††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Magnetic-Free Silicon Nitride Integrated Optical Isolator

Hao Tian  OxideMEMS Lab, Purdue University, 47907 West Lafayette, IN, USA
Junqiu Liu  Institute of Physics, Swiss Federal Institute of Technology
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Anat Siddharth  Institute of
Physics, Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Rui Ning Wang  Institute of Physics, Swiss Federal
Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Terence Blésin  Institute of Physics, Swiss Federal Institute of Technology
Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Jijun He  Institute of
Physics, Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Tobias J. Kippenberg<EMAIL_ADDRESS>Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL),
CH-1015 Lausanne, Switzerland  Sunil A. Bhave<EMAIL_ADDRESS>OxideMEMS Lab,
Purdue University, 47907 West Lafayette, IN, USA

Integrated photonics Thomson  _et al._ (2016) has enabled signal synthesis,
modulation and conversion using photonic integrated circuits (PIC). Many
materials have been developed, among which silicon nitride (Si3N4) has emerged
as a leading platform particularly for nonlinear photonics Moss  _et al._
(2013); Gaeta  _et al._ (2019). Low-loss Si3N4 PIC has been widely used for
frequency comb generation Kippenberg  _et al._ (2018); Gaeta  _et al._ (2019),
narrow-linewidth lasers Gundavarapu  _et al._ (2019); Jin  _et al._ (2021);
Xiang  _et al._ (2021), microwave photonics Roeloffzen  _et al._ (2013); Wu
_et al._ (2018), photonic computing networks Feldmann  _et al._ (2021);
Arrazola  _et al._ (2021), and even surface-electrode ion traps Mehta  _et
al._ (2020); Niffenegger  _et al._ (2020). Yet, among all demonstrated
functionalities for Si3N4 integrated photonics, optical non-reciprocal
devices, such as isolators and circulators, have not been achieved.
Conventionally, they are realized based on Faraday effect of magneto-optic
materials Srinivasan and Stadler (2018); Bi  _et al._ (2011); Huang  _et al._
(2016); Yan  _et al._ (2020) under external magnetic field. However, it has
been challenging to integrate magneto-optic materials that are not CMOS-
compatible and that require bulky external magnet. Here, we demonstrate a
magnetic-free optical isolator based on aluminum nitride (AlN) piezoelectric
modulators monolithically integrated on ultralow-loss Si3N4 PIC Liu  _et al._
(2020a). The transmission reciprocity is broken by spatio-temporal modulation
Yu and Fan (2009) of a Si3N4 microring resonator with three AlN bulk acoustic
wave resonators Tian  _et al._ (2020); Liu  _et al._ (2020b) that are driven
with a rotational phase. This design creates an effective rotating acoustic
wave that allows indirect interband transition Kang  _et al._ (2011); Kittlaus
_et al._ (2018); Sohn  _et al._ (2018) in only one direction among a pair of
strongly coupled optical modes. Maximum of 10 dB isolation is achieved under
100 mW RF power applied to each actuator, with minimum insertion loss of 0.1
dB. The isolation remains constant over nearly 30 dB dynamic range of optical
input power, showing excellent optical linearity. Our integrated, linear,
magnetic-free, electrically driven optical isolator could become key building
blocks for integrated lasers Huang  _et al._ (2019); Xiang  _et al._ (2020);
McKinzie  _et al._ (2021); Xiang  _et al._ (2021); Liang and Bowers (2021),
chip-scale LiDAR engines Riemensberger  _et al._ (2020); Lukashchuk  _et al._
(2021), as well as optical interfaces for superconducting circuits Awschalom
_et al._ (2021).

Integrated photonics Thomson  _et al._ (2016) has enabled chip-scale optical
systems with compact sizes, portable weight, and low power consumption, which
have been translated from table-top research setups to commercial products.
With the advancement of nano-fabrication technology, there have been major
achievements in integrated semiconductor lasers Huang  _et al._ (2019); Xiang
_et al._ (2020); McKinzie  _et al._ (2021); Xiang  _et al._ (2021); Liang and
Bowers (2021), modulators Melikyan  _et al._ (2014); Wang  _et al._ (2018); He
_et al._ (2019a); Tian  _et al._ (2020); Chen  _et al._ (2014); Tadesse and Li
(2014); Shao  _et al._ (2019); Tian  _et al._ (2021), photodetectorsKang  _et
al._ (2009); Beling  _et al._ (2016), as well as ultralow-loss photonic
integrated circuits (PIC) Zhang  _et al._ (2017); Yang  _et al._ (2018); Chang
_et al._ (2020); Xuan  _et al._ (2016); Ji  _et al._ (2017); Liu  _et al._
(2020a); Ye  _et al._ (2019); Puckett  _et al._ (2021). Importantly, the
nonlinearities of ultralow-loss PICs have been explored and harnessed, giving
rise to nonlinear photonics such as optical frequency comb (OFC) generation
Moss  _et al._ (2013); Kippenberg  _et al._ (2018); Gaeta  _et al._ (2019).
Formed in Kerr-nonlinear optical microresonators, dissipative Kerr solitons
(DKS) microcombs Kippenberg  _et al._ (2018); Gaeta  _et al._ (2019); Herr
_et al._ (2013) constitute miniaturized, coherent, broadband OFCs with
repetition rates in the millimeter-wave to microwave domain, and are amenable
to heterogeneous or hybrid integration with III-V/Si lasersStern  _et al._
(2018); Raja  _et al._ (2019); Shen  _et al._ (2020); Xiang  _et al._ (2021).
Major progress has been made in the past decade in developing various
platforms Kovach  _et al._ (2020), ranging from Si3N4Xuan  _et al._ (2016); Ji
_et al._ (2017); Liu  _et al._ (2020a); Ye  _et al._ (2019), LiNbO3Wang:19;
Zhang:19; He  _et al._ (2019b); Gong  _et al._ (2020), AlNJung  _et al._
(2013); Liu  _et al._ (2018a) and AlGaAsPu  _et al._ (2016); Chang  _et al._
(2020). Among the materials developed thus far for integrated nonlinear
photonics, Si3N4 has become the leading platform due to its exceptionally low
loss (down to 1 dB/m), high Kerr nonlinearity, absence of two-photon
absorption, and high power handling capability. Indeed, most of the system-
level demonstrations of microcombs have been based on Si3N4, such as coherent
communication Marin-Palomo  _et al._ (2017); Fülöp  _et al._ (2018); Corcoran
_et al._ (2020), astronomical spectrometer calibration Obrzud  _et al._
(2019); Suh  _et al._ (2019), distance measurement Trocha  _et al._ (2018);
Suh and Vahala (2018); Jang  _et al._ (2021); Wang  _et al._ (2020), coherent
LiDAR Riemensberger  _et al._ (2020); Lukashchuk  _et al._ (2021), frequency
synthesizers Spencer  _et al._ (2018); Liu  _et al._ (2020c), optical atomic
clocks Newman  _et al._ (2019), and photonic neromorphic computing Feldmann
_et al._ (2021); Xu  _et al._ (2021). Meanwhile, high-$Q$ Si3N4
microresonators have also been used recently to create ultralow-noise
semiconductor lasers Jin  _et al._ (2021); Xiang  _et al._ (2021) whose
performance is on par with advanced fiber lasers.

Figure 1:  Principle of the nitride optical isolator. a Schematic and device
rendering. Three discrete AlN piezoelectric actuators are equidistantly placed
on top of an integrated Si3N4 microring resonator (blue solid). Upon coherent
driving these actuators with fixed relative phases
$(\phi_{1},\phi_{2},\phi_{3})=(0^{\circ},120^{\circ},240^{\circ})$, a rotating
acoustic wave (black arrow) is generated that spatial-temporal-modulates the
two co-propagating optical modes (red and green arrows), leading to a indirect
interband transition in only one direction. b Frequency domain representation
illustrating the indirect interband transition. When the two optical modes,
$a$ and $b$, are spaced by the resonant frequency $\omega_{m}$ of the
mechanical mode $c$, i.e.
$\Delta\omega_{ba}=\omega_{b}-\omega_{a}=\omega_{m}$, scattering among modes
$a$ and $b$ happens with a scattering rate of $g=g_{0}\sqrt{n_{c}}$ under a
microwave drive at $\omega_{m}$. c Schematic of $\omega-k$ space showing the
phase matching condition, i.e. the energy ($\Delta\omega_{ba}=\omega_{m}$) and
momentum ($\Delta k_{ba}=-k_{m}$) conservations. Interband transition that
couples two optical modes with the acoustic wave is only allowed in the
direction where phase matching condition is fulfilled, giving rise to
transparency on resonance in one direction (“forward”, as shown in a) and
extinction in the other direction (“backward”). d Fabrication process flow to
suspend the SiO2 membrane containing the Si3N4 microring resonator, to enhance
the optomechanical coupling. The key technique used here is Si isotropic dry
etch using SF6 Bosch process.

Despite of these advances of Si3N4 integrated photonics, non-reciprocal
devices, such as isolators and circulators that are widely used in optical
communications and datacenters for signal routing, multiplexing, and
protecting lasers from reflections, have not been improved via these
developments. Conventionally, optical non-reciprocity is realized in magneto-
optic materials Srinivasan and Stadler (2018) where the Faraday effect induces
non-reciprocal polarization rotation under an external magnetic field.
However, magneto-optic materials are not CMOS-compatible, posing challenges to
integrate using standard CMOS techniques developed for silicon photonics.
Besides, the magneto-optic effect is weak from the near-infrared to visible
wavelength range, therefore requiring a strong external magnetic field applied
at microscopic scale. Nevertheless, successful integration of cerium-
substituted yttrium iron garnet (Ce:YIG) on PIC (e.g., on Si Bi  _et al._
(2011); Huang  _et al._ (2016) or Si3N4 Yan  _et al._ (2020)) has been
demonstrated via wafer bonding or deposition, though these efforts suffer from
significant optical losses and external magnet. In addition, the requirement
of large magnetic field bias makes it incompatible with superconducting qubits
where optical isolation is needed for blocking reflected noises in optical
interfaces for quantum interconnects Awschalom  _et al._ (2021).

To overcome these bottlenecks, magnetic-free schemes have been demonstrated to
break Lorentz reciprocity or time reversal symmetry, through synthetic
magnetic field Fang  _et al._ (2012); Tzuang  _et al._ (2014); Kim  _et al._
(2021); Dostart  _et al._ (2021), optical nonlinearities Grigoriev and
Biancalana (2011); Chang  _et al._ (2014); Bino  _et al._ (2018); Hua  _et
al._ (2016); Yang  _et al._ (2020); Cao  _et al._ (2020); Shao  _et al._
(2020), optomechanically induced transparency Hafezi and Rabl (2012); Shen
_et al._ (2016); Ruesink  _et al._ (2016); Shen  _et al._ (2018); Ruesink  _et
al._ (2018), and stimulated Brillouin scattering (SBS) Kang  _et al._ (2011);
Dong  _et al._ (2015); Kim  _et al._ (2015, 2017). In addition, optical non-
reciprocity has also been demonstrated in atomic systems Scheucher  _et al._
(2016); Zhang  _et al._ (2018); Hu  _et al._ (2020a). However, challenges
remain in all these approaches. For example, optomechanically induced non-
reciprocity requires air-cladded, isolated microtoroids or microspheres that
are difficult to integrate with PICs. In addition, the isolation bandwidth is
primarily limited by the mechanical resonance linewidth that is typically
within kilohertz to megahertz range. Separating the signal from the control
pump laser using common multiplexing schemes is a formidable challenge, as
they are spaced by the mechanical resonance that is below hundred megahertz.
While nonlinear optics can work passively without active modulation, the main
concern is the dynamic reciprocity that forbids the propagation of light in
both directions with limited dynamic range of input optical power Shi  _et
al._ (2015).

Spatio-temporal modulation Yu and Fan (2009); Lira  _et al._ (2012); Sounas
and Alù (2017, 2014); Shi  _et al._ (2018); Kittlaus  _et al._ (2018); Sohn
_et al._ (2018); Kittlaus  _et al._ (2021), which breaks the reciprocity by
coupling two optical modes and prescribing phase matching condition by the
active modulation, stands out in terms of integration and applicability on
nearly all optical materials. Recently, schemes based on acousto-optic
modulation (AOM) has been extensively developed due to the compatibility of
this approach with low-loss PICs (e.g., AlN Sohn  _et al._ (2018) and Si
Kittlaus  _et al._ (2021)). Thus far, only non-reciprocal sideband modulation
is achieved, limited by the modulation efficiency and power handling
capability of the interdigital transducers (IDT) for generating surface
acoustic waves (SAW).

In this work, we demonstrate the first AOM-based optical isolator for Si3N4
integrated photonics. Three AlN piezoelectric actuators are equidistantly
placed along a Si3N4 microring resonator, and generate high-overtone bulk
acoustic resonances (HBAR) to realize AOM Tian  _et al._ (2020); Liu  _et al._
(2020b); Tian  _et al._ (2021). By carefully tuning the relative drive phases
among these actuators, HBAR modes create an effective rotating acoustic wave
that couples two optical modes in the momentum-biased direction. We achieve an
optical cooperativity $C=1$ with 14 dBm RF drive power applied on each
actuator, and maximum isolation ratio of 10 dB and minimum insertion loss of
0.1 dB with 20 dBm RF power. An isolation bandwidth of 700 MHz is obtained,
which is primarily determined by the optical resonance linewidth. The device
performance can be further improved by designing and operating the Si3N4
microresonator in the critical coupling regime.

##  I Results

Device principle and design. Our integrated optical isolator consists of three
AlN piezoelectric actuators on top of a Si3N4 microring resonator, as shown in
Fig. 1a. A Si3N4 PIC was fabricated using the photonic Damascene process
Pfeiffer  _et al._ (2018); Liu  _et al._ (2018b, 2020a), followed by
monolithic integration of AlN actuators as illustrated in Ref. Tian  _et al._
(2020); Liu  _et al._ (2020b). The piezoelectric AlN thin film (green), with a
piezoelectric coefficient Dubois and Muralt (1999) of $d_{33}=3.9$ pm V-1, is
sandwitched between top aluminum (Al, yellow) and bottom molybdenum (Mo,
orange) electrodes. When the electrodes are microwave-driven, bulk acoustic
waves are formed vertically in the substrate (i.e. the HBAR mode) beneath the
actuators. The Si3N4 PIC (blue) is fully buried in thick SiO2 cladding to
preserve the optical losses. In contrast to the previous work Tian  _et al._
(2020); Liu  _et al._ (2020b), here we apply a Si-release process to create
suspended SiO2 membrane where Si3N4 microresonator is embedded. The process
flow is shown in Fig. 1d. This release process enables tight confinement of
HBAR modes inside the Fabry-P$\acute{\text{e}}$rot acoustic cavity formed by
the top and bottom SiO2 surfaces, and thus enhances the acousto-optic coupling
through stress-optical effect, as shown in Ref. Tian  _et al._ (2020, 2021).

The Si3N4 microresonator is designed to support two optical eigenmodes, $a$
and $b$, with frequency difference matching a mechanical/acoustic resonant
frequency, as shown in Fig. 1b. The microwave drive applied on the three AlN
actuators creates acoustic waves inside the mechanical cavity, which scatters
light between modes $a$ and $b$ (i.e. indirect interband transition). Figure
1c illustrates the $\omega-k$ space, where $k=2\pi/\lambda$ is wavenumber (the
photon/phonon momentum is $\hbar k$, with the sign denoting the rotating
direction along the microring, clockwise or counter-clockwise), and $\omega$
is the angular frequency. To induce interband transition, energy and momentum
conservations must be satisfied, known as the “phase matching condition”. With
a nonzero phonon momentum $k_{m}$, phase matching requires
$\Delta\omega_{ba}=\omega_{b}-\omega_{a}=\omega_{m}$ and $\Delta
k_{ba}=k_{b}-k_{a}=-k_{m}$, where the minus sign of $k_{m}$ indicates that the
acoustic wave needs to counter-propagate with the two co-propagating optical
modes, as illustrated in Fig. 1a. We denote this direction, where phase
matching condition is fulfilled, as the “forward” direction in the following
discussion.

While a single vertical HBAR mode from one AlN actuator carries zero in-plane
momentum, an effective acoustic wave rotating along the microring is generated
by driving three actuators coherently with phases of ($0^{\circ}$,
$120^{\circ}$, $240^{\circ}$), i.e. $120^{\circ}$ phase difference between two
adjacent actuators, as illustrated in Fig. 1a. In the forward direction, two
optical modes are coupled via the acoustic wave, which can be described by a
simplified Coupled Mode Theory (CMT) equations Fan  _et al._ (2018):

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-(i\Delta_{a}+\frac{\kappa_{a}}{2})a-igbe^{i\omega_{d}t}+\sqrt{\kappa_{a,\text{ex}}}a_{\text{in}}$
(1) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-(i\Delta_{b}+\frac{\kappa_{b}}{2})b-igae^{-i\omega_{d}t}$ (2)

where $a$ ($b$) and $\kappa_{a}$ ($\kappa_{b}$) are the intra-cavity amplitude
and the total loss rate of mode $a$ ($b$), $\kappa_{a,\text{ex}}$ is the
external coupling rate of mode $a$. These equations of motion are transformed
under the rotating wave approximation (RWA) around the input laser frequency
$\omega_{L}/2\pi$, thus $\Delta_{a}=\omega_{a}-\omega_{L}$
($\Delta_{b}=\omega_{b}-\omega_{L}$) is the laser frequency detuning to the
resonant frequency of $a$ ($b$). $\omega_{d}/2\pi$ is the microwave drive
frequency which can be slightly detuned from the mechanical resonant frequency
$\omega_{m}/2\pi$. Mode $a$ is probed by the input light with amplitude
$a_{\text{in}}$, and its optical transmission/isolation is studied in the
following analysis.

In the forward direction, the two optical modes undergo interband transition
with a scattering rate $g=g_{0}\sqrt{n_{c}}$, where $g_{0}$ is the single
phonon-photon coupling rate describing the optomechanical interaction
strength, and $n_{c}$ is the intra-cavity phonon number. The optical
cooperativity, i.e. the ratio of scattering rate and losses, is
$C=4g^{2}/\kappa_{a}\kappa_{b}$. Strong coupling requires $C\gg 1$, and can
leads to Rabi oscillation and mode splitting. The latter results in a
transparency window on the resonance in the light transmission, which can be
calculated from Eqs. 1 and 2 as follows:

$T|_{\Delta_{a}=0}=\left|\frac{a_{\text{out}}}{a_{\text{in}}}\right|^{2}=\left[1-\frac{2\kappa_{a,\text{ex}}}{\kappa_{a}(1+C)}\right]^{2}$
(3)

In the forward direction, $C\gg 1$ and $T|_{\Delta_{a}=0}=1$. This
transparency can be understood intuitively as the impedance mismatch between
the bus waveguide and the microresonator, which is resulted from the
increasing effective intrinsic loss due to the scattering to the other optical
mode. In the backward direction where the three-wave phase matching is not
fulfilled, interband transition is prohibited, leading to $C\ll 1$ and
$T|_{\Delta_{a}=0}=0$ in the critical coupling regime
($\kappa_{a}=2\kappa_{a,\text{ex}}$). Consequently, the microresonator remains
critically coupled and its light transmission is not affected by the presence
of the acoustic wave. This non-reciprocal transmission between the forward
($T=1$) and backward directions ($T=0$) is the basic of our optical isolator.

Figure 2:  Characterization of optical and mechanical properties of the
isolator device. a False-colored top-view SEM image of the fabricated device.
b Optical microscope image highlighting bus-microring coupling section, the
released SiO2 area, and the relative positions between Si3N4 waveguides (blue
line) and two actuators. Inset: False-colored SEM image of the sample cross-
section, showing the vertical structure of the piezoelectric actuator and
quasi-square Si3N4 photonic waveguide. c FEM numerical simulations of the
vertical stress $\sigma_{z}$ distribution of a typical HBAR mode at 2.9 GHz,
and the optical profiles of the TE00 and TM00 modes of the quasi-square Si3N4
waveguide. White arrows mark the optical polarization directions. d Optical
transmission spectrum showing a pair of TE00 and TM00 modes with frequency
spacing of around 3 GHz. The x-axis is frequency-calibrated relative to the
center frequency of TE00 mode around 1546 nm. Inset shows the schematic of
relative position of the two modes in $\omega-k$ space. e From top to bottom
are microwave reflection S11, and optomechanical response S21 of the actuator
1, 2, 3 (labeled in a), respectively. Red arrows mark the mechanical mode at
2.958 GHz that is used in the experiment.

Meanwhile, in the forward direction, a single modulation sideband is generated
in mode $b$ which is frequency-shifted by $\omega_{d}$ relative to the input
laser $\omega_{L}/2\pi$. The mode conversion efficiency $\eta$ is:

$\eta|_{\Delta_{a}=0}=\left|\frac{b_{\text{out}}}{a_{\text{in}}}\right|^{2}=\frac{\kappa_{a,\text{ex}}}{\kappa_{a}}\frac{\kappa_{b,\text{ex}}}{\kappa_{b}}\frac{4C}{(1+C)^{2}}$
(4)

Equations 3 and 4 are derived assuming that the microwave drive $\omega_{d}$
matches the optical frequency spacing $\Delta\omega_{ba}$, i.e.
$\omega_{d}=\Delta\omega_{ba}$. A detailed derivation of these equations and
general cases with a frequency mismatch, which were used for subsequent
fitting experiments, are provided in the Methods section. This non-reciprocal
sideband generation has been demonstrated in previous studies in the $C<1$
regime Kittlaus  _et al._ (2018); Sohn  _et al._ (2018); Kittlaus  _et al._
(2021).

Experimental realization. Figure 2a shows the false-colored, top-view scanning
electron microscope (SEM) image of the fabricated device with three AlN
actuators integrated on a released Si3N4 microring resonator. The thickness of
Al / AlN / Mo is 100 / 1000 / 100 nm, respectively. The center black hole is
opened for Si-isotropic dry etching using the SF6 Bosch process, to partially
remove the Si substrate and to suspend the 5.4-$\upmu$m-thick SiO2 cladding.
Figure 2b shows the optical microscope image highlighting the bus waveguide
coupling region, the Si3N4 microring with a radius of 118 $\upmu$m buried in
the suspended SiO2 membrane, and two AlN actuators. Figure 2c shows the
simulated stress distribution of one HBAR mode within the SiO2 layer using
Finite Element Method (FEM). It can be seen that the HBAR mode is uniformly
distributed under the AlN actuator and tightly confined in the SiO2 membrane,
allowing direct modulation of the optical mode propagating along the waveguide
through photoelastic and moving boundary effects Balram  _et al._ (2014); Jin
_et al._ (2018); Stanfield  _et al._ (2019).

We use the fundamental transverse electric (TE00, mode $a$) and magnetic modes
(TM00, mode $b$) to realize interband transition assisted with the acoustic
wave. A quasi-square waveguide cross-section ($810\times 820$ nm2) is used, as
shown in Fig. 2b inset. Figure 2c shows the simulated TE00 and TM00 mode
profiles, which also include the slanted sidewall. The transmission of
polarization-tilted light through the optical microresonator, including a pair
of TE00 and TM00 resonances, is shown in Fig. 2d. The TM00 mode frequency is 3
GHz higher than that of the TE00 mode at around 1546 nm wavelength. The
resonance linewidth is 1.16 (0.68) GHz for the TM00 (TE00) mode.

Figure 2e shows the microwave reflection S11, where mechanical resonances are
revealed. Only one actuator’s S11 is shown as the others are similar. Three
strong resonances are found around 3.0, 3.4 and 3.8 GHz, which are due to the
SiO2 mechanical cavity with $\sim$470 MHz free spectral range (FSR, determined
by the SiO2 cladding thickness). Besides, weak resonances with an FSR of
$\sim$19 MHz are observed, due to the HBARs in thick Si substrate formed under
square signal probe pads which were not undercut Tian  _et al._ (2020).
However, only the HBARs confined in SiO2 can efficiently modulate the optical
mode because the HBARs in the Si substrate have negligible overlap with the
Si3N4 waveguide, which can be verified from the optomechanical S21 response
shown in Fig. 2e. S21 measures the ratio between the output light intensity
modulation and the microwave drive power. Three actuators are measured
individually. Since the HBARs are mainly determined by the thickness of each
layer that was well controlled during fabrication and highly uniform over the
full wafer scale, the HBAR frequencies of the three actuators show only sub-
megahertz misalignment. A maximum of -45 dB S21 was achieved, providing an
improvement of 20 dB over a previously reported unreleased Si HBAR AOM Tian
_et al._ (2020). This is due to the significantly reduced mechanical mode
volume and tighter HBAR confinement in the released SiO2 membrane. The SiO2
HBAR at 2.958 GHz is used in the following experiments to match the optical
mode spacing. Additionally, the signal cross-talk between the actuators is
maintained below -60 dB (see Supplementary Note 6) because the HBARs are
tightly confined vertically beneath the actuator, and the center etched hole
prevents the transmission of any lateral mechanical modes.

Figure 3:  Optical isolation and RF phases dependency. a Experimental setup.
Three RF signals are amplified and applied to the AlN actuators with phase
controlled individually. An optical switch is used to control the direction of
input TE light (red arrows). The output TE light (red arrows) and generated TM
sideband (green arrows) are spatially separated and detected. The white arrow
in the device denotes the clockwise rotation of the RF drive in the forward
direction. Amp: RF amplifier. ECDL: external-cavity diode laser. FPC: fiber
polarization controller. F-PBS: fiber polarization beam splitter. PD:
photodetector. b Optical transmission and c converted sideband under phase
sweep of signals 2 and 3 relative to signal 1, i.e. sweeping $\phi_{21}$ and
$\phi_{31}$. Each column is an experimentally measured spectrum under the same
$\phi_{31}$ with spectral span of (-16 pm, 16 pm) relative to the center
wavelength $\lambda_{0}$ (1542.6 nm) of TE00 mode. Both b and c are normalized
to the input TE light power on chip. d Optical transmission spectra with
$(\phi_{21},\phi_{31})=(120^{\circ},-120^{\circ})$ (perfect phase matching)
and $(\phi_{21},\phi_{31})=(-120^{\circ},120^{\circ})$ (largest phase
mismatch). These two phase combinations are marked with black dashed lines in
b.

Optical isolation demonstration. The measurement setup is shown in Fig. 3a.
Three RF signals are amplified and applied to each actuator, and the
amplitudes and phases of each channel are controlled individually by each
signal generator. The TE00 mode is excited by aligning the polarization using
fiber polarization controllers (FPC 1, 2). The light propagation direction,
forward or backward, is controlled by a $2\times 2$ optical switch. The
transmitted TE light and generated TM sideband are separated by a polarization
beam splitter (PBS), which is a key reason why we use two modes of different
polarizations. The laser wavelength is continuously scanned across the
resonance to probe the spectral response around the TE00 resonance.

The RF phases are critical for phase matching. Figure 3b shows the
transmission spectrum of TE light by sweeping the RF phases of signals 2 and 3
relative to signal 1 ($\phi_{21}$ and $\phi_{31}$), while the output RF power
(20 dBm for each actuator) and light input direction are fixed. Note that
reversing the sign of the RF phases changes the rotation direction of the
acoustic wave. Non-reciprocity can be seen from the disparate transmission by
reversing the RF phases with respect to the origin (0∘, 0∘). Strong mode
splitting is induced under ideal phase setting
$(\phi_{21},\phi_{31})=(120^{\circ},-120^{\circ})$, while the original single
resonance is maintained at ($\phi_{21},\phi_{31})=(-120^{\circ},120^{\circ}$),
as shown in Fig. 3d. Intriguingly, non-reciprocity only degrades slightly when
the phases deviate from the ideal value, which allows large tolerance of phase
settings in practical applications. This behaviour and RF phase dependency are
also revealed by Finite-Difference Frequency-Domain (FDFD) simulations Shi
_et al._ (2018), showing reasonable matching with experimental data (see
Supplementary Note 4).

Figure 4:  RF power dependency and anti-Stokes TM sideband generation. a
Optical transmission spectra of the TE light in the forward (red solid) and
backward (gray solid) directions, with increasing RF power from 15 to 20 dBm.
$\Delta\lambda$ is the detuning of the input laser relative to the TE00 mode
$\Delta\lambda=\lambda_{L}-\lambda_{TE}$, with $\lambda_{TE}=1544.1$ nm. b
Generated light of the anti-Stokes TM sideband in the forward direction,
normalized to the input TE light power. The RF power increase is same as a.
The anti-Stokes sideband is blue shifted relative to the input laser by the
modulation frequency 2.968 GHz ($\sim 24$ pm). The fitted transmission using
Coupled Mode Theory (CMT, blue dashed) is also shown in a and b. c Conversion
efficiency of the TM sideband at $\Delta\lambda=0$ and d scattering rate $g$
as a function of RF power. Experimental data (squares) are grouped as low RF
power (red) and high RF power (purple), fitted individually with CMT (solid
lines) with different microwave drive to mechanical resonance detuning
$\omega_{d}-\omega_{m}$. Horizontal gray dashed line in d marks the value of
$g$ (460 MHz) with cooperativity $C=1$. Vertical red and purple dashed lines
in c and d mark the maximum conversion at each detuning. The error bars of
each data point represent the standard deviation from 5 individual
measurements. The dependence of isolation on RF power is shown in c with
experimental data (brown circle) and exponential fitting (brown solid line).
Note: the RF power in all the panels is the power applied on each individual
actuator, thus the total RF power consumption is three times (4.8 dB) higher.

Light transmission of the generated anti-Stokes TM sideband is simultaneously
measured as shown in Fig. 3c. It is normalized to the TE’s input power, thus
can be interpreted as conversion efficiency $\eta$. Figure 3b and c show
similar pattern but with reversed color rendering. Prominent splitting and TE-
TM conversion are found at $(120^{\circ},-120^{\circ})$, while TE-TM
conversion is negligible at $(-120^{\circ},120^{\circ})$. As the measured TM
sideband is resulted from mode coupling and phase matching, it can be used as
feedback signal for tuning and stabilizing the RF phases. From Fig. 3d, the
optical isolation ratio between the clockwise (forward) and counter-clockwise
(backward) directions of the RF drive is calculated as 9.3 dB, which is mainly
limited by the level of critical coupling (-10.1 dB) of the optical microring
(the current device is slightly under-coupling). This can be easily improved
in the future by fine tuning the bus-microring gap in the design and
fabrication. We achieve $83\%$ transmission on resonance corresponding to 0.8
dB insertion loss in the forward direction. We further note another device
with 0.1 dB insertion loss ($98\%$ transmission) which shows higher modulation
efficiency but less backward extinction due to the fact the microring is more
weakly coupled with a wider gap (see Supplementary Note 5).

Figure 5:  Influence of optical mode spacing, time-domain response, and
optical power linearity. a Top panel: Optical transmission spectra in the
forward (red) and backward (gray) directions, with increasing center
wavelength (left to right). The mode spacing ($\Delta\omega_{ba}$) changes
correspondingly due to the different FSRs of the TE00 and TM00 modes. Insets
show the relative positions of the two modes in $\omega-k$ space (not to
scale). Bottom panel: spectra of isolation ratios varying with the center
wavelength. b Transmission of optical pulse trains in the forward (red) and
backward (gray) directions for 100 ns (top panel) and 10 ns (bottom panel)
pulse width. The power is normalized to the maximum power of the forward pulse
in each case. c The isolation ratio decreases with narrower pulse width due to
the finite photon lifetime (1.5 ns) in the optical cavity. Experimental data
(circle) are fitted with an exponential function (solid line). d The isolation
remains nearly constant around 9 dB (red dashed line) over 30 dB dynamic range
of the optical input power.

RF power dependency. The evolution of optical isolation with varying applied
RF power is studied in Fig. 4 with fixed RF phases of
$(\phi_{21},\phi_{31})=(120^{\circ},-120^{\circ})$ and drive frequency of
2.968 GHz. Figure 4a shows the measured forward and backward transmissions of
the TE00 mode, with the RF power (applied to each actuator) increased from 15
to 20 dBm. In the forward direction, initially the resonance depth decreases
and the linewidth broadens with increasing RF power up to 16 dBm, resulted
from the increasing intrinsic loss caused by the scattering to the TM00 mode.
Above 17 dBm, mode splitting appears, creating a transparency window at the
original resonance frequency. In the backward direction, single-resonance
profile remains, however with slightly increasing linewidth due to the weak
mode coupling as predicted and described by the Floquet theorem Shi  _et al._
(2018). Figure 4c shows the isolation at zero detuning relative to the TE00
mode ($\Delta\lambda=0$) which increases exponentially with the applied RF
power.

The mode splitting rate, which is two times of the interband scattering rate
$g$, is extracted by fitting the resonance profile using the generalized Eq. 3
from CMT, with $g$ and $\Delta\omega_{ba}$ being the fitting parameters.
Figure 4d shows that $g$ gradually increases with increasing RF power, and
sharply increases to a higher value at 18 dBm RF power, above which $g$
continuously increases and finally saturates at 20 dBm. This behaviour is
caused by the blue shift of the SiO2 HBARs due to RF thermal heating, as SiO2
has a large positive temperature coefficient of elasticity Rais-Zadeh  _et
al._ (2013) of 188 ppm K-1. The RF drive frequency $\omega_{d}$ is initially
blue-detuned from the HBAR frequency $\omega_{m}$ at room temperature, i.e.
$\omega_{d}-\omega_{m}>0$. As the RF power is increased and the acoustic
velocity in SiO2 increases, $\omega_{m}$ approaches $\omega_{d}$ (i.e.
$\omega_{d}-\omega_{m}\rightarrow 0$) and more phonons are pumped into the
mechanical cavity. This in turn increases the temperature which further blue-
shifts the HBAR. Thus, the thermal nonlinearity leads to an increase in $g$ at
an RF power of approximately 18 dBm. This transition is studied in
Supplementary Note 7 with fine sweep of RF power in 0.1 dBm step.

With RF power above 20 dBm, the interband scattering rate $g$ saturates, which
suggests the efficiency of pumping phonons into the cavity starts dropping.
This is likely because $\omega_{d}$ becomes red-detuned to $\omega_{m}$, i.e.
$\omega_{d}-\omega_{m}<0$. Therefore, for RF power between 18 to 20 dBm,
nearly zero detuning can be derived. In this regime, the single-phonon
optomechanical coupling strength $g_{0}$ is calculated as 208 Hz, by fitting
high RF power data (purple line) with $g=g_{0}\sqrt{n_{c}}$, where $n_{c}$ is
calculated by extracting the electro-mechanical coupling efficiency
$k_{\text{t,eff}}^{2}=0.2\%$ from S11. The low RF power region is fitted with
$\omega_{d}-\omega_{m}\approx 6$ MHz blue-detuning (red line). Another
consequence of the RF thermal effect is the drift of mode spacing
$\Delta\omega_{ba}$. Figure 4a shows that the mode splitting evolves from
symmetric (18 dBm) to asymmetric (20 dBm) with increasing RF power, and
$\Delta\omega_{ba}$ is increased from 3 GHz to 3.3 GHz.

The spectrum of the TM anti-Stokes sideband is shown in Fig. 4b. The
conversion efficiency $\eta$ when the input laser is zero-detuned to the TE00
mode ($\Delta\lambda=0$) is plotted in Fig. 4c. Similarly, each data set with
low and high RF power are fitted individually using Eq. 4 with the same
detuning as used in Fig. 4d. It can be derived from Eq. 4 that the maximum
value of $\eta$ is reached at $C=1$. This can also be seen from Fig. 4b where
the conversion starts to drop at center due to the mode splitting when $C>1$.
Maximum of $8\%$ (-11 dB) of the TE00 mode power is converted to the TM00
sideband, which is mainly limited by the external coupling efficiency of the
TE00 ($\kappa_{a,\text{ex}}/\kappa_{a}=0.34$) and TM00
($\kappa_{b,\text{ex}}/\kappa_{b}=0.24$) modes (see Eq. 4). At zero RF drive
detuning, $C=1$ is achieved with 14 dBm RF power applied on each actuator
(18.8 dBm in total), and the system lies in the strong coupling regime at 20
dBm. On the other hand, the detuning not only lowers $g$ but increases the
required RF power to achieve $C=1$. Finally, it is worth noting that the same
device can work as a non-reciprocal frequency shifter Hu  _et al._ (2020b)
with $100\%$ conversion achievable for device that is strongly over-coupled
(i.e. $\kappa_{\text{ex}}\approx\kappa$). It could serve as a key building
block in photonic quantum computing Kobayashi  _et al._ (2016); Lukens and
Lougovski (2017); Joshi  _et al._ (2020) and microwave photonics Marpaung  _et
al._ (2019).

Table 1: Comparison of monolithically integrated, magnetic-free optical
isolator devices.

Year |  Scheme |  Structure |  Material |  Isolation |  Insertion loss |  Bandwidth |  Power
---|---|---|---|---|---|---|---
2020Yang  _et al._ (2020) |  Nonlinear optics |  Ring |  Si |  20 dB |  1.3 dB |  20 GHz |  No drive
2014Tzuang  _et al._ (2014) |  Synthetic magnetic |  MZI |  Doped Si |  2.4 dB |  |  20 nm |  34 dBm
2021Kim  _et al._ (2021) |  Synthetic magnetic |  Ring |  AlN |  3 dB |  9 dB |  4 GHz |  16 dBm
2021Dostart  _et al._ (2021) |  Synthetic magnetic |  Ring |  Doped Si |  13 dB |  18 dB |  2 GHz |  -3 dBm
2012Lira  _et al._ (2012) |  Spatio-temporal |  MZI |  Doped Si |  3 dB |  70 dB |  200 GHz |  25 dBm
2018Kittlaus  _et al._ (2018) |  Spatio-temporala |  MZI |  Si |  39 dB |  NA |  125 GHz |  90 mWb
2018Sohn  _et al._ (2018) |  Spatio-temporala |  Ring |  AlN |  15 dB |  NA |  1 GHz |  18 dBm
2021Kittlaus  _et al._ (2021) |  Spatio-temporala |  MZI |  Si+AlN |  16 dB |  NA |  100 GHz |  21 dBm
This work |  Spatio-temporal |  Ring |  Si3N4+AlN |  10 dB |  0.1-1 dB |  0.7 GHz |  20 dBm

00footnotetext: Non-reciprocal sideband modulation00footnotetext: Optical
drive

Detuning of optical mode spacing. Since there is 380 MHz difference in FSR
between the TE00 and TM00 modes (see Supplementary Note 2), their frequency
spacing $\Delta\omega_{ba}$ varies from pair to pair for different center
wavelengths. Here, the dependence of isolation performance on
$\Delta\omega_{ba}$ is studied in Fig. 5a under 20 dBm RF power. At
$\lambda_{0}=1542.58$ nm, $\Delta\omega_{ba}$ is nearly equal to the driving
frequency $\omega_{d}$, and the mode splitting is symmetric. Because the
optical resonance linewidths are around gigahertz level, TE and TM modes can
still be coupled even with a frequency mismatch between $\Delta\omega_{ba}$
and $\omega_{d}$ on the order of 0.5 GHz. The mismatch leads to asymmetric
mode splitting. Nevertheless, there is no prominent degradation of isolation
within the measured range. The decrease of maximum isolation for shorter
wavelength is caused by the reduction of extinction in the backward direction.
This is because that shorter wavelength has smaller mode size and thus a
weaker bus-microring external coupling rate $\kappa_{\text{ex}}$ that leads to
under-coupling. Maximum of 9.5 dB isolation is achieved at 1545.55 nm, which
is even larger than the matched symmetric case due to its better critical
coupling. Therefore, the isolator can work simultaneously for multiple center
wavelengths which is important for optical communication using wavelength
multiplexing Marin-Palomo  _et al._ (2017). The wavelength range can be
extended to cover the optical C-band by using pulley coupling scheme Moille
_et al._ (2019) to maintain critical coupling or slight over-coupling, and
engineering the waveguide geometry to reach precise FSR match. On the other
hand, thermal tuning to continuously shift the operating resonance can further
increase the bandwidth.

Isolation of optical pulses. We further evaluated the optical isolator to
demonstrate a unidirectional transmission of optical pulse train to mimic (0,
1) data stream, as shown in Fig. 5b. The 100-ns pulse shows the quasi-static
response, where the backward reflection is only $11\%$ of the forward
transmission. The bump at the pulse edge is caused by the limited bandwidth
(125 MHz) of the photodetector. The dynamic response is tested with 10-ns
pulses, illustrating a vast contrast in the two directions. The isolation
bandwidth can be inferred from the decreasing isolation with decreasing pulse
duration, as revealed in Fig. 5c. A pulse minimum of 8 ns is measured, limited
by the photodetector’s bandwidth. Over 8 dB isolation is maintained for pulses
longer than 20 ns. However, the isolation drops exponentially when the pulse
duration is shorter than 20 ns, which is ultimately limited by the photon
lifetime of $\sim 1.5$ ns, corresponding to 680 MHz linewidth of the optical
resonance.

Optical power linearity. Dynamic reciprocity has been a well-known limitation
for most optical isolators relying on optical nonlinearity, where the
isolation degrades dramatically when light transmits simultaneously in both
directions with optical power exceeds the threshold Shi  _et al._ (2015); Yang
_et al._ (2020). In our work, with only an electrical drive, the optical
linearity is preserved as long as the intra-cavity photon number is smaller
than the phonon number. Since phonon frequency is five orders of magnitude
smaller than photon frequency, theoretically it suggests maximum of 6 kW
optical power for the 20-dBm RF power in the experiment. However, the optical
linearity is finally limited to several Watts due to the Kerr nonlinearity of
Si3N4. The quasi-square waveguide used here has a normal group velocity
dispersion (GVD), thus Kerr parametric oscillation is suppressed. The
linearity is experimentally verified in microwatt to milliwatt range as shown
in Fig. 5d. The optical isolation remains nearly constant within the
measurement range. The large variation at high optical power is mainly due to
the optical thermal nonlinearity as we sweep the laser across the optical
resonance. Theoretically there is no limit on the lower bound of optical power
which can work for photonic quantum computing Kobayashi  _et al._ (2016);
Lukens and Lougovski (2017); Joshi  _et al._ (2020).

##  II Discussion

This work has demonstrated an integrated optical isolator achieved by spatio-
temporal modulation of a Si3N4 microring resonator via three AlN
pieozoelectric actuators. The device has been fully characterized in terms of
RF phases, RF powers, optical spectra, and optical powers, showing agreement
with theoretical models and numerical simulations. Table 1 summarizes recently
demonstrated integrated, magnetic-free isolators (to the best of our
knowledge), in comparison with our work. A comprehensive comparison with other
experimental realizations can be found in Supplementary Note 8.

Although promising progresses have been made using nonlinear optics Yang  _et
al._ (2020) and synthetic magnetic field Kim  _et al._ (2021); Dostart  _et
al._ (2021), most spatio-temporal modulation demonstrations Kittlaus  _et al._
(2018); Sohn  _et al._ (2018); Kittlaus  _et al._ (2021) are still in the non-
reciprocal sideband modulation regime. In this work, the HBAR AOM helps to
boost the optical cooperativity $C$ beyond 1 and enter strong coupling regime,
thanks to its power handling capability and the tight acoustic confinement. We
obtained an isolation of 10 dB under an RF power of 20 dBm. Notably, a record-
low insertion loss of only 0.1 dB is achieved due to the intrinsic low loss of
Si3N4 waveguides. In contrast to nonlinear optics, the electrical drive
largely preserves the optical linearity by separating RF driving and optical
sensing in two different domains.

In the future, the isolation can be significantly improved by optimizing the
bus-microring coupling gap to reach critical coupling or slight over-coupling
Pfeiffer  _et al._ (2017). Furthermore, it has been theoretically predicted
that the RF power consumption can be reduced to a few dBm by reducing the
microring radius to 20 $\upmu$m, which will increase both the electro-
mechanical and optomechanical coupling efficiencies Blésin  _et al._ (2021).
With these improvements, electrically driven, magnetic-free optical isolator
could be reliably incorporated as a key building block into current integrated
opto-electronic systems.

Methods

Device fabrication: The Si3N4 PIC is fabricated using the photonic Damascene
process Pfeiffer  _et al._ (2018); Liu  _et al._ (2018b, 2020a). The
monolithic integration of piezoelectric actuators on top of Si3N4
microresonators is illustrated in Ref. Tian  _et al._ (2020); Liu  _et al._
(2020b). Films of 100 nm Mo and 1 $\upmu$m AlN are sputtered on the wafer
through foundry services (Plasma-Therm). The actuators are patterned by thick
photoresist SPR220-4.5 and dry etched using Cl2 and BCl3 in Panasonic E620
Etcher. The bottom Mo electrodes are patterned by photoresist AZ1518 and dry
etched using Cl2 in the same etcher. Finally, the top 100 nm Al is deposited
by an electron-beam evaporator and patterned using a standard lift-off
process. The SiO2 release process is shown in Fig. 1d. The center release hole
is patterned using photolithography, and SiO2 is dry-etched to expose the Si
substrate. The Si is then isotropically etched using SF6 Bosch process to
undercut and suspend the SiO2 membrane.

Design of Si3N4 waveguides for phase matching: From the microring’s resonant
condition $k=2\pi/\lambda=m/R$, the momentum is related to the azimuthal order
$m$ of the mode and the microring’s radius $R$. As the three actuators cover
the entire microring, the generated rotating acoustic wave has an effective
wavelength of $2\pi R$, and thus an azimuthal order of 1 ($m_{c}=1$). Thus the
phase matching condition requires the azimuthal order difference $\Delta
m_{ab}=m_{a}-m_{b}=m_{c}=1$ between the TE00 and TM00 modes. As the Si3N4
waveguide is surrounded by isotropic SiO2 cladding, a quasi-square waveguide
cross-section ($810\times 820$ nm2), as shown in Fig. 2b inset, is designed to
have slightly different effective refractive indices for the two optical modes
(see Supplementary Note 1). Here, phase matching requires two optical modes to
co-propagate, with counter-propagating acoustic wave. In the final devices
used in the experiment, the azimuthal order difference $\Delta m_{ab}$ is
measured and calibrated (see Supplementary Note 2) to be around 4. Due to the
discrete nature of the spatial modulation, there are higher-azimuthal-order
Fourier components simultaneously excited that can fulfil phase matching at
the expense of lower efficiency Sounas and Alù (2014).

Measurement setup: The electromechanical S11 is measured by detecting the
reflected microwave signal using a vector network analyzer (VNA, Agilent
E8364B), where the electrical signal is applied to the device through an RF
GSG probe (Cascade ACP40-GSG-150). To measure the optomechanical S21,
continuous-wave (CW) light from a diode laser (Velocity Tunable Laser 6328) is
edge-coupled into the chip using a lensed fiber and an inverse taper with
around 50 $\upmu$W power on chip. An RF signal of -5 dBm power is applied from
the port 1 of the VNA to drive the AlN actuator and the light intensity
modulation is detected by a 12-GHz photodiode (New Focus 1544), which is sent
back to port 2 of the network analyzer.

In the main optical isolation measurement, three RF signal generators (Agilent
E8257D) output RF drive for each actuator and are amplified by three RF
amplifiers (ZHL-5W-63-S+) before going into the device. The phases between
them are controlled by their built-in phase controller after they are
synchronized by their internal 10 MHz clock. The same diode laser outputs CW
light whose wavelength is continuously swept around the optical resonance. The
input polarization is controlled by fiber polarization controllers (FPC561) to
align with the TE polarization on chip. The light input direction is selected
by a MEMS (micro-electromechanical system) $2\times 2$ optical switch
(OSW22-1310E). The output TE and TM polarized light is separated by a fiber
polarization beam splitter (PBC1550SM-FC) and measured by two photodetectors
(New Focus 1811), whose signals are recorded by an oscilloscope (MSO8104A).
Note the optical axis of the beam splitter is fixed, thus we have to use a
polarization controller to align with it. During the measurement, TE
polarization is selected by rotating the polarization controller until only TE
mode resonance is observed. This process is repeated for both directions. It
is worth noting that, although the polarization will change as we switch the
MEMS optical switch, the polarization right before the PBS will be the same,
since the optical switch itself is a reciprocal device.

For testing the isolation of optical pulses, the input light before the
optical switch is modulated by an electro-optical intensity modulator (Lucent
2623CSA) and the input pulse is generated by a function generator (Agilent
33250A). In testing of the optical linearity, Erbium Doped Fiber amplifier
(EDFA-I-B) is used before the optical switch for optical powers larger than
100 $\upmu$W.

Derivation of coupled mode equations: The three-wave mixing (two optical modes
$a$ and $b$ and one mechanical mode $c$) process can be described by the
quantum interaction Hamiltonian as:

$H_{I}=\hbar g_{0}(ab^{{\dagger}}c+a^{{\dagger}}bc^{{\dagger}})$ (5)

assuming phase matching is fulfilled and $a$ has smaller frequency than $b$.
Thus, only two processes can be supported: 1. $ab^{{\dagger}}c$, annihilation
of photon $a$ and phonon $c$ and generation of one photon $b$; 2.
$a^{{\dagger}}bc^{{\dagger}}$, annihilation of one photon $b$ and generation
of photon $a$ and phonon $c$. Following similar approach as Ref Fan  _et al._
(2018), the equation of motion can be obtained by assuming resolved sideband
and rotating-wave approximation:

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-(i\Delta_{a}+\frac{\kappa_{a}}{2})a-ig_{0}bc^{{\dagger}}+\sqrt{\kappa_{a,\text{ex}}}a_{\text{in}}$
(6) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-(i\Delta_{b}+\frac{\kappa_{b}}{2})b-ig_{0}ac$ (7)
$\displaystyle\dot{c}$ $\displaystyle=$
$\displaystyle-(i\omega_{m}+\frac{\gamma_{c}}{2})c-ig_{0}a^{{\dagger}}b+\sqrt{\gamma_{c,\text{ex}}}c_{\text{in}}e^{-i\omega_{d}t}$
(8) $\displaystyle a_{\text{out}}$ $\displaystyle=$ $\displaystyle
a_{\text{in}}-\sqrt{\kappa_{a,\text{ex}}}a$ (9) $\displaystyle b_{\text{out}}$
$\displaystyle=$ $\displaystyle-\sqrt{\kappa_{b,\text{ex}}}b$ (10)

where $\gamma_{c}$ and $\gamma_{c,\text{ex}}$ are total loss rate and external
coupling rate of the mechanical mode, and
$c_{\text{in}}=\sqrt{P_{\text{in}}/\hbar\omega_{d}}$ is input microwave
amplitude. Assuming optomechanical back-action (term $g_{0}a^{{\dagger}}b$ in
Eq. 8) is much smaller than the microwave drive, the intra-cavity amplitude of
$c$ at static state is:

$\displaystyle c$ $\displaystyle=$
$\displaystyle\sqrt{n_{c}}e^{-i\omega_{d}t}$ (11) $\displaystyle n_{c}$
$\displaystyle=$
$\displaystyle\frac{\gamma_{c,\text{ex}}}{(\omega_{d}-\omega_{m})^{2}+\gamma_{c}^{2}/4}\frac{P_{\text{in}}}{\hbar\omega_{d}}$
(12)

By inserting Eq. 11 into Eqs. 6, 7, we will get Eqs. 1, 2 in the main text.
Due to the modulation, $b$ is frequency shifted by $\omega_{d}$ in the
rotating frame of $\omega_{L}$. The slow amplitude and fast oscillation of $b$
can be separated by substitute $b$ with $\tilde{b}e^{-i\omega_{d}t}$. At
steady state, Eqs. 1, 2 can be solved by letting time derivatives to be zero,
and after some linear algebra we will arrive at the general expressions for
transmission $T$ and sideband conversion efficiency $\eta$:

$T=\left|\frac{a_{\text{out}}}{a_{\text{in}}}\right|^{2}=\left|1-\frac{\kappa_{a,\text{ex}}}{i\Delta_{a}+\frac{\kappa_{a}}{2}+\frac{g^{2}}{i(\Delta_{b}-\omega_{d})+\frac{\kappa_{b}}{2}}}\right|^{2}$
(13)
$\eta=\left|\frac{b_{\text{out}}}{a_{\text{in}}}\right|^{2}=\frac{\kappa_{a,\text{ex}}}{\kappa_{a}}\frac{\kappa_{b,\text{ex}}}{\kappa_{b}}\frac{4C}{|C+(1+\frac{2i\Delta_{a}}{\kappa_{a}})(1+\frac{2i(\Delta_{b}-\omega_{d})}{\kappa_{b}})|^{2}}$
(14)

where $\Delta_{b}-\omega_{d}=\Delta_{ba}+\Delta_{a}-\omega_{d}$. It can be
seen when $\Delta_{a}=0$, $\Delta_{ba}=\omega_{d}$, Eqs. 13, 14 reduce to Eqs.
3, 4. Eqs. 13, 14 are used for fittings in Fig. 4 and 5. The mode splitting
can be found from Eq. 13 if assuming $\Delta_{a}\gg\kappa_{a},\kappa_{b}$ and
$\Delta_{ba}=\omega_{d}$:

$T\approx\left|1-\frac{(i\Delta_{a}+\frac{\kappa_{b}}{2})\kappa_{a,\text{ex}}}{(g+\Delta_{a})(g-\Delta_{a})+i(\frac{\kappa_{a}+\kappa_{b}}{2})\Delta_{a}}\right|^{2}$
(15)

where the transmission minima is reached when $\Delta_{a}=\pm g$. Thus the
mode splitting is $2g$.

Acknowledgments: This work was supported by U.S. National Science Foundation’s
RAISE TAQS program under grant PHY 18-39164, by NSF QISE-Net under grant DMR
17-47426, by the Air Force Office of Scientific Research under Award No.
FA8655-20-1-7009, by funding from the EU H2020 research and innovation
programme under grant agreement No. 732894 (HOT), and by Swiss National
Science Foundation under grant agreement No. 176563 (BRIDGE). Samples were
fabricated in the EPFL center of MicroNanoTechnology (CMi), and Birck
Nanotechnology Center at Purdue University. AlN deposition was performed at
Plasma Therm LLC.

Author contribution: H.T. and J.L. designed the devices. J.L., H.T. and R.N.W.
developed the process and fabricated the samples, with the assistance from
J.H.. H.T. performed the experiment and simulations, and analyzed the data,
with the assistance from A.S. and T.B.. H.T. and J.L. wrote the manuscript,
with the input from others. S.A.B and T.J.K supervised the collaboration.

Data Availability Statement: The code and data used to produce the plots
within this work will be released on the repository Zenodo upon publication of
this preprint.

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