††thanks: Corresponding author<EMAIL_ADDRESS>
# Light-induced dynamic frequency shifting of microwave photons
in a superconducting electro-optic converter

Yuntao Xu  Department of Electrical Engineering, Yale University, New Haven,
Connecticut 06511, USA  Wei Fu  Department of Electrical Engineering, Yale
University, New Haven, Connecticut 06511, USA  Yiyu Zhou  Department of
Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA
Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA
Mingrui Xu  Department of Electrical Engineering, Yale University, New Haven,
Connecticut 06511, USA  Mohan Shen  Department of Electrical Engineering, Yale
University, New Haven, Connecticut 06511, USA  Ayed Al Sayem  Department of
Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA
Hong X. Tang  Department of Electrical Engineering, Yale University, New
Haven, Connecticut 06511, USA  Yale Quantum Institute, Yale University, New
Haven, Connecticut 06511, USA

###### Abstract

Hybrid superconducting-photonic microresonators are a promising platform for
realizing microwave-to-optical transduction. However, the absorption of
scattered photons by the superconductors leads to unintended microwave
resonance frequency variation and linewidth broadening. Here, we
experimentally study the dynamics of this effect and its impact on microwave-
to-optics conversion in an integrated lithium niobate-superconductor hybrid
resonator platform. We unveiled an adiabatic frequency shifting of the
intracavity microwave photons induced by the fast photo-responses of the thin-
film superconducting resonator. As a result, the temporal and spectral
responses of electro-optics transduction are modified and well described by
our theoretical model. This work provides important insights on the light-
induced conversion dynamics which must be considered in future designs of
hybrid superconducting-photonic system.

††preprint: APS/123-QED

##  I Introduction

A hybrid superconducting-photonic transducer serves an important quantum
interface for future quantum networks [1, 2, 3, 4, 5, 6] where the quantum
information processed in microwave frequency qubits [7, 8] could be
distributed by photonic circuits [9]. In various microwave-to-optical
transduction schemes [10, 11, 12], such as cavity electro-optics (EO) [13, 14,
15, 16, 17, 18], magneto-optics [19, 20, 21], and piezo-optomechanics [22, 23,
24, 25, 26], an intense optical pump is required to uplift the conversion
efficiency. Harnessing the Pockels effect, cavity EO systems are particularly
attractive for achieving direct transduction between microwave photons and
telecom optical photons without involving an intermediate excitation (phonon
or magnon) whose thermal excitation may add noise to the full conversion chain
[27, 28, 29]. Recently, a pulsed-pump microwave-to-optical conversion scheme
has been developed to take advantage of a strong optical pump while reducing
the overall thermal heating effect in the cryogenic environment [16, 14, 23,
22, 24]. In a bulky lithium niobate EO transducer, an impressive, near-unity
cooperativity has been achieved by using watt-scale pump pulses [14].

Moving onto an on-chip platform, the integrated cavity EO converter benefits
from the increased single photon coupling rate ($g_{\mathrm{eo}}$) due to the
reduction of resonator’s modal volume, meanwhile providing high tunability and
scalability towards multi-channel transduction and all-in-one devices. The
compactness of integrated superconducting-photonic resonators, however, comes
at the cost of resonator quality ($Q$) factor degradation and unintended
microwave frequency shift, mainly arising from light-induced quasiparticle
generation in superconductors [18, 16, 17].

In this letter, we present a study on the pulsed-light induced dynamics in the
integrated superconducting EO transducer. An on-chip EO microwave-to-optical
transducer on a thin-film lithium niobate (TFLN)-niobium nitride (NbN) hybrid
material system is investigated as the experimental platform. Here, we observe
a dynamic microwave frequency shifting in the superconducting cavity (“color”
change of microwave photons) resulting from its fast resonance frequency
change induced by the optical pump pulse. The phenomena could be considered as
an analog of the adiabatic frequency conversion in the optical domain, where
the frequency of cavity photons is shifted due to a rapid change of the
refractive index [30, 31, 32, 33, 34]. These explorations reveal that optical
illumination not only leads to static frequency variation and increased loss
in the microwave resonator, it also imposes a fast, adiabatic microwave photon
frequency shifting which must be considered in the design and calibration of
microwave-to-optics converters. While this dynamic effect is regarded as
deleterious for microwave-to-optics conversion, it could be exploited for
"optically" control the microwave photon frequency at the cryogenic
temperature or even shape the microwave-to-optical conversion process in the
time domain.

##  II Method

The integrated EO transducer studied in this work consists of a
superconducting microwave resonator and a pair of racetrack optical
resonators, in which their electrical fields are coupled via Pockels
nonlinearity of LN thin films [27, 28, 29] (Fig. 1(a)). The two racetrack
resonators, patterned from thin film lithium niobate, are strongly coupled to
support a pair of hybridized transverse electric (TE) optical modes: anti-
symmetric (mode $a$) and symmetric (mode $b$) [35]. The microwave-to-optical
conversion efficiency is maximized when a triple resonance condition is
satisfied by tuning their frequency spacing to match the microwave mode
frequency via bias voltage. The superconducting resonator is made of NbN and
supports a quasi-lumped LC resonant mode (mode $c$) that satisfies EO phase
matching.

Previously, it has been shown [18] that the microwave Q of converter devices
were limited by device packaging that introduces extra microwave loss through
coupling to spurious modes. In this work, the microwave signal is coupled to
the microwave resonator through a pair of on-chip feedline couplers connected
to the coplanar waveguides (CPW). This permits the application of the DC bias
voltage through the same set of couplers and avoids the use of separate DC
electrodes used in our previous designs which could introduce extra microwave
loss. The microwave cavity thus can be better shielded by both on-chip
superconducting ground plates and an RF-tight copper box to eliminate the loss
via coupling to spurious modes. With these improvements, we achieve an
intrinsic microwave $Q_{\mathrm{in}}=17.1\mathrm{k}$, which is more than 10
times higher than that of the previous work.

Figure 1:  (a) False colored microscopic image of the EO transducer device.
Two strongly coupled optical racetrack resonators are integrated with a
superconducting microwave resonator. The microwave resonator is linked to a
coplanar waveguide (CPW, not shown) for both RF coupling and DC biasing. The
electrical field direction of the designed microwave mode is marked in the
coupling regime. (b) A schematic of the pulsed-pump measurement setup to
characterize the response of the device in time domain. The microwave tone
($\sim$ $6\text{\,}\mathrm{GHz}$) is synthesized by up-converting a
($50\text{\,}\mathrm{MHz}$) intermediate frequency (IF) signal outputs from a
lock-in amplifier (LIA) through an IQ mixer, while the optical signal is
launched by optical single-sideband modulation (SSBM). The optical pump
(together with optical signal) is pulsed via an acoustic-optic modulator (AOM)
driven by a pulse generator. The microwave output signal is demodulated by the
LIA after being down-converted back to $50\text{\,}\mathrm{MHz}$. PD:
photodetector. HVS: high voltage source.

The packaged device is loaded on the 1K plate ($T=$ $800\text{\,}\mathrm{mK}$)
in a dilution fridge where high cooling power is available. The pulsed pump
measurement setup is shown in Fig. 1(b). We operate the transducer in a blue-
drive (parametric amplification) scheme, in which the higher frequency optical
mode $b$ is excited with a strong pump to stimulate the coherent coupling
between optical mode $a$ and microwave mode $c$ [36, 37]. Though this is not a
quantum-state-conversion, it does not change the physics of the dynamic
process we present in this work. A continuous wave (CW) tunable laser is tuned
to the resonance wavelength (of mode $b$) and is pulsed via an
$80\text{\,}\mathrm{MHz}$ acoustic-optic modulator (AOM) with a pulse width of
$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The optical pulses are sent to and
collected from the device by a pair of grating couplers glued with angled
lensed fiber [38]. The total optical insertion loss is
$-22\text{\,}\mathrm{dB}$ within the fridge. To characterize the coherent
response of the EO system in time domain, we utilize a lock-in-amplifier (LIA)
as the signal generator and receiver. The microwave signal is synthesized from
a $50\text{\,}\mathrm{MHz}$ low frequency signal from the LIA via an IQ mixer,
in which the up-converted frequency is tuned to match the microwave resonance.
This GHz probe tone is then sent to the device either directly as the
microwave input to the transducer, or as the optical input via optical single-
sideband modulation (SSBM). The output microwave or optical signal from the
device is then down-converted to $50\text{\,}\mathrm{MHz}$ and sent back to
LIA for quadrature measurement in time domain.

##  III Result

The dynamic process we study here arise from the fast photo-response of
superconducting cavity which mainly originates from quasiparticle generation
due to superconductor absorption of optical photons [39, 40, 41]. When the
optical pump pulse is turned on, the superconductor’s absorption of light
leads to a microwave resonance shift together with extra microwave loss. In
Fig. 2, we present the microwave reflection spectra $|S_{\mathrm{ee}}|^{2}$
under different peak optical power when the device reaches the steady state in
the presence of the optical pump pulse. As the on-chip peak pump power is
increased from $-7\text{\,}\mathrm{dBm}$ to $14\text{\,}\mathrm{dBm}$, the
microwave resonance exhibits a frequency shift from
$0.25\text{\,}\mathrm{MHz}$ to $13.7\text{\,}\mathrm{MHz}$, with its intrinsic
$Q$ decreasing from 17.1 k to 2.2 k. For each trace shown in Fig. 2, the
repetition rate of the optical pulse is adjusted from $4\text{\,}\mathrm{kHz}$
to $31.25\text{\,}\mathrm{Hz}$ to keep the average power unchanged.

Figure 2: The microwave reflection spectra $|S_{\mathrm{ee}}|^{2}$ under
different peak optical pump pulse power. The right panel shows the
corresponding intrinsic microwave quality ($Q$) factor of the superconducting
cavity. The no-light microwave $Q_{\mathrm{in}}=17.1\mathrm{k}$ is marked as
purple dashed line. Figure 3: (a) Illustration of light-induced dynamic
frequency shifting in a superconducting cavity. When the light-induced
resonance frequency change is faster than the cavity decay, the intracavity
microwave photons is shifted to the new resonance frequency, resulting in a
temporal interference fringe. (b) Experimental results exhibit the microwave
dynamic frequency shifting. Panel (i) shows the time-resolved microwave
reflection spectra $|S_{\mathrm{ee}}|^{2}$ for a pump pulse with
$-1\text{\,}\mathrm{dBm}$ peak power. Panel (ii) and (iii) show the microwave
reflection with input microwave frequency of $f_{0}$
($6.4658\text{\,}\mathrm{GHz}$) and $f_{\mathrm{shift}}$
($6.4644\text{\,}\mathrm{GHz}$), respectively. The microwave reflections
exhibit an exponentially decaying oscillation when the optical pulse is
on/off, indicating a beating between the shifted microwave signal and original
input microwave signal.

We then study the temporal response of the microwave cavity to the incident
light by recording the microwave reflection spectrum at a fixed time delay
with respect to the optical pulse. Here, we observe a dynamic frequency
shifting of microwave photons in the superconducting cavity induced by the
fast resonance frequency variation. A conceptional illustration of this
dynamic frequency shifting process is shown in Fig. 3(a). Before switching on
the optical pulse, the CW input microwave signal with $f_{\mathrm{in}}=f_{0}$
is coupled on resonance to the microwave cavity. When the optical pulse
arrives, the intracavity microwave photons frequency is rapidly modulated
along the cavity mode to a new resonance frequency $f_{\mathrm{shift}}$. The
beating between the emitted microwave cavity photons (at $f_{\mathrm{shift}}$)
and the reflected microwave signal (at $f_{\mathrm{in}}=f_{0}$) thus can be
observed to verify this frequency shifting. This process can be considered as
an analogy of the adiabatic frequency conversion in the optical domain, where
the frequency of light traveling in a cavity is shifted by a fast variation in
refractive index [30, 31, 32, 33, 34]. The requirement for this dynamic
frequency shifting to be observable is that the cavity mode frequency shifts
faster than the cavity photon lifetime. Because the photo-response time of
thin-film NbN was reported to be less than $1\text{\,}\mathrm{ns}$ [39, 40,
41], the resonance frequency transition time is mainly defined by the optical
pulse rise/fall time, which is estimated to be $70\text{\,}\mathrm{ns}$ in our
system. Even with the lowest $Q=2.2$ k, the corresponding cavity photon
lifetime is $\tau_{\mathrm{e}}=$$340\text{\,}\mathrm{ns}$, thus the
requirement is satisfied.

Figure 4:  (a) An optical-to-microwave transduction cascade with a dynamic
microwave frequency shift. (b) Time evolution of the optical-to-microwave
scattering parameter $S_{\mathrm{eo}}$. The peak optical power is
$-1\text{\,}\mathrm{dBm}$ and input frequency to optical SSBM is the shifted
resonance $f_{\mathrm{shift}}=$ $6.4644\text{\,}\mathrm{GHz}$. When the pulse
is turned off, the intracavity microwave photons converted from optical domain
experience a secondary frequency shifting to the original “un-illuminated”
resonance frequency $f_{0}=$ $6.4658\text{\,}\mathrm{GHz}$. A phase delay of
$2\pi\delta t$ can be identified in the demodulated phase to confirm the
frequency shifting, where $\delta=$ $1.41\text{\,}\mathrm{MHz}$ is the change
of resonance frequency.

In Fig. 3(b), we present experimental results that confirm this dynamic
frequency shifting. Panel (i) presents the time evolution of the microwave
reflection spectra $|S_{\mathrm{ee}}|^{2}$ when the device receives a
$5\text{\,}\mathrm{\SIUnitSymbolMicro s}$-long optical pump pulse with peak
power of $-1\text{\,}\mathrm{dBm}$. When the optical pump is turned on, the
resonance frequency shift down from $f_{0}=$ $6.4658\text{\,}\mathrm{GHz}$ by
$\delta=$ $1.41\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to
$f_{\mathrm{shift}}=$ $6.4644\text{\,}\mathrm{GHz}$. The ripples in the time-
resolved spectra indicate the interference between microwave signals with
different frequency in the system. Specifically, with microwave input
$f_{\mathrm{in}}=f_{0}$, the microwave reflection $|S_{\mathrm{ee}}|^{2}$
exhibits a decaying oscillation when the optical pump pulse is turned on, as
shown in panel (ii). The decaying oscillation represents the interference
between the frequency-shifted out-coupling microwave signal
($f_{\mathrm{shift}}$) and reflected input microwave signal ($f_{0}$), which
is an evidence of the frequency shifting. Reversely, microwave signal with
frequency $f_{\mathrm{in}}=f_{\mathrm{shift}}$ could be shifted back to
$f_{0}$ when the the optical pump is switched off (panel (iii)). The
oscillation period of $0.71\text{\,}\mathrm{\SIUnitSymbolMicro s}$ corresponds
to the frequency shift $\delta=$
$1.41\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The exponential decay rates of
$|S_{\mathrm{ee}}|^{2}$ represent the different loss rates of microwave cavity
when the optical pulse is turned on/off. The base level of
$|S_{\mathrm{ee}}|^{2}$ reflects the extinction of the microwave resonance. We
numerically simulate this dynamic process, and the results fit well with the
experimental data. The details of the simulations are given in the Appendix.

We next explore how this dynamic process affects the bidirectional microwave-
to-optical transduction. In the pulsed transduction scheme, the microwave
carrier is typically aligned with the shifted resonance frequency to optimize
transduction efficiency. Figure 4(b) shows time evolution of the optical-to-
microwave scattering parameter $S_{\mathrm{eo}}$. The peak optical pump power
is $-1\text{\,}\mathrm{dBm}$, and the input frequency (to optical SSBM) is
$f_{\mathrm{shift}}=$ $6.4644\text{\,}\mathrm{GHz}$. When the optical pulse is
turned on, $|S_{\mathrm{eo}}|^{2}$ is gradually stabilized as the converted
microwave photons build up in the superconducting cavity. Interestingly, we
observe an optical-to-microwave transduction cascaded with a dynamic microwave
frequency shifting here. An illustration of this cascaded process is shown in
Fig. 4(a). When the optical pulse is turned off, the intracavity microwave
photons converted from optical domain experience another frequency shifting to
$f_{0}=$ $6.4658\text{\,}\mathrm{GHz}$. When these new microwave photons
($f_{0}$) leak out from the cavity, there are no photons with frequency of
$f_{\mathrm{shift}}$ in the cavity anymore, thus no oscillation behavior is
observed here. Instead, by demodulating the output signal with the input
frequency ($f_{\mathrm{shift}}=$ $6.4644\text{\,}\mathrm{GHz}$), the secondary
dynamic shifting is confirmed by the phase delay with a period of
$0.71\text{\,}\mathrm{\SIUnitSymbolMicro s}$ in the lower panel of Fig. 4(b).

A similar cascade process can be further observed in a reverse manner, where
the dynamic frequency shifting is followed with a microwave-to-optical
transduction (Fig. 5). The time evolution of the microwave-to-optical
scattering parameter $S_{\mathrm{oe}}$ is shown in Fig. 5(b). With the input
microwave frequency aligned with the shifted resonance frequency
$f_{\mathrm{shift}}$, $|S_{\mathrm{oe}}|^{2}$ is gradually stabilized as the
microwave field builds up in the cavity (yellow trace). However, when the
input microwave frequency is aligned with the native resonance frequency
$f_{0}$, the cavity is pre-loaded by the CW microwave input before the pulse
arrives. The stored microwave photons are first shifted with the microwave
resonance as the pump pulse turned on, and then converted to optical domain.
These cascaded processes result a transient peak together with a delayed phase
in the demodulated $|S_{\mathrm{oe}}|$ scattering parameter. The pre-loaded
cavity has a higher microwave $Q$ due to the absence of superconductor optical
absorption. As a result, the instantaneous value of $|S_{\mathrm{oe}}|^{2}$
could be higher than the value in a stabilized transduction.

Figure 5:  (a) A dynamic microwave frequency shifting cascade with a
microwave-to-optical transduction. (b) Time evolution of the microwave-to-
optical scattering parameter $S_{\mathrm{oe}}$ with peak pump power
$-1\text{\,}\mathrm{dBm}$. When $f_{\mathrm{in}}=f_{0}$, the preloaded
microwave photons in the cavity are first shifted with the resonance frequency
as the pulse turned on, and then are converted to optical photons. This
cascaded processes exhibit a transient peak of $|S_{\mathrm{oe}}|^{2}$ and a
phase delay in $S_{\mathrm{oe}}$.

##  IV Discussion and Conclusion

The incident light introduces both static resonance frequency variation and
dynamic photon frequency shifting. On the one hand, the light-induced
resonance frequency variation, Q degradation, and added thermal noise
excitations reported previously [17, 16, 18] are major limiting factors of the
EO transducer device performance. Since scattered light from the chip-fiber
interface is the dominant source of superconductor absorbed photons [16],
improving fiber-chip coupling and proper light shielding can be incorporated
to mitigate this problem. On the other hand, the dynamic frequency shifting
studied in this work pose another issue that must be considered and carefully
engineered in the future microwave-to-optical transducer schemes.
Nevertheless, this dynamic photon frequency shifting could be exploited, for
example in the range where the microwave resonator is over coupled for
bandwidth matching and the added noise by light is not a limiting factor.
Meanwhile, how this dynamic photon frequency shifting interacts with the added
microwave noise induced by the pump light would be interesting for further
investigation.

The efficiency of this dynamic frequency shifting is determined by the
microwave photon leakage during the resonance frequency transition time [31,
32]. In the NbN superconducting resonator we used, the cavity microwave photon
lifetime is much longer than the transition time, which is currently limited
by the pulse synthesis timescale in the experiment. As a result, the
intracavity efficiency is close to 100%. However, for superconductors with a
long quasiparticle lifetime, such as Aluminum [42], the resonance frequency
transition time would be much longer, making this dynamic frequency shifting
difficult to be observed. In our current device, the inductance part of the
superconducting resonator have a wire width of
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Under the same optical power, a
much larger resonance frequency variation could be achieved in a lumped LC
resonator with thinner wires and higher kinetic inductance [16, 43], thus the
range of of this dynamic shifting can be expanded.

In conclusion, the optical pulse-induced dynamics in integrated TFLN-
superconductor hybrid platform are experimentally investigated. We describe
and analyze a dynamic microwave frequency shifting in the superconducting
cavity due to the fast resonance frequency variation induced by optical pump
pulse. We further study how this "color change" of microwave photons plays a
role in the bidirectional microwave-to-optical transduction. Our findings not
only help understand the dynamics in the transducer, but also show a possible
method to control the microwave photon frequency using optical pulses, which
may be utilized to expand the transduction frequency range and thus to match
the qubit excitation frequency.

###### Acknowledgements.

This work is funded by ARO under grant number W911NF-18-1-0020. H.X.T.
acknowledges partial supports from NSF (EFMA-1640959). Funding for substrate
materials used in this research was provided by DOE/C2QA grant under award
number DE-FOA-0002253. The authors thanks Michael Rooks, Yong Sun, Sean
Rinehart and Kelly Woods for support in the cleanroom and assistance in device
fabrication.

##  Appendix A Theory of the EO system and device parameters

The triply-resonant EO system is described by a Hamiltonian of [27, 28]

$H=\hbar\omega_{a}a^{\dagger}a+\hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}c^{\dagger}c+g_{\mathrm{eo}}(ab^{\dagger}c+a^{\dagger}bc^{\dagger})$
(1)

where $a$, $b$ and $c$ denote annihilation operators for the lower (higher)
frequency optical modes and microwave mode respectively. The resonance
frequency of $a$, $b$, $c$ are $\omega_{a}$, $\omega_{b}$, $\omega_{c}$
respectively, and $g_{\mathrm{eo}}$ denotes the single photon electro-optical
coupling rate. In the parametric amplification scheme, the mode $b$ is
coherently driven with a strong optical pump, and the system Hamiltonian in
the rotation frame of the optical pump can be simplified to [13, 37]

$H=\hbar\delta_{a}a^{\dagger}a+\hbar\omega_{c}c^{\dagger}c+\hbar
G(ac+a^{\dagger}c^{\dagger})$ (2)

where $\delta_{a}=\omega_{a}-\omega_{p}$ and $G=\sqrt{n_{p}}g_{\mathrm{eo}}$
is pump photon number ($n_{p}$) enhanced coupling rate. Thus the equations of
motion could be written as

$\displaystyle\frac{d}{dt}a=-(i\delta_{a}+\frac{\kappa_{a}}{2})a-iGc^{*}+\sqrt{\kappa_{a,\mathrm{ex}}}a_{\mathrm{in}}e^{-i\delta_{a,\mathrm{in}}t}$
(3)
$\displaystyle\frac{d}{dt}c=-(i\omega_{c}+\frac{\kappa_{c}}{2})c-iGa^{*}+\sqrt{\kappa_{c,\mathrm{ex}}}c_{\mathrm{in}}e^{-i\omega_{c,\mathrm{in}}t}$
(4) $\displaystyle
a_{\mathrm{out}}=a_{\mathrm{in}}-\sqrt{\kappa_{a,\mathrm{ex}}}a$ (5)
$\displaystyle
c_{\mathrm{out}}=c_{\mathrm{in}}-\sqrt{\kappa_{c,\mathrm{ex}}}c$ (6)

where $\kappa_{a}$, $\kappa_{a,\mathrm{ex}}$
($\kappa_{c}$,$\kappa_{c,\mathrm{ex}}$) are the total decay and external
coupling rate for mode $a$ ($c$) respectively. $a_{\mathrm{in}}$
($a_{\mathrm{out}}$) and $c_{\mathrm{in}}$ ($c_{\mathrm{out}}$) denote the
input (output) signals, $\delta_{a,\mathrm{in}}$ and $\omega_{c,\mathrm{in}}$
are the angular frequency of the inputs. The dynamic of the system could be
numerically simulated in the time domain with Eqs. (3)-(6). In our simulation
model, we simply assume $\omega_{c}$ linearly change to the final frequency
within a transition time $\tau$,

$\omega_{c}(t)=\left\\{\begin{aligned}
&\omega_{0}+\frac{t-t_{1}}{\tau}(\omega_{\mathrm{shift}}-\omega_{0})&&t_{1}<t<t_{1}+\tau\\\
&\omega_{\mathrm{shift}}-\frac{t-t_{2}}{\tau}(\omega_{\mathrm{shift}}-\omega_{0})&&t_{2}<t<t_{2}+\tau\end{aligned}\right.$
(7)

where $\omega_{0}$ and $\omega_{\mathrm{shift}}$ are the light-off and light-
on microwave angular frequency, respectively. $t_{1}$ and $t_{2}$ are the
pulse-on and pulse-off time, respectively; and $\tau$ is the optical pulse
rise/fall time. A more complex model including real nonlinear terms is worth
further investigation with more detailed knowledge of the material properties.

By solving the Eqs. (3)-(6) in steady state, we can get the full scattering
matrix of the transducer system. When all resonances are perfectly aligned
($\omega_{a}+\omega_{c}=\omega_{b}$) and $C<1$, the on-chip transducer
scattering parameter is

$|S_{\mathrm{eo}}|^{2}=|S_{\mathrm{oe}}|^{2}=\frac{\kappa_{a,\mathrm{ex}}}{\kappa_{a}}\frac{\kappa_{c,\mathrm{ex}}}{\kappa_{c}}\frac{4C}{(1-C)^{2}},$
(8)

where figure of merit of this parametric amplification process is the
cooperativity $C=4n_{\mathrm{p}}g_{\mathrm{eo}}^{2}/\kappa_{a}\kappa_{c}$.
When $C\geq 1$, the system should be considered with a pump depletion, and
enters a parametric oscillation regime.

The parameters of the device is shown in Table.1. The on-chip transduction
efficiency of the device is also calibrated by the full spectra of the
complete scattering matrix [44], and the cooperativity $C$ is extracted from
Eq. (8) accordingly. Here, a cooperativity of $C=0.107\pm 0.06$ is reached in
this device with on-chip $5\text{\,}\mathrm{dBm}$ pump power, which is more
than twice higher than our previous result under $8\text{\,}\mathrm{dB}$
weaker pump power.

Parameter |  Description |  Value
---|---|---
$(\omega_{a},\omega_{b})/2\pi$ |  optical resonance frequency |  (191.698, 191.704) THz
$(\kappa_{a},\kappa_{b})/2\pi$ |  total optical loss rate |  (197, 238) MHz
$(\kappa_{a,\mathrm{ex}},\kappa_{b,\mathrm{ex}})/2\pi$ |  external optical coupling rate |  (13, 65) MHz
$(\kappa_{a,\mathrm{in}},\kappa_{b,\mathrm{in}})/2\pi$ |  intrinsic optical loss rate |  (185, 173) MHz
$\omega_{c}/2\pi$ |  microwave resonance frequency |  6.4658 GHz
$\kappa_{c}/2\pi$ |  total microwave loss rate |  397 kHz
$\kappa_{c,\mathrm{ex}}/2\pi$ |  external microwave coupling rate |  19 kHz
$\kappa_{c,\mathrm{in}}/2\pi$ |  intrinsic microwave loss rate |  378 kHz
$g_{\mathrm{eo}}/2\pi$ |  single photon EO coupling rate |  500 Hz
Table 1:  Device parameters. Note that the parameters of the microwave
resonator is calibrated in the absence of optical light. The decrease of
intrinsic microwave $Q$ is shown in Fig. 2, while the external coupling rate
only slightly change under different optical power.

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