}_{mn}^{(k)}|_{\text{click}}$, where the index $k$
runs over the daily chunks. This inversion process follows the methods
detailed in Sec. 7 and assumes that no correlations exist between $\hat{H}$
and $\hat{A}$ as well as $\hat{H}$ and $\hat{C}$. We then take a weighted
average to compute the entries of the moments matrix,
$\bar{C}_{mn}=\sum_{k}w^{(k)}\bar{C}_{mn}^{(k)}$ and
$\bar{C}_{mn}|_{\text{click}}=\sum_{k}w^{(k)}_{\text{click}}\bar{C}_{mn}^{(k)}|_{\text{click}}$,
where $w^{(k)}$, $w^{(k)}_{\text{click}}$ denote the fraction of records in
the $k^{\text{th}}$ chunk of the unconditional and conditional datasets
respectively. The moments matrices constructed from this process are shown in
Fig. S9b.

The acquisition rate for the moments of the conditional heterodyne output,
$\bar{S}_{\text{mn}}|_{\text{click}}$ is determined by the optical heralding
rate, $R_{\mathrm{click}}=0.14$ Hz. This is much slower compared to that for
the amplifier noise moments, $\bar{H}_{\text{mn}}$, which can be acquired at
the 50 kHz repetition rate of the experiment and determined with high
accuracy. As a result, the error in $g^{(2)}_{CC}|_{\text{click}}$ is
dominated by uncertainty in $\bar{S}_{\text{mn}}|_{\text{click}}$. To
calculate error in $g^{(2)}_{CC}$ and $g^{(2)}_{CC}|_{\text{click}}$, we
employ bootstrapping with replacement using $10^{5}$ bootstraps to construct
the probability density functions shown in Fig. S10a. Finally, error estimates
were calculated by numerical integration of the distributions such that they
they cover a 34.1% confidence interval above and below the mean. The error
estimates calculated using this process vs. number of bootstraps is shown in
Fig. S10b-c for the conditional and unconditional datasets. We find that the
use of $10^{5}$ bootstraps yields acceptable convergence in the error
estimate.

To mitigate the effects of long term drifts in frequencies of the microwave
modes, we periodically measured the unconditional microwave power spectrum and
monitored the resonance frequencies, $\omega_{\pm}$. This additionally allowed
us to characterize the spectral diffusion of the microwave modes. In our data
analysis, we excluded intervals of the measurement where the mode frequencies
drifted by more than twice the standard deviation associated with spectral
diffusion, which represented $3\%$ of all recorded traces.

Figure S10: a. Probability distribution for $g^{(2)}_{CC}$ and
$g^{(2)}_{CC}|_{\text{click}}$ calculated from bootstrapping. Vertical blue
and red lines correspond to one standard deviation above and below the
distribution mean respectively. b,c. Convergence of the error estimate vs.
number of bootstraps for all traces, and optically heralded traces
respectively. Blue lines correspond to positive error, red lines to negative
error, and black lines indicate the error estimate used in the main text.

##  14 Classical bound on the conditional second order intensity correlation

In the main text, we claim that the conditional microwave autocorrelation
function is expected to satisfy the inequality,
$g^{(2)}_{CC}|_{\mathrm{click}}\geq 1$ for classical microwave-optical states.
This can be proved by rewriting the conditional quantities defined in the main
text in terms of correlators of the joint microwave-optical state,

$\displaystyle g^{(2)}_{CC}(\tau)|_{\text{click}}$
$\displaystyle=\frac{\braket{\hat{C}^{\dagger
2}(\tau)\hat{C}^{2}(\tau)}|_{\text{click}}}{\left(\braket{\hat{C}^{\dagger}(\tau)\hat{C}(\tau)}|_{\text{click}}\right)^{2}}$
$\displaystyle=\frac{\braket{\hat{A}^{\dagger}\hat{C}^{\dagger
2}(\tau)\hat{C}^{2}(\tau)\hat{A}}\braket{\hat{A}^{\dagger}\hat{A}}}{\left(\braket{\hat{A}^{\dagger}\hat{A}}\braket{\hat{C}^{\dagger}(\tau)\hat{C}(\tau)}\right)^{2}},$
(25)

where the above correlators are explicitly written in normal order. Using the
the optical equivalence theorem, we can recast the expectation values of the
above correlators in phase space using the Sudarshan-Glauber P representation
[18, 19] as

$\displaystyle\braket{\hat{A}^{\dagger m}\hat{C}^{\dagger
n}(\tau)\hat{A}^{m}\hat{C}^{n}(\tau)}$
$\displaystyle=\int|\alpha|^{2m}|\gamma|^{2n}P_{\tau}(\alpha,\gamma)d^{2}\alpha
d^{2}\gamma$
$\displaystyle:=\braket{}{\alpha}{{}^{2m}|\gamma|^{2n}}_{P_{\tau}},$ (26)

where $P_{\tau}(\alpha,\gamma)$ is the joint phase space density cprresponding
to modes $\hat{A}$ and $\hat{C}$ with relative delay $\tau$. Applying this
directly to Eq. (14) gives,

$g^{(2)}_{CC}(\tau)|_{\text{click}}=\frac{\braket{}{\alpha}{{}^{2}_{P}|\gamma|^{4}}_{P_{\tau}}\braket{}{\alpha}{{}^{2}}_{P_{\tau}}}{\braket{}{\alpha}{{}^{2}|\gamma|^{2}}^{2}_{P_{\tau}}}\geq
1,$ (27)

which follows directly from the Cauchy-Schwarz inequality.

##  15 Convergence of $g^{(2)}_{AC}$ to the classical bound

The results of pump power dependent measurements of the normalized intensity
cross-correlation function, $g^{(2)}_{AC}$ are shown in Fig. S11.

Figure S11: Maximum value of $g^{(2)}_{AC}$ versus increasing pump power
plotted on the horizontal axis as the peak intra-cavity occupation of the
optical mode, $n_{a}$. Dashed line denotes the classical upper bound of 2.

All measurements were performed with the same pump pulse duration and
repetition rate used to acquire data in the main text. We observe that at
higher pump powers, the value of $g^{(2)}_{AC}$ approaches the classical upper
bound of 2 as expected from the increased thermal noise added to the photon
pair generated in SPDC. The data in this figure was collected from the same
device used for the experiments in the main text albeit after a partial warm
up and cooldown to base temperature. This thermal cycle led to modified device
parameters, which produced an increase in $g^{(2)}_{AC}$ at the lowest pump
power of $n_{a}=0.8$ compared to the result presented in the main text.

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